WATER POLLUTION CONTROL RESEARCH SERIES • 16070 DZV 02/71
ESTUARINE MODELING: AN ASSESSMENT
ENVIRONMENTAL PROTECTION AGENCY • WATER QUALITY OFFICE
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Series describes
the results and progress in the control and abatement
of pollution in our Nation's waters. They provide a
central source of information .on the research , develop-
ment, and demonstration activities in the Water Quality
Office, Environmental Protection Agency, through inhouse
research and grants and contracts with Federal, State,
and local agencies, research institutions, and industrial
organizations.
Inquiries pertaining to Water Pollution Control Research
Reports should be directed to the Head, Project Reports
System, Office of Research and Development, Water Quality
Office, Environmental Protection Agency, Room 1108,
Washington, D. C. 20242.
Tar nle by the Superintendent at Documents, U.S. Government Printing Office
WMhimton, D.C. 20402 - Price StSO
Stock Number 5501-0129
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ESTUARINE MODELING: AN ASSESSMENT
Capabilities and Limitations for
Resource Management and Pollution Control
by
TRACOR, Inc.
6500 Tracor Lane
Austin, Texas 78721
for the
WATER QUALITY OFFICE
ENVIRONMENTAL PROTECTION AGENCY
Project 16070DZV
Contract 14-12-551
February, 1971
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EPA REVIEW NOTICE
This report has been reviewed by the Water
Quality Office, EPA, and approved for publi-
cation. Approval does not signify that the
contents necessarily reflect the views and
policies of the Environmental Protection
Agency, nor does mention of trade names or
commercial products constitute endorsement or
recommendation for use.
ii
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ABSTRACT
This report constitutes a technical review and critical appraisal of present techniques
of water quality modeling as applied to estuaries. Various aspects of estuarine modeling are
treated by a selection of scientists and engineers eminent in the field, and these essays are
supplemented by discussions from technical conferences held during the course of the report's
preparation. Topics discussed include one-, two-, and three-dimensional mathematical models for
estuarine hydrodynamics, water quality models of chemical and biological constitutents including
DO, BOD, nitrogen forms, phytoplankton and general coupled reactants, models of estuarine temp-
erature structure with special attention given the modeling of thermal discharges, and the
principles and applicability of physical models in estuarine analysis. Also included is a
review of solution techniques, viz. analog, digital and hybrid with a detailed discussion of
finite-difference methods, a brief survey of estuarine biota and present biological modeling
activities, and a collection of case studies reviewing several past estuarine modeling projects.
Conclusions about the existing state of the art of estuarine modeling and recommendations for
future research by EPA are advanced throughout the report and are summarized in the final
chapter.
This report was submitted in fulfillment of Project No. 16070DZV between the Environ-
mental Protection Agency and TRACOR, Inc., as part of the National Coastal Pollution Research
Program.
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AUTHORS AND REVIEWERS
William E. Dobbins
Partner
Teetor-Dobblns, Consulting Engineers
MacArthur Airport
Ronkonkoma, New York
John Eric Edinger
Associate Professor of Civil Engineering
Towne School of Civil and Mechanical
Engineering
University of Pennsylvania
Philadelphia, Pennsylvania
William H. Espey, Jr.
Director, Ocean Sciences and Water
Resources
TRACOR, Inc.
Austin, Texas
James A. Harder
Professor of Hydraulic Engineering
Department of Civil Engineering
University of California
Berkeley, California
Donald R. F. Harleman
Professor of Civil Engineering
Ralph M. Parsons Laboratory for Water
Resources and Hydrodynamics
Department of Civil Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
Arthur T. Ippen
Institute Professor
Director, Ralph M. Parsons Laboratory for
Water Resources and Hydrodynamics
Department of Civil Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
Jan J. Leendertse
The RAND Corporation
Santa Monica, California
Donald J. O'Connor
Professor of Civil Engineering
Department of Civil Engineering
Manhattan College
Bronx, New York
Gerald J. Paullk
Professor of Population Dynamics
Center for Quantitative Science in
Forestry, Fisheries and Wildlife
University of Washington
Seattle, Washington
Donald W. Pritchard
Professor of Oceanography
Director, Chesapeake Bay Institute
The Johns Hopkins University
Baltimore, Maryland
Maurice Rattray, Jr.
Professor of Oceanography
Chairman, Department of Oceanography
University of Washington
Seattle, Washington
Robert V. Thomann
Associate Professor of Civil Engineering
Department of Civil Engineering
Manhattan College
Bronx, New York
George H. Ward, Jr.
Ocean Sciences and Water Resources
TRACOR, Inc.
Austin, Texas
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WATER QUALITY OFFICE PARTICIPANTS
Donald J. Baumgartner
Chief, National Coastal Pollution
Research Program
Pacific Northwest Water Laboratory
Corvallis, Oregon
Richard J. Callaway
Chief, Physical Oceanography Branch
National Coastal Pollution Research
Program
Pacific Northwest Water Laboratory
Corvallis, Oregon
Kenneth D. Feigner
Systems Analysis and Economics Branch
Washington, D. C.
W. H. Fitch
Systems Analysis and Economics Branch
Washington, D. C.
Dan Fitzgerald
New England River Basins Office
Northeast Region
Needham Heights, Massachusetts
Thomas P. Gallagher
Southeast Water Laboratory
Athens, Georgia
Howard Harris
Southwest Region
Alameda, California
David R. Hopkins
South Central Region
Dallas, Texas
C. Robert Horn
Middle Atlantic Regional Office
Charlottesville, Virginia
Norbert A. Jaworski
Chesapeake Technical Support Laboratory
Middle Atlantic Region
Annapolis, Maryland
Malcolm F. Kallus
Federal Coordinator, Galveston Bay Project
South Central Region
Houston, Texas
Larry Olinger
Southeast Water Laboratory
Athens, Georgia
George D. Pence
Edison Water Quality Laboratory
Northeast Region
Edison, New Jersey
W. C. Shilling
Robert S. Kerr Water Research Center
South Central Region
Ada, Oklahoma
Roger D. Shull
Pollution Control Analysis Branch
Washington, D. C.
John Vlastelicia
Northwest Region
Portland, Oregon
T. A. Vastier
Estuarlne and Oceanographic Programs Branch
Washington, D. C.
John Yearsley
Northwest Region
Portland, Oregon
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ATTENDANCE AT CONFERENCES
24 June 1969
St. John's College
Annapolis, Maryland
D. J. Baumgartner
R. J. Callaway
W. E. Dobbins
W. H. Espey
R. D. Feigner
D. Fitzgerald
W. H. Fitch
T. P. Gallagher
J. A. Harder
D. R. Hopkins
C. R. Horn
R. J. Huston
N. A. Jaworski
M. F. Kallus
D. J. O'Connor
G. D. Pence
D. W. Pritchard
M. Rattray
W. C. Shilling
R. D. Shull
R. V. Thomann
J. Vlastelicia
G. H. Ward
T. A. Wastler
6 May 1970
Salishan Lodge
Gleneden Beach, Oregon
D. J. Baumgartner
R. J. Callaway
W. E. Dobbins
J. E. Edinger
W. H. Espey
K. D. Feigner
D. Fitzgerald
J. A. Harder
D. R. F. Harleman
H. Harris
D. R. Hopkins
C. R. Horn
N. A. Jaworski
M. F. Kallus
J. J. Leendertse
L. Olinger
G. J. Paulik
D. W. Pritchard
M. Rattray
W. C. Shilling
R. V. Thomann
J. Vlastelicia
G. H. Ward
T. A. Wastler
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FOREWORD
The Water Quality Office of the Environmental Protection Agency has historically
supported research and development efforts for estuarine models, and is continuing to do so.
The purpose of this report is to provide a summary of presently available modeling methods used
in studying the causes of estuarine pollution problems and pollution abatement alternatives.
This current project has been undertaken by the National Coastal Pollution Research
Program as part of an overall mission to expand scientific understanding of the marine environ-
ment with respect to improvement and protection of marine water quality. This mission is
achieved by fundamental and applied studies on (a) physical transport and dispersion, (b bio-
logical, chemical and physical transformations and interactions, (c) water use damages,
(d) restoration of degraded marine waters, and (e) conservation of resources associated with
additions of heat and materials introduced from land, rivers, and the atmosphere.
The nature of estuarine water pollution problems demands consideration of solid wastes
which, through land use and ocean disposal, contribute materials to the estuarine system. Air-
borne materials can contribute to oceanic pollution—which can in turn influence or add to
existing estuarine concentrations. Estuaries act as a focal point for man's environmental
ills, screening, as they do, wastes discharged to inland watersheds and fed by small portions
of all the wastes accumulated in the oceans. With the establishment of the Environmental Pro-
tection Agency there now appears to be a framework for estuarine pollution control consistent
with the physical dimensions of the problem.
It was decided a report of this nature would be most beneficial if it were prepared
by a group of consultants and specialists with considerable experience with one or more modeling
topics. To make sure this assessment was directed to the problems pollution control agencies
were working on now and foresaw as impending problems, members of the Federal Water Quality
Administration regional office technical staffs were included.
This report presents a discussion of capabilities and limitations of both mathematical
and physical models, and of various solution techniques for the mathematical formulations.
Based on first-hand experience and a critical review of published reports by the authors and
participants, recommendations are made for research and development found necessary to over-
come deficiencies in present methods and to provide effective methods for handling more complex
problems in the future.
D. J. Baumgartner
R. J. Callaway
National Coastal Pollution Research Program
Pacific Northwest Water Laboratory
Water Quality Office
Environmental Protection Agency
Corvallis, Oregon
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PREFACE
1C is appropriate to include here some prefatory remarks on what this report is and
what it is not, along with something about how it came to be. The purpose of this report is
to critically appraise the present state of the art of water quality modeling as applied to
estuaries. This purpose subsumes the question of how well do we understand estuarine processes,
and includes, further, the questions: how well can we predict these processes, how well need
we predict them for water quality regulation and management, and how do we best get from where
we are to where we need to be. The answers provided to these questions are not simple, as can
be judged from the bulk of this volume, nor in many cases are they complete or even clearly
defined. But this is a liability of attempting to obtain these answers, quickly enough to be
useful, from such a young, complex and vital field.
Although in a sense this report constitutes a cataloging of models, it is not intended
nor written to be a reference manual for the practicing engineer. Many of the arguments require
an examination of fundamentals, and return therefore to derivations and basic developments.
But assumed throughout is an at least superficial familiarity of the reader with estuarine
processes and their representation. Neither is this report a disjoint collection of technical
papers, but it is, rather, an organic sequence of essays on assigned topics written from a
consistent point of view.
The basic plan by which this report was prepared was to retain a group of consultants
with considerable experience and expertise in aspects of estuarine modeling, some of whom would
author the report and some of whom would provide a critical review. Preparation of the report
was coordinated insofar as the geographical distribution of the consultants permitted, and was
augmented by two general conferences attended by the consultants and those individuals in the
Water Quality Office of EPA with direct responsibility for the development or application of
models. These conferences were truly conferential, in that their purpose was to discuss in as
much technical depth as necessary the various aspects of the report and of water quality models
in general. In addition, throughout Its preparation this report was open for discussions of
the contents and the inclusion of alternate viewpoints. The discussions, both formal and
informal, therefore constitute an integral and substantial part of the report.
The project was formally initiated in June 1969 and the first conference was held
within the month at St. John's College in Annapolis, Maryland. The purpose of the conference
was to agree on the content and scope of the report and to select the principal authors. The
Water Quality Office (then FWPCA) had submitted a suggested outline consisting of five chapters,
(i) mathematical models, (ii) analog and hybrid models, (ill) hydraulic models, (iv) use of
models in evaluating streamflow regulation, and (v) use of models in evaluating physiographic
modifications, which exemplifies the primary concerns of FWPCA at the time. At this conference
the federal programs In estuarine water quality modeling were reviewed, and the report outlined
in essentially its present form. (At the beginning of this work, the responsible agency was
the Federal Water Pollution Control Administration, which subsequently became the Federal
Water Quality Administration and is now the Water Quality Office of the Environmental Protection
Agency. The agency is therefore referred to variously as FWPCA, FWQA and EPA throughout the
report.)
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After careful consideration of the report's intended content, scope and objectives,
the resources available for its preparation were apportioned in the following way: hydro-
dynamics 207., water quality 207., temperature 67., physical models 10%, solution techniques 207,
case studies 107,, biological modeling 27., critical review 127.. While in fact the consultants
contributed more than their allotted time, this distribution does serve, however, to indicate
the relative weight given the various topics.
The authors of this report are men whose varied activities and scholarly pursuits
place a high demand on their time. With such contributors, printing deadlines are propositions
of naive optimism. (Out of similar undertakings has grown many an eloquent lament for the
editor who daily nurses an empty mailbox.) Although it was scheduled that the second conference
be convened in September and the report completed in November, or at least by the end of the
year, it was not until June of 1970 that the second conference met in Salishan Lodge at Gleneden
Beach, Oregon, nearly a year after the first conference. At this conference, the nearly com-
plete report was subjected to a thorough and wide-ranging discussion. Plans were made for
modifications and additions to the text and for its completion.
Editorial policy for the report was somewhat variable. The individual chapters were
only lightly edited. Although notations could be made uniform, duplication eliminated, and
the chapters modified to conform to a consistent style, we felt that in doing so, more would
be lost than gained, and that the overall value of the report would be greatest if the identity
and individuality of each author's work were preserved. The discussions, in contrast, required
heavy editing, as a precise rendering in print of spoken dialog is less a transcript than a
transmogrification. The contents of Chapter IX and the discussions appended to each chapter
have been freely rearranged to achieve a more useful organization. As a general rule, dis-
cussions which involve technical details or which are of specific pertinence to a particular
chapter are placed following that chapter, while discussions of a more general or recommenda-
tory nature are included in Chapter IX. In no instance, however, are statements separated
from the discussion context in which they occurred.
A number of acknowledgements are in order. Foremost are due D. J. Baumgartner and
R. J. Callaway of the National Coastal Pollution Research Program of EPA. Their interest in
the work and their resoluteness in achieving the project objective are in large measure respon-
sible for the quality of this report. We are grateful for their assistance and support
throughout the project.
The assistance and cooperation of our firm TRACOR, Inc., during the project is
gratefully acknowledged. Several of our associates here, especially R. J. Huston and
J. P. Buckner (now with the Coastal Bend Regional Planning Commission, Corpus Christ!), con-
tributed their assistance throughout the study. We wish to thank T. A. Wastler (Estuarine
and Oceanographic Programs Branch, EPA) for his interest and comments, and EPA scientists and
engineers, Norbert Jaworski, Howard Harris, George Pence, Tom Gallagher, Ken Feigner and
John Vlastelecia, for their individual assistance, particularly in the compilation of the case
studies (Chapter VII). E. G. Fruh (University of Texas) and B. J. Copeland (North Carolina
State University) read portions of Chapter VII and their reviews lead to many substantive improve-
ments. We are grateful to Colonel Frank P. Bender, Director of the Galveston Bay Project, for
permitting the use in Chapter VII of results from the Galveston Bay Project. Dr. Garbls H.
Keulegan of the Waterways Experiments Station has followed the preparation of the report since
its inception, although owing to other commitments he was unable to formally review the docu-
ment. We are grateful to him for his comments and for his great interest in the project.
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We appreciate receiving permission from the following publishers and authors to
reproduce copyrighted material: the American Society of Civil Engineers for several figures in
Chapters IV, V and VII which appeared in the Proceedings. McGraw-Hill Book Company for Figure
2.10 from Estuary and Coastline Hydrodynamics, the Water Pollution Control Federation for
Figures 3.11 - 3.17 from the Journal Water Pollution Control Federation, the American Fisheries
Society for the quotation of Prof. Hedgpeth in Chapter VIII from A Symposium on Estuarine
Fisheries. Donald Watts and 0. L. Loucks of the University of Wisconsin for Figure 8.1 from
Models for Describing Exchanges Within Ecosystems, the Controller of Her Britannic Majesty's
Stationery Office for the figures and tables in Chapter VII from Effects of Polluting Discharges
on the Thames Estuary, and the Controller of H. M. Stationery Office and the authors A. L. H.
Gameson and I. C. Hart for Figure 7.29 and Table 7.4 from "A Study of Pollution in the Thames
Estuary" in Chemistry and Industry. The data in Figure 7.29 and Table 7.4 are those of the
Greater London Council.
Finally, we wish to express our gratitude to the consultants, Drs. Dobbins, Edinger,
Harder, Harleman, Ippen, Leendertse, O'Connor, Paulik, Pritchard, Rattray and Thomann, who
retained their patience and generosity through the eighteen long months this report was in
preparation. This is their report. Its value and its merits are due to their integrity, their
industry, and their command of their fields.
G.H.W.
W.H.E.
Austin
November 1970
JCi
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CONTENTS
Abstract iil
Authors and Reviewers - v
Water Quality Office Participants vl
• •
Attendance at Conferences vl1
• • •
FOREWORD D. J. Baumgartner and R. J. Callaaay Vlll
PREFACE 1X
I. INTRODUCTION a. B. Hard and W. S. Septy
1. Estuaries 1
2. Estuarine Models 2
3. Report Content and Organization 4
References 4
II. HYDRODYNAMIC MODELS
1. Three-Dimensional Models D. V. Pritchard 5
1.1 Introduction 5
1.1.1 The Coordinate System 5
1.1.2 The Concept of the Ensenble Average: 5
Deterministic vs. Stochastic'Modes of Motion
1.1.3 The Notation ^
1.2 The Basic Equations 8
1.2.1 Conservation of Mass 8
1.2.2 Conservation of Momentum 10
1.2.3 Conservation of Dissolved Constituents 16
1.2.A Some Possible Further Simplification of the 17
Three-Dimensional Conservation Equations for
Mass, Momentum and Salt
2. Two-Dimensional Models D. V. Pritohard 22
2.1 Introduction 22
2.1.1 The Coordinate System and the Notations to be Used 22
2.2 The Basic Equations for a Vertically Averaged 22
Two-Dimensional Model
2.2.1 Conservation of Mass 22
2.2.2 Conservation of Momentum 24
2.2.3 The Vertically Averaged Two-Dimensional 27
Salt Balance Equation
2.2.4 Summary of the Set of Vertically Averaged Dynamic and 28
Kinematic Equations Suitable to Serve as the Basis of
a Two-Dimensional Numerical Model of an Estuary
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2.2.5 An Alternate Development of a Set of Vertically Averaged 30
Dynamic and Kinematic Equations in an Estuary
References for Sections 1 and 2 33
3. One-Dimensional Models D. R. F. Harleman 34
3.1 Introduction 34
3.2 Mathematical Models in Real Tiwe 36
3.2.1 Mass Transfer Equations for a Non-Conservative Substance 37
3.2.2 Source and Sink Terms 39
3.2.3 Tidal Velocity 40
3.2.3.1 Continuity Equation for a Variable Area Estuary 42
3.2.3.2 Momentum Equation for a Variable Area Estuary 43
3.2.3.3 Solution Techniques for Tidal Hydraulic Problems 45
3.2.4 Longitudinal Dispersion in the Real Time Mass 53
Transfer Equation
3.2.4.1 Longitudinal Dispersion Coefficient in Regions 55
of Uniform Density
3.2.4.2 Longitudinal Dispersion in Salinity 57
Intrusion Regions
3.2.5 Discussion of the Real Time Mass Transfer Equation 61
3.2.6 Solution of the Real Time Mass Transfer Equation 63
in a Uniform Estuary
3.3 Mass Transfer Equations Using Non-Tidal Advective Velocities 67
3.3.1 Mass Transfer Equation: Time Average over a Tidal Period 67
3.3.2 Mass Transfer Equation: Slack Tide Approximation 67
3.3.3 Analytical Solutions of the Non-Tidal Advective 68
Mass Transfer Equations in a Uniform Estuary
3.3.4 Mathematical Models of Salinity Intrusion 70
3.3.4.1 Descriptive Studies 71
3.3.4.2 Predictive Studies 71
3.4 Comparison of Real Time and Non-Tidal Advective 76
Mathematical Models
3.4.1 Continuous Inflow of a Non-Conservative Pollutant 76
3.4.2 Salinity Intrusion 78
3.5 Summary and Conclusions 79
3.5.1 Mathematical Models in Real Time 79
3.5.2 Mass Transfer Equations Using Non-Tidal Advective 81
Velocities
3.5.3 Conclusions 81
References for Section 3 85
Discussion 90
I. WATER QUALITY MODELS: CHEMICAL, PHYSICAL AND
BIOLOGICAL CONSTITUENTS D. J. O'Connor and R. V. Thomann
1. Introduction 102
2. One-Dimensional Analysis 106
2.1 Solution Due to Instantaneous Release 107
2.2 Conservative Substances 109
2.3 Non-Conservative Substances 110
2.4 Consecutive Reactions (BOD-DO) 112
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2.5 Applications Co Dissolved Oxygen Analysis 113
2.6 Multi-Stage Consecutive Reactions (Nitrification) 117
2.7 Data Requirements 119
3. Solution techniques for One-Dimensional Steady State 123
3.1 Available Approaches 123
3.2 Continuous Solution Approach 125
3.3 Finite Section Approach 127
4. Two-Dimensional Steady State 131
4.1 Solution Approach: Finite Difference 131
5. One-Dimensional Time-Variable Models 135
5.1 Solution Approach 136
6. Verification 140
6.1 An Example: The East River (O'Connor 1966) 141
6.1.1 Description of the System 142
6.1.2 Field Data 144
6.1.3 Analysis of Data 145
6.1.4 Waste Water Discharges 147
6.1.5 BOD and DO Deficit Profiles 148
6.1.6 Discussion 151
7. Future Directions 152
7.1 Steady-State Feedback Modeling 152
7.2 Dynamic Phytoplankton Model 156
8. Discussion and Recommendations 163
References 167
Discussion 169
IV. ESTUARINE TEMPERATURE DISTRIBUTIONS J. E. Edinger
1. Natural Temperature Distributions 174
1.1 Similarity Between Salinity and Temperature Distributions 174
1.2 Influence of Surface Heat Exchange on Temperature Distributions 175
1.3 Additional Influences on Temperature Structure 180
2. Analytic Description of Estuarine Temperature Structure 182
2.1 The Vertically Mixed Case 182
2.2 Two-Layered Segmented Models 183
2.3 Continuous Vertical Temperature Structure 185
3. Modeling Temperature Distributions due to Large Heat Sources 187
3.1 The Temperature Excess 187
3.2 Initial Temperature Distributions 188
3.3 Intermediate Temperature Distributions 191
3.4 Large-Scale Temperature Distributions 194
3.5 Physical Hydraulic Modeling of Thermal Discharges 200
References 202
Discussion 207
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V. PHYSICAL HYDRAULIC MODELS D. R. F. Harleman
I. Introduction 215
2. Similitude of Momentum Transfer Processes in Tidal Motion 218
2.1 Physical Boundary Conditions for Tidal Models 218
2.2 Froude Scale Ratios for Tidal Motion 221
2.3 Reynolds Number Effects 225
3. Similitude of Mass Transfer Process 226
3.1 Similitude of Salinity Intrusion 226
3.1.1 Saline-Wedge Estuary 228
3.1.2 Mixed Estuary 230
3.2 Mass Transfer Similitude in Regions of Uniform Density 233
4. Model Verification 238
4.1 Tidal Verification 238
4.2 Salinity Verification 241
4.3 Sediment Transport Verification 241
5. Discussion of the Use of Physical Hydraulic Models in Estuarine 247
Water Quality Studies
6. Sumnary and Conclusions 251
References 252
Discussion 255
VI. SOLUTION TECHNIQUES
1. Analog and Hybrid Techniques j. A. Harder 264
1.1 Introduction 264
1.2 Analog Modeling 265
1.2.1 Analog Simulators and Analog Computers 265
1.2.2 Analog Simulators for Tidal Flows 265
1.2.2.1 Simulating Hydraulic Friction 269
1.2.3 An Analog Computer Element for Pollution Dispersion 270
1.3 Hybrid Computation 271
1.4 Economic Considerations in Analog Computing 272
1.5 System-Response Type Models 273
References for Section 1 276
2. Digital Techniques: Finite Differences J. j. Leendertee 277
2.1 Introduction 277
2.2 Considerations in Design of Computation Schemes 278
2.2.1 Conservation of Mass 278
2.2.2 Stability 279
2.2.3 Dissipative and Dispersive Effects 281
2.2.4 Effects of Noncentered Operations 284
2.2.5 Discontinuities 285
2.2.6 Comments on Higher Order Schemes 286
2.2.7 Use of Conservation Laws to Obtain Nonlinear Stability 287
2.3 Tidal Computations 290
2.3.1 One-Dimensional Computations 290
2.3.2 Two-Dimensional Tidal Computations 291
2.3.3 Numerical Simulation of Tidal Flow in Two Dimensions 295
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2.4 Mass Transport Computations 296
2.4.1 One-Dimensional Water Quality Computations 296
2.4.2 Two-Dimensional Water Quality Computations 298
References for Section 2 301
Discussion 303
VII. CASE STUDIES G. B. Hard and V. tt. Eopey
1. Introduction 310
2. The Thames 311
2.1 Physiography and Hydrography 311
2.2 Polluting Loads 311
2.3 Modeling Techniques 317
2.3.1 Modeling of Temperature Distribution 323
2.3.2 Modeling of Dissolved Oxygen Distribution 325
2.4 Model Results 327
2.4.1 Calculated Profiles and Their Accuracy 327
2.4.2 Model Application 332
3. The DECS Model 341
3.1 The Delaware Estuary 341
3.1.1 Characteristics of the Estuary 342
3.1.2 Model Application and Results 342
3.2 The Potomac Estuary 348
3.3 Hillsborough Bay 354
3.4 Summary 356
4. San Francisco Bay 361
4.1 Characteristics of the Bay-Delta System 361
4.1.1 Physiographic and Hydrographic Features 361
4.1.2 Water Quality 363
4.2 The Physical Model 366
4.2.1 Verification 368
4.2.2 Evaluation of the Effects of Solid Barriers on 371
Hydraulics and Salinities
4.2.3 Application of Dye Releases 377
4.3 Digital Computer Model 383
4.3.1 Theory and Computational Framework 383
4.3.2 Model Application and Verification 390
4.3.3 Comparison of Physical Model and Digital Computer Model 396
5. Galveston Bay 399
5.1 Galveston Bay Project Mathematical Models 399
5.1.1 Hydrodynamic Model 401
5.1.1.1 Model Formulation and Application 401
5.1.1.2 Comparison of Mathematical Hydrodynamic 404
Model with Physical Model
5.1.2 Salinity Model 408
5.1.3 Thermal Discharge Model 411
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5.1.4 Application of the GBP Models to Ecological Parameters 420
|.2 Physical Models 424
References 434
VIII. BIOLOGICAL MODELING IN ESTUARIES: A NOTE G. J. Paulik
1. Characteristics of Animal Communities in Estuaries 438
2. Mathematical Models of Biological Communities 442
3. Use of Numerical Models to Conserve, Exploit and Exterminate 446
Animal Populations
4. Recommendations for Biological Modeling in Estuaries 447
4.1 Key Species Models 447
References 449
DC. DISCUSSIONS
1. Estuarine Hydrodynamic and Water Quality Models 451
2. Computational Aspects 465
3. Data Collection and Verification 470
4. Prepared Remarks 480
4.1 Discussion G. B. Ward 480
4.2 Discussion R. J. Callaway 482
4.3 Some Projections for the Immediate Future J. j.^ Leendertee 484
4.4 Discussion V. B. Bspey 487
5. Review and Comments A. T. Ippen 489
X. CONCLUSIONS AND RECOMMENDATIONS
1. Conclusions 492
1.1 Estuarine Hydrodynamic Models 492
1.2 Estuarine Water Quality Models 493
1.3 Utility of Physical Models 494
1.4 Solution Techniques 494
1.5 Applications 495
2. Recommendations for Future Research 496
2.1 Hydrodynamic Models 496
2.2 Water Quality Models 496
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CHAPTER I
INTRODUCTION
George H. Ward, Jr.
and
William H. Espey, Jr.
Growing concern with the preservation of our water resources, particularly within the
last decade, has precipitated the need for a balanced and astute management of these resources.
Even disregarding the political intricacies, water quality management is an exceedingly
difficult task. The difficulties arise from two essential facts. First, utilization of the
resources of a natural waterbody is subject, almost universally, to conflicts among potential
users, in that one use inherently delimits or excludes another. The problems are compounded
since, typically, each potential use is of some socio-economic importance. Much, if not
most, of the management task ultimately devolves to the resolution of these conflicts in a
satisfactory manner. Secondly, the processes—hydrographlc, chemical, biological—which operate
in a waterbody are extremely complex and often poorly understood. Yet it is these processes
upon which management decisions must be based.
The application of water quality models has proved to be a powerful technique in
water resource management, as models can incorporate the complexity of the relevant processes
in the waterbody into a utilitarian form for management consideration. Through the use of
models both a diagnostic and predictive capability is provided, diagnostic in the sense of
permitting the identification and isolation of specific factors affecting the water quality,
and predictive, in permitting evaluation of future effects of proposed changes in the water-
body, its inputs, or its uses.
1. ESTUARIES
This report is concerned with water quality modeling as applied to a specific type of
waterbody, the estuary. (The general features of estuarine hydrography, viz., the defining
characteristics of estuaries, associated circulation patterns, physiography, etc., are well
known and may be assumed ab inttio. See, e.g., Fritchard 1952, Cameron and Pritchard 1963,
Lauff 1967.) This apparently superficial circumscription perhaps deserves discussion.
Estuaries are, collectively, of singular importance. They are important biologically;
in contradistinction to the nutrient-rich water-poor land and the water-rich nutrient-poor
ocean, estuaries are typically both a nutrient-rich and water-rich environment, highly produc-
tive, constituting the prime habitat for a myriad of species and the nursery and spawning area
for many more. Besides their biological significance, however, estuaries are of specific
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importance to man, as is evidenced by their propensity for stimulating urban development.
Estuaries offer an accessibility to the sea, often with excellent shipping and harbor facilities,
while the feeding rivers provide a source of freshwater for domestic and industrial use, as well
as affording receiving water for municipal and industrial wastes.
Estuaries are similarly quite susceptible to pollution and degradative modifications
in quality. They often receive excessive volumes of wastes both from peripheral discharges and
from upstream discharges carried into the estuary by the inflows, and are often adversely
influenced by upstream freshwater consumption. In addition to affecting the estuary's own
delicate ecology, the effects of these practices nay extend beyond the physiographic bounds of
the estuary itself.
Besides these very pragmatic reasons for a concern with estuaries, the estuary per
_ae poses modeling problems of extreme scope and complexity. Its hydrodynamics, for instance,
are determined by the interaction of the tides from the ocean and the influx of fresh water
from the rivers, modified by complicated boundary stresses due to the semi-enclosed physiography,
and further altered—or even dominated, as in fjords—by gravitational circulations arising from
density gradients. These factors, however, are sufficient to characterize only the transport
of a specific constituent in the estuary: the various processes and reactions to which that
constituent is subjected must be determined as well. These range from simple first-order
kinetics to reactions which depend nonlinearly upon additional variables and are possibly linked
with the biota.
It Is evident therefore that estuaries, besides comprising a water resource of
critical Importance, present modeling problems of intrinsic difficulty. This report attempts
to summarize these problems and assess present methods for their treatment. Although many of
the topics discussed can find application as well In coastal zones, in streams and rivers, or
in lakes, the discussions are inclined to those aspects fundamental to, or peculiar to,
estuarine processes.
2. ESTUARINE MODELS
The meaning of the term "model" as used in this report is not precise, and, indeed,
four distinct meanings of the word as employed with regard to estuarine water quality may be
identified. It is appropriate here to Indicate these usages In a somewhat pedestrian manner,
avoiding any Judgments on their adequacy or correctness as such judgments can entail a lengthy
philosophical digression. (Recent considerations of the meaning and role of models in science
may be found in, e.g., Magel, Suppes and Tarski 1962.)
In the strictest sense, a model is a formalism: the mathematical expression of the
relevant physical processes, e.g., the application of the principles of Newtonian fluid mechanics
with approximations appropriate to an estuary. A different but related usage is that a model
is a conceptual idealization or simplified representation of a physical process. The employment
of such a model is usually motivated by tractability and Is justified by the ability of the
model to reproduce observed results, even though it is ultimately unfaithful to the basic physics.
One example is the representation of the Reynolds flux of salinity (u^'s') by a "model" in-
volving the product of the mean salt gradient and an eddy diffusivity -K •£ . (Another example,
but not related to estuaries, is the representation of surface effects In vapor-drop equilibria
by a "model" of the surface of the drop as a distensible membrane of constant tension—viz.,
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the surface tension—even though the surface Is known to be a transition region of high density
gradient.)
'Model" is also used to mean a simple (in some sense) analog of the real system. In
estuarine studies, the most familiar example is, of course, the physical or hydraulic model, a
highly distorted, small-scale reproduction of the estuary complete with freshwater inflows, a
sump of ocean salinity, and mechanical tide generators. This use of the term could as well
apply to a simulator consisting of electrical components, as described in Chapter VI.
Finally, a "model" may refer to a mathematical formulation (a "model" in either the
first or second sense above) together with a solution technique for the variables of concern.
This usage is implicit in the expressions "analog model" and "digital model," and may in some
cases even refer to a specific computer program, such as DECS-III or the URE Model (the "Orlob
Model"). This usage, which has some obvious faults, does not enjoy general approval by
scientists and engineers involved in estuarine water quality, but nevertheless has become more
frequent in recent years, particularly on the administrative or program level.
In the following pages all four uses of the term can be noted, sometimes in almost
confusing contiguity. Probably the only definition of "model" sufficiently general to encompass
these variations in meaning is the (superficial) statement that an estuarine model is a technique
or device for the representation of some property in an estuary.
The preponderance of present estuarine water quality models is concerned with the
concentration or distribution of a constituent, property or parameter in the estuary.
Accordingly, the processes represented in the model are generally classified as either transport
processes or reaction processes. Transport processes are basically hydrodynamic and include
advection, turbulent diffusion, and, when spatial averaging is involved, dispersion. Reaction
processes encompass the sources and sinks to which the parameter is subjected and may be
physical, chemical or biological, e.g., sedimentation and flocculation of organics, uptake of
oxygen in biochemical degradation of wastes, colifora mortality, algae loss by zooplankton
grazing, etc. The relative importance of transport processes and reaction processes obviously
varies from estuary to estuary, and depends upon the parameter modeled as well as the required
spatial and temporal refinement.
The question eventually arises in the development of an estuarine model as to
whether all of the relevant processes in the estuary have been identified or correctly re-
presented. Classically this is determined by model verification, the comparison of model
predictions with empirical measurements. As might be expected in a state-of-the-art assessment,
the topic of verification pervades this report. But a word of qualification (and, perhaps,
warning) is due, in that 'Verification" is often employed in a wider sense, and may mean the
use of empirical data for anything from calibration of the model to its actual validation. The
context must be considered, for a 'Verified" model may not be all it appears.
3. REPORT CONTENT AND ORGANIZATION
The basic arrangement of this report is to progress from model formulation to
solution to specific application. Such a segregation of topics, of course, cannot be strictly
observed and there is therefore considerable immixture. Estuarine hydrodynamics, which determine
the transport of substances in the estuary, are considered first and organized from the stand-
point of spatial dimensionality. Both advective and diffusive processes are discussed. Here,
also, the discussion of salinity distribution is undertaken, not only because of the value of
-------
salinity as a natural tracer, but because of its considerable influence on estuarine circulation.
Then the modeling of water quality parameters per se is treated, which involves the specifica-
tion of source and sink terms and their incorporation with the transport mechanisms into various
conservation relations. The discussion is divided into a treatment of general chemical and
biological parameters (Chapter III) and a treatment of temperature (Chapter IV). Temperature
is discussed separately for several reasons. It is a parameter which is currently receiving
particular attention because of its profound ecological importance. It is a parameter which,
like salinity, exhibits significant gradients in estuaries due to the mixing of river and
ocean water. Further, because of the dramatic dynamics of thermal discharges, e.g. the large
volumes of flow involved and the effects of buoyancy, their modeling requires special
consideration.
An appraisal of the use of physical hydraulic models in estuarine quality is
presented next (Chapter V) taking advantage thereby of the formulations of mass and momentum
transfer developed in the preceding chapters. Solution techniques are discussed in Chapter VI,
separated into analog and analog-hybrid techniques, which accomplish the integration of the
basic equations continuously (in one independent variable) using electronic components, and
numerical methods, in particular finite-difference techniques which involve integration of the
equations using finite increments in the Independent variables and principally find application
with high-speed digital computers. In the latter topic, consideration is given to accuracy and
stability problems as well as numerical phenomena such as false diffusion and non-conservation
of mass or energy. Several case studies are presented in Chapter VII to provide examples of
recent modeling investigations of estuarine water quality. The selection is not intended to be
exhaustive but rather to exemplify the present state of model application to estuarine problems.
Finally, a brief overview of the estuarine animal populace and present trends in biological
modeling in estuaries is given in Chapter VIII. At present, there is very little effort
toward incorporating or linking the higher life-forms in estuaries to water quality models,
but this is indubitably one of the ultimate directions of estuarine modeling.
In a strict sense, a complete state-of-the-art report on estuarine water quality
modeling would encompass a large part of existing knowledge in chemical and biochemical
kinetics, the dynamics of nonhomogeneous stratified fluids, boundary and transfer processes
at the air-sea Interface, modeling of biological and ecological systems, turbulent transport,
and so on. For evident reasons, such an ambitious approach was not undertaken here. The
emphasis throughout this report is, rather, on the interaction of these phenomena and its
formulation. Indeed, the uniquely complex and difficult estuarine system can probably be best
characterized by that feature, interaction.
REFERENCES
Cameron, W. M. and D. W. Pritchard, 1963: Estuaries. Chapter IS, The Sea. II, M. N. Hill (Ed.).
New York, Interscience Publishers.
Lauff, George H. (Ed.), 1967: Estuaries. American Association for the Advancement of Science,
Washington, D. C.
Nagel, Ernest, Patrick Suppes and Alfred Tarski (Ed.), 1962: Logic. Methodology and Philosophy
of Science. Stanford University Press.
Pritchard, D. W., 1952: Estuarine hydrography. Advances in Geophysics. .1. New York, Academic
Press.
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CHAPTER II
WDROOTNAMIC MODELS
1. THREE-DIMENSIONAL MODELS
D. W. Pritchard
1.1 INTRODUCTION
In this section Che basic time-dependent equations in three spatial dimensions,
expressing the conservation of mass, of momentum, and of a dissolved or finely divided suspended
constituent of the waters in an estuary, are developed. In order to keep this section from
becoming overly long, the development will of necessity be abbreviated, with frequent use of
the phrase "it can be shown." However, some outline of the development of the basic equations
from first principles is desirable, in order to clearly indicate what terms and processes are
neglected in arriving at simplified forms of the equations which offer some possibility of
solution, either in terms of our present knowledge or of knowledge which might be gained in the
foreseeable future.
This development will include arguments for the neglect of certain terms, particularly
certain terms that arise in the averaging process. Again, these arguments must be brief, with-
out extensive observational proof of the statements of relative size of the pertinent terms
being considered.
1.1.1 The Coordinate System
Tensor notation will be employed in this section because of the economy it affords
in respect to the number of terms which must be written down in expressing the complex differ-
ential equations which develop from the first principles to be considered here. Thus, a
right-handed rectilinear coordinate system is envisioned with the three axes x^ (i = 1,2,3)
oriented such that x^ and Xj are in a horizontal plane, and x^ is the vertical axis
directed upwards (i.e., parallel to the force of gravity but positive in the direction of
-g ). The plane of X-, - 0 is considered to coincide with mean sea level.
1.1.2 The Concept of the Ensemble Average; Deterministic vs. Stochastic Modes of Motion
The basic conservation principles will first be expressed in terms of the instan-
taneous field of motion and of the distribution of the pertinent properties of the water in
the estuary such as density, salinity and pressure. It has long been recognized that for the
real, turbulent conditions of a natural waterway these equations are intractable. Some type of
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averaging is required in order to separate what night be called the deterministic nodes of
motion from that part of the motion which is by nature stochastic in character. The method of
averaging has considerable influence on the meaning of the individual terms of the resulting
equations.
The type of averaging to be initially employed here is based on the following con-
ceptual argument. Assume that the instantaneous fields of motion and of the pertinent
properties could be observed, in detail, within an estuary over some time period. During this
time period the external processes, i.e., tidal variation at the mouth of the estuary, fresh-
water inflow, radiation balance and meteorological conditions, are defined—that is, are known
functions of tine. Now assume that the instantaneous fields of motion and of the pertinent
properties in this same estuary were observed over another time period of the same length as
the first during which the time variations of the external processes were the same as in the
first period, and that the time histories of all the external processes prior to this second
period were the sane as those prior to the first period, back as far in tine as is required by
the inherent dynamic "memory" of the system.
If such measurements were possible, it would be found that the fields of motion and
of the distribution of pertinent properties would differ in detail from one period to the
other, despite the equality of the time variations of the external processes. If k such
"identical" periods occurred, the instantaneous field of notion and of the distribution of
pertinent properties for any one period would differ, in detail, from all the others.
A part of the instantaneous field of motion would be common to all such k "ident-
ical" periods, and this part is called the deterministic mode of motion. A part of the
instantaneous field of motion, representing the difference between the instantaneous field of
motion for any one of the k "identical" periods and the deterministic mode, constitutes the
stochastic mode of motion. These "turbulent" departures in the instantaneous velocity field,
at any point in space and at any time relative to the start of each of the k "identical"
periods, are distributed among the k periods in accordance with some probability distribu-
tion—that is, they are "random" in character, though not necessarily "gaussian". A similar
deterministic mode and stochastic mode would exist for each of the instantaneous distributions
of the pertinent properties such as density, salinity and pressure.
Note that each of these conceptual k "identical" periods starts at the same point
in the identical time histories of the external processes; for example, at the same tidal
epoch. If, at a given point in space, the time records of the velocity for each of the k
periods were lined up relative to this starting time, and the average at a given point in tine
were taken across all k records, then the resulting average is called the ensemble average.
Such an ensemble average is time dependent, since it is taken at every point in time over all
the k periods. A similar operation would provide ensemble averages for the pertinent para-
meters such as density, salinity and pressure. If k is sufficiently large, the ensemble
averages may be taken as suitable approximations to the deterministic modes of notion and of
the distribution of the pertinent properties.
It has been more common practice to obtain averages of the instantaneous equations
by taking such averages over a time period AT small compared to the time scale at which one
wishes to resolve the field of motion or the distribution of properties. The problem with
this approach is that the stochastic or turbulent mode of motion encompasses a wide range of
time and spatial scales.
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The ensemble average has the advantage of smoothing out the stochastic mode of
motion at all time scales, while also retaining the variations in the deterministic mode at
all time scales. Furthermore, this is the average used in most theories of turbulence. The
fact that it is only conceptually attainable is not a disadvantage. The predicting equations
which can be developed will in any case provide an estimate of the most probable field of
notion and/or of properties in the estuary, and not the actual instantaneous velocities and
instantaneous values of the pertinent properties which would occur at a real instant of time.
1.1.3 The Notation
The instantaneous velocity at a point x^ and a time t during one of the k
"identical" periods is designated by u^ . We then define
and
u, - . (2.2)
where u^ is the ensemble average of u^ over all k "identical" periods, and u^' is the
departure of the instantaneous velocity for any one of the k periods from the ensemble
average. Note that < >k represents the operation of obtaining the ensemble average. Further
note that
-------
u. « che ensemble average of ui over all k.
s " che ensemble average of "a over all k .
p « Che ensemble average of "p over all k .
p « Che ensemble average of p over all k .
C • Che ensemble average of £ over all k .
< )jc • symbol for Che ensemble averaging operation.
u.' • che "turbulent" departure of Che instantaneous velocity from its ensemble
average.
.s1 - Che "turbulent" departure of Che instantaneous sale concentration from its
ensemble average.
p' - Che "turbulent" departure of Che instantaneous density from its ensemble
average.
p' - che "turbulent" departure of Che instantaneous pressure from its ensemble
average.
(' - Che "turbulent" departure of Che instantaneous concentration of a constituent
from its ensemble average.
1.2 THE BASIC EQUATIONS
1.2.1 Conservation of Mass
The basic statement of Che principle of conservation of mass is simply that the mass
of a moving fluid particle, that is, a particle of fluid whose boundaries move with Che
instantaneous velocity, remains constant. Thus
where M is the mass of the moving fluid particle. This statement leads Co the differencial
equation
As discussed in Sections 1.1.2 and 1.1.3 above, che ensemble average of this
equation is taken, afCer substituting
7 • P +• p1
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resulting in
H + a + s k-° <2-5>
Note Chat while Che ensemble means of u^1 and p' are zero, Che ensemble mean of
Che cross-produce u.' p' is not necessarily zero. However, note that in this case
where vu« Qi designates the correlation coefficient between u^1 and o' . The maximum value
the correlation coefficient can have is 1.0. The root-mean-square velocity deviation is on
the order of 10"J of the ensemble average, and the root-mean-square density deviation is at
most on the order of 10~3 of the ensemble average density. Consequently
< 10-*
and the last term of Equation (2.5) can therefore be neglected, giving finally
<2'6>
(Mote that this argument is essentially the same as Che Boussinesq approximation.)
Making use of the equality
its ft+ ui
Equation (2.6) can also be written as
° <2-7>
It is usually stated that, for an incompressible fluid, -£• = 0 and hence, since the
compressibility of water is very small, the equation of continuity applicable to an estuary
can be closely approximated by
8u.
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that is, the three-dimensional flow field must be non-divergent. This equation might be more
appropriately called the equation of volume continuity. However, more than simply incompressi-
bility must be invoked to obtain (2.8) from (2.7). The density of es marine water is a function
of temperature, salinity and pressure. Thus
dp (*o\ dp , fdo\ da . f*o\ d6
+ +
dp (*o\ dp , fdo\ da . f*o\ d6
at Upv at + UsV at + (n) ar
The statement that a fluid is incompressible means that (5-7 ~ °- However, this
statement does not mean that (^) and (~f) are zero. To obtain Equation (2.8) from
Equation (2.7) it is necessary to invoke a Boussinesq-type approximation; that is, that time
and spatial variations in density may be neglected except in terms in which they are multiplied
by gravity. In any case, Equation (2.8) is in fact a good approximation to the equation of
continuity for use in estuaries.
1.2.2 Conservation of Momentum
Newton's Second Law is the basic statement of the principle of conservation of
momentum; that is, the time rate of change of momentum of a moving elemental fluid particle is
equal to the sum of the forces acting on the particle. The differential equation expressing
this principle is called the equation of motion. For estuaries the instantaneous equation of
motion is given by
~ 8*u
P" + PS +
where ejs, is the cyclic tensor, g, the acceleration of gravity, 11 the kinematic molecular
viscosity, and rt. the component' of the angular velocity of the earth in the x, direction.
Now, substituting for the instantaneous value of each of the variables the sum of
the ensemble average value and a deviation term, that is
u. » u^ + u. '
"p - P + p'
p - p + p'
then the Xj_, Xj and x3 components of the ensemble-averaged equation of motion are
au, 3(11,11,) , a a'u,
a(u_u.)
• — - " ' - - (2.11)
and °- -" 8
10
-------
where f = 2Q, = 20 sin sp is the coriolts parameter. Here 0 Is the angular velocity of the
earth and k - p ) and from
-------
Applying Leibnitz* Rule to the last term of (2.13) gives
To the same approximation as the Bousslnesq approximation, we may replace (p)_ , the density at
the surface, by p , the density at depth z , in the second term on the right side of (2.14).
Hence
(2.15)
"Xa
Designating the x^-component of the slope of the pressure surface at depth
relative to the slope of the water surface by 1, _ _ , then
•••fPlM
The Xj-component of the ensemble-averaged equation of motion can then be written
au, a(u,u.) -,_ , ap_ a'u,
Likewise, the X- component of the ensemble-averaged equation of motion becomes
where the pressure force term - has been replaced by
>.18)
• e -- + B i« (2 19)
pax, 8 8 T u<1^
The relative isobaric slope terms i^ and i.^ are given by
and (2.20)
1 ri an . ,
i~ ~ — I "r dir '
2.P.1 p J^ ax, "^
The density depends on temperature, salinity and pressure. In the estuary, the
pressure dependence is unimportant. The density dependence on temperature and salinity is
12
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conveniently expressed in tables issued by the U. S. Navy Oceanographic Office (1966), in terms
of the oceanographic function sigma-t; i.e.
ot - 103 (o-l)
for p expressed in units of the c.g.s. system. Hence
io-'
OAJ
and
^£_ _ 10-3 __t
ax, ax,
Hence, the relative isobaric slope may be expressed as
and
3o
i
1Q- 3
.
2,P,T, l+10-3ot
3xf *** ' <2-22>
Given the spatial distribution of temperature and salinity, then the spatial distri-
bution of ot can be obtained from tabular compilations such as USNOO (1966) or from appropriate
numerical relationships which have been developed for use in high-speed computers. Equations
(2.21) and (2.22) can then be used to compute the relative isobaric slopes i, and i,
•"••Pf"! ^>PtTt
for use in the horizontal components of the equation of motion.
The slope terms '•inn an^ ^2 D n ar^8e because the vertical distance between any
two isobaric surfaces varies inversely as the mean density in the water layer between these
surfaces. These terms, which remain in modified form even in the vertically averaged, two-
dimensional equations and the sectionally averaged, one-dimensional equation, have been
neglected in numerical models which have been developed to date. In a strong, moderately mixed
coastal plane estuary, typical of those which occur along the Atlantic coast of the United
States, these terms are not negligible. Consider for example the James River estuary. The
mean density of the vertical water column at the mouth of the estuary is approximately
1.025 gm cnT9 , while 30 miles up the estuary the mean density is approximately 1.000 gm cm'3.
The vertical distance between the water surface and the 10 decibar pressure surface (which
occurs at a depth of approximately 10 meters) is therefore about 25 cm (0.82 ft) greater at the
upstream location than at the mouth. The mean longitudinal slope of the 10 decibar surface
relative to the water surface over this 30 mile distance is therefore 5.5 x 10~8. Variations
in this slope with time over a tidal cycle are relatively small. In this same stretch of the
estuary, the longitudinal slope of the water surface trj-j- varies with tidal phase from zero to
maximum values of about ± 1.5 x 10~B .
These relative slopes of the pressure surfaces resulting from horizontal variations
in density, which are significant in estuaries having relatively strong horizontal salinity
13
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gradients, are the primary cause for the characteristic vertical shear in the horizontal current.
In such estuaries the seaward-directed flow during the ebb period typically decreases with depth,
while the flow directed up the estuary during the flood period typically increases with depth.
Consequently there is a net non-tidal circulation with seaward-directed flow in the upper layers
of the estuary and a flow directed up the estuary in the lower layers. The intensity of this
vertical shear in the current pattern is not necessarily related to the vertical gradient in
salinity, as has been pointed out by Hansen and Rattray (1966). That is, vertical mixing may be
strong enough in some estuaries to produce near vertical homogeneity in salinity, but the
horizontal gradients in salinity, and hence density, may still be sufficient to produce signifi-
cant vertical shear in the current. Such a vertical shear can be very important in producing
longitudinal dispersion of an introduced pollutant.
While, in this treatment, the vertical component of the equation of motion has been
reduced to the hydrostatic equation, the vertical velocity remains in certain terms in the
horizontal components of the equation; i.e., in the field acceleration terms and in the
Reynolds stress terms. Thus it is certainly not a_ priori evident that the terms "3 du./dx. ,
and a(u1' u^'^/dx^ are negligibly small.
In order to put Equations (2.17) and (2.18) into a form which offers any hope of
solution, it is necessary to relate the Reynolds stress terms jc and (i^' u.' >. to
the ensemble average field of motion. Except at the boundaries, the viscous frictional term is
generally small compared to the terms involving the Reynolds stresses. Classically, it has
been assumed that the Reynolds stress terms could be replaced by the product of a turbulent or
eddy viscosity and the appropriate component of the deformation tensor. That is,
(2.23)
and
,3u4 3us%
(2.24)
where the eddy viscosity vj» * vi i when t - j and is otherwise zero. Similarly
v7 , • v« . for I - j and is otherwise zero. Equations (2.17) and (2.18) may then be written
3x^,
and
au2 a(u2u) • a,, i
• • + — —- — '•* ' » -g —^ —ft io — ^
oC dx* ox - Zypjfi p
(2.25)
a K.2 (£ + 55)} + si
(2.26)
axj 1V2,3 \ax3 "
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These equations have sometimes been further simplified by: (a) assuming
2 1 °* V2 2
VH
t*le norizont*l "Austausch", or horizontal eddy viscosity;
3 = V2 3 " V3 ' the vertlcal eddy viscosity; and (c) neglecting du^/dx-j
and 3Uj/dX2 in comparison with duj/dx-j . The horizontal components of the ensemble mean
equation of motion then become
vi i " vi 2
(b) assuming
dp
* £U
a r 3un
sij (2vH 3x7}
au,
au
and
au, a(u,u.)
(2.27)
\ i dpa a r /Su2 dui
.p.J " a^T • **! + 3x7 IVH laxY + a^/
(2.28)
There does not exist any clear direct observational or theoretical evidence for
further simplifying these equations, even though some indirect evidence will be used later in
this section in further modifying them. It should also be noted that the coefficients of eddy
viscosity are functions of position and, perhaps, time. They are dependent upon the intensity
of the turbulent velocity fluctuations, and hence probably on the velocity, the boundary
roughness and distance from the boundary, but the relationships are riot known. The vertical
eddy viscosity must also be dependent upon the vertical stability, or more probably the
Richardson Number.
In any case, Equations (2.27) and (2.28), together with Equations (2.21) and (2.22)
and the equation of continuity (2.8), are the equations which one must start with in developing
a three-dimensional dynamic model of an estuary. If the goal is to develop a dynamic model
capable of computing the time dependent surface elevation TI and the three components of the
time-dependent velocity u. , these equations, standing alone, in the form given, are insoluble,
since there are more unknowns than equations.
for overcoming the difficulty.
However, .there are several possible procedures
In Section 2 of this chapter a two-dimensional vertical-averaged model is
developed. In that model the surface elevation TI appears as a dependent variable. Assuming
for the moment that such a two-dimensional model can be shown to provide satisfactory estimates
of the time and space variations in TI , even for estuaries with significant vertical and
horizontal variations in salinity (and hence density) and with vertical shear in the velocity
field, then such a two-dimensional model could be used to determine TI , as well as the vertical
mean values of the two horizontal velocity components. Thus, given TI as a function of time
and position, Equations (2.8), (2.27), and (2.28), together with the auxiliary equations (2.21)
and (2.22) for determining the relative slope terms, could theoretically be solved, by appro-
priate numerical means, to obtain the time-dependent three-dimensional velocity field.
In order to make such an approach possible, however, it will be necessary to develop
empirical and/or theoretical relationships for the coefficients of eddy viscosity in terms of
known or computable parameters.
15
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It should also be pointed out that the slope terms 1, and i, depend upon
" I P 1 M — |P>*1
the spatial distribution of density and hence of salinity. Consequently the dynamic model
cannot be used to compute the field of motion, which then would be used in the determination
of the distribution of salinity. The equations of salt continuity, to be developed in the next
sub-section, must be solved simultaneously with the dynamic equations.
1.2.3 Conservation of Dissolved Constituents
The differential equation expressing the conservation of salt, in three-dimensions,
for instantaneous values of the salt concentration and the velocity, may be written
If --str^i') +Dfxfr (2.29)
where 0 is the molecular diffusivity. Proceeding as in the cases for conservation of mass
and of momentum, the ensemble average salt balance equation is given by
(2.30)
Note that, using the equation of continuity 3u./3x. • 0 , the first term on the right side of
(2.30) may also be written
a ( x _ as
axj^ * 1B' ui ax^
In order to make Equation (2.30) tractable, it is usual to replace the non-advective flux terms
by Fickian-type diffusion terms. That is,
(2.31)
where K. - K^ for •£, - i and otherwise is zero. Then Equation (2.31) is written
If ' - ^ <«!•) +
where the molecular diffusion tern has been neglected compared to the eddy diffusion terms,
and it has been assumed that K^ • Kj " Kg , the horizontal eddy diffusivity.
As in the case of the coefficients of eddy viscosity, there does not exist an adequate
theoretical or empirical basis for relating the coefficients of eddy diffusivity to the ensemble
mean velocity distribution and to the distribution of density.
However, some limited evidence does exist which can be used to at least indicate
further simplifications of the three-dimensional conservation equations, and also a possible
format for relating the eddy coefficients to known or computable parameters. Before proceeding
with such a further treatment, however, consider the general form of the equation expressing
the balance for a dissolved constituent other than salinity, for which there may be internal
sources and sinks.
16
-------
In Equation (2.32), it has been assumed that salt is conserved within the water body;
that is, that there are no internal sources and sinks for salt. Sources and sinks may exist at
the boundaries, however. For example, the vertical diffusion term takes on a boundary value at
the water surface equal to the effective flux of salt across the boundary resulting from the
difference between evaporation and precipitation. In the case of a dissolved substance which
is not conserved within the water body, such as an oxidizable pollutant or a radioactive
material subject to decay, additional terms must appear in the equation.
Designating the ensemble-averaged concentration of a non-conservative material by C ,
then the three-dimensional equation expressing the local time rate of change of concentration
is given by
3x7 - 5x7
<2-33>
where r is the mass rate of generation of substance per unit volume and r. is the mass
rate of decay of substance per unit volume. Again, expressing the non-advective flux »-•" + »^~ IK<» »»-' + r» " r* (2-3/*)
1.2.4 Some Possible Further Simplification of the Three-Dimensional Conservation Equations
for Mass. Momentum and Salt
Studies of one-dimensional and two-dimensional models of conservation of mass,
momentum and salt, and of one-dimensional and two-dimensional water quality models, suggest
the following:
(a) The greater the detail provided by the model with respect to variations
in velocity in time and space, the less significant are the horizontal
eddy diffusion terms, and probably the horizontal eddy viscous terms.
(b) In the vertically averaged two-dimensional equations of motion, the
bottom boundary frictional term is satisfactorily represented by a term
of the form
u(u» + v»)fe
8
V
where C^ is the Chezy coefficient.
(c) In a study of the James River estuary using a tidal mean, lateral-
averaged two-dimensional equation of motion, Fritchard (1956) showed that
the field acceleration term u^ au^/ax-j is small compared to other terms
in the equation.
These observations suggest that the three-dimensional equations expressing conservation
of mass, momentum, and salt might be further simplified as follows:
17
-------
(a) In Che x^-conponent of Che equation of moCion, Equation (2.27), neglect
the field acceleration term u^ du^/ax^ and the horizontal eddy viscous
terms 3/dx^ {2vH du^/ax^} and 3 /ax, {vH(dUjyax2 + du2/dx.) }
(b) In Che x2 -component of Che equation of motion, Equation (2.28), neglect
the field acceleration term u^ a^/ax^ , and the horizontal eddy viscous
terms 3/dx^ {vH(au2/3x^ + au^/axj) } and a/dXj (2vH au2/3x2}
(c) In the salt balance equation, Equation (2.32), neglecC Che horizontal
eddy diffusion terms 3/ax^ (Kg as/BXj] and 8/8X2 [K^ as/3x2}
The x.-component of the equation of motion then becomes
au, au, au, a_ , dp. . , au,,
<2-35>
the Xj-component of the equation of motion then becomes:
.. . ap . f au,,
at + "I ai + U2 3 * -g '8 ^.P.T, - F ~ *1 + 5 K ai^ <2'36>
and the salt balance equation becomes
If ' ' a + 3
3
The auxiliary Equations (2.21) and (2.22) must still be used to compute the relative isobaric
slopes i^ and i2 _ from the horixental and vertical distribution of density (salini
Also, Che equation of continuity in the form given in Equation (2.8), that is
3u.
must be also solved simultaneously with the dynamic equations and the salt balance equation.
The solutions to these equations must satisfy the following vertical integral
conditions:
.TI .du, au, du, -,
J (at1* "i a^+ "2 a4+ 8 h.p.r, ' ^2) ^3
V (2.38)
-''i-h(l
du, au-
ui 55+ U2 aif + 8 i2>pfr,
t (2.39)
18
-------
5x7 J „ ul dx3 + axT j „ U2 d*3 ' - f? <2-*°>
* * '
and
-? xl -5 X2 -s
where -5 is the x^ value at the bottom, (E-F) is the difference between evaporation and
precipitation at the water surface, expressed in units of length per unit time, T , and T
Wy X VJ
are the x^ and x^ components of the wind stress at the surface, and the subscript h
Implies a vertical average. That is,
U2.h •
The solutions to the three-dimensional equations must also satisfy the following
cross- section integral conditions, where in order to simplify the notation it is assumed chat
the coordinate system is oriented such that . x^ is directed along the axis of the estuary,
and the cross section at any point x^ has an area a and is in the plane perpendicular to
and
(E-P)b
A rr
Ft JJ
a rr
ax7 JJC "1
For the coastal plane or drowned river valley estuaries characteristic of the Atlantic
Coast of the United States, which in their upper reaches extend into the fresh water tidal
river, and are bounded at their upper end by the position where the rise and fall of the tide
and the tidal oscillations in the current are reduced to zero, an additional integral condition
can be obtained. If Equations (2.45) and (2.46) are integrated from x^ - (XI)Q where the
integrated flow through the cross -section equals the river flow QR to a cross section in
the estuary proper located at x^ , then
da - QR - |f {j a dxj (2.47)
-f1
<*l>
tt f*i r** f'u-n tt , \ A a rri /rr j \ j i /^ /o^
u,s do - \ —"• s dof dx, - ^ 1 s dof dx. (2.48)
JJ_1 J/_\'- o J*_ J 1 8tL-l/-\'.JJ_ J 1J v '
19
-------
Equations (2.8), (2.35), (2.36) and (2.37), expressing the conservation of mass,
momentum and salt in three spatial dimensions, plus Equations (2.38), (2.39), (2.40), (2.41),
(2.45), (2.46), (2.47) and (2.48) expressing the integral boundary conditions, together with
the auxiliary Equations (2.21) and (2.22), represent a set of equations which, on the basis of
present knowledge, include the most important processes controlling the dynamics and kinematics
of an estuary. As such, these equations offer a suitable basis for a three-dimensional,
numerical model of an estuary.
The dependent variables to be computed from such a model are: (a) the surface
elevation of the estuary n as a function of horizontal position (z^, x2) and of time t ;
(b) the three velocity components, u^, Uj and u~, as a function of horizontal position and
depth (x^, xj, Xj) and of time t ; and (c) the salinity s as a function of horizontal
position and depth (x^, x^, x^) and of time t. The necessary inputs to the model are:
(a) the physical dimensions of the estuary, i.e., the horizontal boundaries and the distribution
of depth at, say, mean low water; (b) the temporal and spatial distribution of atmospheric
pressure and of surface wind stress over the period covered by the computation; (c) values of
the vertical coefficients of eddy viscosity v^ and eddy diffusivity K^ as a function of
position (xp Xj, x~) and time during the period covered by the computations; (d) values of
the water density p as a function of position (x^, x2, x^) and time during the period
covered by the computations; (e) the fresh water inflow to the estuary as a function of time
during the period covered by the computations; (f) values of the difference, evaporation minus
precipitation, at the water surface of the estuary, as a function of horizontal position
(x^, X2> and time during the period covered by the computations; (g) the elevation of the water
surface TJ along the line marking the seaward boundary of the estuary, as a function of time
during Che period covered by the computations; (h) values of the salinity s in the cross-
section marking the seaward boundary of the estuary, as a function of time during the period
covered by the computations; and (i) an initial set of values of the dependent variables at
all positions In the estuary.
The spatial and temporal distribution of density is required for computations of the
relative isobaric slopes i, _ and i0 „ from Equations (2.21) and (2.22). The density
J-jP," *»P,Tl
of the estuarine water depends on temperature and salinity. In estuaries in which the relative
isobaric slopes contribute significantly to the dynamic balance, salinity variations are
generally much more important than temperature variations in determining density variations.
Consequently it should be adequate to assume either a constant water temperature throughout
the estuary, or a time independent spatial distribution of temperature-characteristic of the
period covered by the computations. The initial distribution of salinity would then be used,
together with the assumed temperature, to determine the initial spatial distribution of density.
Equations (2.21) and (2.22) would then be used to compute initial values of the relative
isobaric slopes it _ „ and i, _ „ as a function of position (x,, x-, x,). After each time
•"••P,"! *»P»T1 1. & J
step in the computation, the new distribution of salinity would be used with the assumed
distribution of temperature to determine the spatial distribution of density, from which new
values of i, _ and L, _ „ would be found for use in the next time step.
l.P.f) Z.P.I
The major remaining unknown input terms for the proposed model are values of the
vertical eddy viscosity Vj and the vertical eddy diffusivity K^. Some evidence is available
as to the relationship of K^ to the geometry of the estuary, the velocity shear and the
vertical stability. Presumably the eddy viscosity would depend on these parameters in a
similar manner. However, research is required to develop generally applicable relationships
such that VQ and K, can be determined from known and/or computable parameters.
20
-------
Pritchard (1960) using the results of studies made in the James River (Pritchard
1954, Kent and Pritchard 1959) found that for tidal mean, lateral-averaged data, the vertical
eddy diffuslvity was related to the mean tidal velocity |u| , the water depth h , the
height H , period T , and wave length L of the surface wind waves, and the Richardson
Number R^ by the equation
i H e-2nZ/L (1+BRi)-2 (2
h
where Z is the vertical distance below the water surface (i.e., Z - TI - x^ ) , h is the
vertical distance between the water surface and the bottom, and TI, ?, and B are non-
dimensional constants. For the James River estuary, the numerical value of these coefficients
q -l
were: T) - 8.59 x 10 ; ? - 9.57 x 10 ; and B - 0.276. This equation is based on data
collected during one relatively short time interval in one segment of one estuary. It is
based on tidal mean data. The degree to which these results can be extended to other estuaries
and to conditions of time-varying currents within the tidal cycle is not known.
For an estuary which is vertically homogeneous and for calm wind conditions,
Equation (2.49) reduces to
K3(Z) - 9.59 x 10'3 lu| Z
-------
2. TWO-DIMENSIONAL MODELS
D. W. Pritchard
2.1 IHTRODUCTIOH
The basic time-dependent equations expressing the conservation of mass, momentum, and
of a dissolved constituent of the waters of an estuary, in three spatial dimensions, as
developed in Section 1 of this chapter, will serve as the starting point for the treatment of
two-dimensional models of an estuary to be presented in this section. This section will include
discussions of the significance of terms which are neglected and of simplifying assumptions
which are made in order to develop a model composed of a tractable set of equations. Where
adequate descriptions are already available in scientific publications or technical reports,
existing models will be referred to, but not described in detail.
2.1.1 The Coordinate System and the Notations to be Used
In Section 1 of this chapter tensor notation was employed in conjunction with a
right-handed rectilinear coordinate system, x^ , with the x^ and x^ axes lying in a
horizontal plane coincident with the plane of mean tide level, and the x^-axis directed
upwards, positive in the direction opposite to the acceleration of gravity. In this section
it will generally be more convenient to employ an x-y-z rectilinear coordinate system, and
a vector-scalar notation system. The x- and y-axes are considered to lie in the horizontal
plane coincident with the plane of mean tide level, and the z-axis is directed upwards. Thus
the x-axis corresponds to the x^-axis; the y-axis corresponds to the j^-axis; and the
z-axis corresponds to t*"» x,-axis. The ensemble average velocity components along the x-,
y-, and z-axis will be designated by u, v, and w, respectively. Thus u in the
notation to be employed in this section corresponds to u^ in the notation used in Section 1.
Likewise, v corresponds to u2 , and w to u3 .
Much of the remaining notation used in Section 1 can be carried over directly to this
section. Where confusion might result, special note is made in the text.
2.2 THE BASIC BQUATTOMS FOR A VERTICALLY AVERAGED TWO-DIMENSIONAL MODEL
2.2.1 Conservation of Mass
We start with Equation (2.6), the ensemble mean equation of mass continuity in three
spatial dimensions as developed in Section 1. In the notation used in this section, that
equation is written
22
-------
snl _ o (2.52)
az
Now consider that, at any point (x,y,z), the value of the density and of each of the velocity
components is equal to the sum of the vertical mean value of that variable taken over the water
depth plus a deviation term representing the difference between the value of that variable at
the point (x, y, z) and the vertical mean value of that variable. That is, we designate
P • Ph + Ph'
u ' "h + V
(2.53)
v - vh + vh'
w - wh + wh'
where
dz - h
f? v dz - h
1
Note from above that use has been made of the following notation for the mean value over the
total water depth at time t:
where f is any spatial-dependent variable, r\ is the surface elevation, -5 is the z
value at the bottom, and h - (?+TI) is the total water depth at the time t . Hence, from
(2.53) and (2.54), we note that
h - h - h ' h - ° (2-56)
Substituting from Equation (2.53) into Equation (2.52) and taking the integral of that equation
over the total depth h - (?+T|) gives
h h> + 57 h> ' ° <2'57>
-------
Note that even though (p^1)^ • <°h')h " ("h'^h " ° ' che cr°ss-product terms ("h'Ph'^h and
^h'Ph'^h are not nece88arily zero. However, In this case it is well known that
and
h ^.
vh<>h
i.e., these ratios are on the order of 10 or less. Thus, the last two terms in (2.57) may be
neglected. Expanding the differentials, and using the equality ah/at - dri/dt (since
h - 5 + n and d§/3t - 0), Equation (2.57) can be written
'
As discussed in Section 1, the first term in (2.58) may be neglected for a fluid such
as water which baa a very small compressibility, and for which the Boussinesq approximation is
appropriate. Consequently, the equation of continuity of mass in two-dimensions is reduced to
a(hu.) 3(hv.) ,_
-- + -- ' - <2-59>
2.2.2 Conservation of Momentum
Xhe three-dimensional ensemble-averaged equations of motion in the form given by
Equations (2.17) and (2.18) of Section 1 of this chapter serve as the starting point for the
development of a vertically averaged, two-dimensional model of the dynamics of an estuary. In
the notation used In this section, these equations are (emitting the molecular viscous term)
p dx dx k dy K dz k
(2.60)
and
a"E ax ay ac ~8 ay ~8 y,p,Ti ~ ~p ax T ax ' v 'k ay" k " 5? ^v w 'k
(2.61)
How, we introduce into these equations
v - vh + vh'
w " wh + "h
24
-------
and integrate the equations from the bottom (z - -;) to the surface (z - ri) . The resulting
equations are *
- -g
> J
- e- — •fhdv'v,.' + , >«.]• + ,-i- T - i-i-
E dy I Ti h x 'k hJ EpT w,x hpT
(2.62)
find
- -g
K Sy
(2.63)
TB,x
where T and T , are the x- and y-components of the surface wind stress, while TH
W|X w»y D|j^
and TB are the x- and y-components of the bottom stress. This last term in each equation
has usually been replaced by a term of the form
and
where
is the Chezy coefficient.
These forms of the equation of motion are similar to the equations used by Leendertse
(1967) and by Masch e_t al. (1969) , among others, in formulating a two-dimensional dynamic
model of an estuary, but with two rather significant differences.
Equations (2.62) and (2.63) each include a term which arises from the relative slope
of the pressure surfaces due to the horizontal variation in density (salinity) . The term
h , h and *uh'vhl*h * represent the vertical averages of
the cross-products of the velocity departures at any depth from the vertical average velocity.
25
-------
If Che velocity field has a vertical shear which is predominantly of one sign, then these terms
can be of significant size. In the typical drowned river valley estuary, the vertical shear
»
of the horizontal velocity is predominantly of one sign; i.e., if x is oriented along the
axis of the estuary toward the sea, then du/dz is predominantly positive.
It can be argued that terms of the type ^uh'vh'^h can ^e rePlace<^ by the product of
a coefficient, having the dimensions of kinematic viscosity, times an appropriate component of
the deformation tensor, as was done with the ensemble average terms such as k . That is,
we might assume that
' "
and
x "
and
dvK ra_ •> i 3p0
-h W m -8 (ly + <1y,P,r1>J ' £ W ~ f uh
1 (2.67)
._ Tw,y '8
Equations (2.21) and (2.22) from Section 1 of this chapter must be used as auxiliary
equations to the above forms of the equation of motion In order that the relative isobaric
slopes, 1,, and 1 , be computed from the distribution of density (salinity). Since
xip tM y »PIM
the temporal changes in the distribution of salinity will alter the values of the relative
Isobaric slope, the equation expressing conservation of salt in two-dimensions, to be developed
in the next sub-section, must be solved simultaneously with the equations of motion and the
two-dimensional form of the equation of continuity of mass.
26
-------
2.2.3 The Vertically Averaged Two-Dimensional Salt Balance Equation
Equation (2.30) given in Section 1 of this chapter, expressing the conservation of
salt in three dimensions, serves as a starting point for this development of the vertically
averaged salt balance equation. Omitting molecular diffusion, that equation is written in the
notation used in this section as
- - - - - - k - 7
SZ
k - 57
<2-68)
As was done in the treatment of continuity of mass and salt, we express the dependent variables
as the sum of a vertical average plus a difference term. For example
where
Integrating Equation (2.68) then gives
9(hsh)
-
,
57
a
- SK tWV«hr + k>hJ
(2.69)
Again, this equation contains terms of the type ^uh'si'^h ' wn^c'1 have not normally been
discussed in connection with existing two-dimensional models. These terms arise from the
vertical shear in the horizontal velocity and the vertical gradient in salinity . We might
argue that these terms can be replaced by a Fickian-type diffusion term as was done for terms
of the type (u's') in the three-dimensional case. That is, we might make the substitution
-------
Coefficients such as K* and K* , which arise from terms involving shear in the velocity
field and spatial gradients in the concentration distribution, are sometimes called "effective"
diffusion coefficients. It is to be expected, however, that these coefficients will differ
considerably from the eddy coefficients of diffusion which were introduced in the three-
dimensional salt balance equations. There is also less justification for neglecting the
diffusion terms in Equation (2.71) than for neglecting the horizontal viscous terms In the
two-dimensional momentum equations. It is reasonable to assume, however, that K^* - K* - Kg*
the "effective" horizontal diffusivity, and Equation (2.71) would then be written
-Vh - -vi +
- K* } + k {**** w\ + "h (E-p) <2-72)
The vertically averaged equation expressing the conservation of a dissolved or finely
divided constituent, having sources and sinks in addition to those for salt, would then be
written
ac <2'73)
+ iy K* 3jr} + ^h + hrg.h - ^d.h
where r h is the vertically averaged mass rate of generation of the constituent per unit
volume (such as the discharge of a waste material) and r^ h is the vertically averaged mass
rate of decay of the constituent per unit volume (such as oxidation of an organic waste).
2.2.4 Summary of the Set of Vertically Averaged Dynamic and Kinematic Equations Suitable to
Serve as the Basis of a Two-Dimensional Numerical Model of an Estuary
The vertically averaged equation of mass continuity,
»
together with the somewhat simplified vertically averaged equations of motion, given by
(2.66)
a\k
and
28
-------
ph ay <2^7)
V (Vi_8 + U,_s)^
_. Ji n n
T ~K i ""jj and ,, and (i >. can then be determined from the relationships
X»P»T| n y>P»M n
(obtained by taking the vertical average of Equation 2.20 from Section 1 of this chapter)
TST,
and
h - * ' W ' * (T1+5) I*l (2'75)
The required inputs for numerical solution of Equations (2.59), (2.66), (2.67) and
(2.72) are: (a) the physical dimensions of the estuary; (b) the temporal and spatial distribu-
tion of atmospheric pressure and of surface wind stress over the period covered by the compu-
tations; (c) the fresh water inflow to the estuary as a function of time during the period
covered by the computations; (d) values of the difference, evaporation minus precipitation, at
the water surface of the estuary as a function of horizontal position and time during the
period covered by the computations; (e) an assumed horizontal distribution of vertical-averaged
water temperature in the estuary; (f) values of the Chezy resistance coefficient as a function
of position in the estuary; (g) values of the effective horizontal diffusivity as a function
of position in the estuary and of time over the period covered by the computations; (h) values
of the surface elevation n and of the vertically averaged salinity sn along a line marking
the seaward boundary of the estuary, as a function of time during the period covered by the
computations; (i) an initial set of values of the dependent variables at all positions in the
estuary .
The dependent variables to be computed from the numerical solution of these equations
are: (a) the surface elevation TI ; (b) the vertically averaged horizontal components of the
velocity, uh and vn ; and (c) the vertically averaged salinity sn ; all as functions of
horizontal position (x, y) and of time over the period of the computations. Equations (2.74)
and (2.75) must be used as auxiliary equations of the computations to obtain values of the
29
-------
vertically averaged relative isobaric slopes n and h from the computed
distribution of salinity and the assumed distribution of temperature. The additional integral
condition, obtained from the integration of the vertically averaged equation of continuity across
the estuary and then along the estuary from the landward boundary of the estuary (x - XQ) ,
where the time variation in water surface elevation is zero, must also be satisfied. That is,
b
u
where J dy indicates an integral across the width of the estuary b(x,t) ; and o(x,t) is
b
the cross-sectional area of the estuary perpendicular to the x-axis, which is considered here
to be oriented along the estuary.
2.2.5 An Alternate Development of a Set of Vertically Averaged Dynamic and Kinematic
Equations in an Estuary
Jan J. Leendertse (personal communication) has suggested an alternate approach to the
development of the vertically averaged equations which offers some computational advantages
over the development given above. Briefly, this approach is based on the assumption that the
ensemble mean velocity components, u(x,y,z,t) and v(x,y,z,t) , and salinity s(x,y,
z,t) are related to their respective vertical-averaged values ua(x,y,t) , v^x.y.t) ,
and sh(x,y,t) by the relationships
v - vh {1 + f (z)}
h v (2.77)
and
where fu(z) , fy(z) and ffl(z) are functions of the vertical coordinate z , and also perhaps
of the time t , and describe all the vertical variations in u , v , and x , respectively.
Mote that this manner of relating the ensemble mean values of the variables to their respective
vertical- averaged values is more restrictive than the division given in Equation (2.53), since
Equation (2.77) requires that u, v and s are zero everywhere in the vertical when their
respective vertical mean values uh, vh and sh are zero, a condition not generally true.
The vertical integrals of the functions fu(z) , fy(z) and f (z) must all be zero.
However, cross-products involving these functions do not necessarily have vertical averages
equal to zero. The integral cross-products which appear in the vertically averaged equations
are
u2 dz - u^ J" {1 + f^z)}2 dz
(2.78)
30
-------
J uv dz - u,^ J {1 + fu(z)J [1 + £v(z)} dz
(2.79)
-Vn r u + fu-fv(z)i dz
(2.80)
us dz - u^ {1 + fu(z)} (I + fg(z)} dz
VhJ
_? fs]d« (2.82)
auv " K J^ tl + fu(z)'fv(z)} dz (2.83)
aus " E .^ {1 + fu(z)-f8(z)1 dz (2'84)
and
avs ' E .f {l + MZ>'VZ» dz <2-85>
where auu> auv, aus> and avs are called the distribution functions for the respective
cross-products which define them. Hopefully, the spatial, and perhaps temporal variations in
these parameters are sufficiently regular to be revealed from observations of the vertical
variation in the velocity and salinity in the estuary. The uncertainties in these parameters
substitute for the uncertainties in the effective horizontal coefficients of viscosity and
diffusivity. Hopefully, however, the variations in these in space and time would be more
readily related to known parameters than in the case of the effective coefficients.
In any case, the vertically averaged equation of continuity has the same form in
this development as that given earlier. That is, Equation (2.59) remains applicable. The
vertically averaged equations of motion become, however,
,t , a
h - ^T a5T
(2.86)
31
-------
and
--*$*
(2.87)
o o IL
• fhuh + i:
where the vertical average of the ensemble mean cross-product terms involving the turbulent
velocity deviations, such as Byy* auv>
aus, and ays. Certainly the observational requirements for establishing empirical relation-
ships for the distribution functions are less severe than in the case of the effective
horizontal coefficients of viscosity and diffusivity. On the basis of available evidence, the
distribution functions will, for most estuaries, and for most times during the tidal cycle,
range between 0.5 and 1.5. The more nearly the estuary is vertically homogeneous in velocity
and salinity, the closer the distribution functions will be to unity.
32
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REFERENCES FOR SECTIONS 1 AND 2
Bowden, K. F., 1960: Circulation and mixing in the Mersey Estuary. Publication No. 51,
International Association of Scientific Hydrology, Commission of Surface Waters.
Hansen, Donald V., and Maurice Rat tray, Jr., 1966: New dimensions in estuary classification.
Limn, and Oceanography, 11. No. 3 (July).
Kent, R. E., and D. W. Pr it chard, 1959: A test of mixing length theories in a coastal plain
estuary. J. Mar. Res.. 18. No. 1 (June).
Leendertse, Jan J., 1967: Aspects of a computational model for long-period water-wave
propagation. Rand Corporation Memorandum RM-5294-PR, May, 1967.
Masch, Frank D., 1969: A numerical model for the simulation of tidal hydrodynamics in shallow
irregular estuaries. Technical Report HYD 12-6901, Hydraulic Engineering Laboratory,
Department of Civil Engineering, The University of Texas at Austin.
Pritchard, D. W., 1954: A study of the salt balance in a coastal plain estuary. J. Mar. Res..
.13, No. 1.
Pritchard, D. W., 1956: The dynamic structure of a coastal plain estuary. J. Mar. Res.. 15,
No. 1.
Pritchard, D. W., 1960: The movement and mixing of contaminants in tidal estuaries.
Proceedings. 1st Conference on Waste Disposal in the Marine Environment, E. A. Pearson
(Ed.). Pergaxnon Press.
Sverdrup, H. U., Martin W. Johnson, and Richard H. Fleming, 1946: The Oceans, Their Physics,
Chemistry, and General Biology. New York, Prentice-Hall.
U. S. Naval Oceanographic Office, 1966: Handbook of Oceanographic Tables (Compiled by
E. L. Bialek). Special Publication SP-68, Washington, D. C. (Replaces H. 0. Publ.
No. 614.)
33
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3. ONE-DIMENSIONAL MODELS
Donald R. F. Harleman
3.1 INTRODUCTION
One-dimensional equations for the hydrodynamlc and mass transfer processes in estuaries
have an obvious advantage of mathematical tractability in comparison with their multi-dimensional
counterparts. In addition, the one-dimensional equations utilize and predict information that
is related to available or accessible observational data. This is especially true in dealing
with the mass transfer equations for water quality parameters involving source and sink terms,
such as biochemical oxygen demand and dissolved oxygen.
An important objective of this section is the presentation, comparison and discussion
of the various one-dimensional hydrodynamic and mass transfer models which have been used in
estuaries. Thus it is necessary to consider the most general form of these equations in which
quantities such as concentration and velocity are functions of both longitudinal position and
real time. The general equations are employed as the basis for evaluation of simpler mathemat-
ical models which use non-tidal advective velocities.
The one-dimensional mass transfer equation is derived by a spatial integration of the
three-dimensional equation over the flow cross section. Thus any quantities such as velocity,
concentration or dissolved oxygen are spatial averages over a cross section corresponding to a
specified period of time averaging. The one-dimensional continuity and momentum equations are
derived by considering a material element occupying a cross section of the esjtuary. The one-
dimensional equations are well suited for estuaries displaying vertical and lateral homogeneity;
however, they have been successfully applied to estuaries with varying degrees of sectional
non-homogeneity. Applications to a fully stratified or saline-wedge estuary should be excluded.
The latter problems can be treated by the theory of two-layer stratified flows (Harleman 1960,
Keulegan 1966).
The various quantities, such as velocity, concentration, cross-sectional area, in the
one-dimensional mathematical models are assumed to be functions of a single spatial variable x
and time t . The longitudinal distance x is measured along the axis of the estuary and
cross-sectional areas are normal to the local direction of the axis, as shown in Figure 2.1.
One-dimensional mathematical models encompass a wide range of time-averaging concepts.
This has been the subject of confusion in the literature due to the use of ambiguous terms for
time averaging. Adjectives such as "unsteady", "quasi-steady state" or "steady" are meaningless
unless precisely defined with respect to (i) the duration of the time period to be used in
averaging, and (11) the quantity to which the adjective applies. Careful definition is particu-
larly important in estuarine problems because of the multitude of time-averaging concepts which
have been employed. As shown in Figure 2.2, the term "unsteady" when applied to the section-
mean velocity may Imply as many as four possible periods of time averaging. These are defined
as follows:
34
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X
TI
NATURAL B
a 6
h * -I
* ' i
, 1 * 1
W.^uun,,."
OUNDARY/ """^^r^
REFERENCE DATUM
1 J^ ^^^
rr-rfftf^ -1 FOR TIDAl
-^
JfP^ 1 MEAN WATER LEVEL
— = — MEAN LOW WATER
CAL CHANNEL
. FLOW
TTWWnV^ \ BOTTOM OF SCHEMATIZED CHANNEL
z.
Fig. 2.1 Longitudinal and transverse schematization of an estuary.
Fig. 2.2 Various periods of time-averaging the tidal velocity.
35
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(i) If Atj is of the order of magnitude of a minute, the time average of the velocity
fluctuations associated with the fluid turbulence is obtained. This will be referred to as
the tidal velocity U . Its magnitude varies in real time between zero and the maximum tidal
velocity UL-— > and the direction changes from flood (landward) to ebb (seaward) during each
tidal period. The time of zero tidal velocity will be referred to as the time of slack tide;
high water slack if the velocity change is from flood to ebb and low water slack for the reverse.
The tidal velocity is always an unsteady quantity.
(ii) If At» is approximately six hours, the average magnitude of the tidal velocity during
either the flood or ebb portion of the tidal period is obtained. This will be referred to as
mean flood (or ebb) velocity. It is inherently unsteady because the magnitudes of successive
mean flood and ebb velocities change from day to day and week to week in accordance with the
spring- neap variations of the lunar month.
(iii) If At, is equal to a tidal period (12.4 hours for a semi-diurnal tide), the non- tidal
advective velocity is obtained. Whenever the time rate of change of estuary volume between the
section under consideration and the head of tide is a harmonic function of tidal period, the
non-tidal advective velocity will be equal to the velocity due to freshwater inflow, 0£
(Pritchard 1958) . The non-tidal advective velocity may be steady or unsteady depending on the
time variation in the magnitude of the freshwater inflow.
(iv) If At^ is equal to or greater than 25 hours the mean, non- tidal advective velocity for
the specified period of averaging is obtained.
Time-averaging terms may be applied to quantities other than velocity and care should
be taken to specify the intention. For example, consider the instantaneous or pulse injection
of a tracer into a steady, uniform flow. In this case the velocity is steady, however the con-
centration at a fixed point is a function of time and is therefore unsteady.
The various one-dimensional mathematical models employed in analysis of pollution
problems in estuaries can be classified according to the length of the time averaging associated
with the advective velocity. It is essential that good judgement be used in arriving at the
decision of the type of mathematical model to be used. An important function of a state-of-the-
art review is to provide a sound basis for such decisions. Mathematical models for the hydro-
dynamic and mass transfer processes for various time- aver aging periods related to velocity are
presented in the following sections. Considerable attention will be given to the development
of models using real time in order to provide a basis for comparison with other models using
longer periods of time averaging.
3.2 MATHEMATICAL MODELS IN REAL TIME
If the events in an estuary are to be treated in real time, the tidal velocity and
concentration quantities in the mathematical models are those associated with measuring instru-
ments having integrating periods which are short compared to the tidal period. We consider an
estuary of arbitrary shape and derive the one-dimensional form of the mass transfer equation for
a non-conservative substance. The resulting differential equation describes the temporal and
spatial distribution of concentration of that substance in an .estuary. In addition, the one-
dimensional continuity and momentum equations are derived. The latter pair of differential
equations may be used to determine the tidal velocity which appears in the advective term of the
mass transfer equation.
36
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3.2.1 Mass Transfer Equation for a Non-Conservative Substance
The one-dimensional mass transfer equation can be obtained by direct integration of
the three-dimensional equation over a cross section normal to the axis of the estuary. The
three-dimensional equation for turbulent flow, neglecting molecular diffusion is
"at u ^x v ^y w ^z" " "5x ' x 3x' 3y * y 3y* 3z * z 3z' i
where u, v, w are the time-averaged velocity components associated with turbulent flow, c
is the local concentration, e , e and e are turbulent diffusivities , and r. is the time
' x' y z i
rate of generation (or decay) of substance per unit volume of fluid. We define
u - U + u"
v - v"
w - w" (2.90)
where U - -r | u dA is the real time longitudinal tidal velocity averaged over the cross
A A
section. The double-primed quantities u", v" and w" are spatial deviations of velocity from
the section-mean value due to the vertical and lateral velocity distribution. Note that
f u" dA - 0
JA
and that v" and w" are not zero even though the section-mean velocities V and W are
zero in a one-dimensional flow. In a similar manner, we define
c - C + c" (2.91)
where C - 4 | c dA is the concentration averaged over the cross section and c" is the
A JA
spatial deviation of the concentration due to the variation of concentration over the cross
section. Concentration is defined as the mass of substance per unit mass of solution (or weight
of substance per unit weight of solution) .
Equations (2.90) and (2.91) are introduced into the three-dimensional mass transfer
Equation (2.89), the products of sums are expanded and each term is integrated over the cross-
sectional area A. Inasmuch as the expanded equation contains at least thirteen terms a complete
development will not be presented. Rigorous derivations of this type have been given by Okubo
(1964) and Hoi ley and Harleman (1965) . After simplification, the one-dimensional equation may
be written as
(2.92)
The integral term on the left side involving the cross product of the longitudinal velocity and
concentration deviations represents the mass transport associated with the non-uniform velocity
distribution. This is usually designated as the longitudinal dispersion term. The quantity e~x
37
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is the spatial-mean value of the turbulent diffusivity. The term r^ represents the time rate
of internal addition of mass per unit volume by generation of the substance within the fluid.
The term r is the time rate of addition of mass per unit volume externally across the lateral
boundaries (sides, bottom and water surface). The two latter quantities are usually known as
source and sink terms. As examples of internal terms, a positive r^ would result from oxygen
produced by suspended algae; and r. would be negative for a substance undergoing radioactive
decay or for the consumption of oxygen by suspended matter possessing BOD. In the case of mass
transfer across external boundaries, a positive value of rg could be due to the absorption
of atmospheric oxygen across the free surface of an oxygen deficient stream. A negative rfi
would result from the adsorption of a tracer dye on the solid boundaries.
For steady, uniform flow, Taylor (1954) and Aris (1956) have shown that the advectlve
mass transport associated with the cross product of the spatial deviations u" and c" can be
represented as an analogous one-dimensional diffusive transport. To distinguish this process
from turbulent diffusion, which is associated with the temporal deviations u' and c' , the
transport due to spatial deviations is called longitudinal dispersion. On this basis a coeffi-
cient of dispersion E is defined in terms of the concentration gradient as
J u"c" dA - - AE|| (2.93)
The negative sign indicates mass transport in the direction of decreasing concentration. Taylor
(1954) has shown that E is more than two orders of magnitude larger than ~ex ; therefore it
is convenient to add the two coefficients and refer to the sum as the longitudinal dispersion
coefficient ^ , where
EL - E + ¥x (2.94)
Using Equations (2.93) and (2.94), Equation (2.92) becomes
>+ + (2'95>
Equation (2.95) is the most general form of the one -dimensional mass transfer for a non-
conservative substance. It is assumed to be valid for unsteady, non-uniform flows within the
restrictions of the one-dimensional approximation. The independent variables, tidal velocity
D , cross-sectional area A , and longitudinal dispersion coefficient E, can be functions of
both x and t . In addition, the source and sink terms can be functions of x and t .
The dependent variable, concentration, refers to any substance of interest. In general, one
mass transfer equation of the form of Equation (2.95) must be written for each substance under
consideration. Typically, mass transfer equations are written for salinity, dye or radioactive
tracers, BOD and dissolved oxygen.
The following discussions refer specifically to the various terms in the mass trans-
fer equation.
38
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3.2.2 Source and Sink Terms
The nature of the internal and external source and sink terms in the mass transfer
Equation (2.95) depends on the substance under consideration. Their specification in quantita-
tive terms is probably one of the most difficult problems in water quality modeling; this is
treated in detail in Chapter III.
A substance is called conservative if the source and sink terms r^ and rfi are
both zero. For example, salinity is usually considered to be a conservative substance in an
estuary, in that the source of salinity is specified as a maximum concentration at the ocean
end. The term r. - 0 since no salinity is generated internally within an estuary and re - 0
if there is no influx of salinity along the external lateral boundaries.
The following discussion will serve to illustrate the proper dimensional form for
two typical source and sink terms. A common sink term of the internal type is a decay which
follows a so-called first-order reaction. This would be appropriate if the substance under con-
sideration were a radioactive tracer. The first-order reaction is stated as follows: the time
rate of decay (or generation) of a substance per unit volume (r^) is directly proportional to
the mass of the substance per unit volume present at any instant of time (pC) . Thus, for
decay,
and the sink term in Equation (2. 95) is
- - Kd(pC) (2.96)
T • - Kdc (2'97)
where K, is the reaction coefficient in units of t .
A common source term of the external type arises from the transfer of oxygen from
the atmosphere across the free surface of a fluid having the capacity to increase its dissolved
oxygen content. The following statement expresses the film theory concept (Dobbins 1964) of
oxygen transfer: the time rate of increase in the mass of dissolved oxygen per unit volume
(r ) is directly proportional to the product of the free surface area per unit volume and the
deficit in the mass of dissolved oxygen per unit volume in the liquid phase. This assumes no
vertical gradient in the concentration of dissolved oxygen within the fluid except for a film
of molecular thickness at the free surface. Thus
increase in mass of DO „ , [" As "1 ["mass of DO at saturation _ mass of DPI ,~ go)
(volume) (time) T, L(volumeyj L (volume) " (volume) J *• ' '
where K, is the film coefficient (L/t) of oxygen transfer and Ag is the free surface area.
Multiplying and dividing the right-hand side by the fluid density results in
C8 - C] (2.99)
where C is the saturation concentration of dissolved oxygen. The ratio of surface area to
volume can be interpreted as the reciprocal of the mean depth d and the combination K./d is
usually known as the reaeration coefficient Kfl . Therefore the source term in Equation (2.95) is
39
-------
— - Ra (2.100)
p 35
Both K and K, may be functions of x and t .
a d
3.2.3 Tidal Velocity
The tidal velocity term (U) in the mass transfer equation can be obtained in a
number of ways. The choice depends on available data and the accuracy desired in describing the
advective motion in the estuary. It should be emphasized that the advective motion and the
source and sink terms are usually the dominant quantities in the mass transfer equation. The
dispersion term is related to the spatial variations of the actual advective velocities from
the sectional-mean value U in the one-dimensional equation. Thus the importance of the dis-
persion term decreases as the accuracy of the description of the advective process increases.
This is important because the dispersion coefficient has been one of the most elusive quantities
in water quality mathematical models.
Significant advances in the analysis and calculation of tidal motion in estuaries
have been made in recent years. Therefore it is technically feasible to incorporate the knowl-
edge of tidal motion into the analysis of estuarine water quality. Improvements in the hydro-
dynamic aspects of water quality models are a prerequisite to a systematic investigation of the
magnitude of the various source and sink terms for estuaries.
The tidal velocity can be obtained (i) by direct measurement of tidal velocities
in an estuary, (ii) by measurement of tidal elevations and calculation of tidal velocity through
the continuity equation, or (iii) by a combined solution of the continuity and momentum equa-
tions. The latter requires only the specification of estuary geometry and certain boundary
conditions, and it is therefore the only method which can be used for predictive purposes. The
appropriate one-dimensional continuity and momentum equations are derived in the following
sections.
A cross section of an irregular estuary is shown in Figure 2.3. The x-coordinate
is measured horizontally along the longitudinal axis from the ocean end of the estuary, and h
is the distance to the instantaneous position of the water surface from a horizontal reference
datum. The flow is assumed to be one-dimensional, hence channel curvature and Coriolis effects
are neglected and the transverse water surface is horizontal. The density is assumed to be
constant, and hydrostatic pressure is assumed to prevail at all points in the flow.
Within the Eulerian viewpoint, the one-dimensional conservation of volume (conti-
nuity) and momentum equations may be formulated in either of two ways: the material method or
the control volume method. In the material method the flow characteristics at a section are
obtained by following the motion of a given mass of fluid Am through a small increment of time
it in the vicinity of a fixed section. In the control volume method, the equations are derived
by considering the flux of mass and momentum through a fixed control volume. The one-dimensional
momentum equation is most readily derived by means of the material method, whereas the continu-
ity equation is usually derived by the control volume method. In order to be consistent, the
material method will be used for both developments. Derivations of the continuity and momentum
equations for variable area tidal channels have also been given by Stoker (1957), Dronkers
(1964), Lai (1965), and Harleman and Lee (1969).
40
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HORIZ. DATUM X) X2
Fig. 2.3 Definition sketch - Irregular channel.
INST. WATER SURFACE
REFERENCE WATER LEVEL
FOR STAGE VARIATION
HORIZ. DATUM
Fig. 2.4 Definition sketch - Rectangular channel.
41
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If, as shown in Figure 2.3, Che section has both deep and very shallow portions, the
cross section may be divided into conveyance and storage regions. In this case the tidal flow
is ass uned to be confined to the conveyance area defined by the width bg , and the shallow
portion defined by the width b - b contributes to storage only. The cross-sectional area A
is defined as the area of the conveyance section. Hence, the section- average tidal velocity
U " Q/A , where Q is the instantaneous discharge through the section. The fluid mass Am at
time t is shown in Figure 2.3 by the planes at x^ and x2 . The mass of water in motion at
time t is given by pAAx , in which p is the density of water, A is the cross-sectional
area and Ax - ^ - x^ is the length of the element.
3.2.3.1 Continuity Equation for a Variable Area Estuary The continuity equation is derived
by applying the law of conservation of mass to the fluid element Am . This may be expressed
mathematically by the requirement that the total (or substantial) derivative of Am be equal
to the change in mass of the moving element. Thus
D(oAAx) i_ • f _.••*.
''jCr ' ~ change in mass of moving element
Two factors may contribute to a change in the mass of the moving element: (1) a change due to
storage in the shallow portion equal to
(2) a change due to external lateral flow into the element (caused by tributary inflow, spillage
over a levee or seepage through the channel bank) equal to
p q Ax
where q is the lateral discharge per unit of longitudinal length (positive for inflow and
negative for outflow). Thus the continuity equation can be written
- p(b - bs) &x + p q Ax (2.101)
The total derivative can be expressed in terms of partial derivatives through the
relation
where U is the tidal velocity at the cross section. Considering a homogeneous, incompressible
fluid of constant density, Equation (2.101) can be expanded to the form
-(b- bB)to + qto (2.103)
The term may be treated as follows: at time t ,
Ax - x - X
42
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at time t + At ,
Ax1 = x, + Cv + |^ Ax} fit - (x, + U At)
f. ^ OX ~ J.
or
Therefore,
D^pl . Ax'^- ax , || Ax (2.104)
and Equation (2.103) becomes, after dividing by Ax ,
aA + uaA + A — +(b-b)ah-q-0
or, since the discharge Q - AU ,
M+!xi+
-------
t-«+"fi -
and Equation (2. 107) becomes
PAX CH + u IS) + * II *x • I Fx <2-108>
The possibility of a change in the longitudinal momentum flux due to flow entering or
leaving the main channel from the storage area and due to the lateral inflow q has been dis-
cussed by various investigators. This involves an inherent assumption as to the direction in
which the flow enters or leaves the conveyance section. As pointed out by Dronkers (1964,
p. 194), the momentum effect also depends on whether the water level is rising or falling. It
is generally agreed that the effect on the momentum equation is small, hence in this develop-
ment it will be assumed that the lateral flows enter or leave the main channel at right angles
to the longitudinal axis and that there is no contribution to the longitudinal momentum flux.
The correction to the longitudinal momentum flux due to the non-uniform velocity
distribution at a cross section has also been neglected in this development. The momentum
correction factor is usually of the order of three to five percent. The possibility of sche-
matization of the cross section into conveyance and storage regions is an approximate method
of accounting for highly non-uniform velocity distributions, and further refinements concerning
the velocity distribution are not warranted.
The summation of external forces £ F consists of P , the resultant hydrostatic
pressure force on the vertical cross section; (?w)x > the x- component of the horizontal pres-
sure force exerted by the converging boundaries of the section; (Ff)x > the frictional resis-
tance force exerted by the boundaries. Hence, the three forces, using the notation shown in
Figure 2.3, are
or
The total pressure force on a vertical face is
J
zb
Pg(h - z)b'dz
where b' is the channel width at elevation z . Hence, by the use of Leibnitz1 rule
PSf^A + pgJ (n-^Tx-d2 (2.110)
zb
where the flow area A - J b'dz . The boundary pressure force is
zb
44
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- pg j (h - z) 2^L. ax dz
The average frictional shear stress on the boundary of the element is
where R is the hydraulic radius, R •* A/P_ , P_ is the wetted perimeter and Sg is the
slope of the energy gradient. The frictional force is therefore
F£ - To(Pr6x) - pg AS£ Ax (2.112)
The x-component may be taken equal to F£ since cos awl. The slope of the energy gradient
is evaluated from the Chezy equation in which
S..-21SL
and Ch - i^2 R1/6 („ - Manning roughness coefficient) in ft-sec units. Equation (2.112)
becomes
-------
of a technique to determine the tidal advective velocity U in the mass transfer Equation (2.95).
The choice depends upon the availability and type of field data and upon the desired accuracy.
Harleinan and Lee (1969) have given a summary and evaluation of one- dimensional solution tech-
niques for estuaries and sea- level canals. The various methods can be grouped in the following
categories.
(i) Computation of Tidal Velocity from Simultaneous Solution of the Continuity and Momentum
Equations
This method depends upon simultaneous solutions of the continuity and momentum equa-
tions. It requires a minimum amount of field data and it is capable of describing nonlinear
tidal effects. The continuity and momentum equations can be put into more usable forms by
means of the following substitutions. From Figure 2.4
h - ZQ + d + T) (2.117)
where r\ is the instantaneous water surface elevation with respect to a reference water level.
( T) may be positive or negative.) The bottom elevation ZQ and the depth d are functions
of x but not of time, hence
||- M (2.118)
and
dh. _ o , dd . dii /«
-"- + - +
Substituting Equation (2.118) into the continuity equation (2.106) yields
b f£+|§- q -0 (2.120)
The momentum equation (2.116) contains both the tidal velocity and the discharge; it is con-
venient to eliminate the velocity in favor of the discharge by the relation U - Q/A , to
replace 3h/3x by Equation (2.119), and dQ/dx by Equation (2.120). After dividing each
term by the area A , we obtain
(2.121)
The term S— SA may be shown to be of negligible importance, unless a tidal bore is developed.
As 8x
Equations (2.120) and (2.121) constitute a pair of first-order differential equations in which
the water surface elevation TJ and the tidal discharge Q are the unknowns. The equations
are nonlinear and their solution is most readily accomplished by means of finite-difference
techniques. Details of such solutions have been provided by several authors including Dronkers
(1964 and 1969), Shubinski et al. (1965) who discuss branching estuarial networks, Baltzer and
Lai (1968), Balloffet (1969), Liggett and Woolhiser (1967) and Harleman and Lee (1969). These
investigations include finite-difference applications of explicit, implicit and characteristics
techniques. All such methods require transverse and longitudinal schematization of the estuary
46
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into a finite number of segments and the specification of roughness parameters (Chezy or Manning)
along the estuary. In addition to specifying the geometry and roughness, two boundary condi-
tions and an initial condition are required. The observed tidal elevations at the ocean entrance
of the estuary, i.e., r\ - f(t) , almost invariably forms one boundary condition. If the mathe-
matical model spans the entire tidal region, another boundary condition at the landward end is
required.
The majority of tidal problems can be grouped into three types according to the physi-
cal characteristics of their boundaries and the availability of physical data to be used as an
input to the mathematical model.
An Estuary Having a Well-Defined Head of Tide or a Canal Closed at One End. The
estuaries in this category are those with a well-defined head of tide such as a fall line or
critical flow section. An example is the head of tide at Trenton on the Delaware estuary. The
appropriate boundary conditions are
ocean end: r\ ~ f(t)
head of tide or closed end: Q = 0
At the ocean end, the ocean tide normally can be obtained from tide tables or from prototype
measurements by tide gages. It should be emphasized that the ocean tide need not be sinusoidal;
any single tidal cycle record or continuous records may be used. At the head of tide, the con-
dition of no upstream momentum transfer results in the boundary condition Q « 0 . The fresh-
water inflow of the river into the last upstream segment of the tidal portion is introduced as
a lateral inflow.
An Estuary with an Open End. An open end estuary is one in which the tidal region
merges gradually with the upstream river. It is characterized by a progressive reduction of
tidal range in the upstream direction. The Savannah estuary is an example of this category.
The appropriate boundary conditions are
ocean end: TI - f(t)
open end: Q - f(t)
or Q « constant = freshwater discharge of upland
river
The ocean end conditions are the same as in the previous section. The open end boundary con-
dition depends on available data. For example, discharge measurements may be available at an
upstream location at which the current is always in the ebb direction, however there may be
a cyclic variation in the magnitude and Q - f(t) is known. A location farther upstream may
be found at which the cyclic variation disappears; in this case the open end boundary condition
is Q - constant (equal to the upland river discharge). The location of this point and there-
fore the length of the segmented tidal channel will change seasonally with variations in the
upland discharge. If no measurements are available, it is necessary to assume a length for
the tidal channel and to set Q equal to the river discharge. A check on the assumption may
be made by determining whether the tidal range at the upstream end becomes small enough to be
negligible.
47
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An Estuary Embayment or Canal Connecting Two Bodies of Water. In this category the
two bodies of water may be two oceans, as in the case of a sea-level canal, or an ocean and a
large lake without tidal fluctuations. An embayment may also be connected to the same ocean
through two separate openings. The appropriate boundary conditions are
water body (1): r\^ - f ]_(t)
water body (2): r\2 " f2(O
Other types of boundary conditions may arise in special cases, for example: the specification
of discharge at both ends of a channel; the conditions to be applied at a junction of three
tidal channels (summation of discharge equal to zero and equal values of T) at the junction).
The latter is discussed by Dronkers (1964, Chapter X).
As mentioned earlier, it is necessary to assume initial values of r\ and Q through-
out the tidal region in order to begin the numerical solution. Fortunately, as the solution
proceeds forward in time, the property of the hyperbolic partial differential equation is such
that the effect of the assumed initial condition diminishes rapidly. In the absence of any
other information it may be assumed that i\ • 0 at t - 0 and Q - 0 at t = At . Using the
known boundary conditions for one tidal cycle, the finite-difference solution proceeds to the
end of the first tidal cycle T + At (where T is the tidal period). The values of n at
t - T and Q at t » T + At become the new initial conditions and the tidal cycle is repeated
with the same boundary conditions to the time level t = 2T + At . Thereafter, the computation
is repeated to the (k + l)th tidal cycle. The repeated computation ends when the tidal eleva-
tions obtained in the (k + l)th cycle differ by a small specified amount from those obtained in
the (k)th cycle. The solution obtained at the (k + l)th tidal cycle is referred to as a "quasi-
steady state solution." This solution is independent of the magnitudes of the assumed initial
conditions. In other words, if the values of TI at t - 0 and Q at t = At had been chosen
as other than zero, the same quasi-steady solution would have been obtained. The only difference
would be in the number of cycles necessary to reach the quasi-steady state. If the assumed
values are close to the final values, the number of repeating cycles would be reduced. Mathe-
matically, it is a convergent solution for the given boundary conditions for one tidal cycle.
The Manning coefficient must be specified as a function of x . In an existing estuary
or canal, past records of tidal elevation should be used to determine n by matching the com-
puter solution to the field observations. In the case in which no field data is available, n
must be assumed on the basis of experience with similar estuaries or canals. Meyers and Schulz
(1949) have collected a table of resistance values for some rivers and canals in North America.
Dronkers (1964) gives values of the Chezy coefficient for some estuaries in the Netherlands.
Baltzer and Lai (1968) have made some detailed computer studies of tidal resistance for short
reaches in certain estuaries.
When the computation yielding T) - f(x,t) and Q = f(x,t) has been completed for
the tidal region, any available field data for tidal amplitude and phase can be used to check
the geometric schematization and assumed roughness. If appreciable differences occur, the
roughness magnitude and its spatial variation can be adjusted until agreement is obtained. This
approach is particularly useful in the prediction of the effect of proposed geometric changes,
such as dredging, on tidal motion in estuaries. The tidal discharge Q (or velocity U )
48
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obtained in this manner is a direct input in the advective term of the mass transfer equation
(2.95). Hie following figures taken from Harleman and Lee (1969) illustrate the results of a
finite-difference, digital computer calculation of tidal motion in the Delaware estuary. Fig-
ure 2.5 shows a comparison of the computed and observed mean tidal range along the Delaware from
Miah Maull (in Delaware Bay) to Trenton (the head of tide for this closed-end estuary). Fig-
ure 2.6 shows an important feature of the nonlinear solution. The observed and computed tidal
elevations versus time are plotted at Miah Maull, a middle section and at the Trenton end of
the estuary. The progressive distortion of the tide curve from a sinusoidal shape in Delaware
Bay to a highly nonsinusoidal curve at Trenton is clearly indicated. The computed values of
the maximum flood and ebb velocities are plotted as a function of distance along the estuary in
Figure 2.7. The corresponding maximum discharges are given in Figure 2.8 for fresh water inflow
of 11,650 cfs at Trenton. Figure 2.9 shows the time variation of the tidal velocity and dis-
charge at two sections through one tidal period. Again, the nonlinear tidal advection effects
are shown by the nonsinusoidal shape of the velocity curve and by the fact that the duration of
flood velocity is 5.2 hours, whereas the duration of the ebb velocity is 7.2 hours at a section
58 miles from Miah Maull.
MIAH MAULL
(FT.)
TREHTON
LEGEND
EXPONENTIALLY VARYING
WIDTH CHANNEL
I
50
100
MILES
PROTOTYPE, 1951 CHANNEL
(C. & G.S. TIDE TABLES-
ATLANTIC COAST
COMPUTER NONLINEAR
SOLUTION, 1951 CHANNEL
(HARLEMAN AND LEE 1969)
COMPUTER NONLINEAR
SOLUTION, EXPONENTIALLY
VARYING WIDTH CHANNEL
(HARLEMAN AND LEE 1969)
Fig. 2.5 Delaware Estuary — mean tidal range.
49
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AT 0 MILES
AT 59 MILES
AT 112 MILES
O AT 0 MILES FROM MIAH MAULL
V AT 59 MILES FROM MIAH MAULL
D AT 112 MILES FROM MIAH MAULL
— COMPUTER NON-LINEAR SOLUTION
(HARLEMAN AND LEE 1969)
Fig. 2.6 Delaware Estuary (1951 Channel) — Tidal variations in
elevation at 0, 59, and 112 miles from Mlah Maull.
MIAI
MAX. LANDWARD VELOCITY
TON
LEGEND:
A LINEAR SOLUTION FOR EXPONENTIAL
CHANNEL (HARLEMAN 1966)
O COMPUTER NON-LINEAR SOLUTION
MAX "FLOOD" VELOCITY
(HARLEMAN AND LEE 1969)
D COMPUTER NONLINEAR SOLUTION
MAX "EBB" VELOCITY
(HARLEMAN AND LEE 1969)
Fig. 2,7 Delaware Estuary (1951 Channel) — Maximum tidal velocities.
50
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MlAH MAULL
3X10
2X10
!i ixio" -
TRE
-2X10
•3X10
APPROX. 10% OF
KAX. DISCHARGE
AT ENTRANCE
MAX. DISCHARGE
SEAWARD
(TON
LEGEND:
O LINEAR SOLUTION FOR
EXPONENTIAL CHANNEL
(HARLEMAN (966)
A CALC. BY CUBATURE
(WICKER 1955)
COMPUTER NON-LINEAR
SOLUTION (HARLEMAN
AND LEE 1969)
=1=
-11 ,650 C.F.S.
FRESHWATER INFLOW AT
TRENTON
Fig. 2.8 Delaware Estuary (1951 Channel) — Maximum tidal discharges.
e i
3 X 10°
2 X !0°
•5
o
- I X 10° •
o
_l
2
x io6 "
- -2 X 10°
Q AT 0 MILE
Fig. 2.9 Delaware Estuary (1951 Channel) -- Tidal velocities
and discharges at 0 and 58 miles from Mi ah Maull.
Harleman and Lee (1969).
51
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(ii) Computation of Tidal Velocity when the Variation of Tidal Amplitude and Phase are Known
This method depends upon the existence of field data specifying the distribution of
tidal amplitude and phase along the estuary. Thus if T) = f(x,t) is known, the continuity
equation (2.120) can be integrated with respect to x to find Q •= f(x,t) . This is known as
the method of cubature. It was originally developed in France and it has been applied to the
Delaware (Pillsbury 1956, Wicker' 1955). The estuary is segmented into a number of finite reaches
ji which h - f(t) is known at the end of each reach. The continuity equation (2.120) is writ-
ten in finite-difference form and Q • f(t) is found by numerical integration. The momentum
equation is not used, hence the method cannot predict the effect of changes in estuary geometry
or other conditions. Figure 2,8 shows a comparison of a tidal discharge obtained by cubature
rfith the finite-difference solution described in the previous section.
Analytical methods for the determination of tidal discharge (or velocity) when r\ =
f(x,t) is known have been developed by several investigators (Redfield 1950, Ippen and Harleman
1958, Harleman 1966). These are based on the work of Fjeldstad (1918-1925) on damped, co-
oscillating tides. Both the continuity and momentum equations are used. However, the quadratic
,000
1,200,000
I,000,000
<
I
u
- 800,000
Q
600,000
-------
frictional term in the latter is replaced by a linear term. In addition, it is assuned that
the tidal motion is harmonic throughout the tidal portion. The problem of specifying the estuary
roughness is avoided by making use of the field data on tidal amplitude and phase throughout the
estuary. A comparison of the maximum tidal discharges in the Delaware computed by the method of
cubature and damped co-oscillating tide is shown in Figure 2.10 (Harleman 1966). It should be
emphasized that if the tidal velocity is assumed to be harmonic, the non-tidal advective velocity
due to freshwater inflow must be added as a separate term. Otherwise the flushing effect of
the upland discharge would be lost. This is discussed in the following section.
(iii) Direct Measurement of Tidal Velocities
There are probably few cases where sufficient field measurements in estuaries are
available to determine U » f(x,t) directly. In situations where observations of maximum
tidal velocities can be made at certain sections, it may be sufficient, as a first approxima-
tion, to utilize an ad hoc equation of the form
U - Uf(x) + UT(x) sin [ at - F(x) ] (2.122)
where U^ is the velocity due to non-tidal (freshwater) inflow, UT is the maximum tidal
velocity, o » 2n/T ( T - tidal period) and F(x) is a function describing the change in the
phase of tidal velocities along the estuary. Equation (2.122) is the simplest representation
of the advective velocity term in the mass transfer equation (2.95) in real time.
3.2.4 Longitudinal Dispersion in the Real Time Mass Transfer Equation
The objective of this section is to summarize various methods of determining longi-
tudinal dispersion coefficients. The first approach is an analytical one considering the fluid
mechanics of dispersion in an oscillating flow. The second is an empirical one in which the
dispersion coefficient is determined by comparing a solution of the mass transfer equation with
the measured concentration distribution of some substance in an estuary. The same dispersion
coefficient is then used to predict the concentration distribution of some other substance. In
practice, the empirical determination of dispersion coefficients is limited by the requirement
that all source and sink (decay) terms for the substance must be known with reasonable accuracy.
Therefore, the second method is restricted to naturally conservative substances such as salinity
or to artificially introduced tracers for which decay and absorption rates can be determined.
In the past, observational data from either the prototype or a hydraulic model have been used
in the empirical method. Chapter V includes a discussion of certain difficulties in the determi-
nation of dispersion coefficients from observations in distorted hydraulic models.
A physical understanding of longitudinal dispersion can be gained by considering a
steady, uniform, turbulent flow in a pipe. Assune that a finite amount of a tracer substance
is injected instantaneously and uniformly across the entire flow section. Because of the veloc-
ity distribution the tracer near the center of the pipe moves downstream much faster than the
tracer near the walls. Lateral turbulent mixing maintains uniform concentrations at various
sections; however, there is a large longitudinal spreading due to the relative velocity between
adjacent layers. After a certain time, the longitudinal distribution of concentration of the
tracer approaches a Gaussian distribution about the peak. The advective term in the one-
dimensional mass transfer equation contains the average longitudinal velocity at the section
and it is therefore unable to account for the longitudinal spreading due to the velocity dis-
53
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ESTUARY
HEAD OF TIDE
RIVER
SALINITY INTRUSION
REGION
FRESH WATER
TIDAL REGION
MSL
Fig. 2.11 Mean current and salinity distribution in a mixed estuary.
tribution. The mass transfer due to the velocity distribution is combined with that due to
turbulent diffusion and the total effect is represented by a longitudinal dispersion coefficient.
Qualitatively, the more non-uniform the velocity distribution, the larger the dispersion coef-
ficient. On this basis we should expect to find much larger dispersion coefficients within the
salinity intrusion portions of estuaries in comparison with the non-saline tidal regions. It
is well known that in the salinity intrusion region the tidal velocities are modified by the
gravitational circulation set up by the longitudinal density gradients. A typical estuary can
be divided into a salinity intrusion region and a fresh water tidal region as shown in Fig-
ure 2.11. The line of demarcation is not fixed since the longitudinal extent of salinity intru-
sion varies during a tidal period by a distance equal to the tidal excursion. In addition, the
intrusion distance depends on the magnitude of the fresh water inflow into the estuary.
At the present there is no analytical method of predicting longitudinal dispersion
coefficients within the salinity intrusion region of an estuary. We must therefore rely on
empirical determinations based on observed salinity distributions. Fortunately, salinity is
relatively easy to measure and it usually may be considered a conservative substance. The
empirical determination of longitudinal dispersion coefficients in the fresh water tidal region
is much more difficult because of the absence of any natural, conservative substance. Field
tests, involving the injection of artificial tracers and subsequent measurement of time and
space concentrations are difficult, time consuming and expensive. Fortunately, considerable
progress has been made analytically and it now appears possible to predict longitudinal dis-
persion coefficients in tidal regions containing water of uniform density, with an accuracy
sufficient for practical purposes. It should be noted that a tidal region containing water of
uniform salinity can be considered in the same category as a freshwater tidal region. A quanti-
tative discussion of longitudinal dispersion in the uniform density and salinity intrusion
regions is given in the following sections.
54
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3.2.4.1 Longitudinal Dispersion Coefficient in Regions of Uniform Density
(i) Dispersion in Steady. Unidirectional Flow. Taylor (1954) appears to have been the first
to determine a numerical value for the longitudinal dispersion coefficient E, . He considered
steady, uniform, axisymmetrical turbulent flow in a long, straight pipe and assumed that the
concentration c at any point in the pipe could be written as (c + c ) where c was a
x r x
function of x only and cr was a function of the radial coordinate r only. He assumed the
form of the radial velocity distribution. The turbulent (mass) diffusivity was assumed to be
isotropic and equal to the eddy viscosity. Considering the asymptotic solution to the axisym-
metric mass balance equation, Taylor determined that the dimensionless longitudinal dispersion
coefficient for a pipe should be
EL - 10.1 rQu* (2.123)
where CQ is the pipe radius and u* - VTO/P is the shear velocity ( TQ is the shear stress
at the wall).
Elder (1959) carried out a computation similar to Taylor's for a steady, uniform,
two-dimensional, infinitely wide open channel flow with a logarithmic velocity distribution in
the vertical direction. Elder's result is
EL - 5.9 d u* = 5.9 d VgDSE (2.124)
where d is the depth of flow. The predicted longitudinal dispersion coefficients given by
Equations (2.123) and (2.124) have been confirmed experimentally by injecting tracers into
turbulent flow in uniform pipes and wide open channels.
Taylor's equation (2.123) can be rewritten in terms of the channel flow and resistance
parameters by noting that
where f is the pipe friction factor and U is the average velocity. The Chezy coefficient
relates f to the Manning roughness n and the hydraulic radius R^ ,
C - jig ..LM p 1/6
h V f n ^i
Thus, replacing rQ by 21^ for a pipe, Equation (2.123) becomes
EL « 77 n U Rh5/6 (in ft-sec units) (2.125)
Fischer (1967 and 1968) analyzed concentration distributions produced by tracers
introduced into non-tidal rivers and concluded that Equation (2.125) predicts longitudinal
dispersion coefficients which are too small, in some cases by two orders of magnitude. He
attributes this to the fact that longitudinal dispersion in rivers is produced primarily by
velocity variations in the lateral direction rather than by velocity variations in the vertical
direction.
55
-------
(li) Dispersion In an Oscillating Flow. Hoiley et al. (1970) and Fischer (1969) have investi-
gated the Implications of the above conclusions with regard to dispersion associated with tidal
motion in constant density regions of estuaries. They have shown that dispersion in an oscil-
lating flow depends on the ratio of the period of oscillation (T) to a time scale for cross-
sectional mixing T . Because estuaries are relatively wide and shallow, two cross-sectional
mixing time scales can be defined: one for transverse mixing and one for vertical mixing.
Using values typical of the upper Delaware estuary, it is estimated that the time scale for
transverse mixing is approximately 200 times longer than the tidal period. On the other hand.
the time scale for vertical mixing is approximately 15 times shorter than the tidal period. On
this basis, velocity variations in the lateral direction would contribute very little to the
longitudinal dispersion and it is concluded that the velocity distribution in the vertical direc-
tion is of primary importance. Therefore, the open channel modification of Taylor's equation
(2.125) would be appropriate in most estuaries.
The remaining question is the velocity which should be used in the modified Taylor
equation when it is applied to an oscillating rather than a unidirectional flow. If the absolute
value of the tidal velocity were used, E^ would vary with time throughout the tidal cycle with
two peaks as shown in Figure 2.12. Hoiley and Harleman (1965, see also Harleman et al. 1966)
conducted dispersion experiments in a longitudinal oscillating flow and concluded that the dis-
persion coefficient could be assumed to be constant throughout a tidal cycle if the time average
of the absolute value of the tidal velocity were used in the Taylor equation. If the temporal
variation of the tidal velocity is approximately sinusoidal, the time average of the absolute
value is 2/n times the maximum tidal velocity umax • In an estuary it is expected that bends
and channel irregularities will cause an increase in the longitudinal dispersion in comparison
with a straight channel. On this basis it is suggested that the modified Taylor equation (2.125)
be increased by a factor of 2 ; the proposed equation, in terms of the maximum tidal velocity,
is
100 n
-------
The longitudinal dispersion coefficient is frequently expressed in square miles per day rather
than in square feet per second. (1 sq. mile/day - 320 ft /sec.) Thus Equation (2.126) can be
written in mixed units as
in sq mi/day
"max «h
5/6
Umax and *h *n ^c~sec
(2.127)
In the freshwater, tidal portion of the Delaware near Philadelphia the following
values are appropriate (Harleman and Lee 1969):
Manning roughness, n - 0.028
maximum tidal velocity, Umax -2.0 ft/sec
hydraulic radius, R^ - 20 ft
The order of magnitude of the longitudinal dispersion coefficient near Philadelphia, given by
Equations (2.126) and (2.127), is E^ - 70 ft2/sec or 0.2 sq mi/day .
The validity of the modified Taylor equation for the longitudinal dispersion coeffi-
cient in a constant density tidal region can be tested by comparing the predicted value with a
dispersion coefficient determined from a field test. The necessary prototype measurements are
very expensive and time consuming, and only a few such tests have been conducted. Hetling and
O'Connell (1966) have reported the results of a dye dispersion test in the freshwater tidal
portion of the Potomac. Rhodamine dye was discharged continuously for a period of 13 days at
a point below Washington. Dye concentration observations were made throughout the upper 25
miles of the estuary for a period of 34 days following the release. The observed value of EL
in the vicinity of the release point was reported to be 0.2 sq mi/day. Using the following
geometric and tidal quantities assumed to be characteristic of this reach of the Potomac:
n = 0.035
Umax " 1'6 ft/sec
Rh - 13 f t
the calculated value from the modified Taylor equation (2.127) is E_ " 0.14 sq mi/day which
is in good agreement with the reported value.
3.2.4.2 Longitudinal Dispersion in Salinity Intrusion Regions Longitudinal dispersion is a
result of spatial variations in the longitudinal tidal velocities at a section. In the constant
density tidal region the velocity distribution is due to frictional shear exerted at the section
boundary. In the region of salinity intrusion, the longitudinal density gradients associated
with the transition from ocean water to fresh water cause significant changes to the boundary
shear velocity distribution. This is due to the superposition of the frictional effect and a
large-scale density-driven horizontal circulation in which near-bottom velocities are augmented
during the flood phase of the tide and retarded during the ebb phase. The salinity circulation
is shown schematically in Figure 2.13. A set of tidal velocity observations (as a function of
depth) at ten percent intervals throughout a tidal period is shown in Figure 2.14. The data
wets taken in the salinity intrusion flume at the Waterways Experiment Station (Harleman and
Ippen 1967 and 1969) at a station within the salinity intrusion region. The fact that the ebb
velocity distributions are much more nonuniform than the flood velocity distributions is evident.
57
-------
MSL
VERTICAL ADVECTION + V
x HORIZONTAL ADVECTION - U
VERTICAL
/"DISPERSION
OCEAN
II
t{ IK
FRESH
WATER
NULL POINT
Fig. 2.13 Schematic diagram of estuary salinity circulation.
EBB
U, FT/SEC
Fig. 2.14 Horizontal velocities throughout tidal cycle, Test 14,
Sta. 40. From Harleman and Ippen (1967).
58
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The distortion of velocity distribution by the salinity intrusion is even more apparent when the
velocity distributions are averaged over a tidal period. A set of such velocity distributions
at various distances from the ocean entrance is shown in Figure 2.15. the time-average tidal
velocity at each depth is normalized by dividing by the magnitude of the freshwater velocity.
The upstream limit of salinity intrusion clearly lies between stations 160 and 200.
The longitudinal dispersion coefficient in the salinity intrusion region will be desig-
nated by EL' in order to distinguish it from the dispersion coefficient in the constant density
tidal region.
The physical interrelationships between the tidal motion and salinity intrusion can be
demonstrated by means of field and laboratory observations in estuaries of uniform geometry. We
consider the situation in real tidal time by coupling the one-dimensional equations of continuity,
momentum and the mass transfer equation for salt.
If the width b is constant and z (Figure 2.4) is zero, the instantaneous depth
is h . Thus the cross-sectional area is
The continuity equation (2.106) becomes
A - bh and the tidal discharge is given by Q - bhU
ab+ulJi + hlfi- o
&t OX OX
(2.128)
where the lateral inflow q is also assumed to be negligible.
into Equation (2.116), the momentum equation becomes
3U
THE
ulul _ 0
VR
Substituting Equation (2.128)
(2.129)
y/h
-8
Fig. 2.15 Vertical variation of time-averaged tidal
velocity relative to freshwater velocity:
Test No. 16. Harleman and Ippen (1967).
59
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In a similar manner, the mass transfer equation^can be simplified for an estuary of uniform
width. By expanding the derivatives and making use of the continuity equation (2.128) the mass
transfer equation (2.95) for salt (a conservative substance for which t^ and rg are zero)
can be written as
„ ac _ i a /v _ i ac ~N
u alT ~ IT ax" C.hEL axV
The three equations can be solved for the three unknowns, h » f(x,t) , U - f(x,t) and
C - f(x,t) provided that the proper initial and boundary conditions are given. In addition,
the roughness (Chezy or Manning) and the longitudinal dispersion coefficient EL> must be
specified as functions of x and t .
Stigter and Siemens (1967) have carried out a finite- difference solution using an
implicit scheme. The boundary and initial conditions for the tidal calculations have already
been discussed in Section 3.2.3.3. The observed salinity variation with time at the ocean
entrance of the estuary was used as the salinity boundary condition. Both C^ and E^ were
assumed to be constant in time and the roughness C^ was also assumed to be constant with
respect to x . The value of C^ chosen was the one giving the best agreement between the
calculated and observed tidal elevations. The dispersion coefficient E^' - £(x) was adjusted
until a best fit was obtained between the calculated and observed salinity concentrations. The
results are shown in Figure 2.16 for both the temporal and spatial distribution of salinity.
The observed salinities are field measurements from the Rotterdam Waterway. This is a tidal
estuary of almost uniform depth and width, carrying a portion of the freshwater discharge of
the Rhine, and joining the North Sea at Hook of Holland. The longitudinal dispersion coeffi-
cient was determined to be a function of x of the form
3
EL' - 13,000 C1 - r) • <£t2/sec> (2.131)
where L is the length of the entire tidal region of the estuary. The magnitude of E^' at
the ocean entrance (x - 0) is 13,000 ft2 /sec or 40 sq mi/day. Under the same tidal condi-
tions, in the absence of density gradients due to salinity intrusion, the longitudinal disper-
sion coefficient predicted by the modified Taylor equations (2.126) and (2.127) is 175 ft /sec
or 0.5 sq mi/ day. This illustrates the large longitudinal variation in the magnitude of the
dispersion coefficient between the constant density tidal region and the salinity intrusion
region of an estuary. The analysis of Stigter and Siemens (1967) indicates that the time varia-
tion of E.1 within a tidal period is not of great Importance. Lee (1970) has used a similar
approach to determine E.1 as a function of x for variable-area estuaries. In this case the
general continuity equation (2.120), monentum equation (2.121) and salt balance equation (2.95)
must be used and the computations are carried out in finite-difference form. An important
difference between the work of Stigter and Siemens (1967) and Lee (1970) is in the treatment
of the ocean salinity boundary condition. In the latter case it is not necessary to know the
salinity variation with time at the ocean entrance to the estuary. The computational program
is arranged such that during the flood tide the salinity entering the estuary is equal to the
ocean salinity. During the ebb phase the salinity leaving the estuary is a function of the
longitudinal salinity distribution within the estuary. Lee's (1970) results show that E^ is
essentially a linear function of x for both the Delaware and Potomac estuaries under various
freshwater discharge conditions. At the upstream limit of salinity intrusion the value of E^'
approaches the Taylor value appropriate to the nonsallne region (see Equation 2.126). The max-
imum value of E, ' is reached near the ocean entrance.
The analyses presented above are examples of methods of determining the longitudinal
dispersion coefficient as a function of x from salinity observations during one tidal period.
60
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1.0
Fig. 2.16 Comparison of calculated and observed temporal and
spatial salinity distributions In the Rotterdam
Waterway. Stlgter and Slemons (1967).
3.2.5 Discussion of the Real Time Mass Transfer Equation
The foregoing developments have been primarily concerned with the hydrodynamic aspects
of one-dimensional mathematical models for estuarine water quality control. The central relation-
ship is the mass transfer Equation (2.95) in which the real time variables are related to the
tidal motion.
+ "
D
(2.95)
61
-------
The hydrodynamic inputs are the independent variables of tidal velocity (U) and longitudinal
dispersion (E^ or Z^') as discussed in Sections 3.2.3 and 3.2.4. The dependent variable is
the concentration of a particular water quality parameter, C - f(x,t) . The internal and ex-
ternal source and sink terms, r^/p and rfi/p , take various forms depending on the nature of
the water quality parameter. The estuary geometry [A - f(x,t)] and appropriate initial and
boundary conditions must also be specified in order to complete the formulation of the problem.
A simpler form of the one-dimensional mass transfer equation, also in real time, can
be used if two conditions are satisfied: the lateral inflow of water (due to local tributaries
or subsurface inflow along the estuary) is negligible compared to the primary inflow at the head
of the estuary, and the tidal motion is effective over the full cross-sectional area at all
sections. Under these conditions q - 0, b - b and the continuity equation (2.105) can be
written as
|A + ^L (AD) - 0 (2.132)
since Q - AU . By combining Equation (2.132) and Equation (2.95), the following form of the
one-dimensional, variable-area mass transfer equation is obtained
-^ + -JT (2.133)
Additional simplifications can be obtained under the following assumptions:
(i) The width (b) of a tidal waterway is assumed to be constant. In this case, the depth
h - f(x,t) and Equation (2.133) becomes
ir + TT (2.134)
(ii) Both the width and the mean depth (d) (see Figure 2.4) are constant, and variations in
depth due to tidal motion are small compared to the mean depth. This is equivalent to assuming
that the cross-sectional area (A) is constant. Equation (2.133) becomes
-pi + lT (2.135)
(ill) The cross-sectional area is assumed to be constant and the longitudinal dispersion coeffi-
cient is assumed to be independent of x . This approximation may be reasonable in the constant
density portion of certain estuaries in which the magnitude of the tidal velocity at a given time
is essentially independent of x . In this case Equation (2.133) becomes
w >r >ar r* r-
ft+Ufx:~ELlj?"l"'p~(? (2.136)
In all of the approximate forms of Equation (2.95), Equations (2.133) - (2.136), II
represents the tidal velocity which, in general, is a function of x and t . Certain physical
properties of the real time mass transfer equations are demonstrated in the following sections.
62
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3.2.6 Solution of the Real Time Mass Transfer Equation in a Uniform Estuary
Analytical solutions of the real time mass transfer equation for estuaries of simple
geometry are useful: (i) in providing a graphical visualization of the advection and dispersion
processes in tidal motion; (ii) as a basis of comparison with mathematical models involving longer
periods of time averaging; (iii) as a basis for determining the accuracy of finite-difference
formulations.
The essential features of the real time solution can be demonstrated by considering
an idealized estuary of constant cross-sectional area; the tidal velocity is a function of time
and independent of x and the longitudinal dispersion coefficient is constant. The concentra-
tion distribution due to the introduction of a substance at a certain section is to be determined
as a function of x and t . The substance is non-conservative and undergoes internal decay in
accordance with a first-order reaction given by Equation (2.97). There are no external sources
or sinks other than at the section of injection. Under these conditions the general mass trans-
fer equation (2.95) reduces to the form of Equation (2.136),
OC ^ |» _pC _ rf O C y f+ /O 1 ^7^
where U - f(t) and ^ and Kd are constants.
Initially it is desired to find the solution of Equation (2.137) in an infinite channel
for an instantaneous release of W pounds of pollutant at x • 0 at time t » T . The analyt-
ical solution can be obtained by introducing the following changes of variables. Let C be a
new concentration variable such that
C - £e'Kd(t ' T) (2.138)
Let x be a new longitudinal coordinate distance measured with respect to a cross-sectional
plane moving with the velocity U(t) . Therefore
x - x + J* U(t) dt (2.139)
Introducing the above into Equation (2.137), the mass transfer equation can be written as
which is the elementary form of the diffusion equation known as Tick's second law. The solution
for an instantaneous release (delta function) is
C -
where Y is the specific weight of the receiving fluid. Equation (2.141) can be written in
terms of the original concentration variable by substituting Equation (2.138), therefore
C -- " exp [- n-L^. - Kd(t-r)] (2.142)
63
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The solution for a continuous lnput« of pollutant at a rate w (pounds per second) can
be obtained from Equation (2.142) by considering the continuous release to be made up of a con-
tinuous series of incremental Instantaneous releases of magnitude dW where
dW - w dr (2.143)
The Increment of concentration resulting from each increment of instantaneous release is given
by Equation (2.142) with C replaced by dC and W by dW . Integrating Equation (2.142)
with respect to T sums the effect of the continuous series of instantaneous releases. The
concentration distribution at time t for a continuous release beginning at t - 0 is given by
c -
(t-T)
Finally, it is desirable to return to the original variable x in place of x by
means of Equation (2.139). This requires the specification of U(t) . We can, without loss of
generality, demonstrate the essential features of the real time model by assuming that the tidal
velocity is a harmonic function of time of the form
U(t) - Uf + UT sin at (2.145)
Equation (2.145) is a special case of Equation (2.122) in which Uf , the velocity due to fresh-
water inflow, and Up , the amplitude of the oscillating component of the velocity, are constants.
In addition, the phase lag F(x) - 0 and o - 2n/T (T - tidal period). Substituting Equation
(2.145) in Equation (2.139) and eliminating x in Equation (2.144), we have
/
fA/4nEL(t-T)
r °T n2
x - U-(t-r) + -± (cos at - cos ar)
- Kd
dr (2.146)
If the injection rate w is a constant, Equation (2.146) can be written in dimensionless form
by defining a reference concentration
Co"W" (2.147)
Therefore,
Uf ( [x - Uf(t-T) +-^ (cos at - cos ar)~f
t ___ ) _ _t r a J_ v ft._.\ \j. (2,148)
The integral of Equation (2.148) cannot be evaluated analytically; however, it can be programmed
readily for numerical evaluation on a computer.
Calculated concentration distributions for a continuous injection, as given by Equation
(2.148), are shown by the.curves in Figure 2.17 for a conservative substance in which Kd - 0.
The dlmenslonless concentration C/C is plotted as a function of the dlmensionless longitudinal
distance x/X , where X is the tidal excursion. The tidal excursion is the distance traveled
-------
by a particle of water in half the tidal period. For the tidal velocity given by Equation (2.145)
(2.149)
UTT
Concentration distributions at four different times are shown in terms of a dimensionless time
variable
N - t/T
(2.150)
where N is the number of tidal periods from the beginning of injection at x = 0 and t = 0 .
Values of N equal to an integer correspond to times of high water slack (i.e., N = 30 and 400)
and the half-integer values (i.e., N - 30.5 and 400.5) to times of low water slack. Within
each tidal period, the pollutant is advected upstream and downstream of the injection section
at x - 0 , and the concentration increases slowly until a quasi-steady state is reached at
about 400 tidal periods.
1.0-
2.0
Fig. 2.17 Continuous injection dispersion test showing comparison of data with Equation (2.148).
65
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Laboratory experiments conducted by Harleman et al. (1966) in a uniform, oscillating
flow system Indicate good agreement with the theoretical curves. The longitudinal dispersion
coefficient was calculated from the modified Taylor equation (2.126). It is important to note
that the theoretical concentration distributions given by Equation (2.148) are relatively insen-
sitive to changes in the magnitude of the dispersion coefficient. The dominant mass transfer
mechanism is the advection due to the tidal velocity. Additional laboratory experiments on dis-
persion in an oscillating flow have been reported by Disko and Gedlund (1967) and Awaya (1969).
Numerical evaluations of Equation (2.148) have been carried out by Lee (1970) for a
non-conservative substance (first-order decay) using freshwater flow, tidal parameters and a
dispersion coefficient typical of the Delaware estuary near Philadelphia. The quasi-steady
state concentration distributions, after 150 tidal periods, are shown In Figure 2.18 for both
high and low water slack conditions. The tidal excursion X equals 28,500 feet and the upstream
limit of detectable concentration is seen to be of the order of 30,000 feet upstream of the in-
jection station. The distribution curves of Figure 2.18 are relatively insensitive to changes
in the magnitude of the dispersion coefficient, both upstream and downstream of the injection
station. The concentrations upstream of the Injection station are almost entirely due to tidal
advection, while in the downstream direction the decay term Kd is the dominant quantity. The
reference concentration
case of non-conservative substances.
C0 defined by Equation (2.147) has no physical significance In the
1.6
1.1.
1.2
o 1.0
3.000
—H.W.S.
L.W.S.
AT IATH » 3WH
""TIDAL PERIOD
. AVERAGE OVER ONE
* TIDAL CYCLE
GEOMETRIC DATA:
CONSTANT CROSS-SECTIONAL
AREA
tX - 2.500 FT
VELOCITY DATA:
(UT)MAX - 2.0 FT/SEC
- 0.1 FT/SEC
T I
TIDAL DATA:
T - 44.712 SEC
COEFFICIENT:
k - 0.034/DAY
E - 65 FT'/SEC
TIME INCREMENT:
*T - 1,863 SEC
-100.000 -50.000
50,000
DISTANCE (FT)
100.000
I 50,000
200,000
250,000
Fig. 2.18 Quasi-steady state solution of Equation (2.148) for non-conservative substance at high
water and low water slack.
66
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3.3 MASS TRANSFER EQUATIONS USING NON-TIDAL ADVECTIVE VELOCITIES
Because of the complexity of the unsteady, nonuniform flow In an estuary, it Is not
surprising that many investigators proposed mathematical models for the longitudinal distribution
of pollutants which ignored the tidal advective velocity. Prior to 1950, the analysis of pollu-
tion in estuaries was based on the tidal prism exchange concept. This is illustrated by the
work of Phelps and Velz (1933) in New York harbor and by the later work of Ketchum (1951a and
1951b). Ketchum divided the estuary into segments and assumed complete mixing within each
segment during each tidal period. The state of knowledge as of 1950 was summarized in the pro-
ceedings of a colloquium on flushing of estuaries (Stommel, 1950).
3.3.1 Mass Transfer Equation; Time Average Over a Tidal Period
Arons and Stommel (1951) and Stommel (1953) were among the first to recognize the
limitations of the tidal prism approach. They proposed the use of the one-dimensional mass
transfer equation in which each term of Equation (2.133) is averaged over a period of time equal
to the tidal period. Thus, the time-averaged mass transfer equation becomes
-^— f U«* ^— *" — T— I AT** T— J T "~~ ~I" \2« 151)
at f ax j ax V/TJ axy p p
where C is the concentration averaged over one tidal period, Uf is the non-tidal advective
velocity due to freshwater inflow Q-/A , and ^ is the time-averaged longitudinal dispersion
coefficient. 75 is the time-averaged area and the source and sink terms are also time-averaged.
Stommel (1953) wrote the steady-state form of Equation (2.151) for a conservative sub-
stance by setting aCVat " 0 and assumed that the freshwater discharge Q£ = AU^ was a
constant, thus
(2.152)
The steady state refers to the fact that ~C does not change from one tidal period to the next
if Q£ is a constant and if the boundary conditions for C are also constant. Inasmuch as all
effects of mass transfer by tidal advection have been eliminated from Equations (2.151) and
(2.152), such effects must appear in the time-averaged dispersion term. Stommel used Equation
(2.152) in finite-difference form to determine E^ as a function of x from measurements of
the salinity distribution in the Severn Estuary. He then calculated longitudinal concentration
distribution curves for a pollutant introduced at an arbitrary section of the estuary. Pritchard
(1952, 1954, 1958, 1960) developed similar mathematical models for conservative substances using
time averages over a tidal period for both one- and two-dimensional conditions.
3.3.2 Mass Transfer Equation: Slack Tide Approximation
Mathematical models, based on a different non-tidal advective concept, have been formu-
lated by O'Connor and DiToro (1964) and O'Connor (1965) for non-conservative substances. These
studies have been concerned with the longitudinal distribution of BOD and dissolved oxygen re-
sulting from instantaneous and continuous discharges of pollutants. It is important to note
that O'Connor states that his mathematical models apply to concentration distributions at slack
67
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tide conditions rather than to concentrations which have been averaged over a tidal period.
This is probably related to the fact that observations in many estuaries are available only at
times corresponding to slack tide conditions. The terminology high water slack is used to indi-
cate the change from flood to ebb and low water slack indicates the change from ebb to flood.
In a strict sense, the time of slack tide is the time at which the advective velocity is zero.
However, it was recognized that the elimination of the advective term in Equation (2.133) at
slack tide would result in a loss of the entire flushing action. O'Connor, therefore, retained
the non-tidal advective velocity due to freshwater inflow Qf/A . This type of mathematical
model will be called the slack tide approximation. Thus the appropriate form of Equation (2.133)
is
ir + -T (2.153)
The subscript s on the concentration variable and the longitudinal dispersion coefficient is
a reminder that the concentration distribution Is at slack tide. Equation (2.153) Is the basic
differential equation for O'Connor's mathematical models. The freshwater discharge Q£ and the
longitudinal dispersion coefficient at slack tide ES are assumed to be constant and independent
of x . O'Connor obtained analytical solutions to Equation (2.153) for Instantaneous and con-
tinuous injection of a pollutant for both constant and variable area estuaries, the latter for
cases in which the area change is a linear, quadratic or exponential function of x .
The existence of two non-tidal mass transfer equations (i.e., the time average over
a tidal cycle and the slack tide approximation) has resulted in a certain amount of confusion
in the literature. This is due to a failure on the part of some investigators to discriminate
between the two approaches. One reason for the confusion is that the mass transfer equations
(2.151) and (2.153) are similar in that both contain an advective term due to freshwater inflow.
The difficulty arises from the fact that the magnitudes of the longitudinal dispersion coeffi-
cients, E, in the real time equation (2.133), TZ, in the tidal-period-averaged equation
(2.151) and Eg in the slack tide approximation equation (2.153), are not identical.
3.3.3 Analytical Solutions of the Non-Tidal Advecttve Mass Transfer Equations In a Uniform
Estuary
Mass transfer equations based on the non-tidal advective concept are of two general
types: the time average over a tidal period as given by Equation (2.151) and the slack tide
approximation as given by Equation (2.153). There is no way for analytically predicting the
dispersion coefficients in the'non-tidal advective equations in a manner analogous to the modi-
fied Taylor Equation (2.126) or (2.127) in the real time system. This Is one of the serious
disadvantages of the non-tidal mathematical models. Whenever these models are employed it is
necessary to determine the dispersion coefficients empirically for the particular reach under
study by comparing solutions of the non-tidal mass transfer equation with observed concentration
data. If the observed concentrations are for a non-conservative substance, such as a dye, BOD
or DO, the solutions will contain one or more rate constants which depend on the nature of the
source and sink terms for the substance whose concentration Is being measured. These rate con-
stants, together with the unknown dispersion coefficient, appear in the solution for the con-
centration distribution. Therefore there are several degrees of freedom, or combinations of
parameters, which can be varied. It is not surprising that such multi-parameter models can be
adjusted to match a given set of observations. It is difficult to claim that dispersion-
coefficients or rate constants determined in this manner have any general validity.
68
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Expensive and time consuming field tests, involving the injection and measurement of
tracer substances, have been undertaken in the non-saline portions of estuaries for the purpose
of determining dispersion coefficients. Similar tests have also been made in physical hydraulic
models o£ estuaries. (See Chapter V, Section 3 for a discussion of certain similitude problems
associated with dispersion in vertically distorted physical models.)
An understanding of the physical significance of the dispersion coefficients in the
non- tidal mass transfer equations can be gained by comparing their solutions with the solution
of the real time mass transfer equation under identical physical conditions. We consider the
idealized estuary of constant cross-sectional area described in Section 3.2.6. The real time
solution for a continuous injection of a non-conservative substance (first-order decay) is given
by Equation (2.148), and the quasi-steady state concentration distributions are plotted as a
function of x in Figure 2.18. In the following paragraphs the corresponding solutions for
the non-tidal mass transfer equations are developed for a uniform estuary. The comparisons with
the real time solutions are given in Section 3.4.
We consider initially the slack tide approximation. The appropriate form of Equation
(2.153) for a constant-area estuary in which Uf , ES , and Kd are constant is
ac
ac
a»c
The solution for a continuous injection at x - 0 at the rate w (pounds per second), beginning
at t = 0 , is given by Equation (2.146) with ttj, - 0
exp
ft-T)
-[X - CF(t-T)]
— - Kd(t-T)
(2.155)
This integral can be evaluated analytically in terms of error functions and exponentials
(80 -
where n - VU-a + 4K.E . Where a choice of sign is indicated, the minus sign applies to values
of x >0 and the positive sign to values of x < 0 .
The steady-state spatial distribution of concentration is found by letting t -• <° and
Equation (2.156) becomes
c. -
<2-157>
Equation (2.157) reduces to very simple steady-state expressions for a conservative substance,
K - 0 and 0 - U ,
69
-------
(2'158)
for x > 0 (downstream of injection section) , and
<2-i59>
for x < 0 (upstream of Injection section). Equation (2.158) Indicates a constant concentration
downstream of the Injection section and Equation (2.159) an exponentially decreasing concentra-
tion upstream of the Injection section In a uniform estuary.
We consider next the mass transfer equation time-averaged over a tidal period. The
appropriate form of Equation (2.151) for a constant-area estuary in which U^ , TZ^ and Kd
are constant Is
From a mathematical viewpoint, Equation (2.160) and the slack tide approximation Equation (2.154)
are Identical. The difference is only in the meaning of the quantities TJ and E^ as opposed
to C and E . The steady-state spatial distribution of concentration for the time-averaged
mass transfer equation follows immediately from Equation (2.157)
-- Of
*
where A - ^UfZ + 4KdE^< . A comparison of Equations (2.157) and (2.161) with the real time
solution is given in Section 3.4.
3.3.4 Mathematical Models of Salinity Intrusion
A large number of Investigators have used the non-tidal advectlve equations for the
study of one-dimensional salinity intrusion in estuaries having varying degrees of sectional
homogeneity. In dealing with salinity or chlorinity as the concentration variable, it is usually
permissible to assume that the substance is conservative. This eliminates the source and sink
terms and the associated rate constants in Equations (2.151) and (2.153). For an estuary in
which certain salinity data is available and in which A and Qf are known functions, the above
equations can be used to determine dispersion coefficients. The choice of Equation (2.151) or
(2.153) will depend on the nature of the salinity data.
A complete discussion of the many one-dimensional salinity investigations is beyond
the scope of this report. Even a concise summary is difficult because of the different objec-
tives of the various Investigators. In general, the objectives may be classed as descriptive
or predictive. Under the category of descriptive studies we include the use of salinity data
and the conservative mass transfer equation to determine the longitudinal variation of the dis-
persion coefficient within the salinity intrusion region. A second mass transfer equation can
be solved for the longitudinal distribution of a non-conservative pollutant introduced at a
70
-------
section within the salinity region. Under the predictive category we include investigations
whose primary objective is the prediction of variations in the longitudinal salinity distribu-
tion due to changes In freshwater inflow, estuary geometry, or tidal range. A few examples are
given in the following paragraphs.
3.3.4.1 Descriptive Studies The conservative form of the time-averaged mass transfer equation
(2.151) can be integrated with respect to x (in the seaward direction) to obtain
A ft ***
I. (2.162)
XP"
A "ax
The evaluation of Equation (2.162) can be carried out in finite-difference form to determine
E- as a function of x for the period of time during which the salinity concentration data was
obtained. Equation (2.152) is a special case of Equation (2.162) in which aC/at - 0 . An ex-
pression for ES similar to Equation (2.162) can be obtained by integrating the slack tide
approximation Equation (2.153). Numerical evaluations of dispersion coefficients have been
carried out by Kent (1960) in the Delaware, Burt and Marriage (1957) in the Yaguina (Oregon),
Lawler (1966) in the Hudson, Cox and Macola (1967) in the Sacramento-San Joaquin Delta, Glenne
and Selleck (1969) in San Francisco Bay, Dorresteln and Otto (1960) and Eggink (1966) in the
Eems (Germany), and Boicourt (1969) in upper Chesapeake Bay. The dispersion coefficients obtained
are characterized by large variations longitudinally with increasing values in the seaward direc-
tion. For example, in the northern arm of San Francisco Bay, Glenne and Selleck (1969) calculated
values of ^ varying from 300 ft3/sec (1 mi*/day) to 18,000 ft*/sec (56 mia/day).
3.3.4.2 Predictive Studies The principal difficulty in the development of predictive models
for salinity intrusion is in the formulation of the boundary conditions for salinity. If x is
measured in the landward direction from the mouth of an estuary, the upstream boundary condition
for salinity as x becomes large, can be stated as
C - 0
(2.163)
In the seaward direction if is obvious that the salinity must approach the constant value of
salinity in the ocean; the difficulty arises because the longitudinal location at which the
salinity becomes constant is a variable depending on freshwater inflow and tidal range.
Ippen and Harleman (1961) (see also Ippen 1966) analyzed a series of salinity distri-
bution tests in a tidal flume at the Waterways Experiment Station. The experiments were conducted
in a rectangular flume of constant cross section having a length of 327 feet. The upstream end
of the flume has provision for the introduction of fresh water at a controlled rate. The down-
stream end is connected to a large tidal basin in which a constant salinity and a harmonic tide
can be maintained. Observations of the salinity distributions were made for various combinations
of tidal range and freshwater discharge.
Salinity distributions at high water slack and low water slack for three tests having
the same basin tidal range are shown in Figure 2.19. The freshwater discharge for Test No. 2
is twice that of Test No. 16; and for Test No. 11 it is 2.8 times that of Test No. 16. The
71
-------
s/s.
s/s.
s/sn
0.05 FT Uf =0.02 FT/SEC -
S0=29.2PPT G/J - 50
HW
TEST 2
V
S =25.0 PPT G/J - 29
a =0.05 FT Uf -0.0'« FT/SEC-
TEST 11
0.05 FT Uf =0.056 FT/SEC
S - 26.4 PPT G/J - 20
0 20 Uo 60 80 100 120 1M) 160 180 200
X IN FEET
Fig. 2.19 Comparison of experimental and
analytical salinity distributions
for series I, tests 16, 2, and 11.
Ippen and garleman (1961).
change in the longitudinal position of the salinity distribution due to the increasing freshwater
discharge is evident. Figure 2.19 also shows the validity of the upstream salinity boundary con-
dition given by Equation (2.163) and the difficulty of specifying the seaward boundary condition.
It was shown that: the salinity distribution at low water slack could be advected upstream by a
distance equal to the tidal excursion and that this distribution (shown by the solid lines in
Figure 2.19) agreed with the salinity observations at high water slack.
72
-------
Harleman and Abraham (1966) made use of this property and developed a mathematical
model for the salinity distribution in a constant-area estuary based on the slack tide approxi-
mation. Under conditions of constant freshwater inflow and tidal range, Equation (2.153) reduces
to
dC,0 . ., dC...
where the subscript ts indicates the distribution at low water slack and x is measured in
the landward direction from the mouth. Equation (2.164) may be integrated with respect to x ,
since Uf is a constant,
dC
-Uf Cts ' Ets Tx* + cl <2'165>
The integration constant c^ is zero from the upstream boundary condition given by Equation
(2.163). An equation for the salinity distribution is obtained by integrating Equation (2.165)
a second time with respect to x . This requires that a function E, - f(x) be specified, and
that a boundary condition for the seaward end be formulated for evaluation of the second inte-
gration constant. The seaward boundary condition is obtained by assuming that the salinity
distribution curve at low water slack can be extended in the seaward direction by a distance B
to a point on the negative x axis where the salinity is equal to the ocean value CQ , thus
C. = C at x - - B (2.166)
vo o
The dependence of E/s on x is assumed to be an inverse function of x of the form
B
(Oo
where (E, ) is the dispersion coefficient for low water slack at x - 0 . Under the above
assumptions, Equation (2.165) can be integrated and the constant of integration evaluated from
Equation (2.166),
C _ Uf(x + B)2,
LS-expr. f - - (2.167)
If the salinity is known at low water slack for at least two values of x , the parameters
(E. ) and B can be determined from Equation (2.167).
•trS O
Figure 2.20 shows the parameters determined from Equation (2.167), and the agreement
between the computed and observed salinities at high water slack after displacement of the low
water slack distribution by a distance equal to the tidal excursion. The following empirical
correlations were determined for the series of tests conducted at the Waterways Experiment
Station:
- °-008 Uo T (!)2'5 C <2'168>
73
-------
where
B - 0.11
UQT
(2.169)
U •
tidal velocity at x - 0
- tidal amplitude at x - 0
- mean depth
- cross -sectional area
a
h
A
PC " tidal prism (volume of sea water entering on flood tide)
Qf - freshwater discharge
T " tidal period.
The salinity distribution predicted by Equation (2.167), using the parameters specified in
Equations (2.168) and (2.169), has been verified by field measurements in the Rotterdam Waterway
(Harleman and Abraham 1966) . Salinity data for both 1908 and 1956 were used in the verification
as shown in Figure 2.21.
1.0
LOW WATER SUCK
— — — HIGH WATER SUCK
EXPERIMENTAL OBSERVATION
Uf- 0.017 m/SEC
IE,,).- 0.1 J85 m*/SEC
B - 17.4m
Fig. 2.20 Determination of salinity distribution parameters (Eis)o and B by
comparison of Equation (2.167) and experimental data (Test No. 11).
74
-------
s/s
0
0.8
0.6
0.1*
0.2
0
1 0
0.8
fi
'
0.1.
0.2
0
1 ~T*>» 1 1 1 " 1
NX CALCULATED L.W
\SLACK CURVE
\
~ \
\
\ •
— V
\
\
\
\
\
—
22 JULY 1908
B - 10,700 M
D0 - 690 M2/SEC
•CENTER LINE OF RIVER
1 1 L 1 I L
-12 -10 -8 -6 -1* -2 C
ST. 1032. 7
SI
r^ i i i i
N, CALCULATED L.W.
"VSLACK CURVE
X
~~ X
\
\
X
X
X
X
\
_
26 JUNE 1956
B-1300 H
D0 - 1 ,500 M2/SEC
•CENTER LINE OF RIVER
I NEAR BANKS OF RIVER
1 1 1 1 1 I
-12-10-8 -6 -1* -2 1
ST.1
' 1 ^S. 1 I 1 1 1
Jv TRANSLATED L.W.
\SLACK CURVE
X • v>
\ <
\ ^
L., = 12 ,800m \ 3
X o
\H.W.SLACK _ —
\ Z
\ °
\ I—
\ *->
\ z
X =
L.W. SLACK \ ~> —
\ \
\ x-
N
N^
! -*..
11(1 fl f-, 1 1
) 2 k 6 8 10 12 ! U 16 1
ST. 1025 ST. 1017
.1029 ST. 1021
^r^j i j i i i
^V TRANSLATED L.W.
^XSLACK CURVE ^
x °
\ H.W.SLACK ^
^v
L,,»l3,500m \^ g
-
_
K i
!>--.. -
1 1 1 1 1 1 & — r=fr-
) 2 i* 6 8 10 12 1<> 16; T
1 1
030 ST. 1023 ! ST. 101
ST. 1026 ST. 101 5
8
y /L_\
*. \IU!)/
8
3
km)
Fig. 2.21 Salinity data, Rotterdam Waterway.
75
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Studies In the prediction of salinity Intrusion changes In variable-area estuaries
have been undertaken by several Investigators. Harleman and Hoopes (1963) have proposed a method
for predicting the effect of freshwater discharge on salinity distributions In the Delaware
Estuary. Dl Toro (1969) has developed a mathematical model for salinity distribution based on
a Markov chain representation of mixing In discrete segments of a variable-area estuary. How-
ever, the method Is not truly predictive inasmuch as salinity observations are required In order
to establish the boundary conditions. Wastler and Walter (1968) have undertaken a statistical
analysis of a large number of field observations of salinity in the Cooper River - Charleston
Harbor estuarine system.
3.4 COMPARISON OF REAL TIME AMD NOR-TIDAL ADVECTIVE MATHEMATICAL MODELS
The objective of this section is a quantitative comparison of concentration distribu-
tions produced by the real time and by non-tidal advective mathematical models under identical
conditions. The essential features can be illustrated by comparing solutions for the following
conditions: (1) the continuous inflow of a non-conservative pollutant into the non-saline tidal
portion of an estuary, (2) the Intrusion of salinity in the seaward portion of an estuary. In
order to use the previously developed analytical and numerical solutions, an estuary of constant
cross-sectional area is assumed in both cases.
3.4.1 Continuous Inflow of a Non-Conservative Pollutant
The problem description and the solution of the real time mass transfer equation are
given In Section 3.2.6. The discharge of a non-conservative pollutant, having a first-order
decay constant K^ - 0.034/day , occurs at x - 0 at a constant rate w (Ibs/day). The cross-
sectional area Is constant and the injection point is well above the limit of salinity intrusion.
The velocity in the hypothetical tidal channel is assumed to be given by Equation (2.145),
U " Uf + UT Sll> TT
where
Uf - 0.1 ft/sec
UT - 2.0 ft/sec
T - 44,712 sec (12.4 hrs).
The tidal excursion Equation (2.149) is X - 28,500 ft. The constant longitudinal dispersion
coefficient given by the modified Taylor equation (2.126) and (2.127) is 65 ft'/sec (0.2 sq ml/
day) for n - 0.028 . The reference concentration is
Co
The real time solution is given by Equation (2.148) and the quasi-steady state concentration
distributions at high and low water slack are plotted in Figure 2.18. The corresponding steady-
state solution for the slack tide approximation is given by Equation (2.157). This equation can
be written in terms of the reference concentration CQ as follows:
76
-------
C U- .- .,
^-#«p[^C«f*o
(2.170)
where the negative sign applies to x > 0 and the positive sign to x < 0 and 0
Vu^
The comparison of the real time solution and the slack tide approximation is shown in
Figure 2.22, using the same numerical values in both solutions. According to the slack tide
approximation, there is essentially no concentration distribution upstream of the injection sec-
tion (x » 0). At x - -3000 ft. , or ten percent of the tidal excursion, the concentration pre-
dicted by Equation (2.170) is Cs/CQ = .01 which is too small to show in Figure 2.22. The real
time solution predicts measurable concentrations over a distance of 30,000 feet or approximately
6 miles upstream of the injection section. In the downstream direction, the slack tide approxi-
mation falls essentially between the high and low water slack distribution of the real time
solution.
Inasmuch as the dispersion coefficient E in the slack tide approximation is not
s
predictable in advance, we may inquire whether it is possible to find a constant value of E
S
which will result in a reasonable agreement with the real time solution, both upstream and down-
stream of the injection section. The result of two trial values of Eg of 1200 ft*/sec (3.75
sq mi/day) and 2250 ft8/sec (7 sq mi/day) is shown in Figure 2.23, the latter value being 35
times larger than the dispersion coefficient used in the real time solution. It can be seen that
the shape of the concentration distributions are quite different and no single value of Eg will
bring the two mathematical models into agreement. The same difficulty arises with respect to the
time-averaged mass transfer equation since Equation (2.161) is of the same form as Equation
(2.170).
16
o JO
t—
$
08
06
UJ
02
INPUT
1 .
2.
3.
4.
GEOMETRIC DATA
CONSTANT CROSS-SECTION
AREA
VELOCITY DATA
UF = O.I FT/SEC.
TIDAL DATA
T = 44,712 SEC
COEFFICIENT
K - 0.03WDAY
E =• 65 FT2/SEC
H.W.S.
-------
3.4.2 Salinity Intrusion
The example used for this comparison is described in Section 3.2.4.2 and is based on
salinity intrusion data for the Rotterdam Waterway. The agreement between the observed salinity
distributions and calculated values using the real time equations is shown in Figure 2.16 where
- 13,000 (1 - x/L)3 , [ft2/sec]
(2.131)
It is desired to compare the longitudinal dispersion coefficients obtained from the non-tidal
mathematical models, using the same salinity data, with the above values of E^' in the real
time model.
The dispersion coefficient in the time-averaged mass transfer equation can be deter-
mined as a function of x from Equation (2.162) by using the time-averaged salinity distribution
data of Figure 2.16. In the Rotterdam Waterway, the cross-sectional area is essentially constant
and the salinity data was obtained during a period of time when dC/8t - 0 . Therefore, Equation
(2.162) reduces to
dC/dx
(2.171)
C/C0
0.6 -
CONSTANT CROSS SECTION
mat
0.1 FT/SEC
0.03WDAY
,800 SEC
REAL TIME- LWS
EL'6SFT2/SEC
REALTHC-HWS
E,«€5FT2/SEC
SLACK TIDE
APPROXIMATION
0.1* -
0.2 -
-100.000
+100,000
Fig. 2.23 Comparison of real time solution, Equation (2.148) [EL - 65 ft»/sec], with
slack tide approximation, Equation (2.170) [Eh- - 1200 and 2250 ft"/sec],
for continuous inflow of a non-conservative pollutant.
78
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The corresponding form for the slack tide approximation is
(2.172)
It is apparent that two sets of E - f(x) can be obtained from Equation (2.172) depending on
whether the salinity data for high or low water slack is used.
The results of applying Equations (2.171) and (2.172) to the Rotterdam Waterway data
are shown in Figure 2.24 where t 'and E (high and low water slack) are plotted as functions
of x . In addition, the real time dispersion coefficient E." given by Equation (2.131) is
plotted as a function of x for comparison. It is apparent that significant differences exist
between the real time and the non-tidal dispersion coefficients in salinity intrusion regions.
3.5 SUMMARY AND CONCLUSIONS
The basic mathematical framework for estuarine water quality is a mass transfer equa-
tion, alternatively known as a conservation of mass equation, a mass balance, a convective-
diffusion or a convective-dispersion equation. This section of Chapter II has dealt with the
one-dimensional form of the mass transfer equation with particular emphasis on the hydrodynamic
aspects of the mathematical model. The general form of the one-dimensional equation is given by
Equation (2.95). The hydrodynamic aspects, as here defined, are contained in the first three
terms of this equation which relate to the temporal and spatial variation of area, velocity and
longitudinal dispersion coefficient. The water quality aspects are contained in the remaining
source and sink terms in the mass transfer equation. The dependent variable is the concentration
of a substance in the water mass whose distribution in space and time is to be determined. The
concentration of substance may refer to chemical and/or biological components. In general, a
coupled set consisting of N mass transfer equations is required in order to define the water
quality state of an estuary in terms of N water quality parameters.
The section is divided into three parts; Section 3.2 treats the mathematical models
in real time (i.e., following the tidal motion), Section 3.3 deals with the mass transfer equa-
tions using non-tidal advective velocities, and Section 3.4 presents a comparison of these two
basic types of water quality models. A brief summary of the mathematical models is given below.
3.5.1 Mathematical Models in Real Time
The real time mass transfer equation requires some knowledge of the tidal motion in
the estuary. The input to the mass transfer equation is the specification of the cross-sectional
area and velocity as functions of x and t . Considerable latitude is possible in the com-
pleteness and accuracy in which the tidal information is supplied. In general, the analytical
efforts are in inverse proportion to the amount of field data available. Three approaches are
described in Section 3.2.3.
(i) Calculation of the tidal characteristics by simultaneous solution of the continuity and
momentum equations. This approach requires a minimum amount of field data; techniques involving
simultaneous solution by finite-difference equations are well developed.
(ii) Calculation of the tidal characteristics by cubature. This approach requires extensive
field data on the distribution of tidal amplitude and phase. The continuity equation is solved
79
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60,000
1,000
Fig. 2.24 Comparison of longitudinal dispersion coefficients
obtained from analysis of Rotterdam Waterway salinity
data (Stigter and Siemens 1967).
80
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by finite-difference techniques to obtain the tidal velocity.
(iii) Assumption of a sinusoidal or harmonic representation of the tidal velocity in the form
of Equation (2.122). Field data on the maximum values of tidal velocity are required and no
analytical techniques are employed.
The first two approaches may contain important nonlinear information on the tidal motion which
is characteristic of many estuaries. Such nonlinear!ties take the form of unequal magnitude and
duration of ebb and flood velocities. In general, the trade-off between the time and expense of
tidal analysis versus field measurement of tidal quantities is heavily in favor of analysis.
Longitudinal dispersion in the real time mass transfer equation is discussed in Section
3.2.4. It is important to distinguish between longitudinal dispersion in estuarine regions of
uniform water density as opposed to regions in which density gradients exist due to salinity
intrusion. A modified form of the Taylor dispersion equation (2.126) and (2.127) is used to
determine the magnitude of the longitudinal dispersion coefficient E^ in regions of uniform
density. The magnitude of E. depends on the local value of the maximum tidal velocity, hence
any of the techniques for determining the tidal hydraulics provides the necessary information
to find the dispersion coefficient in the uniform density region. The determination of the
longitudinal dispersion coefficients in the salinity intrusion region requires field data on
salinity distribution. The magnitudes of E^' are found by fitting the solution of the conser-
vative substance mass transfer equation to the observed salinity data. The tidal hydraulic
solution previously obtained provides the advective velocity input to the mass transfer equation
for salinity.
3.5.2 Mass Transfer Equations Using Non-Tidal Advective Velocities
Two forms of the mass transfer equation using non-tidal advective velocities are dis-
cussed in Section 3.3. In one case the real time mass transfer equation is simplified by time
averaging over a tida.1 period. All concentrations refer to averages over the tidal period, and
the advective velocity reduces to a small value associated with the non-tidal discharges such
as freshwater inflow. In the other case, a simplified mass transfer equation is obtained from
the real time equation by considering concentration distributions only at times of slack tide.
The advective velocity is arbitrarily set equal to that associated with non-tidal discharges.
The two models are mathematically similar although the concentration distributions derived from
them are quite different. Each of these simplifications of the real time mathematical model has
the effect of incorporating the mass transfer effects associated with the advective tidal motion
into the dispersive term. The dispersion coefficients in the non-tidal mass transfer equations
must be determined empirically, both in the constant density and salinity intrusion regions.
3.5.3 Conclusions
A state-of-the-art review should have as a primary objective the establishment of
guidelines for future developments in the field. Within the inherent limitations of one-
dimensional mathematical models for estuarine water quality, two choices appear to be open: (i)
to continue development of the non-tidal advective models; (ii) to continue development of the
real time models. Water quality models based on the non-tidal advective concepts were developed
in the pre-computer era. They were logical extensions of the Streeter-Phelps approach for streams
and rivers. The difference is in the inclusion of a dispersion term which is the only mechanism
by which mass can be transported upstream in a non-tidal model. The non-tidal mass transfer
differential equation can be solved in a straightforward manner, at least for estuaries in which
81
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the geometry Is a simple function of x . No analysis of the estuary tidal motion is required
and the time scale, of (he order of a tidal period, seemed adequate for many water quality in-
vestigations. There are, however, numerous examples of estuarine problems in which the analytical
solutions requiring constant coefficients and simple geometric relations are inadequate. In
these cases, finite-difference computer solution techniques have been developed. It is important
to question whether the continued development of the non-tidal advective models is a reasonable
exploitation of the capabilities of the computer era.
The longitudinal dispersion coefficient in the non-tidal mathematical models is an
elusive quantity requiring an ad hoc empirical determination for each situation under investi-
gation. Its determination has been the subject of many expensive and time consuming studies
involving dye injection or other tracer observations both in the field and in physical hydraulic
models. Thus the analytical simplifications gained by ignoring the tidal motion are more than
offset by the difficulties of determining an appropriate dispersion coefficient. This is espe-
cially important in tidal regions where salinity cannot be used as a conservative natural
substance. In this context Figure 2.23 shows the essential point of the argument as applied to
the constant density tidal region. The dispersion coefficient for the real time mathematical
model is predicted by the modified Taylor equation; however, the transport of mass upstream of
the injection point is almost entirely by the mechanism of tidal advection. The concentration
distribution produced by the real time model is relatively insensitive to variations in the pre-
dicted dispersion coefficient. In the non-tidal mathematical model an inflated dispersion
coefficient of the order of 35 times the real time value is required .to produce an upstream
transport, which is at best a poor approximation to the real time distribution. It is concluded
that the importance of longitudinal dispersion decreases as the accuracy of description of the
advective motion increases.
Further ambiguities in the non-tidal-advection dispersion coefficients have been
illustrated in the salinity intrusion region. Figure 2.24 shows the wide variation in the magni-
tudes of non-advective dispersion coefficients obtained by analyzing salinity data (i) at high
water slack, (ii) at low water slack, and (iii) as a time average over the tidal cycle. It is
apparent that dispersion effects, and especially dispersion coefficients, cannot be discussed
in an abstract manner. They are essentially defined by the formulation of the mathematical model
in that the effective dispersion always depends on the way in which the advective motion is
described.
Technical developments in the field of tidal hydraulics have provided the capability
of treating the tidal advection in the mass transfer equation in a satisfactory and unambiguous
manner. The use of real time water quality models eliminates uncertainty associated with arbi-
trary dispersion coefficients in the non-tidal models. This should permit future developments
to center on the important and difficult problems relating to generation and decay of non-
conservative substances and the interaction of chemical and biological substances in simultaneous
mass transfer equations.
The above considerations suggest that the time scale associated with the mathematical
water quality model is not necessarily determined by the time scale of interest in a water quality
problem. The small time scale of the real time models should be considered as a necessary solu-
tion technique which avoids the ambiguous coefficients associated with the non-tidal advective
models. The choice is again one of a trade-off between the time and expense devoted to analysis
as opposed to the time and expense of the additional field work, such as special dye dispersion
tests, required by the large time scale, non-tidal water quality models. In addition, increasing
concern with the total ecological system makes it difficult to ignore the real "time of travel"
associated with the tidal motion.
82
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The real time water quality model makes it possible to consider estuary pollution
control by time-dependent effluent discharges. For example, a source of pollution located within
one tidal excursion of the estuary mouth could make use of a six hour detention basin in order
to discharge at twice the average rate during the period of ebb tide.
Although these conclusions have a strong bias in favor of the real time mathematical
model it is not proposed that all existing or future water quality models be reformulated on
this basis. There will continue to be a place for the simpler non-tidal advective models in
the analysis and prediction of estuarine water quality. For example, in a major estuary study,
both the real time and the non-tidal advective models could be formulated initially. Solu-
tions for both models could be obtained for a single effluent source undergoing a simple
first-order decay. The real time model could then be used to verify the hydrodynatnic aspects
of the non-tidal model through the choice of non-tidal dispersion coefficients to a desired
degree of agreement with the real time solution. Thus the formulation and solution of the real
time problem is an "analytical dye dispersion test" which replaces the prototype or hydraulic
model "dye dispersion test" previously used to verify the non-tidal mathematical model. In
the remainder of the study, simultaneous non-tidal advective models could be used to consider
the interaction of water quality substances and multiple sources of pollution.
Solutions from the real time model may also be used to check finite-difference formu-
lations of water quality models. The finite-difference technique can be checked by applying it
to an estuary of uniform geometry for which analytical solutions in real time are available.
The details of finite-difference solution techniques are not within the scope of this
chapter. However, reference is made to several studies involving the formulation of the real
time mass transfer equation in finite-difference schemes. Bella and Dobbins (1968) presented
the results of finite-difference calculations for BOD and DO profiles in the constant density
portion of a hypothetical constant-area estuary having a sinusoidal tide. Their results are
compared with solutions obtained from the steady-state, non-tidal advective mass transfer equa-
tions. Dornhelm and Woolhiser (1968) developed a finite-difference scheme for a hypothetical
estuary (whose area varies as a linear function of x ) with a sinusoidal tide. Certain diffi-
culties with boundary conditions were encountered. In both of the above investigations the
numerical values of the real time dispersion coefficients were arbitrarily assigned. In other
words, the link between the magnitude of the dispersion coefficient and the tidal motion was not
considered. Orlob et al. (1967) and Orlob (1968) have studied water quality problems in
San Francisco Bay and the Sacramento-San Joaquin Delta area. The problem includes two-dimensional
components due to lateral variations in embayments such as San Pablo and Suisun Bays. The
computational scheme consists of a link-node network configuration of uniform flow channels.
The real time finite-difference model considers tidal advection and dispersion in the uniform
channel links between nodes. All non-advective and dispersion aspects of the mass balance such
as decay and surface transfer are assumed to be concentrated at the nodal junctions. The dis-
persion coefficients and the rate constants of the various source and sink terms must be adjusted
to match field data. Callaway et al. (1969) have proposed the application of this type of model
to the lower portion of the Columbia River.
An extensive study of the real time mass transfer mathematical model has been under-
taken by Lee (1970). Finite-difference formulations of the continuity, momentum and mass transfer
equations have been developed for variable-area estuaries of arbitrary geometry. This permits
inclusion of nonlinear tidal advection, multiple pollution sources, and time-dependent freshwater
inflow. The real time longitudinal dispersion coefficients are related to the tidal motion,
through salinity observations in the salinity intrusion region and through the modified Taylor
dispersion equation in the constant density tidal region. Water quality parameter observations
83
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In the Potomac, Delaware and James estuaries are used for comparison with the analytical results.
Preliminary results have been published by Harletnan et £l.(1968) for the upper freshwater tidal
reach of the Potomac and compared with the field observations of Hetling and O'Connell (1966).
This is essentially a demonstration of the validity of an "analytical dye dispersion test".
In conclusion, it is suggested that the state-of-the-art is such that ambiguities
regarding the hydrodynamic (i.e., advective-dispersion) aspects of mass transfer in estuaries
should no longer be accepted in one-dimensional water quality models. Undoubtedly, future
developments will continue both in the real time and In the non-tidal advective models. The
rapidly increasing capability of high-speed computers suggests less reason for future reliance
on the non-tidal models. When they are used, it is hoped that their limitations and their rela-
tion to the real time models will be understood.
-------
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Harleman, D. R. F., and J. A. Hoopes, 1963: The prediction of salinity intrusion changes in
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86
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Harleman, D. R. F., and C. H. Lee, 1969: The computation of tides and currents in estuaries
and canals. Technical Bulletin Ho. 16, Committee on Tidal Hydraulics, Corps of
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DISCUSSION
FRITCHARD: In this first section we deal with the basic equations In a development that might
serve as a three-dimensional model. Now I went to some extent philosophically here to dis-
cuss some questions concerning averaging that I feel, If not properly treated, can get us
off on the wrong foot with the basic equations. Also, I spent some time In trying to start
with first principles and show what Is neglected. I know that many of the engineers are
concerned with conservation of these properties over a cross section or seaward element
extending from surface to bottom, and apply the basic principles to that finite element. I
find that I sometimes miss terms when I do that. So my approach to the bookkeeping starts
with the basic principle which was really developed to be applied to a moving particle of
water properly defined, and then Integrating these equations over space, and I think it may
be that one is less likely to miss some important terms.
We note a couple of things about these equations. One is that we have these stress
terms which we need to find out how to treat. And then what I have done with respect to the
pressure term is to try to put it into a form that people who have used one-dimensional and
two-dimensional models would feel more at home with, that Is, to put the pressure term in
terms of the slope of the sea surface. When we do this, we've got to recognize that in
what I consider real estuaries, estuaries that have freshwater inputs sufficient to give
a field gradient in salinity, there Is a distribution of density, both vertical and hori-
zontal, and that this distribution of density leads to an internal slope in the pressure
field. An internal slope In the pressure field introduces Into the pressure force term an
additional slope, the slope of the pressure surfaces relative to the sea surface, which I
have labeled here as i, meaning the slope of a pressure surface In the x^ direction
•*-t Pt1!
referred to the surface, the surface being n .
These equations are then coupled with the equation of continuity and with the ver-
tical equation which can be used to compute the internal slope of the pressure field. This
kind of computation is a classical one In oceanography. It's done every time oceanographers
compute geostrophic currents to determine essentially the slope of the isobaric surfaces.
It is not stated that way, but the technique for carrying out the integral given on the right
side of (2.21) and (2.22) is readily available in oceanographic texts. So those equations
must be used as auxiliary equations.
Now, these equations as they stand have more unknowns than there are equations.
For one thing, you've got to know the time-dependent density field to compute the Internal
slope of the pressure surfaces, so that it's necessary in any case, not only to solve simul-
taneously the conservation of mass and the conservation of momentum, but you must also
simultaneously solve the time-dependent distribution of salinity. You've got to feed that
back In, as density is dependent on salinity, to compute the density field and to compute
the internal pressure field.
What I've done then is to attempt to argue for some further simplification of these
equations. That leaves us with the Equations (2.27) and (2.28). We really have no direct
evidence that these equations can be further simplified. They represent equations with three
velocity components, as a function of time and three spatial coordinates, and the surface ele-
vation, as a function of position and time, being the things that we want to get out of them.
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Then I have given a development of what you might call the salt balance equation
in three dimensions. Now it has been argued from studies that have been done in two dimen-
sions that the more one knows about the details of motion, the less concerned one has to be
with the diffusion terms. I think that this is probably true in the horizontal components.
These terms arise out of the cross products of the horizontal velocity deviations, the
gradients of these terms, and they arise out of averaging the advective terms, so that if
you know more about the advective terms it's obvious that the diffusion terms become less
important. I think that if you are going to deal with three dimensions, you can't treat
the vertical term in the same way. * It becomes important when we have to talk about the
vertical third dimension. But I have argued that in both the momentum equation and the salt
balance equation the horizontal eddy viscous terms and the horizontal eddy diffusion terms
can probably be neglected as long as we are talking about ensembles and the ensemble-averaged
equations.
Finally, one then arrives at Equations (2.35) and (2.36) which are the two horizontal
components of the equation of motion in three dimensions reduced about as far as I think we
have any basis for reducing them at the present time, and the salt balance equation (2,37)
also reduced about as far as I think we have any basis for reducing it. 1 want to point out
that these equations still contain a vertical turbulent viscous term, a vertical diffusion,
and also that the equations of motion contain this term which involves the internal variation
in the pressure field because of the distribution of density. 1 want to point out also that
the size of that term is not revealed by looking at the vertical distribution, as Rattray
has shown. This term is a term that leads to, in what I would call strong estuaries, there
being a shear positive towards the ocean. The net effect is that there is essentially a
shear in the time-dependent vertical velocity field leading to a tidal-averaged motion which
is essentially two-layered, with motion towards the ocean and towards the land. I want to
emphasize that you can't be fooled into using the vertically averaged equations in which
that part of the pressure term is omitted simply by saying that this is a we11-mixed estuary,
because salinity hardly varies from top to bottom. You may have an estuary which is suffi-
ciently well-mixed vertically, as far as salinity goes, but which will not be well-mixed as
there is a shear in the velocity field. There still will be this two-layered flow. A good
case of this is the study of the Mersey by Bowden which was considered as a classical case
of a well-mixed estuary, but he showed it very definitely had a vertical shear in the velocity
field, and had a flow pattern as I have described.
Then in order to complete the ensemble of equations, you've got to impose certain
boundary conditions, among these certain integral conditions, that is, conditions on the
equations integrated vertically and across the cross section. These integral forms of the
equations provide additional terms, for instance Equation (2.40) provides an additional
relationship which allows the computation of TI coupled with the other equations, and so on.
We still end up in this situation having some unknown terms, for instance the
vertical eddy viscosity which I've labeled va and the vertical eddy diffusivity which I've
labeled Kg . I have talked about some evidence for the vertical variation in the eddy
diffusivity based on some data that I got in the James, and compared it with some data that
was taken by Bowden in the Mersey.
I'd like to point out that this is one of the really big unknowns, and this is an
area in which we really need to have some work done. And that is: how do we relate the ver-
tical eddy viscosity and the vertical diffusivity to known or computable parameters? And
to me it is a very important unknown.
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Equation (2.58) is a vertically-averaged equation of continuity expressing the
conservation of mass, which can also be shown to reduce to the fact that the horizontal
divergence of flow must equal the time rate of change, the rise and fall, of the sea sur-
face, that is, Equation (2.59). It represents continuity in two dimensions. Then I've
treated the conservation of momentum in the same way, but in this case you can't a priori
drop the cross-product terms. We treat the two-dimensional equations of motion, in which
we get some terms which involve the vertical mean average of cross products of the devia-
tions, like u,' v. ' and 0^')* > which depend on the vertical shear of the motion. In
estuaries in which the mean internal pressure slope is important, there's going to be a
vertical shear and these cross-product terms are likely to be important. One has boundary
values for the vertical stresses. One of these is the wind stress, TW x and TM v , and
Tw,y •
sed in 1
the other is the bottom stress which I have accepted as being reasonably expressed in terms
of a Chezy-type equation. I've argued about the size of these viscous terms (or these terms
that have the appearance of a Reynolds stress terra but which really involve the fact that
you are vertically averaging a vertically varying velocity field) and said that, well, maybe
you can express these in the same way you expressed the Reynolds stress terms before, writ-
ing the vertical-mean value of these velocity deviations in terms of a component of the
deformation tensor and some sort of an eddy coefficient. But these eddy coefficients now
are quite different from those in the three-dimensional equation. They are not simply the
average of a cross-product of turbulent deviations, but are averages which involve the fact
that the real mean motion, the deterministic part of the motion, varies through the depth.
I don't know how to treat these. I don't know whether we know enough about them. Don
[Harleman] has some ideas from some of his flume studies, but for the moment I'm just saying
that I hope they're small enough to be neglected.
Let's consider the case where they're small, Equation (2.66) and (2.67), which says
that the acceleration, both local and field, of the vertical-averaged velocity field is
equal to the sum of a pressure term which involves gravity times the gradient of the slope
of the surface plus the vertical-mead value of the slope of the isobaric surfaces relative
to the sea surface, a term involving an atmospheric pressure gradient, the coriolls term,
wind stress term and the bottom stress term.
One can't really argue that these terms, the average of the internal slope of the
pressure surface, are small, even in this averaged case, even in this two-dimensional case.
In the James, for instance, this kind of process represents about 36% of the mean of the
absolute magnitudes of the surface slope over a tidal period. This is a term that doesn't
vary much with the tide, that is, the tide causes both of these things to vary, the surface
of the water and the pressure surface. Over the tidal cycle this internal slope doesn't
vary, so that it's there at all phases of the tide. So I don't think you can neglect it,
even in the vertically averaged equations for most real estuaries. Now, there are bays,
lagoons, and other kinds of waterways in which the input fresh water is sufficiently low so
that salinity doesn't have much horizontal variation, and they're shallow, so that the type
of vertical average that has been proposed by Jan Leendertse here probably is valid. But
I don't think it applies to the estuaries along the east coast of the United States.
RATTRAY: I'd like to reinforce the statement about the importance of the internal density
field in governing the motion, and particularly the rate of dispersion of any quantity intro-
duced. I think this, as far as I have been able to see In many of the studies that have been
going on in the recent past, has been overlooked, and the effect seems to be important in
almost all cases where they do have salinity distributions, estuaries for example. Again,
It is not true that you can look at the vertical salinity profile and on the basis of that
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alone say anything about the vertical profile, on the average. Looking at the salinity
distribution won't uniquely give you the mean velocity distribution nor the other way around,
so that one has to look at it in a little more detail. Particularly the depth is important:
the same density difference will drive a stronger circulation as the estuary becomes deeper.
So in these very shallow situations the density fluctuations due to the salinity may not be
very important, but as you get into deeper bodies of water it will be. The Mersey, for
example, was a case in point. Certainly Puget Sound is a case in point.
In the equations that have been derived here, I'm very happy to see the detail that
has been gone into, because 1 think this is an important starting point which will show quite
clearly the shortcomings of various models that have been proposed, and where they might be
improved.
I have maybe a question, Don. I haven't really been able to digest it this fast,
but with what you've done with two-dimensional estuaries, it doesn't seem that you have a
closed set of equations—now that you've put in some of the terms that should have been
included—equations that you can't really solve without going three-dimensional.
PRITCHARD: Well, I think it is closed from the standpoint that there are some integral boundary
conditions that you've got to take just as you do in the three-dimensional case, to give you
some additional terms which allow you to close the solution for n , you see. Because n
is not dependent on three dimensions; it's only dependent on two effective dimensions.
RATTRAY: Well, I guess I see what I am really after. You use Reynolds stresses for the
vertically integrated difference terms. I wonder if you think there is any such real eddy
coefficients that one can reasonably define and then use from one situation to another, as
the river conditions and whatever might change.
,<
PRITCHARD: You notice that in the next paragraph I say 1 don't know how to treat these and 1
am going to treat the case where you can forget about them. But now I think Don [Harleman]
might have some comments on this from his work at Vicksburg, where he puts fresh water in
one end of a flume and salt water in the other, then vibrates a frame of what you might call
roughness elements in the flume, and gets something of a convective transport picture.
RATTRAY: That's part of it. The other is that you have internal pressure fields. How do
you get those in a vertically integrated model?
PRITCHARD: That's the i and i averaged over h . The way you get this is to
X»PITI yjp»M
have a simultaneous solution of the vertically averaged salinity balance equation. Again
what do you do about diffusion in that case? Even if you might neglect the horizontal terms
involving these vertical averages of the velocity deviations that arise out of vertical
shear—because, even If those terms individually might be strong, their gradients may not
be large, as they appear in the dynamic equations or the equations of motion—I don't think
the same kind of arguments can be made necessarily about the cross-products involving velocity
and salinity. You can't a priori say just because this kind of process is small in one
equation then it is in all of the others. So that you do have the question of what can you
do about these diffusion terms. At least, 1 think that it is not a question of how fine
you take the velocity field because you are not taking the vertical variation in the hori-
zontal velocity fine at all in this vertically averaged equation.
So I think we have problems here, but let's assume that through experiments and
such evidence, we get some way of relating the diffusion processes to known or computable
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terns, and we can solve these equations simultaneously in the two dimensions. This means
that you have to take in each time step the salinity you get, and compute a slope term.
You'll get a value for this vertical mean of the internal slope, simply by taking the ver-
tical-mean density field using the salinities you computed each of these time steps, and
using the conventional arguments about how thick that column of water must be to have the
same pressure at the bottom.
RATTRAY: How do you get the horizontal advection of salt correct, which is usually a signifi-
cant fraction of the horizontal salt flux, in the vertically averaged set of equations? Don't
you have to know what the vertical profile velocity is?
PRITCHARD: I agree with your problems. They are problems that have worried me. But what I
am saying is if we are going to try to go on, quite frankly as I've developed it now, I think
we are going to the three-dimensional equations.
RATTRAY: That's what I wanted to get you to say.
PRITCHARD: After all, we're writing this as a state-of-the-art report, and we've got to have
in here relationships which can be compared to models that the federal people have to look
at In terms of having been presented as ways to solve their problems. And I wanted to get
the equations in the forms that are nearest to the ones that they have been faced with, in
evaluating and perhaps using, and then say: well look, these equations miss these terms.
That's the main purpose of the exercise here. I think that it will be well worthwhile to
try some two-dimensional models with this mean of the Internal pressure field in them just
to see what difference it makes in those kinds of equations. But ultimately I think we have
got to go to the three-dimensional equations, which means a lot more computer time.
LEEHDERTSE: I would like to make a comment on this slope term. Assuming now that it doesn't
change very much, let's say if you have a certain amount of fresh water to salt water, a
tidal excursion at the inlet of, say, five miles or so, you only get very small time varia-
tions so that effectively this gives a fixed value of this slope In your computations. Now
you never know in actual computations what the different levels really are between the gauges
which you have, so you may not be able very often to distinguish this accurately enough to
insert in your computations. So part of this term would become indistinguishable.
PRITCHARD: Become indistinguishable from the uncertainties in the slope of the surface?
LEENDERTSE: Yes. I don't think that it is very difficult to put it in, to go to the salinity
equation and compute it as a time-varying function also. 1 expect that it doesn't make very
much difference.
PRITCHARD: I don't think that it varies very much over the tidal cycle. I think that we are
talking about longer term variations.
HARLEHAN: The objectives of this third section are as follows. First, to provide a broad,
critical review of the literature relating to one-dimensional models which have as their
purpose some phase of a water quality problem In estuaries. Secondly, I have tried to set
down the basic equations of one-dimensional continuity, momentum, and mass transfer in an
unambiguous form relating to time-averaging concepts. A great deal of the emphasis here is
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on this problem of what period of time you are looking at when you write one-dimensional
equations. (I have used the terminology in here of "real time". I am not very happy with
that, and I think it probably should be changed because "real time" has certain other conno-
tations, as most of you are aware.) Thirdly, I've attempted to clarify the nature of the
assumptions made by existing mathematical models by referring back always to this concept
of the period of time averaging. Fourthly, 1 have tried to indicate what I feel are the
directions of future research and applications of one-dimensional models. 1 have a very
strong feeling that we will still be involved with one-dimensional models for a period of
time. Hopefully, we will be able to move into the more complete modeling of two- and three-
dimensional problems. But this is not going to occur very rapidly, and problems that have
to be handled today and tomorrow are probably going to continue to be handled in the one-
dimensional framework.
The emphasis also here is on the development of predictive models, in the sense of
prediction of everything except what I call water quality parameters, which are basically
source and sink terms and which 1 feel are really the most difficult problem of water quality
modeling. The real crux of the problem is the source and sink terms. For too long we have
been concerned with the application of simplified hydraulic-hydrodynamic approaches which
have complicated and confused the picture, and have, I think, delayed getting on with the
real job of dealing with the source and sink terms and the multiple interactions that have
to be considered. Certainly in these areas of interacting equations where one sink becomes
a source for another term and so on, much of the important work remains to be done. These
problems are so complex in defining these sources and sinks that 1 think this will be another
reason why the one-dimensional approach will have to continue to be used. We simply don't
know enough about these in even the most elementary framework to begin putting them into a
more complex situation.
To summarize, then, I think that it is perfectly clear that on one-dimensional
basis, if we define the geometry of an estuary, if we define the boundary conditions in
terms of the tidal amplitude at the ocean and the conditions at the head of tide or whatever
the upstream conditions are, and if we make some assumption about the roughness, we can then
proceed to calculate the complete tidal motion without any great difficulty. We can then
modify the roughness assumption by comparing these calculations with observed tidal ampli-
tudes. Once that is achieved, we can be fairly certain that the velocities computed by this
model are reasonably accurate.
So we have the technique. We have the capability of computing one-dimensional tidal
motion. This, then, becomes an input to the conservation of mass equations which are dis-
cussed and shown at various places in here, for example, Equation (2.95), which I call the
real time mass transfer equation. This expresses the concentration of any substance, con-
servative or non-conservative, in a one-dimensional variable-area estuary, in which the
advectlve term U is the instantaneous tidal velocity that we have already calculated by
solving the continuity and momentum equations.
Now we have to distinguish between the saline portion and the non-saline portion.
I have not been careful here to use the word "estuary" in its precise formulation which
refers only to saline regions. I consider an estuary the extent of tidal motion. Here we
have to consider the part of estuaries which are non-saline or estuaries which are entirely
saline, in other words in the absence of salinity gradients. By computing the tidal velocity,
we can make an estimate, a fairly good one, of the longitudinal dispersion coefficient E^
based upon the work of Taylor. Now this is a very large extension of Taylor's original work.
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We have done some laboratory work In oscillating flows to confirm some of these extensions,
but I am not really going to argue about precise numbers of this longitudinal dispersion.
I'm simply going to say that it is primarily related to the instantaneous magnitude of the
tidal velocity, to the cross section, and to the roughness. The typical equation is Equa-
tion (2.127), which gives what I think is a reasonable estimate of the magnitude of E^ in
the non-saline regions of estuaries. 1 don't really care whether this is off by a factor
of two, five, or perhaps ten, because by including the instantaneous tidal velocity in the
conservation of mass equation, as has already been pointed out, these longitudinal dispersion
effects become much less important.
Figure 2.18 is an example of a real time solution for a continuous injection of a
non-conservative substance at a point in an estuary. This is for a constant area estuary
with sinusoidal tidal motion. What you see here, then, are the real time solutions shown
at high water slack and at low water slack and as an average over the tidal period. The
important result of this calculation is that you find concentrations existing upstream of
the point of injection, roughly to a distance of 30,000 feet, primarily by virtue of the
tidal motion. We then ask the question: what do we have to do to the non-tidal mathematical
model for the same problem in order to replicate the real-time solution? And you see in
Figure 2.23 that what you have to do is include a grossly inflated dispersion coefficient.
I've plotted here solutions for 1200 ft*/sec and 2250 ft"/sec and neither of these is a very
good representation of the real time solution. The effect of dispersion is shown relative
to the real time value of 65 ft3/sec which gives a distribution of concentration at high water
slack extending again about 30,000 feet upstream, in comparison with the pseudo-dispersion
of the non-tidal or slack tide models.
To summarize, what I have tried to show here is that the hydrodynamic problem,
given the decay coefficient, is a simple exercise in water quality prediction. The hydro-
dynamic problem can be solved exactly. It can be solved in terms of an analytical solution
for the constant-area estuary, and it certainly can be solved in terms of finite-difference
solutions for a variable-area estuary. The real time model obviously requires more calcu-
lations. It requires knowledge of the tidal motion in the estuary, and it certainly requires
that you know the geometry of the problem which you are trying to deal with. I think the
basic philosophical difference between the two kinds of mathematical models devolves to
this: the non-tidal model is a simple mathematical model, but it includes an arbitrary,
unknown, pseudo-dispersion coefficient which hydrodynandcally we cannot say anything about,
because it takes up all the slack of ignoring the tidal motion. Therefore, a great deal of
time and energy and effort and expense has been devoted to the'determination of this type
of dispersion coefficient by field tests, by dye dispersion tests, and by doing the same
thing in hydraulic models (which opens up a whole new can of worms). The simplification
(so-called) of a simple mathematical model' then introduces a great complexity of how you
find the arbitrary coefficient which this simple mathematical model introduces. So my point
is that by going to a more sophisticated--and not really very sophisticated—mathematical
model, one in a sense eliminates this unknown parameter, because you can make a reasonable
estimate of its value. Therefore, I do not consider that we have achieved any simplification
by oversimplifying the hydraulic problems. In fact, it's a great complication, to my way
of thinking, because it does not really make use of present day capabilities.
I've only mentioned the dispersion in the non-saline region. There is a considerable
discussion in here on how one determines the dispersion coefficient in the salinity intrusion
region, and this again can be done in real time models using available data for the salinity
intrusion, which is one of the most satisfactory things to measure in an estuary. The present
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state-of-the-art, even in terms of one-dimensional models, implies that we have to have some
knowledge of the salinity distribution in order to calculate the dispersion in these areas.
So what it boils down to, in terms of dispersion, is that if you are in the saline region we
use the salinity measurements, and if you are in the non-saline region then we use the modi-
fied Taylor estimate.
So in summary, I would say that I think the state-of-the-art of one-dimensional
modeling is such that the hydrodynamic aspects of mass transfer in estuaries should no longer
be accepted as an unknown parameter, and we should stop devoting major efforts to the deter-
mination of this hydrodynamic parameter by field tests. We should get on with the much more
difficult and -much more important problems of making field tests which will enable us to
determine the non-conservative parameters of interest in the water quality models.
RATTRAY: I'm glad to see some of your statements on this horizontal dispersion in the strati-
fied situation, which is the one I'm more familiar with. There's a lot yet to be known about
what to put in there. I think this is a fair warning.
One thing we've already mentioned is that there is a vertical shear due to the
stratification, and this augments any kind of a dispersion of this kind. The other effect,
that we seem to find is more noticeable maybe in the ocean, is in any stratified system the
turbulent diffusion tends to be non-isotropic. That is, there will be a much larger hori-
zontal diffusivity, without any tidal motion, than there is vertical, due to the resistance
of the stratification for vertical turbulence just like for any other vertical motion. It
may very well be, then, that the effects of dispersion due to shear in the vertical are not
going to be dominant over horizontal diffusion in all cases. But it will be very much larger,
orders of magnitude, than will be vertical diffusion. The Taylor work was done with hori-
zontal and vertical diffusion being the same order of magnitude. I think that these are the
concerns that are represented here about what happens when you go into a stratified set-up.
The other limitation on this is that Taylor's work, and a lot of this here, assumes that the
concentration is relatively uniform over the vertical. Not being a biologist, I would sus-
pect many biological forms and maybe solid forms that one has to worry about in water quality
do not have uniform distributions in the vertical, and that a dispersion coefficient or
anything based upon a salinity distribution may be orders of magnitude off and not represent
the rate of horizontal movement of those sorts of distributions.
HARLEMAN: Well, I think what you're saying is that if things aren't one-dimensional, you
shouldn't use one-dimensional equations.
RATTRAY: I think we all know that, and I think you have decided that there are these limi-
tations, and all I can do is sort of say the same thing.
HARLEMAN: If you have strong gradients in the lateral or vertical directions, then you
shouldn't expect to get good results from one-dimensional equations.
FRITCHARD: Well, I night say that I've been sort of a reluctant admirer of the success of the
one-dimensional equations, both the dynamic equation and what you night call the mass balance
equations or salt balance equations, even in systems which I always thought had some varia-
tion across the cross section. I don't want to belabor the point too long but I think it's
of interest to try to decide why certain integrals disappear. That is, if I take the three-
dimensional equation of motion and salt continuity and integrate them over the cross-section,
I can get forms that look exactly like the forms that Don has shown here, plus some additional
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terms, the additional terms being terms which involve the cross product of the velocity
deviation terms in the cross section-referred to the cross-sectional means. In the salt
balance (or the mass continuity equation, the conservation of a substance) there also appear
some of these additional terms, which involve the cross product of the deviation in the
velocity times the deviation in the salinity from their sectional means. Where the system
is reasonably homogeneous, these terms are probably small, but where there is a shear, either
laterally or horizontally, the deviation in the horizontal velocity with position in the
cross section and the deviation in the salinity with position in the cross section from their
respective cross-sectional means must be highly correlated, so that these cross products must
really be real fluxes. It's amazed me that in estuaries like the Delaware one appears to be
able to treat it as one-dimensional. I guess it must be that even though the fluxes may be
large, the gradients must be small. It's just phenomenal. As I say, I've been a reluctant
admirer of its success. I would just point out that I think that people using these relation-
ships certainly must not jump too readily to the conclusion that a one-dimensional situation
is going to work everywhere. And I think this has been said.
RATTRAY: On this one-dimensional model in general, this thing has sort of been raised again.
It is quite evident, to me, that no purely advectlve model will give any dispersion. There
has to be some diffusive term in there. That's true even if you include turbulence, that
is, you wouldn't have diffusion by turbulence unless there was molecular diffusivlty working
also. The reason you normally can neglect the molecular diffusivity part is that you're
working in large Reynolds numbers, so that there is a lot of energy in the turbulence and
there is a very small transfer being done by molecular processes. The turbulence is in very
large gradients for very rapid diffusion by molecular processes. This same thing has to be
going on when you use tidal motion. It must give you large gradients which are diffused by
something in the model or you would never have a one-dimensional model with any diffusion
at all.
FRITCHARD: There has to be diffusion by the turbulence, diffusion which in turn then really
goes back to molecular. If you follow the theory all the way, you would have to include in
the physical description molecular diffusion or molecular viscosity.
RATTRAY: But I would like to make the further analogy that the thing that would be important,
then, is that the tidal terms themselves would give the major part of the diffusion if the
effective Reynolds number were large. It would be independent of the turbulent terms for
large Reynolds number type behavior. But if you have the case where the effective Reynolds
number, using a turbulent eddy diffusivity, is not large, then of course it starts to depend
upon the eddy turbulent diffusivity, just like if in turbulent flow you go towards the molec-
ular limit. You can get to the point where both diffusions are significant. This has to
be worried about. Apparently in most cases we don't reach that, just judging by the success.
Maybe you don't even have to put it in. Maybe you get enough turbulence in roundoff errors
in a numerical model. But it has to be there. It shouldn't be forgotten that it's there
somehow.
THOMANN: On use of tidal models with tidal variability in the velocity term (and it's fairly
large in most of these estuaries), there really is a question, for example, in some of your
plots that are shown here, Dr. Harleman, about whether there would really be any difference
between those and what you would have if you had set that dispersion coefficient to zero.
HARLEMAN: However, when you use finite-difference forms that's a more difficult problem. We
want some dispersion in the model.
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THOMANN: Why?
RATTRAY: You need It.
THOMANN: Why? Actually you'll introduce some right off the bat in a finite-difference model,
and that may be more than you need.
HARLEMAN: You don't have to worry about that. If you are going to have a continuous model
that goes from the non-saline region to the saline region, you want to have some mechanism
for introducing dispersion. So if you are going to have dispersion in the salinity region,
which you definitely need, then there's no reason why you shouldn't include it in the non-
saline region.
THOMANN: Yes, but all I am pointing out is that it may be quite insensitive.
HARLEMAN: Well, that's what I am saying.
THOMANN: But where it does become important, and one can show this, is when the input at the
boundary is fluctuating with a certain frequency, and then one has to be very careful about
incorporating even small amounts of dispersion, and the difference between the straight
advective model and the dispersive model can be quite large. But in the analysis that
Dr. Harleman presents, would those profiles be substantially different if E , which is
about 60 ft3/sec, were set at zero?
FRITCHARD: It would not be very different in the numerical output, but remember you've got
dispersion in the numerical equation.
THOMANN: Yes, right. And all I am saying is that when you incorporate the tidal velocity
as a time variable in the equation, for many practical purposes the inclusion of the dis-
persion coefficient E may be unnecessary.
HARLEMAN: Unless you get into the salinity gradient.
THOMANN: Yes.
HARLEMAN: But if you are going to include it in the salinity region, then there is no reason
not to include it in the non-saline region even though it's small.
PRITCHARD: Do I understand you, that when you move into the salinity region or into the region
in an estuary where the salinity gradients are high, that ^ term, even with the real time
system, becomes larger than the Taylor E, ?
HARLEMAN: Yes, it becomes quite large, and quite a function of x .
FRITCHARD: And even larger if you average over the tide. But even if you don't average over
the tide, if you use a real time solution, you still have an E^ which you've got to relate
to something else besides the Taylor equation.
HARLEMAN: We are relating it now to the local salinity.
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PRITCHARD: Right. But if you go back to your flume work, you related it to the difference in
the salinity of the river input and the ocean. If you go to some of those relationships,
you come up with the pertinent factor involving the ratio of a Froude-type number and a
velocity at the surface to the tidal velocity relationship.
RATTRAY: Well, again I come back to the fact that you have to have two parameters and that
the salinity differences do not give you the velocity profiles uniquely. It's the velocity
profiles that cause the dispersion and not the salinity differences.
HARLEMAN: We're only talking about one-dimensional models so we can't Include velocity pro-
files. The momentum and continuity equations give you the tidal velocities.
RATTRAY: Well, in terms of the dispersion coefficient, the salinity profile in the saline
region must be very important, and I think you've said that. But the way of getting at it
is more complicated.
HARLEMAN: The way of getting at it is to simply take the measured salinity and match it, in
a one-dimensional framework. Then you have included everything that is there in the one-
dimensional model, because you have matched the existing salinity distribution.
PRITCHARD: I think where we would have a problem is if we intended to predict everything from
the boundary values and the hydrodynamics, that is, including the salinity distribution.
Then we want to have some independent way of estimating these things in order to compute the
salinity, as well as the distribution of properties. As you say, for many practical problems,
in the near future reduced-dimensional models may still be desirable. Hence we would like
to get at ways to independently predict the terms that enter some of these averaged equations,
even recognizing the fact that it is not truly one-dimensional or two-dimensional.
HARLEMAN: Well, this section of Chapter II does include some discussion of predicting salinity
intrusion changes on a one-dimensional basis, which is a requirement for what you say.
PRITCHARD: Yes, that's what I'm getting at. The results of some of your flume studies which
you then apply to the Delaware. And as I say, the success of that has been remarkable.
DOBBINS: I think there is one other question I'd like to raise here. In these steady-state
models, if you determine a value of E based on longitudinal salinity distribution, if you
make a dye dump, say, an instantaneous dump, and determine E , and if you, let's say, deter-
mine E for a continuous dump, are you going to get the same value? I don't think you are.
And therefore it ia not valid to measure E from a salinity distribution and apply that to
determine the fate of a pollutant. I don't think this is being recognized in many cases.
That is, everybody thinks it's the same E , but it isn't.
HARLEMAN: They are not the same initially, but they approach each other rather quickly in
time.
PRITCHARD: You mean as the scales of the phenomena get to be Identical?
HARLEMAN: If you were concerned with an instantaneous release, this difference would be quite
large; but not if you are concerned with continuous releases.
DOBBINS: Most of the dye dumps that are made are, after all, Instantaneous releases.
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HARLEMAN: But I'm saying we shouldn't make them, because it isn't what we want to solve. If
you are Interested in solving the problem of the dye dump, okay. But why do a dye dump to
solve some other problem?
DOBBINS: Well, that's my point.
RATTRAY: The point is, isn't it, that in many cases you are using a mean salinity to solve
other problems. It's not valid.
DOBBINS: Right. Also, I think this is a question of whether you're interested in local
effects. Take, for example, the city of Poughkeepsie, which takes its water supply out of
the Hudson River just about the head of the tide. You are going to be very, very much con-
cerned about the diurnal variation of the water quality there. In fact they've run into
severe problems of treating this water because of the diurnal variation in the quality.
PRITCHARD: I should bring in a future paper which shows a comparison between a distribution
of temperature from a thermal discharge averaged across a section as computed from coeffi-
cients gotten from the salinity distribution, and show you the comparison with the real
distribution. They are very close. I am now talking about scales which are clear across
the estuary, averaged in a section. This was an input which essentially was not local, and
it worked.
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CHAPTER III
WATER QUALITY MODELS:
CHEMICAL, PHYSICAL AND BIOLOGICAL CONSTITUENTS
Donald J. O'Connor
and
Robert V. Thomaxm
1. INTRODUCTION
Any natural body of water nay be viewed as a mathematical system, composed of a
number of complex Interacting subsystems. The system receives, on the one hand, a series of
external inputs such as rainfall, solar radiation, runoff, and winds which determine the
natural quality of the water. On the other hand, the system Is also subjected to a variety
of man-made inputs such as wastewater discharges, water diversions, runoff from urban and land
developments. The response of the system to each of these Inputs is the spatial and temporal
distribution of the concentration of various substances which affect water use. Such substances
include various chemicals, dissolved oxygen, nutrients (nitrogen and phosphorous), bacteria and
algae concentrations, and dissolved and suspended solids. The system is composed of a number
of parts, with physical characteristics and corresponding mathematical descriptions. Physically,
the concentration of these substances is determined by the dispersion and advection character-
istics of the water body and by the various physical, chemical, biological or radiological
reactions which affect the substance. Mathematically, the system is described by a set of
partial differential equations, with variable coefficients, each term of which corresponds to
one of the basic characteristics. Modifications of these equations comprise a water quality
mathematical model.
The purpose of a model of any system is to reproduce in some way the observable
phenomena which are of particular significance. From a scientific viewpoint a model provides
greater understanding and Insight of the phenomena, and from an engineering perspective it
provides a means of prediction and, therefore, control of the system. In both cases, there
is posed a specific question or problem to which the construction of the model is directed.
Fundamental to the development are the time and space scales of the phenomena involved and/or
the question posed, which, to a large extent, determine 'the nature and complexity of the model.
Relating these concepts to the modeling of water quality in natural systems, three
criteria may be applied which determine the nature and characteristics of the model.
(1) The water quality problem. The time scale of the problem Is the most Important element
which determines the type of model to be employed. In this respect it is frequently possible
to isolate a singular scale within which the analytical framework of the solution may be
structured. For example, is the problem concerned with the relatively short-term variation
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of water quality, due to, for example, a storm water overflow in an estuary, or with the
long-term accumulation of a conservative material in a tidal bay or harbor? In the first case,
the time scale is in the order of hours or days, while in the latter, it may be of years or
decades.
(2) The characteristics of the water body. Each body of water is characterized by its own
geomorphological structure and a set of hydrological-meteorological inputs which determine to
a large extent the hydrodynamic regime of the system. Each regime has its own characteristic
scale in both time and space. In the case of estuaries, a most predominant time scale from
the hydrodynamic point of view is the tidal period, and, therefore, may be an important factor
to include in water quality modeling. However, it is not necessarily a significant fac-
tor. Referring to the example above, it is significant in the case of the storm water over-
flow where the time scales are of the same order. Since it is necessary to account for
hour-to-hour variation in water quality, it is obvious that the hour-to-hour variation in the
tides must be included. On the other hand, if the accumulation of a pollutional material in
a tidal bay is in the order of years, the hour-to-hour variation in the tides may not be
significant and may frequently be neglected without introducing significant error.
(3) The reaction time of the pollutants. This factor is determined largely by the nature of
the substance and the physical, chemical and biological reactions to which it responds. In
similar fashion to that described above, the time scale can range from very short, almost
instantaneous reactions to those which persist for such long periods that the substances
affected may be regarded as conservative. Again, there may be a spectrum of time scales for
a particular parameter. For example, one may be interested in modeling algae dynamics for a
short-term bloom over a period measured in hours or days, or over the year in which the
seasonal variations are apparent and the time scale is in the order of weeks, or over a
eutrophication period of decades for which the scale would be based on units of a year.
Relating the scale of these phenomena to the hydrodynamic scales, the tidal variations are
the most significant in the short-term bloom. In the seasonal variation of algae, the
hydrologic factors such as the freshwater flow and the temperature assume primary importance,
while the tidal effects, although significant, may be incorporated in a relatively simplistic
manner. In the long-term buildup, the tidal factors are even less significant, while the
year-to-year water balance, as directed by long-range weather patterns, is the most influential
hydrodynamic factor.
Fundamental to the analysis of the problem is the continuity equation. The point
form of this equation is
|r ' v ' J ± £ S (3.1)
at
in which
c • concentration of water quality variable
t - time in tidal cycles
S = sources and sinks of variable c
j - flux - E |°- - Uc
dn
E - tidal dispersion coefficient
U = freshwater flow velocity
The continuity equation expresses a relationship between the flux of mass and the sources and
sinks of mass where the flux of mass is considered to occur from tidal cycle to tidal cycle.
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For within-tidal-cycle descriptions, the velocity U is interpreted as the tidal velocity (see
Chapter II). The term Uc is the flux due to advection by the fluid containing the mass. The
term E ac/dn is the flux commonly ascribed to dispersion in the "n" direction. The flux due
to dispersion is assumed to be proportional to the gradient of concentration in the direction
of decreasing concentration. Any mass of pollutional substance is transferred by this mechanism
from a zone of high concentration to one of low concentration. The flux is determined by the
hydrodynamics of the system,! which is related to the hydrology, meteorology and geomorphology
of the areas. The sources and sinks are the results of the various reactions of a physical,
chemical or biological nature. Many factors contribute to the total spread of mass, and the
term tidal dispersion ratherj than diffusion more appropriately describes the phenomenon. The
coefficient E includes not only the diffusion associated with turbulent mixing but also the
dispersion due to velocity gradients and density differences (Harleman 1964). As such, this
coefficient incorporates the major features of the tidal mixing phenomena in estuaries and
provides a useful approximation for many water quality studies. The relative advantages and
disadvantages of finer time scaling of the model to describe hour-to-hour changes in some water
quality variables have been discussed (Callaway et al. 1969 and Chapter II, Section 3 of this
report). Generally the trade-off involves longer computer time versus Introduction of the
inter-tidal dispersion coefficient. In particular, the dispersion factor is affected in streams
by freshwater flow, in estuaries by the tidal translation and the freshwater flow, and in the
oceans by the various currents. The velocity coefficient U is likewise determined by many
of these parameters. These hydrodynamic coefficients, as well as the source term S , may be
both time and space variables.
While the flux term defines the material moving in and out of an elemental volume,
the term Z S represents the sum of the various sources and sinks of material within the
volume. Characteristic sources and sinks are reactions of a physical, chemical or biological
nature which occur in natural waters. Many of these reactions may be represented by first-order
kinetics, i.e. the rate of the reaction is proportional to the concentration of the substance.
The coefficient K , which is the proportionality constant in the first-order reaction, is
characteristic of many reactions in water pollution. In some cases it fundamentally defines
the mechanism, as in the case of radioactive decay or gas transfer; in other cases it is an
empirical approximation of the phenomenon, e.g. the decay of waste material and the bacterial
die-away.
The three dimensional form of Equation (3.1) is
A- fE 1£U *_ (EZ 1£
ay \y ay/ az V z az
at ax
- |-(uc> - |-(vc) - |-(wc) ± I S (3.2)
oX ay oZ
in which the subscripted E refers to the dispersion coefficient along each of the three axes.
Equation (3.2) may be simplified to steady state in many cases, particularly at that period of
year when the freshwater flow and temperature are approximately constant and the waste water
discharges may be assumed so. The analysis of some estuaries is further simplified by virtue
of the fact that the concentration of various substances over the cross-sectional area has been
observed to be uniform, thus reducing the problem to one dimension. It is the steady-state
one-dimensional form of Equation (3.2) which has found the most extensive application in the
analysis of estuarlne pollution.
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Longitudinal tidal dispersion is most important in the saline portion of the estuary
where a number of factors contribute to the intrusion of the salt into the estuary (see Chapter II) .
the concentration of other substances which are of concern in water quality in estuaries is
affected in a manner similar to that of the salt. In the tidal but nonsaline sections of the
river, the dispersion although not as pronounced as in the saline section is still a signifi-
cant factor in the analysis of water quality. Upstream of the tidal influence, the effect of
longitudinal mixing is much less and in many cases may be disregarded, depending on the relative
magnitude of the dispersion, advection and reaction.
As previously indicated, the sources and sinks refer to the various reactions and
waste water inputs which affect the concentration of material in the estuary. In terms of the
pertinent water quality variables, at one end are the conservative (no decay) substances such
as the chloride concentration, and at the other end such quantities as dissolved oxygen, various
nitrogen forms and phytoplankton, which are the result of a series of coupled reactions. It is
convenient to first consider material which is conservative, secondly a single decaying species,
followed by a consecutive two-stage reaction, and lastly a multi-stage reaction.
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2. ONE-DIMENSIONAL ANALYSIS
The form of Equation (3.2) which is appropriate for the one-dimensional analysis in
estuaries is (O'Connor 1960 and 1965)
2£ - I L. (EA^£) - i^- (QC) - Kc (3.3)
at A Sx v 3xy A dx
where A is the cross-sectional area of the estuary, Q is the freshwater inflow and K is
the first-order decay coefficient. This equation is the one-dimensional estuarine model for
the dispersion and advection of a non-conservative substance subjected to a first-order reaction.
It should be noted again that this equation is assumed to apply over a time interval of at least
a tidal cycle. Many substances of practical interest are susceptible to physical, chemical or
biological decay of this type including collform bacteria and biochemical oxygen demand. In
those cases in which the substance is conservative, the solutions for the non-conservative case
usually hold with K-0. Although the diminution of concentration may be effected by many dif-
ferent mechanisms and factors, the assignment of a first-order reaction is most appropriate and
practical, from both theoretical considerations and experimental observations. Most of the
common pollutants are subject to a decay of this order. However, as discussed in Section 7 of
this chapter, complex nonlinear reaction kinetics can be incorporated in the more advanced
models of algal productivity.
In general, the solution to an instantaneous waste discharge (a delta function input)
provides the key to the solution, of the transient and the steady-state conditions, all three of
which have practical value. The delta function input, which is referred to as 6(t) , is that
of an instantaneous release of mass M uniformly distributed over the cross-sectional area A
at the origin x - 0 . By integrating the solution of Equation (3.3) with a delta function
input with respect to time, the solution of the continuous source is obtained. If the dis-
charge is given at a rate W(t) , starting at time t - 0 and' continuing to time t , then the
following equation gives the concentration c(x,t) at time t and location x :
t
c(x,t) - J W(t') cfi (t - t')dt' (3.4)
where c. (t - t') is the solution of Equation (3.3) due to the instantaneous input. If W(t)
o
is constant, the expression is simplified and the solution for the steady state for this
condition is
' f
- t/>dt/ <3'5>
Alternately the steady-state solution may be determined from the original equation (3.3) by
letting Jt£ « 0 and seeking a solution of the ordinary differential equation. Solutions of
various spatial distributions are similarly found by integration of the appropriate equation
with respect to distance. In principle, any type of spatial- or temporal input of a waste water
source may be formally evaluated if the solution to the delta function is available.
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2.1 SOLUTION DUE TO INSTANTANEOUS RELEASE
Consider an estuary of constant cross-sectional area and constant flow. For a non-
conservative substance, Equation (3.3) simplifies to
S£ = E l?c _ D ac . Kc 6)
at %x* ax
in which the longitudinal tidal dispersion E , the velocity U , and the reaction K , are
constants. If a mass of material is released instantaneously at time t - 0 and uniformly
over a cross section whose location is designated as the origin, then the solution- of
Equation (3.6) is
exp = . Kt (3.7)
i
The mass M is released at t - 0 over the area A , located at x - 0 . As the material
moves downstream its concentration is decreased by the characteristic reaction and its mass
is spread upstream and downstream from its center of gravity, which moves at the net downstream
velocity of the estuary.
Since Equation (3.7) has the same mathematical form as the normal or Gaussian proba-
bility density function, it may be seen that the spatial distributions are symmetrical for a
specific tidal cycle. The peak concentrations of the spatial distributions occur at x = Ut
and the dispersion about the peak may be described by
C ' C°CXP &
(3.8)
-x 2
c « cDexP ( ' '
in which
u
= peak concentration at time t of a conservative substance
Cp = coexp(-Kt) » peak concentration at time t of the non-conservative substance
Xp = distance measured from the peak concentration
These equations may be used to describe the spatial distribution about a point which moves
with the mean velocity of flow.
The temporal distribution, on the other hand, for a fixed point in space is asymmetri-
cal and the degree of skewness depends upon the magnitude of Ut as compared to (4Et)* . The
spatial and temporal distributions, of the delta function solution are shown in Figure 3.1.
If a substance in solution or suspension is released continuously, its concentration
will increase in time at a fixed location until an equilibrium value is achieved, as shown in
Figure 3.2. The manner in which this increase occurs for a conservative substance may be
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DOWNSTREAM
RELEASE POINT, INSTANTANEOUS DISCHARGE (M)
X' "
—.— CONSERVATIVE
SUBSTANCE
NON-CONSERVATIVE
SUBSTANCE
•*• X
TIME - I (TIDAL CYCLES)
Fig. 3.1 Tenporal and spatial distributions of water
quality due to Instantaneous discharge.
RELEASE POINT
CONTINUOUS DISCHARGE AT RATE W
DOWNSTREAM
EQUILIBRIUM CONCENTRATION
TIME -t (TIDAL CYCLES)
Fig. 3.2 Illustration of build-up of substance C to equilibr
concentration.
108
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determined at a particular station by the operation indicated in Equation (3.4). The resulting
expressions are not always directly integrable and arithmetic integrations are frequently
required. In effect, this build-up time to an equilibrium depends on the interaction between
the tidal dispersion phenomena, freshwater flow and reaction coefficient. For rivers where E
approaches zero, time to reach equilibrium is almost instantaneous. For estuaries, equilibrium
times may be on the order of days or weeks.
2.2 CONSERVATIVE SUBSTANCES
The saline intrusion of the ocean waters into the river provides a classic example
of a conservative substance in an estuarine system. For some problems, it is useful to consider
other variables such as the total nitrogen or total phosphorous to also be conservative. In
these cases, overall mass balances between waste water inputs and land runoff can be analyzed.
Haste waters from various sources may contain substances not indigenous to the natural runoff.
Such substances (e.g., sulfates) may act as tracers of such sources, since their concentration
is easily detectable by contrast to the amount present from natural sources.
The form of Equation (3.3) which is appropriate for conservative substances under the
steady state is
Integration once and application of the appropriate boundary condition yields, without loss of
generality concerning the spatial variation of the parameters,
0 = ££. - 2_ dx (3.10)
c EA
which upon integration gives
c = c0exp [J 2_ dx] (3.11)
The boundary condition CQ at x = 0 may apply in practical cases to a point where the con-
centration is known or may be established, e.g. at the mouth of the river for the saline
intrusion analysis or at the location of a waste water source of a conservative substance.
In either case, and in many other practical situations, the concentration in the ground water
is insignificant by contrast to the concentration in the tidal river and may be usually
neglected. Constant values of Q , A and E characterize the simplest tidal river and
provide a model which may be used as an approximation of prototype conditions. The differen-
tial equation for constant parameters is
0 = E~ - U— (3.12)
dx2 dx
and its solution, which may also be written directly from Equation (3.11) is
c - c0exp (jx) (3.13)
0
where 3 = AE
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A conmon application of.Equation-(3.13) is made in the evaluation of the dispersion coefficient
E from observations of the salinity or chloride concentration in estuaries. In this applica-
tion, a semi-log plot of chloride concentration with distance is prepared. According to
Equation (3.13), the slope.of'a straight line through these data is given by Q/AE from which
E can be obtained. Typical values of this overall tidal dispersion coefficient for estuaries
range from 1-15 square miles/day in the non-saline sections and 5-25 square miles/day in the
saline.
In many cases, the parameters Q , A and E , and particularly the latter two, vary
with distance. Various functional forms may be substituted in Equation (3.11) and integrated.
Alternately the estuary may be segmented as discussed in subsequent sections.
2.3 NON-CONSERVATIVE SUBSTANCES
In the steady state and for constant coefficients, Equation (3.3) reduces to
0-E^f-U^-Kc (3.14)
dx2 <**
The form of the general solution of this ordinary differential equation is
c - B exp (gx) + C exp (jx) (3.15)
in which
B and C are constants to be evaluated from the boundary conditions. The exponents are the
positive and negative roots of the quadratic expression which contains the coefficients of the
differential equation, E , U and K . It is readily shown that for an infinitely long
estuary into which a pollutant is being discharged at a constant rate, the solution may be
generally expressed as
c - CQ exp(gx) for x < o (3.16a)
c - CQ exp(jx) for x > o (3.16b)
Equation (3.16a) applies to the region upstream of the discharge and Equation (3.16b) to the
downstream section (positive x in the downstream direction). The concentration CQ and the
exponents g and j -take on particular values, depending on the magnitudes of E , U and
K . If these parameters are greater than zero, it can be shown that
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W U
- rr (3.17)
A + 4KE
where
These expressions apply to the distribution of a non-conservative substance in a river with
significant dispersion and freshwater flow. If a conservative substance is considered the
expressions become
c . !L . H
0 AU Q
(3.18)
It is apparent that the concentration downstream from the outfall is constant, since j - 0 .
The concentration upstream, however, decreases due to the back dispersion which balances the
advective component downstream.
If the freshwater flows are zero, the advective component drops from the basic equa-
tion and the expressions become
C°"~?~
(3.19)
•-J-7F
Since the exponents are equal, it is apparent that the upstream and downstream distributions are
symmetrical about the discharge point.
If the dispersion coefficient is zero and the advective component greater than zero,
the basic differential equation simplifies to a first-order form and represents the non-
dispersive case that is often used in nontidal rivers and streams. The solution for the
concentration is
c - c0 exp(-^p) (3.20)
where
which can be recognized as the equation used to describe the BOD in streams. The significant
dimensionless number is P = KE/U2 . One limit, P = « is characteristic of highly dispersive
or estuarine systems; more specifically, a system in which the dispersion is so great by contrast
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to the advection that the latter may be neglected. The other end of the spectrum, where P = 0 ,
represents strictly advective systems, as characterized by freshwater streams, in which the
reverse is true. As one moves downstream from the freshwater sections through the deep river
stretches and thence to the tidal portions and finally to the saline outlet at the ocean, the
value of "m" in Equation (3.17) increases.
The importance of the reaction coefficient must not be overlooked in the structuring
of the model. From the physical or hydrodynamic point of view, it is the ratio of the dispersion
and advection which determines the most realistic model. From a chemical or biological point
of view, it is the magnitude of the reaction coefficient which is the determining factor. For
many estuarine situations, the water quality response tends to be more sensitive to the reaction
coefficient than to the tidal dispersion coefficient.
2.4 CONSECUTIVE REACTIONS (BOD-DO)
A common example of a consecutive reaction is the dissolved oxygen deficit produced
by the concentration of oxygen -demand ing material through biological action or chemical oxida-
tion. Under the steady-state constant-coefficient condition, the differential equations for
concentration of the two reactants are
d2c-i dci
O - E — 5^ - U — i - KllCl (3.21)
dx2 dx Li L
d2C9 dc2
0 = E — T-=- - U — - - K22c2 + K12Ci (3.22)
dxz dx " ' "
where K^ is the decay coefficient associated with the concentration cj and acts as
a sink in a typical first-order decay phenomenon, K.^2 *s a reaction coefficient and
together with c^ acts as a source for c2 > and K22 is the decay coefficient associated
with variable c2 , For continuity, K^£ 2 KJ* . For Che case of dissolved oxygen, c^
represents the carbonaceous or nitrogenous BOD, K-^ represents the BOD decay rate (which
may include oxidation and settling), K^2 is the deoxygenation coefficient, K22 is the
reaeration rate, and c2 is the dissolved oxygen deficit. The solution of Equation (3.21) is
shown by Equations (3.16a) and (3.16b) for a single discharge of waste water Wj (mass/unit
time). Substitution of these equations into (3.22) yields an expression for the concentration
of the second reactant. Substitutions are made for both the negative and positive values of
x , i.e. both Equations (3.16a) and (3.16b). The solution of Equation (3.22) given Equation
(3.16a) and (3.16b) as input is
K12 ,exp( jn x ) exp( J22 x )^ „
Ct) " v - v - V - ™ -- - ™ - / n (3.23)
2 K - K \ m m
where
Hi
2j [l - Jl + 4K22 E/U2]
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- /I + 4KU E/U2
iiijj - /I + ^Kjj E/U2
for x > 0 . For x < 0 , replace jj^ and ^22 w*-t*1 8ll an<^ 822 as Per Equation (3.16a).
Note that for this coupled system the reaction coefficients of the equations act as multipliers
in the solution.
The coordinates of the maximum value c^ are obtained as follows. The distance xm
to the maximum value of c.% is evaluated by differentiating Equation (3.23) and setting the
result equal to zero. After simplification, the distance is
The value of the maximum c2m may be determined by substituting Equation (3.24) in Equation
(3.23); or, alternatively, by taking the second derivative of Equation (3.23), substituting in
Equation (3.22), setting the first derivative term equal to zero and evaluating the result.
The expression for the maximum value of ^ is
25)
»••«;
where e^ - K^/K^ , am « >/l
and * j ' C1 ' V1 + k*&\&& ) C1 *
2.5 APPLICATIONS TO DISSOLVED OXYGEN ANALYSIS
Equation (3.23) merits particular attention in view of its wide application in the
analysis of water quality data. The most significant example of its application is in the
analysis of the spatial distribution of the concentration of dissolved oxygen in tidal rivers
and estuaries. Graphical presentations of Equation (3.16) and Equation (3.23) are shown in
Figure 3.3. The former represents the concentration of a substance which utilizes oxygen,
such as the respiration of bacteria or algae, the biochemical oxygen demand (BOD) or a chemi-
cal oxidation. The mass rate of discharge is indicated by W and the freshwater flow by Q .
The latter equation describes the distribution of dissolved oxygen deficit due to the concen-
tration of the former. The coefficient of oxidation or deaeration is identified by the symbol
Kio ant^ the reaeration by K££ • The solid line represents the mean tide of the ebb or flood,
While the dotted and dashed lines are the profiles at ebb and flood slack, respectively.
The primary purpose of such an analysis is to assess the effect of waste waters on
the dissolved oxygen levels and, in particular, to determine the efficacy of various engineering
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OCEAN
W LBS/OAYBOO
BOO
DO
DEFICIT
SATURATION
Fig. 3.3 Estuarine BOD, deficit and DO profiles.
alternates in restoring water quality or in preventing deterioration of quality. On the one
hand, the analysis is used to predict dissolved oxygen levels given degrees of treatment for
the various waste constituents. On the other hand, the analysis may be used to determine the
degree of treatment required of the various waste constituents to restore or maintain water
quality of appropriate levels.
The information required for such an analysis may be divided into the following
categories. The assumption of the steady-state condition is frequently valid if the waste
discharge is constant during the low-flow, high-temperature period of the late summer and
early fall.
(1) Waste Water Discharges. It is necessary to know the nature and quantity of the various
constituents of waste waters which use dissolved oxygen. It is the oxidation of these sub-
stances, either biologically or chemically, which causes a depression of the oxygen levels in
estuaries. Most waste may be divided into carbonaceous and nitrogenous components, each of
which biochemically utilize dissolved oxygen at different rates. Furthermore, the suspended
and dissolved fraction of each component of each waste should be assessed. The former fraction
may be susceptible to settling, producing typical benthai deposits and sludge banks which
cause an additional claim on the oxygen resources. Some industrial wastes contain components
which may be oxidized chemically such as the ferrous or the sulfite ion. The nitrogen part of
wastes may be oxidized biochemically by nitrifying bacteria or assimilated by phytoplankton,
which in turn use dissolved oxygen in their respiratory processes.
114
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(2) Physical. Chemical and Biological Reactions. The waste discharges, described above, act
as sinks of dissolved oxygen insofar as the various constituents of these wastes are oxidized
in estuarine environment. The rates at which these reactions progress are essential to the
analysis. They are functions of the characteristics of the biophysical environment. Tempera-
ture is usually the most predominant factor, although sunlight, winds, velocity and the nature
of the estuarine channel, among others, may also be factors. The reactions are influenced by
the chemical or biological nature of the constitutent, by its carbon, nitrogen, and other
components, and by the state of solution or suspension. Each factor affects the dissolved
oxygen in its own fashion and should be evaluated separately. In general, each type of
waste—municipal, industrial, agricultural and natural—should be broken down into the fol-
lowing categories for dissolved oxygen sinks:
1. Carbonaceous component
2. Nitrogenous component
3. Potential for sludge deposits
4. Chemical oxidation
5. Algae respiration
In addition to the reactions which cause the depletion of oxygen, account must be taken of
those reactions which act as sources of oxygen, such as atmospheric aeration, photosynthetlc
production and anaerobic reduction. Each of these are also functions of temperature and other
biophysical conditions. The specification of the reactions, which act as either sources or
sinks of dissolved oxygen, is a most critical step in the modeling effort.
(3) Hydrodynamic Factors. While the reactions described above determine the increase or
decrease of dissolved oxygen within a segment of the estuary, the hydrodynamic factors fix
the transport of oxygen in and out of the element. The flux consists of dispersive and advec-
tive factors due to tidal phenomena, circulation and freshwater flow, as discussed in other
chapters of this report. Within the context of this chapter, these factors are the dispersion
coefficient, fAich includes a tidal translation, and an advective coefficient, usually asso-
ciated with the freshwater flow. It is also necessary to know the geometry and shape of the
estuarine channel, since it affects these and other parameters.
With these data, the modeling effort proceeds as follows:
(1) Segmentation of the system. The estuary is divided into sections, the size of which
depends on the computational scheme involved in the analysis of the water quality problem.
In any case, the geometry, the hydraulic characteristics, tributaries, islands, inflow and
tidal factors are the criteria which establish the bases for segmentation. Each segment has
its own equation, either a finite-difference form of the differential equation or the general
analytical solution of the differential, with its own set of coefficients. The segments are
connected by flux concentration conditions which depend on the computational procedure.
(2) Coefficients of each segment. Each segment is characterized by a set of geometric,
hydrodynamic and reaction coefficients. The latter describe the various sources and sinks of
dissolved oxygen. The former relate to the transport terms of dispersion and advection, which
are usually evaluated by the longitudinal distribution of salinity.
(3) Oxidation coefficients. The reaction coefficients of biochemical oxidation may frequently
be assigned by order of magnitude and checked by comparing the calculated profile with obser-
vations of the particular characteristics, i.e. carbonaceous BOD or TOC for the carbon, and
115
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nitrogenous BOD or ammonia-nitrite for the nitrogen. In view of the usually low concentration
of these substances and the inherent accuracy of the tests, such a comparison does not offer a
method of exact refinement of these coefficients but indicates an approximate value, which may
require further adjustment in view of the next step.
(4) Reaeratlon coefficient. This parameter may be evaluated by various formulae, which
include the velocity and depth of the estuary. The temperature also is accounted for in the
calculation of the reaeration coefficient.
(5) Comparison of computed profiles and observed data. With the information and coefficients
available, the dissolved oxygen profile is computed and compared to observations. Because of
the approximations and inaccuracies of both the parameters and data, it usually is not justi-
fied to employ a technique of best fit. A visual comparison is usually sufficient. If the
comparison is favorable, the set of coefficients are tentatively accepted. If the computed
profile does not agree with the observations, two factors are most frequently the cause: the
sources of wastes and the oxidation coefficient. A thorough check on the various sources of
waste water should be made and the approximation and estimates made in assigning magnitudes
to these sources should be reviewed. Background values can sometimes be significant. If the
BOD test is used, the ratio of the five-day to the ultimate value should be reanalyzed. If
these are found to be satisfactory, the next coefficient to be adjusted is that of the oxida-
tion process. Adjustment of these factors will usually account for discrepancies between
calculations and observations. If more than one source of pollution exists, the principle of
superposition applies; the effects of each are computed individually and added to determine
the total.
(6) Evaluation at other conditions of model. Comparisons between observation and calculation
should be made for various combinations of waste water discharge, temperature, freshwater flow
and tidal conditions. The coefficients associated with the various processes may then be cor-
related to these parameters. Consistency in the relationships between the parameters of
temperature, flow and tides on the one hand, and the coefficients of dispersion, advection,
oxidation and reaeration, on the other, is the crux of the verification procedures. The
greater the range of conditions evaluated and measured, the greater the certainty in inter-
polation and the greater the probability of realistic extrapolation. It 'is, of course,
this process of either interpolation or extrapolation which provides the basis of the
prediction procedure.
(7) Application of the model. Given the above, the model may be tentatively assumed to be
verified, or at the very least adequate for purposes of prediction. Various engineering
alternatives may now be quantitatively investigated and assessed, such as varying degrees
'and types of treatments, low-flow augmentation, flow diversions, and waste water relocations.
In each case changes in the coefficients must be accounted for, particularly variations in
the oxidation coefficient for various levels of waste treatment.
(8) Verification of model. The model may be considered verified if it has successfully
predicted the change in water quality, after a significant change in the inputs to the system
or a change in the estuarine environment itself has been made. Further modifications may be
necessary and further data may have to be collected after changes are effected. It is this
interplay between observations and modeling which yields the most fruitful approach to the
development of water quality models.
116
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2.6 MULTI-STAGE CONSECUTIVE REACTIONS (NITRIFICATION)
The BOD-DO case discussed in Section 2.4 is an example of a two-stage reaction.
There are also several phenomena of importance in water quality management that are represented
by a sequence of several forward reactions. These can be termed multi-stage consecutive reac-
tions. If one is willing to accept a first-order kinetic assumption, bacterial nitrification
can be thought of as a set of forward reactions which result in the conversion of various forms
of nitrogen. The assumption of first-order kinetics is generally warranted if there are a
large number of nitrifying bacteria present in the water. Nitrification is an important phenom-
enon in water quality management because of the significant amount of oxygen that may be utilized
during the oxidation of ammonia nitrogen and nitrite nitrogen. In addition, several nitrogen
forms may serve as a nutrient source for algal growth which upon death release nitrogen back into
the system thereby completing the cycle. The incorporation of this feedback phenomenon is dis-
cussed in Section 7.1. In order to place the presentation of the multi-stage reaction in a
specific problem context, nitrification of nitrogenous material will be used as the example,
with no algal influences or feedback.
If Nj_ , Nj , Nj and N^ represent organic, ammonia, nitrite and nitrate nitrogen
respectively, the bacterial nitrification phenomena can be represented as shown in Figure 3.4.
Note that each of the nitrogen forms may also have direct waste inputs given by W^ , W, ,
Wj and W^ . Each block incorporates the advection and dispersion in the estuary as described
in the previous sections. If one is interested in describing the effect of nitrification on
the DO resources of the estuary, the ammonia and nitrite outputs, as a first approximation,
can be stolchiometrically converted to oxygen demands. The DO balance equation would there-
fore have two additional sink terms: the DO depletion due to the oxidation of NHg to NH»
through the source of DO deficit given by 3.43 K-oN- , and the depletion due to oxidation of
N02 to N03 through the deficit source of 1.14 KOAN, . Again, the sensitivity of these
consecutive systems to the reaction coefficients is readily apparent in the form given in
Figure 3.4.
The steady-state differential equations for this four-stage system are given in the
context of nitrification by
i tk (** 5r) - i IT - KuNi <3-26a)
0 = I fj- (EA —^) - i f- (QH ) - K N + K N (3.26b)
A dx v dx' Adx i 22 Z 12 1
" A 1 (** a?) " A &
0 ' ** - (QN> - KN + KN
A dx v dx ' A dx
444 343
The coefficients Kj^ represent the first-order decay of substance i and combine
oxidation effects and settling effects. The coefficients Kji are the forward reaction coeffi-
cients and represent the source term of substance i feeding into substance j . Equations
(3.26a)-(3.26d) therefore permit determination of the individual nitrogen components. As shown
117
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In Figure 3.4, the forcing functions for the DO deficit are 3.43 K23N2 and 1.14 K34N3 i the
first representing amnonia oxidation and the second nitrite oxidation. These inputs are then
incorporated in the DO deficit equation and appropriate solutions obtained for each of the
nitrogen forms.
ORGANIC
NITROGEN
( REACT.-Kn)
N,
AMMONIA
NITROGEN
(REACT.-*22)
*
NITRITE
NITROGEN
( REACT. -K33)
— 1
*
NITRATE
NITROGEN
( REACT. -K^)
\
00 DEFICIT
NH3-N02
(REAERATION)
DO DEFICIT
N02-H0j
(REAERATION)
Fig. 3.4 Consecutive reactions, nitrification phenomena.
An application of this feedforward multi-stage system was made for the Delaware
Estuary (Thomarm et al. 1970) assuming bacterial nitrification predominated in the system.
For the Delaware, it Is estimated that about 110,000 pounds/day of oxidizable nitrogen (organic
and ammonia) is discharged from fifteen municipal and industrial sources (Hydroscience, Inc.,
1969). The method of computation used was the continuous solution approach for the steady-
state first-order kinetic case. Verification analyses were applied to a series of longitudinal
profiles of the various nitrogen forms collected during different seasons. Results of one such
118
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verification run are shown in Figure 3.5. The ammonia profile was verified by utilizing a
greatly reduced reaction rate in the reach from mile 100 to mile 85, a stretch of the estuary
that coincides with low dissolved oxygen (less than 2 mg/1). The hypothesis is that the
nitrification phenomenon is inhibited at these low concentrations. Verification of the
relatively high levels of ammonia observed was not possible without this assumption. Other
survey periods were also analyzed to provide a consistent set of reaction coefficients for
each of the nitrogen forms. The nitrogen analyses then formed a basis for estimating the
effects of nitrification on the dissolved oxygen deficit. The DO analysis indicated that with
favorable conditions for nitrification, no nitrogen removal, and assuming that all oxidizable
nitrogen was utilized in the bacterial nitrification stage, as much as 2.5 mg/1 DO deficit
could occur.
It should be noted that the classical BOD-DO equations described previously are a
special case of these multi-stage reactions. For arbitrary coefficients, useful analytical
solutions to Equations (3.26a)-(3.26d) are not available. For many purposes successive solu-
tions can be obtained for constant coefficients. These solutions have forms similar to the
BOD-DO solutions, Equation (3.23), and can be expressed in terms of a series of exponentials.
This can readily be seen by recognizing that at any step in the consecutive reaction, the
input from the preceding substance will be in terms of an exponential. For an exponential
forcing function, the response of any equation of the form of Equations (3.26) is another
exponential. The problem consists essentially of adequately accounting for boundary conditions
and reaction coefficients. Alternatively, Equations (3.26) can be expressed in finite-difference
form and the resulting set of algebraic equations can then be solved directly. Both solution
techniques are discussed in Section 3.
2.7 DATA REQUIREMENTS
The previous discussions have generally indicated the relevant variables and parameters
that must be examined in many estuarine water quality problems. It is appropriate to summarize
here the data that are necessary for analyzing the effects of water quality control programs.
There are obviously a wide variety of estuarine water pollution problems ranging from the
relatively simple salt distribution problem to the more complex and highly interrelated algal
dynamics problem. However, there are certain basic data requirements which can be stipulated
and which will aid in placing relative weights on the modeling effort.
Data are most often lacking on the magnitude and characteristics of the waste water
inputs. These inputs include direct discharges from municipal and industrial sources, storm
water and combined sewage runoff and agricultural drainage. Determining the spatial,and
temporal variations of characteristics of these waste inputs is an expensive and time-consuming
process. The only problem context In which this data requirement is not of paramount concern
is when the loads do not yet exist and a simulation is being performed on the possible effect
of some future load development. Many problem contexts, however, have required determination
of both the present and future waste loads. Necessary sampling should include:
(a) Waste water physical characteristics
1. Flows, diurnal and seasonal variability, by-pass frequencies, combined
overflows, land runoff
2. Temperature
3. Solids
119
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g
o
2.0
1 .0
0.0
3.0
2.0
1.0
0.0
BACKGOUND °
ASSUMED DUE °
TO PLANKTON
I O
I 1
COEF. K22
o JULY 30, 1964
o AUG 10, 196 <•
• AUG 31. '964
DWPC DATA
ALL LWS
FLOW-JOOO CFS
TEMP-26-27°C
2.0
0.0
120
no
100 90 80
DISTANCE-MILES
70
60
50
Fig. 3.S Comparison of observed and computed (solid line) organic,
amoonia, nitrite and nitrate nitrogen - Delaware Estuary,
August 1964.
120
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(b) Waste water quality characteristics
1. Carbonaceous BOD and nitrogeneous BOD (long-term BOD studies)
2. Nitrogen forms: organic, ammonia, nitrite, nitrate
3. Phosphorous forms: total, dissolved
4. Bacteria: total and fecal coliform, fecal streptococci
5. pH, acidity, alkalinity
6. Other variables that may be at potentially toxic levels
Not all analyses need be done on every waste source since problem contexts will differ.
However, many estuarine water quality problem situations now demand a greatly expanded list of
pertinent waste water characteristics to be determined. In terms of the number of samples, an
attempt should be made to obtain data throughout a year on possibly a once-a-week daily-composite
basis. Shorter term hour-to-hour waste water fluctuations may be important in some cases. As
indicated previously, lack of definitive information on the waste load inputs may lead to poor
approximations of load magnitude and variability with subsequent weakening of the model effort.
Data on the hydrologic and water quality characteristics of the estuary itself are
also required and in some instances are more readily available than the waste water input data.
In terms of sampling programs, data on the estuary itself are collected to provide some basis
for the modeling effort including verification analyses. Generally such programs are relatively
expensive and care must be taken that the problem context has been properly analyzed to insure
that the proper time and space scales are being examined. Sampling generally includes:
(a) Estuarine hydrologic and physical characteristics
1. Meteorological variables, e.g. precipitation, wind
2. Solar radiation
3. Tidal range and velocities
4. Cross-sectional area
5. Water depth
6. Freshwater flow: time and space distribution
7. Salinity, chlorides
8. Temperature
9. Light extinction properties
(b) Estuarine water quality characteristics
1. Dissolved oxygen
2. BOD determination: oxygen uptake rate studies
3. Nitrogen forms
4. Phosphorous forms
5. Bacterial levels
6. pH, acidity, alkalinity
7. Other special variables related to specific waste discharges
121
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(c) Estuarine biological characteristics
1. Total algal counts: species determinations and counts
2. Chlorophyll forms
3. Zooplankton
4. Vertebrates
5. Toxicity determinations
Again not all variables must be determined to the same degree for all problem con-
texts. However, many water quality estuarine situations require determinations of the water
quality characteristics listed above together with the biological information, and it is
precisely these characteristics that have demanded the greatest share of time and expense in
estuarine sampling studies. Most of the physical parameters are available or are readily
obtained.
Occasionally, it may be desirable to carry out dye dispersion tests to determine the
order of the intertidal dispersion coefficient. However, as pointed out previously one can
obtain such estimates from salinity or other conservative tracers in the estuary. Alternatively,
one can turn to numerical nonsteady-state models which do not require an estimate of the tidal
dispersion coefficient (see Chapter II).
Estuarine water quality modeling, therefore, requires a complex sampling program
which must obtain data on waste water inputs and receiving water quality. There are no rigorous
rules for the time and space scales that must be examined; each problem context (including the
available budget) determines the magnitude and sequencing of the number of samples that must be
obtained and the analyses that must be performed.
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3. SOLUTION TECHNIQUES FOR ONE-DIMENSIONAL STEADY STATE
3.1 AVAILABLE APPROACHES
Although the use of analytical solutions to the one-dimensional steady-state equation
is helpful in gaining insight into the behavior of water quality in an estuary, for a number
of practical problems complete analytical solutions are not always available. For certain
functional forms of the system parameters analytical solutions may be obtained as indicated
above, but as the size and complexity of the problem increases the equations become intractable
and/or the solution becomes too cumbersome. Because of this difficulty, two approaches have
been utilized to solve one-dimensional problems in estuaries.
The first approach, which can be called the continuous solution approach, makes use
of the solutions to the constant-coefficient steady-state equation (see Equation 3.15). Most
estuaries have hydraulic and geometric characteristics, such as cross-sectional area, depths,
widths, and flows, which vary along the longitudinal axis of the channel from upstream to the
ocean. In order to represent these variations mathematically, it is necessary to divide the
system into a number of individual segments, each of which is characterized by its own physical
and hydraulic parameters. The segments are then joined in a mathematical fashion, by means
of boundary conditions relating concentration and flux of pollutant at each face of the junction.
In this manner, changes in geometry, tributaries, dams, inflows, may all be incorporated into
the mathematical definition of the system. Figure 3.6 illustrates the segmentation. Solutions
are written down in terms of unknown coefficients which may be evaluated by means of a sequence
of mass balance and concentration equalities. Continuous solutions are therefore obtained
throughout each reach. A medium-size computer is usually required.
The second approach, which can be called the finite section approach, in essence
replaces the derivatives in the steady-state equation with difference approximations. The
estuary is divided into a number of sections (usually many more sections than in the continuous
solution approach). It is assumed that each section is completely mixed so that continuous
solutions are not obtained within each section. With each section the steady-state equation
is therefore replaced by a series of algebraic equations, one equation for each finite section.
Solution is obtained by matrix inversion or relaxation techniques. Depending on problem size,
a medium to large computer is required.
The relative advantage of the continuous solution approach is that longer lengths of
the estuary can be examined without resorting to a finite approximation. As such, the size of
the matrix to be inverted is equal to or less than 2n where n is the number of reaches.
In contrast, the finite section approach, because of the completely mixed approximation, requires
more sections and the matrix to be solved becomes larger. As an example, where comparisons have
been made, the continuous approach required solution of a 15 x 15 matrix while the finite section
matrix was twice that size. In other situations, the disparity may be greater.
On the other hand, the logic surrounding the continuous solution approach tends to
be complex and generalized systems are difficult to handle. The logic of the finite section
approach is simple and for one-dimensional estuaries, the matrix is of a special form (tri-
diagonal) allowing rapid computation. All possible analytical solutions must be programmed
123
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DAM
CHANGE IN CROSS-
SECTIONAL AREA"
WASTE
INPUT
TRIBUTARY
CHANGE IN
DEPTH
DISTANCE
Fig. 3.6 Illustration of segmentation.
for the continuous solution approach Including zero freshwater flow cases and uniform load
cases. In the finite section approach, only one set of equations is programed. The funda-
mental difference is that the continuous solution approach deals with solutions to the dif-
ferential equation while the finite section approach deals with an approximation to the
differential equation. Each will be considered here for some simplified cases.
124
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3.2 CONTINUOUS SOLUTION APPROACH
A detailed review of the continuous solution approach is given elsewhere (Hydro-
science, Inc., 1968 and O'Connor 1962). The general complementary solution to Equation (3.14)
(constant parameters, steady state) is Equation (3.15), repeated here:
c - B exp(gx) + C exp(jx)
(3.27)
where B and C are unknown coefficients to be evaluated and g and j are given following
Equation (3.16). If there is a uniformly distributed (in space) load entering the system, then
the general solution is
c - B exp(gx) + C exp(jx) + £—
AK
(3.28)
These equations form the building blocks from which complicated systems can be handled. A simple
case will illustrate the use of these equations.
X=0
-oo
-no
Fig. 3.7
A two-reach estuary.
Figure 3.7 shows an estuary with an incoming point waste load W at x = 0 .
Upstream from this point, one or several of the system parameters (for example area and K )
are different from the parameters downstream of the discharge point. The estuary is then
divided into two reaches: reach 1 upstream and reach 2 downstream, as shown. All system
parameters are then subscripted depending on reach location. Equation (3.28) is applied to
each reach to give
Cx exp(j]
C2 = B2 exP(82x^ + C2 exP^2x>
(3.29)
125
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Note that g and j are also subscripted by reach. In order to obtain the complete solution
to Equation (3.29), four boundary conditions are available:
a) c, - 0 at x - —
b) c_ = 0 at x = +">
c) Cj^ - Cj at x - 0
d) mass in - mass out at x » 0
The first boundary condition gives C. - 0 or
GI = BL exp(glx) (3.30)
This is obtained by recalling that g is positive and j negative. Thus for c^ - 0 at
x » -« , exp(Jjx) would be infinite unless C^ - 0 . The second boundary condition gives
B2 " 0 or
c2 - C2 exp(J2x) (3.31)
The third boundary condition yields at x •= 0
B1 - C2 (3.32)
The fourth boundary condition can be written as
W - - QC + EA + Qc - EA = (3.33)
Evaluation of the derivatives at x • 0 is easily accomplished from Equation (3.30) and
Equation (3.31). Substitution into Equation (3.33) and grouping yields
Bl + C2^2 ~ E2A2^2> * W (3'34)
where Q^ is the flow from the upstream reach to x » 0 and Q2 is the flow out from x * 0
into the downstream reach. Positive Q is directed into the plane x « 0 and negative Q is
directed outward from the plane x - 0 . For many estuarine situations Qj - Q2 which simpli-
fies some of the elements.
Equations (3.32) and (3.34) can now be solved for the unknown coefficients B^ and
B2 wtiich together with Equations (3.30) and (3.31) provide the complete solution.
In matrix form,
(3.35)
ii r i i
ElAl8l ' Ql
The complete solution is
"- - W , . . TT- exp(g.x) (3.36)
- E2A2J2 + Q2 1
126
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and
c2 exp(J2x) (3.37)
ElAlgl ' Ql " E2A2J2 + Q2
For the special case of constant parameters everywhere: E, = E, = E; Ai - Ao = A; Qi = Q? ™ Q>
and KL - K2 - K . Equations (3.36) and (3.37) reduce to Equations (3.16) and (3.17). This
example illustrates the use of the continuous solution approach for the simple case of two
reaches with differing parameters extending upstream and downstream from a point waste source.
If at some distance downstream, a new reach is necessary because of a change in one
of the system parameters or because of a new waste load, a similar procedure is followed.
In this case, new unknown coefficients are introduced and the matrix becomes larger. There will
now be three equations, one for each reach as
G! - B! exp(gjx) + G! exp(jjx)
c2 - B2 exp(g2x) + C2 exp(J2x) (3.38)
C3 - B3 exp(g3x) + C3 exp(J3x)
Concentration equalities are now formed and, together with the mass balance equations, will
provide four equations in four unknown coefficients. Although there are six unknown coeffi-
cients in Equation (3.38), C^ and 63 are zero because of the minus infinity and plus
infinity boundary conditions. The situation is now more complex because one of the coefficients,
either B2 or C2 , for the middle reach cannot be eliminated. The distance from x = 0
(point of discharge) to' the beginning of the next downstream reach is designated XQ . It
can be shown that four equations in four unknowns (B^, B2, C2 and C3) result from the appli-
cation of the boundary condition. Simultaneous solution provides the values of the unknown
coefficients for use in Equations (3.38) (Thomann 1970).
This approach is a powerful technique for dealing with one-dimensional estuarine
systems. The system can be made as complex as desired. The logic of constructing the elements
of the matrix must be carefully followed and appropriate boundary conditions must be used.
Infinity conditions are not always possible (as in the case of dams) in which case special
boundary conditions can be utilized.
3.3 FINITE SECTION APPROACH
This approach (Thomann 1963), instead of employing various solutions to the steady-
state equation (3.14), considers the differential equation directly. Continuous space is
replaced by finite volumes that are assumed to be completely mixed. Material balances are
then written around each finite length of estuary. The finite section approach is essentially
a finite-difference approximation to an ordinary differential equation.
Figure 3.8 shows the length of the estuary divided into N sections. The first
section is usually placed at the head end of the estuary or sufficiently far upstream from the
last major waste outfall. There are two similar derivations that may be used to determine the
127
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HEAD OF
TIDE
OCEAN
'
2
...
H
i
W
...
(H
n
Fig. 3.8 Finite section segmentation
for one-dimensional estuary.
necessary computation formulas. Both derivations give the same result. The first derivation
constructs a model on physical reasoning by writing a mass balance equation around a finite
volume similar to the technique used in constructing Equation (3.14). The second derivation
starts with Equation (3.14) and approximates each of the derivatives.
Using the first derivation, one proceeds by writing a mass balance around section 1
of the estuary. Each section is considered to be a mixed volume. No gradients are permitted
in the sections. The time rate of change of material c^ in section i is then given by
dc4
- KiVici
(3.39)
W
... N
where
. - volume of segment i
, - bulk dispersion coefficient - EjjAij
adjacent sections
. - dimensionless mixing coefficient; f)
• sources and sinks of c
where E is the average length of
1 - a
The parameter c^, is, in mathematical terms, a finite-difference weight. For a - 1/2 , a
central difference approximation of the derivative is utilized while for a - 1 , a backward
difference approximation is used. In physical terms, .a can be related to the ratio of the
dispersion to advective forces and it can be shown that a > 1 - E '/Q (Thomaim 1970) . If all
terms in ci_i . ci and ci+^ are grouped on the left side, one obtains
ai,i-li-l aiici a
as the general equation for the 1th section where a^j are functions of V , Q , E' , a
and K . For the upper and lower boundary sections, a similar derivation Indicates that the
incoming and exit boundary concentration can be incorporated into W^ and Wn . The n equa-
128
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tions can be written in matrix form as
[A](C) - (w) (3.40)
where [A] is an n x n matrix and (c) and (W) are n x 1 vectors. The solution vector (c)
is then obtained by multiplying through by the inverse of the A matrix or
1-1
(c) - [A]'1 (W) (3.41)
The problem then of determining the steady-state one-dimensional distribution of waste material
in an estuary with spatially variable parameters reduces to solving n simultaneous algebraic
equations (Equation 3.40) or inverting an n x n matrix (Equation 3.41).
The matrix [A] has a particular form for the one-dimensional estuary. This form
is known as a tri-dtagonal matrix where only the main diagonal and the diagonals above and
below the main diagonal appear in the matrix. All other elements are zero. This is a feature
which permits special efficient computing programs for determination of the inverse [A]"1 .
One can of course use other methods of solution of simultaneous equations (e.g. Gauss-Seidel
technique) to obtain the concentrations in each section.
The inverse matrix [A]"1 is termed a steady-state response matrix and represents
the responses in c due to the discharge of material of a unit amount into each section.
This can be seen by
M H - M
or (3.42)
[c]
where [I] is the identity matrix and [c] is now an n x n matrix. The first column of [c]
then represents the response over all sections due to a unit steady input into the first sec-
tion, the second column of c represents the response over all sections due to a unit steady
input into the second section, and so on.
For two-stage consecutive reactions, as in the case of carbonaceous BOD-DO, a similar
procedure is followed. A matrix [B] is generated; the only difference between [A] and [B]
is the reaction coefficients on the main diagonal. Thus let D stand for DO deficit and L
for BOO. The matrix equation for DO is
[B] (D) - (=)
where (£) is the vector of sources and sinks. If only the BOD sink of DO is considered then
[B] (D) - (vKdL)
where Kj is the deoxygenation coefficient. Multiplying by [B]"1 gives
(D) - [»]-! (vKdL)
129
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But since
Chen
(D) - [B]-l [VK,] [A]-l (W) (3.43)
where [VKj] is an n x n diagonal matrix. Equation (3.43) indicates the method of solution
for two stage consecutive reactions. (See also Bunce and Hetling 1966 and Hetling and
O'Connell 1966 for other discussions and applications of this approach.)
The solution technique in terms of the finite section approach for multi-stage con-
secutive reactions is now clear. Thus, consider the case of the four nitrogen forms designated
as Nj - organic nitrogen , N2 - ammonia nitrogen , Ng - nitrite nitrogen and N4 - nitrate
nitrogen . The matrix equations are
[A2] (H2) - (W2)
(3.44)
[A3] (N3) - (W3) + (VK,^)
[A4] (N4) - (W4) + (VK34N3)
where K^. are the "forward" reaction coefficients and the [Aj] differ only on the main
diagonal by the differences in the reaction coefficients K^ . The solution for (N-j) is
given as an example. Substitution of successive solutions of (N,) and (N-) into the
equation for (No) gives the solution as
- Mr1 W - Ml Ks) M1 M
(3.45)
Again, the solution vector need not be obtained by matrix inversion, but rather more efficient
routines for solution of simultaneous equations can be used.
This methodology can be applied to the multi-stage consecutive reactions described
above and forms the basis for the feedback model described in Section 7.1.
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4. TWO-DIMENSIONAL STEADY STATE
The steady-state mass balance equation for single-stage non-conservative substances in
two dimensions is given by
(3.46)
where u and v are the velocities in the x- and y- direction and similarly Ex and £„
are the tidal dispersion coefficients in the x- and y- direction. These two directions can
be interpreted in terms of either the horizontal plane (x - length, y - width) or the vertical
plane (x - length, y - depth). General analytical solutions of Equation (3.46) for arbitrary
coefficients are not available. Hence, in water quality modeling, various assumptions are
usually made in the system parameters or for variable parameters a finite section approach can
be used. Forms of Equation (3.46) have been applied to the vertical distribution of salt in
salt water wedge estuaries and fjords (Hansen 1967, Rattray and Hansen 1962). Other applica-
tions in water quality studies of wide estuaries or bays have generally been limited. A finite
section approach was used in studying the lateral distribution of salinity in Raritan Bay and
Hillsborough Bay (Hagan and Gallagher 1969). Other work is being done on the application of
a finite-difference form of Equation (3.46) to two-dimensional distribution of such water
quality variables as bacteria, BOD and DO in Boston (Hydroscience 1970) and San Juan Harbors.
A pseudo-two-dimensional approach has also been used in the San Francisco Bay area (Orlob
et al. 1967) and the Columbia estuary (Callaway et al. 1969). In these cases, the two-
dimensional aspects are introduced by coupling one-dimensional channel sections. This requires
an a_ priori specification of the directions in which gradients will occur. A complete two-
dimensional nonsteady-state analysis has been used in Jamaica Bay in New York City (Leendertse
1970). This work requires solution of the two-dimensional equations of motion together with
water quality modeling. It probably represents one of the more advanced truly two-dimensional
analyses to date. As increased attention is turned to the bay and harbor case, one can expect
further application of two-dimensional models.
4.1 SOLUTION APPROACH: FINITE DIFFERENCE
One approach to solving Equation (3.46) is to utilize the notion of a sequence of
completely mixed sections discussed in terms of the one-dimensional estuary in Section 3. For
the multi-dimensional steady-state case, a mass balance around a finite section k surrounded
by segments j is given (Thomann 1963) as (see Figure 3.9)
131
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J
J
•
3
Fig. 3.9 Segmentation in two dimensions.
dc,.
- ck>]
(3.47)
where all terms have been defined previously and the summation extends over all j segments
bordering on segment k . This equation also results from a formal finite-difference approxi-
mation to Equation (3.46) with a variable weight given to the advective term. If all terms
involving the dependent variable c^ are grouped on the left hand side, one obtains
where
akk '
akkck + akjcj " "k *
Vk
(3.48)
a '
akj
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The flow convention is positive leaving the section. Note that
and
QjkBjk
For sections on a boundary where the flow between the boundary and the section is designated
(positive leaving the section) ,
akk -
Vk
and the forcing function is
where c_ is the boundary concentration. For
(negative) ,
entering the section from the boundary
akk '
and
Wk ^ Wk
The n equations with suitable incorporation of boundary conditions can be represented
in matrix form as
"12 '
S21 a22
a2n
(3.49)
or
7
[A] (c) - (w)
where [A] is an n x n matrix of known coefficients that depends on the system parameters.
For most applications, a relatively large number of the elements of [A] are zero. The multi-
dimensional matrix can be compared to the tri-diagonal form for one-dimensional estuaries.
It is important to note that the use of this approach requires an £ priori specifica-
tion of the net flow vectors and tidal dispersion fields. For some cases, it may be possible
to route the freshwater flow component without resorting to complex hydrodynamic analyses.
133
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In general, however, it will usually be necessary to model the two-dimensional hydrodynamic
equations (see Chapters II and VI) to obtain an estimate of the distribution of the net advec-
tive flow. This requires integration of the hour-to-hour tidal velocities to determine the
net flow patterns. For steady-state analyses, if one has estimates of the distribution of net
advective flows (Qifc) then the salinity or chloride two-dimensional surface can be used to
obtain estimates of the intertidal dispersion coefficients. Alternatively one can run a time-
variable two-dimensional model with the hydrodynamic equations and water quality mass conserva-
tion equation (Leendertse 1970) on an hour-to-hour prototype time scale thereby eliminating,
for most purposes, concern with the tidal dispersion coefficient (Efci) • Tw° difficulties
arise: (1) computation time is increased greatly at those small time scales, and (2) the
transition from that time scale to scales of seasons and years is not clear.
In any case, the steady-state multi-dimensional water quality problem (note that in
principle a third dimension is included in the above development) reduces to solving n simul-
taneous equations or inverting matrix [A] in Equation (3.49) provided an estimate of the net
flow and dispersion fields is available. There are several advantages to obtaining the inverse
matrix of [A] . Each column of the inverse indicates the water quality response in all sec-
tions due to a unit load in a specific location. This often proves useful for rapid assessment
of load location or magnitude changes. During verification analyses, however, relaxation
techniques for solving simultaneous equations are much more efficient. It can be shown that
for a solution vector (c) that is positive everywhere, all elements off the main diagonal
of [A] should be negative. This provides a bound on the advection coefficient a which
relates to the dispersion and advection forces (see discussion of Equation 3.39). Various
algorithms are readily available for solving large systems of equations. For example, opera-
tional programs have been written for the IBM 1130 using disc storage to handle multi-
dimensional water quality problems of up to 120 sections. Larger machines permit even greater
numbers of sections up to at least 500 sections and with special programming techniques up to
about 1,000 sections if necessary.
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5. ONE-DIMENSIONAL TIME-VARIABLE MODELS
The equations for the one-dimensional time -variable water quality model for two -stage
reactions, such as BOD and DO, are given by
ft - -J fx-(Qc) + i fe(EA £> + Vv> - V- + p - * - B <3-51>
Equation (3.50) can obviously be used to describe a single conservative variable such as chlo-
rides where Kp « 0 .
A temporal variation of the following parameters or inputs is usually present:
freshwater flow, temperature, waste water discharge, and certain biological, chemical, and
physical reactions. Furthermore, there are various time scales over which these variations
may be significant. For a particular situation, a prescribed period of time is established
for the analysis, and a unit of time is assigned. For example, if it is desired to study the
quality variations of a natural body over the period of a year, it is probable the unit of time
would be a day, perhaps a week or a month. If on the other hand, the period of concern is a
day, obviously the time unit would be in the order of minutes or perhaps hours. The purpose
of the analysis, the nature of the phenomena, and the order of the temporal variations con-
tribute to the assignment of the scale and unit of time. Finally the variations observed can
be thought of as varying in a statistically and random fashion or in a deterministic and pre-
scribed manner. Thus, natural phenomena such as the temperature variations of the year and the
sunlight over the day may be approximated by a cyclic function on which is superimposed a
random variation. The discernible periodic fluctuations are usually evident. However, over
smaller or larger time scales, the random nature of some phenomena may be more significant
and without any apparent periodic form. One of the first steps in the analysis, therefore, is
the assignment of the time scale of interest and unit of analysis, from which the choice of
various functional forms of the temporal variations follow and the approximate methods of
solution employed in the calculation. Equations (3.50) and (3.51) as they stand are quite
general in their time scale of application. Intratidal time scales have been used to describe
short-term and locally important quality gradients. Intertidal time scales extending out to
seasonal variations have also been used to describe longer term fluctuations and larger scale
spatial gradients.
Analyses of water quality in natural waters discussed previously in this chapter have
concentrated on steady-state solutions. One of the critical questions which arises in this re-
gard is the time interval required to establish the steady state. This period varies depending
on the nature of the system, which includes the reaction, dispersion and advection coefficients.
Thus from the hydrological point of view, in a freshwater stream the steady state may be quickly
established and just as readily disrupted, while in the estuary a longer period is required be-
cause of the presence of dispersion. In the former case, the interval may be hours or days,
while in the latter weeks or months.
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5.1 SOLUTION APPROACH
Several approaches are presently available for dealing with the time-variable
estuarlne water quality model depending on the time and space scale of application (see
Orlob et al. 1967, Leendertse 1970, Jeglic 1966, Pence et al. 1968). For many problem con-
texts, the minimum time interval is on the order of tidal cycles, although for other problem
contexts shorter time scales are necessary. One method of solving the time-variable model
with arbitrary inputs and coefficients is to replace the partial differential equations with
a finite-difference scheme. Continuous space is then replaced by discrete finite spatial
elements which are assumed to be homogeneous. Thus the finite section approach discussed
previously is used, with the exception that the time derivative is not set to zero. For the
BOD and DO systems, this results in a sequence of linear ordinary differential equations with
time as the independent variable. One equation for each finite reach represents the waste-
load-to-estuarine-BOD transformation, and the second equation for each reach represents the
BOD-to-DO transformation. For n reaches, then, there are 2n differential equations to
be solved.
This time-variable problem, requires that a numerical integration scheme be used.
A variety of numerical schemes are available, and care must be taken that a scheme is adapted
that is appropriate for the problem context. Short-term models have been used in Jamaica Bay
(Leendertse 1970) and San Francisco Bay Studies (Orlob et al. 1967). Longer term models have
been used in the Delaware (Jeglic 1966) and Potomac estuaries (Bunce and Hetling 1966, Hetling
1968). An extended discussion of differencing schemes is given in Chapter VI. A brief review
is given here of the approach used in the Delaware and Potomac work.
Equation (3.50) will be used as an illustration of a differencing scheme that can be
used in time-variable problems. The region of interest along the length of the estuary is
segmented into n finite volume segments. The region of L(x,t) in the vicinity of the k th
segment will be examined and an approximation to Equation (3.50) in this region can be developed.
Figure 3.10 shows the notation, where V is the volume of the segment and x the length of
the segment.
One finite-difference approximation to the first derivative is given by
/ay \ LL- lrj.1 " L|r 1 lr
I OLt 1 *** IvTJL It A., R. e o c O \
'•ax'jf AX
The second derivative approximation is given by the derivative of (3.52) evaluated at x = k or
the advective term on die right side of Equation (3.50) can then be written as
Similarly, the difference approximation for the second term, the dispersion term, gives
136
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L(X)
Q
O
Ct>
k-l
k-fl
DISTANCE
SEGMENT
Fig. 3.10 Segmentation of longitudinal direction.
12_rFA ah« * TF A n
A te(Eta} . ^2 |.Ek,k+iAk,k+i(Lk+i"
Equation (3.50) can then be rewritten in terms of finite spatial coordinates, and hence as an
ordinary differential equation in time, as follows:
dLk
vk dE OL>k,k+i - k-i,k]
(3.53)
Wk
where Vfc = AfcL .
Equation (3.53) represents an approximation to the continuous form of Equation (3.50)
and can be used for those cases involving arbitrary coefficients. One must guarantee physically
realizable results in terms of solutions that are positive everywhere. One method of accom-
plishing this is to introduce a variable finite-difference weight for the advective term in
137
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the equation. Thus, an expression for the values of L at the boundaries of the segments can
be written as follows:
"'k.k+lH: + ^'^k.
and (3.54)
Ht-l.k " Vl.kH-l + <1-(x>k-l,kLk
where a physically represents that fraction of the upstream concentration L advected into
the downstream section. It can be shown that for solutions that must be positive everywhere
(the physical condition required here) a > 1 - EA/Qto . This provides a type of changing
difference approximation to the advective term. For low values of E , the a > 1 - EA/QAx
difference approximation shifts to a greater weighting of the upstream concentration, and at
zero dispersion the approximation is entirely a backward difference, a desirable trait of the
approximation. A lower bound of a - % for one-dimensional systems is usually used, which
indicates that the difference approximation to the advective term continually shifts from a
central-difference approximation to a backward difference, depending on the ratio of the
dispersion to the advective forces. Once the criterion of a > 1 . EA/Qto is met, the solu-
tions are highly stable and relatively insensitive to the weighting factor.
Substituting Equation (3.54) into (3.53) gives
vk dt
" Qk-l.kl'Vl.kMc-l^k-l.k1*) ' Qk.k+lfck.k+l^^k.k+lk+l
(3.55)
where QJJ ..• Ajjujj , and EJJ - EjjAjj/Ax and ^ - 1 - a . Equation (3.55) expresses the
time rate of change of L^ in terms of the concentration in the downstream segment Lk+1 .
This equation also results when a simple material balance is taken around each segment (see
also Section 3.3). The first term then represents the amount of L advected into the segment.
the second term represents the amount of L advected out of the segment, the third and fourth
terms represent the dispersion effect, proportional to the concentration difference be-
tween the two segments, and the fifth term represents the decay of L in segment k .
It should be noted that in Equation (3.55), the time scale is arbitrary and the numerical
schemes that have been developed (Orlob et al. 1967, Leendertse 1970, Jeglic 1966) can be
applied equally to within-tidal analyses or longer time scales. The choice is left to the
analyst after careful examination of the problem context.
It should be recognized that the use of finite section approximations introduces a
numerical or psuedo-dispersion effect into the model. This can readily be seen when it is
realized that each segment is assumed to be completely mixed. An impulse input is therefore
Immediately mixed in the volume not because of any physical effects but solely because of the
numerical procedure. The amount of dispersion so introduced is given by ufix/2 which provides
an estimate for determining the relative amount of numerical dispersion compared to "true"
dispersion. A detailed analysis of this effect is given in Bella and Dobbins (1968). For
short time scale applications in estuaries, this numerical dispersion may be severe and lead
to distorted results. For "low frequency" studies in turbulent estuaries where time scales
of days to seasons are important as opposed to hours, this effect is not as significant.
138
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In a simplified form, the equations to be solved can be written as
dt
and
dck
Basically, the computational procedure used in Jeglic (1966) begins from a set of
initial values at time zero and in all segments, k - l...n . Numerical integration is
utilized to pull the solution one time step forward. After each step is completed, the new
values of L^ and c^ are downgraded to become the initial values of the next step. The
process is continued until the entire time span that is originally specified has been com-
pleted. Jeglic (1966) controls truncation errors by a halving and doubling procedure which,
while sacrificing some computional time, does not require the user to specify an integration
interval.
The computer program used in the Delaware case (Pence et al. 1968) for solution of the
general time-variable estuary program is relatively large and designed primarily for machines
of the size of an IBM-360, Model 65, or CDC 6600. In the application of the program to the
Delaware Estuary the river was divided into thirty reaches. Primary emphasis was placed on
the longer time scales (seasonal effects) so that the equations were used with a day as the
basic prototype time unit. The dispersion coefficients used represented the tidal mixing
effects, from one tide stage to the next. The integration interval was between four steps
and fifteen steps per prototype day. The average problem for the Delaware required about four
computer minutes per simulated month on an IBM-7094. This included seasonal variations in
temperature and freshwater flow. Jeglic (1966) has given a rough gauge for overall running
time as
Each integration time (milliseconds) - 20 • number of estuary sections
Shubinski et al. (1965) and Orlob et al. (1967) have used a similar approach to the
one-dimensional diffusion-advection equation in applications to water quality problems in the
San Francisco Bay area. In that work, primary attention is given to the shorter term, within-
tidal cycle oscillations. As such, the tidal mixing phenomenon, which is incorporated above
in the dispersion term, is of lesser consideration and primary interest centers about the
advection term. Several forms of difference approximations were considered so that numerical
mixing and stability factors were balanced. A "quarter-point" approximation to the advective
term was finally used. This is in contrast to the time-dependent model used in the Delaware
where a variable difference approximation to the advective term is used. In that case,
guarantee of positive solutions is maintained by continually checking the ratio of dispersion
to advection.
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6. VERIFICATION
In the discussion above, the assumption was made that the system parameters and inputs
are known. However, it is usually difficult to specify the values of the system parameters
a_ priori. Although the order of magnitude of all of the parameters and usually most of the
inputs is known, from either empirical correlations or fundamental analyses, the means for
more precise independent specification for many of the terms in the equations are not presently
available. Consequently, for coupled systems such as the DO problem, Equations (3.16) and
(3.23) in conjunction with field data may be used to evaluate one or all of the coefficients.
The nontidal advective component U is most readily determined by the hydraulic continuity
relation, regardless of the small difficulty involved in obtaining representative values of
the flow and area. Conservative tracers, such as salt from the ocean or a stable chemical
from a waste source, may be used in Equation (3.13) to determine the tidal dispersion coeffi-
cient, if the advection due to freshwater inflow is known. With these two coefficients, the
first reaction coefficient may be obtained from Equation (3.16) and sequentially the second
from Equation (3.23).
If all the coefficients must be determined in this manner, the predictive ability of
the model is greatly reduced. For each coefficient which may be determined independent of con-
centration data, the predictive capacity increases one degree. In the example of the dissolved
oxygen deficit, all of the coefficients are known at least in order of magnitude and a more
exact value is specified by analysis of the data. Furthermore, one of the coefficients (in
this case the reaeration coefficient) may be evaluated from the hydraulic characteristics of
the system, independent of the dissolved oxygen concentrations. Thus, each set of data is
used to extract a value of a coefficient, but the final profile of dissolved oxygen deficit
can be calculated independently. Comparison of the calculated profile to observed data indi-
cates, first, the general validity of the model, and, second, what adjustments, if any, are
required in the empirical assessment of the coefficients to achieve a closer agreement between
model output and observed data.
Verification of any mathematical model is a necessity. The conceptualization of the
phenomena in mathematical terms must be quantified and compared to observed data. The rela-
tive confidence in the use of the model rests on this comparison. Several difficulties arise,
however, in the verification of estuarine water quality models. These include (a) gaps in
available data, (b) degree of verification required, and (c) the nature of the verification.
In terms of water quality, it is difficult to obtain representative data over time and space.
While great strides have been made in the continuous measurement in time of certain variables
such as DO, temperature, and conductivity, it should be noted that these are point measurements.
The adequate determination of spatial detail is required. Continuous recording of other
variables such as coliform bacteria, chlorophyll, algae, or various pesticides is not available.
There are therefore many problem areas especially in estuarine situations where only grab
sample data are available. In terms of utilizing mathematical models, all data (assuming it
is accurately measured) is of some use. However, data availability governs, to a large extent,
the extent of the verification that is possible.
Given these data difficulties, present model verification consists essentially of
judgmental comparisons between observed and computed data. This comparison should be conducted
140
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for a variety of combinations of freshwater flow, waste water inputs and temperature. Where
possible, "independent" verifications are carried out in the sense that the system parameters
used for one verification are subsequently used to compare observed and computed data for a
second set of survey data. This process is continued until a "consistent" set of system
parameters is obtained. The determination of the reasonableness of the "fit" is left to the
analyst. The state of the art has not yet progressed to the point where rigorous statistical
comparisons are made between observed and computed data. This is primarily due to the relative
uncertainty of the observed data itself. For example, in the one dimensional models discussed
previously, homogeneity is assumed in the cross-section. Data collected during a survey must
be related to the total cross-section, and the variability in the observed data must be taken
into account. This is especially significant for estuaries where the tidal oscillations occur
and where the variability of models in use refer to average tidal conditions. In other words,
observed data are only estimates of the "true" nature of the water quality variable. In some
cases, the observed data are only very crude estimates of the "true" value and wide limits can
be placed around the observed data. In the light of these difficulties in data acquisition and
interpretation, present verification procedures rely heavily on the judgment of the analyst as
to when a verified model has been obtained.
It should also be recognized that even with the procedure Just outlined, the model
has been verified only on the basis of past observed events. In that sense, the models are
verified on a hindcast basis rather than a forecast basis. It is rare that one is in a situa-
tion to immediately test the ability of a model to forecast some future level of water quality
given water quality levels up to the present time. The true test of the validity of a model is
its ability to forecast future water quality within some acceptable limits. In large estuaries,
the problem is compounded by the relatively small water quality responses expected from rela-
tively large removals of waste. For example, in the Delaware Estuary, removals of 100,000 Ibs
of BOD per day are estimated to result in DO improvements of about 0.5 mg/1 in the lower part
of the estuary. The determination of whether, in fact, 0.5 mg/1 improvement occurred after
removal of 100,000 Ibs/day is difficult and requires an extensive sampling program.
These problems in the verification of estuarine water quality models must be recog-
nized and appreciated in evaluating the usefulness of any mathematical trial. Notwithstanding
all these problems, verification analyses must be performed to provide some basis for predic-
tion. Models that are suggested for use and which have not been compared to observed data at
any level are of little use to the practitioner. If no data are available, it behooves the
analyst to insist that some data be collected before his models are used in the water quality
improvement program.
6.1 AN EXAMPLE: THE EAST RIVER (O'Connor 1966)
The East River is part of the complex tidal network of the New York metropolitan area.
It is essentially a long, narrow strait connecting the Upper New York Harbor, its southern
boundary, with Long Island South, its northern limit. Its Junction with the Harlem River is
located approximately midway between these points. It receives no freshwater flow except
for the waste waters and the periodic storm overflows. These wastes contain large quantities
of organic matter which cause significant depletion of dissolved oxygen. This condition is
particularly acute at the end of the summer and early fall when the water temperature is maxi-
mum and a steady-state condition may be assumed.
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6.1.1 Description of the System
A map of the East River showing the major elements of the system is shown in
Figure 3.11. The locations of the various sources of pollution from both treated effluents
and untreated waste waters are indicated. The major point sources are the existing treatment
plants: Ward's Island, Bowery Bay, Tallman's Island, Hunt's Point in New York City and the
City of New Rochelle. There are also a number of small plants not shown. The untreated sewage
is discharged from Manhattan on the west bank and Brooklyn and Queens on the east bank. The
locations of proposed treatment plants are also shown. In certain areas, the discharges con-
tain significant quantities of industrial wastes. Harlem River and Newtown Creek, as well as
the periodic discharges from storm overflows, also contribute pollutional matter to the East
River. Sampling of a limited extent has indicated the presence of sludge deposits along the
bed of the river. The survey stations of the New York City Department of Public Works are also
indicated on Figure 3.11.
LEGEND
O NYCDPW STATIONS
A EXISTING TREATMENT
PLANTS
A PROPOSED TREATMENT
PLANTS IN DCSIGN OR
CONSTRUCTION
— DRAINAGE AREAS OF
PLANTS
tSU AREAS NOT PRESENTLY
SERVED BY PLANTS
Fig. 3.11 Location map. Reprinted with permission
from Journal Water Pollution Control
Federation, Vol. 38, No. 11, Page 1814
(November, 1966), Washington, D. C. 20016.
142
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The area of the Lower East River from the Battery to Harlem River was assumed constant,
as the arithmetic average of the individual plottings taken every 500 feet. Although there is
considerable variation about the mean, this assumption when incorporated in the mathematical
model yielded a reasonable approximation to the observed data. The Upper East River from the
Harlem River to Long Island Sound is characterized by a cross-sectional area which increases
as the square of the distance from a hypothetical origin.
A schematic representation of the system is shown in Figure 3.12. A point source of
pollution U at x « a in the Lower East River is considered. The elements of the system
for which solutions are required are designated by numerals. The solution requires the appli-
cation of appropriate boundary conditions of concentration and flux to the set of equations
representing each element of the system. The actual system contains the Hudson River and the
Kill Van Kull. However, in order to simplify the analysis, as a first approximation the
effects of these bodies are included in the freshwater flow Q and the mass input WQ . The
latter term also includes the effluents from the various treatment plants which discharge
directly into the harbor. Furthermore, the assumption is made that the effect of the Harlem
River is not significant with respect to the BOD distribution in the East River. The flux of
the East River into the Harbor is due only to diffusion, since the advective term is negligible.
This contribution may be significant depending on the magnitude of the concentration gradient
dt^/dx in the river at its confluence with the Harbor. The gradient is determined from the
solution of the equation from zone II of the Lower East River and the input is equal to
EA
0
At steady state, an equilibrium concentration is achieved and the
output from the system is therefore Q-t^ in which -t^ equals the equilibrium concentration.
When the concentration of organic matter (BOD) or any non-conservative substance is considered,
a sink is the overall reaction or removal. If first-order kinetics are assumed to describe
this reaction, this term is K£ . Furthermore the loss from the harbor to the ocean by tidal
exchange may be expressed by R(t - •£,„) in which R is the volumetric exchange coefficient
and I the concentration in the ocean. If the concentration in the ocean may be assumed to
o
be zero, then the exchange may be added directly with the reaction. A summary of the equations,
the boundary conditions and the general solutions for the point source is presented in
Table 3.1. The constants, AJ, and B,, , may be evaluated by application of the boundary
conditions to the various equations.
OCEAN
Fig. 3.12 Schematic representation of system. Reprinted
with permission from Journal Water Pollution
Control Federation. Vol. 38, No. 11, Page 1817
(November, 1966), Washington, D. C. 20016.
143
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TABLE 3.1
Formulae Used in the East River Model
Zone Equation
o - "o + EAO sr
ax x-o
.,2
II, III O - E
dr
IV 0-E^4 + !^-K-t
Boundary Conditions
x - <•> 4=0
x - b S ~ 44
r - r0 t3 = t4
x»b aj-do-0
x ~ a lj " ^3
x-a ij-do-W
x - 0 *1 = *2
Zone General Solutions
I tx - (wo + EA0 (-JA2 + JBj))^ + (K+R) v)
III £- ~ A- e + B_ ej
IV 14- ^
6.1.2 Field Data
The East River survey stations of the New York City Department of Public Works are
Indicated in Figure 3.11. Surveys are regularly conducted by the Department once or twice a
week during the simmer months. Various phases of the tidal cycle are therefore encountered at
each station. Both surface and bottom samples are collected for temperature, dissolved oxygen,
chlorides, biochemical oxygen demand and coliform bacteria. A typical set of data of both
surface and bottom samples is shown on Figure 3.13. The broken lines are drawn through the
data to indicate the trend, rather than a functional relationship. Vertical stratification
in temperature and concentration may be observed. This characteristic is particularly
144
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dissolved oxygen Is relatively high, in the order of 3 or 4 mg/1 at the Battery. It then falls
to a value of approximately 1 mg/1 at the time of high temperature and then rises to about
2 mg/1 at the end of the summer. The period of maximum temperature is usually in August and
is probably representative of steady-state and equilibrium conditions. Steady-state conditions
are approximated during the period just before and after, roughly during July and September.
The period of June is least representative of steady state, since the time gradients are
relatively steep and transient conditions are in effect. Furthermore, the concentration of
the various substances during this period is probably affected by the freshwater flow of the
Hudson River, as may be observed in the salinity measurements. The relative constancy of the
salinity during the remaining periods is indicative of the steady state, at least with respect
to the effect of the Hudson River flow. The analysis, based on the steady state, is therefore
restricted to the periods of July, August and September. The actual delineation of these
periods was determined by observation of the temperature and concentration data. In summary,
the steady-state solution may be applied to the East River since:
(1) The freshwater flow is substantially zero and the waste water inputs are relatively
constant, yielding a constant concentration of BOD.
(2) Temperature variations during the equilibrium period are minimal and the average value
may represent the steady state.
ISC 1959
15
12
80
70
60
50
40
30
ao
10
0
SJUXBEFCMEEM
SLACK KFOIC FLOOD
t I t I
MI235 689K>IIQBMSei7BS20
SAMPUNG STATIONS
Fig. 3.14 Spatial distribution of chlorides and
DO at slack water. Reprinted with
permission from Journal Water Pollution
Control Federation, Vol. 38. No. 11.
Page 1824 (November, 1966), Washington,
D. C. 20016.
146
-------
(3) Tidal variations are eliminated by considering slack water measurements or average values
over a significant period, as described above.
The individual data shown in these figures are based on samples taken at various
times over the tidal cycle and therefore reflect the effect of tidal translation. This factor
accounts, in large measure, for the deviations of the observed data from the average trend.
The tidal effect is well borne out by the data collected by the Interstate Sanitation Commission
in the summer of 1959, during which period a short-term, intensive survey was conducted. Dis-
solved oxygen, temperature and salinity were measured hourly every day for a three-week period.
The spatial distribution of the average values at high and low water slack are shown in
Figure 3.14. The variation over the tidal cycle is minimal in the vicinity of the low point
on the DO profile where the gradient is flat, but significant in portions of the Upper East
River where the gradient is steep. A typical set of spatial DO data for one season is shown
in Figure 3.15.
6.1.4 Waste Water Discharges
The BOD input data from the treatment plant effluents and average water temperatures
are given in Table 3.2 for the month of July, 1960, the period selected for analysis.
1 •
S 4
Q
SI 3
t-EOEND JC
- l«t PERIOD
'ra
4 6 8 10 12 14 16 18 20 22
DISTANCE FROM BATTERY-miles
Fig. 3.15 Spatial distribution of DO over
simmer period. Reprinted with
permission from Journal Water
Pollution Control Federati-onT
Vol. 38, No. 11, Page 1825
(November, 1966), Washington,
D. C. 20016.
147
-------
TABLE 3.2
TREATED BOD SOURCES
Treated Point Sources Pounds Per Day
Ward's Island 37,600
Bowery Bay 50,000
Hunt's Point 14,700
Tallmans Island 5,600
UNTREATED BOD SOURCES
Distributed Sources Pounds per Day - Mile Distance (Miles)
Manhattan 14,200 7.0
Brooklyn-Queens 17,900 5.2
Storm Overflows 7.5% of the total BOD from drainage areas receiving
treatment
Newtown Creek 30,600
Harlem River 6,600
Average Water Temperature 22°C
The BOD input for the untreated areas is based on population estimates, assuming a
unit discharge of 0.17 pounds BOD per capita-day. This figure was determined from an average
of the July and August data of the Ward's Island and Bowery Bay treatment plants for the
period from 1956 to 1960. The influx of the Newtown Creek is 30,600 pounds per day, and of
the Harlem River is 6,600 pounds per day.
6.1.5 BOD AND DO DEFICIT PROFILES
The average value of the cross-sectional area AQ of the lower river is 80,000 square
feet, which was determined from plots of the area taken every 500 feet from the Battery to the
Harlem River. The dispersion coefficient E was taken as 10 square miles per day for all
elements of the system, and the reaction coefficient K was taken as 0.23 per day at 20°C
for all sources. In order to convert to the prevailing temperature, a value of 1.04 for 9
was used. The coefficient K , which reflects the overall reaction of BOD removal, is not
necessarily equal to that of deaeration. In tidal bodies, however, they are usually of the
same magnitude, provided settling is not a significant factor. This assumption is made in
this analysis. An example of the individual profiles and the total profile of BOD is shown
in Figure 3.16 with the observed 5-day BOD values for the period of July 1960. Dividing this
value by the bottom area-volume ratio yields a numerical value of 0.18 mg-day/1.
An approximate value of the reaeration coefficient Ko was calculated by the formula
n „ 1/2
>
,.3/2
148
-------
Q
O
CD
to
OS
OS
O4
02
00
IO
OB
O6
Q4
O2
00
e
02
00
08
OS
0.4
02
OO
06
O4
O2
OO
OS
04
02
OO
Ofi
a*
02
oo
4O
C?
O
O
I
JO
20
to
00
BROOKLYN QUEENS
I7.9OO*/OAY4M.
MANHATTAN
14,200 «/DAY-MI
NEWTOWN CREEK
MM 3O.6OO*/OftY
HARLEM RIVER
W« 6,600»/DAY,
WARDS ISLAND PLANT
W» 37,600*/DAY
HUNTS POINT PLANT 8 TALLMAN ISLAND PLANT
W»I4,700»/OAY W*5,|600*/DAY
I i 1i
STORM OVERFLOW
W» 7,7OO»/OAY-MI_
OBSERVED DATA
TOP
AVERAGE
BOTTOM
1
2O2 4 6 S O 12 14 16 18 20 22
DISTANCE FROM BATTERY-miles
Fig. 3.16 Individual and total BOD profiles. Reprinted
with permission from Journal Water Pollution
Control Federation. Vol. 38, No. 11, Page 1827
(November, 1966), Washington, D. C. 20016.
in which I>2 = coefficient of diffusivity of oxygen in water
Uo - average tidal velocity
H « average depth.
Hie average tidal velocity is the mean between the average ebb and flood tide for the lower
and upper sections. The average depth was determined from the cross-sectional area data.
Values of this coefficient at 20° are 0.08 per day and 0.11 per day for the lower and upper
rivers respectively. In order to convert to the prevailing temperature, the value of the
coefficient 6 was taken as 1.02. Samples of the individual DO deficit profiles and the
total deficit profile for the period of July 1960 is shown in Figure 3.17. These profiles of
DO deficit and BOD are typical. In general the deficit profile is a better fit than that of
the BOD, particularly in the upper East.
149
-------
JULY I960
o>
LO
Q5
OO
20
I.S
LO
OS
OO
I.O
05
00
10
OS
00
\JO
OS
00
LO
OS
00
1.0
05
OO
1.5
1.0
05
OO
6
BROOKLYN-OUeENS
MANHATTAN
r . . . .
I
Lu
LU
O
*
HARLEM I^VER a WARDS ISLAND PLANT
HUNTS POMT PLANT 8 TALLMAN ISLAND PtANT
STORM OVERFLOW
SLUDGE DEPOSITS
OBSERVED DATA
TOP
AVERAGE
BOTTOM
i
4 6 8 10 12 14 16 IS 20 22
DISTANCE FROM BATTERY- miles
Ftg. 3.17 Individual and total DO deficit profiles.
Reprinted with permission from Journal
Water Pollution Control Federation)
Vol. 38, Ho. 11, Fage 1»ZB (NovemBer, 1966),
Washington, D. C. 20016.
150
-------
6.1.6 Discussion
The distribution among the carbonaceous matter, nitrogenous material, storm overflows
and sludge deposits should be determined in each estuarine analysis. The relative contribution
of each of these factors is significant, not only with regard to present water quality, but more
important, to future conditions when treatment facilities are installed. Treated effluents may
have a more pronounced effect on the oxygen resources of the system than has been appreciated.
Although the carbonaceous matter is effectively removed in high rate biological treatment
plants, the nitrogenous components of this material may pass through relatively unaffected.
This condition gives rise to an environment which is conducive to the growth and development
of the nitrifying bacteria. The effect of nitrification is apparently not as pronounced in
the untreated discharges. Thus, that portion of the dissolved oxygen depression which is the
result of nitrification under present conditions of little or no treatment, may be much more
significant under future conditions of partial or secondary treatment. It is most important
that the relative effects of each component in both raw and treated effluents be more fully
investigated and assessed.
151
-------
7. FUTURE DIRECTIONS
7.1 STEADY-STATE FEEDBACK MODELING
There are several phenomena in water quality problems that are characterized by
"feedback" effects. Examples include the loss of nutrient material to sediments with subse-
quent feedback into the overlying water, and the utilization of nutrients by algae with
subsequent release of the nutrients upon algal death. Figure 3.18 shows the major features
of the nitrogen cycle and indicates several possible feedback loops. This figure can be com-
pared to Figure 3.4 which illustrated only the feedforward phenomena of nitrification with
accompanying oxygen utilization. Section 2.6 discussed the importance of the nitrogen compo-
nent of waste discharges.
WASTE
SOURCES
WASTE
SOURCES
NITRIFICATION
N,
WASTE
SOURCES
ORGANIC
NITROGEN
—HYDROLYSIS =•*
AMMONIA
NITROGEN
DEATH
-NITROSOMONAS-"
NITRITE
NITROGEN
-NITROBACTER-*
NITRATE REDUCTION
PHYTOPLANKTON
UTILIZATI
PLANT
AND
ANIMAL
NITROGEN
~L
Fig. 3.18 Major features of the nitrogen cycle.
For some problem situations, then, it is useful to have available a generalized
steady-state model which can incorporate feedback (or feedforward) loops in a multi-dimensional
water body with N consecutive reactions. For preliminary studies to determine general order-
of-magnitude responses, such a model provides the user with the ability to analyze any system
configuration, the model therefore represents the limit of complexity of estuarine models
under steady-state and first-order kinetics assumptions.
152
-------
Hie use of the finite section (difference) approach for consecutive reactions was
given previously in matrix form in Section 3.3. In addition, it was shown in Section 4 that
in principle the addition of lateral or vertical water quality gradients results in a general
matrix of coefficients rather than the tridiagonal form as for one-dimensional estuaries. All
that remains is to incorporate any first-order kinetic feedback loop from any substance in the
consecutive reaction sequence to any other substance. A complete description of the model is
given in Manhattan College (1970).
The matrix equations with all feedforward and feedback loops are then given by
JA2](C2) - (W2
(3.58)
+ ... +
[AN](cN) - («*) + [VKIN](C1)
As before, the matrices [A ] are of size n x n where n is the number of spatial
segments. The superscript i refers to the itn substance in an N-stage consecutive reaction
sequence. The matrices [A*-] differ only in the K^^ on the main diagonal. The vector (c*)
is an n x 1 vector representing the spatial distribution of the output of the tth reaction;
[VK*J] , i <• j , represents the feedforward loops as an n x n diagonal matrix and [VK1^] ,
i > j , represents the feedback loops, both given by the vector of first-order reaction coeffi-
cients (K) ; and (W*) is an n x 1 vector of input of waste material of the ith reaction.
In matrix form, the matrix equations (3,58) can be written
[A1] [-VK21] . . . [-Vl"1]"
[A2] . . . [-VKN2]
or
-VK1*] [-VR2N] . . . [AN]
[a] (c) - (W)
(3.59)
(3.60)
153
-------
where [*] Is an nN x nN matrix, (6) Is an nN x 1 solution vector and (tf) is an nN x 1
waste load input vector. The solution vector is given formally by
(3.6D
The steady-state multi-dimensional, N reaction system with any feedback-feedforward
configuration water quality problem is therefore given by solution of Nn equations or in-
verting matrix [a] as given by Equation (3.61). This approach can be quite useful for a
variety of water quality problems providing one is willing to accept first-order kinetics.
Although the solution technique is computationally direct (e.g. matrix inversion or solution
of simultaneous equations) special care must be taken in computer algorithms to ensure proper
problem set-up and data input preparation. A Gauss-Seidel relaxation technique is used for
simultaneous solution of the equations with generation of the matrix inverse as an option.
The steady-state feedback model was used in a preliminary analysis of steady-state
water quality profiles in the Potomac Estuary (Thomann et al. 1970). The objective of the
analysis was to obtain greater insight into the effect of feedback and to determine the use-
fulness of the approach in sequential reactions such as nitrification.
The major waste source in the Potomac Estuary is the effluent of the Washington, D. C.
secondary waste treatment plant. Water quality problems include low DO in the vicinity of the
District of Columbia discharge and high algal concentrations downstream. A steady-state feed-
back model which considers organic, ammonia and nitrate nitrogen forms was constructed. A
feedback loop is incorporated which represents ammonia and nitrate nitrogen utilization by
algae with the subsequent nitrogen release upon death recycled to the organic nitrogen form.
The loop therefore represents the utilization of nitrogen by algae with subsequent nitrogen
release upon death. The four system model (organic, ammonia, nitrate and nitrogen found in
living algae) can therefore be used to obtain information on the behavior of the nitrogen
cycle in the Potomac. Figure 3.19 shows the results of an analysis of data collected during
July - August 1968.
The first-order reaction coefficients for the verification analysis shown in
Figure 3.19 are given in Table 3.3.
TABLE 3.3
Reaction Coefficients - Potomac Estuary, July-August 1968
Temperature - 28°C
Reaction Reaction
Coef Coef
Nitrogen Step (I/day) Nitrogen Step (I/day)
Decay of Organic-N 0.1
Organic-N. - NH3-N 0.1 N03-N - Algal-N 0.1
NH3-N - N03-N 0.28 Algal-N - Organic-N 0.12
NH3-N - Algal-N 0.02
These coefficients represent the end result of many runs which tested the effects of various
interactions and levels of coefficients. There is no a priori reason for assuming this set of
coefficients to be unique. The order of the reaction rates is, however, similar to that which
154
-------
k.O
3. 5
3.0
2. 5
12.0
1 .5
1 .0
0.5
0
JULY - AUG. 1968
Q - 2 ,000 CFS
LEGEND:
COMPUTED WITH
ALGAL FEEDBACK
COMPUTED WITH
NO FEEDBACK
APPROX. RANGE
OF OBSERVED DATA
10
15
20
25
30
35
23456 89 10 II 12 13 14 15
B | 9 | 10 | II
SESMENT NO.
CHLOROPHYLL'^'
10
15 20 25
MILES FROM CHA IN BRIDGE
SEGMENT NO.
30
35
Fig. 3.19 Comparison of observed and computed (solid lines) nitrogen
and chlorophyll - Potomac Estuary, July-Aug. 1968.
155
-------
gave meaningful results in an analysis of nitrogen in the Delaware Estuary. Although the set
of coefficients is not unique, this does not imply that any arbitrary set of coefficients will
yield similar results. Indeed, with sequential-type reactions with feedback, one is limited •
in the type of reaction changes that can be imposed. One is faced with obtaining "good" agree-
ment of observed and computed data of four (or more) system variables simultaneously.
It can be noted that organic nitrogen is "settled out" of the system, because of the
difference between the decay coefficient of organic nitrogen and the conversion of organic
nitrogen to .ammonia nitrogen. This was justified on the basis of bottom sampling which indi-
cated significant deposits in the vicinity of the Washington, D. C., outfall. Ammonia nitrogen
followed two paths: (a) utilization in the algal nitrogen loop (K - 0.02) , and (b) oxidation
to nitrate (K - 0.28) . This split allowed a proper spatial profile to be maintained. Nitrate
was recycled to algal nitrogen all of which was allowed to decay to organic nitrogen.
Figure 3.19 compares the observed data on nitrogen forms to computed values generated
by the model with and without the feedback of ammonia and nitrate nitrogen to organic nitrogen.
For the uppermost curve only Kjeldahl nitrogen observed data were available. The effects of
the feedback loop are to increase all profiles in a nonlinear spatial manner. The relative
downstream shift of the various nitrogen forms is interesting and reflects the sequential
nature of these types of reactions. Steady-state analyses such as shown In Figure 3.19 can
provide a basis for estimating the effects of environmental changes on nitrogen distribution
in addition to the effects of nitrification and algal utilization on the oxygen regimes. For
more detailed work where the assumptions of steady-state and first-order kinetics are dropped,
a general dynamic model is appropriate. This is discussed below.
7.2 DYNAMIC PHYTOPIANKTON MODEL
In order to obtain increased understanding of the dynamics of algal growth and
subsequent nutrient utilization, a model was constructed which incorporates several of the
important features of this phenomenon. Mass balance equations are written for the phytoplankton
population as measured by its chlorophyll concentration, for the zooplankton population, the
next higher trophic level, as measured by its carbon concentration, and for the nutrient level as
measured by total inorganic nitrogen. It was assumed that for the situation being considered
this was the only nutrient in short supply. The model in principle, however, is not restricted
to this nutrient. A detailed discussion and derivation of the model is given in DiToro et al.
(1970).
The following coupled set of differential equations are used to describe the dynamics
of algal growth:
' ~ + G(N>1'T)
P ' VZ'T) P - - <3'62>
t0 * v t0
n
m - + G2(P) Z - DZ(T) Z - L. (3.63)
inf" !r " b GP(N'I)T) p '-
156
-------
where P Is phytoplankton chlorophyll, C_(N,I,T) is Che phytoplankton growth rate as a
function of inorganic nitrogen concentration (N) , solar radiation (I) and temperature (T)
Dp(Z,T) is the death rate of phytoplankton due to zooplankton (Z) grazing and endogenous
respiration; Z is the zooplankton concentration; GZ and Dz are zooplankton growth and
death rates, respectively; b is the nitrogen yield coefficient (mass nitrogen/mass algae)-
and W is the input nitrogen from waste discharges or land runoff. A schematic of the type
of system under investigation is shown in Figure 3.20. The form given in Equations (3.62) -
(3.64) corresponds to a completely mixed water system with tQ as the detention time, equal
to the volume divided by the flow, and Pg , Zg and Ng are respective influent concentra-
tion. For situations where spatial gradients exist, additional terms are necessary to incor-
porate the advection and tidal dispersion. However, the incorporation of spatial gradients
does not alter the conceptual model of the dynamics of the system, but it does increase compu-
tational complexity.
FLOW
TEMPERATURE
SOLAR
RADIATION
ZOOPLANKTON
PREY
GRAZING
PHYTOPLANKTON
NUTRIENT
LIMITATION
NUTRIENT
USE
NUTRIENTS
MAN MADE
INPUTS
Fig. 3.20 Interactions: environmental variables and
the phytoplankton, zooplankton, and nutrient
systems.
157
-------
In may application, the general expressions for the growth and death coefficients
must be made explicit. Phenomena such as nonlinear growth as a function of nutrient concentra-
tions (using Michaelis-Menten expressions) and solar radiation, light extinction, self-shading,
and temperature are included in the growth and death terms for phy top lank ton. For the case of
one limiting nutrient, say total inorganic nitrogen, the growth expression for the rate in
the j th segment of the water body is given by
K T e-ij-e-oj __ (3.65)
'
Here K^ is the saturated growth rate of phytoplankton, I* is water temperature, H, is
water depth, f is the photoperiod, Nj is total inorganic nitrogen concentration, Kg, is
the Michaelis constant for total inorganic nitrogen and
kej - ke'j + h(Pj)
aoj ' VXs
where k'. is the extinction coefficient, h(P.) is the extinction due to phytoplankton
( self -shading) , Ia is the average incident solar radiation intensity during the photoperiod
and Is is the light saturation intensity for phytoplankton. The inclusion of the self-
shading phenomena h(Pj) is Incorporated linearly as
h(Pj) - 0.17Pj
for h in meters'1 and P. , the phytoplankton concentration, in mg/1 of dry weight.
The phytoplankton death rate is due primarily to endogenous respiration and grazing
by herbivorous zooplankton.* A search of the literature indicated a linear equation between
respiration and water temperature was justified. The grazing by zooplankton requires a knowl-
edge of the filtering rate by zooplankton and is most readily included in the death rate by
assuming a direct relationship between the death rate and the population of zooplankton. The
proportionality is given by the grazing rate of zooplankton.
The death rate in the j th section for phytoplankton is thus given by
DpJ - K2T + CgZj (3.66)
Incorporation of the growth rate, Equation (3.65), and death rate, Equation (3.66), into the
phytoplankton growth equation (3.62) completes that equation in terms of the primary variables,
light, temperature, nutrient concentrations and zooplankton. However, the nutrient and
zooplankton concentrations must be obtained simultaneously in order to solve the entire inter-
dependent system given by Equations (3.62) to (3.64).
158
-------
For the zooplankton system, the source of zooplankton biomass is
Szj - ZJ
where GZJ and DZJ have units of day"1, and Z. Is the concentration of zooplankton carbon
in volume element Vj . The growth rate Gzj is a function of the phytoplankton concentra-
tion, since at high phytoplankton counts the zooplankton do not metabolize ail the phytoplankton
that they graze, but rather excrete a portion of the phytoplankton in undigested or semi-
digested form. A convenient choice for a functional relationship is a Michaelis-Menten form,
so that the growth rate for zooplankton is given by
where Azp is the utilization efficiency which relates phytoplankton biomass ingested to
zooplankton biomass produced (approximately 0.6), C_ is the filtering rate, and !(„,_ is the
Michaelis constant, set at 60ug chlorophyll/liter in this work.
The death rate for zooplankton depends on both predation by higher trophic levels
and endogenous respiration. In this model, the death rate was given by
Dzj ' K3T + K4
where K^ is an empirically determined constant.
The nutrient system can now be readily written down since the sink tern for phyto-
plankton, zooplankton excretion and mortality sources have been formulated. In addition to
these sources and sinks of total inorganic nitrogen, the direct sources W as shown in
Equations (3.64) must also be included. Thus the net source-sink term for nitrogen is
(3.67)
+ b K2TP., + AnzK3TZj
All growth, death and source-sink terms have now been formulated. It should be
noted that although the expressions all include various coefficients, the literature provides
laboratory and/or field estimates for the majority of the parameters. Indeed in the applica-
tion of the model all the parameters were required to vary only within reported literature
values .
The equations discussed above were applied to phytoplankton and zooplankton popula-
tions observed at Mossdale Bridge on the San Joaquin River in California during the two years
1966 and 1967. The San Joaquin River, located in the center of Western California, in the
vicinity of its confluence with the Sacramento River, forms a delta which is an extremely pro-
ductive agricultural region. The combined discharge from these rivers flows through the Delta
network, Sulsun and San Pablo Bays through the North San Francisco Bay and then to the ocean.
159
-------
For an initial application of the nonsteady-state algal growth model, the region
near Mossdale was selected. This location is on the San Joaquin River, about thirty miles
from its confluence with the Sacramento. The river in this region is quite shallow and
although it is turbid, the region is nevertheless productive. For simplicity, a reach of the
San Joaquin in the vicinity of Mossdale was considered to be a completely mixed system.
The solution to Equations (3.62) - (3.64) with nonlinear time-variable coefficients
as given in Equations (3.65) - (3.67) requires machine computation. For this study, the equa-
tions were integrated on an IBM 1130 computer using a Continuous System Modeling Program (CSMP).
This software package incorporates a system simulation language which allows the user to
specify equivalent analog components which are then Implemented digitally. Time-variable
solar radiation, inflows, temperature and concentration inputs can all be handled in this
manner. At each integration step, the equations are integrated numerically using a second-
order Runge-Kutta approximation. In this approximation, a half-step integration is performed
to give
* h
v w v + fit v i
and this value of y is used to estimate the slope at the midpoint of the interval which is
then used as the slope of the straight-line approximation
yx - y0 + hf (t0 + £, y*)
The environmental variables used as input for the two-year period of interest and the
straight-line approximations used in the numerical computations are shown in Figure 3.21. A
comparison between observed data over two years (1966-67) and the calculated solution is shown
in Figure 3.22. The growth and death coefficients in Equations (3.65) - (3.67) were computed
internally as nonlinear functions of temperature, nutrient concentrations, and light, using
the appropriate functional forms. Net growth rates (growth minus predation and death) con-
tributing to exponential phytoplankton growth during the spring varied from 0.0 to about
0.3/day. The various coefficients and parameters in the equations were then adjusted to
achieve an acceptable fit of the data. However, In no case did the assigned values for the
coefficients exceed the boundaries of reported levels. There is a marked difference in the
phytoplankton population which developed during the two years. In 1966 a large population
formed in the spring, thereby reducing the inorganic nitrogen concentration markedly. This
population then remained stable throughout the summer and fall. By contrast in 1967, no
significant population developed during the spring and summer, and not until the late fall
did a population begin to develop. The cause for the differences in these two years is due
to the variation In hydrographs. In the spring and summer of 1967, the freshwater flow was
very large resulting in a wash-out of the population during this period. Only during the late
fall did the flow decrease sufficiently so that a population could develop. The model calcu-
lations behave in a similar way and the agreement obtained IB quite encouraging.
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0.
2:
30
20
10
120
20,000
360
120
360
TIME - DMS
240
360
1967
Fig. 3.21 Temperature, flow and mean dally solar
radiation. San Joaquin River, Mossdale,
1966-1967.
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100,000
80 ,000
60 ,000
2o 40.000
X
Q.
20,000
120
240
360
120
o
zz
o
-JO
o —
1.0
0.8
0.6
O.k
0.2
240
120 21*0 360 120 240
1966 TIME - DAYS 1967
360
360
Fig. 3.22 Phytoplankton, zooplankton, and total Inorganic
nitrogen. Comparison of theoretical calculations
and observed data. San Joaquin River, Mossdale,
1966-1967.
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8. DISCUSSION AND RECOMMENDATIONS
On the basis of the quality models outlined in the preceding sections of this
chapter, it is appropriate to compare and contrast, in a general manner, hydrodynamic and
quality modeling. In a fundamental sense, both utilize the same basic principles of conser-
vation of mass. The distinction between the two modeling efforts lies in the fact that the
principles and mechanisms which underly most hydrodynamic phenomena are reasonably well under-
stood and quantified. The mathematical relationships which describe the hydrodynamics of a
system are scientifically more valid than those which are used to describe most biological
and some chemical reactions in natural waters. (Note, for example, that in all three sections
of Chapter II, one can immediately write down the equations of motion as a starting point.
Such is not the case with water quality problem contexts where the phenomenon itself is not
yet clear.) The nature of these biological and chemical phenomena are known, in a qualita-
tive fashion, but in most cases have not been described in a quantitative manner. In the
case of some simple, single-stage reactions, significant progress has been made and fundamen-
tal relationships are available, comparable to those in the hydrodynamic realm.
However, in the case of consecutive, complex reactions, which are more relevant to
water quality problems, the relationships are not available and mathematical simplifications
are therefore assumed in order to make the equations more tractable. While the assumed rela-
tions are effective in fitting prototype observations, it is recognized that they may not be
fundamental, and they are utilized more for their simplicity than their validity.
Furthermore, historical data of water quality, particularly of a biological nature,
is generally less available than that of hydrodynamic observations such as tides and currents
in estuaries and bays. This lack of quality data imposes further restrictions on the extent
to which water quality modeling can be developed. In view of these factors, it must be
accepted that quality modeling is greatly restricted by contrast to hydrodynamic modeling.
One of the most difficult assessments to make in this regard, therefore, is the extent to
which the more advanced hydrodynamic modeling should be incorporated in the less sophisticated
quality modeling. Part of the answer is found in applying the criteria described in the
beginning of this chapter, particularly with respect to appropriate time and space scales.
In general, hydrodynamic developments should be incorporated to a level of significance with
respect to the water quality problem under investigation and not beyond. More advanced hydro-
dynamic modeling offers no further advantage from the quality viewpoint. This is not to imply
that research should be curtailed in this respect; it is simply stating that for some cases
it may not be necessary to employ the most advanced hydrodynamic techniques for the analysis
of certain water quality problems. On the other hand, the use of these techniques in analyzing
the distribution of conservative substances and in some problems, e.g. simple single-stage
nonconservative variables, is obviously appropriate. The success of this approach in such
problems as the time and space distribution of salinity is noteworthy. However, with respect
to the majority of the problems which involve nonconservative substances, the application of
advanced hydrodynamic modeling techniques yields little additional information on the phenomena
occurring because of the lack of knowledge of the complex water quality reactions. Until the
gap in fundamental understanding between the two fields is narrowed, the excellent work of the
hydrodynamicist, particularly over the past decade, is of value for only a limited number of
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estuarine quality problems. The majority of the water quality problems involving nonconservative
substances with consecutive reactions, feedback, and higher-order life forms are only marginally
improved in understanding by the use of advanced hydrodynamic modeling.
In summary, for conservative substances and single-stage non-conservative substances
with simple reaction kinetics, water quality modeling in estuaries is well advanced for both
spatial and temporal analysis. In this case, the problem is primarily a hydrodynamic one and
this field is sufficiently developed to provide fundamentally realistic and useful models of
water quality. On the other hand, systems that include non-conservative substances with a
series of consecutive and feedback reactions are less well understood. Some modeling efforts
have been initiated which appear to be fruitful. The complexities of most of these systems,
however, outweigh the hydrodynamic effects and attention should be directed to the understand-
ing and modeling of these phenomena in which the hydrodynamics play a minor role (in the sense
of understanding the phenomena). Between the two extremes lie a variety of practical problems
in water quality, in which both the reactions and transport may be equally significant. The
extent to which each is important and therefore the type of model to use must be carefully
assessed in each particular case.
It is therefore recommended that future research efforts be directed to an under-
standing and development of the reactions occurring in estuarine waters relating to physical,
chemical and biological parameters. The latter two categories are particularly complex,
because of the recycling and feedback mechanisms which produce pronounced nonlinearities in
the equations. Support should be directed to the analytical development and solution of such
models as well as field surveys to provide the necessary data for verification and evaluation.
More specifically, the following recommendations are offered with regard to future
directions that should be taken on the modeling of estuarine water quality, and reflect a need
to investigate model structures that are relevant to water quality management problems. The
recommendations are not presented in any order of priority.
(1) Fhytoplankton Models. Continual and expanded efforts should be devoted to the additional
application of available approaches to areas of specific Interest and to the development of new
models which will adequately reflect interactions between several nutrient constituents,
phytoplankton and zooplankton. The effect of seawater intrusion in estuaries and bays on the
species composition and subsequent effects on the total phytoplankton population is one example
of phenomena that are yet to be modeled.
(2) Dissolved Oxygen Models. Contrary to widely held opinions In the field, the dissolved
oxygen phenomena in estuaries are not completely understood. This lack of understanding applies
especially to the role that nitrogeneous discharges play in oxygen utilization, e.g. the "com-
petitive" use of ammonia by algae and nitrifying bacteria. In addition, the use and generation
of oxygen by phytoplankton and the subsequent use of oxygen by bacteria in oxidizing algal
carbon still require analysis and mathematical model construction. There is an obvious inter-
play here between the phytoplankton models and the DO models.
(3) Nutrient Models. These models form a subset of models that generally must be solved
together with models of phytoplankton growth and dissolved oxygen. Included here are various
nitrogen and phosphorous forms, interactions and significant sources and sinks. Bottom sedi-
ment exchanges and seawater intrusion effects are particularly important.
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(4) Models of Intermediate Levels of Food Chains. A variety of estuarine situations revolve
about interactions between water quality, phytoplankton and higher levels of the food chain.
There are virtually no models available which relate in a meaningful way the effects of waste
discharges on intermediate levels of estuarine flora and fauna. A typical example in this
regard is the lack of any quantitative model relating environmental conditions to the distri-
bution of opossum shrimp (Neomysis awatschensis), an important representative of the inter-
mediate level of the food chain in the San Joaquin Delta in California.
(5) Fishery Resource Models. There are significant water quality problems that interact
directly with Juvenile and adult fish populations. An example is the effect of dissolved
oxygen levels on the passage of anadromous fish such as the American Shad. Verified models
are required to describe the effects of waste reduction programs on the population dynamics
of such populations. Included in these models should be the effect of regional fishing pres-
sures, "natural" environmental variations (temperature, salinity, etc.), as well as the effects
of time-variable waste discharges and river flows.
(6) Chemical Kinetic Models, Group I. The modeling of complex reaction kinetics in estuarine
systems has rarely been attempted. The full nonlinear kinetic interactions superimposed on the
transient buffering capacity of seawater represents an area of water quality modeling that can
pose formidable conceptual, mathematical, and computational problems. In this first group, a
need exists to model such variables as iron, manganese, and various sulfur forms all of which
nay play important roles (such as interactions with bottom sediments) in the total water quality
distribution in estuaries.
(7) Chemical Kinetic Models, Group II. In this group, are included the acidic or alkaline
discharges from concentrated industrial wastes. These types of discharges interact with the
carbon dioxide system which is considered in this grouping. The C02 system also interacts
with the phytoplankton models discussed in (1) above. Again, these chemical systems involve
complex nonlinear equations, whose behavior is poorly understood. In addition, some of the
chemical reactions accompanying the discharge of acids, alkalis or other wastes are associated
with precipitates which may produce water use problems. The deposition of solids and partlcu-
late matter must therefore also be modeled in a workable, meaningful fashion (see also (9)
below).
(8) Near-Shore Waste Disposal. There is a complete absence of any unified mathematical model
approach toward describing the effect of a waste discharge in the near-shore environment
(generally the continental shelf limits). The increasing tendency to utilize the oceanic
environment for the disposal of municipal and Industrial waste, sludge, toxic chemicals, dredge
spoil, and solid wastes requires models of shelf circulation patterns, assimilation capacity in
terms of organic material and nutrient enrichment. These models will require a close inter-
action with viable hydrodynamic circulation models and biological and chemical models. The
problem context is generally multidimensional and will pose significant computational (and
conceptual) problems.
(9) Particulate Matter Models. As indicated in (7) above, some discharges of waste into the
estuarine or near-shore environment are accompanied by chemical precipitates, the distribution
and settling patterns of which must be modeled. The settling characteristics of dredge spoil
in the estuarine or shelf areas determine the ultimate bottom distribution of such spoil and
hence the effects on bottom flora and fauna. As a final example, the zone of Influence of
ocean outfalls on bottom life due to the distribution of sewage or sludge solids must be
modeled.
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(10) Gas Transfer Models (Air-Sea Boundary Phenomena). Several of the variables indicated
above, including oxygen, {X>2 and nitrogen all ultimately involve some type of gas transfer
phenomena at the air-sea boundary. Submodels are necessary to relate these exchanges to such
variables as wind speeds, wave conditions and salt content. Successful formulation of such
models would provide an important input for the more macroscopic models discussed above.
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REFERENCES
Bella, D. A., and W. E. Dobbins, 1968: Difference modeling of stream pollution. Proc. ASCE.
94, No. SA 5 (October), pp. 995-1016.
Bunce, Ronald E., and L. J. Hetling, 1966: A steady-state segmented estuary model. Tech.
Paper No. 11, FWPCA, U. S. Dept. of the Int., Mid. Atl. Reg., Charlottesvilie, Va.
Callaway, R. J., K. V. Byram, and ,G. R. Ditsworth, 1969: Mathematical Model of the Columbia
River from the Pacific Ocean to Bonneville Dam, Part I: Theory, Program Notes,
Programs. U. S. Dept. of the Int., F.W.Q.A., Pacific N.W. Water Lab., Corvallis, Ore.
DiToro, D. M., D. J. O'Connor, and R. V. Thomann, 1970: A dynamic model of phytoplankton
populations in natural waters. Env. Eng. & Sci. Prog., Manhattan College, Bronx,
N. Y. June, 1970.
Hagan, J. E., and T. P. Gallagher, 1969: A systematic approach to water quality problems--
a case history. Paper presented at 2nd Nat. Symp. on San. Engr., Research,
Development, and Design. SED, Ithaca Section, ASCE, Cornell University, July, 1969.
Hansen, D. V., 1967: Salt balance and circulation in partially mixed estuaries. Estuaries.
G. H. Lauff (Ed.), Pub. No. 83, Washington, D. C., AAAS.
Harleman, D. R., 1964: The significance of longitudinal dispersion in the analysis of pollution
in estuaries. Proc.. 2nd Int. Water Pollution Res. Conf., Tokyo, Japan.
Hetling, L., 1968: Simulation of chloride concentrations in the Potomac Estuary. Department
of the Interior, FWPCA, CB-SRBP Tech. Pap. No. 12, Mid. Atl. Reg.
Hetling, L. J., and R. L. O'Connell, 1966: A study of tidal dispersion in the Potomac River.
Water Resources Research. 2, No. 4, pp. 825-841.
Hetling, L. F., and R. L. O'Connell, 1968: An 02 balance for the Potomac Estuary. Chesapeake
Field Station, FWPCA, Dept. of the Int., Annapolis, Md. Working paper, unpublished.
Hydroscience, Inc., 1968: Mathematical models for water quality for the Hudson-Champlain and
Metropolitan Coastal Water Pollution Control Project. Prepared for Fed. Water Poll.
Control Admin. Hydroscience, Inc., Leonia, N. J. April, 1968.
Hydroscience, Inc., 1969: Nitrification in the Delaware Estuary. Prepared for Delaware River
Basin Commission. Hydroscience, Westwood, N. J. June, 1969.
Hydroscience, Inc., 1970: Interim report, development of water quality model, Boston Harbor.
Prepared for Mass. Water Resources Commission. Hydroscience, Inc., Westwood, N. J.
Jeglic, J., 1966: DECS III, mathematical simulation of the estuarine behavior. Analysis
Memo. No. 1032, Gen. Elec. Co., Phila., Pa.
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Leendertse, J. J., 1970: A water-quality simulation model for well-mixed estuaries and coastal
seas: Volume I, Principles of computation. Memo. RM-6230-RC, The RAND Corp.,
Santa Monica, Calif.
Manhattan College, 1970: Nitrification in natural water systems. Env. Eng. & Sci. Prog. Tech.
Kept., Manhattan College, Bronx, N. Y.
O'Connor, D. J., 1960: Oxygen balance of an estuary. Proc. ASCE. 86. No. SA 3 (May),
pp. 35-55.
O'Connor, D. J., 1962: Organic pollution of New York Harbor: theoretical considerations. J^
WPCF. 34, No. 9 (Sept.), pp. 905-919.
O'Connor, D. J., 1965: Estuarine distribution of non-conservative substances. Proc. ASCE,
91, No. SA 1 (Feb.), pp. 23-42.
O'Connor, D. J., 1966: An analysis of the dissolved oxygen distribution in the East River.
J. WPCF. 38, No. 11 (Nov.), pp. 1813-1830.
Orlob, G. T., R. P. Shubinskl, and K. D. Feigner, 1967: Mathematical modeling of water
quality in estuarial systems. Proc.. National Symposium on Estuarine Pollution,
Stanford University, California, Aug. 1967.
Pence, G. D., J. M. Jeglic, and R. V. Thomann, 1968: Time-varying dissolved oxygen model.
Proc. ASCE. 94, No. SA 2 (April), pp. 381-402.
Rattray, M., and D. V. Hansen, 1962: A similarity solution for circulation in an estuary.
J. Marine Res.. 20, pp. 121-33.
Shubinski, R. P., J. C. McCarty, and M. R. Lindoy, 1965: Computer simulation of estuarial
networks. Proc. ASCE. 91, No. HY 5 (Sept.), pp. 33-49.
Thomann, R. V., 1963: Mathematical model for dissolved oxygen. Proc. ASCE. 89, No. SA 5
(Oct.), pp. 1-30.
Thomann, R. V., 1970: Systems Analysis and Water Quality Management. Stanford, Conn.,
Environmental Sci. Serv. Pub. Co. To appear, 1970.
Thomann, R. V., D. J. O'Connor, and D. M. DiToro, 1970: Modeling of the nitrogen and algal
cycles in estuaries. Presented at 5th Int. Hater Poll. Res. Conf., San Francisco,
Calif., 1970.
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DISCUSSION
THOMANN: I think many of the points that have been raised already to some extent underlv the
philosophy of our approach. What we have done is to try to place an emphasis on sumnarizing
some of the more important kinds of water quality problems. We proceed from general steady-
state equations. These, to my mind, are still extremely valid approaches to many problems
that we face in water pollution control which are of a longer term, capital-improvement,
capital-intensive type problem. That is, people ask questions such as: if we want to
improve our level of treatment to a certain degree, what do you think is going to happen
over the long haul? So steady-state models still, I think, have a large part to play in
water quality modeling. We proceed from one-dimensional analyses, from constant-area
steady-state solutions (some one-dimensional analytical impulse-response solutions) into
conservative substances and some non-conservative substances (we emphasize again the fact
that the non-conservative nature of the system can be important) , and then go into con-
secutive reactions, which form the basis, of course, for the classical BOD-DO type models.
We indicate here, in a very general way, the interaction between one species, in
this case Cj in Equation (3.21), and how that acts as a source into a second species,
C2 in Equation (3.22). That kind of interaction is then built upon in the following
section on multistage consecutive reactions. The idea here is to try to display the inter-
action between different water quality variables without, at this stage anyway, intro-
ducing the complexity of time-variable phenomena or detailed types of dynamic considerations.
Equation (3.26), for example, is an illustration of a four-stage consecutive reaction
system in the context of nitrification, species N^ being organic nitrogen, N2 being
ammonia nitrogen, 1*3 being nitrite, and N^ being nitrate. Again, here we just simply
introduce the coupling between these equations as closeup simple first-order reaction kinetics.
The order of magnitude of those rate coefficients governs greatly how the system
responds and to what extent the feedforward reactions occur. These equations have no feed-
back in them at all (and later on in this chapter we discuss some of the possibilities of
introducing steady-state feedback). Essentially we know very little about these reaction
coefficients. For the simply coupled system, BOD-dissolved oxygen type system, we have a
fairly good idea of what is happening. We are not completely in the dark. We know at
least some order of magnitude of the possible responses that may occur due to carbon
sources, that is, oxidizable carbon being discharged, and the resulting dissolved oxygen
deficits. In the case of nitrogen, the only place I've really seen any substantial work
done in this area in the past has been in the Thames, Where they attempted to measure some
of the rates of ammonia oxidation. We have since done a fair amount of work on some
modeling on the Delaware and the Potomac where we tried to pin down some of those reaction
rates using available field data.
We have tried to place an emphasis on the kinds of interactive reaction kinetic
systems that are of importance in describing water quality problems. I think I'd like to
draw your attention to Equation (3.42), and subsequently Equations (3.43) and (3.44), to
highlight the importance of these reaction kinetics. Equation (3.43) has a diagonal matrix
introduced with a reaction term Kj , which is the classical deoxygenation rate. You see
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it's a multiplicator factor. It ends up multiplying everything that appears to the
right-hand side of that. Now in those matrices [A] and [B] is a conglomeration of
dispersion and advection effects, and that total matrix product that you see there responds
most sensitively to those reaction kinetics. There is an illustration of where it certainly
is of some importance.
Equations (3.43) and (3.44) display the kinds of systems that result from
consecutive-type reactions without any feedback, nitrification being an example, shown
in Figure 3.4. Now there's a four-system model, outputs of which are then fed into a
dissolved oxygen model to track four different forms of nitrogen, and the utilization of
oxygen in the oxidation of ammonia and nitrite. That is a fairly simple model of the
nitrogen system at this stage. As I said, it incorporates no feedback loops, nor does it
incorporate the fact that any of these nitrogen forms, especially the ammonia and the
nitrate, are nutrient sources for phytoplankton growth. But as a sort of a framework for
describing nitrification effects due to the discharge of large amounts of oxidizable
nitrogen, it has proven to be useful in describing the effects on the dissolved oxygen
deficit. And that, as has been pointed out, can be very, very substantial. The little
circles there are the reaction coefficients. Those are the ones that are somewhat diffi-
cult to pin down and, we believe, really require a fair amount of additional work.
We also make the point that we don't believe that the state-of-the-art at this
stage is up to rigorous statistical comparisons, generally, between output from some of
these water quality models and observed data. More often than not, our observed data are
somewhat limited, and there is undoubtedly at this stage an amount of judgment that is
still rampant in our analysis of whether, in fact, we have a valid model.
RATTRAY: I guess I have one concern, and that's about making a simplified, one-dimensional
hydrodynamic model and then putting in a whole bunch of time-dependent interactions, and
saying, "Aha! Our limitation in predicting what happens is due to the lack of knowledge
of reaction rates." And we've forgotten that that's the part that hasn't been simplified
to have just one number. You forget the fact that you're talking about an oversimplified
hydrodynamic model for many of the kinds of problems you might be concerned with. If you
put in a two- or three-dimensional hydrodynamic model, then the final result might be quite
dependent upon some of the parameters in that kind of a model. What would happen with an
oil spill Interacting with the upstream migration of a biological population which stayed
in the lower layer? You could know the reaction rates perfectly, but if you didn't know
how much the populations overlapped, you couldn't tell how they interacted. So there are
cases where you have to know how much of the time these different kinds of concentrations
are overlapping, and at what concentrations they are overlapping. 1 think we can't neglect
the fact that some of the hydrodynamics will be important even though we need to know
reaction rates, too. So this is just adding a little bit, I think, to what was said.
THOMANN: I think that that reinforces the point that when one runs into the kind of situation
you just mentioned, one most obviously has to go to a more complicated hydrodynamic situa-
tion, taking cognizance of vertical or lateral gradients.
RATTRAY: Agreed.
THOMANN: I guess what we are responding to, again, is the general order of most of the water
quality problems that seem to be in existence today. Oil spills, there is no question that
it's a water quality problem. I don't know whether we've ever developed a state-of-the-art
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to deal with that kind of situation, since this was a state-of-the-art report. There are
no models that we know of for dealing with oil spills.
HARLEMAN: Basically, the problem of the nonadvective tidal models, which is one of the
problems which hasn't been brought up, is the longitudinal fogging which these introduce.
That is, by in a sense ignoring the tidal motion you somehow automatically introduce a
longitudinal fogginess which is of the order of magnitude of the tidal excursion, which
is of the order of magnitude of five to ten miles in most estuaries. When we come down
to talking about local problems, certainly this kind of fogginess is going to introduce
difficulties. We get one thing if we talk about low water slack, and we get something
else if we talk about high water slack. Again, when you come to the question of sampling,
the problem of what you sample and how you sample must first be determined by, obviously,
what kind of model you are going to use the data in. And in many ways a short period of
sampling, fairly intense, is much more adaptable to the so-called real time models because
you can deal with quantities taken over a relatively short time and hopefully make some
sense out of it. But if you are going to use a freshwater advection model then you need
to really do a lot more sampling because of this longitudinal fogginess, which is really
in there. 1 think in some ways this has been largely overlooked. If you ignore the tidal
motion then, really, how can you pin down quantities within a distance of a tidal excursion?
And at the same time, I don't follow really Dr. Thomann's argument that when you are deal-
ing with large reaction coefficients, that is, when things are pretty much controlled by
large decay terms, that you can then dispense with worrying about the tidal motion, because
it seems to me that then you need to be more concerned with real time behavior. Just think
about time of travel. The particle of water is moving up and down the estuary very large
distances every tidal period.
THOMANN: What I meant to imply is that with higher reaction rates the system is more sensi-
tive to those reaction rates than it is to dispersion.
HARLEMAN: It is also sensitive to real time and not twelve-hour averages.
THOMANN: But all of our problems need not be solved on a real time basis, or what we are
calling a real time basis. There are any number of problems with which we are confronted
that are the kinds of seasonal-type phenomena that don't require that kind of time scaling.
1 think that's the point that we are getting at. In those kinds of seasonal, quasi-steady-
state situations, we can turn that dispersion knob an awful lot in one direction or the
other. For higher reactive systems it is not going to make a significant difference. That
was the point.
PRITCHARD: Well, with regard to this discussion of being concerned with what happens in time
and space over scales of the order of a tidal excursion or less, or whether or not we can
be concerned with a grosser picture, again I've got to draw on my own experience in a
system where quite frequently it depends on whether we are looking at close-in distribu-
tions or distributions well away from the source. If you happen to be in a small waterway
where twelve miles is most of the waterway then you ought to be concerned with the tidal
time everywhere. But if you're concerned with a large waterway, there you may only be
interested in real time in a close space within a certain distance up and down from the
outfall.
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THOMANN: What we seem to be saying is that when the problem demands it, one goes to a small
time and small space model. I think my point is that many of the problem contexts in which
we find ourselves as far as pollution control is concerned are generally not of that type.
They are generally more of the sort of longer term, capital intensive. If we do this
generally to our waste load discharges in this area, what do we expect to have happen?
FRITCHARD: Well, I think that depends on where you are in the system decision. Maybe that's
what FWQA has to consider overall, but not in a day-to-day response to an actual decision.
For instance, in my case, does the state approve water use of this particular plant? I've
got to know what it does locally as well as what it does on a large scale.
HARLEMAN: I still think there's a fallacy of assuming in going to more computation that this
is only a question of whether it's warranted from the whole time span. I still maintain
that by closing down on the time scale one gets a better computational scheme and less
adjustable coefficients. Even if your interest extends over a year, it may well pay off
in the end to spend more time on the computer to generate a lot more time data than you're
really every going to use, because by doing this you avoid introducing arbitrary unknown
coefficients which take up the thing that you haven't spent the time computing. It's not
really a question of just deciding on the time span. It may well pay off to do it on a
small time scale and spend more time on the computer and only use the results once a day
that you get out.
THOMANN: Yes. I think, as you pointed out, that it's essentially the same. I still see it
as a trade-off as to whether in fact it's worthwhile in any particular problem context to
throw the computational time associated with the real time model into a dispersion coeffi-
cient, an over-a-tidal-cycle dispersion coefficient, on which the problem solution doesn't
depend to a large extent anyway. That's the point. It's not completely dependent on that
coefficient. And it may very well prove worthwhile to make that trade-off.
I think part of the problem revolves around how one looks at the time scales that
are of importance. If there is a local, within-tidal time phenomenon, as just pointed
out, then, obviously, yes. There are a number of problems, however, where seasonal effects,
yearly effects, and we might be concerned with running these models for periods of ten,
fifteen or twenty years. Then working on an hourly time frame, it seems to me, would not
be the way to go. Suppose I want to estimate the effects over a fifteen or twenty year
flow hydrograph of ways to control,'and what I expect to happen to the water quality.
Then would we do that on an hourly'time step? Probably not. At least I don't think so.
PRITCHARD: Well, if the dynamics required it, I think you'd have to.
THOMANN: Well, the point there is that the time gradients, in a sense, of the seasonal
fluctuations and weekly fluctuations, are probably orders of magnitude greater than what's
happening over a tidal cycle. For example, take dissolved oxygen, which fluctuates from
fourteen, twelve, thirteen mg/1 down to zero. Over a year you don't get those kinds of
time fluctuations over a tidal cycle.
WASTLER: Well, you could get a fair percentage of them over a tidal cycle.
THOMANN: Only if you advect from complete saturation and go right past a zero DO location.
That would be the only way, and that would be an extremely sharp spatial gradient.
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PRITCHARD: There are a lot of regions, eutrophic regions, where you have die-off at night and
the DO goes from supersaturation to pretty low values overnight.
THOMANN: In which case one runs that kind of model.
PRITCHARD: Yes. Agreed.
RATTRAY: We've got to worry a little bit about differentiating space scales and time scales.
Tidal motion has not only different time scales, but space scales. In some cases the
problem is the fact that they are different space scales and in others it's time scales.
DOBBINS: The things that need more work than anything else are the mechanisms, the assimi-
lation of the various types of inputs. We don't know enough about these inputs, the
interrelationships between these inputs, and I think, if we are going to use these things
for predictive purposes, it is very important that we clarify some of these processes.
These linear equations assume that if we take the individual effect of any particular
input and we remove that input from the system then its effect will be removed. The only
trouble is nature doesn't necessarily act that way. And when we remove some inputs, we
change the ecological system and make it trigger new inputs. A very crude example. If
you take the silt out of a polluted river, the water doesn't turn clear. It turns green.
These are problems for which we need some answers before long, if we are going to avoid
spending billions of dollars and finding out that we're not accomplishing what we hoped
to accomplish. The City of New York, for example, has under design treatment plants that
will cost over a billion dollars to build. A very serious question in my mind is whether
or not this expenditure is going to make the changes some people are currently anticipating.
It's a very, very serious question.
And so we are interested in the water quality and not just the hydrodynamics of
these systems. The assimilation mechanisms.
PRITCHARD: It seems to me that we are a lot further ahead in the physics than we are in the
biology.
DOBBINS: That's right.
PAULIK: As a biologist, I'm happy to see engineers at least recognize the problem. This
should receive high priority in the future. We should begin to look at the sources and
sinks, and the non-conservative properties of the system.
DOBBINS: It's not so much a question about the reaction rates; we don't even know what
reactions take place. The type of cycles. We can't even draw diagrams.
THOMANN: Well, we're not completely ignorant. It's Just that we're not nearly as sharp as
we are in the hydrodynamic area. I think there is an order of magnitude of difference
between the level of understanding there and in the water quality.
173
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CHAPTER IV
ESTUAR1NE TEMPERATURE DISTRIBUTIONS
John Eric Edinger
1. NATURAL TEMPERATURE DISTRIBUTIONS
Water temperature observations are an integral part of almost every type of estuarine
study. Gas and salt solubility, biological and chemical reaction rates, and density and vis-
cosity are all functions of water temperature. Water temperature has been an ancillary
variable. Despite the wealth of data that might be available, a systematic description of the
spatial and temporal distribution of temperature has not, until recently, been an important
part of estuarine hydrography.
Salinity dominates the density structure of an estuary. The density difference due
to a gradient of 1 o/oo of salinity is about equal to that produced by a 7°F temperature dif-
ferential. In the formulation of large-scale estuarine circulation, the density and buoyancy
terms in the equations of motion are adequately described by relation to the salt concentration
and its conservation expression. It has not been necessary, fortunately, to develop the heat
budget and consequently the spatial and temporal temperature distribution for inclusion in the
momentum balance of an estuary. The influence of temperature on density structure becomes
important locally, in the vicinity of advective heat sources such as steam electric power plant
condenser cooling water discharges and in the description of buoyant convection near the water
surface particularly as related to surface heat exchange.
Circulation and mixing within an estuary are described and classified by the salinity
distribution resulting from the Interaction of a freshwater inflow at the head of the estuary
and a high salinity source at the ocean end (Pritchard 1955). This description is related to
the dominant advective and dispersive terms in the salt conservation equation. A classification
of estuarine temperature distributions can be developed within the classification scheme of
estuaries based on salinity by assuming that the advective and dispersive processes that are
dominant in the salt conservation equation for a particular estuary are also dominant in the
heat conservation relation describing the temperature distribution. In addition to the dominant
advective and dispersive processes, it is necessary to include surface heat exchange and atten-
uation of short-wave radiation with depth in the heat conservation equation.
1.1 SIMILARITY BETWEEN SALINITY AND TEMPERATURE DISTRIBUTIONS
When surface heat exchange and artificial heat sources are a small fraction of the
estuarine heat budget, the temperature distribution, like the salinity distribution, is
174
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controlled by the temperature of the freshwater Inflow at the head of the estuary and conditions
at the ocean end. Salinity is generally higher at the ocean end of the estuary than at the
freshwater end, but temperature can be higher at the freshwater end than at the ocean end, as
during a summertime condition, or lower at the freshwater end than at the ocean end, as during
a wintertime condition. When the freshwater inflow temperature is less than the ocean tempera-
ture, the temperature will increase from the head of the estuary to the mouth and increase from
the surface downward as does salinity. When the freshwater inflow temperature is greater than
the ocean temperature, temperature will decrease from the head of the estuary to the mouth and
decrease from the surface downward.
The similarity between the salinity distribution and temperature distribution can be
expressed in terms of a temperature differential where the base temperature is taken as the
freshwater inflow temperature Tr such that zero temperature differential corresponds to zero
salinity. The ocean salinity S0 would correspond to the maximum temperature differential,
which is T0 - Tr where T0 is the ocean temperature. The relationship between the tempera-
ture T(x,y,z) and the salinity S(x,y,z) for any point in the estuary can be stated at least
as a proportionality as
[T(x,y,z) - Tr] / [T0 - Tr] .. S(x,y,z)/S0 (4.1)
which applies to either condition of the freshwater inflow temperature being greater or less
than the ocean temperature. This similarity relation is stated for steady-state conditions
because the freshwater inflow temperature Tr will vary independent of the circulation within
the estuary. Simultaneous temperature and salinity distributions presented by Owen (1969) and
reproduced here in Figures 4.1 and 4.2 qualitatively demonstrate the similarity relation.
Figure 4.1 shows a wintertime condition with Tr < TQ and Figure 4.2 shows a summertime condi-
tion with Tr > T0 .
Any difference between an observed temperature distribution and a temperature
distribution determined by similarity to salinity should be attributable to surface heat
exchange, attenuation of short-wave radiation, groundwater inflow and advective sources other
than freshwater inflow. Similarity between salt and conservative temperature distributions
may also differ if the diffusivities for salt are not the same as for heat.
1.2 INFLUENCE OF SURFACE HEAT EXCHANGE ON TEMPERATURE DISTRIBUTIONS
The freshwater inflow and ocean water mixed to the surface layers of an estuary are
heated or cooled by exchange of heat across the water surface. Inclusion of surface heat
exchange in the description of estuarine temperature distributions complicates in particular
the longitudinal distribution of surface temperature and, consequently, the vertical tempera-
ture distribution.
Heat is exchanged across the water surface by shortwave solar and longwave atmospheric
radiation HS + Ha , radiation reflection Hsr + Har , longwave back radiation Hjjr , evapora-
tion He , and conduction Hc . The net rate of surface heat exchange H,, is defined as
Hn - (H. + H. - H.r - Har) - (Hbr + Hg + H,.) (4.2)
175
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£ '5 —
ZI| HOUR SALIHIIY (%>.) AVERAGE
DEC 3-7. 1952
VERTICAL EXAGGERATION: 900 TIHES
30
35
18 !6 Hv
PATUXENT RIVER
DISTANCE FROM MOUTH IN NAUTICAL MILES
I I
10
E is
20
25
30
35 -I -I I
2k HOUR TEMPERATURE AVERAGE
DEC 3-7. 1952
VERTICAL EXAGGERATION: 900 TIHES
I I I I .iv I I t
I .... J
30
28
26 24
22 20 IB 16 14' 12 10
PATUXENT RIVER
DISTANCE FROM MOUTH IN NAUTICAL MILES
Fig. 4.1 Mean Salinity and Temperature in
Patuxent Estuary: Winter. From
Owen (1969).
176
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10
15
£ 20
bj
a
25
30
24 HOUR SALINITY (°/v>) AVERAGE
JUNE 23-28. 1952
VERTICAL EXAGGERATION: 900 TIMES
35 *-t ti
i
r- I §
30 28 26 2k 22 20 18 16 1U 12 10
PATUXENT RIVER
DISTANCE FROM HOUTH IN NAUTICAL MILES
to
" 15
Ul f
20
25
30
2I» HOUR TEMPERATURE AVERAGE
JUNE 23 - 29, 1952
VERTICAL EXAGGERATION: 900 TIMES
*.r..:...;..JL i
-'\
I
i -r
JO 28 26 2
-------
When back radiation, evaporation, and conduction are expressed in terms of their dependency on
water surface temperature TS and meteorological variables, the net rate of surface heat
exchange can be expressed as
Hn = -K(Tg - E) (4.3)
where K is the coefficient of surface heat exchange and E is the equilibrium temperature
(Edinger and Geyer 1965, Edinger et al. 1968). A discussion of the properties of the equilib-
rium temperature is given by Edinger (1969) and recent simplified formulations for it based on
shortwave solar radiation, the surface heat exchange coefficient and dew point temperature
have been presented by Brady et al. (1969).
In order to determine the influence of surface heat exchange on the temperature
structure of an estuary, it is necessary to consider each case of the relation between fresh-
water inflow temperature Tr , ocean temperature To , and the equilibrium temperature E .
The six possible relationships between these three variables are shown in Figure 4.3. Each
case approximates a different season if it is assumed that the ocean temperature remains rela-
tively constant compared to the freshwater inflow temperature. The equilibrium temperature
will follow a seasonal cycle because of the strong dependency on shortwave radiation and air
temperature. The freshwater inflow temperature will lag the seasonal cycle of equilibrium
temperature, and can be less than the ocean temperature in the winter period and greater in
the summer period.
Longitudinal temperature profiles resulting from the inclusion of surface heat
exchange in the description of temperature distributions for a salt-wedge estuary are shown
in Figure 4.3. The salt-wedge case is the simplest to consider since the lower layer tempera-
tures would be almost uniform longitudinally and determined by the ocean source. The dominant
vertical transport mechanism is an advective flux from the lower layer toward the surface.
Beginning in the early fall (Tr > E > T0) , the freshwater inflow tends to cool due
to mixing with the colder ocean water and due to surface cooling as it approaches the equilib-
rium temperature. Since the lower layer temperature is below the equilibrium temperature it
will tend to increase as ocean water is advected vertically to the surface. The result is a
temperature minimum at the river end of the estuary due to combined cooling and mixing and a
temperature maximum due to heating near the mouth of the estuary. In late fall and winter
(Tr > TQ > E and To > Tr > E) both the freshwater inflow and the ocean water tend to cool and
tend to approach equilibrium temperature. The net result is a temperature minimum in the longi-
tudinal distribution of the surface temperatures. In early spring (To > E > Tr) the surface
temperature would tend to be continuously increasing from the head of the estuary to the mouth.
This results from the freshwater inflow heating up to approach equilibrium and the ocean water
tending to cool. In late spring and summer (E > To > Tr and E > Tr > To) both the freshwater
inflow and the ocean water tend to warm up. This leads to a temperature maximum within the
estuary.
The partially mixed estuary is characterized by a longitudinal gradient of salinity
in the lower layer. This is due to vertical flux from the lower layer to the upper layer by a
combination of vertical advection and turbulent exchange of salt. The lower-layer temperatures
decrease from the mouth of the estuary to the head of the estuary when T < TQ and increase
from the mouth to the head when Tr > T0 . When the lower layer is wanner than the surface
layer there is a vertical turbulent flux of heat upward in the same direction as the turbulent
178
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RELATIVE ORDER
ANOE
POSSIBLE
TIME OF
OCCURANCE
LONGITUDINAL PROFILE OF
UPPER AND LOWER TEMPER-
ATURE DISTRIBUTIONS (FOR
TYPE-A SALTWEDGE ESTUARY)
INFLUENCE
ON VERTICAL
STRATIFI-
CATION
EARLY FALL
(-0
LOWER
HEAD
LONG.WST.-» OCEAN
Tr>le>E
LATE FALL
HEATT
LON6.DIST.-w OCEAN
WINTER
W
»-
\
HEATT
j- LOWER
™
LONG. DIST.-fr
OCEAN
LOWER
To>E>Tr
EARLY
SPRING
HEAD
LONG. OIST.-» OCEAN
E>Tr>T0
LATE
SPRING
MS:
LONG. OIST.— HCEAK
E> Tr
SUMMER
HEAD
OCEAN
Fig. 4.3 Tendency of Estuarlne Upper and Lower Temperature
Distributions When Influenced By Freshwater Inflow
Temperature (Tr); Ocean Temperature (To); and
Equilibrium Temperature of Surface Heat Exchange (F).
salt flux. This condition exists when T0 > Tr or for the winter to late spring sequence of
Figure 4.3. The tendency is for less heat to be transported vertically near the head of the
estuary than at the mouth, hence the river end of the estuary responds more to the equilibrium
temperature (i.e., surface heat exchange) than to the ocean temperature.
When the surface layers are warmer than the bottom layers in the partially mixed
estuary the turbulent vertical flux of heat is downward and opposite to the vertical flux of
179
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salt. The vertical flux 01 heat downward tends to decrease the longitudinal gradient of tempera-
ture in the lower layer. This condition can exist for all but the winter and early spring
periods shown in Figure 4.3. The times of possible temperature maxima and minima in the longi-
tudinal distribution of surface temperature should be no different for the partially mixed
estuary than for the salt-wedge estuary.
The vertical stability of an estuary is increased due to heating, over that resulting
from the salinity distribution alone, from the late spring period through the early fall period.
The trend toward cooling during the late fall through the early spring tends to decrease verti-
cal stability. The differences in surface and bottom salinity for the salt-wedge estuary are
quite large and it is unlikely that the density difference due to salinity will be negated by
the density difference due to temperature. The vertical salinity differences are smaller for
the partially mixed estuary than for the salt-wedge case, hence there might be an effect of
temperature differences on stability, particularly during the winter and early spring periods.
1.3 ADDITIONAL INFLUENCES ON TEMPERATURE STRUCTURE
Two Important factors that must be considered to fully describe the vertical
temperature structure of an estuary are attenuation of shortwave radiation within the water
column and buoyant convection (or vertical motions due to density instability) at the water
surface. The net rate of heat exchange across the water surface must be related to the
processes by which heat is transferred up or down the water column. When the water column can
be considered vertically mixed throughout the total depth, or even through a finite identi-
fiable depth in a stratified waterbody, the details of these two processes can be ignored.
They must, however, be considered in a proper formulation of stratified temperature structure.
Shortwave radiation is attenuated in the water column such that the rate at which it
passes through a unit-area horizontal plane decreases exponentially with depth and is described
by a modification of Beer's Law as
H8(x3,t) - Hs(t) • (1 - 0)exp(-ox3) (4.4)
where Hs(t) is the rate at which the net shortwave radiation crosses the water surface and is
a function of time depending on the angle of the sun and cloud cover, a is the attenuation
coefficient and is a function of turbidity and chemistry of the water, Hs(x3,t) is the rate
at which shortwave radiation passes a plane at depth x3 , and 6 is the fraction of Hs(t)
immediately absorbed at the surface (Dake and Harleman 1969). The rate of heat absorption due
to shortwave radiation attenuation in a unit volume of fluid at depth x3 below the water
surface is
Mx3,t) - -|^ Hs(x3,t) - Hs(t) • a • (1 - B)-exp(-c«3) (4.5)
Normally In an estuary the attenuation coefficient a is thought to be so large that most of
the shortwave radiation is absorbed near the surface. However, if a is small and if the
estuary is sharply stratified, as for example a salt-wedge estuary, only a small portion of
the shortwave radiation absorbed in the deeper layers is transported back to the surface. The
lighter surface layer can create a greenhouse effect which results in a very high temperature
in the lower layer relative to the surface temperature during the winter months.
180
-------
In order to relate surface heat exchange to a continuously varying vertical tempera-
ture structure it is necessary to have a boundary condition at the surface accounting for the
vertical heat flux into the water column. The necessary condition can be stated as
- P)H8(t) = pcpK3 - I -bc (4.6)
where H,, - (1 - 6)Hs(t) is the difference between the net rate of heat exchange at the water
surface and shortwave radiation absorbed within the remaining water column, and represents the
amount of heat that must be transported away from or to the water surface by vertical disper-
sion and by a vertical heat flux bc due to buoyant convection. If buoyant convection at the
surface is Ignored in computations of vertical temperature structure accounting for surface
heat exchange, then very steep surface temperature gradients would be produced when H,, > Hs(t).
For the condition that % < Hs(t) there would exist a positive temperature gradient at the
surface (i.e., the temperature would increase with depth below the water surface). Evaporation
at the water surface tends to increase the surface salt concentration, hence surface density,
and takes place during either net heat gain or net heat loss across the water surface. Phillips
(1966) details possible directions toward a description of the buoyant convection process at
the water surface and indicates that formulation of the problem is far from complete.
Dake and Harleman (1969) have formulated the vertical temperature structure for
fresh water resulting from surface heat exchange, attenuation of shortwave radiation, vertical
dispersion and vertical convection. The convectlve process is accounted for by introducing a
computational procedure that amounts to an internal heat balance through a finite depth below
the water surface. The procedure requires that just enough heat be brought up to the surface
for the temperature at one depth to always be equal to or greater than the temperature at a
lower depth.
181
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2. ANALYTIC DESCRIPTION OF ESTUARINE TEMPERATURE STRUCTURE
Derivation of a three-dimensional time-varying heat balance per unit volume of fluid
produces a continuity relation which is identical to the salt balance in the storage, advection,
and dispersion terms. An additional source term must be included to account for radiation
attenuation below the water surface and to indicate that heat is generally non-conservative
relative to the advection and dispersion processes. The differential expression for tempera-
ture distribution thus Is
3T 3T ST a 3T d_K 3T _ 3 5T
(4.7)
where T is the temperature at point x^.Xj.x^ and time t ; u^ , u2 , and u-j are the
velocity components; Kj , K2 , and Kj the eddy coefficients for heat which may differ from
those for salt; p and cp are the specific weight and heat of the fluid; and hp(xi,x2,x3,t)
is the internal source or sink term. Spatial averaging and application of the concept of
homogeneity can be applied to the heat balance in the same manner as for a salt balance.
The problem of including heat as a non-conservative property in estuarine models is
one of relating the waterbody temperature structure to surface temperature, hence to surface
heat exchange. Three models of progressively increasing complexity in vertical structure are
chosen here to illustrate how they might be adapted as temperature models. First is the verti-
cally homogeneous and sectionally homogeneous temperature distributions as adapted from the
similar salt models of Pritchard (1958). Second is the two-layered segmented model (Pritchard
1967). Third are models that might be used to describe a continuously varying structure such
as developed by Hansen and Rattray'(1965).
2.1 THE VERTICALLY MIXED CASE
The assumption of vertical homogeneity leads directly to an expression relating
internal temperature structure to water surface temperature. Thus at a point (xj,x2) in
the surface coordinate plane, the vertically homogeneous heat balance can be written as
-— —• 0X1 vXO 0X1 » 0X1 * QXo \ QAO * P n
where u^ and u2 are the vertically homogeneous longitudinal and lateral velocity components
and KI and K2 are the vertically homogeneous longitudinal and lateral dispersion coeffi-
cients. The term K(T-E) is the net rate of surface heat exchange where K is the coeffi-
cient of surface heat exchange, E is the equilibrium temperature, and T is the surface
182
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temperature which is the same as the temperature described in the storage, advection and
dispersion terms. As for the vertically homogeneous salt distribution, given the velocity
components and dif fusivities, and necessary boundary and initial conditions, Equation (4.8)
can be used to determine a temperature distribution. It should not be implied, however, that
an estuary in which the salt distribution is adequately described by a vertically homogeneous
relation will also have a vertically homogeneous non-conservative temperature distribution.
The sectionally homogeneous description also provides a direct relation between
internal temperature structure and water surface temperature. The time-varying longitudinal
temperature distribution for the sectionally homogeneous case including surface heat exchange
can be written as
where the surface heat exchange term includes the surface width of the estuary w(xpt) as a
function of longitudinal distance and time. The coefficient of surface heat exchange K and
the equilibrium temperature E are both functions of time and distance along the estuary. The
longitudinal temperature profile T(x^,t) for the sectionally homogeneous case is determined
by integration of Equation (4.9) after specification of the geometry, o^(x1,t) and w(x^,t) ;
the sectionally homogeneous velocity u^(x^,t) ; and a description of the longitudinal dis-
persion coefficient, the coefficient of surface heat exchange, and the equilibrium temperature.
The boundary conditions are T - Tr(t) at the head of the estuary and T = T0 E > T0) .
The surface heat exchange condition of the vertically and sectionally homogeneous
cases also holds for the segmented form of these vertically mixed models.
2.2 TWO-LAYERED SEGMENTED MODELS
Incorporation of surface heat exchange in a two-layered segmented model as used by
Pr it chard (1967) is possible if it is assumed that the heat transfer process of back radiation,
conduction, and evaporation are a function of the upper layer segment temperature. The salt
balance is extended to a heat balance by adding to the relationships of the upper layer the
terms for surface heat exchange. Following the notation of Pritchard (1967), shown here in
Figure 4.4, the steady-state heat balance for the upper-layer n th segment is
fun-l+un\ . ,_ , f untnl . „
nl 2 > n
,_ . f(Tu)n+(Tu)n+n VJWf.. . _l
-------
n-t
INII
'
'°u >
u ium
v
(CL )
(Tu
(1>
n*ui-
)„ -
Vn -
)n -
'n "
E,, -
K.
VOLUME RATE OF NET FLOW FROM THE (n-1)
SEGMENT TO THE nth SEGMENT . UPPER LAYER.
VOLUME RATE OF NET FLOW FROM THE nth SEGMENT
TO THE (n+l) SEGMENT, UPPER LAYER.—
VOLUME RATE OF NET FLOW FROM THE (n+l )
SEGMENT TO THE nth SEGMENT , LOWER LAYER.
VOLUME RATE OF NET VERTICAL FLOW FROM THE
LOWER LAYER TO THE UPPER LAYER , nth SEGMENT .
VERTICAL EXCHANGE COEFFICIENT, nth SEGMENT.
TEMPERATURE OF UPPER LAYER, nth SEGMENT .
TEMPERATURE OF LOWER LAYER, nth SEGMENT .
EQUILIBRIUM TEMPERATURE AT nth SEGMENT.
COEFFICIENT OF SURFACE HEAT EXCHANGE AT nth
SEGMENT. —
A,, - SURFACE AREA OF nth SEGMENT .
Fig. 4.4 Notation For TWo-Layered Segmented
Temperature Model. After Prltchard
(1967).
and for the lower-layer n th segment the heat balance is
(4.11)
Rearranging the upper-layer segmented heat balance in such a form that the segmented tempera-
tures are separated as linear variables gives
' 2
-------
Conceptually, the advection terms (Qu)n_ifll > (Qu)n,tH-l and ^v^n can be determined from
the salt distribution and then utilized in the temperature equations. Specification of the
boundary temperatures at the head and ocean end of the estuary, along with Equations (4.10)
and (4.11) gives 2n + 2 equations from which to determine the same number of unknown segmented
temperatures. The vertical mixing coefficient Vn can be changed for heat from that found for
salt.
One feature of the two-layered segmented salt balance is that it requires no speci-
fication of the segment geometry, although it does require judicious selection of the salinity
values that apply at the segment boundaries when evaluating the advection terms. Extension to
a temperature model requires specification of the surface area of the upper layer segments.
The influence of surface heat exchange on the two-layered temperature structure can
be determined by comparison of the term K^Ajj/pCp numerically with Vn and with either
(Qu)n.1)It or (Qu)njn+i in Equation (4.12). If K^/pCp is small in comparison with both
of these terms then the temperature distribution is controlled by the end conditions, as is
salinity, and the similarity relation between temperature and salinity can be employed. The
lower n th segment heat balance. Equation (4.11), can be extended by including a term for
absorption of shortwave radiation, and the vertical advection term (Qv)n ma7 require modi-
fication to account for buoyant convection.
2.3 CONTINUOUS VERTICAL TEMPERATURE STRUCTURE
In each of the cases discussed above, the surface temperature could be explicitly
related to a temperature in the advective and dispersive terms. This is not possible in cases
where the temperature structure varies continuously through the vertical.
Relating the source term hp in Equation (4.7) to the rate of heat absorption due
to attenuation of shortwave radiation, Equation (4.5), gives the heat balance for a volume
element below the water surface (xj > 0) . Integrating Equation (4.7) over the depth of the
water column h(x^,x.,t) and utilizing Equation (4.6), where the buoyant convection term bc
is related to the vertical advective flux at the water surface gives
h(x1,x2,t)
PC I IS + u a_ + _ . _ (K _, . _ (K _)& . -K(T-E) (4. 13)
z
1,2,
I (IS + u ar_ + ar_ . ^_ (K «_, . ^_ (K *!_
J ldt L axj^ z ax2 ax^ L axj ax2 z ax2
where TS is the surface temperature. Equation (4.13) is the most general heat budget that can
be written relating the vertical temperature structure to surface heat exchange and attenuation
of shortwave radiation below the water surface. Applying the concept of vertical homogeneity
to Equation (4.13) reduces It to Equation (4.8).
The integral condition of Equation (4.13) requires knowing the internal temperature
structure for its application; yet it must be satisfied to determine the temperature structure.
At the same time the temperature gradient below the water surface must satisfy the buoyant flux
relation of Equation (4.6), and attenuation of shortwave radiation, Equation (4.7). This sug-
gests that determination of a continuously varying vertical temperature structure which is
strongly related to surface heat exchange will require an iterative computation.
185
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The iterative computations required to determine the internal heat balance of an
estuary can be initiated from a conservative temperature distribution based on similarity to
the salt distribution. Vertical salinity structure as predicted, for example, by the relation-
ships developed by Hansen and Rattray (1965) would provide an initial temperature distribution.
Relaxation of the conservative temperature distribution to satisfy Equation (4.13) could proceed
using an empirical computation scheme similar to that of Dake and Harleman (1969).
It is obvious that the temperature structure of an estuary can be predicted in no
more detail than the salinity structure. Further, the detail with which the vertical tempera-
ture structure can be predicted in an estuary is clearly limited by the ability to account for
buoyant convection near the free water surface.
186
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3. MODELING TEMPERATURE DISTRIBUTIONS
DUE TO LARGE HEAT SOURCES
The modeling of the effects of large heat sources may be conveniently classified
according to three spatial scales. First is the localized region in the immediate vicinity of
the discharge and intake structures, which encompasses on the order of hundreds to thousands
of feet. The temperature distribution in this region is principally controlled by the dis-
charge configuration which can assume many forms depending upon the design constraints and the
desired effects. The second scale for heated discharges could be named the intermediate region
or transition region. This is the region in which currents and eddies within the receiving
waterbody mix the heat from the localized region of the discharge to where its distribution is
governed by the larger scale circulation, and typically extends over thousands of feet to tens
of thousands of feet. The third scale considers the influence of the heat source on the large-
scale temperature distributions, as discussed in Sections 1 and 2.
The present (1970) semiemplrical basis for analytically describing the temperature
distributions in each of these three regions is discussed. It is necessary to first introduce
the concept of temperature excess as developed from the heat conservation relations. Also, a
brief discussion of the application of physical hydraulic models to the study of large thermal
sources is given.
3.1 THE TEMPERATURE EXCESS
The temperature excess can be defined for a single heat discharge as the difference
between the temperature distribution with the discharge and the temperature distribution with-
out the discharge (Fritchard and Carter 1965) . This definition is made more precise from an
analysis of the general heat conservation relation, Equation (4.7). Let T(x1,x2,x3,t) be
the temperature related to the internal heat source term h-Cxj^Xj.Xjjt) and a given set of
boundary and initial conditions. An additional heat source will result in a new total internal
source of h'(x^,X2,X3,t) and temperature T'(x1,x2,x3,t) . The temperature excess is then
defined as
9(xi,t) - T'(xlft) - T(Xl,t) (4.14)
Equation (4.7) would hold for T'(xj,t) as well as for T(x^,t) by inclusion of the additional
heat source, as long as the velocity field U£(xi,t) and the dispersion coefficients Kj(xj,t)
remain unchanged. Then the expression for the conservation of excess temperature becomes
ae j- „ ae j. „ 86 a (K ae
1x + U2 w + U3 - -1K1 S
(4.15)
187
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Only the spatial and temporal distribution of the additional heat source (hp - hp) is required
to Integrate Equation (4.15), and the details of radiation attenuation and surface convection
can be. ignored.
When the definition of excess temperature is applied to the sectionally homogeneous
relation, Equation (4.9), it yields
I + (U| 06) —("Di ——;I ~ - fl (4.16)
Lat 8xj L axj» •*• axj'-* PC-
Equation (4.16) is Independent of the equilibrium temperature E and consequently much of the
detailed meteorological data required for evaluation of surface heat exchange processes.
The temperature T(x^,t) can be considered the ambient temperature relative to the
temperature excess 8(x£,t) . The conservation expression for excess temperature, Equation
(4.15), is valid in a region of the waterbody for which the spatial gradients of ambient
temperature dT/dx^ are small in comparison to the spatial gradients of excess temperature
38/axj . This condition is usually satisfied in the immediate vicinity of a large heat source
where the consistency of velocity field and dispersion field is violated. Linearity of the
equation for temperature excess permits the separate determination of the temperature excess
distribution for each individual thermal discharge and superposition of the results.
3.2 INITIAL TEMPERATURE DISTRIBUTIOHS
The temperature distribution in the immediate vicinity of a discharge is a function
of the discharge structure design. The advective and dispersive field transporting heat in
this region is determined by the momentum and buoyancy of the discharge, as these dominate the
advective and dispersive properties of the receiving waterbody. Currents and stratification
in the receiving waterbody eventually dictate the limits of initial mixing.
A classification of initial «<»
-------
Table 4.1 Classification of initial mixing conditions and
representative studies.
DISCHARGE CHARACTERISTICS
Surface
Submerged
CHARACTERISTICS
ERBODY
CS
"3
I
No
Currant
Curnnt
B«^nt
Jen et al. (1966)
Taaal et al. (1969)
Harlenan and
Stelzenbach (1967)
Rlgter (1970)
Hada (1967b)
Honbuoymt
Yevdjevlch (1966)
Carter (1969)
(Buoyant)
Albcrtsm at al. (1948)
Abraham (1965)
Fan and Brook* (1966)
Fan (1967)
Bart (1961)
Fan (1967)
discharges are of relatively low volume compared to a thermal discharge. A practical summary
of momentum jet relationships applicable to the study of initial mixing has been given by
Wiegel (1964). Utilizing the method of dynamic similarity (Townsend 1956), it can be shown
that for a jet expanding in two directions (laterally from the axis and downstream) the charac-
teristic velocity will decrease with the square root of distance along the axis, and that for
a jet expanding in three directions (laterally and vertically (or radially) and downstream)
the characteristic velocity will decrease linearly with distance.
A fundamental property of momentum jet relations is that they are scaled directly
to the geometry of the discharge orifice or structure. Thus, the following scaling relations
can be established:
(i) Conservation of momentum along the jet axis
Q(x)U(x)
(4.17)
where U(x) is the characteristic jet velocity at a distance from the discharge, Q(x) is the
flow within the jet at a distance from the discharge,
is the discharge flow rate.
U_
is the discharge velocity, and
189
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(ii) Dynamic similarity of the mean velocity field and the root-mean-square (turbulent) velocity
field, which dictates that for a two-dimensional plume
MX) rx *
-tr't^j (4-18)
and for a three-dimensional jet
U,(x) x__
-!— --2L (A.19)
where 1)2(x) signifies the two-dimensional longitudinal characteristic velocity and xQ2 the
two-dimensional virtual source position; and UjCx) signifies the three-dimensional longi-
tudinal characteristic velocity and XQJ the three-dimensional virtual source position.
Combining the characteristic velocities with the conservation of momentum in the jet,
Equation (4.17), leads to the dilution flow relations of
Q'(X> ' ' ~ (4.20)
for a two-dimensional plume, and
^l-^_ (4.21)
Qo xos
for a three-dimensional plume. For jet discharges, the virtual source distance is about two to
ten times the diameter or width of the discharge orifice. An important result of the above
analysis is that the dilution ratio Q(x)/Q0 is constant with distance. Therefore, doubling
the discharge rate through the same discharge structure doubles the entrainment and should lead
to the same dilution at the same distance from the discharge.
Dynamic similarity does not dictate the shape of the velocity profile perpendicular
to the jet axis. This shape must be inferred from observation and it has been the practice
based on laboratory experimentation to assume a Gaussian function. Pritchard (1970) indicates
that a rectangular or "top hat" distribution may be more appropriate for large volume flow rate
thermal discharges. Extension of the dynamic relations to contaminant distributions is based
on the assumption that contaminant concentrations will follow the same distribution axially and
radially as the velocity distribution. Concentrations will therefore decrease with the square
root of distance for a two-dimensional jet and decrease linearly with distance for a three-
dimensional jet.
Buoyancy of a horizontal surface momentum discharge tends to retard the rate of
vertical mixing and results in greater lateral spreading than for a nonbuoyant horizontal dis-
charge (Jen et al. 1966 and Tamai et al. 1969). Consistent with the dimensional relations of
dynamic similarity, it is assumed that the axial concentration for a buoyant jet decreases
linearly with distance as for the three-dimensional nonbuoyant case. It has been necessary,
therefore, to incorporate the influence of buoyancy empirically into the lateral distribution
function.
190
-------
The influence of ambient currents and receiving water-body stratification on surface
jet discharges presents a large number of possible situations. Pritchard (1970) suggests that
if a longshore current is large compared to the normally directed velocity of a jet, then the
plume can be bent toward the shoreline and entrainment of the shoreward side of the plume would
be inhibited. When a highly developed thermocline exists it is possible that vertical entrain-
ment will be inhibited to such an extent that a three-dimensional jet discharge will degenerate
to a two-dimensional laterally spreading configuration. In the case of an estuary with a strong
halocline, it is possible that a condenser temperature rise will produce a density only slightly
less than exists for the surface waters with the result that a plume will plunge below the
surface after limited entrainment. In addition, Abraham and Kondstall (1969) have presented
data showing the influence of surface winds in distorting surface temperature distributions.
Dilution by a momentum discharge subjected to an oscillating current must also be
considered. Momentum of the jet relative to the receiving waterbody would be large during
slack water, but the reverse may be true during ebb and flood current conditions with the
result that jet entrainment may be ineffective during these periods. The domination of ebb
and flood current momentum over discharge momentum can lead to large "globs" of heat (scaling
tens of feet) being broken away from the plume. Large ebb and flood currents will, however,
provide the "new" dilution water necessary for entrainment. It appears insufficient to say
that momentum mixing terminates where jet velocity matches current velocity. Rather, it
appears necessary to match jet momentum to waterbody momentum including not only the mean
current velocity but also the current shear.
Submerged buoyant jets have been studied extensively for application to waste
discharges, and relations dictating their major properties are well established (Fan and
Brooks 1966). They are characteristically three-dimensional up to the point of interference
between individual jets. Comparison of submerged buoyant jet relations to the high volume
flow rate discharges of thermal power plants has not yet been accomplished. The submerged
buoyant jet, as well as the surface jet, provides the source distribution of temperature
relative to larger scales of mixing.
3.3 INTERMEDIATE TEMPERATURE DISTRIBUTIONS
Temperature distributions outside the region of momentum mixing are governed by the
advective and dispersive field of the receiving waterbody. The temperature distribution In
the intermediate region can be estimated no more precisely than the detail with which the
velocity field and dispersive field can be specified for inclusion in the heat conservation
relation, Equation (4.15). The velocity field is no longer dictated by discharge characteris-
tics but must be inferred from the equations of motion applied to large segments of the
waterbody.
Relative to the intermediate temperature distributions, the initial temperature
distribution determined by momentum mixing acts like an extension of a thermal plant condenser.
The distribution resulting from initial mixing, either by surface or submerged discharges,
gives the source distribution to be used as one boundary condition in the heat continuity
relation. When a large fraction of dilution is achieved by momentum mixing, there is little
to be accomplished in the intermediate mixing processes.
191
-------
In order to provide relationships for establishing the relative size of plant
discharges to waterbody characteristics, EdInger and Polk (1969) have sumnarlzed solutions to
the heat conservation relation assuming lateral, and lateral and vertical dispersion of heat
from a point source into a uniform current. These relationships are represented as the surface
area within a given temperature excess contour, in order to provide a rapid method for analyzing
large amounts of observational water temperature data. With such an oversimplified representa-
tion of the velocity field, dispersion, and boundary conditions, the surface area within con-
tours is about as detailed a description of the temperature distribution as is practical. For
a three-dimensional distribution the scaling relation between surface area and temperature
excess is approximately
r a .-3/2
0.496 (J-) (4.22)
Here A is the surface area contained within a contour of temperature excess 9 , 9p is the
temperature excess after momentum mixing, and A,,} is a scaling area given as
3/2
in which Qp is the dilution flow after jet entrainment, U is the uniform current velocity,
Dy is the lateral dispersion coefficient, and DZ is the vertical dispersion coefficient.
In the case where there is no momentum mixing at the discharge, Q_ and 8 would be the
plant pumping rate and condenser rise respectively. For a two-dimensional, distribution which
is vertically mixed the scaling relationships become
A -0.168f2-Y"3 (4.24]
where the scaling area A^ • Is defined as
-0.715 l-rr (4.25)
where 9_ , Q , D_ , and U are as previously specified and d is the thickness of the
vertically mixed water column. In the above scaling relations, it is significant to note that
the area increases with the 3/2 power of the plant pumping rate for the three-dimensional case
and with the cube of the plant pumping rate for the two-dimensional case.
The vertically mixed case was extended to include the influence of surface heat
dissipation and it was found that there was little reduction in area for contours of tempera-
ture excess greater than 0.2 6_ . A similar conclusion on the relative influence of surface
cooling during initial and intermediate mixing stages has been reached by Pritchard (1970).
The three-dimensional intermediate mixing relation was extended to include the cross-
sectional area within a temperature excess contour as a function of distance from the discharge.
These relations indicate that the section area within an isotherm should increase as the
192
-------
ln(8p/e) at each cross section, which agrees with the data plots of Lawler and Leporati (1968)
for a thermal discharge on -the tidal reaches of the Hudson River.
Description of surface area within a temperature excess contour represents the present
empirical detail for scaling large heat sources to receiving waterbody size. A unique relation-
ship between initial and intermediate mixing relations can be developed for the two-dimensional
cases which shows the aret Am and the temperature excess 8^ where the two match. Beginning
with Equation (4.20), the "width" of the two-dimensional jet can be defined as
) (4.26)
the distance from the discharge, to a 6(L) contour is given by
L ' xo, <8L>2 <4'27>
and the surface area in this contour becomes
= J b(x) dx = %-Sp- (B /9L)
(4.28)
and eL/8p = (^Qg/^d)* A£* (4.29)
Now, Equation (4.24) gives the intermediate surface area for a two-dimensional distribution.
Equating (4.24) to (4.29), the area A^ where jet dilution matches intermediate mixing is
found as
A^ = (2.07 x 10'4) Q9, U3, D~4 U'8d'9x~3 (4.30)
The temperature excess around this area, from (4.29), would be
Thus the area at which two-dimensional jet mixing matches two-dimensional intermediate mixing
increases as the ninth power of the plant pumping rate 0^ and as the cube of the jet velocity
U which are two variables at the control of the designer. This makes the question of design-
ing a condenser cooling water system for low 0^ and high 9 or high QQ and low 9 one
one of the most important facing the industry today. The biologist concerned about organism
entrainment and pump damage would probably prefer a low (L. (!•• L. Jensen, personal communi-
cations) . Similiar relations can be developed for the three-dimensional cases once the form of
the lateral temperature equation is assumed.
Other analytic studies of intermediate mining have been conducted. Wada (1967) has
provided a numerical integration of the heat conservation relation for an assumed uniform cur-
rent, a temperature-density dependency on the dispersive terms, a specified ambient temperature
193
-------
stratification downstream of the discharge, and rectangular discharge boundary configuration.
Pritchard and Carter (1965) have predicted the temperature excess distributions based on in situ
dye measurements by assuming similarity between the conservation of dye and conservation of
excess heat, these predictions were later compared to prototype conditions by Carter (1968).
The amount of mixing that can be achieved in a waterbody of finite extent is
determined by large-scale circulation. In the simple case of a river, for example, the upper
limit to dilution is set by the river flow. In tidal and estuarine situations the detail with
which the fully mixed conditions can be defined depends on the detail with which the large-
scale tidal circulation is modeled.
3.4 LARGE-SCALE TEMPERATURE DISTRIBUTIONS
Problems of formulating large-scale temperature distributions in estuaries have been
discussed in Sections 1 and 2 of this chapter. This discussion concerns the superposition of
temperature excess due to a single source in large-scale models. It is at this large scale
that dissipation of excess heat to the atmosphere becomes important. However, depending on
the relative size of a thermal discharge to large-scale circulation, the fully mixed tempera-
ture excess will usually be negligible compared to the temperature excess in the initial and
intermediate mixing regions.
Analysis of large-scale temperature distributions presumes the existence and verifi-
cation of a water quality model for other water quality parameters such as salt, dissolved
oxygen or BOD. The spatial and temporal detail with which excess heat can be modeled is pre-
determined by the nature of the existing water quality model. There is no independent tempera-
ture distribution model since heat obeys the same conservation relations, at least in the
advective and dispersive processes, as the other water quality parameters. Exceptions have
been discussed in Sections 1 and 2 where it is indicated, for example, that a vertically homo-
geneous or sectionally homogeneous salt distribution does not necessarily imply a_ priori a
similar vertical distribution of temperature.
Explicit, but simplified, continuity relations for a heat source relative to large-
scale distributions can be written for the sectionally homogeneous and the two-layered
segmented cases discussed in Section 2. This is possible because the advective terms in these
models can be inferred. Empirical requirements for inserting heat sources in segmented models
are examined in this section. For geometrically complex large-scale models, the excess tempera-
ture distribution after intermediate mixing becomes the source distribution, and if the large-
scale model includes the details of velocity and dispersion over the scale of intermediate
mixing then the source distribution is determined by initial mixing conditions.
Consider the sectionally homogeneous case. The excess heat balance is given in
Equation (4.16). Assuming that the excess heat balance is averaged over a time period so
that the local heat storage term is negligible, the "steady-state" heat balance for a short
segment of the estuary becomes
qli - ^("(aD]) Ml--"™ (4.32)
• •- *- -l *
o^ w **~ P n
where Q is the advective flow through the segment found by averaging the product of cross-
194
-------
sectional area and velocity over the same time period that makes the heat storage term
negligible. Q is often taken as the freshwater inflow. It is implicit that additional cross-
product terms «Uj9'» are now incorporated in the longitudinal dispersion coefficient as a
result of this averaging. If it is further assumed that Q and erDj are constant over a
segment near the source (located at x = -t-s ) , then the time -averaged sectionally homogeneous
heat balance equation has the solution near x = is of
e = e^-*-' + e/'*"^ <4.33)
where
a = %[q/aD! '+ ,/(Q/°I>l>2 + (AWK/pCpODi) ] (4 . 34)
and
- 7(Q/oD!)2 + ^WK/pCpQDi.)] (4.35)
and where 6L and 82 are constants of integration. For x > ts , Q^ - 0 for 6 to decrease
with distance and
8 = 8seb(x'ts) for x > ts (4.36)
where 6S is the fully mixed sectionally homogeneous temperature near x = ts which is yet
to be evaluated. Similarly for x < ts , 82 - 0 for 6 to decay away from the source,
hence
9 = e.."0"4^ for x < *s (4.37)
The rate at which heat is discharged at a source Hs must be Just equal to the rate at which
heat is advected and dispersed upstream and downstream, assuming no heat dissipation over the
area required to achieve complete mixing. Since there is no net advection of heat through the
sectionally homogeneous segment, the heat balance relative to the source becomes
x=t:
5 a
and using Equation (4.36) and (4.37) to evaluate the upstream and downstream gradients, then
Hs = pCpDj^ 6s(a - b) (4.39)
Qs S (4.40)
pcpQ/l + (4WoD1K/pcpQ2)
which relates source strength H8 and fully mixed temperature excess 9S . In this case, the
195
-------
fully mixed temperature excess 6& represents the limit on mixing that can be achieved by the
initial and intermediate process. Thus, in the end, the initial and intermediate mixing
processes cannot be completely independent of surface heat exchange.
Sectionally homogeneous relationships, similar to those above but modified for net
downstream advection and upstream boundary reflection, were applied by Edinger and Geyer (1968)
to a thermal discharge in a narrow tidal embavment. Results of this analysis, shown in Figures
4.5-4.7, are presented in terms of dimensionless parameters for ease of scaling to different
plant and embavment conditions. The advective flux could be explicitly identified with the
plant pumping rate since the plant intake was from a source other than the embavment receiving
the discharge. The temperature rise was measured from the equilibrium temperature rather than
the temperature distribution before addition of excess heat because of the significant change
in the advective field due to the discharge. It was found that the "steady-state" sectionally
homogeneous heat balance matched mean monthly temperature averages of weekly observations
during the July period for two different plant loadings. Considering the number of assumptions
involved in arriving at the sectionally homogeneous relations and the uncertainty of the time-
averaging procedure involved, Equation (4.40) can be considered at best a crude estimate of
fully mixed temperature excess.
SCALE (FT)
Fig. 4.5 Location map, power plant discharge area.
From J. E. Edinger and J. C. Geyer, Analyzing
Steam Electric Power Plant Discharges, Proc.
ASCE. 94, No. SA 4 (August 1968), pp. 6TTT23.
Used wTth permission of the American Society
of Civil Engineers.
196
-------
1 .0
0.8
0.6
o.u
0.2
o
OBSERVATIONS FOR 200
OBSERVATIONS FOR 400
J I
; t
0.2 0.4 0.6 0.8
NONDIMENSIONAL UPSTREAM DISTANCE. -£-
OBSERVATIONS TOR 200 HW
OBSERVATIONS FOR 400 MW
l l
(a) Upstream temperature distributions.
NONOIMENSIONAL FLOW RATE, o ,Jb ,/m* 04*
(b) Initial temperature rise ratio as
function of plant discharge rate.
•OBSERVATIONS FOR 200 MW
a OBSERVATIONS FOR WO MW
0.2 0.^ 0.6 1
IOC
0.1
kO 60 100
VULUE OF r-- —
W D,d
(c) Downstream curves of constant temperature rise
ratio as function of nondimensional coordinates.
Fig. 4.6 Nondimensional forms of one-dimensional analysis.
From J. E. Edinger and J. C. Geyer, Analyzing
Steam Electric Power Plant Discharges, Proc. ASCE,
94, No. SA 4 (August 1968), pp. 611-623"! Used
with permission of the American Society of Civil
Engineers.
197
-------
Table 4.2
Mean monthly temperatures at Intake and In
receiving embayment (degrees Fahrenheit).
After Edinger and Geyer (1968).
Location
Intake, Ti
Station A
Station B
Station C
Station D
Station E
Station F
Station G
Plant Loading, 200 Mw
23 x 106 cu ft per day
May
70.5
68.8
69.0
70.0
70.0
69.0
69.0
68.5
June
77.8
80.0
80.0
81.5
80.8
79.8
79.8
79.0
July
78.1
81.3
81.8
83.4
82.8
81.4
81.3
80.0
August
77.5
81.0
80.9
81.4
82.4
81.3
80.9
78.9
September
70.5
73.5
73.3
74.3
75.5
74.7
73.5
72.5
Plant Loading, 400 Mw
46 x 106 cu ft per day
May
61.8
70.5
70.5
71.8
71.5
68.3
68.0
66.5
June
74.3
79.5
80.2
81.5
81.8
80.0
79.0
76.0
July
78.2
81.8
82.0
83.5
85.2
84.7
83.8
82.0
August
76.0
79.3
76.6
81.9
83.2
82.7
80.5
79.7
September
66.5
68.0
68.5
72.3
74.0
71.3
71.3
70.0
10
•M f
v> 6
oc
UI
•FITTED CURVE FOR 200 MW
•FITTED CURVE FOR 400 HW
PREDICTED CURVE FOR 400 MW FROM 200MW DATA.
• OBSERVATIONS FOR 200 MW
A OBSERVATIONS FOR 400 MW
I
I
I
I
I
1 2 34 5
DISTANCE FROM CLOSED END OF EMBAYMENT
678
(THOUSANDS OF FT)
Fig. 4.7 Longitudinal temperature distributions for case study.
From J. E. Edinger and J, C. Geyer, Analyzing Steam
Electric Power Plant Discharges, Proc. ASCE. 94, No. SA 4
(August 1968), pp. 611-623. Used with permission of the
American Society of Civil Engineers.
198
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Table A.3
Parameters from fitted data. After
Edinger and Geyer (1968).
Parameters
Plant pumping rate
Q (cu ft per day)
Distance to closed end
L (ft)
Thermal exchange coefficient
K (Btu ft per day per °F)
Upstream apparent cliff us ivity
D2 (sq ft per day)
Downstream apparent diffusivity
D. (sq ft per day)
200 Mw
operation
26 x 106
3,900
155
7.0 x 106
3.76 x 106
400 Mw
operation
52 x 106
5,100
144
10.0 x 106
4.6 x 106
Magnitudes of fully mixed temperature excess given by Equation (4.40) for the
Delaware River-Estuary based on values of longitudinal diffusivity and river geometry tabu-
lated by Pence et al. (1968) are shown in Figure 4.8. The result is given in terms of °F rise
per 10^ Btu hr"1 of waste heat rejection and is presented here only to indicate the magnitude
of heat sources relative to a particular waterbody, which has been the subject of discussion
elsewhere in this report. The transition from dilution by river flow alone to that provided
by dispersion and increased geometry with distance is illustrated. A more satisfactory esti-
mate of fully mixed concentrations would be based on Equation (2.148) of Chapter 11 which
accounts for velocity variations within a tidal cycle and eliminates the difficulty of inter-
preting the temporal averages of Q and D^ .
For segmented and multilayered models, as discussed in Section 2.2 of this chapter,
Equations (4.10) and (4.11), the source flux ( Hs/pcp in these equations) is added to the
heat balance for the segment at the point of discharge in a manner similar to that indicated
by Pritchard (1967). Implied are the conditions that (1) initial mixing and intermediate
mixing are complete within this segment, (2) momentum entrainment only influences circulation
within the segment but not across segment boundaries, and (3) intake-to-discharge circulation
takes place within the same segment. The first condition might be satisfied empirically if
the surface area of the segment is larger than required to give the average segment tempera-
ture rise based on the intermediate distribution relationships of Section 3.3. The second and
third conditions are related to modification of the overall circulation pattern by the large
volume flow rate of a thermal source. An imputed circulation pattern based on the continuity
of salt cannot be readjusted by mass continuity relations alone, but will require additional
approximations based on momentum conservation. However, the second condition might be satis-
fied if it can be assured that the flow required to dilute the condenser temperature rise to
199
-------
1.00
0.9
* 0.7
Ul
x 0.6
u.
o
T 0.5
Z O.k
-------
A detailed comparison of model results and analytic descriptions of initial and
intermediate mixing processes of a thermal discharge have been presented by Harleman and
Stolzenbach (1969). The model is scaled according to a densimetric Froude number criterion,
a beach slope and discharge channel aspect ratio criterion based on the work of Wiegel (1964),
and a surface cooling criterion based on the results of Hayashi and Shuto (1967). The Froude
1/9
criterion establishes the velocity and depth scale (Vr - h~ ) , and the surface cooling cri-
terion used in this study establishes the length and depth scale (Lr - hr'2) . With distortion
in the model the slope criterion of Wiegel cannot be satisfied. Results of the model are com-
pared to the initial mixing behavior of a three-dimensional jet (where dilution increases linearly
with distance) and it is found that jet mixing dominates to a distance of thirty times the dis-
charge width. Beyond this distance, where temperature rise is about two-tenths the condenser
rise, it is concluded that surface ccoling begins to be important.
Future progress in the quantification of the complex temperature distributions
surrounding thermal discharges requires a combination of hydraulic and analytic modeling
coupled with prototype studies. In the past, analysis has dictated the scaling of hydraulic
model results to prototype conditions, but it appears that the future direction must be rein-
forcement and verification of analytic solutions (which are now becoming quite complex) by
hydraulic modeling and then extension to more variable prototype conditions through analytic
methods.
201
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Proc. ASCE, 91, No. SA 2.
Frazer, W., D. I. H. Barr, and A. A. Smith, 1968: A hydraulic model study of heat dissipation
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Goedjen, M., R. M. Collins, and J. P. Roche, 1959: 305,000 KW extension to the Fisk Steam
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Hansen, D. V., and M. Rattray, 1965: Gravitational circulation in straits and estuaries.
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Hayashl, T., and N. Shuto, 1968: Diffusion of warm cooling water discharged from a power plant.
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Hayashl, T., N. Shuto, and K. RawakamL, 1967: Basic study of the diffusion of warm water jets
discharged from power plants into bays. Coastal Engineering in Japan, Japan Soc.
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Jeffrey, L. J., and I. M. Gardiner, 1965: Studies in field and model of cooling water
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Jeffers, F. J., J. C. Geyer, and L. C. Neale, 1969: Design of condenser cooling water system
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Jen, Y., R. L. Wiegel, and I. Mobarek, 1966: Surface discharge of horizontal warm-water jet.
Proc. ASCE. 92, No. PO 2.
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Silberman, E., 1967: Discussion of surface discharge of horizontal warm-water jet. Proc. ASCE.
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Proc. ASCE. 86, No. PO 3.
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Tamal, N., R. L. Wiegel, and G. Tornberg, 1969: Horizontal surface discharge of warm water
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DISCUSSION
EDINGER: My approach was to begin with the assumption that there is no single water
temperature model for an estuary, and proceed to look at how many different formulations
can be developed from continuity relations. It is recognized that we cannot determine the
temperature distribution in an estuary in any more detail than we know the velocity field,
both temporally and spatially, hence the temperature models are very dependent on dynamic
models. In the first two sections are discussed large-scale natural temperature distribu-
tions with ambient conditions at a river and ocean end and surface heat exchange but no
major sources in the middle. So the argument is that, first of all, we consider that the
heat conservation relation and the salt conservation relation will be the same in the advec-
tive, storage and dispersive terms, and this will lead to some similarity between salinity
distributions and temperature distributions, and any deviation in this similarity will be
related then to surface heat exchange and a number of other factors. The fact is here that
you have the same transport relationships, and the boundary condition of zero salinity at
the river end can be made compatible to zero enthalpy at the river end by using the river
end temperature as a base temperature from which to make your similarity arguments. The
next step is to superimpose upon this, then, what we think might happen with incorporating
surface heat exchange. This then lets us intuit or sketch out what we might expect for
longitudinal temperature distributions. These are shown on Figure 4.3.
We then come to two very important problems that make heat difficult to handle
in the analysis of temperature distribution. The first is the problem of attenuation of
shortwave radiation with depth and through the water column. Most of our knowledge on the
relationships of shortwave radiation attenuation and vertical temperature structure is
presently derived from freshwater lake type models. The fact is, though, in some situations,
with high salt concentrations in deep layers and even in winter months, we can have high
temperatures in lower layers due to absorption of shortwave radiation.
The second problem is what do we do with the surface boundary condition necessary
to account for attenuation of shortwave radiation through the surface In a vertically vary-
ing temperature structure. I have written In Equation (4.6) a fairly general statement to
show that the rate at which heat has to be transported away from the water surface is
related to the vertical dispersion at the water-surface, as well as some buoyant convection
which is related to the velocity field. This, I might point out, is following the discus-
sions that Phillips brings to bear on the problem of vertical convection at the surface.
It becomes clear that If for any reason we can assume heat uniformly mixed through the
vertical in the water column, there is no problem as far as temperature prediction is con-
cerned. The question is whether this is valid. It seems to me that we are going to have
to be incorporating in a temperature model, particularly when we want detail in the verti-
cal temperature structure, some type of computational process, similar to that used by Dake
and Harleman, to account for convection of heat near the water surface. In the case of the
estuary, we are going to have to make the convective computations dependent on salt con-
centration as well as temperature, but it won't be too much of a problem, because I do not
think we are going to affect the salt distribution that much in the initial calculations.
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The third section of this chapter deals with temperature distributions due to
large heat sources. Generally we have discovered that we have to consider three spatial
scales in classifying present analytic techniques. One is an initial mixing region near
the discharge which is a function of discharge design. Secondly, there is an intermediate
region where the heat is tending toward the larger scale distribution. Third, there is the
large-scale temperature distribution which we might consider to be the longitudinal and
vertical temperature distribution resulting from the large heat source.
These three regions then, right at this stage, are described by independent
analytic methods. I think we are in a situation where we need some simple scaling relation-
ships to tell us how big or small each region is in a particular type of water body. We
find that most of the research that has been done and published in this field has not been
directed explicitly towards high volume, high momentum, low concentration discharges like
we have with heat, that is, for initial mixing of discharges, most of the work has been
directed towards sewage discharges which have low volume, very high concentrations, that
require dilutions in the order of a hundred or a thousand to one, and with heat we are not
looking for much more than five or ten to one in the way of dilution. Also not much of the
work has been done in the field directly with power plants themselves.
RATTRAY: The part of this that Z have had a chance to look at here, I go along pretty
well with. These segmented models bother me, but I'm not the only one.
EDINGER: They do me, too. Well, this is a fairly common one that is used, and the thing
that I am pointing out here is that it is not too difficult a trick to turn this into a
temperature model if you want. We don't have to derive a new segmented model if we accept
this one, you see.
RATTRAY: There is one place I had trouble on something that seems to be fairly commonly
understood and maybe I don't understand. And that is in salt-wedge estuaries. It is in
here and other places too, about net upward advection out of the salt wedge. I just don't
really have a very good feel for that. I suspect the salt exchange between the wedge and
the fresh water is a practically negligible quantity. That's what I am asking.
PRITCHARD: Depending upon what you define as a salt-wedge estuary. If you take something
that doesn't exist and call it a salt-wedge estuary, you might be right. But if you take
the real systems which we classify as salt-wedge, like the Mississippi, then you have the
water surface, the bottom, and the wedge coming in here as a fairly well-defined gradient
in salinity. The main difference between a salt-wedge and a moderately mixed estuary is
that there is no dilution of water in the wedge. That is, this is ocean concentration all
the way up the mouth and is defined almost like a front, but an exchange does occur with
a transport of salt into the fresh water, and you can see that gradient, the salinity
increasing.
From his flume work, Keulegan showed at this Interface there is a velocity dif-
ference between the underlying wedge and the top layer, and you get interfacial waves, and
these interfacial waves break. They always break upwards, so even that though this looks
like turbulence you are really transporting both salt and water. You find the wedge is
constant for constant river flow, and since you are passing salt upwards you've got to have
flow inwards. Admittedly, this is very much smaller. It's like equal to the river flow
or something like that, rather than several tines the river flow. That is, the volume of
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water coming in at the ocean end in the wedge is not an order of magnitude larger than the
river flow as it is many moderately mixed estuaries.
RATTRAY: I Just wonder how good the data are on that. I have tried to get Mississippi
River data.
PRITCHARD: Most of this is from the Mississippi model, in fact.
VLASTELICIA: Have these waves been observed in the model breaking?
PRITCHARD: They've been observed in the flume. The original work was done by Keulegan, his
study of instability in stratified flows. And it depends upon your terminology. I just
don't know of any natural environment in which the salt-wedge comes in completely so that
you have no gradient in salinity in the upper layer. Mow it's true that for very low
river flows, that salt-wedge goes up the Mississippi for a hundred-some miles and you are
taking out fresh water out over the top of it. If you measure the salinity of it, it Is
slowly Increasing seaward over the wedge.
RATTRAY: The rate of that Increase is what really worries me, because if you talk about
it as an advective flux, it should be fairly rapid. If it is a molecular diffusive flux,
it should be very slow.
PRITCHARD: It's a matter of definition. And I define an advective flux as where you pass
both water and salt, as distinct from where you just transport salt.
RATTRAY: It is possible to work out a balance that works very well in predicting salt-
wedge shape and intrusion with nothing but molecular friction stresses at that boundary.
So I don't know, because I haven't seen good data, but I raise the question.
EDINGER: The point of it is that I don't think it depends upon the reliability of the
model, or whether we believe the existence of the salt-wedge estuary or not, but that there
is a similarity between temperature and salt. The fact of it is that we certainly shouldn't
be surprised to have longitudinal temperature distributions in an estuary, of what magni-
tude I am not sure yet, but I think at least we should expect it to vary seasonally. There
will be times when there will be maximum temperatures and minimum temperatures throughout
the longitudinal length, and this is mainly what I am trying to spell out. As a matter of
fact, the same arguments about mixing colder water toward the surface would hold for the
partially mixed estuary.
PRITCHARD: You just wouldn't have the bottom the same temperature as the ocean there.
EDINGER: That's right. The main thing Is to have some type of framework In which to
summarize it.
VLASTELICIA: What about extending the plume into the real time model at all in an estuary,
in the distribution and its movement?
EDINGER: The thing that I point out here is that the status of the art is such that we're
basically picking up simple descriptions of the sizes of these plumes case-by-case and
then overlapping these. I don't think that we have any general model that will cover all
these scales yet, and, even numerically, I don't think we will. I don't have a computation
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on this but my feeling is that we'll probably have as much computational difficulty
determining the local temperature distribution right off the discharge as we will the
large-scale distributions. I feel that the state-of-the-art is such that the problem of
knowing when a momentum discharge condition ends and the larger scale dispersion takes
over is not well-defined, and I don1t think it will ever be well-defined because it is not
that clear as to what it should be analytically. We only have a vague feeling about when
we reach, shall we say, the sectionally homogeneous or laterally homogeneous condition.
Anyway, each almost has to be done separately.
DOBBINS: What about whether you release it at the surface or release it at the bottom?
EDINGER: Yes, it's going to make a difference in the distributions. Certainly. There's
no doubt about it.
PRITCHARD: I think the point that needs to be emphasized is that from a practical standpoint,
you're now talking about much larger volume rates of flow. You're talking about thousands
of cubic feet per second, which is very large compared to even a big city's municipal dis-
charge. And the kinds of problems you face now become different. Natural diffusion and
the natural mixing processes near the source are the things that, I would say, decrease
that sewage discharge. The tidal oscillations, etc., that make the plume around a sewage
discharge a relatively small volume have very little effect for some time on the distribu-
tion of temperature from such an outfall, because essentially you're starting with such a
large volume source.
Schematically, if you look at a continuous point source release, you're starting
with a delta spike in the distribution curve so that there is a high gradient for the dif-
fusion to work on. So you see diffusion acting Instantaneously to decrease the concentra-
tion curve as you move down the plume. But when you start with a large volume source—one
way to picture it Is a line source in which essentially the distribution is square—diffu-
sion, even large-scale diffusion, can only act on die edges where there is a concentration
gradient. You've got to move well down the plume before natural diffusion can affect the
concentrations in the center.
So what you have to do is design so that you promote mechanical mixing, or augment
the nature here. In other words, you need to find a site where there Is high dilution
volume available, and then find a way to use it by the design of the discharge. Now, of
course, one design is Just to make a large-scale diffuser that is the type that Brooks
designs for the ocean. Generally, this is difficult to design for this kind of a waterway,
so what you do is to depend on a single port of high velocity dilution—a high velocity
discharge and momentum entralnment. And the amount of dilution you can get this way is
still very large.
From a practical standpoint, your concern is with an area which is dominated by
Che dynamics of the discharge itself. You Just never phase Into the large-scale computer
model except Insofar as you want to know how one system might interact with the other in
a large scale. Then I think you can sort of treat this case after the initial mechanical
dilution has done some nixing, if you take multi-sources as already have been mixed in part
and look at how they overlap with one of these larger scale, reduced-dimensional models. But
the real problem that faces us today is what Is the distribution around the outfall. I'm
talking now within a tidal excursion of the outfall. This is where the decision is going
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to be made whether a plant meets the criteria or whether you're going to allow It to be In
this estuary or not. It's not the large-scale distribution, but the distribution within a
tidal excursion.
EDINGER: One of the problems is that in this intermediate region which I would consider to
be the size of a tidal excursion or less, so to speak, the distribution you get depends
solely upon your assumptions concerning the details of the velocity field. That is,
if we know a two-dimensional time-varying velocity field in this region, that's the detail
to which we can predict it. Whether we plug that amount of velocity detail into, say, a
finite-difference calculation or not makes no difference: that's it. There is no sense
in saying that different types of models are going to replace this.
PRITCHARD: I'd say except in the vicinity of the system where the dynamics are determined
by the flow velocity of the jet itself.
EDINGER: That's right. This is what I've called the initial mixing region.
PRITCHARD: That probably occupies a sizable dilution zone, so that a significant part of
the dilution is controlled by the dynamics of the discharge itself. There, of course,
what you get out of it depends on what kind of model you take for the distribution of
momentum within the jet.
CALLAWAY: In the third part there, you have something on temperature excess. How do you
determine that?
EDINGER: Basically, temperature excess is defined as the difference between the temperature
you have with a source and the temperature without it, both spatially and time varying.
Within the Immediate vicinity of a discharge, it doesn't seem to make much difference
because your longitudinal temperature distributions probably wouldn't be that large.
PRITCHARD: Sometimes it does. For instance, I just Iboked at a Great Lakes situation where
the temperature about 6,000 feet offshore varied from 48° onshore to 42° offshore. Your
predictive technique.predicts temperature excess, but then how do you apply it to this
varied concentration offshore? It becomes a problem. In this case, one assumption one
could make would be that the water being entrained into the jet has the mean ambient
temperature between shore and the distance out to the point where you're making the pre-
diction. Then you have to add your excess temperature to the mean of the sloping tempera-
ture curve between the discharge and the point at which you're making the prediction.
But it does get very difficult to talk about what is the base temperature, the
ambient temperature. Particularly, if you just look at the real world, with no thermal
plants on the waterway, and look at the temperature distribution spacewise and timewise
in an estuary, they vary significantly. The problem of saying what is the base temperature
if the plant weren't there, once it's there, is an extremely difficult thing. We've gone
out and measured temperature distributions from heated discharges, and our biggest problem
was to say how much of the temperature we're seeing is a rise above an ambient, because
we couldn't pick the ambient very well. In the Patuxent River you can say to just put the
measuring devices far enough away. But if you put them far enough away they vary very much
anyway in time and space. Besides that, you can show that the plant affects the whole
thirty miles of the estuary anyway. So it gets pretty difficult to do.
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BAUMGARTNER: In answer to Dr. Dobbins' question, we, In another effort, have made a report on
what models are available for ocean outfall design. We talk about the temperature models
there. As Dr. Prltchard points out, you can come up with larger diffusers, but It's by no
means technically or economically Impractical to .use them.
I have a question for Dr. Edlnger. I just wonder In this chapter or some kind of
summary, if you'll say something about how well these models can be applied to the types
of situations we're concerned with now. Like if we had a model of Biscayne Bay for tempera-
ture distributions, or Fuget Sound, or the Columbia River, or any other place you can
think of. How well can we do that?
EDINGER: It would be my opinion that again it's only as good as the experience of the
fellow who has worked with this particular set of calculations. I don't think any general
rules can apply. Our state-of-the-art in terms of what formulations we have, or what few
cases we can identify, isn't that good.
The one thing you can do roughly is that if you have an existing discharge, with
some field observations, you can make some approximations about scaling that discharge to
a different plant pumping rate and to a different heat rejection. This is within bounds,
but won't be very precise. In other words, we can get from the continuity relations a few
power scaling relations. There are many things we don't know about the coefficients on
these. When I say we haven't had enough experience, it depends some days on how you feel,
what you take for the scaling area or something.
BAUMGARTNER: You're saying you don't know how to measure a or bc?
EDINGER: No, a and bc are not the problem in local distributions. We are looking at
entirely different types of relations. We're looking at relationships that involve some
knowledge of the mean advectlve field that we haven't measured or predicted a priori.
This we can at least get a picture of from an existing discharge. All our unknown con-
stants, so to speak, are lumped in one place. So you can make a number of approximations,
depending on what you want to assume for the type of discharge condition you expect, at
least empirically, whether it's two-dimensional or three-dimensional, whether your entrain-
ment will be limiting.
One Important fact to keep in mind is that you can't entrain more water into the
plume than you get to the edge of the plume. If anything limits the amount of water that
can come in, it won't entrain any more. It will Just keep on mixing on itself and you end
up with a blob that disperses on through. These are embarrassing things to find out.
I think the state-of-the-art is that analysis is almost site-by-site. You can
make some approximations as to whether a waterbody is large or small compared to a power
plant, or vice versa, whether a power plant is large or small compared to a waterbody.
Now, essentially we're doing this today for plants that will be looked at in detail seven
years hence. If our guess is bad today, we're going to be in more Biscayne Bays seven
years hence than we're looking at today. I would imagine the same thing has happened in
many situations. The history of power growth has been that thousand-megawatt plants hit
us almost overnight, with manufacturers coming out with a package unit within three or
four years. We Just weren't geared up to even know much about siting them, much less
about the waterbodles In which they'd go. I see it not in terms of having a fixed formu-
lation for all conditions. I see, rather, building up within individual utilities trained
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people who can look more critically at the siting problems and be familiar with waterbody
hydrography. The better ones are doing this. And along with this, I think there is going
to have to be some type of larger scale, more organized effort to pick the sites for the
larger plants. After all there's going to be a finite number of them, and they're going
to have to go someplace.
BAUMGARTNER: Is this an assessment that you mean to apply only to, say, the small scale or the
intermediate scale problem? If you mean it to apply to the large-scale distribution, I
think that's a materially different assessment than what 1 understand for, say, any other
water quality constituent.
EDINGER: I think the thing of it is, if your power plant is so large in an estuary that
it's going to influence the large-scale distribution, then the plant's probably too big.
PRITCHARD: I would agree that if it's really a problem on a large scale, if it's such a
large problem, on a small scale it's impossible. It would be readily available with rather
crude methods of prediction.
I'm more optimistic about what can be done now. I think that one can set general
criteria for design, in terms of size, so that you would have said right off the bat that
the size of the plant on Biscayne Bay is too big. And secondly, if you will accept not a
detailed picture of the isotherms but such things as the area within having excess tempera-
tures less than stated values, then, in fact, given a discharge design, the characteristics
of the water, something about the longshore current distribution, one can predict the area
contained within given isolines of excess temperature. You must decide whether you're
going to predict a characteristic distribution or a 90% worst case or something like this.
But for having made that statement, I think you're good within 257, or something like that
on the average on the areas.
BAUMGARTNER: One last point. There has been some discussion that perhaps the water quality
administrators cannot be satisfied with a worst case or a 90% prediction value, and they
may want to have time distributions of temperatures, which might require joint probability
distribution functions for evaporation effects, currents, winds, and variations on the heat
loads. You may have seen in this first report that we talked a little bit about stochastic
processes. I wonder if you would tend to include anything about that.
EDINGER: No, I don't, because I don't feel that this has developed far enough yet that we
know enough about the statistics of water temperatures. A number of years ago, in '65, I
set up the field studies for the Hopkins-EEI studies, and this was based upon the premise
that the one thing that we could get at many different types of sites and the one thing
that was needed was time-varying data. Unfortunately, it takes three years to get three
years of time-varying data; and it still takes a few years and some people who are inter-
ested in time-variation statistics to start looking at it. I don't even think we know what
type of statistics, so to speak, to crank into our model. I'm not sure what the status of
the art is on this business; whether we have a deterministic model and can start cranking
in time-varying conditions and get the calculated statistics out, or whether we should go
back and redesign a model that's designed specifically to be a stochastic model. It's a
rough question. It's an important one.
THOMANN: But I think that the point on stochastic aspects is very well taken, that you can
run these models in a simulation-type framework with input statistics that generally
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represent what is occurring in the field environment. I think a fair amount of progress
has been made in characterizing the statistics of some of these time series. Combinations
of a few simple harmonics plus a regressive scheme can pretty well characterize some of the
fluctuations. You can grind these into the deterministic model and take a look at the
probability distributions of your output to answer those kinds of questions.
BAUM5ARTNER: That answers one of them. You can simulate the conditions, if. But I think that
you need to know something about the joint probabilities if you forecast.
EDINGER: Right, I think.
PRITCHARD: A comment. Ibis is what has been observed in some records, not exceptionally
long. But there has been some tine series records taken in a way that one can sort of see
what happens to an annual temperature curve, or the day-to-day temperatures over an annual
cycle, with or without the plant there, by having recording thermistors at a location
unaffected by the thermal plume--and, as I say, it's hard to know how far away you have to
go. But we have some of these, which show, say, the annual pattern at that site removed
from the plant site, watching the daily temperatures over a year's time. If you look at
the distribution in the vicinity of the plant, what you see is the larger scale features,
just the annual curve displaced. What one finds is in the spring the difference is some-
what greater than in the simmer and winter, and during fall somewhat less. (Because during
summer and winter the temperature is near equilibrium. During spring the equilibrium
temperature is higher than the ambient temperature, and hence the cooling driving term is
less than the difference between the heated temperature and the ambient. During fall the
reverse is true.) But the major statistics of the two curves are the same, except for the
mean value. All the higher moments than the first are essentially the same. So I'm just
wondering whether one really needs to look at this. You want to know the stochastic charac-
ter of the natural temperature variations from the biological standpoint. The imposition
of the added heat term or added source term doesn't significantly change any of the moments
of the probability distribution above the first.
EDINGER: What you're saying is pretty interesting in one sense, in that if you're talking
about the statistics in natural temperature records, the best thing to analyze, by far, is
the natural temperature record. But if all you're doing is playing the game and saying, "Well,
I don't have a natural temperature record but I have a lot more meteorological data, so
let's crank this in and then crank our flows into a temporal-spatial model and start crank-
ing out answers on this," I'm. still not convinced that there wouldn't be an awful lot of
noise showing up in this calculation routine itself. An awful way to get water temperature
records is to calculate them from meteorological data. We do everything we can to stay
away from that.
214
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CHAPTER V
PHYSICAL HYDRAULIC MODELS
Donald R. F. Harleman
1. INTRODUCTION
The earliest tidal models were built in Europe in the period 1875-85. In France,
Fargue constructed a model to study the Garonne estuary from Bordeaux to the sea. In Britain,
Osborne Reynolds used a tidal model to study shoaling and channel formation in the Mersey
estuary. Reynolds' model had a horizontal scale ratio of 1/31,800 and a vertical scale ratio
of 1/960, a distortion of 33 in the vertical direction. This was the first model in which the
time scale for the tidal motion was related to the Froude condition for dynamic similitude for
gravity and inertial accelerations. A summary of the history of tidal models, including a
discussion of the important contributions of Osborne Reynolds, has been given by Ippen (1968).
In the United States, one of the first large tidal models was built in 1936 at M.I.T.
by K. C. Reynolds (1936) for the Corps of Engineers. This was a model of the sea-level Cape
Cod canal. The length scales were 1/600 horizontal and 1/60 vertical. This seems to have
established the American practice of employing a vertical distortion of 10 in tidal models.
Because of its responsibility for navigational facilities, the Corps of Engineers has played a
leading role in the development of hydraulic model techniques at the Waterways Experiment
Station, Vicksburg, Mississippi. During the last three decades, models of many of the estuaries
of the East, Gulf and West coasts of the United States have been constructed. Motivation for
the models has usually been problems associated with navigation improvements. In recent years
many of these models have been used for the study of other problems including pollution and
water quality. This has been due to an increasing awareness that navigation improvements may
be detrimental to the multiple uses of an estuary, wicker (1969) has classed estuary problems
in the following categories:
Estuarine Problems
Due to Navigation or
Navigation Improvements
Due to Factors Other
Than Navigation
Channel Dimensions
and Layouts
Shoaling
Disposal of Dredge Spoil
Salinity Problems
Effects on Shorelines
Effects of Land Fills
Effects of Bridges
Hurricane Surge and
Tsunami Problems
Effects of Modifications
of Upland Discharges
Effects on Shorelines
215
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Changes In Regimen Changes in Regimen
Pollution Pollution
Problems Concerned with Problems Concerned with
Estuary Ecology Estuary Ecology
Aquatic Weeds
Physical hydraulic models have been used for the investigation of almost all of the problems
listed above.
A tidal model is a much more complex and costly undertaking than its counterpart in
the area of river and hydraulic structure models. Instrumentation is required for the transient
measurement of water depths, velocities, and concentration of salinity, dye and other tracers.
The reproduction of tidal motion at the model boundaries requires accurate control systems, and
the boundary conditions are further complicated by the necessity of reproducing density
differences associated with fresh water and saline ocean water. Light weight materials are
often introduced to study scour and deposition patterns. A comprehensive survey of the use of
hydraulic models in tidal waterway problems has been given by Simmons and Lindner (1965) and
Simmons (1966 and 1969).
The reproduction of natural phenomena in a physical hydraulic model has an analytical
basis in the theory of similarity. Traditionally, the derivation of the conditions necessary
to achieve similarity is based on the concepts of dimensional analysis. This is a mathematical
process of generating, as an output, a sequence of dimensionless groups from a set of input
quantities which the engineer decides are important to the problem at hand. An objection to
this approach is based on the fact that there is no well-defined procedure to guard against
the inclusion of too few or too many input quantities. An alternative method, known as
inspectional analysis (BIrkhoff 1955), is recommended as a more rigorous approach to the theory
of similitude. Inspectional analysis is based on the concept that there are certain physical
laws which describe the flow processes. Usually they are stated in the form of differential
equations related to momentum, heat and mass transfer processes. In a water quality model the
set of differential equations would include the momentum equation of motion and a conservation
of mass equation for each water quality parameter identified by concentrations of dye, salinity,
BOD or dissolved oxygen.
Inspectional analysis is not concerned with the solution of the differential equations.
Once the governing differential equations have been Identified, the inspectional analysis is
carried out by rewriting each of the differential equations in dimensionless form. This
procedure will be illustrated in a later section. At this point it is only necessary to state
that this results in certain dimensionless groups of physical quantities which appear as
coefficients of various terms in the dimensionless differential equations. The principle of
similitude can be started in the following manner: the fluid processes in two different
systems, which are geometrically similar, will be identical if the dimensionless equations
governing the fluid processes in the two systems are likewise identical. Since the governing
equations are in dimensionless form, it is only necessary that the coefficients be numerically
the same in model and prototype.
In physical estuary models, a difficulty arises from the requirement that the two
systems (model and prototype) be geometrically similar. Non-geometric similarity, in the form
of vertical scale distortion, is a characteristic feature of all physical estuary and tidal
models. The necessity for vertical distortion becomes apparent when it is considered that the
216
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ratio of length to depth in a typical estuary may be of the order of 25,000. Thus a model depth
of only 0.2 feet would require an undistorted model having a length of one mile. A model having
the same depth and a vertical distortion of 10 results in a feasible length of approximately
500 feet. There is no exact analytical approach to dynamic similitude for distorted models.
However, the need for vertical distortion is well recognized and non-controversial. In
addition to feasibility in terms of space and operating personnel, vertical distortion is a
means of avoiding prolonged periods of laminar motion during the tidal cycle, the increased
water depths in the model result in better accuracy of depth and current velocity measurement,
and undesirable capillary effects in the model are reduced.
The disadvantages of distortion may be summarized by observing that dynamic similarity
is sacrificed to a certain extent, therefore reliability of the model results are more subject
to question. It follows that the process of model verification, whereby the model is adjusted
by trial to reproduce phenomena which were observed in the prototype, is a vital and necessary
step. Physical models are built in order to study the effects of a proposed change in proto-
type conditions and it is Important that the field data used for verification be closely related
to the phenomenon involved in the future change. The validity of a distorted estuary model can
only be assured for certain specialized phenomena which have been established through the
verification process. Unfortunately there is a tendency to extend the usage of such models for
predictive purposes beyond the scope of the physical processes used in the verification studies.
This Is especially true in the area of water quality investigations on physical models which
were designed and verified for the study of navigation improvements.
The above remarks are not intended to discount the usage of physical models in
certain water quality investigations. Rather it is a plea that the careful verification
procedures used for the original purpose of the model be extended into the verification of
flow phenomena which are uniquely associated with water quality problems.
217
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2. SIMILITUDE OF MOMENTUM TRANSFER PROCESSES IN TIDAL MOTION
In this section the basic similitude conditions for the advective motion are developed
from the principles of inspectional analysis. This results in the Freudian scale ratios for
velocity, discharge, and time. In addition, boundary conditions and roughness relations for
physical hydraulic models are discussed.
2.1 PHYSICAL BOUNDARY CONDITIONS FOR TIDAL MODELS
The first consideration in the design of a tidal model is the determination of the
physical limits of the model. Usually an estuary model includes a portion of the ocean area
in the form of a basin containing the ocean tide generator and provisions for maintaining a
constant salinity representative of the ocean area. The shape and extent of the ocean basin
depend on the complexity of tide and current conditions at the ocean entrance of the estuary.
The tide generator may be required to produce sinusoidal tides with variable range, represent-
ing spring and neap conditions, or it may be designed to produce a complex mixed tide con-
sisting of diurnal and semi-diurnal components.
Figure 5.1 shows the seaward boundaries of the model of the Rotterdam Waterway at
the Delft Hydraulics Laboratory (Delft Hyd. Lab. 1969). The ocean basin has independent tide
generators at the northeast and southwest boundaries of the basin representing the offshore
area in the North Sea. In addition to the tidal oscillation, a current parallel to the coast
may be produced. Vertical rotating cylinders have been used to simulate the deflection of the
ebb flow from the estuary due to the rotation of the earth.
The landward extremity of a tidal model generally coincides with the upstream limit
of tidal motion. In estuaries having a well-defined head of tide, such as a dam or natural
fall, this location is readily determined. In some estuaries the tidal motion gradually dimin-
ishes in the upper reaches of tributary rivers. In this case the extent of tidal motion depends
upon the ocean tide range and the magnitude of the freshwater inflow. In either case the
landward boundary requires the reproduction of the temporal variation of freshwater inflows.
Figure 5.2 shows the limits of the Delaware Estuary model which is typical of an estuary having
a well-defined head of tide (at Trenton). The scale ratios are 1/1000 in the horizontal direc-
tion and 1/100 in the vertical direction. Figure 5.3 shows a similar view of the Savannah
estuary model which is typical of the open-end type. The scale ratios are 1/800 horizontal and
1/80 vertical. Both models are of the fixed-bed type.
Where it is desirable to study only a limited portion of a tidal waterway, it is
possible to terminate the model within the region of tidal motion. This requires the pro-
vision of a second, independently controlled tide generator at the inland boundary. This
limits the use of the model since the inland tide generator can reproduce only those tidal
stages and currents which have been observed in the prototype. If longitudinal salinity gra-
dients due to salinity intrusion exist, it is usually not possible to terminate the model
within this region because of the difficulty of controlling salinity variations at the inland
boundary. Figure 5.4 shows the model limits of Absecon Inlet on the New Jersey coast. This
model had a primary tide generator in the Atlantic ocean basin and a secondary generator at
218
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10 km
Fig. 5.1 Model of the Rotterdam Waterway - Delft Hydraulics Laboratory (1969).
F
'^.TRENTON
PENNSYLVANIA
PHILADELPHIA
WILMINGTON
o TIDE GAGE
F FRESHWATER INFLOW
IOO 1000-FT CHANNEL STA
t\'
N
5\?S =
B\**« x.
OCCAM
Fig. 5.2 Model of the Delaware Estuary - Waterways Experiment Station (1956).
219
-------
EBENEZER
LOG
SAVANNAH HARBOR MODEL
COASTAL
HIGHWAY BRIDGE
MOOLE
SUGAR REFtCRY,
/ &
KINGS ISLAND;
Fig. 5.3 Savannah Harbor Model - Waterways' Experiment Station
220
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ATLANTIC OCEAN
Fig. 5.4
Absecon Inlet Model - Waterways Experiment Station.
the model limit in the inlet. In addition, surface waves were produced by a wave generator in
the ocean basin. The scales were 1/500 horizontal and 1/100 vertical and the model had a move-
able bed composed of fine sand.
2.2 FROUDE SCALE RATIOS FOR TIDAL MOTION
Tidal motion may be broadly classified into one-dimensional and two-dimensional cate-
gories. In the one-dimensional category, the lateral boundaries of the system constrain the
tidal motion to a direction which roughly coincides with a centcrline drawn between the adjacent
boundaries. Tides in long, narrow gulfs or bays, fjords, estuaries and sea-level canals can
generally be treated in the one-dimensional category. Two-dimensional considerations are
generally necessary in treating tidal motion in broad bays, the mouths of certain estuaries on
the continental shelf, wide straits, gulfs and enclosed seas. In these cases, the influence
of the earth's rotation, through the Coriolis terms, must be considered. Aside from local
221
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simulation of Coriolis effects through the use of rotating cylinders, a complete simulation
requires that the model be mounted on a rotating table. This technique has been used for
hemisphere models of the ocean circulation (Von Arx 1962), small models of the Great Lakes
(Harleman et al. 1964, Rumer and Robson 1968), and for a tidal model of a portion of the
English Channel (Bonnefille 1969). It is generally not feasible for estuary models because
of their large size.
One-dimensional tidal motion in an estuary is described by two differential equations,
one expressing the conservation of volume (continuity equation), and the other expressing the
dynamic equation of motion in the longitudinal direction (momentum equation). The governing
equations have been developed in Section 3.2.3 in Chapter II, Equations (2.106) and (2.116).
Continuity equation:
Momentum equation:
-i /* •* rv ^ it ^t. /* I n I
0 (5.2)
The scale ratios relating quantities in model and prototype are developed by the
method of inspectional analysis (Daily and Harleman 1966, Chapter 7) which requires that the
governing differential equations be written in dimensionless form. Since the tidal model is
assumed to be vertically distorted, it is necessary to specify two geometric and one kinematic
parameter as reference quantities. These are a horizontal length L, a vertical length Y,
and a reference velocity UQ . The following set of dimensionless variables (identified by a
superscript o ) are defined:
x
,0
x/L t° - tU /L
o'
b° - b/L A° - A/LY
(5.3)
h° - h/Y Q° - Q/LYUC
o
U° - U/U0 R° - R/Y
The dimensionless form of the continuity equation (5.1), obtained by substituting the above, is
bo 3h +_ . „ (5
at0 ax°
In a similar manner the dimensionless fora of the momentum equation is obtained
at0 ax° ax° ax c"Y AR
(5.5)
where F « Uo / Vgf is the Froude number.
222
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The model and prototype will be dynamically similar if the two coefficients of
Equation (5.5) are numerically the same in the two systems. Equality of the first coefficient
is obtained by equating the Froude numbers
where the subscripts m and p refer to model and prototype. Equation (5.6) implies that
<5.7>
where the subscript r indicates the ratio of corresponding quantities in model and prototype.
Since gr - 1 the velocity ratio is given by
and the discharge ratio is
The time ratio is
L_ L_
Ur - Yr1/2 (5.8)
Qr - ArUr - LrYr3/2 (5.9)
r
r
1/2
Yr
Equality of the second coefficient in Equation (5.5) is given by
(5.11)
or
C 2 Yr
The Chezy coefficient can be expressed in terms of the Manning roughness n,
r> _ 1.49 Dl/6 /c 1O\
Cw R ^J.U^
or
1/6
Rr
In a wide, shallow channel the hydraulic radius is essentially equal to the depth, therefore,
with Ji - Yr and combining Equations (5.12) and (5.14),
2/3
223
-------
In a typical model in which Lr - 1/1000 and Tfr - 1/100, the velocity ratio Ur - 1/10,
the discharge ratio Qr - 1/106, the time ratio tr - 1/100 and the roughness ratio nf • 1.46.
The latter indicates that the model roughness should be approximately 50 percent greater than
the prototype.
The scale ratios for velocity, discharge and time which are derived from the Froude
number equality should be strictly maintained, however the roughness indicated by Equation (5.15)
should be regarded as a first approximation. It will be necessary to adjust the model roughness
during the verification phase. This is due to the following factors: (i) uncertainty in the
magnitude of the prototype roughness and its spatial distribution; (ii) the assumption of
hydraulically rough flow in the model is not correct since there will be periods of non-
turbulent flow near the times of slack tide; (iii) the assumption that the hydraulic radius
is equal to the depth is not correct in the distorted section of the model. The form of the
model roughness elements which are used to achieve the high degree of resistance indicated by
Equation (5.15) is an important factor. The required energy dissipation can be obtained by
blocks or stones attached to the model bottom, or by vertical rods or strips extending almost
the full water depth. The choice of resistance elements will be discussed in a later section.
In this connection it will be helpful to determine the ratio of the rates of energy dissipation
per unit mass of fluid in model and prototype. The rate of energy dissipation is given by
vQSgix, where Y is the specific weight, Q the discharge and S£ the slope of the energy
gradient. The mass of fluid is given by pAix where p is the density and A the cross-
sectional area. The rate of energy dissipation per unit mass of fluid, G , is given by
(5.16)
x»
and the model-prototype ratio is
Qr(SE)_
r E r - - ur(sE)r
The scale ratio for the slope of the energy gradient (Sg)f ~ ¥r/Lr and Ur is given by Equa-
tion (5.8), therefore
3/2
(5.17)
For the frequently used scales of Lr • 1/1000 and Yr • 1/100, the magnitude of Gr is unity.
Thus, in a distorted model having these scale ratios, the rates of energy dissipation per unit
mass of fluid in model and prototype are equal.
The scale ratios derived in this section provide all the necessary relations for the
control of tidal motion in the model. For example, if the prototype has a semi-diurnal tidal
period of 12.4 hours, the tidal period of a distorted model having Lr - 1/1000 and Yr - 1/100
as given by Equation (5.10) is 7.5 minutes. The tidal range in the ocean basin is scaled from
prototype range in accordance with the vertical scale ratio Yr . Tidal velocities in model
and prototype are related by the velocity ratio given by Equation (5.8) and freshwater tribu-
tary discharges are related by Equation (5.9).
224
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2.3 REYNOLDS NUMBER EFFECTS
The foregoing development contains no similitude condition involving the fluid vis-
cosity or its associated dimensionless parameter known as the Reynolds number. This is due to
the fact that the quadratic friction term characteristic of hydraulically rough turbulent flow
was included in the longitudinal momentum Equation (5.2). The quadratic assumption is justified
for the prototype, and in view of the high roughness distortion it is a reasonable assumption
for the model provided the flow is predominantly in the turbulent regime. When the same fluids
are used in model and prototype it is impossible to have equality of both Froude and Reynolds
numbers. Therefore it must be accepted that the Reynolds number of the model will always be
considerably less than that of the prototype. The Reynolds number ratio is given by
UY,.
*r--£-£ (5.18)
The velocity ratio must satisfy the Froude similitude condition of Equation (5.8) and since
vr-l,
- Y3/2 (5.19)
r
If Y - 1/100, the model Reynolds number will be 1/1000 of the prototype.
225
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3. SIMILITUDE OF MASS TRANSFER PROCESSES
In this section the basic similitude conditions for mass transfer processes are
developed from the principles of inspectional analysis (Harleman et al. 1966). The similitude
of mass transfer implies that the concentration of a certain substance measured in the model at
a fixed point and time is identically equal to the concentration in the prototype at a
geometrically similar point and kinematically similar time. In an estuary model, where geo-
metric distortion is always a necessity, the geometrically similar points are related by the
horizontal and vertical scale ratios. Since the advective processes (tidal motion and fresh-
water inflow) must satisfy the Froude relationships developed in Section 2, the model-prototype
times are related to the geometric scales by Equation (5.10). If the mass transfer similitude
conditions can be satisfied, in addition to the Froude conditions required by the advective pro-
cesses, the scale ratio for concentrations should be unity since concentration is a dimension-
less quantity (mass or volume of substance per unit mass or volume of solution). This statement
implies that all boundary conditions relating both to the advective processes and to the
introduction of the substance into the estuary are correctly reproduced in the physical
hydraulic model.
To determine the conditions under which mass transfer similitude can be achieved, it
is necessary to postulate the form of the differential equation governing the mass transfer
process. In water quality applications we are generally concerned with the distribution and
effect of one or more effluents discharged into an estuary or tidal waterway. If the effluent
is discharged into the salinity intrusion region, the mass transfer will depend on whether the
estuary is stratified or mixed. The governing equations in the two cases are quite different
and the correct reproduction of the salinity intrusion process will be an essential feature of
the physical model. If the effluent is discharged into a tidal region consisting of entirely
fresh or homogeneous saline water, the correct reproduction of concentration distribution will
depend on the ability of the reduced scale model to simulate dispersion associated with the
tidal advection.
3.1 SIMILITUDE OF SALINITY INTRUSION
The majority of existing estuary models were originally constructed for the purpose
of studying shoaling processes. It is well known that the intrusion of saline water is a major
factor in the distribution of sediment in estuaries (Wicker and Eaton 1965, Ippen 1966b and
1966c, Harleman and Ippen 1969). In hydraulic model practice the reproduction of salinity
intrusion in both the horizontal and vertical dimensions has been largely a result of empirical
adjustments during the second stage of model verification. The first stage of model verification
is concerned with the tidal motion and involves the reproduction of local tidal range, phase,
current distribution and magnitude. These are based on the Froude scale ratios derived in the
previous section and verification is obtained by local adjustment and distribution of roughness
elements. Experience has shown that the form of the resistance elements should be chosen in
relation to the type of salinity intrusion which is to be reproduced. The salinity intrusion
can be broadly grouped into two categories: the saline wedge and the mixed estuary. In
Pritchard's (1955) classification these correspond to type A and B respectively. Harleman and
Abraham (1966) suggest the use of a dimensionless estuary number as a means of classifying
estuaries in terms of vertical mixing. The estuary number is defined in the following manner:
226
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estuary number
(5.20)
where P = tidal prism or volume of sea water entering the estuary on the flood tide
FO = Uo//gh, where Uo is the maximum tidal velocity at the estuary mouth and h
is the mean depth
Q, = freshwater inflow rate
T = tidal period
Estuaries of the saline-wedge type appear to have values of this parameter smaller than 0.001,
whereas the value approaches unity for the highly mixed types. This suggests that the saline
wedge occurs with a combination of small tidal range (small PC and F ) and large freshwater
discharge. There are relatively few classical saline-wedge estuaries in the United States.
One of the best examples is the mouth of the Mississsippi where the freshwater flow almost
completely dominates the circulation. Profiles of the salt water interface for various rates
of freshwater flow in the Mississippi River are shown in Figure 5.5.
CJ
z 10
g 20
>- 30
UJ
Ul
LL
E 1*0
z
o
^ 5C
6'J
DATA WERE TAKEN ON CENTER LINE OF SHIP CHANNEL
SALT-WATER INTERFACE IS 10,000 PPM
30 ,000 CFS
_- •
60,000 CFS
^____—-""Too ,000 CFS
1*00,000 CFS
300,000 CFS
200,000 CFS
u\s.
12 13 1 ** 15 16 17
MILES BELOW HEAD OF PASSES
18
19
20
21
22
Fig. 5.5 Profiles of salt-water interface at mouth of
Mississippi for various freshwater discharges.
After Rhodes (1950).
227
-------
3.1.1 Saline-Hedge Estuary
The salinity and velocity distributions in a saline-wedge estuary are shown schemati-
cally in Figure 5.6. A summary of experimental and analytical investigations on this type of
estuary has been given by Keulegan (1966). An equation for the slope of the interface in a
general two-layer stratified flow can be developed by writing the one-dimensional equations of
motion and continuity for both the upper (fresh water) and lower (salt water) layers (Schijf and
Schonfeld 1953, Harleman 1960 and 1969). For the special case of an arrested saline wedge in
which the mean velocity of the salt water layer is zero, the differential equation for the slope
of the interface has the following form:
rfh f V 2
un.0 EJ „ r,,
(5.21)
where f^ is the mean friction coefficient for the Interfacial shear stress and the other
quantities are as defined in Figure 5.7. FH is the densimetric Froude number given by
(5.22)
Equation (5.21) can be written in aimensionless form by defining characteristic horizontal and
vertical lengths L and Y, such that
1- O |- /y
h— n^/it
H° - H/Y
x° - x/L
then
^o --*|-H*o- T—^ <5-23>
FfiLl
Similarity of the two-layer stratified flow requires that the coefficient [~$-J and the
densimetric Froude number F be numerically the same in model and prototype. Thus
m. - m.
or
r " VLr (5'24)
CFH]p
228
-------
SEA
RIVER
FRESH
SALT ^ ^
S.
\\\\\\\\\\\\\\\\\\\\\\\
SEA
RIVER
\\\\\N\\\\\\\ \\V\\\\\\\\\\\
Salt wedge estuary: Above - section along Partially mixed estuary with entrainment and
estuary; Below - typical salinity and mixing: Above - section along estuary; Below
velocity profiles. typical salinity and velocity profiles.
Fig. 5.6 Circulation in salt wedge and partially mixed estuaries.
Fig. 5.7 Arrested saline wedge near the ocean entrance of a freshwater channel.
229
-------
Ur -(*) Yr* (5.25)
- 'r
In addition to the similitude conditions expressed by Equations (5.24) and (5.25), the flow in
unstratified portions must also satisfy the free surface Froude condition given by Equation
(5.8),
Ur - Yr* (5.26)
Equations (5.25) and (5.26) can be satisfied simultaneously only if
. 1 (5.27)
which implies that the density differences in model and prototype must be identical.
To satisfy the similitude requirement of Equation (5.24), an assumption as to the
nature of the interfaclal shear stress In both model and prototype must be made. One possible
assumption is that both follow the smooth- turbulent resistance law in the form of the Blasius
equation
ft - (5.28)
ft*
where R is a Reynolds number characteristic of the flow. From Equation (5.28) we find
(ft)r - i/ ar1A (5.29)
and since Bf - Yf3/2 (Equation 5.19)
r " V*r3/8 (5.30)
Equations (5.24) and (5.29) can be satisfied simultaneously by choosing a model distortion such
that
Yr - Lr8/11 (5.31)
For example, if Lr - 1/1000 , the appropriate vertical scale is Yr - 1/150.
The procedures developed above have been used by Harleman and Stolzenback (1967) in
the design of a model to investigate thermal stratification associated with the discharge of
heated condenser water from a power plant. In this study an additional similitude requirement
is Introduced by the dissipation of heat from the water surface.
3.1.2. Mixed Estuary
A mixed estuary is one in which the tidal motion is sufficient to induce vertical
mixing In spite of the stabilizing influence of the vertical density distribution. The majority
of United States estuaries are classified as mixed estuaries with varying degrees of vertical
mixing. The range of mixed types is illustrated in Figure 5.8 showing laboratory data from the
230
-------
0
0.1
0.2
0.3
0.
-------
Waterways Experiment Station series of tests in a rectangular estuary of constant cross-sectional
area. Test No. 14, the most well-mixed of the series, had an estuary number (Equation 5.20)
equal to 0.34, corresponding to the maximum tidal range and the minimum freshwater inflow.
Test No. 16 had the same freshwater discharge, however the tidal range was half that of Test
No. 14. The corresponding estuary number was 0.08. Test No. 11 is the most stratified of the
series, having the same tidal range as Test No. 16 and a freshwater discharge increase by a
factor of 2.8. The estuary number was 0.03.. In all three tests, the roughness, mean depth
and ocean basin salinity were identical. The longitudinal stations are measured from the ocean
basin.
The one-dimensional mass transfer equation for salinity distribution in a variable-
area estuary is obtained from Equation (2.133). Let s represent the salinity concentration
and designate the dispersion coefficient in the salinity intrusion region as E1 . The source
and sink terms are zero since salinity is conservative. Equation (2.133) becomes
la + u 4^ • 4 4- (AE' <|£) (5 32)
at dx A ax v
-------
V • YrLr <5'36>
In a typical estuary model in which Lr - 1/1000 and Yr - 1/100 , the ratio of dispersion
coefficients in the salinity intrusion region is
Er' - 1/10,000
In hydraulic models of mixed estuaries, there has been a fairly high degree of success in
reporducing the salinity distributions observed in the prototype during the model verification
process. In practice, the fresh-ocean water density difference has usually been equal in model
and prototype, although there is no strict necessity for density equality in models of mixed
estuaries.
3.2 MASS TRANSFER SIMILITUDE IN REGIONS OF UNIFORM DENSITY
We consider an effluent discharging into a tidal region consisting of fresh water or
homogeneous saline water in which the effluent substance is undergoing a first-order decay.
In the absence of longitudinal and vertical salinity gradients, the dispersion is related to
the tidal advection (through the depth and shear velocity) by the Taylor-Elder dispersion
equation (2.124). The differential equation for the mass transfer process, Equation (2.133),
with re - 0 and r^/e - - KdC , is
<5-37>
With the reference quantities L , Y , CQ , UQ and EQ , the dimensionless form of
Equation (5.37) becomes
Ao .o 1C . \ co (5.38)
i£ . (!° U -JL (
ax° \V-/A° ax° V
at0 ax° \UOL/A° ax° V T- ax°
Concentration distributions in model and prototype will be similar only if the two
coefficients of Equation (5.38) are numerically equal in the two systems. Equality of the
first coefficient requires that
^ - 1 (5.39)
or, since Ur - Yr1/2 ,
Er - LrYr1/2 (5.40)
In a uniform-density tidal region, the ratio of model and prototype dispersion is
given by the Taylor-Elder equation (2.124), thus
The depth ratio d - Y , g - 1 and the ratio of energy gradient slope in a distorted model
233
-------
is (Sg)r - Vr/Lr . Therefore Equation (5.41) becomes
Er - Yr2/Lr1/2 (5.A2)
(Harleman et al. 1966).
It is apparent that there is a direct conflict between the similitude requirement
for the ratio of dispersion coefficients as expressed by Equation (5.40) and the ratio of
dispersion coefficients predicted from fluid mechanics considerations as expressed by
Equation (5.42). It should be noted that this conflict does not arise in the salinity
intrusion region, provided that the hydraulic model is made to agree with the prototype through
the process of verification of the salinity distribution. In a distorted model with
Lc • 1/1000 and Yr « 1/100 , the ratio o£ dispersion coefficients required by Equation (5.40)
is 1/10,000 , whereas the predicted dispersion coefficient ratio is 1/316. This means that
the dispersion coefficient in the uniform-density tidal region of a distorted hydraulic model
is much too large. For the above horizontal and vertical scale ratios, the dispersion
coefficient is too large by a factor of approximately 30. The physical explanation of
dispersion, as given in Section 3.2.4 of Chapter II, points out that the magnitude of the
dispersion coefficient is governed by the nonuniformlty of the velocity distribution.
Figure 5.9(a) shows a schematic velocity distribution in a wide, shallow estuary cross section
and Figure 5.9(b) shows the same section with a distortion ratio of 10. It is apparent that
the velocity distribution in the distorted section is relatively much more nonunlform than in
the wide, shallow section; therefore, the dispersion is distorted.
(b)
Fig. 5.9 Comparison of velocity distributions in
channels having a 10 to 1 distortion.
234
-------
Quantitative results substantiating the above conclusion are given in the following
manner:
(i) Assume that, because of verification, the dispersion coefficients in the salinity intru-
sion region in model and prototype are in accord with the similitude requirement given by
Equation (5.36). It follows that
nl_n|»JL/£v / C / -I \
Em EP Yr Lr <5-43>
(ii) Assume that the ratio of dispersion coefficients in model and prototype in the uniform-
density tidal region are in accord with the Taylor relation given by Equation (5.42). It
follows that
Em " Ep Yr2/Lr (5.44)
The division of Equation (5.43) by Equation (5.44) results in an expression for the relative
magnitude of dispersion coefficients in the salinity intrusion and uniform-density regions in
model and prototype,
3/2
S-=J_f^£\ (5.45)
(iii) The magnitudes of E ' and E for the Rotterdam Waterway, as given in Section
3.2.4.2 of Chapter II, are E ' - 13,000 ft2/sec and E = 175 ft2/sec. Thus the prototype
ratio of
indicates a very large increase in dispersion due to the large-scale advective circulation
induced by salinity gradients.
(iv) A series of salinity intrusion tests conducted in a model estuary at the Waterways
Experiment Station was analyzed by Ippen and Harleman (1961, see also Ippen 1966a). The
salinity in the model tidal basin varied between 5 ppt and 30 ppt (ocean salinity) in various
tests, and for each test the magnitude of Em' was determined. In addition, tests for each
tidal condition were made in which the ocean basin contained fresh water with a dye tracer.
These tests established the magnitude of the uniform-density dispersion coefficient Em . The
ratio E '/L is plotted in Figure 5.10 against a modification of the estuary number, Equation
(5.20), defining the degree of vertical mixing. For the model tests in which the basin
salinity was equal to 30 ppt, the increase in the dispersion coefficient due to salinity
circulation is of the order of 4. Harleman and Abraham (1966) have pointed out that some of
the U.E.S. salinity tests may be interpreted as models of the Rotterdam Waterway having a
distortion Lr/Yr • 1/10 .
(v) The above data may be used to check the validity of Equation (5.45). If E '/E =75 and
Lr/Yr - 1/10 , the calculated value of Em'/Em =2.5 . This ratio compares favorably with the
results given in Figure 5.10. It is concluded that the increase in dispersion due to salinity
in the model is small because the dispersion in the uniform-density region has been magnified
by the distortion of the geometry.
235
-------
1*0
20
10
8
6
l«
W.E.S. DATA
D SERIES I (a/h-0.1)
A SERIES II ( o/h-0.15)
O SERIES III (o/h-0.2)
10
100
1000
V
Fig. 5.10 Ratio of dispersion coefficients with and without
salinity induced circulation in a distorted tidal
model at the Waterways Experiment Station.
We return to Equation (5.38) to consider the second requirement for similarity which
involves the- decay rate constant. The equality of the second coefficient in Equation (5.38)
is given by
(5.46)
or
(5.47)
and since U =
it follows that
(Kd)
(5.48)
Equation (5.48) could have been established independently by noting that the decay constant
K, has the dimensions of
t~* , therefore the ratio
In a distorted model in which
r
is the inverse of the time ratio
given by Equation (5.10). In a distorted model in which Lr = 1/1000 and Yf = 1/100 , the
ratio (Kj)r " 100 . Thus the decay constant for a substance introduced into the model should
be one hundred times the prototype decay constant. The possibility of finding a model tracer
substance satisfying this requirement is remote since the commonly used dyes such as Rhodamlne
have small decay rates which depend in part on the absorption of the dye on the boundary
surfaces of the model. Thus, if tracer dyes are used in physical hydraulic models, the
observed concentrations in the model must be corrected to equivalent concentrations for a
236
-------
K.t
conservative substance by multiplying the observed concentrations by e , where t is
measured from the start of the dye injection. The determination of the appropriate value of
Kd for the model is in itself a difficult task. It must be accomplished by periodically
integrating the total mass of dye remaining in the fluid portions of the model and comparing
this mass with the known quantity introduced.
237
-------
4. MODEL VERIFICATION
Model verification is a process by which a hydraulic model is tested to determine its
ability to reproduce certain phenomena which have been observed in the prototype. Whenever
possible the model is adjusted by trial until reasonable agreement with field observations is
obtained. The need for verification is due to the fact that the only precise scale ratios for
a distorted model are those for velocity, discharge and time. Flow processes involving mixing,
diffusion and dispersion are influenced by the geometric distortion of the cross section and its
effect on velocity distributions, the low Reynolds number of the model in comparison with the
prototype, and the arbitrary form and distribution of model roughness and mixing elements. The
validity of model observations is strongly dependent on there being a close relationship between
the phenomena observed in the model and the prototype phenomena used in the verification
processes. For example, because of the marked difference in the mechanism of dispersion in
salinity-gradient regions and in uniform-density tidal regions, the verification of salinity
distributions in the model is no guarantee that dispersion in the uniform-density region is
correctly reproduced in the model.
Model verification is a painstaking and time consuming process which may occupy a
period of one or more years in a major estuary model. Adequate model verification requires the
collection of extensive field data over a significant prototype time span. Very little general
information has been published on verification techniques. They are highly dependent on the
experience of the model operator and it is difficult to formulate general rules. Summary
descriptions of some American tidal models and their verification have been prepared by
Simmons (1966) and Simmons and Lindner (1965). The following sections illustrate model verifi-
cation by showing comparisons of model-prototype data in certain studies.
A.I TIDAL VERIFICATION
The first phase of model verification consists of checking the reproduction of all
aspects of the tidal motion. This includes local water surface elevations, local time phase of
tidal motion, and the vertical and lateral distribution of tidal currents. Adjustments are made
in the model roughness which may consist of stones or rough stucco on the model bed in shallow
portions. In deeper portions, vertical strips of thin metal (3/4" in width) extending to the
low water elevation are commonly used. During the verification process, the strips, which are
placed at the tine of construction, may be cut, bent down or twisted to achieve the desired
local effects.
Figure 5.11 shows a comparison of model and prototype data for the Delaware estuary
(WES 1956) for tidal range, mean tide level and phase. The verification of tidal velocities is
a much more difficult task. For one reason, it is known that the salinity intrusion has a
marked effect on the distribution of tidal velocities. Therefore, It is usually necessary to
maintain the ocean basin salinity and to Introduce freshwater inflows, corresponding to condi-
tions under which the prototype velocity measurements were made, prior to undertaking tidal
velocity verification. The complexity of tidal velocity and salinity interaction can be seen
by referring to field data for a section of the Savannah estuary near the mouth (Sta. 190)
(Rhodes 1950). Figure 5.12 shows the temporal variation of salinity, tide height and current
velocity for one tidal cycle at three different depths for a freshwater inflow of the order of
238
-------
TIDE STATIONS
Sz u
O -I
SUI ^ «
u O O
O Z Z VI
I- O U — W
OC »- O «
o v> o. u
I— O LJ W
o * **j •-
< O UJU OC
£ ui zoc
i—i—i—i—i—i
i i—i—i—i—i—i—i—i—i—i—i—i—i—i
RANGE OF TIDE
5.0 U 1 1 1 1 1 ' 1 ' 1 " 1 1 1 1 1 1 1 1 1
I I I I 1 1 1 1 1 1 1 1 1 1
MEAN TIDE LEVEL
z
HIGH WATER LUNAR INTERVAL
LOW WATER LUNAR INTERVAL
-150
-100 -50 0 50 100 ISO 200 250 300 350
1000-FT CHANNEL STATIONS FROM ALLEGHENY AVENUE, PHILADELPHIA
<»00
MODEL TEST DATA
TIDE MEAN
FRESH WATER DISCHARGE...20,200 CFS (MEAN AT CAPES)
OCEAN SALINITY 26.000 PPM (15.460 PPM CHLORINE)
LEGEND
MODEL
—— PROTOTYPE
Fig. 5.11 Model-pro to type verification for tidal range and phase
Delaware Estuary Model.
239
-------
30
28
26
2k
22
20
16
SALINITY
-V
3 FT BELOW SURFAC
•AT MID-DEPTH
3 FT ABOVE BOTTOM
TIDE HEIGHTS
e>3
o
S
CURRENT VELOCITIES
1
I
•3 FT BELOW SURFACE
•AT MID-DEPTH
•3 FT ABOVE BOTTOM -
8
10 11 12 13
TIME IN HOURS
§ISCHARCES AT CLYO GAGE
CT 18, 1935. 3.700 CFS
OCT 19. 1935. 3.W)0 CFS
OCT 20. 1935. 3.300 CFS
OCT 21. 1935. 3.600 CFS
OCT 23, 1935. 3.800 CFS
15 16 17 18 19 20
SAVANNAH HARBOR. GA.
CURRENT - SALINITY DATA
STATION 190
23 OCTOBER 1935
Fig. 5.12 Vertical distribution of salinity and tidal current -
Savannah Harbor. Qf - 3,800 cfs. After Rhodes (1950).
240
-------
3600 cfs. The data for the same station given in Figure 5.13, for an inflow of approximately
70,000 cfs, shows a marked change in both salinity and velocity. Figures 5.12 and 5.13 refer
only to changes with depth at a fixed location. In addition, there are lateral variations as
shown in Figure 5.14 at Station 155 in the Savannah estuary.
In the Delaware estuary, prototype velocity data for model verification were available
at 74 locations. In about one-half of the locations velocities were measured at three
different depths, while in the remainder only mid-depth values were available. Two velocity
verification results are shown for the Delaware model. Figure 5.15 was selected as representa-
time of the better verifications, and Figure 5.16 is representative of the poorer verifications.
In the latter case the maximum flood velocity near the bottom was 2.1 ft/sec in the prototype
and 3.3 ft/sec in the model, or a difference of approximately 501. The velocity verification
consisted of local rearrangements of resistance elements with an attempt to keep the total
resistance changes to a minimum. This is necessary because major changes would alter the local
tidal elevation and phase which had been adjusted previously.
4.2 SALINITY VERIFICATION
The second phase of model verification involves the comparison of model and prototype
salinity distributions. In the Delaware model it was reported that verification of the salinity
distribution was achieved without further adjustment of the resistance rods. Figure 5.17 shows
a comparison of surface and bottom salinity in model and prototype throughout a tidal cycle,
and Figure 5.18 shows the advance of salinity into the estuary during a two-month period of low
freshwater inflow. The salinity scale in this model was unity and the agreement in each case
is reasonably good.
Another aspect of the salinity verification is the necessity to determine empirically
the proper initial location of a barrier in the model. At the beginning of model operation the
barrier is used to separate the salt and freshwater regions. The barrier is removed when the
tide generator is started and a periodic steady-state salinity distribution is usually obtained
within 10 to 20 tidal cycles.
4.3 SEDIMENT TRANSPORT VERIFICATION
Many estuary models have been designed to study shoaling and sediment transport
effects. This involves the introduction into the model of special light-weight sediments which
are capable of being moved by the tidal currents in the model. Because of the lack of geometric
and dynamic similitude and the low Reynolds number of the model, there is no direct way of
scaling rates and amounts of sediment deposition from model to prototype. This is an example
of the necessity of developing ad hoc verification procedures. This consists of finding an
empirical time scale for sediment transport. The time required for a change in sediment distri-
bution in the model is observed and compared to the known time required for a similar change to
have occurred in the prototype. This time scale may be considerably different from the Freudian
time scale Equation (5.10). For example, in the Absecon Inlet model, the time scale for sedi-
ment transport was 1/675 while the Freudian time scale for the model was 1/50. The prototype
period for verification was three years.
241
-------
SALINITY
3f
30
26
o 22
i l8
r^
Ul
a.
I 6
2
n
1 1 I 1
— ***-"3T"~"*»
/'"""~*"~^
- l«
1 *%
* • •
u
1 «
1 1
%
~"X
— • t
«i A
Mi V\
UsS^^J
1 — 1 K^l 1
1 1
1 1 1 1 1 1 1
^m~.m*~—~"~'
/' /--
I
1 1
J /v
/
i
-
-
*
/
/
/ 3 FT BELOW SURFACE
/ — — AT MID- DEPTH
/ 3 FT ABOVE BOTTOM -
>H — r~r-H — i — i
\ ~
TIDE HEIGHTS
Z 6
2
z
o
.- 0
I
CURRENT VELOCITIES
I,
g '
O
o
S o
-------
S 2
Ul
I/I
a: 0
UJ
o.
_ 2
NORTH SIDE OF CHANNEL
I I
I I
I 1 I
•5 FT BELOW SURFACE
•12 FT BELOW SURFACE
20 FT BELOW SURFACE
13 JUNE I93
-------
5-0
4.0
§3.0
o
t.O
.0
1.0
g2.0
"3.0
4.0
5l°0
Ul
Q.
^
Sen
1 4.0
'§3.0
|32.o
5 i-0
IU w'w
a.
- 1-0
Egz.o
5 3.0
o 5-0
Ul
.U
4.0
§3.0
o
if 2.0
1.0
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1.0
a 2.0
S3.0
4.0
5.0
F
1
f
t
1 1 1 1 1 1 1 f 1 1 1 1
v /7^~^
^^v. s f'
^^-^, ^— ^x'
^,
i i i i i i i i i i i i
1 23 4 5 6 7 8 9 10 11 12
TIME - HOURS AFTER MOON'S TRANSIT OF 75IM MERIDIAN
MIOOEPTH
1 1 1 1 1 1 1 1 1 1 1 i
- ^Ss,^ yX
^"7 _ r^ ~
_ --«..- »— - _
1 1 1 1 1 1 I 1 1 1 I i~
) 1 2 3 4 5 6 7 8 9 10 11 12
TIME - HOURS AFTER MOON'S TRANSIT OF 752£ MERIDIAN
BOTTOM
_ 1 1 1 1 1 1 1 1 1 1 1 l_
I y^^-
V^,^
3.0 "
4.0
.O
lAI
ft.
>•
5n £
4.0 |
3.0 |.
2.0 if
u
°-° 5
1.0 ^
2 0 a*\u
3-0 "**
4.0 £
5.0 g
Ul
4.0
3.0§
o
2.0^
1.0
1.0
2-° S
3-0"
4.0
5f)
....
VELOCITY OBSERVATIONS. STATION 15
Fig. S.1S Model-prototype tidal velocity verification - Deli
Estuary Model, Station 15-F.
244
-------
SURFACE
5.0
4.0
a
| 3.0
^ 2.0
1.0
0.0
1.0
2.0
S j.O
1*1
-------
MODEL OPERATED FOR MEAN CON-
DITIONS WITH SUMP SALINITY
-15,
-------
5. DISCUSSION OF THE USE OF PHYSICAL HYDRAULIC MODELS IN-ESTUARINE WATER QUALITY STUDIES
The use of physical hydraulic models for the investigation of estuarine water quality
problems has an obvious attraction, compared to field studies, in terms of economy and speed of
data collection. Physical models can be operated under controlled conditions and changes in
effluent characteristics, location and discharge rates can be made very easily. The growing
concern with water quality, coupled with the fact that physical models of many of our major
estuaries are already in existence, has led to a rapid increase in model usage for water quality
problems.
The undisputed success of physical models of complex estuaries in providing engineer-
ing solutions to important problems concerned with navigation has generated a high degree of
confidence in these models. This confidence has been built up over a period of several decades
by careful model verification procedures based on large amounts of field data directly related
to the original purpose of the model. A review of the literature dealing with the application
of physical models to water quality studies indicates that there is some danger of transposing
confidence well beyond the demonstrated validity of such models.
Relatively few attempts have been made to explore the theoretical basis for the simili-
tude of mass transfer processes in model and prototype. The important points are that estuary.
models are geometrically distorted (usually by a factor of 10 to 1) and that only gravitational-
inertial similitude is guaranteed by the Freudian scale ratios for velocity and time. In most
cases the only mass-transfer-related process subjected to model verification is the salinity
distribution. The ability to reproduce salinity distributions in physical models is undoubtedly
related to the fact that gravitational effects play a dominant part in the salinity-induced
circulation. Mass transfer processes associated with initial dilution, turbulent diffusion,
and dispersion depend primarily on internal fluid shears related to local velocity and turbulence
distributions, none of which are assured by Freudian similarity.
The discussion .in Section 3.2 of mass transfer similitude is based on an inspectional
analysis of longitudinal dispersion. It is concluded that there is a large distortion of the
one-dimensional dispersion process in the uniform-density regions of estuary models. Further-
more, the Freudian scales relating model-prototype times are such as to preclude the rate process
scaling of non-conservative substances. These considerations suggest that water quality studies
in physical models be approached with considerable care in recognition of potential sources of
difficulty.
The earliest investigations of water quality problems in estuary models were concerned
with the empirical determination of one-dimensional flushing and dispersion characteristics.
The Delaware estuary model was used extensively for this purpose beginning in 1952 (Pritchard
1954, Interstate Comm. 1961, O'Connor 1962). One of the objectives of this type of study was
the determination of dispersion coefficients for use in the one-dimensional, non-tidal advective
mathematical models (see Sections 3.4 and 3.5 of Chapter II). In the Delaware model, dye was
injected in the upper freshwater tidal portion in the vicinity of Philadelphia. In the light
of present-day knowledge, both the validity and the need for such tests is highly questionable.
The validity is questionable because inspectional analysis indicates distortion of the disper-
sion effect. Furthermore, the model had not been verified for mass transfer processes above
the region of salinity intrusion; and in addition, the model dye concentrations must be corrected
247
-------
for dye loss In the model. The need Is questionable because the real time mathematical model
(Sec. 3.2 of Chapter II) relates the dispersion characteristics to the tidal velocities. There-
fore, one-dimensional concentration distributions can be calculated without resort to field or
hydraulic model dispersion tests.
A second group of water quality investigations in physical hydraulic models is con-
cerned with variations in concentration distributions brought about by changes in the location
and geometry of an effluent outfall structure. Potentially this is one of the most important
areas of application of the physical model. The concentration distributions are three-
dimensional and their prediction by means of a mathematical model is generally not feasible
unless the geometry and current patterns can be greatly idealized. Beginning about 1956, the
Delaware estuary model was used for the study of the dispersion of effluents from two industrial
plants (WES 1957a and 1957b). In each'case it was desired to investigate the benefits to be
obtained from changing from a shoreline outfall to a submerged outfall extending laterally to
a deeper portion of the estuary cross section. The site of one plant, near Philadelphia, was
above the region of salinity intrusion, while the other was just below the limit of intrusion
for normal freshwater inflows.
Recently an outfall location study was undertaken on the model of the James River
estuary (Bobb and Brogdon 1967). Tests were made of three submerged sewage outfalls located
4000, 8000 and 12,000 ft. from the shore, the latter being in the middle of the estuary section
above Newport News. The model extends from the Atlantic Ocean at Cape Henry to the head of tide
at Richmond, Virginia. The usual scales of Lr - 1/1000 and Yr - 1/100 were used and the entire
model is 550 ft. long and 130 ft. wide at the widest point. The difficulties associated with
determining the effect of a total lateral change in outfall location of 8 ft. in the model are
readily appreciated. No local verifications were made for either the James or Delaware outfall
model studies. It is therefore difficult to assess the quantitative significance of the local
concentration distributions observed in the model. The state-of-the-art is such that we are
not able to evaluate the effect of a 10 to 1 distortion of the side slopes of the estuary on
the initial dilution and entrainment of an outfall structure.
The Savannah estuary model (Simoons and Rhodes 1965) has been used to determine the
effectiveness of several proposed closure dams in reducing the amount of pollution entering
the Wilmington River from the City of Savannah sewer outfall located in Savannah harbor. A
continuous release of dye simulated the sewage effluent and sampling was performed periodically
for 50 tidal cycles in the model to determine the resulting concentrations throughout the study
area. There were no verification tests related to the effluent discharge and the model results
were not adjusted for dye decay, loss in the ocean basin, or adsorption on boundaries of the
model. Such model results are very useful in a comparative sense.
• An extensive series of water quality related model tests have been performed in the
San Francisco Bay model in recent years. The original series of dye dispersion tests began in
1960 for the purpose of determining the pollution effects of proposed salinity control barriers
within the bay area (O'Connell and Walter 1963). In these tests dye was released at various
locations during one tidal period in the model. Computational procedures were suggested for
correcting the measured dye concentrations for dye loss in the model and for estimating the
"conservative" concentrations to be expected for a continuous release in the model. A uniform
prototype decay rate constant was then applied to the corrected model data for the purpose of
simulating prototype BOD distributions. No verification results were reported for these tests.
Later model tests (Bailey et al. 1966, Bailey 1966) were conducted to study the pollution in
248
-------
the northern arm of the Bay resulting from the discharge of the proposed San Joaquin Valley
Master Drain. Again, model data were corrected by estimating dye losses due to decay, adsorp-
tion and absorption. Field studies were made for model verification purposes by means of dye
injection and extensive sampling in the prototype. The primary difficulty is in the determina-
tion of the appropriate dye loss correction factor for the field data. In the absence of an
accurate means of evaluation of this factor, a normalizing procedure was used in which ratios
of prototype-to-model concentrations were determined for several locations at various times.
The final comparison of model-prototype data indicates a sizeable variability as shown in Fig-
ure 5.19. In general, the prototype concentrations are lower than those predicted by the model;
1000
100
6
k
10
100
"
- 6
12
ce.
o 10
100
6
10
6
STATION 1 (CHIPPS ISLAND)
I I _
-MODEL
BOTTOM
CORRECTED
- PROTOTYPE' PROTOTYPE
x 108
AVERAGE
I I I I I I I I I I I 1 I I I I
I >>.
STATION 3 (RYER ISLAND)
ill i i i i i i i i i i
-..CORRECTED
PROTOTYPE
I I I I I I I I I I I I I I I I III
STATION 6 (PORT CHICAGO)
) I I
-MODEL
x-HOULL
'V- AVERAGE pROTOTYPE x 108
I I I
6 8 10 12
TIME IN TIDAL CYCLES
20
Fig. 5.19 Verification of the San Francisco Bay Model
by comparison with prototype dye dispersion
patterns. From T. E. Bailey, C. A. McCullough,
and G. G. Gunnerson, Mixing and Dispersion
Studies in San Francisco Bay, Proc. ASCE, 92.
No. SA 5 (October 1966), pp. 23-45. ITsed
with permission of American Society of Civil
Engineers.
249
-------
however, the order of magnitudes are reasonable. This represents one of the few attempts at
direct model verification of water quality parameters and it provides an excellent illustration
of the difficulties involved. The fact that the dye injection locations were within the region
of salinity intrusion undoubtedly was an important factor in the ability to achieve some degree
of verification.
Recently, the San Francisco Bay model has been used for effluent disposal tests in
the southern arm of the Bay (Lager and Tchobanoglous 1968). No verification studies were con-
ducted. However, an attempt was made to compare the results of the hydraulic model with the
predictions from a mathematical model. The results, as shown in Figure 5.20, are inconclusive
as they differ by an order of magnitude. It appears that the mathematical model is over-
simplified in that it ignores all advective motion and considers only dispersion and a first-
order decay term (Stover and Espey 1969).
10.0
MATHEMATICAL HO
- HYDRAULIC MODEL
DOWNSTREAM: GOLDEN GATE
UPSTREAM: SAN JOSE
Fig. 5.20 Comparison of computed BOD distributions for
measurements in San Francisco Bay Model with
results from a mathematical model. From
J. A. Lager and G. Tchobanoglous, Effluent
Disposal in South San Francisco Bay, Proc.
ASCE, 94, No. SA 2 (April 1968), pp. 713^236.
Used with permission of American Society of
Civil Engineers.
250
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6. SUMMARY AMD CONCLUSIONS
The similitude conditions for tidal motion in distorted hydraulic models are well
established through the Froude scaling laws. Experience with the large number of tidal models
constructed has shown a good reproducability of the essential features of tidal motion.
Although Reynolds numbers are of the order of 10 less in the model than in the prototype, the
boundary resistance is primarily in the rough-turbulent regime and similarity is achieved
through roughness adjustments in the model.
The analytical basis for similarity of salinity intrusion in both stratified and
well-mixed estuaries is discussed. < For the stratified case (the classical saline wedge), the
requirements are (i) equal density difference in model and prototype, and (ii) a condition
relating the ratio of interfacial shear stresses to the horizontal and vertical scale ratios of
the model. By making an assumption as to the nature of the interfacial stress in model and
prototype, it is shown that the similarity condition is satisfied for only one value of Y ,
if L is fixed. For the well-mixed case, it is shown that the ratio of pseudo-dispersion
coefficients describing the one-dimensional gravity-induced circulation is given by
E * - Y L_ within the region of salinity intrusion. There is no strict necessity for
density difference equality in model and prototype although the practice has been to make the
density differences equal. Similarity of the salinity intrusion can be guaranteed only by
verification of the model with field data. The verification process is assisted by employing
vertical strip resistance elements which promote vertical mixing in the model.
The analytical basis for the similarity of dispersion processes in tidal regions
consisting of fresh or homogeneous saline water (no longitudinal density gradients) is examined.
o 1/9
It is shown that the ratio of dispersion coefficients, given by Er - Yf /Lr ' , is incompatible
with the dispersion coefficient ratio Er' in the well-mixed salinity intrusion region. It is
concluded that the dispersion process in hydraulic models of uniform-density tidal regions is
distorted by virtue of the physical distortion of the cross-sectional geometry. ' Because of the
Froude time scale requirement, it is not possible to simulate directly the distribution of
non-conservative substances in physical models.
The necessity of model verification, using field data closely related to the
phenomena to be studied, is discussed. Examples of model-prototype verifications and the
associated difficulties are presented.
Examples of the use of physical hydraulic models of estuarine systems for problems
related to water quality control are presented. In view of the very limited experience in
direct verification of model-prototype water quality studies, generalized guidelines are
difficult to formulate. Physical model investigations are of value in certain comparative
studies of three-dimensional concentration distributions. The use of physical models in the
determination of one-dimensional concentration distributions appears to be of limited value in
comparison with available computational techniques. Additional attention should be given to
the need for model verification of water quality parameters, although it is recognized that
quantitative verification is difficult due to the necessity of independent dye loss corrections
in both model and prototype.
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Bailey, T. E., 1966: Fluorescent-tracer studies of an estuary. Journal Water Pollution
Control Fed.. 38, No. 12 (December).
Bailey, T. E., C. A. McCullough, and C. G. Gunnerson, 1966: Mixing and dispersion studies in
San Francisco Bay. Proc. ASCE. 92. No. SA 5 (October).
Birkhoff, G., 1955: Hydrodynamics; A Study in Logic. Fact and Similitude. New York, Dover.
Bobb, W. H., and N. J. Brogdon, 1967: Dye dispersion patterns for three outfall locations for
the Warwick River Sewage Treatment Plant. Misc. Paper No. 2-951, Waterways Experiment
Station, Corps of Engineers, Vicksburg, Mississippi.
Bonnefille, R., 1969: Theoretical and experimental contribution on the study of the tidal
processes. Bulletin de la Direction des Etudes et Richerches, Electricite de France,
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Daily, J. W., and D. R. F. Harleman, 1966: Fluid Dynamics. Reading, Mass., Addison Wesley.
Delft Hydraulics Laboratory, 1969: Rijnmond tidal model. Hydro Delft. No. 14 (January).
Harleman, D. R. F., 1960: Stratified flow. Ch. 26, Handbook of Fluid Dynamics. V. Streeter
(Ed.), New York, Mc-Graw Hill.
Harleman, D. R. F., 1969: Mechanics of condenser water discharge from thermal power plants.
Ch. 5, Engineering Aspects of Thermal Pollution. Parker and Krenkel (Ed.), Vanderbilt
University Press.
Harleman, D. R. F., and G. Abraham, 1966: One-dimensional analysis of salinity intrusion in
the Rotterdam Waterway. Delft Hydraulics Laboratory, Publication No. 44 (October).
Harleman, D. R. F., R. M. Bunker, and J. B. Hall, 1964: Circulation and thermocline development
in a rotating lake model. Proceedings 7th Conference on Great Lakes Research.
Publication No. 11, Great Lakes Research Div., University of Michigan.
Harleman, D. R. F., E. R. Holley, and W. C. Huber, 1966: Interpretation of water pollution
data from tidal estuary models. Proceedings 3rd International Conference on Water
Pollution Research. Section III, Paper No. 3, Munich. Pergamon Press.
Harleman, D. R. F., and A. T. Ippen, 1969: Salinity intrusion effects in estuary shoaling.
Proc. ASCE. 95, No. HY 1 (January).
Harleman, D. R. F., and K. Stolzenbach, 1967: A model study of thermal stratification produced
by condenser water discharge. M.I.T. Hydrodynamics Laboratory Technical Report
No. 107 (October).
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Interstate Commission, 1961: Dispersion studies on the Delaware River Estuary Model and
potential applications toward stream purification capacity evaluation. Report to
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Ippen, A. T., 1966a: Salinity intrusion in estuaries. Ch. 13, Estuary and Coastline
Hydrodynamics. A. T. Ippen (Ed.), New York, McGraw-Hill.
Ippen, A. T., 1966b: Salt-water fresh-water relationships in tidal channels. Proceedings 2nd
Annual American Water Resources Conference, Chicago, Illinois.
Ippen, A. T., 1966c: Sedimentation in estuaries. Ch. 15, Estuary and Coastline Hydrodynamics.
A. T. Ippen (Ed.), New'York, McGraw-Hill.
Ippen, A. T., 1968: Hydraulic scale models. Osborne Reynolds Centenary Symposium, University
of Manchester.
Ippen, A. T., and D. R. F. Harleman, 1961: One-dimensional analysis of salinity intrusion in
estuaries. Technical Bulletin No. 5, Committee on Tidal Hydraulics, Corps of
Engineers, U. S. Army, Vicksburg, Mississippi.
Keulegan, G. H., 1966: The mechanism of an arrested saline wedge. Ch. 11, Estuary and
Coastline Hydrodynamics. A. T. Ippen (Ed.), New York, McGraw-Hill.
Lager, J. A., and G. Tchobanoglous, 1968: Effluent disposal in South San Francisco Bay.
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O'Connell, R. L., and C. M. Walter, 1963: Hydraulic model tests of estuariel waste dispersion.
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O'Connor, D. J., 1962: Analysis of the dye diffusion data on the Delaware River estuary.
Evaluation of diffusion coefficients. Report to U. S. Public Health Service,
Regional Office, Philadelphia, August, 1962.
Pritchard, D. W., 1954: A study of flushing in the Delaware model. Chesapeake Bay Inst.,
Johns Hopkins University, Technical Report No. 7.
Pritchard, D. W., 1955: Estuarine circulation patterns. Proc. ASCE, 81, Separate No. 717
(June).
Reynolds, K. C., 1936: Report on model study of Cape Cod Canal and approaches. M.I.T.,
Department of Civil Engineering, Cambridge, Massachusetts.
Rhodes, R. F., 1950: Effects of salinity on current velocity. Evaluation of Present State of
Knowledge of Factors Affecting Tidal Hydraulics and Related Phenomena. Report No. 1,
Comm. on Tidal Hydraulics, Corps of Engineers, Vicksburg, Mississippi.
Rumer, R. R., and L. Robson, 1968: Circulation studies in a rotating model of Lake Erie.
Proceedings llth Conference on Great Lakes Research. International Association on
Great Lakes Research.
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Schljf, J. B., and J. C. Schonfeld, 1953: Theoretical considerations on the motion of salt
and fresh water. Proc. Minnesota Int. Hyd. Convention (ASCE-IAHR). p. 321.
Simmons, H. B., 1966: Tidal and salinity model practice. Ch. 18, Estuary and Coastline
Hydrodynamics. A. T. Ippen (Ed.), New York, McGraw-Hill.
Simmons, H. B., 1969: Use of models in resolving tidal problems. Proc. ASCE. 9_5, No. HY 1
(January).
Sinmons, H. B., and C. P. Lindner, 1965: Hydraulic model studies of tidal waterways problems.
Ch. IX, Evaluation of Present State of Knowledge of Factors Affecting Tidal Hydraulics
and Related Phenomena. Report No. 3, Conn, on Tidal Hydraulics, Corps of Engineers,
Vicksburg, Mississippi.
Simmons, H. B., and H. J. Rhodes, 1965: Results of model investigations, Section 5: Wilmington
River pollution studies. Savannah Harbor Investigation and Model Study. Vol. III.
Technical Report No. 2-580, Waterways Experiment Station, Corps of Engineers,
Vicksburg, Mississippi.
Stover, J. E., and W. H. Espey, 1969: Discussion of reference 40, Proc. ASCE. 95. No. SA 3
(June).
Von Arx, W. S., 1962: Introduction to Physical Oceanography. Reading, Mass., Addison Wesley.
Waterways Experiment Station, 1956: Delaware River model study: Hydraulic and salinity
verification. Report No. 1, Technical Memo. No. 2-337, Corps of Engineers,
Vicksburg, Mississippi.
Waterways Experiment Station, 1957a: Dispersion of effluent in Delaware River from New Jersey
Zinc Company Plant. Technical Report No. 2-457, Corps of Engineers, Vicksburg.
Mississippi.
Waterways Experiment Station, 1957b: DuPont plants effluent dispersion in Delaware River.
Misc. Paper No. 2-222, Corps of Engineers, Vicksburg, Mississippi.
Wicker, C. F., 1969: Special analytic study of methods for estuarine water resource planning.
Technical Bulletin No. 15, Committee on Tidal Hydraulics, Corps of Engineers,
Vicksburg, Mississippi.
Wicker, C. F., and R. 0. Eaton, 1965: Sedimentation In tidal waterways. Ch. Ill, Evaluation
of Present State of Knowledge of Factors Affecting Tidal Hydraulics and Related
Phenomena. Report No. 3, Comm. on Tidal Hydraulics, Corps of Engineers, Vicksburg,
Mississippi.
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DISCUSSION
HARLEMAN: Hydraulic models, that is, physical hydraulic models, have been around for a long
time, roughly since the 1880' s. However, their use in terms of water quality problems is
much more recent, almost, perhaps, within the last ten years. I began their discussion by
considering similtude of momentum transfer processes in tidal motion, because it seems
quite clear to me that if we are going to deal with water quality processes in estuaries
in terms of a physical hydraulic model, that first of all that physical hydraulic model
must do a good job of simulating the basic tidal motion. As has already been emphasized
many times today, the importance of the advective motion cannot be overlooked. And this
basically boils down to a concern with Froude scaling. Also we should observe that we are
always talking about distorted models, because I think it goes without saying that the
possibility of building a geometrically undistorted model of an estuary is rather remote.
They are all distorted, usually by a factor of ten to one, almost a universal factor, so
we are not talking about small distortions but.large ones.
The consideration of Reynolds number effects in relation to the tidal motion turns
out to be not very important for the simple reason that the energy dissipation of both the
model and the prototype is primarily in the rough turbulent regime, and is therefore very
little Influenced by the fact that the Reynolds number of the physical model is very much
smaller than that of the prototype. In fact, for the typical scale ratio of 1/100, 1/1000,
the model Reynolds number will be a thousand times less than the prototype. However, this
is not of great importance simply because the frictional effects which are important in the
tidal motion are not strong functions of the Reynolds number. They are primarily functions
of the geometric roughness, and one goes to distortions of the geometric roughness in the
model in order to achieve this type of similarity.
Then I went tttrough the same type of operation discussing the similitude of mass
transfer processes in models. This is one which has received surprisingly little funda-
mental attention, considering the number of times we have undertaken to study mass transfer
processes in physical hydraulic models. We begin by considering the problem of similitude
of salinity intrusion, and, to keep the problem simple, consider only two extreme cases.
First, the saline wedge--and this is the simple saline wedge in which the upper layer remains
fresh and the lower layer remains salt, without considering mixing or advection between
the two layers. If you follow the process through again in terms of inspectional analysis,
or conditions necessary for similitude, you find again that there are two Froude number
conditions, the densimetric Froude number and the normal free-surface Froude number, which
must be satlsified simultaneously. This can be done if you keep the density ratios the
same in model and prototype, as indicated in Equation (5.27). Secondly, making some
assumptions about the nature of the interfacial shears, which in these cases are probably
strongly dependent upon Reynolds number rather than geometric roughness (since we are
talking about a fluid interface rather than a physical rough boundary), I have simply
made some speculation on the fact that if you choose the distortion ratio properly as
indicated by Equation (5.31) that you could achieve similitude. It is interesting to see
that if you take a length ratio of 1/1000, based upon these assumptions the appropriate
vertical scale is 1/150, which is fortuitously close to the thing that has been evolved
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over many years of deciding that generally this order of 1/1000 horizontally and roughly
1/100 vertically seems Co work. This indicates a physical reason why this works in the
saline-wedge estuary.
Then we go to the other other extreme, the fully mixed estuary. We go through the
the process of taking the simple salinity equation for the variable-area estuary, Equation
(5.32), putting it in a dimensionless form and looking at the magnitude of the coefficient
which appears as this one-dimensional dispersion coefficient for salt divided by velocity
and a length, and conclude that numerically they must be identical in model and prototype
if we are to have similitude of the mixed estuary kind of salinity intrusion.
This is not of any great analytical consequence because, as most of you probably
know, the achievement of similitude of the salinity intrusion regions (assuming now that
we are talking about a mixed estuary) has generally been one of brute force. You cut and
try, and adjust and snip and bend wire strips, and put in a great deal of blood, sweat, and
patience, extending over a period of a year or two, of simple adjustment-verification until
you make the model duplicate or replicate as well as you can, and you decide that now it's
good enough and stop. So it is essentially a forced verification. You always have field
measurements and you always adjust until you make the model replicate those as best you
can.
I conclude from this that here is a mass transport process which people that
operate models generally have succeeded fairly well in replicating in these small-scale
distorted models. However, I will observe that it is a process which is strongly dependent
upon the density effects. The whole intrusion of salt water, even in the mixed estuary, is
critically dependent on the density effects, and therefore it's probably that the ability
to duplicate this is related to the fact that we are achieving similitude with respect to
gravitational effects, basically in the tidal motion. Since the salinity intrusion is
also a gravitational-type circulation process, it is possible to duplicate this. But I
think that the greatest fallacy that we have fallen into in the use of physical hydraulic
models, is to assume, because we have replicated the salinity Intrusion, that all other
mass transfer processes are therefore going to be correct. That assumption is by no means
valid.
We only have to proceed now to consider the similitude for mass transfer processes
in regions of uniform density. By that I mean I don't care whether It is all salt water or
all fresh water, but we're not going to have any density gradients. In going through that
and looking at the scaling relationships for the one-dimensional dispersion, a la the'
Taylor-type equations, we see there is a vast difference in the scaling for this type of
dispersion as opposed to the salinity-intrusion type of mass transfer. It turns out that
the effect of the distorted hydraulic model in the non-salinity-Intrusion regions is to
introduce a very high distortion of longitudinal dispersion effects. The reason behind
this, I think, is shown in Figure 5.9. This is a simple picture of the cross section of an
estuary represented by Section A, and the cross section of that estuary as represented by
Section B which Is what you have in a ten-to-one distortion. It is quite obvious that the
dispersive effects we are talking about now are primarily related to the details of the
velocity distribution in the cross section.
Be that as it may, the question then comes to the use of physical hydraulic
models in estuarine water quality studies. Now there is obviously a great attraction to
the use of the physical hydraulic model, especially since so many of them already exist.
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They have been built to study shoaling and sedimentation problems, and they are there, and
so there is obviously the Important question: why not use them for water quality studies?
I have tried to present hopefully a fairly objective viewpoint of this. But the general
opinion that 1 nave is that you must be extremely careful in doing this whenever the pro-
cess that you are looking at has involved a mixing or dispersion-diffusive process,
because we now come back to the Reynolds number difference and the fact that we are talking
about Reynolds numbers in the models which are three orders of magnitude less than those
in the prototype. It isn't just a question of saying that since we had salinity intrusion
agreement, everything else is going to be fine. I have given some examples of some of the
things that have been done, and simply raised some questions.
I do not believe that we should at.the present time use physical hydraulic models
for the purposes of determining one-dimensional longitudinal dispersion coefficients, for
the simple reason that the distortion effect is there. Furthermore, 1 think we can
adequately estimate these, and we have already shown that if we use them in real time
models they are not so important. So I think we should not use physical hydraulic models
for the purposes of measuring one-dimensional dispersion any more than I think we should
make field tests to do the sace thing.
Potentially the most Important area of application of the physical model is in
determining concentration distributions which are three-dimensional. And this is
obviously the point that is most difficult to do analytically, and so it would be nice if
you could do this in a physical hydraulic model, especially if it already exists. As an
example I quoted the outfall location study undertaken on the model of the James. The
entire model is 550 feet long and 130 feet wide at the widest point. The outfall
studies that they were doing involved moving the outfall a total of eight feet in that
model. I think you have to raise very serious questions whether you have replicated the
velocity structure in this ten-to-one distorted model ( which means that you have distorted
the slope of sides of the estuary by a factor of ten) sufficiently to study the effect of
moving an outfall a total of eight feet out Into the section. No verifications were done
of this type of thing. It was simply a faith operation and everyone must draw their own
conclusions.
I have also cited the rather extensive series of water quality model tests which
have been undertaken in the San Francisco Bay model. Now here you come to the usual
technique of making dye injections into the model. These have been made as instantaneous
(so-called) injections, some of these so-called instantaneous injections extending in time
over one model tidal period, and then others being so-called continuous injections.
It is pointed out in here that in dealing with a non-conservative substance in
the model, due to the time scale involved, there is no possibility of duplicating the
time scale of decay events, unless you happen to have a substance which had a decay rate
exactly, say, one hundred times that of the prototype. If you have 1/1000 horizontal and
1/100 vertical it requires a decay rate of 1/100, a hundred times greater for the model
substance. Well, dyes In the model don't decay a hundred times faster than Rhodamine in
the prototype obviously, so that you have to unravel from these model tests the decay in
the model, which is, in many cases perhaps, largely absorption of the dye on the boundaries
of the model. This is extremely difficult if you are losing mass to the so-called ocean
basin. In that case I really know of no way that you can truly unravel and determine the
model decay rate precisely, because what you would presumably have to do is determine the
model decay rate, correct your model dye data to a conservative substance, scale that up
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to the prototype, and apply prototype decay rates. This becomes an extremely difficult and
messy operation, and I don't think It should be undertaken lightly without major considera-
tion of what you're getting Into. The attempts at verification with field data, I think,
are somewhat inconclusive, basically because of these difficulties of really unraveling
what is really decay and what is simply loss of substance at the ocean boundary, in many
of these cases.
In summary, I suppose it appears somewhat pessimistic, but I think it Is Important
to emphasize the factors which have to be considered, and I think that what has been
largely lacking in the literature is just this type of being a little pessimistic. I do
not make the statement what we should clearly give up all of these, but I think that we
have too often simply proceeded too much on faith, without a real attempt to sit down and
say what are the problems and whether it is worth trying to overcome them. I'm certain in
many cases it may be, but It is by no means a cut-and-dried question.
PRITCHARD: I certainly would agree with Don. He was the first one to point this out. There
has been considerable argument for many years, but I think that it is generally valid that
in this matter of the freshwater region, the uniform-density part of the model, I think
that we certainly agree that we shouldn't use the model to compute one-dimensional flux
or dispersion characteristics. (Remember when we say uniform density, we're talking about
horizontal as well as vertical, and not just vertical. That is, the advective effects will
still be there even though they are pretty thoroughly mixed vertically.) Much of the work
that I've seen done on these models is again dealing with practical problems where one is
concerned primarily with what happens within roughly a tidal excursion of the discharge,
and admittedly one Is left with considerable uncertainty as to how well the system
matches.
In my experience with verifying these models, or what's called verification
which is really adjustment, the practice is to insert roughness elements, and these
roughness elements are inserted primarily on the basis of having the tidal dissipation
correct. There are empirical relationships with which one could build a model now and
put in almost the right number of roughness elements. How, if you have a number of tidal
observations so that you know the phase and amplitude of the tide at a large number of
places in the bay, you use these in verifying the model or adjusting the model so that you
get the right tidal height, the right tidal phases through the system. Then the next
step is to look at observations of velocities in the field and in the model. Now here we
get into a problem of considerable greater magnitude, from the standpoint of the difficulty
of measuring a velocity versus time in the field and in the model, and the fact that
there's going to be considerable more variation in the current structure due to differences
in winds, etc., in the prototype from time to time, so that the comparison between these
is more difficult. But the general tendency is to now redistribute the roughness elements
locally without changing the total amount of roughness, that is, the total energy loss,
to reproduce the currents, or to better fit the current velocities that are observed.
This is all done with fresh water In the model, not trying to simulate the vertical
shears due to density difference at that stage, but primarily the lateral and longitudinal
distribution of the vertically averaged current.
Once the model adjuster says, "This is as far as I can go. I'm tired," then he
puts seawater in the ocean sump, and lets river water come in at the proper rate, and
looks at what the salinity distribution is. My experience has been that at that stage
there is very little further adjustment to the model. The model does behave correctly
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salinity-wise, except, as you have pointed out, there are some times when they have
pointed out that the comparison is pretty good, and there are some details that look
suspicious.
HARLEMAN: In general it's pretty good, I would agree.
PRITCHARD: But, in general, the salinity distribution is pretty good without special
adjustment for salinity. The adjustment is for tidal height and tidal velocity.
HARLEMAN: But there is still an empirical factor there. They could have gotten that
adjustment with stones on the bottom, but by using strips which give the same energy
dissipation they have evolved also getting the salinity. It is not automatic because you
could do the tidal replication against the stones on the bottom and you would not have
any vertical exchange, which the strips introduce. So it's empirical.
PRITCHARD: Right. In any case, once they've done that with the strips they end up with,
apparently as far as the scale of phenomena that controls the salinity, the right kind of
vertical mixing. You see, you not only have to have the gravitational effect, you have
got to have the vertical mixing between the counter-flowing layers, counter-flowing now
with respect to a tidally moving reference system. My point would be that for those
problems of dispersion, or of concern with the distribution of properties, if you're in
a section of the estuary in which these processes of horizontal advection and vertical
mixing are the important processes as far as an introduced pollutant is concerned, then
I would expect that the three-dimensional distribution of the pollutant would be similar.
Now the problem is that there has not been adequate verification, adequate comparison
between model and prototype, that would lay this question to rest. I think that this is
an unfortunate situation. So my argument would be here, not that I would disagree with
Don's cautions, and I don't say this from the standpoint of promoting the construction of
hydraulic models solely for water quality studies, but since there does exist models for
other purposes, it behooves us to lay this question to rest, or at least to get enough
field data, duplicate sets of observations in model and prototype, on the matter of the
distribution of an introduced material somewhere within the estuary. I'm now talking
about within the part of* the estuary in which there is a variable salinity distribution
with distance.
HARLEMAN: I agree with you. I think the problem is that whenever you do this, that is,
you put dye in the model and dye in the field, you have 'these correction factors which
somehow always seem to obscure the answer finally as to whether really it is verification
or it's really an inability to determine these two different decay factors.
PRITCHARD: Yes. I have somewhat of an opinion about the decay, these so-called dye decay
factors. One thing is that one can select dyes which are much less absorbent than some
that have been used, both in the model and in the prototype. And second, I think that
the reason for loss of dye in many cases, particularly in the field, is the fact that
one loses dye outside, because it goes to levels of concentration below the background.
It's very similar to the problem that has arisen with respect to people using field
studies of thermal distributions to deduce that the cooling coefficients in nature are
much higher than those that either theoretically or empirically come from cooling ponds.
Essentially a good deal of the heat is mixed to such low concentrations that one says
that it is zero concentration. And it is not. You don't draw a zero concentration line.
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We've done field measurements for a month at a time, introducing dye into an
estuary of considerable size and telemetering the distribution, and we have found 110Z
of the dye. Now this Indicates the uncertainty in these kinds of processes. It also
indicates probably some variation in the background. But in other cases we have not been
able to find all of the dye, and I think we can readily show by simply a very slight
adjustment to what we call the background (there always is a background here) that we can
account for all of it.
I'm just not convinced. I think that I would rather make the comparisons
directly, at least if I pick dyes that I have pretty good faith in as far as a rate of
decay. I agree that that has been a problem. I think it need not be a problem.
HARLEMAN: I think one can make a good case for getting reasonable results especially within
the salinity intrusion region if things are fairly one-dimensional, but I think I would
point out that it is precisely those problems which we can calculate, and we don't need
models or field tests to do. It Is the three-dimensional local problems that we can't
calculate that we would like to do with models, and I simply have to come back to the
James River where the thing was 130 feet wide and they moved the outfall a total of 8
feet. To ask the real question, is this, in a ten-to-one distorted model, providing the
three-dimensional answers that you think you are getting?
FRITCHARD: Do you mean whether you can really tell the difference between the discharges?
HARLEMAN: That's right.
PRITCHARD: That I would certainly question.
HARLEMAN: If it's one-dimensional, we don't need to do the model tests. We can calculate
it.
FRITCHARD: The question is that the model may not be able to tell the difference between
those two sites. But what about, let's say, the absolute prediction with respect to
what the distribution was from one side, compared to how well you can produce that
three-dimensional distribution any other way?
HARLEMAN: That seems to me the place where the model has the greatest potential, and
where we don't have really any good information on how good it is.
FRITCHARD: Yes, that is what I am saying. Since inherently the model is three-dimensional
to begin with, although distorted three-dimensional, we ought to not abandon it, until
we are sure it has to be abandoned, for those problems in which its unique features would
be valuable.
WASTLER: I would like to add something to this. On the Savannah, we were doing an
enforcement study In 1963, and we did duplicate dye releases in the prototype, and the
results were quite good. But it was not possible to make a mass balance on that because
of'the characteristics of the kind of system, which is half marsh. And then when we were
working in Charleston, we had a half-scale model on the Charleston system which the
Corps built and verified against their original data. We did not do dye tests in that.
However, we had five field surveys in which we had taken time series data of some 30
samples at three- or four-hour intervals over the course of three-quarters of a year.
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What we did was get the Corps to run the hydrograph in the model, and then we sampled the
model precisely as we had sampled the prototype. We found that the results were extremely
good at the high flows but they fell apart at around 10,000 cfs, which we never reported
because what we were worried about was down below that and we had to reach our conclusions
by other ways. But at least as far as the high flows, that half-scale model worked fine.
HARLEMEN: What do you mean by half-scale?
WASTLER: It was 1/1600, scaled down to half the physical size of the Savannah model.
That was one reason that we tried to do this type of verification, because we weren't at
all convinced that half-scale would work. As it turned out it didn't. Also, we found
that the quality of the hydraulic model data went up considerably on the Savannah and
the Delaware and a couple of others. Also we convinced the Corps of Engineers that
seconds were important in their sampling procedure in the model.
HARLEMAN: With a time scale of ten minutes per tidal cycle, it can be very important.
WASTLER: I stop-watched them a few times, and they were anywhere within 10 seconds, 15
seconds, in their normal thing. We got it down to little flashing lights, to take it
right then, and the data improved tremendously.
PRITCHARD: Interesting. Just one comment. Don had pointed out the fortuitous results
that scales of 1/1000 and 1/100 seemed to be right for certain scaling features that he's
described here, one where it came out 1/1000 and 1/150 as being the right scale for one
of the requirements he had. It is interesting to note that for a one-to-one similarity
in cooling coefficients across the surface, 1/1000 and 1/100 is the proper scale. If
you have other scaling, if you're dealing with thermocouples, you have a correction to
the cooling coefficients. It involves the depth scale and time scale, but it involves
the depth scale in a certain combination that comes out to be 1/1000, 1/100. It's not
just a ten-to-one distortion; the actual scaling comes out right. And if you work in a
bigger model, a model like 1/300 and 1/30, your cooling coefficients are off by about a
factor of two.
VLASTELICIA: Can you actually model thermal exchange problems in a hydraulic model?
HARLEMAN: Yes. It's mentioned briefly in here, one example, but certainly not given any
great prominence.
PRITCHARD: Don and one of his students have done a model of a thermal release in Cape Cod
Bay in which I think he has some reasonable confidence. There are models built of
segments of the estuaries, which I have always been concerned about because they weren't
complete estuarine models, but there is one done at Chalk Point. The data taken in the
model was not very good, but, to the extent that can be compared, certainly was grossly
different from what we have observed since the plant has been built, as far as the
temperature distribution in the vicinity of the plant goes.
HARLEMAN: I think we come back to the point again that in most of the real problems it's
the local dilution rather than the cooling that's the important factor.
PRITCHARD: Yes, and besides that, in the real problems, it's related to the momentum
entrainment in the discharge, and this is a Froude-scale phenomenon which can be
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properly scaled in the model. The cooling term probably Is incidental. It is interesting
that it's Che same, but it is not the important term. It's the momentum entrainment in
the jet discharge, and that needs to be densinetrically Froude scaled so that this can be
done in the model and prototype.
RATTRAY: There are a number of things here. I certainly agree with the basic thesis, that
there is some concern about modeling the dispersion of any kind of thing in these
hydraulic models. If I go back to an old paper I did, I did come up once with a relation
for the roughening required for models with certain ratios of roughness lengths to depths
and distortions. So there's a relation that can be determined. Actually, it turns out
that in the model we have for Puget Sound, the toughness elements are effectively large
compared to the depth of the system, the distortion is 1/40, which is larger, and we get
very reasonable tidal representation with no additional artificial roughening.
HARLEMAN: Osborne Reynolds had, I think, distortions of 1/300 and he got pretty good tidal
representations, too. Tidal representation I think you can get.
RATTRAY: It can be understood why and when you need to put artificial roughness in, and
how much. What I guess I'm saying is if you could really model it, you wouldn't have to
do the empirical tests.
The other thing is, In the things we've been talking about, really what the
roughness elements are doing is generating turbulence, and through the distorted model
and distorted roughness elements, you're generating turbulence that's different in the
model from that in the prototype. Basically, Its linear dimension will be distorted in
the vertical over what it would be in the prototype. Now as far as the mean salinity
distribution, If It is the largest scale of turbulence generated that Is responsible for
that (in addition to there being in flow which Is Froude-scaled and tidal flow which is
Froude-scaled) then with a density ratio of one, you can reproduce the salinity distribu-
tion automatically. This is essentially what has been found out. For other scales,
however, smaller scales, you have to worry about what happens to turbulence In these
smaller scales, and there is a tendency towards isotropy as you go down the scale of
turbulence, so you would no longer expect to have the same distortion in smaller scales
as you would have in the larger scales. Again this was a concern that was brought out
that I think is understandable. Basically, I think that the points brought out here are
very important. There are limitations and some uses.
THOMANN: I have sort of a general overall content. Once a few years ago Dr. Harleman so
aptly pointed out the problems of utilizing the physical model, in the freshwater portions
especially, for determining dispersion coefficients. I am much more pessimistic since
that time about the use of such models for water quality work. I think the tendency In
much of our conversation over the last hour or so on physical models has been to equate
water quality with only such variables as salinity, dye or some kind of tracer. If you
look at decay and Interactive or coupled water quality variables, I can't for the life of
me see how you'd ever be able to utilize a physical model for that, even for very simple
water quality variables. I think if we list the dozen or so variables In which we would
be Interested In water quality, there are only relatively few that we could really even
hope to make any Inroads with the physical models. I'm much more than pessimistic In the
use of physical models for water quality. I don't see how we can use them, for example,
In any coupled system of equations. And as soon as we introduce feedback, it's even
worse. The whole issue of reaction rates, scaling down bacteria. You just can't do
those kinds of things.
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VLASTELICIA: Something that would be Important, I think, to us would be some coverage in this
section about very small scale models and highly distroted models that maybe we could
afford to build. We couldn't afford to build the kind the Corps does. But Dr. Rattray's
model has shown some net circulation patterns. It would be valuable to have a management
tool like that. And you could maybe afford to build one, if it is highly distorted and
small scale.
HARLEMAN: Nell, I simply worry about its use in local effects. If you want to use it to
study the kind of problems that involve the gross effects, I have the opinion that maybe
you can do it by calculation.
RATTRAY: I even go further than that. I think Puget Sound may be the only body of water
in the continental U. S. where it could be possible to even make one of that scale.
VLASTELICIA: I think some of the Alaskan estuaries might also eventually, I would suspect,
be worthwhile modeling that way, to make some management decisions. I know an instance
where a model showed that dyes put in at a pulp mill outfall location actually moved up
at mid-depth into a valuable spawning area for bottom fisheries. And there's something
I don't think you could measure ahead of time.
RATTRAY: We knew that was going to happen, but we couldn't convince others until we did
it in the model, even though the model didn't represent it accurately.
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CHAPTER VI
SOLUTION TECHNIQUES
1. ANALOG AND HYBRID TECHNIQUES
James A. Harder
1.1 INTRODUCTION
Digital computers are now sufficiently familiar so that there is no need to give an
introduction to their characteristics. Their very real advantages in many areas of computation,
their essential role in some areas, and the vigorous selling programs maintained by computer
manufacturers have tended to obscure the somewhat older computational techniques of analog
computing, analog simulation, and even hydraulic modeling. Analog simulation and analog com-
puting have had to retreat, in a figurative sense, to those particular areas where they offer
an advantage in convenience or cost over digital computing. The kinds of problems where their
specialized advantages show up in the strongest way have the following general characteristics:
(1) There are only one or two dependent variables.
(2) The accuracy with which prototype measurements can be made does not warrant a high degree
of accuracy in the computations.
(3) A principal problem is that of constructing a workable model, which has coefficients or
parameters that can be adjusted so that a satisfactory degree of agreement in behavior is
obtained between the model and its prototype.
(4) The number of adjustable coefficients or parameters is sufficiently large so that the in-
tegratlve function or 'intuition' of the human mind can be brought to bear with advantage.
While there is no theoretical limitation on the number of independent variables that
can be linked together in analog computations, certainly the advantage of digital computers is
strongest when many variables must be handled at the same time. No one today would realisti-
cally suggest that analog computers be used to solve sets of linear equations.
Accuracy Is a characteristic of digital computers, whereas analog equipment must work
with errors of between .01 and 1 percent. A high cost premium obtains when errors must be kept
at the lower of these figures. However, the accuracy with which tidal flows and pollutant con-
centrations can be measured in prototype estuaries is an order of magnitude less favorable than
these figures, and it does not seem reasonable to insist that overall computational accuracy
exceed that of prototype measurements by more than a factor of two or three.
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The true accuracy of a water quality model, or other model simulating estuarine
processes, lies not so much in the accuracy of the individual computations that go into its
operation as in the accuracy with which it can duplicate the existing behavior of its prototype
and the estimated reliability with which it can extrapolate that behavior to inputs and other
conditions not yet experienced. "Goodness of fit," in the last analysis, must be subjective,
though, of course, a degree of subjective judgement can be built into an objective evaluation
by the use of weighting factors on errors. Differences in behavior between model and prototype
will be distributed in both distance and time, and, of course, our toleration of error will be
distributed too, but perhaps in a different way.
After a model has been built and tested, these errors can be graphed at leisure; cer-
tainly they will be instructive. However, it is during the testing and verification process
that something may be done, in terms of adjusting the model parameters, to reduce these same
errors. Anything that contributes to the facility with which the model can be adjusted to best
duplicate its prototype's behavior is surely of the highest importance. It is in this phase of
verification and adjustment that analog simulation and computation can make their most important
contribution.
1.2 ANALOG MODELING
1.2.1 Analog Simulators and Analog Computers
Hie concise difference between analog simulators and analog computers is that simula-
tors solve equations implicitly, in the sense that both the simulator (model) and the prototype
are governed by the same equations (within simplifying assumptions), and thus the behavior of
one is similar to that of the other. Analog computers, on the other hand, solve equations
explicitly, in the sense that the equations must be developed and subsequently can be solved
without regard to the setting up of a secondary model system. A more obvious difference is
that the computer makes extensive use of operational amplifiers that perform addition, sign
changing, and integration functions.
These differences will be made more explicit by two examples, each of which is of
interest to those who would model estuarine processes of tidal flows and pollution dispersion.
Experience shows that analog simulators are very useful in modeling tidal flows, whereas they
have less success in modeling dispersion when convective terms must be included. In this latter
case, the incorporation of some analog computing elements into a finite-difference scheme shows
promise.
1.2.2 Analog Simulators for Tidal Flows
In constructing simulators, one looks for electric circuit elements that behave, rela-
tive to their own dependent variables, in a way similar to that of the prototype with respect
to Its dependent variables. The general criterion of similar behavior is that the two systems
obey the same equations and that the equations are similar term-by-term. Very often this cri-
terion is relaxed somewhat when it can be shown that some terms that have no counterpart in the
parallel system can be neglected or compensated for.
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We consider now the equations governing one-dimensional flow in tidal channels, noting
that even in the process of setting up the equations, we have made assumptions that may have to
be reexamlned later. Particularly we here assume that water velocities can be averaged over a
cross section, and that this average value can be used in the equations. We take flow as one
of the dependent variables, for this is a quantity that is conserved; the other dependent vari-
able is the water surface elevation. The independent variables are distance along the channel
x and the time t . With these definitions, the equation governing the flow is as follows
(a more complex derivation may be found in Harder and Masch 1961 and Paschkis and Ryder 1968):
n j. O m /A 1\
gn at " ~7I at ' _72 ax " ~73 ax c ax '
gA gA gA
The first four terms of this equation derive from the fact that the total derivative
of velocity, required by the force-acceleration relationship, involves both the distance and
time coordinates, and that the velocity is the ratio of discharge Q , and the cross-sectional
area A . Since each of these latter variables are functions of distance and time, four terms
result. The third and fourth terms constitute the "convective acceleration," and, in terms of
energy, represent the convection of kinetic energy along the channel by virtue of the water
velocity. The first of these four acceleration terms is by far the largest in tidal estuaries,
and is ordinarily the only one simulated in analog models. Thus, some attention must be paid
to the significance of omitting the next three. This significance will be developed in terms
of a physical, rather than a mathematical sense. It is well known that shallow water waves, of
which tide waves are examples, exhibit a wave velocity that depends on channel properties. This
wave velocity, or celerity, is with respect to the water. Thus, the celerity is additive to
the water velocity, and the tide wave generally travels faster relative to the banks than one
would calculate from the formulas of elementary shallow water wave theory. The means of com-
pensating for this effect will be discussed below, after a further discussion of the similarity
relationships.
When we retain only the first term of Equation (6.1) there remains
where A is the cross-sectional area, g is the gravitational acceleration, Q is the dis-
charge, Se is the friction slope, and 3z/9x is the water surface slope. In terms of a
finite length of channel Ax the equation is
(6.3)
A second important equation expresses continuity of flow
g+Bft-0 (6.4)
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where B is the surface width and other variables are as defined above.
form this equation is
In finite-difference
4* at (6.5)
Note that time derivatives are not put in the form of finite differences, for in the analogs to
be discussed, time remains continuous.
E+AE
I + At
Fig. 6.1 LCR circuit for simulation of one-dimensional tidal flow.
Now consider the equations governing the lumped parameter electrical circuit shown
in Figure 6.1. the equation governing voltage drops is
AE
(6.6)
where I is the current, L is the inductance, t is time in the electrical system, R is
the circuit resistance, and AE is the voltage drop across the circuit. The equation govern-
ing the continuity of current is
AI
£= - 0
(6.7)
where C is the electrical capacitance and other variables are as defined above. From these
equations, scale factors relating the hydraulic and electrical parameters are easily derived
and the following table of corresponding variables and parameters may be confirmed.
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TABLE 6.1
Hydraulic Variable or Parameter Electrical Variable or Parameter
1. Q, discharge I, current
2. z, elevation E, voltage
3. Ax, section length (implied in capacitance and
inductance per section)
4. A, cross-sectional area 1/L, reciprocal of inductance
per section
5. B, surface width C, capacitance per section
6. Se, friction slope RI, energy loss per section
7. (B/gA)*Ax, time of travel (LC)S time constant per section
per section, frictionless
We return now to a discussion of the error introduced by neglecting the minor accele-
ration terms. It will be recalled that the actual wave speed relative to the banks is generally
somewhat greater than the value that would be calculated from the channel properties because of
the additive effect of the water velocity. Thus a frictionless tide wave advances through an
estuary with the theoretical velocity u + (gA/B)^ , where u is almost totally determined by
the wave itself, so we are not calculating independent effects. For example, the water velocity
in a frictionless progressive wave is given by u - y (gB/A)^ , where y is the departure of
the water surface from its mean value. Friction and reflections will alter the phase relation-
ship of u and y , and friction will tend to reduce the wave speed.
When u is a small fraction of the wave speed, as is usually the case in estuaries,
a reasonable compensation is to alter the value of the reciprocal inductance, and thus the im-
plied cross-sectional area, to achieve the correct wave speed. This seems particularly appro-
priate when one considers that the cross-sectional area that must be used in the formula is not
the gross cross section of the channel, but only the active part of the cross section where
inertia is important; shoal areas that contribute storage, but where there is little longitudi-
nal flow, must be excluded. This is equivalent to incorporating a factor that depends on the
velocity distribution and which relates the actual momentum in the cross section to that which
would be calculated on the basis of an average velocity. Estuary channels are often so irregu-
lar that little can be inferred directly about the value of the momentum correction factor.
It can only be determined indirectly through an observation of the wave velocities.
Although friction has an effect on the wave velocity, its principal effect is that of
attenuation. Unfortunately, at present we have no way of directly determining friction factors
in estuary channels, for it is not possible to set up a steady flow condition from which we
could determine Manning's n values from discharge and slope measurements. Instead we must
infer friction coefficients from prototype measurements of wave attenuation. From these few
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remarks it should be apparent that one cannot take the simple-minded view that prototype meas-
urements of cross-sectional areas or estimates of friction factors can be inserted directly into
either equations or analog models with the expectation that good results will obtain. There
is no shortcut around the verification process. A corollary is that the achievement of a cor-
rect wave velocity and attenuation in the model far overshadows in importance the inclusion of
the minor acceleration terms; the latter can be compensated for, while without the first, no
confidence can be placed in subsequent measurements in the model.
1.2.2.1 Simulating Hydraulic Friction The second principal term, the friction slope Se ,
is nonlinear in two important ways. If we neglect bank friction, which is reasonable in the
case of tidal estuaries, the equation relating Se to the other variables is best given by
the Manning equation
Se - KQ|Q|/F(z) (6.8)
where F(z) depends on about the 3.3 power of (z - ZQ) and where the absolute value sign is
necessary for one of the Q factors so that the sign of Se changes with that of Q . z0
is the elevation of a local datum approximating that of the bed. The factor K is introduced
to account for bed roughness, assumed a constant. A nonlinear circuit element constructed of
transistors has been devised that exhibits a voltage drop proportional to the current squared,
with the correct sign convention, and having the requisite dependence on an external voltage
proportional to (z - ZQ) . This dependence on z is over a small range, so it is not diffi-
cult to achieve. This "square law" resistor has been described in Paschkis and Ryder (1968)
and Harder and Masch (1961). The latest design is described in Paschkis and Ryder (1968).
An examination of Equation (6.8) will show the several effects of nonlinear friction.
From the numerator we see that the frictional forces, proportional to Se , depend strongly on
the magnitude of Q ; a "square law" element correctly alters the friction in each channel
element as the flows change during modifications of the model in the investigative stage. A
secondary result is that waveform distortions are duplicated in the model. A third effect is
caused by the simultaneous dependence of the friction on Q and on the water surface elevation.
This produces what has been called "tidal pumping." Note that in Equation (6.8) both Q and
z are functions of time. If we assume that there is no steady flow (from river discharges)
the time average of Q over a tidal cycle must be zero. However, positive values of Q (the
flood tide) are correlated with larger values of z , and negative values of Q (the ebb) are
associated with smaller values. Thus the average of Se over a tidal cycle is non-zero, even
in the absence of a steady flow component, whenever there is a progressive wave. (A standing
wave, in which Q and z are 90 degrees out of phase, does not produce this effect.) Thus,
the so-called mean half tide, i.e., the average water surface elevation, increases in the
direction of travel of the wave. If the mean water surface elevation at two points is con-
strained to be equal, however, the result is that there is a net advection of flow in the
direction of wave propagation. This effect of tidal pumping is simulated by analog models
that incorporate the correct form of square-law friction.
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1.2.3 An Analog Computer Element for Pollution Dispersion
The simplest type of diffusion equation with a convection term has the form
(69)
where
-------
2h
K
. _ "
2h
fW-
Fig. 6.2 Analog computation for Equation (6.13),
In this formulation, multiplication by a constant involves only a potentiometer
setting, and one can see that by ganging three potentiometers the value of the constant K can
be altered for all three inputs. If one separates positive and negative values of qp , a single
setting can likewise alter the value of u through the use of double-gang potentiometers.
The simplest representation of a section of channel would require two sets of ganged
potentiometers, two operational amplifiers (one each for sign inversion and integration) and
one capacitor (as a part of the integrator). Inputs to the section would be an initializing
setting (an electronic switch to a variable voltage source) to establish initial conditions for
the integration, values for the constants K and u , and values of qp from adjacent nodes.
In this formulation values of
-------
section can alter scale factors. This feature is useful in setting up certain kinds of problems
at a considerable Increase in cost over hand-set potentiometers.
It appears, however, that the principal use of the digital section is the saving of
results and their convenient recovery for subsequent insertion back into the analog section or
subsequent print-out. It is possible to use the digital section to perform multiplications and
table-lookups, but it would seem to the writer that analog multipliers and function generators
could adequately perform these functions in most instances.
It is significant that in the practical application of analog computers to estuary
dispersion problems there seems to be no convincing argument for the need for hybrid computing.
However, the flexibility that some digital computation might give should not be entirely dis-
counted; its usefulness will be a function of the ingenuity of the operator.
1.4 ECONOMIC CONSIDERATIONS IN ANALOG COMPUTING
Taking the basic operational amplifier as a measure of cost, prices may range from
nearly $1,000 per unit down to as little as $20 per unit, or even less for some integrated cir-
cuit types. What one pays for in the highest priced units is long-term drift immunity, tempera-
ture stability, and the possibility of higher accuracy to a limit of about 0.01 percent. Less
expensive units can be expected to perform at the 1.0 percent level, and may have to be used in
a temperature-controlled environment to reduce drift. However, the provision of constant
temperature cabinets is less of a problem considering the very compact nature of the small
operational amplifier units. The smallest are no larger than a normal sized transistor in a
TO-5 can.
Electronic multipliers are more expensive than amplifiers, but even here, if one is
satisfied with 1 or 2 percent accuracy, integrated circuit versions can be bought for $25. The
point is that the data usually available from pollution studies are rarely known with anything
like one percent accuracy; and even if one takes into account that the cascading effect can
increase the errors resulting from inaccurate components, it seems that the less expensive
components would serve very well.
However, those firms that rent the use of analog computing equipment must ordinarily
design their equipment for the most exacting of their customers, and this means expensive equip-
ment and a correspondingly high rental figure. If it appears that the kinds of problems en-
countered in estuarine pollution studies can be profitably studied using analog computing
equipment, one has two choices: rental of equipment, or purchase and in-house use of equipment.
From a consideration of the accuracy requirements, it would seem that even a very large capacity
analog computer could be assembled for no more cost than a nominal amount of rental of the high
cost equipment normally available. The limitation would then be the skills of the personnel
operating the equipment. However, high skill in this aspect of the operation is really neces-
sary whether one rents or buys, for the rental firm cannot be expected to be experts in every
field where their analog equipment is used.
It seems to the writer that under the assumption that analog equipment can be shown
to be useful, an agency would be better advised to gather a small group of skilled operators
and electronic experts, and let them have an in-house computer available to them all of the
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time. Very likely such an operation could handle all the problems In the U. S. and consist of
a central laboratory facility.
1.5 SYSTEM-RESPONSE TYPE MODELS
If the water quality aspects of an estuary are considered to be a system in which
inputs (in terms of nutrients, waste products, etc.) are related to outputs (such as BOD, algae
levels), models can be constructed that do not make explicit assumptions about the physical
processes that are taking place. This is often advantageous in those instances where we have
little information about the actual processes.
A primitive form of such models is one in which one assumes a linear relationship
between inputs and outputs; this is the same as assuming that each input acts independently of
every other. This assumption is hardly realistic, but does offer the advantage that very simple
mathematical techniques can be used. When one introduces the possibility of nonlinearity, the
models become much more realistic, but the computational aspects become enormously complex, and
tax the capabilities of even the largest computers. However, the trend towards ever faster and
sophisticated computers will enable us to use system-response type models that are an ever more
realistic means of understanding the process being modeled. The understanding, it must be em-
phasized, is in terms of predicting the response of the system to different levels and combina-
tions of inputs, not in terms of understanding the actual physical processes themselves.
The feasibility of incorporating nonlinearity into systems analysis was advanced
markedly by the work of Weiner (1958) and his colleagues at MIT (Bose 1956, Brilliant 1958,
George 1959). Further work has been done by a group at the University of California in
Berkeley (Amorocho 1961 and 1963, Jacoby 1962 and 1966). Recently Harder and Zand (1969) have
brought the technique to a practical application. The general approach will be described in
the following paragraphs.
In the process- to be described, which has been called "the identification of non-
linear systems," the solution sought is the "system response function" by which an output
function (usually a function of time) can be predicted from one or more input functions. In
terms of water quality, the output function could be BOD at a particular location, algae popu-
lation in a particular section of an estuary, etc. In one of the example problems examined by
Harder and Zand (1969), the output function was the salinity at a certain location in an estu-
ary, and the input function was the freshwater flow at the upper end of the estuary.
The system response function is obtained from an analysis of the past behavior of the
system and is independent of any knowledge of the actual physical processes involved. Thus it
does not use all the information available if the information includes such data as actual
rates or processes, etc. On the other hand, it does not use any misinformation about what
might be going on. It is a useful tool for predicting new behavior that is within the bounds
of past behavior; it is not so reliable for extrapolating behavior.
The following explanation will use the example of salinity prediction as developed by
Harder. The available input information was a record of freshwater Inflow to the upper end of
the San Francisco Bay estuary (Suisun Bay and the contiguous channels) over a period of ten .
years. This was discretized into averaged values for each one-third month period, giving 360
samples. Likewise, the salinity at Benicia (part way down the estuary) was discretized into a
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similar series of averaged values. Through a preliminary analysis it appeared that the "memory
time" or the period after which the input influence on the output became negligible was about
three months or nine periods. This meant that there were 351 (360 minus 9) sets of input vari-
ables; within each set there were nine sequential values of freshwater flow. Each of these sets
was assumed to influence the value of salinity in the one-third month period immediately fol-
lowing the three-month period spanned by the set. Had a linear analysis been applied to this
problem, nine weighting coefficients could have been attached to each of the nine input values
preceding an output value (and distinguished by the lag time between each input value and the
output value). Then a least-squares determination of these coefficients would lead to an equa-
tion from which the salinity (the output variable) could be predicted from a linear combination
of the flows (the input variable) in the nine preceding periods. The error, while a minimum in
the least-squares sense, is not the smallest that can be obtained.
Further reductions in the error of estimate can be obtained by including additional
significant variables. In a nonlinear system these additional variables can be obtained by
making use of squares and products of the original variables. In the present example, for
instance, we could use nine squares and thirty-six distinguishable products, for a total of 54
linear and quadratic variables. This approach, while possible, is not necessarily the most
efficient.
Harder and Zand (1969) made several improvements to this approach. First the input
data sets were subjected to an orthogonal transformation so that the input values in each set
could be represented more compactly; instead of nine values, perhaps six would be adequate.
This transformation is linear, and tends to remove higher frequency components that might not
be important. It has an additional function of allowing a weight factor to be introduced so
that more distant (i.e., earlier) input variables can be given less weight than the more recent.
The output of this linear economizing process is the input to the second stage in the analysis,
the nonlinear process. Here products, squares, cubes, etc., are produced; these are called
"super-variables" and are in turn entered into phase three, the stepwise linear regression
processor.• In this phase each of the super-variables is examined, and a determination is made
of which one reduces the variance (between the measured and predicted output) the most. Pro-
vision is made at the present time (1970) to examine 74 variables at a time. At each succeeding
step all the remaining variables are examined and the one which reduces the variance the most
is included in the working set. At the same time, variables which have been included earlier
are also examined, and those which became ineffective in reducing the error are dropped. The
stepwise analysis continues until no addition or subtraction of a variable is effective in
reducing the standard error of estimate by more than a stipulated minimum amount. At this
point the associated super-variables and their coefficients are assembled into a predicting
equation for use with new data. This mathematical construct constitutes the indentification
of the system response function, and can be used to predict the response of the system to new
input sequences.
In the Suisun Bay salinity analysis, the root-mean-square error between measured and
predicted values was 15.9 percent. This figure included field measurement errors in both salin-
ity and flows, and the error from ignoring other influences on the salinity such as evaporation,
monthly variations in the mixing intensity as influenced by changing tide patterns, etc. As a
check, a completely deterministic mathematical model of the estuary was constructed which ex-
hibited approximately the same time constants and steady-state response as the real system.
This artificial salinity system was then subject to the same input function as the real system
(it was "forced" by the input function) and a "true" output obtained. The difference between
the predicted output, from a system response analysis of the artificial system, and the "true"
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output constituted an error that could only be attributed to the analysis process itself. This
error turned out to be 8.1 percent. By inference, were the various errors random and uncorre-
lated, the error in the real system that is not due to the analysis itself would be 13.7 percent.
In a second practical example in which two input functions were of importance, the
inclusion of the second input, together with its interactions with the first, reduced the error
from 7.4 percent to 4.4 percent.
In conclusion, it should be pointed out that system-response type models are in an
active state of development and offer a promising new approach to certain kinds of water quality
problems. Their further development should be encouraged.
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REFERENCES FOR SECTION 1
Amorocho, J., 1961: Nonlinear analysis of hydrologic systems. Ph.D. dissertation, University
of California, Berkeley.
Amorocho, J., 1963: Measures of the linearity of hydrologic systems. J. Ceophys. Res., ji8 (8),
pp. 2237-2249.
Bose, A. G., 1956: A theory of non-linear systems. Res. Lab. Electronics Rept. 309,
Massachusetts Institute of Technology, Cambridge.
Brilliant, M. B., 1958: Theory of the analysis of nonlinear systems. Res. Lab, Electronics
Rept. 345, Massachusetts Institute of Technology, Cambridge.
George, D. A., 1959: Continuous nonlinear systems. Res. Lab. Electronics Rept. 355,
Massachusetts Institute of Technology, Cambridge.
Harder, J. A., and F. D. Masch, 1961: Non-linear tidal flows and electric analogs. Proc. ASCE.
JJ7, No. WW 4 (November), pp. 27-39.
Harder, J. A., and S. M. Zand, 1969: The identification of non-linear hydrologic systems.
Report HEL-8-2, Hydraulic Engineering Laboratory, University of California, Berkeley.
Jacoby, S. L. S., 1962: Analysis of some nonlinear hydraulic systems. Ph.D. dissertation,
University of California, Berkeley.
Jacoby, S. L. S., 1966: A mathematical model for nonlinear hydrologic systems. J. Geophys.
Res.. _71 (20), pp. 4811-4824.
Paschkis, V., and F. L. Ryder, 1968: Direct Analog Computers, New York, John Wiley & Sons,
pp. 95-113.
Welner, N., 1958: Nonlinear Problems in Random Theory. New York, John Wiley & Sons.
276
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2. DIGITAL TECHNIQUES: FINITE DIFFERENCES
Jan J. Leendertse
2.1 INTRODUCTION
One of the most powerful methods in the integration of partial differential equations
involves the replacement of the differential equation by one or more equivalent difference
equations and solving the latter numerically. This approach has become increasingly more
important with the growth in computation speed of computers and the increasing size of com-
puter memories.
The difference equations used are approximations of the partial differential equations.
Many different approximations can generally be made for each term of the partial differential
equations, thus resulting in a wide selection. One would expect that one approximation might
be better than another and this is generally true. The choice may also be governed, depending
on the scope of the investigation, by the processes described by the equations, which may vary
widely for the same equation. For example, in a computation of the tide in an estuary, we may
select a scheme that places the emphasis on a good representation of tidal height in the area,
while the mechanism of the dissipation of the wave energy by bottom friction, which in such
cases is generally considerable, is not considered important. The scheme which we might use
can even dissipate some energy just by the computation, and the rest of the dissipation is by
some expressions in the finite-difference equations. Another scientist studying ocean circu-
lation will be integrating the same long-wave equations, but is interested in the momentum
transfer mechanism and the energy balance in the ocean. He will need a scheme that accurately
describes the momentum fluxes, and in addition conserves energy. Another example is computa-
tion of the dispersion, of pollutants. In working with rapidly decaying substances, a small
decay by the computation may be acceptable, which would.not be the case if the substance
is conservative.
Not only have we to try to give the difference equation certain properties, we have
to assure ourselves before all the effort is made to write a program, which may be the work
of many years, that the computation can actually be made. In certain instances, the computa-
tion simply breaks down as errors start to grow and overshadow the solutions. Setting up and
designing the computation requires considerable skill. The experienced worker in this field
generally will be extremely careful in setting up his computation and will try to make predic-
tions of the behavior of the computation. If certain detrimental effects do occur unexpectedly,
he will try to trace the source of them and eliminate them, rather than arbitrarily make
changes in the approximations. In the next sections, some of the methods which are used in
these computations will be discussed.
From the above discussion it will be clear that the engineer or scientist who is in
charge of an investigation must have experience in designing mathematical models and must know
exactly the computational procedures used in order to assess the applicability and character-
istics of the model which he uses in his investigation. As different approximations of the
277
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partial differential equations may lead to different representations of certain effects in the
physical processes, the use of "box" or "cell" modej.8 is not advisable. The use of such models
fixes inmediately the finite-difference equations without regard to computational effects.
2.2 CONSIDERATIONS IN DESIGN OF COMPUTATION SCHEMES
2.2.1 Conservation of Mass
In a numerical computation, the mass in a bounded system should be conserved when
account is made for the loss or gain* of mass through the boundaries and for the sources and
sinks in the system. For example, the continuity equation or the long-wave equation can be
written
o (6.14)
at ax
where C - the distance between the water surface and the reference level,
H » temporal depth, the distance of the surface to. the bottom, and
u - the velocity (averaged over the depth H) .
Integration of Equation (6.14) in time and space between boundaries Tj , T2 , L^ , and L2
yields zero.
/2 |T2 /-T2
C dx + /
"I Tl JTl
(Hu) dt - 0 (6.15)
A difference scheme approximating Equation (6.14) and which satisfies the conserva-
tion of mass can be written, for example,
AX
(6.16)
)
-
where C ~ water level at time t and at location
m
h - depth from reference plane downward.
Numerical integration of (6.16) over one time step At for the range m-1 to M results in
M M .
ZV1 \ L.
C* ' to 2, 4 + " Uh% + KC1* + Cl * Co+ + Co)l u%
m-1 m-1 /
V ) (6.17)
278
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This shows that over one time step the mass Is conserved; the latter two terms in the equation
are the expressions for the inflow over the boundaries of the region. All other terms in the
mass-fluxes cancel each other, even with varying velocity. (Equation (6.16) is difficult to
solve because of the quadratic terms, and has been used here only as an example.)
It will be noted that Equation (6.14) can also be written
S£ = 0
at dx ax
With this representation it is very difficult to set up a difference scheme such that the
numerical integration cancels all the finite-difference approximations of the fluxes in the
field between the boundaries.
From a careful analysis of the finite-difference equations, an experienced engineer
or scientist will rather rapidly determine if the conservation laws are satisfied. Sometimes
an experimental investigation has to be made. For example in a tidal computation, the varia-
tion of the balance of the content of the bay minus the inflow over a tidal cycle should be
small compared to the tidal prism. In an advective and diffusive transport computation which
is coupled to such a tidal computation, a computation of the mass balance of a conservative
substance should be performed. In addition, a computation should be made with constant mass
concentrations as initial conditions and the same constant mass concentration on the boundaries.
All values in the Whole system should stay constant during computation over a tidal cycle.
2.2.2 Stability
In the finite-difference equations employed for approximating the differential
equations, it is also required that numerical errors introduced in the computation do not
amplify in an unlimited manner.
One approach toward the investigation of stability (O'Brien et al. 1950) is to follow
a Fourier expansion of a Une of errors as time progresses. Let us assume to have a finite-
difference approximation
(6.18)
Suppose that a line of errors 8C(x) exists at a particular time t , which we set conven-
iently at t - 0 . It is possible to make a finite Fourier decomposition of these errors as
follows:
6C(x) •
In our computation a finite number of points N exists in the x direction and the number
of terms n of this decomposition equals N .
We consider u to be constant, so that a linear system is being considered. Con-
sequently, the behavior of only one term of the Fourier series needs to be followed. The
279
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components A,, are time -dependent, and, in order to satisfy the expression of the particular
error wave at t - 0 , the components must take the form
An ' ^ en <6-20>
where a,, and Bj, are constants. Thus the expressions for the errors at (t,x) of the mass
concentration C should satisfy the form
i8_t iox
«C(x,t) = ¥n e ^* e (6.21)
It is assumed that the errors are perturbations imposed upon the solutions of the linear sys-
tem. If we subtract the exact solutions from the solutions of the difference equations with
the perturbations, we obtain among the error components a set of relations that are identical
to the relations for the components of C , as we are dealing with a linear system. Intro-
ducing Equation (6.21) into the set of equations which represent the relation of the errors
and using only values of x and t on the grid results in relations between 8 and a .
Using X - e1Bit then
(x - Dl + u |f(etoix - e-lofix)a- - T^ — " (eiaAx - 2 + e~ia^)J - 0 (6.22)
2Ax ^ (fix)2
from which we find for nontrivial solutions
(6.23)
1 . iu 4* sin(cax) - 4 - - sin2(oax/2)
n
In order that the original error, with components A,, e , shall not grow as t increases,
it is necessary that
Thus, it is necessary for stability that
|X| s 1 (6.25)
Therefore from Equation (6.23) and Equation (6.25) we can derive the condition which is nec-
essary for stability. For example If u - 0 , then
(Ax)"
Even for this simple linear case, the derivation of the necessary conditions for
stability is a considerable task, and it becomes increasingly more difficult if systems of
equations are involved, such as the long-wave equations in two directions. In the latter case,
necessary conditions for stability are found by a similar process as described above, but now
with vectors and matrices, and the stability condition will finally be derived from the eigen-
values of the so-called amplification matrix. In the simple case (Equation 6.18), the amplifi-
cation matrix has only one component, which is its eigenvalue X .
280
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Up to this time we have concerned ourselves only with linear equations and it may be
noted that in the discussions the conditions we have derived are conditions necessary for sta-
bility. Conditions which are also sufficient for stability are much more difficult to obtain
(see Richtmyer and Morton 1967).
One particularly troublesome phenomenon is the generation of nonlinear instability.
The analysis described above is applicable for linear systems. In fluid flow computations,
this is generally never the case; the equations are nonlinear or have variable coefficients.
If we have a term of the type u du/dx in a two-dimensional system, the velocities are sums
of Fourier terms
u* cos (a, x + a, y)
^ zm
The expression u 3u/3x then yields expressions with sums and differences of wave numbers
Cj and
-------
dispersive effects will change Che shape of the distribution by the differences in the speed
of propagation in the computation. The field becomes deformed; the fluctuations with shorter
wavelengths are generally overtaken by those with larger wavelengths.
As an example, the one -dimensional advective transportion equation will be used,
•2S.+U.S- - n 23L • 0 (6.26)
at ax *2
for which we now use the approximation
n " (rt+l *rt+1 j. pt+M n if. n\
- DX Cm+l - 2C,,, + C^,,; - 0 (6.27)
Following the derivation in Section 2.2.2, we obtain the eigenvalue
+ *£iu sin(oix) + 4DT sin2(oAx/2) • -^-^-1 (6.28)
It can be expected that computation with this scheme will meet no difficulty. The absolute
value of the eigenvalue is smaller than unity, thus the necessary condition is met and the
computation is stable, assuming u is a constant. Any spatial fluctuation or wave will decay
in amplitude. This fact is also true if the dispersion or diffusion coefficient is zero. In
the latter case, the computation introduces dissipation and this dissipation is dependent on
the time step, the spatial grid size, the velocity, and the wave number a . The fluctuations
with short wavelengths are particularly affected, and this computation acts as a filter on
such short waves.
The dispersive effect can be investigated also by use of the eigenvalues. To illu-
strate this effect again the assumption is made that the dispersion coefficient in Equation
(6.26) is zero. The propagation speed of the Fourier components is naturally the velocity u .
The following relation between frequency and wave number exists:
IB/O • u (6.29)
For the finite-difference equation (6.27) we find, following the explanation in Section 2.2.2,
for the eigenvalue
X- elBI'At - l/(l +^£iu sln(aix) + 4 -5—- sin2(ffAx/2)) (6.30)
to (Ax)Z
where m' - frequency of the computational system. As an eigenvalue smaller than unity was
obtained, this frequency ta' is a complex number, of which the imaginary part is related to
the dissipative effect. If we set
A - ||u sin(oAx) (6.31)
282
-------
then for Dg • 0 , Equation (6.30) can be written
(6.32)
from which we find for the real part
Re [ID '4t] - tan"1 (A)
(6.33)
With an expansion for the inverse trigonometric function, this becomes
10 'At - A - A3 + i A5
or
(6.34)
ci>'At - ^ u sin(CTAx) fl - i A2 + i A4 - . . .1
Ax L 3 5 J
(6.35)
and by also replacing sin(aAx) by its power series then
3!
4!
(6.36)
Both series are smaller than unity. Thus for all values of At u/Ax , the computed wave
propagates slower than the analytic solution. When Ax - 0 and At - 0 both series approach
unity and, as we would expect, the computed wave speed converges to the speed of the analytic
solution, as shown in Figure 6.3. It will be noted that when the representation is on a grid
with less than ten points per wavelength, the representation becomes very poor. The represen-
tation in the spatial dimension is more important than the discreteness in time.
100
WAVELENGTH L/AX
Fig. 6.3 Ratio of the velocity in the model and velocity in
prototype as function of the spatial representation.
283
-------
In the analysis of a computational scheme, the dissipative effects can be expressed
as the ratio of the amplitude of the computed solution and the amplitude of the theoretical
solution after the time interval that the Fourier component travels over its wavelength. For
Equation (6.26) and DX • 0 , the amplitude of any component of the analytic solution stays
the same. For the computation using (6.27) with each time step, the amplitude decreases ac-
cording to the absolute value of X .
For the time step At , the number of times that we have to compute for the time
interval that the analytical solution propagates over its wavelength is
n - -t- - 2n/uito (6.37)
uAt
The ratio of the amplitudes becomes
|T(ofix)| - [l/ 7l + A2]" (6.38)
This ratio is called the modulus of the propagation factor (Leendertse 1967). By plotting
this modulus as a function of the number of points per wavelength, a clear insight is obtained
into the filtering effects of the computational schemes.
The dissipative and dispersive effects also manifest themselves in tidal computations.
The physical process in itself generally is rather strongly dissipative by the bottom frictions,
which then are used to adjust tidal heights in the verification of the model. With a coarse
grid, the higher frequencies in the tidal curves are generally dampened or disappear, and the
phase of the tidal curves cannot be matched.
2.2.4 Effects of Noncentered Operations
For conceptual reasons (Thomann 1963) or for reasons of stability, the advective term
is sometimes taken noncentered in space. Again, this will be studied using the linear advec-
tive transport equation (6.26) for which we now use the following approximation
Cm
- *>ce ,1
' m-l-l
(6.39)
Cm-l] -
where a is a dimensionless weighting factor. This factor is generally assumed to be in the
range of -1 < a. < 1 . The form (6.39) is similar to a form in which a so-called advection
factor (?) is used, where we have the relation
, 1.1
5-2+2*
Equation (6.39) can also be written
284
-------
rt+l _t At_ ft t "I
m m + 2Ax LCm+l ' Cm-lJ
' t t t \
C - 2C + C ) = 0
^ nH-1 m m-l/
It will be clear that the off-centered differencing is in effect an increase or
decrease of the dispersion coefficient. Following the LeLevier method, as described by
Richtmyer and Morton (1967), so-called "upstream" differencing results in an increase of sta-
bility of the computation and an effective increase of the dispersion coefficient. Such up-
stream differencing uses, then, the values for a ,
a < 0 if u > 0
a > 0 if u < 0
The artificial dispersion introduced in this manner makes the actual values of the dispersion
coefficients used meaningless if we do not account for these effects. Dispersion coefficients
are generally found by trial and error until observed values match the computed values, the
use of off-centered operations will now make the apparent dispersion coefficient a function of
the grid size.
The forward implicit scheme (Equation 6.27) which has been discussed previously,
showed considerable artificial dispersion caused by the noncentered forward stepping. It was
shown also that the forward-stepping scheme (explicit) showed just the opposite, as for small
wavelengths the system becomes unstable. In other words, the dispersion term is not able to
diminish some components.
In view of this, it can be expected that the alternating use of the explicit and
implicit scheme might lead*to better results, and this can easily be derived from above dis-
cussions. In this manner multioperation schemes are formed, in which two (or more) approxi-
mations are used for one partial differential equation. Since implicit schemes are often
difficult to solve, particularly if several spatial dimensions are present, a multioperation
approach will give flexibility in computations, as spatial gradients can be taken alternatingly
in the forward and backward fashion, the equations become, then, implicit in one dimension
only.
2.2.5 Discontinuities
Considerable difficulties are encountered if the system, which we represent by
finite-difference equations, has discontinuities. At such discontinuities the Fourier decompo-
sition will present large contributions of very short wavelength, which we are not able to
represent. One of the most frequently used ways, first used by Von Neumann (see Richtmyer and
Morton 1967), Is to introduce artificial viscocity or dispersion at the location of the dis-
continuity, thus locally smoothing the solution. An expedient way is to use local upstream
differencing. For example, for the advectlve transport equation with a source at location m ,
the finite-difference expression becomes
285
-------
t+1 t at ..
Sn ^n + TIT u
2 Ax
(6.40)
If u > 0 , where Sg, is the contribution of the source per tine interval. To conserve mass,
the equation which ia upstream of the point m , in this case at m - 1 , has been written
_t+l rt at /7rt+l rt+l
Cm-l • Cm-l + TAV uV2Cm-l ' Cm-l '
(6.41)
With this approach, the influence of the source at location m is not felt upstream other
than through the contribution of the dispersion term. If the normal finite-difference equa-
tion is used at the source, the upstream point generally becomes severely suppressed, and in
actual computations a dampened spatial oscillation appears.
A more elegant method than the one described above is the introduction of artificial
dissipation in such a form that the contribution is negligible if the adjacent concentration
values are nearly equal, but is the required additional dispersive effect if adjacent values
are widely different (Richtmyer and Morton 1967). For example, the disperson coefficient m +
can be expressed
1 +
where K is a constant, which has to be determined experimentally.
2.2.6 Comments on Higher Order Schemes
In mathematical modeling, the spatial representation or discreteness is often a
limiting'factor, particularly in two-dimensional computations, as large arrays have to be kept
in active memory of the computer. First the representation of depth or volume is generally
somewhat inaccurate as hydrographlc surveys present only data In lines through the area, and
secondly the finite-difference approximations which are used give a rough approximation. Con-
sequently, higher order approximations In the spatial dimension seldom give a noticeable In-
crease In accuracy.
For the time integration, higher order schemes have been used, for example the
solution of the one-dimensional long-wave equation by Collins and Fersht (1968), and the solu-
tion by Jeglic (1966) of the advective and diffusive transport of pollution constituents of
286
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the model by Thomaim (1963). Both Collins and Fersht (1968) and Jeglic (1966) use a Runge-
Kutta equation. Very high accuracy can be obtained with this method for the solution of
ordinary differential equations. This is not the case for the solution of partial differen-
tial equations.
The time-stepping procedure uses higher order derivatives in time and space. For
example, In the first mentioned scheme, in the time step of the transport Q in the momen-
tum equation, a fourth-order spatial derivative of Q is used. On the other hand, in the
equation of continuity the spatial representation of 3Q/dx has only a third-order accuracy.
A Taylor series expansion of the finite-difference approximation will show that such fourth-
order spatial derivatives are truncated! Consequently these methods have then only an
apparent high accuracy. The time step is noncentered and introduces a dlssipative effect,
which offsets the inherent instability of the forward-stepping scheme, such as the one
analysed in Section 2.2.2. As direct forward-stepping methods which are centered in time
are available, a much more efficient computation can now be achieved with the same or
better accuracy.
2.2.7 Use of Conservation Laws to Obtain Nonlinear Stability
In Section 2.2.2 it was indicated that the nonlinear terms or term with variable
coefficients have a tendency to introduce nonlinear Instabilities. The conservation law can
be used for the design of schemes such that these instabilities are suppressed without making
the computation strongly dlssipative. The principle upon which schemes are designed will be
illustrated by use of the advection equation
0 (6.42)
at ax
The diffusion term which normally appears in the computation of water quality characteristics
has been omitted as it would make the discussion more complicated. Let us now consider that
the region of computation is. bounded and has no inflow, which then reduces our problem to con-
centration distributions in a closed-off channel In which the substances are moved advectively
by nonsteady currents.
If we apply the conservation law to the mass conservation of the substance c , as
described in 2.2.1, the total mass does not change. For example, in Figure 6.4a, the area
under the solid line, representing the total mass at time tj , is equal to the area under
the dashed curve at time t2 , as
*2
I* c dx • constant
which means that the first statistical moment of the variable c is conserved, or, in other
words, the mean value of c is conserved. In numerical computation, it is generally impera-
tive that this law be satisfied in order to obtain meaningful results. In numerical compu-
tation, the area of integration bounded by sectional lines over distances Ax should stay
constant after computation over a certain number of time steps (Figure 6.4b). This does not
guarantee that the computation has stability; the results may show a typical instability
and still satisfy the conservation law of the mass of the substance (Figure 6.4c).
287
-------
C I k
(b)
(c)
••M-l
Fig. 6.4 Spatial concentration profiles for
successive time levels exhibiting
conservation of mass (see text).
To design a nonlinear stable scheme, one can also conserve, in addition to the
first statistical moment of the variable c , the second statistical moment of the variable,
or, in other words, conserve the mean variance. If Equation (6.42) is multiplied by c and
integrated over the region from l± to -C^ with closed boundaries, we obtain
r'r2
c i£ dx + I c
at J.
dx - 0
(6.43)
This equation can also be written
1 T«2 3u j,, , 1 P 9(ue
— c •i= dx + — I '
2 J 8x 2 J 3x
(6.44)
As no inflows at the boundaries are assumed, the third term becomes zero and Equation (6.44)
reduces to
288
-------
2 at
Thus, the finite-difference scheme has to satisfy also Equation (6.AS). By Imposing this
restriction, the number of schemes which are now available is severely reduced, and solutions
are obtained which cannot obtain the pattern shown in Figure 6.4c.
A finite-difference approximation of Equation (6.42) can be written
/ t+1 t\,. rr t+1 t t+1 . t\f t+1
(cm - Cm)/At + L(cm+l + cm+l + ^ +
(6.46)
Following Section 2.2.1, it can be seen that numerical integration over one time step At for
the range m-1 to m-M, with uF - uj+ - ujj ^ - UM_I - 0 , results In
X 4+1- X Cm-° <6'47)
m-1 m-1
The conservative property of the variance of c in Equation (6.46) can be demonstrated by
multiplying both sides of Equation (6.46) by
'm (6-48)
after which we find
t?
«W
(6.49)
For the summation of all points in the field we find from (6.49)
289
-------
M ,
V (f t+l\2 ( t\2i
L Ucm ) - (cm) I
n-1
M , (6-50>
t+1 t N / t+1 t
V
// t+i
Uum+% + u
Mote that the last two terms of (6.49) are cancelled in the summation by the contributions of the
adjacent points. Equation (6.50) represents a finite-difference form which is analogous to the
integrated differential equation (6.45).
2.3 TIDAL COMPUTATIONS
2.3.1 One-Dimensional Computations
An extensive literature exists concerning one-dimensional computations. A review of
these methods is given by Dronkers (1964, 1969). Of particular importance to water quality
modeling is the expression for the continuity equation
32 + b ^I - 0 (6.51)
9x dt
where Q denotes the discharge through a cross section, b the width and 5 distance of water
level to reference plane. In this formulation, it is assumed that the fluid density is constant.
If the tidal computation is used in conjunction with water quality computation, both computa-
tions should be compatible. This will be discussed under water quality modeling. Most one-
dimensional tidal computations have not been designed especially for water quality modeling,
consequently their applicability should be analyzed carefully, particularly as to the conserva-
tion of mass.
In computations of estuaries, considerable difficulties are generally experienced in
estuary networks. There is a tendency to use an explicit computation, where all new values at
t + At can be obtained directly from previous information, but stability limits on the time
step makes such computations of a long duration. Dronkers (1969) indicated an implicit scheme
for an upward branching network. A multioperation method can be very useful in such cases.
One operation is then explicit, the second implicit (Leendertse 1967). With careful layout,
one part of the system is computed with the explicit step, the other part implicit. With the
next .operation, the first-mentioned part is computed Implicit and the second part now explicit.
Im Im Im Ex. Ex .Ex
Im
Fig. 6.5 Diagrammatic representation of multioperation technique.
290
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Computation of the Implicit part is relatively fast by use of a sweep method (Leendertse 1967,
Dronkers 1969). This method has no limit for stability reasons on the time step.
2.3.2 Two -Dimensional Tidal Computations
The partial differential equations used for estuaries with vertical homogeneity are a
set of equations with velocity components and water level as the dependent variables (see
Chapter II, Section 2, Dronkers 1969, Hansen 1962, Leendertse 1967).
|H + U2H + VM + gii - fv + R(x) - F(x) (6.52)
at 3X ay ax
|y. + u|y. + v|y. + g|l + &. + R(y) - F(y) (6.53)
at ox ay ay
M + a[(h+g)u] + a[(h+g)y] + s . o (6 54)
3x ax ay
The velocity components u and v are averaged in the vertical, and
f - Coriolis parameter,
R(x) , R(y) • expressions for bottom friction
F(x), F(y) • forcing functions by wind and barometric pressure differences
Rather than the vertical -averaged velocity components, some investigators use as the variable
the transport over a section with unit width (Reid and Bodine 1968, Masch et al. 1969).
(6.55)
g H l +R(y) -F(y) (6.56)
ay
(6.57)
at ax ay
where Qx and QT are the components of transport per unit width. Masch et al. (1969) omit
the convectlve terms, which are relatively unimportant in his Investigation.
The finite-difference approximations for each of the two sets generally use a network
of the type shown in Figure 6.6. Computation is very favorable, as for each point at which the
computation Is advanced in time (for example u in Equation 6.52), the spatial gradient of the
other important variable ( 5 in Equation 6.5.2) is present centered in space. Gates (1968),
integrating the long-wave equations for an investigation of ocean circulation, has an addi-
tional network, of the type shown in Figure 6.6, which is located with shifted coordinates
over distances half the grid size in both directions x and y . In the grid scheme, both
components of the velocity are available at the same location.
291
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K -I- t
K - 1
+ WATER LEVEL (£) AND
MASS DENSITY (f^)
O DEPTH (H)
I
X-VELOCITY (U)
Y-VELOCITY (V)
J - 1
J + 1
Fig. 6.6 Example of space-staggered scheme.
the time operations which are presently most widely used in two-dimensional
computations are either of the so-called "leap frog" type, or are an implicit multioperation
method. Both methods are time-centered difference schemes for the main terms in the equations.
For discussing these stepping methods, the partial differential equations (6.55) through (6.57)
are now written
at
+ F. + F
(6.58)
- 0
(6.59)
3?
at
+ H, + H. - 0
(6.60)
The terms F^ , ?2 ' F3 represent the remaining group of terms in Equations (6.55) through
(6.57). For example, we can take
F, - g H|i (6.61)
and
a(Q~/H) 9(Q Q /H)
- - ^ - + - 5_Z - -
ax
ay
R(x) -
(6.62)
292
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In Che leap frog computations are used the finite-difference equations
- -2 At Gj (6.64)
.t+At -t-At t ., ,..
? - 5 - -2 At H (6.65)
where F? , G? , and H are the finite-difference expressions for all other terms of Equa-
tions (6.55) through (6.57). For example, F, now represents the finite-difference expression
at time t for
3(QXQ /H)
• fQ + R - F(x) (6.66)
y x
x
This scheme has a high accuracy but requires two levels of storage of all values Q
Q_ and ? , namely at time t and time t-At . Such a computation is made by Gates (1968).
To reduce this storage requirement, time -staggered computations are made with Qx and Qy at
the odd tine levels (t-1) and the water levels at the even time levels t . Such computations
are made by Hansen (1962) and Masch (1969). (Masch indicates, in the time notations, that the
water levels are at the same level as the transports, but a careful review will reveal that
they are always taken at the intermediate level, namely centered in time.)
For the computations of the latter, we obtain, neglecting bottom friction in our
discussion,
Q+1 - Q*"1 - 2 At (FI + Fj"1) (6.67)
x
<£"• - Qy - 2 At (GJ + Gj ) (6.68)
t+2 t t+1
? - ? - 2 At Ht
where F, - -g H-^ at time t ( '*' means 'is the finite-difference representation of )
1 ox
F^" = fQ - F(x) at time t-1,
G, " -g H|i at time t,
1 ay
G-"1 = fQx - F(y) at time t-1, and
t+1 . &QX 3QV
H - —- +• —*- at time t+1.
ax 3y
The scheme is thus practically completely centered except for the effect of the earth's rota-
tion. The forcing functions which are input are Indicated at t-1 , but the system cannot
distinguish the difference and assumed these at the level t , thus centered. The time step
is limited by a stability condition
293
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(6-69)
For small grid size and deep estuaries the time step becomes very small.
Leendertse (1967) used two approximations of Equations (6.52) through (6.54) which
are used in succession. For the first set is used
_
t+At . A|. Gt (6 n)
- At H£ (6.72)
where Ft+At i u aw + gll ac time t+At (part of the first, nonlinear term is taken at the
1 3x dx
lower level as sufficient data is not available other than through a pre-
dictor) ,
F$ - -fv + v^H + R(X) at time t
Z ay
t+At ^ a,,
G, i fu + u^Z + V5Y. + R(y) at time t+At
1 ax ay
G« i gii at time t
z 3x
H « a^Hu) at time t (part of this nonlinear term is taken at a lower time level
1 dx
as sufficient data is not available other than through a predictor) ,
Ht i aiHvi
2 ay
For the second set is used
t+2At
t+2At t+At ,,t+2At ^ „ _„
v - v " -At G - At G (6.74)
•J
- At H / (6.75)
where r!+24t ^ -fv + u— +v— + R(x) at t+2At (Parts of these nonlinear terms are taken
3 9x ay
at lower time levels)
•T ' «ff - —
Qt+ At ^ ^ dv +gli at t+2At (Part of the nonlinear term taken at a lower time level)
J ay ay
294
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G - fu + u|^ + R(y) at t+flt
t+2At . d(Hu)
H3 • jx at t+2At (Part of the nonlinear term taken at a lower time level)
3
t
4 ay
Ht+" i at t+At
This scheme is also nearly completely centered if multiples of 2 At are considered. The
exceptions are some of the nonlinear terms. The time step is not limited by a stability condi-
tion, but is limited by considerations of accuracy.
This multioperation method requires simultaneous solution of a system of equations.
In each finite -difference equation (6.70) and (6.72), three unknown values of u and ?
exist, which are all situated on a row running from one side of the computational field to the
other side. Consequently, systems of equations have to be solved, each system representing
one row. Since each equation of such a system has only three unknown values, the system can
rapidly be solved by recursion formulae, as is discussed by Leendertse (1967). Equation (6.71)
can be solved explicitly for each point as all information is given. In the second operation,
the Equations (6.74) and (6.75) are solved implicitly, but not in rows perpendicular to those
of Equation (6.70) and (6.71). Consequently, the implicit solutions are always obtained in
the direction of the velocity components which are to be solved in the stepping procedure.
In all two-dimensional computations, the description of boundary conditions is a
considerable task, which becomes increasingly more difficult when the size of the computations
increase. To reduce the programming effort of complicated fields, the FORTRAN program pub-
lished by Leendertse (1967) determines the boundaries in the computation from a simple method
of data insertion.
2.3.3 Numerical Simulation of Tidal Flow in Two Dimensions
At present the use of large two-dimensional tide models is still limited to direct
computation. The available data are inserted in the program and a computation is made. The
results, which are generally now in graphical form, represent flow and water level information
for a limited period of time. The investigator will then compare observed data with the com-
puted results and will find it necessary to adjust his input data, for example values of
friction coefficients, in order to obtain better agreement. A next computation is made and
the adjustment process is repeated. The rate at which a tide model can be adjusted to resemble
the prototype depends on the extent of the field data, but a very important additional factor
is the experience of the investigator and his access to the computer. An experienced investi-
gator will make the adjustments far more rapidly than a novice in the field. A good under-
standing of the physics of the phenomena and an intimate knowledge of the behavior of the
computational code with respect to physical phenomena are naturally requirements for any inves-
tigator using numerical models.
Models using several thousand computational points may require thirty to fifty
adjustments, and the number of adjustments will increase with the number of grid points in
the model. Thus, the access to the computer becomes critical for the progress of the inves-
tigation. Assuming that no code development is made, a rapid turn-around is essential.
Unfortunately, the tide models with the associated advective-dispersion models have extensive
295
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computer memory requirements. The program queueing in computation centers generally discrimi-
nates against these large core-requiring computations, and results in an inefficient use of
the Investigator's time and of the time of his assistants. This is manifested, for example,
in low ratios of computer cost to investigators' salaries.
Another serious impediment for the progress of the investigation is the handling of
the numerical data, which have to be changed from experiment to experiment during adjustment
of the model. With presently used models, which have arrays of only several thousand points,
this is already a considerable task. The development of computers is still progressing at
rates of an order of magnitude per five years for speed and memory size. Consequently, alter-
nate ways of data handling have to be found. This data handling problem is a common occurrence
in computation and has led to hardware and software developments which enable transfer of data
from graphical form into digital form, and digital form into graphical form (Brown and Bush
1968, Davis and Ellis 1964). These developments make interactive simulation possible. How-
ever, considerable program development is still required. Such interactive simulation enables
the investigator to follow the progress of the computation on cathode ray tubes on which com-
puted data (e.g. tide levels and velocities) are presented simultaneously with measured data.
The investigator can store the computation, change variables and continue, or backstep and
restart from a previous time, and retain previous results on the scope together with results
of the repeated computation and the measured data.
2.4 MASS TRANSPORT COMPUTATIONS
2.4.1 One-Dimensional Water Quality Computations
The balance equation for the mass concentration of a dilute substance moving in a
channel of width b can be written
b IffiEl + liSSl + b *(*** ^ + S - 0 (6.76)
at ax ax
where H - temporal depth, averaged over the cross section
Q • transport
c - mass concentration (mass per unit volume)
DX - dispersion coefficient
S - the local source or sink of the mass per unit length
At present, this equation has not yet been solve . numerically in the so-called conservation law
form as discussed in Section 2.2.1. However, solutions in the nonconservative form have been
obtained and are described in other chapters of this report (see, e.g. Chapter II, Section 3 and
Chapter III).
In this discussion here, the Q will be assumed changing at a rather rapid rate by
the tide. The finite-difference approximation ««hich now will be used will be expressed upon
a grid system that has the location of the Q situated between the grid point locations for
the water levels f and the mass concentrations. The averaged depth up to the reference level
of the water surface is taken at the same location as the Q , thus between the water level
points ? and the points of the mass concentration c . The latter are indicated with even
values of m , and the depth and Q with odd values.
296
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Consider a finite-difference equation at location m for the mass concentration
cm - hm-l + Vl2 + « Cm
fc ^ t fc\ t+1 / t+1 t+l
l (cm+2 + cm) ' Vl (cm + cm-2
i ( w + 0) °x (4« - 4) - K.-I + i (
- 0 (6.77)
It will be noted that in this equation for the location m , only three unknown values of the
mass concentration appear, namely 0^2 > cn+ ' cm^2 • Consequently we have to solve a
system of simultaneous equations, each with three unknown values of the concentration, except
at the upper and lower ends where the equations include only two unknown values. The third
which appears in the equation is a point in the boundary and should be given. A sweep method
can solve such a system very rapidly.
As stated earlier, the computation of the tide has to be compatible with the expres-
sion for the conservation equation of the mass of the substance c . The required finite -
difference expression for the equation of continuity can be obtained by introducing a unit
value for all mass concentrations. This results in
h (C1 - 4) * &
which is a central implicit expression that can be solved simultaneously with momentum equa-
tions. Both Equations (6.77) and (6.78) are conservative, and numerical integration in the
spatial region will show no loss or gain in mass, except for the influxes at the boundaries,
but these equations do not satisfy the conservation of quadratic properties. The dissipative
and dispersive properties are already discussed in Section 2.2.3.
The concentrations generally are slowly varying in time, and the propagation speeds
of the phenomena are much lower, an order of magnitude lower, than the propagation speed of
the tidal wave. One would have the tendency to consider a computation with a much larger step
for the water quality computation than for the tide computation, thus reducing the number of
computations. It is difficult, if not Impossible, to design the computations and have com-
patibility between the continuity equation of water and the mass balance equation of the
substances in the water quality model, with different time steps in the models. In the first
decennium of digital computation, restrictions in computer memory size and slow computation
speeds forced investigators to use separation of tide and water quality computation, by storing
297
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the tide information on magnetic tape and subsequently solving the advective transport equa-
tions. At present, the computation of the tidal equations can be made in a shorter time than
the transferal of data from magnetic tape, and this separation is no longer necessary.
2.4.2 Two -Dimensional Water Quality Computations
Two-dimensional computations can be made by use of the finite -difference approxima-
tion of the mass -balance equation derived in Chapter 11
ajHgl + a(Huc) + a(Hvc> + MB, H + &™y If + s . 0
at ax ay 3x 3y
The finite-difference equations used by Leendertse (1970) are two different sets which
are solved simultaneously with the two sets of finite-difference equations of the tidal
motion. After computation of the continuity equation, which is solved simultaneously with
the momentum equation for the u velocity (6.70) and (6.72), the first mass balance equation
is solved. The mass balance equation is computed using the same grid as the tide computation.
Tide level 5 and the concentration c are situated on the same location, while the u
velocity is situated between the water level grid. points in the eastward direction and the
v velocities are situated midway between the grid points in the northward direction. The water
levels and concentration are at integer indices (m, n) . The scheme uses water depth at loca-
tions in the center of four water level grid points , thus both indices indicating its location
have Integer and one-half values.
The first continuity equation used is
t+1 t \ I-/ t t \ t+1
+1 t \ I-/ t t
,m ' SQ, J AtIA?n,m-l + Sn,«
t+1
"n,
' L(?n-l,m + ^n.m + V%,m-% + V%,m+%) vn-%,m
- 0 (6.80)
after which is solved the following equation:
t+l\
r
c
hm-%,n+%
,t t \ t+i / t+i
5n,m-l + Si.m + hn-%,m-% + ^.n^V un,m-% Vcn,m-l
t+i\
298
-------
* ? j. t,
n,m + ?n,m+l + hn-%,m+%
- (?n,m + ?n+l,m + hn+%,m-%
r(?t + ?fc + h
L^ n-l,m n,m
(?n,m + Sn+l,m
t+l f t+1 t+1 M,
un,m+% lcn,m + cn,m-l)JMix
t / t t \
vn-%,m icn-l,m + cn.ij
n+%,m (Cn+l,m + Cn,m
t+l / t+l t+l
Dx , Vcn,m ' cn,m-
">m~?
y ,
TL-*
C , (ci+l,m ' cn,m)]/2Ax + Sn,
t+1
t+1
In this equation only three unknown values of c occur, namely c ., cC
t+l ' n,nri-l ' n.m '
cn m-1 • "IU8 Equation (6.81) can be written
t+1 t+1 t+1
cn,nH-l + *m cn,m + Cm cn,n,-l ' Dm
(6.82)
**ere Ajn > Bm » cm > and Dm can be derived from Equation (6.81). Thus for each row n ,
a system of equations of the type shown by Equation (6.82) exists, which can be solved directly
by a sweep method.
For the second operation in the stepping order, Equation (6.75) is used for the equa-
tion of continuity. This formula is very similar to Equation (6.80), except that now the
stepping goes from t+1 to t+2 , and the terms with the V velocity are taken at the time
level t+2 , thus
t+2 t+l
t+1 t+1
/ t+ t+} r-f t+ t+
l?n,m - Sn.nJ At ' IA?n,m-l + ?n,m + hn-%,m-%
t+1
t+1 t+1
t+1
Vf t+2 t+2 s t+2
" LV ri-l,m + ?n,m + hn-%,m-% + hn-%,m+%J vn-%,m
(6'83)
/ t+2 .t+2
- (5n,m + §n+l,
t+2
This equation is solved simultaneously with Equation (6.75). The second advective -diffusion
equation is
299
-------
,n-% + Vfc.n+fc + hm+%,n-% + hm+%,n+%
~ cm+1 (hm-%,n-% + hm-%,n+% + hmt%,n-3| + hm+%,n+% + *
r/ t+1 t+1 . \ t+1 / t+1 t+l
-15 , + ? + h L ,+ h , , I u , (c , + c
L\ n,m-l n,m n-%,m-% n+*s,m+%' n,m-% \ n,m-l n,m
/.t+1 t+1 \ t+1 f t+1 t+1
- (5n,m + ?n>nrt-l + V%,m+% + hn+%,»+%) un,nrt% (cn,m + cn,m-
t+1 t+1 \ t+2 / t+2 t+2\
n-i>m + ?n,m + hn-%,m-% + hn-%.»t%^ Vn-%,m ^Cn-l,m + cn, J
.t+1 . . \ t+2 f t+2 t
vn+%,m icn+l,m + cn,m
r/t+l t+1 \ t+1 / t+l t+1
- I V ' 1 + ? "*""!_ L + " I. !_/ D (c ~ C
•- ^ n,m~l n,m n~^,m~^ n+5|m+^j' x i » n,m n,m^
/.t+1 _t . \ t+1 / t+1 t+1
Vcn,m+l ~ cn,n
(6.84)
r/ t+1 t+1 v t+2 t t+2 t+2 \
L \ n»l 9m n^m n~^jin~^ n~^jm+Tj/ y i \ n^in ti~l |in/
Again a double sweep method is used to solve the system of simultaneous equations, of which
each have three unknown values c*"™1 , c , c ,
n+J.,m ' n,m n-l,m
The mass balance equation of the substance c is compatible with the continuity
equation. Tests have shown that with a constant c , the error is of the ratio of 10"-" after
a few thousand time steps using a field with several thousand points. The mass concentrations
are conserved, but the water mass is not completely, due to a lower order approximation, as
indicated-in Section 2.3.2. Near sources, the equations are somewhat changed according to the
principles presented in Section 2.2.5 to suppress the spatial oscillation which otherwise may
occur at the upstream side of the source.
300
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REFERENCES FOR SECTION 2
Brown, G. D., and C. H. Bush, 1968: The integrated graphics system for the IBM 2250.
RM-5531-ARPA, The RAND Corporation, October 1968.
Collins, J. Ian, and Samuel N. Fersht, 1968: Mixed technique for computing surges in channels.
Proc. ASCE. 9[4, No. HY 2 (March).'
Davis, M. R., and T. 0. Ellis, 1964: The RAND Tablet, a man-machine graphical communication
device. RM-4122-ARPA, The RAND Corporation, August 1964.
Dronkers, J. J., 1964: Tidal Computations in Rivers and Coastal Waters. Amsterdam, North-
Holland Publishing Co;
Dronkers, J. J., 1969: Tidal computations for rivers, coastal areas, and seas. Proc. ASCE,
95, No. HY 1 (January).
Gates, W. L., 1968: A numerical study of transient rossby waves in a wind-driven homogeneous
ocean. Journal of Atmospheric Sciences, 25, No. 1 (January).
Granmeltvedt, Arne, 1969: A survey of finite-difference schemes for the primitive equations
for a barotropic fluid. Monthly Weather Review, 97, No. 5 (May).
Hansen, W., 1962: Hydrodynamical methods applied to oceanographic problems. Proc. of the
Symposium on Mathematical-Hydrodynamical Methods of Physical Oceanography, Institut
fur Miereshande der Universitat Hamburg.
Jeglic, John M., 1966: DECS III, Mathematical simulation of the estuarine behavior. Report
No. 1032, General Electric ReEntry Systems Department (December), Revised July 1967.
Leendertse, J. J., 1967: Aspects of a computational model for long period water wave
propagation. RM 5294-PR, The RAND Corporation.
Leendertse, J. J., 1970: A water-quality simulation model for well-mixed estuaries and coastal
seas: Volume I, Principles of computation. RM-6230-RC, The RAND Corporation.
Lilly, Douglas K., 1965: On the computational stability of numerical solutions of time-
dependent non-linear geophysical fluid dynamics problems. Monthly Weather Review.
93, No.. 1 (January).
Masch, Frank D., J. J. Shankar, M. Jeffrey, R. Y. Brandes, and W. A. White, 1969: A numerical
model for the simulation of tidal hydrodynamics in shallow irregular estuaries.
Tech. Report HYD12-6901, Hydraulic Engineering Laboratory, The University of Texas.
O'Brien, George G., Morton Hyman, and Sidney Kaplan, 1950: A study of the numerical solution
of partial differential equations. J. Mathematics and Physics, 29, pp. 223-251.
301
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Reid, R. 0., and B. R. Bodine, 1968: Numerical model for storm surges In Galveston Bay.
Proc. ASCE. 94, No. WW 1 (February).
Rlchtmyer, Robert D., and K. W. Morton, 1967: Difference Methods for Initial-Value Problems.
Second Edition. New York, Intersclence.
Thomatm, Robert V., 1963: Mathematical model for dissolved oxygen. Proc. ASCE. 89. No. SA 5
(October).
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DISCUSSION
HARDER: In my introduction to Section 1 of this Chapter I have tried to convey some of the
conventional wisdom regarding digital computers, analog simulators, and analog computers.
I have distinguished, for those who are not intimately familiar with them, between analog
simulators and computers. Analog computers are going out of style, somewhat; they are
certainly less popular now (1970) than they were before digital computers were important.
But they have an advantage that digital computers have not quite yet attained in terms of
the immediate interaction that's possible between the operator and the computer. I have
indicated that as a considerable advantage, but one which is considerably offset by the
disadvantage that analog computers generally can operate with only one dependent variable.
I'm sure that you will forgive me for not trying to indicate the solution techniques
for each one of the equations of the other chapters. I've had to be content with giving an
illustration of how one could apply analog simulators to a simple problem of tidal flow.
In addition, I've given an indication of how one could apply analog computers (by which I
mean a system in which we are trying to solve the equations explicitly), when one knows
what the equations are, using integrators, adders, multipliers and other analog components.
I have done that by making a brief application to the one-dimensional diffusion equation,
by showing how one builds up the elements of such a system, after having broken the one-
dimensional channel into segments. Then each segment can be simulated on the basis of the
equation by, as I indicate, two sets of ganged potentiometers, two operational amplifiers,
and one capacitor.
I made a few remarks on hybrid computation which won't be welcomed by the manu-
facturers of hybrid computing equipment. I think after some examination of the efforts
that have been made that they have a principal advantage in saving intermediate results
and recovering these results for subsequent insertion back onto the analog section. That
is to say, you can save, through an analog-digital conversion, intermediate results in the
digital form, and then reinsert them.
I then remarked on the economics of analog computation and pointed out that the
generally available analog or hybrid-analog computing equipment is dreadfully expensive
because of the exceedingly tight tolerances that are usually put on errors, typically one-
hundredth of a percent. But it's not at all necessary or needed when the accuracy of our
field measurenents may be no better than 5 percent. Analog computing components are very
cheap now, no more than $25 for integrators, maybe $35 for multipliers, if you're satisfied
with a 1 percent accuracy.
THOMANN: Well, I'd just like to relate the worst experience I ever had in my life, trying
to run diffusion-advection equations In coupled systems on three one -hundred-and-twenty
amplifier, slaved analog computers. It was a horrible experience. We just couldn't get
any kind of reproducibility, and I think you've talked about that in here. It was really
hard to get that doggoned thing to work. Especially if the parameters are poorly behaved
functions, and the function generators are the mechanical type that we have. Really quite
an experience. It Just didn't have the convenience of the digital computer.
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I think the point that one can Interact with the machine rapidly while he's doing
verifications is not purely the domain of the analog. It may be that in the installation
you have, you're In a situation with the digital computer that you have to wait a fair
amount of time. That needn't be the case. You can interact.
HARDER: Well, I'm not suggesting that direct communication is the monopoly of hybrid or
analog systems, nils can be developed for digital systems. I think Jan knows of it. You
can put the output on an oscilloscope directly from a digital computer.
PAULIK: Well, how about the use of time-sharing systems and the cathode ray display? Some
of these are portable. They essentially use the telephone lines, to which you hook up an
acoustical coupler and hook up a time-sharing terminal to a computer and display the results
on a scope.
HARDER: I'm sure that these things are going to be even further developed in the future,
and this is the reason I'm reluctant to recommend even considering analog computing equip-
ment, but the advantage of analog computing equipment is that people who wish to work
slowly can have the whole system spread out in front of them, and they are not chewing up
valuable computer time as they are playing around with it.
I know that there are some things which you can do on digital computers which are
not economical to do. The advantage of analog equipment is that the first cost is pretty
nearly all of the cost. You can take your time about achieving verification, whereas, if
you're paying so much an hour, you may not only get tired but also bankrupt by the time you
finish your verification.
WASTLER: The models we have now are designed as digital models, basically. Would you care
to comment on, for analog use, would there be merit in dropping all the way back to the
basic equations and taking an entirely new tack which would be appropriate for an analog?
HARDER: Well, I can Imagine that If you were doing analog computation, trying to carry
forth seven dependent variables, that you'd have essentially seven parallel analog elements
and these may be connected lengthwise to different reaches of the channel. Then there'd be
cross connections between these seven variables in which each one of the variables would
influence the rate reaction of each one of the others. Now this would be a tremendous
amount of patching, but when you did it you would have a visual picture of exactly what
you were doing. Now this you might not have so well if you'd had to write it out in
FORTRAN. So there is certain visualization even in setting up the equations with analog
type computers.
THOMANN: I think we should mention the use of some software that Is around, provided by
groups like IBM, like CSMP, Continuous System Modeling Program, where one essentially writes
down the model In analog form but the entire program is executed digitally. In other words,
you write down what Dr. Harder is talking about, but the program is Implemented digitally.
You don't have a real analog computer. You have a picture of what he's talking about. But
the entire thing is carried out in a digital framework. In fact, in the early nonlinear
nutrient-algal-zooplankton models that we looked at, we in fact modeled with CSMP, just to
get the feeling of how the dynamics of the system behaved in simple spatially distributed
models.
LEENDERTSE: The prime thing in this regard is that you can do all of these things on a digital
computer. You can have all the interactions you want. You can have the graphical outputs
304
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immediately, if you have the proper equipment available. Even if you don't have, still
there is the typewriter and printed output. You have to spend some time on writing the
program for getting the output you want to have.
THOMANN: As far as the state-of-the-art is concerned, I don't really know anybody now that
is using, In any sort of routine fashion, analog computation for water quality work.
BAUMGARTNER: Well, we have the experience that you've had with it.
WASTLER: We did a little bit on streams with the Streeter-Phelps analysis, but that was
sort of a simple-minded approach «:o the thing.
THOMANN: I would say that the state-of-the-art now is that it has pretty well been dis-
carded.
HARDER: It is possible, Bob, that the whole of the difficulty that you had was not with
the computational scheme itself but with the problem. I mean not with the means of solving
the problem but with the problem itself. You might have had the same problem with digital
computers.
THOMANN: No, it was hardware problems.
LEENDERTSE: Hie subject of this second section is basically finite-difference schemes. It
is evident that if you have a complex differential equation, you can have many, many approxi-
mations. The way to approach this is not by just starting to take finite differences, but
to sit back and look very carefully and ask: what is the best way of approaching, and the
best way of finding the approximations? Now we have, unfortunately I would say, the formu-
lation of the problem in partial differential equations' and the formulation of the dif-
ference equations going hand-in-hand. It is quite noticeable from the work that has been
done up to now in formulating these, and you find it in the previous sections also, that
it has been cued quite a bit to the analytical thinking that people have. I should say we
have to be careful that we don't go along those lines, in order to avoid quite serious
shortcomings in our solutions.
In setting up our computational schemes we have to satisfy the same things, or
we try to satisfy the same things, as when we set up our partial differential equations.
We have conservation of mass, conservation of momentum, and conservation of constituents.
I gave an example here of how you look at conservation of mass; how you integrate over
time and space in the partial differential equation, and how you do that in the finite-
difference approximation. By the way, every scheme which we are working with right now
actually doesn't conserve mass. Only but very few schemes really do that.
Another problem is the stability of computation. You will find in the literature
that people are generally quite concerned about it. This is not necessary. If you really
make an analysis of the problem, in the way I said at first here, you will find that you
can Immediately design a scheme which is stable. There's one exception, and that is the
problem of the nonlinear terms, for example, advection terms. They very often generate
instability just by the fact that you get product terms of very short periodic fluctuation,
which then start to be amplified.
305
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CALLAWAY: Do you get that In the small scale grid that you are using, too?
LEENDERTSE: No, due to the fact that I use filtering techniques. I have explained here how
those terms come into existence, and the way you can get rid of them is actually by taking
one of the nonlinear terms in a manner that you filter out the very short wavelengths
throughout.
Some other problems incurred in computations are the dissipative and dispersive
effects. Now "dispersive" in tills particular subject maybe is not a very good name for it
because you might confuse this with dispersion, but this is the way the term has been used
in mathematics. That a system is dispersive means that the velocity of propagation of the
phenomena becomes dependent upon the mode of computation. The dissipative effect is like
when a certain periodic phenomenon, which is not itself dissipated, becomes dissipated
by the scheme, in other words, you get a kind of numerical dissipation. It is generally
possible to have no dissipative effect in your computations. I think that's the right way
to look at our problems. However, you cannot get rid of the dispersive effects. So certain
phenomena will not propagate properly in our computations. I have given some of the analy-
sis here and presented some of the ideas on how you handle that, and how you can compute
how much this dispersion is. In other words, you set up a scheme, a finite-difference
approximation, and in addition you look at the grid sizes. From that and from the infor-
mation provided here you can get a pretty good idea what the computational solutions will
look like and how they will behave.
Discontinuities are another problem we have. Those occur, of course, at the
points of injection, and there is really not a very good way of handling these. All our
computational methods, all our finite-difference methods, are based upon fairly slowly
varying phenomena. This is also slowly varying in the spatial sense. If you have a dis-
continuity at a discharge, you get computational effects in the results. You have to be
aware of this.
Then in Section 2.2.6, I have some comments on higher order schemes. All the
modeling which has been done, as far as I can see, has been approached in the way ordinary
differential equations are approached. However, we have partial differential equations,
and as a consequence we do not have to worry too much about the accuracy of the say, time-
varying concentrations in relation to the accuracy of the other terms. Section 2.2.7 treats
the use of conservation laws to obtain nonlinear stability. I indicated earlier that
linear stability you can handle rather rapidly, and nonlinear stability you can do some-
thing about, but there is now a rather clean-cut way, by imposing another condition upon
your finite-difference scheme, to assure yourself that you get no nonlinear instabilities.
As far as I know this has not been applied yet anywhere except in computations of the advec-
tion terms in momentum equations.
Now in 2.3.1, I really didn't give a detailed example of that for the following
reason. I went through the literature, and there is not really one scheme which conserves
mass in one-dimensional computations. 1 feel this is rather a shortcoming in our present
state-of-the-art. We should have a very good model of one-dimensional computations which
is conservative.
In two-dimensional computations, there are several schemes available. He do not
have as many problems with two-dimensional computations as with one-dimensional computations.
In this sense the one-dimensional computation is more difficult than the two-dimensional
306
-------
computation. I explain how to get to your method, something along my own method. I may
point out, however, that in order to conserve energy, for instance, or momentum, you do
have to write the equation in the form of (6.55). We obtain the two-dimensional computa-
tion in very much the same way. We have very much the same problems as with adjustment in
tidal models. It is a very time consuming task to look at all the outputs that you have
and to decide where to put a little more resistance in or where to put a little more depth
in.
Then Section 2.4 discusses mass transport. I have given here a scheme (I think
it is also rather new) which conserves the mass as far as I can see. It does not conserve
the quadratic terms, the mean variance of concentrations, very well. Then the discussion
goes into the two-dimensional water quality computations and describes the formulas which
I have been using.
WARD: For the record, could you discuss your comment that the use of box or cell models
is not advisable.
LEENDERTSE: The use of such methods fixes immediately the finite-difference equations without
regard to computational effects. 1 think that is just to write the answer for it. So you
force upon yourself a solution technique with all the possible shortcomings you have. That
is not the way to handle it. You should first look at the partial differential equation,
or maybe an integro-differential equation if you want to be more sophisticated, and then go
from there and design your scheme. If you design your scheme Immediately, you make take
lower order approximations, force upon it lower approximations, which aren't necessary.
BAUMGARTKER: Earlier in the day when we were talking about some bays or harbors where things
might be well-mixed. 1 thought there was some indication that this might be a case where
we use something that would turn out to be a box model. I don't know if I recall your
shaking your head or which way it was going, but I felt otherwise that we had a pretty
general consensus that this might be an acceptable way to handle it.
LEENDERTSE: In that case, it may be so that you can use it, that you can make a box model,
but it just happens to Be so. You just came out all right. Maybe the advective transport
and the diffusion was not really important. But that is not the way you should approach
it. You should approach it from the partial differential equations, and then have a care-
ful look at what are the various contributions.
WARD: I think that's very important.
RATTRAY: Don't compromise with reality before you have to.
WARD: Right. Stick with the differential equation.
RATTRAY: And even in that case of having a well-mixed bay, it seems to me that you have
one problem in matching the boundary conditions. The fluxes of water and the other concen-
trations aren't too realistic. Even though the box itself might well represent a well-mixed
bay, representing the exchanges as pipelines might not be so good.
PRITCHARD: I'd just like to argue this case a little, from the standpoint that when you make
a finite-difference statement of a differential equation, you are, in fact, building a box
model. And so what we are talking about is the size of the grid, for instance in the
307
-------
Thomann model. There is considerable discussion in Orlob's papers about the problems of
how you take the finite differences, because you are dealing with real steps and these are
not real linear gradients. This is where you get in trouble with the box model. It is the
problem of having to have assigned values to the boundaries, or in terms of how you evaluate
the parameters in the boxes, and how you take the differential.
LEENDERTSE: In this particular case, the scheme that is basically used there is likely un-
stable, just by this assumption that is being made. If you have a four part stepping scheme,
so you go from one time level and you compute, from the information which is available,
forward eight times, that scheme is basically unstable, always unstable. You can make it
stable by doing something to the gradients, that's described in Section 2.2.2 or 2.2.3, so
then you start matching and you do something that is really not correct. If you want to
really build a scheme you go from the partial differential equations, and you are not con-
cerned with the box concept at first.
PRITCHARD: When you set up the grid of known information, you are taking finite steps. Now,
you introduce errors with that grid if the parameters really are not linearly varying in
space, and you estimate the differential by delta-over-delta approximately.
LEENDERTSE: But it is not the only way. You see, you have many possibilities as to how you
can take the differences. So you want to design your scheme in such a manner that you get
the right characteristics of the computation or the most preferred. Of course, you want to
have stability in the computation, but you don't want to take a scheme which is so stable
that it diffuses everything.
PRITCHARD: I think I am following that. But what I am really perhaps saying is that I think
there has been a number of so-called computer mathematical models which in effect are box
models in that the time steps and space steps are sufficiently large so that there is a real
question of how you take the difference.
What I am concerned with is the scale of the input information. Just from the
standpoint of the amount of data that we can get, we're limited in space in what kind of
differences we can formulate, so that I think we are down to having to make some approxima-
tions which essentially become almost box-like models. This is my point. Not that one
wants to do this if you could solve the things, but I am saying in the meantime, until we
can, I think we are faced with having to make our best estimates at the present time on the
models that we have.
LEENDERTSE: Yes. Still I would first look at it from the other way. Go from the partial
differential equations and do it in that manner, and then prefer to make your choice not
in terms of, let's say, u
-------
RATTRAY: And you have an explicit expression for the boundary exchanges, instead of
estimating it some other way. That's different from a box-model.
PRITCHARD: Well, all right. But you still come out with dividing the system up into a
finite number of steps. That's my point.
LEENDERTSE: But what you can do is, if you would look at the box-model, you can take the
integral equations and integrate over the boxes and maybe do some corrections on that.
PRITCHARD: That is what I am doing. Starting with the differential equations and integrating
over finite segments,
LEENDERTSE: Yes. Hell, that's justified.
HARDER: In defense of Jan's criticism of the box approach, I think you might find if you
used a simple-minded box approach that you wouldn't by that means itself be able to dis-
tinguish the advantages and disadvantages of implicit, explicit and mixed-mode types of
computation.
PRITCHARD: Yes. When you do have a system of equations, there are advantages to implicit
solutions.
HARDER:
That you would not see from a box approach.
LEENDERTSE: With reference to implicit or explicit scheme, I am very much convinced that the
multioperational approaches are the way that we have to go. Also, in the three-dimensional
case. The multioperational method, where you have more than one approximation to the par-
tial differential equations, gives much more flexibility in the mode of computation. If
you have an implicit scheme, it is very often difficult to solve. If you take a multi-
operational approach for which you have more than one approximation for the partial
differential equations, you have a certain amount of choice for, let's say, making it
implicit in one direction, and in the next operation you take it implicitly in the other
direction. The mathematical way of handling it is available.
HARDER: As an example, let me point out that, if you have a dynamic equation and a con-
tinuity equation which have to be satisfied at all times, you can use an explicit scheme
for the dynamic equation and then turn around and use an implicit scheme for the continuity
equation, and then step alternately in this direction. It helps very often in accuracy and
stability. But it has to be done with a good deal of sophistication, as Jan has applied
to it.
309
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CHAPTER VII
CASE STUDIES
George H. Ward, Jr.
and
William H. Espey, Jr.
1. INTRODUCTION
While the emphasis of the preceding chapters has been on the principles of estuarine
water quality modeling, the concern of this chapter is with the practice of modeling as exem-
plified by specific estuarine modeling projects. This chapter consists therefore of a collec-
tion of surveys, with the stress upon implementation of models and their adequacy in predicting
water quality. The attempt is made to present accounts of these projects without injecting any
judgments on the work, so that each case study is, in effect, a summary of the available docu-
mentation on that project. The selection was made so as to encompass a variety of modeling
techniques as well as to represent different types of estuaries. The choice was limited largely
to major estuarine systems or major modeling projects where, in general, sufficient resources
were available to exploit the capabilities of the models.
As this chapter is concerned principally with those applications offering insight
into the present ability of models to reproduce nature, a number of topics are not discussed
in detail, such as the employment of modeling for management decisions, e.g. least cost opti-
mization, allocation of waste loading, alternative treatment levels, or the delineation of
quality standards. Two applications of current Interest, viz. modeling of the effects of
streamflow regulation and of the effects of physiographic modification, have been given par-
ticular prominence. Existing capabilities for modeling these effects are, of course, limited
by the adequacy of present models as delineated in the preceding chapters, especially with
regard to hydrodynamlc phenomena. However, the examples in this chapter serve to illustrate
the extent and success with which existing techniques have been applied to these problems, as
well as to others. Ultimately, it is in the applications that the practical capabilities and
the practical limitations of present estuarine modeling are demonstrated.
310
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2. THE THAMES
The study of pollution In the Thames Estuary, which was undertaken by the Thames
Survey Conmittee through the Water Pollution Research Laboratory and which culminated with the
Water Pollution Research Laboratory (1964) report, had the objective of developing predictive
mathematical models upon which the future management of the estuary could be based. The study
received Its principal Impetus from public complaints about the production of malodorous gases,
principally hydrogen sulfide, arising from anaerobic reaches of the Thames. The basic problem
in the estuary was the concentration of dissolved oxygen, and therefore DO was the parameter of
principal interest in the study.
2.1 PHYSIOGRAPHY AND HYDROGRAPHY
The physical characteristics of the Thames are summarized in this section. The reader
is referred to WPRL (1964) for details. The estuary proper from the mouth to Teddington Weir,
the limit of tidal influence, is depicted in Figure 7.1. The variation of depth and cross-
sectional area is shown in Figure 7.2. A typical tidal range below London Bridge is 15 feet,
decaying rapidly to zero above the bridge (Figure 7.3).
The principal source of Inflow is the river Thames, and this flow is gauged at
Teddington Weir (which is below the major municipal withdrawals) to be nominally 1,000 MGD.
Seasonal trends in the gauged inflow are displayed in Figure 7.4, which shows the monthly
inflows averaged over many years of record. The daily records for a given year of course
exhibit considerably more irregularity. Additional freshwater Inflows below Teddington Weir
are contributed by the tributaries, storm sewers, and municipal and industrial effluents.
Figure 7.5 shows the total net seaward flow below Teddington Weir for four distinct inflow
conditions. The curves are numbered with the ordinate value 20 miles above the Bridge, viz.
the gauged flow at Teddington Weir. Combining these results with the cross-sectional areas
of Figure 7.2, one can estimate the net velocity due to inflow (Figure 7.6).
The Thames is vertically well-mixed and is typically only slightly stratified. The
vertical distribution of salinity at high water (spring tide) is given in Figure 7,7.with the
equivalent distribution for the Tees (see also Stommel and Farmer 1952, Section 7.21). The
degree of vertical salinity stratification is further Indicated in Table 7.1. Some typical
longitudinal distributions of salinity are shown in Figure 7.8.
2.2 POLLUTING LOADS
The predominate source of pollution in the Thames is municipal effluents and most of
the significant loads are discharged below Teddington Weir. The distribution of BOD loads for
two different sampling periods according to the nature of the source are given in Table 7.2.
It is evident that municipal sewage contributes nearly 80% of the total BOD load, the remainder
being divided nearly equally between industrial discharges and freshwater inflows. Typical
(five-day) BOD distributions are shown in Figure 7.9, and in Figure 7.10 are displayed some
typical longitudinal distributions of DO, nitrogen compounds and salinity. The flow at
311
-------
SOUTHSND-ON-SEA
S3
S3
Fig. 7 1 Thames Estuary. Numbers on Axis are Distances (s.m.) from London Bridge.
After WPRL (1964).
-------
MEAN DEPTH
CROSS-SECTIOMAL AREA
Id-—0 HO
ABOVE BELOW
500
ZOO
100
50
20
to
20 1CH—0 »10 20 30 <«0
ABOVE BELOW
Fig. 7.2 Physical characteristics of Thames as a function
of distance (s.ra.) from London Bridge. After WPRL
(1964).
20
10 » 0
ABOVE BELOW
10 20
MILES FROM LONDON BRIDGE
30
Fig. 7.3 Range of water level during average tidal cycle
In absence of surface disturbance. After WPRL
(1964).
o
_J
_J
s
O«O 1883-1917
o—a 1918-1952
A A 1953-1962
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
Fig. 7.4 Monthly average values (over indicated period)
of gauged flow at Teddington. After WPRL (1964).
313
-------
a. 4000
3000
™ 2000
i 1000
< o
20
3000
-1500.
I
10 0 10 20 30
ABOVE BELOW
MILES FROM LONDON BRIDGE AT HALF-TIDE
40
Fig. 7.5 Estimated total net land-water flow vs.
distance along estuary. After WPRL (1964).
SO
20
10
52 0.5
| 0.2
0.1
0.05
10
MILES FROM LONDON BRIDGE AT HALF-TIDE
40
Fig. 7.6 . Net seaward velocity due to Inflows of
Figure 5 (miles from London Bridge at
half-tide). After WPRL (1964).
TEES ESTUARY
12 10 8 6 4
MILES FROM MOUTH OF ESTUARY
THAMES ESTUARY
T
31
40-
60 £
_l °
45
15
10
505
ABOVE BELOW
10
15
20
25 30
MILES FROM LONDON BRIDGE
35
Fig. 7.7 Isohallne contours (o/oo) at high water of a
spring tide in the Tees and Thames Estuaries.
After WFRL (1964).
40
314
-------
TABLE 7.1
DISTRIBUTION OF SALINITY (o/oo) IN THAMES ESTUARY
From WPRL (1964)
POSITION
(Miles Below
London
Bridge)
0.4
3.1
4.9
7.0
9.3
11.7
13.8
16.5
18.7
21.6
23.9
26.2
27.7
29.4
31.1
33.2
36.0
38.7
40.9
43.1
26.2
28.4
30.5
33.2
36.0
38.7
40.9
43.1
DEPTH
(ft)
SURFACE
SALINITY
Low water.
27
27
26
27
28
29
38
38
41
39
43
39
38
49
39
__
34
38
37
42
53
52
52
53
53
51
51
51
0.18
0.55
0.92
1.73
3.91
5.54
7.88
9.69
11.40
13.57
15.19
16.46
17.36
18.44
19.70
22.05
23.77
26.02
27.20
28.55
SALINITY NEAR
MID-DEPTH
Depth
Salinity
SALINITY NEAR
BOTTOM
Depth 1 Salinity
SALINITY
DIFFERENCE
BETWEEN
NEAR-BOTTOM
AND SURFACE
neao tides. 6-7 April 1949
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
0.18
0.46
1.19
1.93
3.91
6.26
8.24
9.69
11.40
13.75
15.37
16.64
17.90
18.35
20.07
22.68
24.13
26.56
27.20
28.64.
26
26
25
26
27
28
30
30
30
30
30
30
30
30
30
30
30
30
30
30
High water, spring tide, 27 April 1949
23.86
26.02
27.11
29.27
30.17
30.99
31.44
32.25
25
25
25
25
25
25
25
25
23.40
26.92
28.01
29.27
30.53
31.26
31.44
32.34
50
50
50
50
50
50
50
50
0.18
0.64
1.19
2.38
4.00
6.44
7.97
10.41
12.30
14.47
15.55
16.82
18.53
19.07
21.33
23.95
25.12
26.65
27.38
29.36
24.94
26.74
28.55
29.27
30.90
31.26
31.26
32.52
0.00
0.09
0.27
0.65
0.09
0.90
0.09
0.72
0.90
0.90
0.36
0.36
1.17
0.63
1.63
1.90
1.35
0.63
0.18
0.81
1.08
0.72
1.44
0.00
0.73
0.27
-0.18
0.27
315
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°: 20 -
FEBRUARY
MARCH
MAY
JULY
•-• SEPTEMBER
»"* OCTOBER
Q--O NOVEMBER
•"• DECEMBER
5 0
ABOVE BELOW
10 20 30
MILES FROM LONDON BRIDGE AT HALF - TIDE
Fig. 7.8 Average salinity distribution along Thames for
selected months in 1953. After WPRL (1964).
30
25
x 20
Q.
i 15
o
d 10
•
o
5
0
0 10 20 30 <*0
MILES BELOW LONDON BRIDGE AT HALF - TIDE
Fig. 7.9 Quarterly average distributions of B. 0. D.,
April 1953.to March 1954. After WPRI. (1964).
316
-------
FIGURES IN PARENTHESES ARE PERCENTAGES OF TOTAL
SEWAGE EFFLUENTS
STORM SEWAGE
DIRECT INDUSTRIAL DISCHARGES
FRESH-WATER DISCHARGES
TOTAL
1950-53
331 (79)
8.5 (2)
52 (12)
28* (7)
420 (100)
1960-62
270 (78)
11 (3)
31f (9)
36 (10)
348 (100)
*DATA MAINLY FOR 1952-53.
tDATA FOR 1962.
TAble 7.2 Estimated average B. 0. D. loads (tons/day)
discharged to estuary from Teddington to 32 miles
below London Bridge during two periods. From
WPRL (1964).
Teddington on 31 August 1954 (Figure 7.10a) was representative of summer conditions, ca.
400 MGD, while that on 23 February 1954 (Figure 7.10b) was ca. 2500 MGD, representative of
wet winter conditions. The variation of these parameters through the anaerobic reach should
be especially noted. The increase in inorganic nitrogen 10 miles below the Bridge is due to
effluent discharges in this area. The decay of nitric nitrogen (nitrate) indicates that it is
reduced in the anaerobic area, and the decrease in ammoniacal nitrogen establishes that free
nitrogen rather than ammonia is the reduction product.
The Thames is also subject to large discharges of heated effluents, averaging 3 x lO^
BTU/day of which 757. is due to cooling water from power stations, the remainder being the com-
bined effects of freshwater discharges, municipal and industrial effluents, and biochemical
activity. Some typical longitudinal distributions of temperature are given in Figure 7.11.
2.3 MODELING TECHNIQUES
The approach to the mathematical modeling of quality parameters which was employed
in the Thames study possesses some features that differ from the approaches discussed in pre-
ceding chapters and thereby deserves a relatively detailed description. The incorporation of
the effects of tidal mixing is particularly noteworthy in that the advective-diffusion equation
was abandoned in favor of a probabilistic approach.
After one tidal cycle, the water originally within a small interval S£ of a point 0
(measured on the longitudinal axis) is represented as being spread either side of 0 according
to a distribution curve associated with the point 0. Each point on the estuary axis is thought
of as possessing such a distribution curve which completely characterizes the mixing at that
point. The particular form assumed for the distribution is not important so long as its first
and second moments agree with the "real" distribution, because after several successive tidal
cycles approximately the same dispersion pattern results. This is demonstrated graphically by
Figure 7.12. For this reason a rather simple form of the distribution is assumed, viz. that
after a given number (one, say) of tidal cycles, a fraction Pj_ of the water originally at 0
317
-------
• DISSOLVED OXYGEN
A AMMONIACAL NITROGEN
O NITRIC NITROGEN
a TOTAL INORGANIC NITROGEN
• SALINITY
10 0 10 20 30
MILES FROM LONDON BRIDGE AT HALF - TIDE
Fig. 7.10 Distribution of dissolved oxygen, inorganic nitrogen
compounds, and salinity in Thames on (a) 31st August
and (b) 23rd February 1954. After WPRL (1964).
22
20
G'8
1
i
12
10
I
I
APRIL
O 6TH
O I3TH
* 20TH
27TH
MAY JUNE
4TH 1ST
11TH 8TH
18TH 15TH
25TH 22ND
10 0
ABOVE BELOW
10 20 30
MILES FROM LONDON BRIDGE AT HALF - TIDE
Fig. 7.11 Results of weekly measurements of temperature in Thames
Estuary, April-June 1954. After WPM. (1964).
318
-------
is spread uniformly over a distance L downstream from 0 , P_ is spread uniformly L
upstream from 0 , and 1 - P. - P remains within the original interval 6-t of 0 (see
Figure 7.13). the distance L is called the "mixing length" (but bears no relation to
Prandtl's theory). The value of L is not thought to be especially crucial and is therefore
taken to be an integral number of miles "rather less than the average tidal excursion" and is
assumed invariant along the estuary. Characterization of the mixing now reduces to the problem
of finding P, and P~
judgment is required in that if P,
and if P, + P_ is "excessively small" L must be decreased.
Computation of the mixing constants P-^ and P2 is accomplished by the use of two
conservation principles:
(i) The flux of salt upstream across any cross section in a given time period is equal to the
total salt above the cross section at the end of the time period less the amount of salt present
initially (neglecting saline discharges, as was found permissible for the Thames).
(li) Mixing alone does not effect a net transfer of water across any cross section.
During one tidal cycle the salt carried upstream past a point x is
L
J S (x+t) A(x+t) P2(x-U) — d*
0
where S Is the average salinity over the tidal cycle. Similarly the total salt carried down-
stream past x is (since I < 0)
0 _
J S5 (x+t) A(x-Ht) Px (x-H) Mi At
Li S
-L
The net accumulation of salt AW above x in one tidal cycle is therefore
L 0
AW - £ {J S(x+t) A(x-M) P2(x-K) [L-*] dt - J ?(x+*) A(x-l-t) P^ (x44) [L-K] dt} - Q(x) S(x)
0 -L
where Q(x) is the net discharge above x in one tidal cycle and the overbar denotes the mean
value over the tidal cycle. It was found (WPRL 1964, p. 401) that AW could be neglected. Using
this fact and writing X - AP, and Y - AP» for convenience, the first conservation principle
becomes
L _ 0 _
i {J S(x-U) Y(x+t) CL-*] dt - J S(x-t-t) X(x-W-) [L-U] dt} - Q(x) S(x) - 0 (7.1)
L 0 -L
The second conservation principle, by the same reasoning, is stated
L 0
J Y(x-K) [L-t] dt - J X(x-K) [L+t] dt - 0 (7.2)
0 -L
319
-------
/
f
/
/'
1
\
\
\
\
\
\
1
\
\
\
\
\
(TIDE
2 TIDES
3 TIDES
Fig. 7.12 Distribution of water after 1, 2, and
3 tides using three representations of
symmetric mixing, each with same
second moment. After WFRL (1964).
P2
1-P,-P2
P1
Fig. 7.13 Representation of distribution after
one tidal cycle of unit mass of water
originally at 0, neglecting net
displacement due to freshwater inflow.
After WPRL (1964).
320
-------
Calculation of X and Y is equivalent to calculation of the mixing parameters P, and P,
X and Y were calculated for the Thames using measured average salinities and discharges
together with numerical integration of (7.1) and (7.2) for oner-mile increments along the
estuary. L was taken to be 6 miles (for a mixing period of one tidal cycle), and the mixing
parameters were computed to be as displayed in Figure 7.14.
Once P, and
as functions of x have been determined, the salinity distribution
can be calculated from Equation (7.1) for a given inflow conditions;
and P are also used
to compute the effect of tidal mixing on other parameters as described below. Representation
of the mixing by this method was applied to average tides, so that the difference between, say,
spring and neap was effectively neglected.
1*00
0 10 20 30 1*0
MILES BELOW LONDON BRIDGE AT HALF-TIDE
50
0.8
0.7
0.6
gO. 5
X
ro.2
O.I
I
J_
I
1
0 10 20 30 l»0
MILES BELOW LONDON BRIDGE AT HALF-TIDE
50
Fig. 7.14 Calculated mixing parameters for Thames
assuming mixing length of 6 miles and
period of one tidal cycle. After WPRL
U964) .
321
-------
The distribution of substances other than salinity in the Thames is determined by
accomplishing the calculation in two parts: the first is the computation of the effect of net
displacement due to freshwater discharge, sources of the substance from the periphery of the
estuary, and the natural sources and sinks of the substance within the estuary waters; the
second is the computation of the effects of tidal mixing. Thus for each time step T , a sub-
stance is represented as being first displaced without tidal mixing, and simultaneously
Incremented by the cumulative sources and sinks, then mixed without displacement or the effect
of source and sink processes. The time step T must of course be chosen sufficiently small
to prevent a degradation of accuracy, and this is dependent upon the reaction rates being suf-
ficiently slow. In the Thames work, T was on the order of a tidal period.
We denote the concentration of a constituent at position x and time t by C(x,t)
Boundary conditions are values of C' at the head and mouth of the estuary, and the initial
distribution C(x,0) is given. The problem is first to determine C*(X,T) , where C* is
the concentration after a period T without tidal mixing. This problem is pursued from a
Lagrangian standpoint. In time t a parcel of water originally at x will be displaced a
distance s given implicitly by
t - J ds . J AIx±sl ds
• u(x+s) • Q(x+s)
In addition its concentration will be incremented by the net addition I(x) and source/sink
process - ds (7.3)
If the total displacement from x after T Is denoted a , then C*(x+a,r) can be determined
from (7.3), and since o is a continuous function of x , this gives C* as a function of
distance along the estuary.
Given C*(X,T) the next step is to evaluate the resulting concentration C(X,T) with
tidal mixing. The contribution to the new concentration from those points lying seaward of x
is , approximately
L
— i— f C*(X+*,T) Y(x+t)dt
A(x)L J
o
since Y(x-H) « P2(x-U) A(x+*) and we assume A(x) is approximately the average area from
x+t-L to x-K. . Similarly, the contribution from those points x-M. lying landward from x
(t<0) is
V)
—^— J C*(x-K,T)
AlXJt, J
-L
322
-------
Finally, Che contribution to the new concentration due to that portion l-Pj-P, that remains
at x is
C*(X,T)
the new concentration, after mixing, is therefore
C(X,T) - — - — < C*(x-rt,r) Y(x+t)dt + C*(x+t) X(x+t)dt
LC*(x,T) [A(x) - X(x) - Y(x)] 1
(7.4)
Equations (7.3) and (7.4) were solved numerically with a spatial increment of two
miles. The specific techniques used are straightforward applications of linear algebra so
there is no need to review them in detail: Equations (7.1) through (7.4) are sufficient to
exhibit the underlying methodology.
2.3.1 Modeling of Temperature Distribution
The temperature as a function of position in the Thames is expressed as the sum of
a basic temperature, which is that temperature which would result if there were no heating in
the estuary other than natural sources, and a temperature excess. Although the basic tempera-
ture can, in theory, be calculated from various meteorological parameters, in the Thames study
it was instead calculated by statistical regression equations using the long temperature
records available (extending back to the mid-nineteenth century). Regression forms were
obtained for stations at Kingston (above the Wier), Greenwich and Crossness, relating the
water temperature T^ to the air temperature T& and to the difference between Ta and its
value the preceding month, denoted A . Most of the significant variation was associated with
Ta , and all three regression equations were quite similar. For example, the regression form
for Greenwich was determined'to be
Tw - 1.101 Ta - 0.151 A + .41 (°C)
the basic temperature for the estuary seaward from Crossness was obtained from measured data
for the period 1915-1921, which was assumed to be relatively free from artificial heating.
This data was expressed in terms of its excess over the temperature at Crossness for the four
seasons. Thus a complete longitudinal distribution for the basic temperature was obtained by
calculating the temperatures at Kingston, Greenwich and Crossness, passing a smooth curve
through these, and extending this curve from Crossness to the mouth using the appropriate temp-
erature difference curves for the season under consideration. Figure 7.15 shows the temperature
profiles used to predict the basic temperature along with measured temperatures. The effect of
artificial heating in the 1953 example is evident.
The calculation of the temperature excess 8 was effected by application of the
methodology described In Section 2.3 above. The known sources of heated effluents form the
basic Input, viz. the expression I(x) , and a first-order sink was assumed
323
-------
14.5
14.0
O
1- 13.5
UJ
cc
2 13.0
S
UJ
t: 12.5
UJ
12.0
YEAR
1920
15
20
25
30
35
45
MILES BELOW LONDON BRIDGE AT HALF - TIDE
Fig. 7.15(a) Average temperature in Thames seaward of
Crossness for April - June quarters of
1916-21. After WPRL (1964).
C/> (/I
UJ t/>
UUI
25
UJ
O. I—
-0.6
-0.8
-1.0
-1.2
2ND
I
15 20 25 30 35 40
MILES BELOW LONDON BRIDGE AT HALF - TIDE
Fig. 7.15(b) Average excess over that at Crossness for same
quarter of 1915-21. After WPRL (1964).
16
i '2
u 10
8
20
20 10 0 10 20 30
ABOVE BELOW
MILES FROM LONDON BRIDGE AT HALF - TIDE
Fig. 7.1S(c) Comparison between observed temperature (solid
curves) of estuary during October - December
quarters of 1919 and 1953 and basic temperature
predicted (broken curves and encircled points)
for same quarters. After WPRL (1964).
324
-------
where g is a heat exchange coefficient and z is the depth. The coefficient g was deter-
mined from measured data by relating d9/dt to the known rate of addition of heat and was
found to be 4.0 cro/hr.
2.3.2 Modeling of Dissolved Oxygen Distribution
In the Thames work, the sinks of dissolved oxygen found to be of principal importance
are the oxidation of organic carbon and the oxidation of ammonia. The principal sources are
reaeration and, in the anaerobic reach, the reduction of nitrates and sulfates. Accordingly,
modeling of the DO distribution required modeling of the distributions of carbonaceous demand,
organic nitrogen, ammoniacal nitrogen, and nitrates for basic inputs to the DO model.
It was found empirically that the simplest treatment of the carbonaceous demand
reasonably consistent with the data was to regard the demand as being the sum of two components:
a "fast" constituent and a "slow" constituent, the adjectives referring to the rates of oxida-
tion. The oxidation rate of the "slow" carbon was approximately one-fifth that of the "fast."
Therefore, the total carbonaceous oxygen uptake in a sample after a period t becomes
Xt - Ec [l - (1-p) exp(-Kt) + p exp(-Kt/5)]
where p denotes the proportion of "slow" carbon in the sample. The value of p depends upon
the character of the effluent and was evaluated separately for each major pollutant load on the
Thames. Although in several cases it could be assumed that p - 0 , the value of p in other
cases ranged as high as 6/7. Ec is the "effective" total carbonaceous demand, a number some-
what less than the value obtained from stoichiometry by assuming the complete oxidation of all
organic carbon. The oxidation rate K was taken to be .234 day"1. The effective demand E
can be related to the five-day BOD (assuming no nitrification) by setting t - 5 days in the
above equation, whereupon
c .69 - .48p
For p - 0 this becomes Ec • 1.45 BODj . The total carbonaceous demand was obtained by the
method of Section 2.3, computing separately the fast and slow demands with a first-order sink
with rate K « .234 day"! for t^e fast carbon and K - .047 day"! £or t^e siow< i^e sources
I(x) were determined by analysis of effluent data with the above considerations.
The nitrogenous oxygen demand is chiefly due to the oxidation of ammoniacal nitrogen,
which is introduced in the estuary both from direct discharges of ammonia and from the hydroly-
sis of organic nitrogen. Organic nitrogen was assumed to be hydrolysed at the same rates as
the organic carbon; it was therefore represented as the sum of a "fast" and a "slow" organic
nitrogen exactly as the carbonaceous demand was treated. As only the fraction E /U of the
total organic carbon is effectively oxidized (where Uc is the theoretical oxygen demand for
the total oxidation of the organic carbon, so that Uc - 2.67 C), the ammonia produced from
the hydrolysis of organic nitrogen is ECNQ /Uc , where NO is the content of organic
nitrogen in a particular effluent. The sum of the fast and slow organic' nitrogen distributions
325
-------
becomes a source In the amnoniacal nitrogen equation. Anmonia was assumed to decay according
to a simple first-order expression with rate constant K = .102 day"1 at 20° C when the DO con-
centration is greater than ST. When the DO concentration falls to 5% of saturation, nitrifica-
tion was considered to proceed at a reduced rate (see below) so as to maintain the DO at 5% as
long as possible. If the oxidation of organic carbons is sufficient to drive the DO below 5%
of saturation, even in the absence of nitrification, then nitrification was assumed to cease,
and the reduction of nitrates to occur and thereby maintain the level of DO at 5% of saturation.
In addition to effluent discharges and freshwater inflows, nitrates are introduced
into the estuary as a result of the oxidation of ammonia. Except for those reaches in which
the DO is at 5% of saturation and nitrification has ceased, nitrates were considered to be
inert. For convenience the nitrate was expressed as the equivalent oxygen produced in the
reduction (to nitrogen) of the nitrate, and thus available for oxidation of the organic carbon.
This is five-eighths the amount of oxygen required to produce the same amount of nitrate by oxi-
dation of ammonia.
The source of oxygen by diffusion from the atmosphere was represented by the classical
expression
*•
where z is the depth, D is the oxygen deficit, and f the exchange coefficient. For the
Thames work, it was concluded that a value of f - 5.0 cm/hr throughout the estuary was
sufficiently accurate for initial calculations. (See Section 2.4 and Table 7.3.)
Calculation of a DO profile is accomplished by first computing the carbonaceous BOD
(the sum of the "fast" and "slow" demands) and the nitrogenous demand assuming aerobic condi-
tions, i.e. that nitrification occurs at the unrestricted rate K . The distribution of
nitrate is obtained assuming production through the oxidation of amnonia and the contributions
from various discharges, but' no sinks of nitrate in the watercourse. The resulting DO deficit
due to the combined sinks of carbonaceous and nitrogenous oxidation is calculated. The tempera-
ture profile is a basic input to this calculation in that it affects the solubility of oxygen
and the reaction rates of the biochemical oxidation processes. If the resulting deficits are
less than .95 C8 (C8 denoting the saturation concentration of DO) then the resulting profile
is the final solution. If, however, there are reaches in which D > .95 Cg , the profile must
be recomputed with allowance made for anaerobic processes. This is performed by an essentially
trial -and-error technique. The carbonaceous demand profile is assumed to be correct, whatever
the DO concentration may be, therefore it is not modified. However, the ammonia concentration
will have been underestimated in* certain reaches because, in fact, restricted nitrification
should have been assumed. Therefore, a function U(x) is added to the amnonia concentration
to compensate for this error; the nitrate equivalent - -g U(x) is added to the nitrate con-
centration to correct for the overestlmation of nitrates. U(x) is determined by trial
calculations of the deficit profiles. There is an obvious upper limit to the value of U(x) ,
namely the total increase in nitrate (at x ) due to nitrification, and setting U(x) equal
to this maximum is equivalent to specifying no nitrification, i.e. K » o . Even with no
nitrification. It is evident that there may remain reaches in which the deficit still exceeds
.95 C . For these areas, nitrate reduction is assumed, so that the concentration of nitrates
is further reduced by the addition of the function -V(x) , which is also determined by trlal-
and-error. The T*-!""" permissible value of V(x) is that value which reduces the total
nitrates to zero. Should there still remain reaches with deficit in excess of .95 C and for
which the nitrate concentration is zero, the deficit profile is further modified by a function
326
-------
-W(x) representing the reduction of sulfates. The maximum value of this function is that which
balances the carbonaceous demand, i.e. that which brings the DO concentration up to zero.
The finite-difference solution to the relevant equations was performed using 35 points
along the estuary at 2 mile intervals beginning at Teddington Wier. In general, the solutions
obtained were for steady-state conditions, so that the calculations were advanced in time with
constant inputs until an equilibrium solution resulted. For boundary conditions at the mouth
of the estuary it was assumed that the concentrations of polluting substances were zero. At
the head, the concentrations of the polluting substances were measured values or (for pre-
dictions) estimated values. For dissolved oxygen, the sampling point furthest up the estuary
was below the River Brent at Kew, some 6 miles below the head of the estuary. Accordingly,
the boundary value for DO at the Wier was taken to be that value which brought the model value
at Kew into agreement with the measured value.
2.4 MODEL RESULTS
2.4.1 Calculated Profiles and their Accuracy
The methods described in Section 2.3 et seq. were applied to calculations of the
profiles of DO and organics using measured flows and loadings for the period 1950-1961 and the
computed results were compared with observed concentrations in the estuary. Calculations were
performed by dividing each year into quarters, averaging the inputs (inflows and loadings) over
each quarter, and obtaining the steady-state profiles for these averaged inputs. These results
were then compared with the averaged empirical data for that quarter.
Figures 7.16 and 7.17 are examples of the intermediate calculations of temperature,
and organic carbon and nitrogen (oxygen equivalent), respectively. Both the calculated and
observed temperature profiles are shown in Figure 7.16, as well as the computed basic distri-
bution, i.e. that profile which would result in the absence of artificial heating. In Figure
7.17, the calculated organic carbon and nitrogen are separated into their "fast" and "slow"
components.
Figure 7.18 displays the results of calculations for 1951, 1953, 1954, 1957, 1960
and 1961 for the parameters dissolved oxygen, amnonlacal nitrogen, and oxidized nitrogen
(nitrates). Much of the discrepancy in the DO profiles was found to be attributable to large
variations in the inflow during the quarter, hence producing significant departures from the
assumed steady state. This was partially compensated for by subdividing the quarter into
thirteen weekly records, averaging the inflows for each week and computing the resulting
steady-state solution. The thirteen profiles so obtained were then averaged to arrive at a
revised quarterly average. Although no allowance was made for week-to-week variations in the
loadings of temperatures, nor for smaller time scale variations in inflow, this calculation
produces noticable improvements in the refined profiles. Note, for example, the results for
1954 of Figure 7.18. The change in temperature due to the same kind of averaging is demon-
strated by Figure 7.19 (cf. Figure 7.16).
Figure 7.20 shows the predicted and observed profiles of DO obtained by averaging by
quarters the empirical concentration for the period 1950-1955 and by averaging the correspond-
ing predictions after allowance is made for weekly variation in freshwater inflow. The sys-
tematic discrepancies between observations and predictions suggested that a seasonal variation
327
-------
12
10
16
12
S 8
I6
I **
ui
-16
12
16
12
10
OBSERVED
CALCULATED
BASIC
1954 4TH QTR.
..2S74M.G.C
1951 2ND QTR.
2238 M.G.D.
1954 1ST QTR.
1347 M.G.D;
r—-t. 1954 2ND QTR.
^993 M.G.D.
1953 4TH QTR.
670 M.G.D.
I
I
1953 2ND QTR.
594 M.G.D. -
r---.rr
1951 3RD QTR.
57<» M.G.D.
1954 3RD QTR.
506 M.G.D.
1952 3RD QTR.
286 M.G.D.
1953 3RD QTR.
225 M.G.D.
I
I
I
20
18
16
14
21»
22
20
18
22
20
18
16
22
20
18
-10 0 10 20 30 0 10 20
MILES BELOW LONDON BRIDGE AT HALF - TIDE
30
Fig. 7.16 Examples of computed and observed temperature
distributions In the Thames. Heat exchange
coefficient taken to be 3.7 cm/hr. Average
flows at Teddington are Indicated. After VPRL
(1964) .
Z 20
a.
ZO-ln
^J^v*' "
5< 5
UIU
(9
X
O
(B)
i
o.
I— O.
0 ' '—
20 -*-0-* 20
ABOVE BELOW
MILES FROM LONDON BRIDGE AT HALF -TIDE
20 *- 0-*- 20
ABOVE BELOW
x
O
Fig. 7.17 Example of calculated distribution of organic
carbon oxygen demand and organ, nitrogen distri-
bution (oxygen equivalent). P indicates "fast"
oxidation-hydrolysis rate: Q Indicates "slow."
After WPRL (1964).
328
-------
1951
NOXD
'953
100
o 60
20
10
NAMM6
2
1954
-20 0 20 40 0 20 40 0 20 40 0
MILES BELOW LONDON BRIDGE AT HALF-TIDE
20 40
Fig. 7.18 Examples of Calculated DO, Ammonia (N ) and
3 mm
Nitrates (NQX(j) Profiles Compared with Measured
Data.
First through fourth quarters of each year
beginning with the first (January-March) at left.
Solid curves: Calculated
Broken curves: Calculated profiles with
variation of inflow
Plotted points: Measured concentrations
(London County Council data)
Dotted curves: W.P.R.L. data
DO in percent saturation,
After WPRL (1964) .
and N
^
in p. p.m.
329
-------
1957
NOXD
I960
1961
NOXD k --_-VS
-20 0 20 kO 0 20 <*0 0 20 kO 0 20 kO
MILES BELOW LONDON BRIDGE AT HALF-TIDE
Fig. 7.18 Continued.
330
-------
13
5 12
11
10
10 — 0 — 10
ABOVE BELOW
20
30
4o
MILES FROM LONDON BRIDGE AT HALF - TIDE
Fig. 7.19 Effect of variations in fresh-water flow during
quarter on predicted temperature distribution
(for fourth quarter of 1954). A, observed;
B, basic; C, predicted to be in equilibrium with
mean flow during quarter; D, predicted allowing
for variations in flow during quarter. After
WPRL (1964).
-20
20
0 20 40
4™ QUARTER
-20
0 20 40
100
80
60
40
20
0
-20 0 20 40
MILES FROM LONDON BRIDGE AT HALF-TIDE
Fig. 7.20 Observed and predicted DO profiles (per cent
saturation) in Thames for corresponding
quarters of 1950 to mid-1955, and for entire
period. Solid curves, observed data. Broken
curves, calculated. After WPRL (1964).
331
-------
in the exchange coefficient might be in evidence. On the basis of extensive field and labora-
tory investigations, the parameters most important to the value of the exchange coefficient f
were found to be temperature, contaminants (e.g. dissolved solids, sewage, surface impurities),
and wind. f was modified using the results of the experimental studies (some of which were
quite tentative). The resulting exchange coefficients are tabulated in Table 7.3. In compari-
son, the value of f required to bring the calculated results into agreement with the observed
is also shown. The large discrepancy between this value and the "theoretical," particularly in
the second and fourth quarters, is attributed to phytoplankton (i.e. increased photosynthesis
in the spring and phytoplankton decay in the fall). The values adopted for f for use in
model predictions are given in the last line of Table 7.3.
TABLE 7.3
SEASONAL VARIATION IN EXCHANGE
COEFFICIENT f (cm/hr) , From WPRL (1964)
Expected values of f from
experimental work with
allowance for seasonal
variation in:
(a) temperature
(b) temperature and
contaminants
(c) temperature, contaminants
and wind
Calculated from discrepancy between
observed and predicted DO
Final values used in predictions
1st
Qtr.
4.5
5.0
5.4
5.3
5.2
2nd
Qtr.
5.1
4.8
4.7
5.5
5.3
3rd
Qtr.
5.5
5.1
4.8
4.2
4.3
4th
Qtr.
4.9
5.0
5.1
4.3
4.5
The effects on the model calculations of temperature and inflow for the four seasons
are evident from Figure 7.21. These distributions were obtained by assuming the polluting
loads to be the average for 1951-54, a constant inflow in the indicated amount, and a constant
temperature along the estuary. The plotted data are individual measurements from the period
1951-54 grouped according to the temperature at Crossness and the flow at Teddington. The data
were sorted by temperature into groups ±2%°C from the model temperatures and into the flow
ranges 400-600 MGD, 1250-1750 MGD, and 2500-3500 MGD. All other data for the period were
excluded.
2.4.2 Model Application
The principal application intended for the Thames model was to serve as an aid to
management in providing a -tool for the evaluation of the effects of various outfalls and for
the evaluation of management alternatives. Some examples of these types of applications are
given in this section.
332
-------
500 M.G.D.
MtTM.
"OXD
-20 0 20 1(0-10 0 20 40 -10 0 20 40-10 0 20 40
MILES BELOW LONDON BRIDGE AT HALF - TIDE
1500 H.G.D.
'OXD
-20 0 20 40 -10 0 20 40 -100 20 40-10 0 20 40
MILES BELOW LONDON BRIDGE AT HALF - TIDE
M.G.D
AMM
N
OXD 2-
y20 0 20 40 -10 0 20 40 -10 0 20 40 -10 0 20 40
MILES BELOW LONDON BRIDGE AT HALF - TIDE
Fig. 7.21 Observed and calculated effects of temperature and
fresh-water flow on distributions of dissolved
oxygen (per cent saturation), anmoniacal nitrogen
(p.p.m.) and oxidized nitrogen (p.p.in.). Solid
curves, calculated distributions for average
polluting loads in 1951-54, and flows at
Teddington and temperatures as indicated. Plotted
Points, observed concentrations. After WPRL (1964).
333
-------
Figure 7.22 exemplifies model calculations which show the effects of relocating the
Northern Outfall and the Southern Outfall (below Crossness) for simmer conditions and esti-
mated 1964 loadings. These results are compared with the same calculations using the average
loadings of 1950 to mid-1955. An example of the use of the model to ascertain required treat-
ment levels is shown in Figure 7.23. This is the result of trial-and-error determination of
level of treatment required to permit the passage of migrating salmon. Curves A, B, and C are,
respectively, the DO profile with 1964 loadings, the profile obtained from assuming secondary
treatment at the Northern Outfall, and the profile resulting from a more extensive reduction
in polluting loads (see WPRL 1964, pp. 527-528). Salmon were assumed to require 30% saturation
during the second quarter, the period in which migration occurs. Accordingly, curves A, B, and
C were obtained for second quarter conditions assuming 400 MGD at Teddington. In comparison,
curve D was calculated using the same loads as curve C but for summer conditions with 170 MGD
at Teddington.
z 100
Ul
2 -80
Xl-Z
ozo ,n
ui— 60
Our-
ui
2 ko
€/>•— < 20
ut ut
5 0
16
12
8
k
0
10--0 —10 20 30
ABOVE BELOW
rOi-0-*-TO 20 3C
ABOVE BELOW
MILES FROM LONDON BRIDGE AT HALF-TIDE
Fig. 7.22 Effect of moving discharge points to (a) 30 miles,
and (b) 40 miles below London Bridge. Outfalls in
present position (dotted), Northern Outfall relocated
(dashed), Northern and Southern Outfalls relocated
(solid). Average loadings for 1950-55 (left) and
estimated loadings for 1964 (right). After WPRL (1964).
334
-------
100
10 0 10 20 30
ABOVE BELOW
MILES FROM LONDON BRIDGE AT HALF-TIDE
50
Fig. 7.23 Calculated DO profiles assuming various
loading conditions in estuary (see text).
After WPRL (1964).
The period 1950-1961 is of interest in demonstrating the efficacy of the Thames model
for predicting the results of improved treatment due to various alterations at the Northern
Outfall Works. For the period 1950 through mid-1955, the polluting loads at the Northern Out-
fall were relatively constant, discharging 190 MGD of which 21% received secondary treatment.
In mid-1955, a new sedimentation plant became operable, thereby reducing the load. From then
until 1960, the average flow was 195 MGD of which 187. received secondary treatment. Beginning
in 1960, a new aeration plant was operated further reducing the load, and through 1961 the
average flow was 212 MGD with 457. given secondary treatment. Model calculations were performed
for the first and third quarters using the average loads and temperatures for these three
periods. Empirical curves for the same three periods were obtained from observed data (using
the interpolation scheme to "normalize" the inflows) and compared to the predictions. The
observed and predicted DO profiles are shown in Figure 7.24, and the corresponding ammoniacal
nitrogen and nitrates are shown in Figure 7.25. The predicted curves only take into account
the variations in loads at the Northern Outfall; all other changes in polluting loads are
neglected. In the vicinity of the Northern Outfall (mile point 11) this is not important, but
elsewhere, particularly upstream, the effects of other sources can become significant. The
large observed changes in DO above the Bridge were attributed to such variations in the loads.
Extensive predictions were made of the quality in the Thames to be expected in 1964.
At the time of issuance of the report on the Thames work (WPRL 1964), extensions and modifica-
tions in two of the larger municipal treatment plants, Southern Outfall and Mogden, were
underway and scheduled for completion in 1964. The effects of these changes were incorporated
in the predictions by employing the expected effluent characteristics. In addition, increased
discharge in the Northern Outfall (above the River Roding), and some diversion of wastes
through the Northern Outfall were also considered. Figure 7.26 shows the results of the model
calculations for the worst case of summer conditions and low inflow (the legislated minimum).
For comparison, curves derived from empirical data for 1920-29 and 1950-59 are shown. These
curves were obtained by compensating the observed measurements for variation in flow using
empirical "rating curves." In contradistinction, Figure 7.27 displays the same curves for
335
-------
100
5 8o
£
5260
oa.
S~20
100
80
°- tn
/iw 20
i r i
-20
20
40
20
HUES BELOW LONDON MIDGE AT HALF-TIDE
Fig. 7.2A Predicted (above) and observed (below) DO
profiles during first quarter (left) and third
quarter (right), following improvements in
treatment at Northern Outfall. 1961 (solid),
mid-1955 to 1959 (dashed), 1950 to mid-1955
(dotted). After WFRL (1964).
o -20
x
o
0 20 40 0 10
MILES BELOW LONDON BRIDGE A.T HALF-TIDE
Fl(. 7.25 Anooniacal and oxidized nitrogen profiles
corresponding to DO profiles of Fig. 7.24.
After WFW. (1964).
336
-------
20 1C--0-MO 20 30 40
ABOVE BELOW
50
20 10---0-HO
ABOVE BELOW
30 40 50
MILES FROM LONDON BRIDGE AT HALF-TIDE
Fig. 7.26 Predicted profiles of DO, Ammoniacal Nitrogen and
Oxidized Nitrogen for third quarter (summer) condi-
tions, 170 MGD flow at Teddington, and estimated
loads for 1964. Broken curves are derived from
observed data for indicated periods. After WPRL
(1964).
x
o
a
UJ
>
__i
O
ut
o
20
10*-0—HO 20 30 40
ABOVE BELOW
10---0—HO 20 30 40 50
ABOVE BELOW
MILES FROM LONDON BRIDGE AT HALF-TIDE
Fig. 7.27 Predicted profiles of DO for 1964 loadings with
profiles obtained from observed data, (a) First
quarter, inflow 4000 MGD at Teddington.
(b) Second quarter, inflow 1200 MGD at Teddington.
After WPRL (1964).
different seasons and inflows. In Figure 7.28 are given the steady-state predictions for
1964 for all four quarters under various Inflows. For these predictions and those of
Figures 7.26 and 7.27, the exchange coefficient was assumed to vary seasonally as specified
in Table 7.3.
It is of particular interest to compare these predictions with the concentrations
actually measured after the improvements in the Mogden and Southern Outfall plants were com-
pleted. In Gameson and Hart (1966) this comparison was made using data from both 1964 and
1965, and is displayed in Figure 7.29. The model results were obtained from the predictions
of Figure 7.28 by interpolation to the measured inflows. The comparison is somewhat dis-
appointing overall. Lack of agreement in DO values in the upper reach can be attributed to
the fact that the Mogden load was in fact much larger than the load used in the predictions,
337
-------
as Indicated in Table 7.4. However, the Southern Outfall and Northern Outfall BOD loads were
overestimated and In spite of this the predicted DO's are consistently higher than the measured
values. Using the correct loads for the third quarter of 1964, Figure 7.29(a), did not improve
the DO correspondence, although the ammonia and nitrate model profiles are brought into better
agreement with those observed.
The sources of these errors are not apparent. It was suggested that the anmonical
nitrogen, and hence the nitrate, calculations may be adversely affected by using what appears
10 0 10 20 30 W) 50 100 10 20 30 40 50
ABOVE BELOW ABOVE BELOW
HILES FROM LONDON BRIDGE AT HALF-TIDE
Fig. 7.28 Predicted profiles of DO (solid), amnoniacal (dashed),
and oxidized (dotted) nitrogen for estimated 1964
loads. Flows at Teddington in MGD are shown. After
WPRL (1964).
338
-------
100
1st QTR 1697mqd 2nd QTR 1392 mgd 3rd QTR 3l7mgd
4»i QTR 327 mgd
-20 0
}0
0 30 0 30
MILES BELOW LONDON BRIDGE AT HALF-TIDE
(a) I964 DATA
100
IstQTR 786ntgd 2nd OTR 440 mgd 3rd QTR 365 mgd 4th QTR 163 4 mgd
-20 0
0 30 0 30
MILES BELOW LONDON BRIDGE AT HALF-TIDE
(b) 1965 DATA
Fig. 7.29 Observed concentrations of DO, Amm. N and Nitrates compared
with profiles predicted from assumed loads (Table 7.4).
Broken curves in (a) are the profiles predicted from the
measured loads. After Gameson and Hart (1966). Data collected
by Greater London Council.
339
-------
to be too low a temperature coefficient on the nitrification rate, thus overestimating the
ammonia concentrations during sunnier months and underestimating during the winter. This is
consistent with conclusions in WFRL (1964), but probably does not completely account for the
errors. Another possible source of error suggested by Gameson and Hart (1966) is a departure
in the measured data of 1964 and 1965 from the steady-state condition assumed in the model
computations.
TABLE 7.4
LOADS (tons/day) OF BOD, AND OF AMMONIACAL AND OXIDIZED NITROGEN DISCHARGED
FROM MOGDEN (Hog.), NORTHERN OUTFALL (N.O.), AND SOUTHERN OUTFALL (S.O.)
SEWAGE WORKS IN EACH QUARTER OF 1964 AND 1965, AND VALUES
PREVIOUSLY ASSUMED IN LABORATORY'S PREDICTIONS
From Gameson and Hart (1966)
1964, 1st Qtr.
2nd Qtr.
3rd Qtr.
4th Qtr.
1965, 1st Qtr.
2nd Qtr.
3rd Qtr.
4th Qtr.
Average
Assumed
BOD
Hog. N.O. S.O. Total
27.1 90.9 15.4 133
16.1 83.3 10.4 110
6.4 72.5 9.8 89
11.6 108.5 11.3 131
12.3 105.7 14.6 133
11.4 98.4 15.7 126
7.8 91.2 10.7 110
12.6 103.1 15.1 131
13.1 94.2 12.9 120
3.8 110.5 17.0 131
Ammoniacal Nitrogen
Mog. N.O. S.O. Total
11.7 23.9 12.6 48
11.6 22.6 16.1 50
8.9 19.2 14.0 42
11.2 24.9 14.5 51
11.2 25.5 14.7 51
10.7 20.6 14.5 46
8.8 17.6 13.4 40
9.6 19.4 14.2 43
10.5 21.7 14.2 46
6.1 24.1 16.2 46
Oxidized Nitrogen
Mog. N.O. S.O. Total
0.5 1.4 0.4 2.3
0.7 0.6 0.3 1.6
1.5 0.5 0.2 2.2
1.0 0.5 . 0.1 1.6
0.2 1.1 0.1 1.4
0.1 1.0 0.1 1.2
2.4 1.2 0.2 3.8
2.0 3.5 0.3 5.8
1.1 1.2 0.2 2.5
5.7 1.4 nil 7.1
340
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3. THE DECS MODEL
One of the most Important and comprehensive estuarine management studies to have been
undertaken in this country was that performed by the FWPCA in cooperation with the Delaware
River Basin Commission and various state and local agencies in Pennsylvania and New Jersey,
viz. the Delaware Estuary Comprehensive Study (DECS). The overall objectives of this study
included the following (FWPCA 1966):
(1) determine the cause and effect relationships between pollution from any source
and the deteriorated quality of water in the estuary;
(ii) develop methods of water quality management including the development of
techniques of forecasting the variations in water quality due to natural or
man-made causes;
(iii) prepare a comprehensive program for the improvement and maintenance of water
quality in the estuary including the waste removals and other control devices
necessary to manage the quality of water in the estuary for municipal, indus-
trial and agricultural water use, and for fisheries, recreation, and wildlife
propagation.
The modeling approach employed in this study was the application of the technique of finite
differences to the steady-state and time-dependent (tidal-averaged) conservation of mass equa-
tions. (Several program codes therefore exist, corresponding to the steady-state and time-
varying models in various stages of evolution, but since their unifying feature is the
one-dimensional segmentation for finite-difference solution, these programs are referred to
collectively as the DECS model throughout this chapter.) As the details of the model are
well-documented (Thomaim 1963, Pence et al. 1968, Jeglic and Pence 1968, and see Chapter III,
Sections 3 and 5), it is the purpose of this section to review its application. The model
has not only been applied fruitfully to the Delaware Estuary but to other estuaries as well. In
particular, quite extensive application has been made to the Potomac, and the model was adapted
for two-dimensional computations for use on Hillsborough Bay.
3.1 THE DELAWARE ESTUARY
The Delaware, which drains the vast Pennsylvania-New Jersey urban development, is
one of the most important estuaries on the New England Seaboard. The estuary serves the
Pennsylvania-New Jersey area as a source for municipal water and as receiving water for the
wastes of the urban complex. The latter has resulted in seriously degraded water, so that
quality in the Delaware has become a problem of increasing magnitude in recent years. Recrea-
tional uses of the estuary have been seriously truncated. The commercial yield of shad,
sturgeon, white perch and a number of other finfish has declined to the point of being eco-
nomically negligible. (This, of course, is not solely attributable to degraded water quality.
Also, in contrast, the menhaden has become of great economic importance, due to broadening
industrial application of the catch.)
341
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3.1.1 Characteristics of the Estuary
The Delaware River extends from Its headwaters in the south-central area of Mew York
State to the Atlantic Ocean at the coastal limits of Delaware and Mew Jersey. The estuarine
area is the reach between Trenton, N. J., and the ocean (see Figure 7.30), and the Delaware
Estuary Comprehensive Study was principally concerned with 86 miles of the river between Trenton
and Listen Point. Figure 7.30 shows the segmentation employed in the DECS model. (These sec-
tions offer a convenient means of referring to specific reaches of the estuary and will be used
for this purpose hereafter.) The estuary at the ocean has a total drainage area of approxi-
mately 13,500 square miles. The major tributary to the Delaware Estuary is the Schuylkill
River which joins the main stream in the vicinity of Philadelphia. The mean annual discharges
of the Delaware and Schuylkill Rivers are on the order of 12,000 cfs at Trenton and 3,000 cfs
at Philadelphia, respectively. The total length of the Delaware Estuary between Trenton and
the limit at Capes Hay and Henlopen is approximately 135 miles.
Tidal elevation changes in the Delaware River are observed from the mouth to Trenton.
Mean water stage varies from a minimum of 5.5 feet at Reedy Point to a maximum of 6.8 feet at
Trenton under average flow conditions. Tidal current reversals occur along the river above
Burlington, approximately 10 miles below Trenton. Average mmrlnnm current velocities within
the estuary are approximately 3.0 fps. The Schuylkill River exhibits tidal stage changes and
current reversals to the location of the Fairmount Dam, eight miles above its mouth. Salinity
intrusion in the Delaware Estuary generally extends to the area between Wilmington and Marcus
Hook for mean freshwater discharge conditions, although during drought periods significant
chloride concentration is observed near Philadelphia.
Waste discharges to the Delaware Estuary are both municipal and industrial in origin,
with the municipalities generally representing the largest waste sources. The spatial
distribution of discharged loads in 1964 is exhibited in Table 7.5. In general, the water
quality at the head of tide at Trenton Is good, but begins to deteriorate downstream. From
Torresdale, Pa., Segment 7 (see Figure 7.30), to below the Pennsylvania-Delaware state line,
Segment 19, the deterioration is extreme; as a result of waste discharges, dissolved oxygen
is almost completely depleted in some locations and production of gases from anaerobic organic
deposits occasionally occurs. The concentration of coliform bacteria resulting primarily from
unchlorinated municipal wastes is very high in the same stretch of river. Acid conditions due
to industrial waste discharges have been observed for several miles above and below the
Pennsylvania-Delaware state line.
3.1.2 Model Application and Results
An extensive comparison has been made of observed DO concentrations of 1964 with DO
concentrations computed from the DECS model (Pence et al. 1968). The Delaware was segmented,
as Indicated in Figure 7.30, into thirty sections each of which was assigned a (mid-tide)
volume and cross-sectional areas at the junctions from physiographic data of the Corps of
Engineers. Basic inputs were the river flows measured at Trenton and water temperatures
measured by the USGS. Water temperature was assumed to be uniform throughout the estuary but
to vary with time. Plots of water temperatures and inflows with time are given in Figure 7.31.
The average carbonaceous loads for 1964 (Table 7.5) were used in the DECS calcula-
tions. The decay rate for carbonaceous BOD was taken to be .23 day" at 20°C. Nitrogenous
342
-------
PENNSYWANIA
DELAWARE
NEW JERSEY
10
Fig. 7.30 Map of Delaware Estuary showing segmentation for DECS
Model. River Mile 0.0 is at the Mouth of the Estuary.
343
-------
70,000
60,000
£ 50,000
-------
TABLE 7.5
1964 Waste Loading In the Delaware Estuary
Pounds Per Day of Carbonaceous Biochemical Oxygen Demand
Section
(Fig.
7.30)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Totals
Municipal
3,570
2,100
3,380
2,720
900
1,075
0
520
795
128,610
720
0
62,080
174,520
3,330
158,070
14,575
10,185
1,820
0
87,400
0
1,870
0
0
0
660
1,730
0
0
660,630
Industry
0
2,750
1,635
2,850
1,400
435
0
0
35
7,550
1,570
0
13,925
19,670
39,550
25,650
42,420
14,535
64,360
0
8,480
116,755
0
370
0
2,500
0
0
0
0
366,440
Tributary
2,869
4,107
982
1,078
2,047
5,798
1,875
4,309
3,095
3,146
3,189
1,105
1,800
1,566
17,649
3,761
8,678
6,003
1,668
1,071
6,848
294
306
421
855
1,416
322
9,078
5,011
5,509
174,719
Storm
Water
Overflow
1,360
230
1,580
8,570
4,390
16,780
4,480
7,410
2,080
18,860
1,950
8,320
76,010
Total
BOD Load
7,799
8,957
5,997
6,648
4,347
7,308
2,105
6,409
12,496
143,695
22,259
5,585
85,215
197,836
79,389
187,481
67,623
30,723
67,848
1,071
111,048
117,049
2,176
791
855
3,916
992
10,808
5,011
5,509
1,277,799
loads were handled by a separate computation, and the dependency of the nitrogenous decay rate
on temperature was expressed as
K,, - .00475 - .00184 + .000326 T2 - .0000170 T3 + .000000479 T4
Reaeration rates were calculated from the O'Connor-Dobbins (1958) formula (see Chapter III,
Section 6.1.5). Dispersion coefficients were obtained from observations of salinity intrusion
to Torresdale (Segment 7), and ranged from 4 square miles/day at Trenton (Segment 1) to 7 square
miles/day at Liston Point (Segment 30). In Figure 7.32 is shown a comparison of measured and
computed DO concentrations for a one year period at three segments in the Delaware (Pence et al.
1968).
During the 1965 drought, when the intrusion of salinity threatened the water supply
of many of the upstream municipalities, the DECS model was operated to predict salinity profiles
345
-------
i
g
s
o
(a) BRIS10L-SCC1 ION k
U
12
I !
10
9
8
7
6
5
U
3
2
1
I I
I I
(bl IORRESMLE-SECTION 7
JtN TEI HAD if« HA1 JUN JUL »UG StP OCI KOV DEC
[el DELAWARE MEMORIAL 88IOCE-SEC!ION 22
Fig. 7.32 Comparison of Observed DO and DECS
Model Calculations for 1964 data.
346
-------
150
100
3 50
9/23-
9/8-
JUN JUL AUG
1965
TORRESDALE (SECTION 7)
SEP
OCT
600
500
400
300
200
100
9/8
e/25
JUN
JUL
AUG
1965
FORT MIFFLIN (SECTION 15)
SEP
OCT
1*000
3000
2000
1000
JUN
JUL
SEP
AUG
1965
DELAWARE MEMORIAL BRIDGE (SECTION 22)
OCT
Fig. 7.33 Thirty-day forecasts of clorinity in Delaware
estuary. From Jeglic and Pence (1968).
347
-------
Example computations from this application are shown in Figure 7.33 along with the envelope of
observed salinities. These results were useful in flow regulation during the drought.
The DECS model has been employed extensively in the evaluation of water quality
management strategies for the Delaware estuary, as described in, e.g., Jegllc and Pence (1968)
and reported in FWPCA (1966). This has involved, for example, prediction of DO profiles for
various inflows (e.g., Figure 7.34) and for different loadings corresponding to treatment
levels. The model computations have served as basic input for optimization calculations using
a variety of management alternatives and system constraints (Smith and Morris 1969).
£ 8
o
2 6
-SECTION 10
SECTION 13
SECTION 15
SECTION 20
J I
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
Fig. 7.3A Calculated dissolved oxygen profiles for 1 In 25 year flow
and 1964 loading (Table 7.5). From Jegllc and Pence (1968).
3.2 THE POTOMAC ESTUARY
The DECS model has also been applied to the estuary of the Potomac (Figure 7.35),
one of the principal rivers discharging into Chesapeake Bay, and which drains the metropolitan
area of the District of Columbia. The tidal reach of the river extends 116 miles from
Chesapeake Bay to the head of tide at Little Falls in Washington, D. C. The Potomac Is an
Important source of fresh water for the Washington area; in 1967 the Potomac was supplying
over 707. of the fresh water needs of this area. The segmentation employed in the modeling of
the Potomac is shown in Figure 7.35. There are 28 segments varying in length from 2 miles in
the upper portion of the estuary, where finer spatial resolution was required, to 6 miles near
the mouth.
Dispersion coefficients for the first twelve segments were obtained from a dye diffu-
sion study conducted in the Potomac in the summer of 1965. In this experiment 341 pounds of
Rhodamine WT was injected into the river over a 13-day period and concentrations measured with
a fluorometer during the period of release and 21 days thereafter. A segmented model (with
slightly finer segmentation than that of Figure 7.35) solved on an analog computer was used
348
-------
WASHINGTOND.C.
/
HALLOWING
POINT
• MAJOR WASTE TREATMENT PLANT
. GAGING STATION
POTOMAC RIVER AT WASHINGTON D.C.
A, DISTRICT OF COLUMBIA
B. ARLINGTON COUNTY
C. ALEXANDRIA SANITARY AUTHORITY
D. FAIRFAX COUNTY - WESTGATE PLANT
E. FAIRFAX COUNTY - LITTLE HUNTING CREEK PLANT
F. FAIRFAX COUNTY - DOGUE CREEK PLANT
Fig. 7.35 Potomac River showing model segmentation.
349
-------
to determine the dispersion coefficients by a trial-and-error process. These coefficients
ranged from .05 mi^/day at Chain Bridge to 1.0 mi2/day at Hallowing Point, the lower boundary
of the study area. The remaining dispersion coefficients were determined from trial-and-error
calculations of chlorine concentration using the segmented model of Figure 7.35. Thus at
Indian Head, the upper limit of the region of saline intrusion, the dispersion coefficient was
found to be 1.5 mi2/day, increasing to 6.0 mi2/day at Maryland Point and to 10.0 mi2/day at
Segment 22. Example spatial and temporal profiles of chlorides, both measured and computed,
are shown In Figure 7.36 (Hetling 1968).
Computations for BOO and DO in the Potomac have been performed (Hetling and O'Connell
1968) using the same basic model. The six treatment plants shown in Figure 7.35 constitute
the main waste loads to the estuary; estimates of these loads for 1966 are given in Table 7.6.
These loads Include both carbonaceous and nitrogenous components. A single decay coefficient
was used, calculated from the standard formula
K - 0.23 (1.047)20'T (day"1)
More complex expressions, e.g., those used for the Delaware work, were investigated, but the
DO calculation did not seem to be especially sensitive to differences in the formulae. Reaera-
tion rates were calculated from the O'Connor-Dobbins (1958) formula. Explicit values for the
oxygen uptake by bottom sediments were used in the model calculations. These ben thai loads
were obtained from measurements by a benthal respirometer supplemented by COD values. The
values of benthal uptake used in the computations of DO are shown in Figure 7.37.
Example calculations of DO are shown in Figures 7.38-7.40 in which were used the
loads of Table 7.6 and the appropriate inflows. The major source of error is thought to be the
effects of photosynthesis in the estuary, which are not represented in the model.
TABLE 7.6
Estimated 1966 Ultimate Oxygen Demand Loads to Upper Potomac
Estuary. From Hetling and O'Connell (1968).
Plant
Arlington County
District of Columbia
Alexandria
Fairfax County: Westgate
Fairfax County: Little Hunting Creek
Fairfax County: Douge Creek
Flow (MGD)
17.2
232.9
14.1
9.2
2.4
1.2
UOD (Ib/day)
43,400
318,000
11,700
18,400
2,000
1,200
350
-------
4000
3500
3000
| 2500
u 2000
la
K 1500
o
5 1000
500
0
INDIAN HEAD - 1930
i MEASURED VALUES
DAILY RANGE OF CHLORIDES FOUND
IN ESTUARY OVER TIDAL CYCLE
-CALCULATED BY MODEL
_J L_
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
MARYLAND POINT - 1965
6000
5000 —
J2 3000
o
cc.
o
u 2000
1000 —
MEASURED SURFACE CHLORIDES
• CHESAPEAKE FIELD STATION
CHESAPEAKE BAY INSTITUTE
^DISTRICT OF COLUMBIA
CALCULATED BY MODEL
APR
MAY
JUN
JUL
AUG
SEP OCT
NOV
DEC
Fig. 7.36(a) Comparisons of measured chlorides and calculations
using chloride model. From Hetling (1968).
351
-------
10000
9000
8000
7000
^ 6000
I
« 5000
oc
o
1*000
3000
2000
1000
0
I I I I
OMEASURED VALUES
10
20
CALCULATED BY MODEL
I
ASSUMED
BOUNDARY
CONDITIONS
I
30 <»0 50 60 70 60 90 tOO
MILES DOWNSTREAM FROM CHAIN BRIDGE
110 120 130 11*0
Fig. 7.36(b) Comparison of measured chlorinities and model calculations,
Potomac Estuary, 8 October 1965 (Hetling 1968).
10
9
8
7
6
o
MEASURED POINT
CORRECTED TO 25°C
o o
\ Av III I I " I I I I I I I
0 2 k 6 8 10 12 1U 16 18 20 22 2k 26 28 30 32
MILES DOWNSTREAM FROM CHAIN BRIDGE
Fig. 7.37 Benthal uptake in Potomac Estuary
(Hetling and O'Connell 1968).
352
-------
DAILY MEASURED VALUES
CALCULATED BY MODEL
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV
DEC
Fig. 7.38 Comparison of calculated and measured DO
values for Potomac. From Hetling (1969).
i r
DAILY MEASURED VALUES
CALCULATED BY MODEL
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
Fig. 7.39 Comparison of model calculations with 1965 DO
data at Memorial Bridge, Segment 2, Potomac
Estuary. Hetling (1969).
353
-------
1 C.
10
8
! 6
o
o
4
1 t 1
1 1 1 1 1 1 1 1 1 1 1 1 1
0 ©MEASURED VALUES
— o —
8
o
o° o 0 o
— oo ofo
o
8
9
0
0° <
Po
— o
o o
1 1 2 I3l4
1 1 1
5
. r-H 6 ' ' ' ° L—
0 r-rr-_J ° o -
0 0
o ° °
« o ^^^
3° ° 1 0 0
y *r •—
1 8
o
I ! o "^ — CALCULATED BY MODEL
p
6 7 8 9 10 111 1 12 1 13 1 14 I 15 1 16
I I I I 1 1 I I 1 1 1 1 1 1 1
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 4
MILES FROM CHAIN BRIDGE
Fig. 7.40 Comparison of measured and computed DO, Potomac Estuary,
10 August to 1 September 1965 (Hetling and O'Connell 1968).
3.3 HILLSBOROUGH BAY
Hillsborough Bay is located on the Gulf of Mexico side of South Central Florida. The
bay is a natural arm of Tampa Bay and is approximately eight miles long and four miles wide
with two major freshwater tributaries, the Hillsborough and Alafia Rivers (Figure 7.41). The
Interbay Peninsula separates Hillsborough Bay from Old Tampa Bay, and is bounded by the city
of Tampa on the west and north. The surface area of Hillsborough Bay (including the harbor
area, Port Sutton, and McKay Bay) is 39.6 square miles and the total volume is 8.3 x 10 cubic
feet at mean low water. Hillsborough Bay is a typical Gulf Coast estuary, generally shallow
with very weak tides and current. Tides are of mixed type, the diurnal tidal range being
2.8 feet. Average depth of the bay at mean tide is approximately 9 feet. Hillsborough River
is normally the largest source of freshwater inflow into the Bay.
In 1965 FWPCA undertook a water quality study of Hillsborough Bay at the request of
state and local agencies because of odor problems along the western shore and general poor
water quality conditions of the bay. The water quality conditions in Hillsborough Bay had
degraded because of a combination of various factors: (1) Inadequately treated sewage efflu-
ents, (2) high concentrations of nutrients, and (3) high oxygen demand from bottom sediment
deposits. All these factors contributed to excessive algae growth and poor water quality
conditions in certain areas of the bay.
One of the conclusions of this study (FHQA 1969) was that the obnoxious odors along
the western shore of Hillsborough Bay were the result of the death and decay of the marine
algae Cracilarta which constitutes 9&1 of the attached algal crop in Hillsborough Bay (in
contrast to 2Z in nearby Tampa Bay). Death of Gracilaria was found to be precipitated by
354
-------
HILLSBOROUGH BAY
27 . . ^—- 3J
30
TAMPA BAY
MILES
Fig. 7.41 Hillsborough Bay model segmentation.
355
-------
high freshwater Inflows (I.e., in excess of 2,400 cfs) which lowered the chlorinity below 6 ppt,
the salinity stress point for Gracllaria. When the freshwater inflow exceeded 2,400 cfs, the
Gracilarta standing crop was substantially reduced, oxygen was depleted, and sulfides were
produced.
In order to quantitatively evaluate various alternative plans for improvement of the
water quality conditions in the bay, a mathematical model was developed based on the DECS model
but extended to two dimensions to better represent the geometry of Hillsborough Bay. The seg-
mentation is shown in Figure 7.41. These segments were determined principally from depth
variations in the bay; in each segment the depth is approximately uniform.
Initial approximations of dispersion coefficients were determined from a consideration
of dye tracer and current velocity datp using the relations in Bunce (1967). Coefficients
ranged from 0.70 mi^/day to 6.0 mi^/day with a majority of the dispersion coefficients less
than 2.0 mi^/day. Verification of these coefficients was accomplished with winter chlorinity
data, when freshwater inflow was low so that the advective effects could be assumed negligible
in comparison to dispersion. Advective coefficients were obtained by reproducing chlorinity
distributions obtained during high flow periods. Figures 7.42 and 7.43 shows comparisons of
calculated and observed chlorinities for both inflow conditions.
Using the developed model, an analysis was made of the proposal by the U.S. Corps of
Engineers to divert water out of the Hillsborough River Basin into the Sixmile Creek-Palm River
Basin. Maximum diversion was 1,200 cfs out of Hillsborough River, and the corresponding dis-
charge from Palm River was 2,000 cfs. The flow for the Alafia River was assumed to be 1,500 cfs.
Shown in Figure 7.44 is the predicted distribution of chlorides for these Inflow conditions.
Comparison of Figure 7.44 and Figure 7.43 indicates that for the Hillsborough flow approxi-
mately the same, a significant increase in the Palm River flow (169 cfs to 2,000 cfs) results
only in a slight decrease of chlorides along the western shore of Hillsborough Bay. Since
chloride concentrations along the western shore are highly sensitive to flow variations in the
Hillsborough River, the diversion of the inflow to Palm River appears preferable, since it
would result under these conditions in only minor fluctuations in chloride concentrations in
the critical area. Therefore, from the standpoint of odor production and water quality in
Hillsborough Bay, the flow diversion alternative of the Four Rivers Project was considered
desirable.
3.4 SUMMARY
The DECS model is perhaps not as sophisticated, particularly in its hydrodynamic
terms, as many of the computational programs which have been developed more recently, and its
coarse space and time scales render inaccessible the modeling of many estuarine phenomena. On
the other hand, because of its relative computational efficiency and its capability for modeling
the more salient water quality parameters, the DECS model has proved to be extremely useful in
water pollution management. These applications have included the following:
(i) Analysis of the factors affecting DO and the sensitivity of the estuary to
each;
(ii) Evaluation of alternatives such as relocation of outfalls, inflow augmentation,
and levels of waste treatment to determine their efficacy in improving
estuarine DO;
356
-------
NOQD OBSERVED VALUES
OOOCALCULATED VALUES
TAMPA BAY
MILES
Fig. 7.42 Observed and calculated chloride concentrations (ppt)
December, January, February, and March, 1967-68.
357
-------
. OOO OBSERVED vw_Utb
000 CALCULATED VALUES
^
/ n
HILLSBOROUGH BAY
109 /las I
103
,,.4' '« "4I
TAMPA BAY
MILES
Fig. 7.A3 Observed and calculated chloride concentrations
(ppt) August, September, and October, 1967.
358
-------
TAMPA BAY
MILES
Fig. 7.44 Predicted effect of proposed flow diversion
on chloride concentrations.
359
-------
(ill) Providing basic input to various cost-benefit optimization programs for
determination of the least-cost approach to obtaining specific water quality
objectives;
(iv) Computation of salinity distributions under various flow conditions, for
determining those reaches feasible for emergency water supply (in the Potomac),
for aiding flow release decisions during drought (in the Delaware), and for
evaluating proposed freshwater diversions with regard to salinity-sensitive
aquatic plants (in Hillsborough Bay);
(v) Modeling of other parameters such as pH-alkallnity and nutrients;
(vl) Providing quantitative information for the planning, execution and analysis
of estuarine field studies.
In addition to its extensive application by FWQA, the model has been employed by various state
and federal agencies in water quality planning studies.
Certainly an important factor to the widespread application of the DECS model is the
excellent documentation of the computer programs, which enables personnel to set up and operate
the model without requiring detailed knowledge of its formulation. The DECS model should serve
as a paradigm in this respect; continued development of water quality models not only is depen-
dent upon advances in theory and computational techniques, but also upon making these advances
available to the practicing engineer and water resource manager in an accessible and utilitarian
form.
360
-------
4. SAN FRANCISCO BAY
The San Francisco Bay and its associated systems (i.e., San Pablo Bay, Suisun Bay,
and the Sacramento-San Joaquin Delta) comprise one of the major harbor and estuarine systems
of the continental United States. Over sixty cities are situated on its periphery, presently
discharging an estimated 660 million gallons per day of municipal-industrial wastewater.
Demands are increasing on the freshwater inflows into the system, from municipal consumption,
export to other areas of the state, and agricultural use in the Central Valley and Delta area.
Salinity intrusion into the Delta during periods of low freshwater inflow has been a major
problem to the farmlands and industries of the region.
The San Francisco Bay system, the Delta system and the Central Valley River system
have been the subject of many modeling investigations, addressing their hydrology, hydro-
dynamics, salinity distribution, and quality indicators, and the methods employed encompass
all of the techniques discussed previously. Of this profusion of studies, two estuarine models
of particular interest are selected for discussion, the physical model of the U.S. Corps of
Engineers (1963) and the numerical hydrodynamic-water quality model of FWQA developed under
contract by Water Resources Engineers, Inc., (1968) (see also Feigner and Harris 1970, Orlob
et al. 1967, Kaiser Engineers 1969).
4.1 CHARACTERISTICS OF THE BAY-DELTA SYSTEM
4.1.1 Physiographic and Hydrographlc Features
San Francisco Bay per se is an irregular, elongated bay of some 50 miles length nearly
landlocked with its only connection with the ocean at the Golden Gate (Figure 7.45). At its
northern tip San Francisco Bay joins San Fablo Bay, and from this juncture the system extends
eastward through Suisun Bay to the confluence of the Sacramento and San Joaquin Rivers and the
sprawling Delta. The Sacramento and San Joaquin are the major sources of inflow to the Bay
system. A number of smaller streams discharge into the Bay, but their combined annual flows
are (nominally) less than 3% of the inflow from the Delta.
San Francisco Bay is generally quite shallow, averaging 20 feet in depth. The deepest
region of the Bay is the Golden Gate where the depth is 382 feet at mean sea level. In contra-
distinction, however, about 50% of the total Bay area is less than 10 feet deep and about 68%
is less than 20 feet deep. Due to this general shallowness, waves and currents cause con-
siderable resuspension of sediment which results in highly turbid water, even during periods
when little turbidity is being brought in by the major rivers. The southern part of
San Francisco Bay below Dumbarton Bridge is characterized by extensive tidal flats and meander-
Ing sloughs. The total area of these tidal flats exposed at mean lower low water is some
57% of the high tide area (10,700 acres) of the Bay south of the Bridge. The total area of
the sloughs is about 22% of the total tidal area.
The upper bays, San Pablo and Suisun, are shallower than San Francisco Bay. Suisun
Jay, like South San Francisco Bay, is a shallow body of water with meandering tidal channels
361
-------
* TIDE STATIONS FM 5*1* OF
FIGURE T.«l
tit. 7.45 Location up, Sm FrxwUco B.y nxj S*cr«wnto - S«n Jo«)uln D*lt« Sjr.t™.
-------
and extensive tidal flats, and is almost completely surrounded by salt marshes. These marshes
are traversed by a system of tidal sloughs which overflow at higher tide to flood the marshes.
Carquinez Strait, the waterway joining San Pablo and Suisun Bays, varies in width from 2,700 feet
to 6,900 feet and has depths ranging from 40 to 80 feet on the western end, 108 feet in the
central portion, and 67 feet on the eastern end.
Tides in San Francisco Bay are of the mixed type and have a mean range (i.e., mean
high to mean low) of 3.99 feet with a mean great diurnal range of 5.72 feet. The greatest mean
tidal range is in the southern part of the Bay. The mean range in South Bay is about 6 feet
and at Alviso the mean range reaches 7.22 feet. The range on the western shore is slightly more
than that on the eastern shore of South Bay due to the Coriolis effect. The mean ranges in
San Pablo Bay and Carquinez Strait are 4.45 feet and 4.40 feet. Suisun is evidently more
subject to wind tides and the variations of inflow, and seems to average 4.0 feet. Influence
of the tide is observed far inland and during low flow can extend well over a hundred miles
from the Golden Gate. For example, during low flow, the mean tidal range at Sacramento,
110 miles from the Golden Gate, is 2.5 feet.
San Francisco Bay is typically well-mixed, becoming slightly stratified upstream from
the Carquinez Strait during periods of low freshwater inflow. During high inflow, slight
stratification may be evidenced in San Francisco Bay proper and pronounced stratification
obtained in the Carquinez Strait. This is exemplified by Figure 7.46.
The Delta is an area of approximately 1,100 square miles of highly developed, irri-
gated land lying between Sacramento on the north and Tracy on the south, and extending westward
from Stockton to the confluence of the Sacramento and San Joaquin Rivers near Collinsville.
The Delta was at one time an extensive marshy area and possesses very fertile peat soils.
During the last century practically the entire region has been drained and reclaimed into more
than 50 islands now used principally for agriculture. Nearly all of the islands lie below sea
level, some over 18 feet.
A number of rivers enter the Delta region along its eastern boundary. The major
streams are, of course, the Sacramento and San Joaquin Rivers; others include the Cosumnes
River, the Mokelumne River, and the Calaveras River. The Delta is interlaced by more than
700 miles of meandering channels and waterways which have a total surface area of more than
75 square miles and a volume more than twice that of Suisun Bay. The varying inflows to the
Delta from the tributary rivers, the influence of tides in the western Delta, the consumption
of water within the region, and the varying amounts of water exported from the Delta all con-
tribute to extremely complex patterns of flow within the area. In many of the smaller channels,
and even in some reaches of the San Joaquin River, the direction of flow may be reversed at
certain times of the year.
4.1.2 Water Quality
The principal types of discharge in the Bay system are municipal and industrial
effluents and agricultural drainage. An appreciation of the volumes involved can be gained
from Table 7.7. Water quality zones are delineated in Figure 7.47. DO levels throughout
most of San Francisco Bay proper are high, consistently above 80% saturation. There is a
significant DO depletion problem in the region below Dumbarton Bridge, which has been studied
using a steady-state finite section model (Consoer, Townsend and Associates 1968). A problem
throughout the Bay is toxicants associated with wastewater discharges. Increasingly in recent
363
-------
J«
.
30 25
20
AVERAGE SALINITY P.P.T.
15 10 5 0
35 30 25 20
:
-
-:
B
- 100
NORTH LINE
21 - 22 SEPT. 1956
13 - H< FEB. 1958
SOUTH LINE
FRESH WATER INFLOW
16.000 CFS - SEPT. . 1956
195.000 CFS - FEB.. 1958
(a) OBSERVED SALINITIES
(b) STAT ION LOCATIONS
Fig. 7.46 Average salinities in San Francisco Bay.
I ,
I
•
I
I
i
«
364
-------
Fig. 7.47 San Francisco Bay-Delta area showing water quality zones.
Water Quality
Zone
(Fig. 7.47) Waste Source
1, 2 & 3 Municipal
Discrete Ind.
S. Bay Total
4 Municipal
Discrete Ind.
Cent. Bay Total
5, 6 & 7 Municipal
Discrete Ind.
N. Bay Total
8 & 9 Municipal
Discrete Ind.
Delta Total
TOTAL: Mun. and Ind.
Flow
(MGD)
273
4
277
78
1
79
55
18
73
120
36
156
585
BOD
(106 Lb/Yr)
384
-
384
90
-
90
52
65
117
120
21
141
732
Total
Nitrogen
(106 Lb/Yr)
95
-
~95
22
-
~2T
13
23
36
29
-
29
182
TABLE 7.7
Annual (1965) Average Characteristics of
Untreated Municipal and Industrial Uastewaters.
From Program Staff (1969)
365
-------
years, low DO concentrations have been observed locally in parts of the Delta. The two
principal contributing factors to this condition are the BOD from municipal and Industrial
wastes, and excessive algal growths.
Biostimulation resulting from excessive nutrients is becoming an increasingly serious
problem throughout the system, but particularly In the Delta. Recent studies have indicated
that nitrogen and phosphorous concentrations were from ten to one hundred times greater in the
Delta than those reported necessary for substantial growths of algae. The higher concentra-
tions tended to occur in the San Joaquin River, particularly In the vicinity of Stockton.
This condition has precipitated extensive algae blooms in the Delta, particularly during the
summer. Evidences of enrichment in San Francisco Bay are observed mainly along the shores and
In the tidal reaches of some of the tributaries to the Bay. Algal populations in the Bay are
generally smaller than those found in the Delta system. Nevertheless, total nitrogen and
phosphorus concentrations in the waters of San Francisco Bay are substantially higher than the
levels where either nitrogen or phosphorus might be growth-limiting. It should also be
remarked that in San Francisco Bay there is algal suppression due to the light-limiting effects
of the high inorganic turbidity.
The problem of salinity intrusion into the Delta region was noted previously.
Salinity intrusion is now partially controlled as a part of the Bureau of Reclamation's opera-
tion of its Central Valley Project by the release of additional water during the dry season
from reservoirs in the Sacramento River Basin to the north. These controlled releases of stored
water have largely eliminated the very low flows of the past when chlorides intruded well up
into the Sacramento River. In the future there will be increasing competition between demands
for water for salinity repulsion and for export for Irrigation and other consumptive uses.
With increased withdrawals of fresh water from the south end of the Delta as more water is
exported to water-deficient areas in other parts of the State, the problem of salinity intru-
sion can be expected to be seriously aggravated. The object of many barrier plans has been
to Isolate fresh water from salt water, and thereby conserve water now required for salinity
repulsion.
4.2 THE PHYSICAL MODEL
The San Francisco Bay estuary model developed and constructed by the U. S. Corps of
Engineers (1963) is a physical hydraulic model which represents all of San Francisco Bay,
Suisun Bay, the lower reach of the Delta to the confluence of the Sacramento and San Joaquin
River, and 17 square miles of the Pacific Ocean beyond the Golden Gate (Figure 7.48). The
model is located in Sausalito, California, and covers an area of approximately an acre. The
vertical-to-horizontal scale distortion is 10:1 and the .time scale is 1:100. Specific data
on the model-prototype scale relations are given in Table 7.8,
The effect of the tides is simulated by means of a primary tide generator located at
the seaward end of the model and a secondary tide generator located at the upstream limit of
the model at the head of Suisun Bay. This secondary unit is synchronized with the primary in
order to reproduce proper ebb and flood flows at the upper limit of the model. Salinity is
regulated to the exact concentration required by mixing salt water and tap water in the sump.
Freshwater inflow points are located around the model to simulate stream flow. The model has
been applied extensively to studies of shoaling in the Bay, and to evaluation of the effects
of proposed barriers on the Bay system.
366
-------
NAP A I
r
PITTSBURG
• TIDE STATIONS
A VELOCITY 8 SALINITY STATIONS
ALVISO
SCALE IN FEET
10000 O 3OOOO «0000
PWTOTYfl it | i I i i I
•ODE I 10 0 (0
Fig. 7.48 San Francisco Bay physical model
367
-------
TABLE 7.8
Relation of Model Properties to Prototype Equivalents
Property
Depth
Length or Width
Discharge
Velocity
Time
Salinity
Prototype
100 feet
1,000 feet
1,000,000 cfs
10 ft. per sec.
24 hours and 50 minutes
1
Model
1 foot
1 foot
1 cfs
1 ft. per sec.
14.9 minutes
1
4.2.1 Verification
Several surveys were performed in San Francisco Bay for the purpose of obtaining
verification data. The first of these was conducted 21-22 September 1956, a period of low,
fairly steady freshwater inflow. Eleven control stations were used (see Figure 7.48) located
over the deep-water channels. Current velocity and salinity measurements were made at 6 or 7
points in the vertical simultaneously at each of the 11 boat stations, velocities being meas-
ured at half-hourly intervals, and salinities at hourly intervals. Suspended sediment samples
were collected at 6 of the boat stations. Tide observations were made concurrently at 24 shore
stations. During the period of measurement, a sustained freshwater inflow of 16,000 cfs was
being discharged into Suisun Bay from the Sacramento and San Joaquin Rivers. Flow from the
other tributaries was practically nil.
To obtain data representative of high inflow conditions, two field surveys were con-
ducted in 1958, on 13-14 February and 3-4 March. On the former date, the combined discharge of
the Sacramento and San Joaquin Rivers into Suisun Bay was approximately 180,000 cfs. Discharges
from other tributaries to the Bay were as follows: Napa River, 5,000 cfs; Petaluma River,
5,000 cfs; Alameda Creek, 3,500 cfs; and Mountain View Slough, 1,500 cfs. Thus, the total
freshwater discharge into the Bay amounted to approximately 195,000 cfs. On the latter date
the total discharge was about 169,000 cfs. These surveys were less ambitious than the one of
1956. In tile February run, current velocity measurements were made at boat stations F, G, H,
1 and J in the upper part of the Bay system; and salinity measurements were made at boat sta-
tions A, F, G, H, I and J, and shore stations 18 and 20 (see Figure 7.48); observations of
tide were made at 11 shore stations. The March survey included current velocities and salini-
ties measured at only 4 boat stations, G, H, 1 and K.
Model reproduction of prototype tidal ranges, elevations, and phases was considered
satisfactory in both cases. For the 1956 survey, the maximum error observed in the model at
the 23 gages was 0.4 feet prototype (0.004 feet, model). Current velocities were satisfactorily
reproduced at the 11 control stations. The maximum difference between model and prototype
absolute velocities was somewhat over one ft/sec. Examples of the low inflow verification are
shown in Figure 7.49.
With the same roughness elements as the 1956 case, and an inflow of 195,000 cfs, the
model results for 13-14 February were somewhat low in salinity and exhibited higher velocities,
particularly in Suisun and San Pablo Bays, This was compensated for by reducing the inflows
368
-------
TIDAL ELEVATIONS
VELOCITIES
jMO
1 8
2
o
5 6
u 2
n
-J 0
STATION 8
NEAR STATION E -
I I I l l l I
z ioi—r
2
8 12 16 20 21*
STATION 2
NEAR STATION A -
I I I I I I I I I I I I
-• 0 4 8 12 16 20 2<(
STATION E
S1A1ION A
06
u,S
So5
uo
uil,
SURFACE
I* 8 12 16 20 21|
o- 0
I I I I I I I I I I I I
I I I 1 I
i i i i i r
MIDDEPTH
BOTTOM
SURFACE
MIOOEP1H
BOTTOM
48 12 16 20 24
1 8
ui -2
i 0
STATION 7
NEAR STATION C -
I I I I I I I I l j l l
8 12 16 20 2
-------
SALINITIES
STATION E
*°l I I I I I I I I I I I 1
en
020
a.
£30
oe
s
z20
>-^0
<•
to
Ill III
III III
1 1 1 1 1
1 1 1 1 1
20
NOTE: NO PROTOTYPE BOTTOM
SALINITY SAMPLES OBTAINED
k 8 12 16 20 2*t
SURFACE
MIDDEPTH
BOTTOM
35
025
3
§15
s«
-15
STATION A
I I
15
-
1 1 1 1
1 I 1 1
STATION C
1 1 1 1 1 1 1 1 1 1
SURFACE
1 1 I 1 1 1
MIDDEPTH
1 '_!. BOTTOM
1 1 1 1 1 1 1 I
0 4 8 12 16 20 2k 0 i« 8 12 16 20 2k
TIME IN HOURS - AFTER MOON'S TRANSIT
OF 122° 28' MERIDIAN PLUS 12 HOURS
Fig. 7.49 Continued
370
-------
to 185,300 cfs and making slight alterations in the roughness elements in the navigation
channel of Suisun and San Pablo Bays. The alteration in the inflow was considered permissible
due to the errors in estimating the prototype inflows, especially in that the effects of evapo-
ration, transpiration, and consumption are not well accounted for. Examples of model-prototype
comparisons for the high inflows are given in Figure 7.50.
4.2.2 Evaluation of the Effects of Solid Barriers on Hydraulics and Salinities
Much of the work performed by the Corps of Engineers and reported in USCE (1963) was
directed at the evaluation of the influence of proposed barriers on the San Francisco Bay Sys-
tem. The barriers investigated and their approximate locations are shown in Figure 7.51. The
barrier studies were conducted with two kinds of simulated tides: the approximate mean tide
of 21-22 September 1956, for which the model was verified, and the spring tide of 13 September
1954. The former exhibited considerable diurnal inequality and the latter practically no
inequality. The freshwater flows of the Sacramento and San Joaquin Rivers dicharged into
Suisun Bay on 21-22 September 1956 and 13 September 1954 were considerably below the mean daily
freshwater inflow (45,000 cfs) into the Bay system being 16,000 cfs and 7,000 cfs, respectively
The purpose of the tests of solid barriers was to determine the effects of the barriers on the
TIDAL ELEVATIONS
STATION 6
10
2 6
a
o
<
i i i i
30
UJ
8 12 16 20 24
to
-2
STATION 11
8 12 16 20 21*
TIME IN HOURS - AFTER MOON'S TRANSIT OF 122° 28' MERIDIAN
TEST CONDITIONS
TIDE OF 13-14 FEB 1958
FRESH-WATER INFLOWS (CFS)
AT CHIPPS IS.-175,000 ALAMEDA CR-1,800
NAPA RIVER-3,500 PETALUMA RIVER-4,000
MOUNTAIN VIEW SLOUGH-1 ,000
OCEAN SALINITY AT BOTTOM-33.5
LEGEND
• PROTOTYPE
• MODEL
Fig. 7.50 Example of physical model verification for
high inflow conditions.
371
-------
STATION F
V /' "*V^
\ .,'/ MIDDEPTH
• i \fcry i i i i i i i i r
8 12 16 20 24
BOTTOM -
i i i i i
8 12 16 20 24
TIME IN HOURS - AFTER HOON'S TRANSIT OF 122° 28' MERIDIAN
STATION A
SALINITIES
STATION H
STATION F
30
10
£20
ElO
SURFACE _
i i i i i
i I i i
MIDOEPTH .
10
i I I i I i r
BOTTOM .
I I I I I
20
10
0
20
10
0
25
15
I I I I
SURFACE
i i I i i r
MIDDEPTH
8 12 16 20 2k
V I I
^' 12 16 20 2lf
20
10
0
30
20
10
30
20
10
0 It 8 12 16 20 24
TIME IN HOURS - AFTER MOON'S TRANSIT OF 122° 28' MERIDIAN
Fig. 7.50 Continued
372
-------
NAPAl
ITS BURG
CHIPPS ISLAND
SIERRA POINT TO
MULFORD LANDING
• TIDE STATIONS
A VELOCITY a SALINITY STATIONS
DUMBARTON
ALVISO
SCALE IN FEET
10000 o
WOTOTYPl ,.,
•woet
soooo eoooo
11111
Fig. 7.51 San Francisco Bay physical model,
location map of Carriers.
373
-------
hydraulic and salinity regimens of the Bay on the downstream side of the barriers. To
accomplish this, it was first necessary to determine the hydraulic and salinity characteristics
throughout the model for existing conditions, i.e. without barriers. A test in which no bar-
rier was installed in the model is referred to as a "base" test. Tests conducted with barriers
installed in the model are designated "plan" tests.
The effect of each barrier was studied individually, with only that barrier imple-
mented in the model. The barriers were simulated in the model as bank-to-bank closures,
representing conditions with all gates closed. No salinity measurements were made for the
Point San Pablo and Reber Plan Barriers. The basic operation of the tests was to employ a
salt water gate across San Pablo Strait, and with the gate initially closed to flood the model
below the gate and above the gate to the same level with different salinity water. The gate
was opened as the primary tide generator Mas started and the model operated until a steady
salinity regime was obtained. The salinity of the water below the gate at initiation was
30 o/oo. For the 1956 tide (16,000 cfs inflow), the water above the gate was fresh; for the
1954 tide (7,000 cfs inflow) the water above the gate and at the secondary tide generator was
5 o/oo salinity. This latter was necessary since under such low inflow, saline water intrudes
well into the Delta which is beyond the bounds of the model. Therefore, the inflow had to be
at an increased salinity in order to reproduce realistic salinity contours throughout the Upper
Bay.
With either the Chipps Island Barrier or the Dillon Point Barrier implemented, no
inflow reaches the Bay from the effective source, viz. the Sacramento and San Joaquin. This
imposes some difficulty in operating the physical model as well as interpreting the results.
The stable salinity distribution would produce ocean salinity below the barrier and an inordi-
nate amount of tidal cycles would be required for the model to acquire this result. The
physical relevance of this result would also be somewhat in doubt. Accordingly, the results
reported are those of a transient state after twenty-odd tidal cycles, and are of qualitative
value only.
Results of the solid barrier studies are summarized below. The reader is referred to
USCE (1963) for details of the measurements and their interpretation. Study of the effects of
the Reber Plan Barrier was confined to the area between the barrier and the Golden Gate and
the ocean beyond, and therefore are not reported.
In Table 7.9 are indicated the effects of the Chipps Island Barrier on currents in
the Bay for the 1956 tide. The effect of the barrier was to create a stagnant pool immediately
downstream from the barrier and to reduce current velocities to Station I at the westerly end
of Carquinez Strait. Average flood and ebb velocities throughout the depth at the barrier,
Station R, were reduced about 97% (to less than 0.1 ft/sec); at Station J in Suisun Bay, they
were reduced nearly 50% (to 1.2 to 1.7 ft/sec); at Station I in Carquinez Strait, they were
practically unchanged (3.5 to 3.7 ft/sec); and from Station I to the Golden Gate, Station E,
they were generally increased from 0.2 to 0.9 ft/sec. Maximum surface currents were found in
San Pablo Strait (6.5 ft/sec) and in the Golden Gate (5.2 ft/sec). The effects of the barrier
on salinities are summarized in Table 7.10 for both tides. These results, as noted above,
are transient.
The effect of the Dillon Barrier on currents was to create a stagnant pool immediately
downstream from the barrier and to reduce current velocities for a considerable distance down-
stream. Its effects on salinities are indicated in Table 7.11; once again, these represent a
transient condition.
374
-------
TABLE 7.9
Effects of Chipps Island Barrier on Current Velocities
Tide of September 21-22, 1956
(Average Velocities Throughout Depth in ft/sec)
Location
Golden Gate
West of Treasure Island
San Pablo Strait
San Pablo Bay
Carquinez Strait
Suisun Bay
Barrier Site
Station
E
C
G
H
I
J
R
Without Barrier
Flood Ebb
4.0
3.3
3.2
2.3
3.8
2.0
2.6
4.3
3.4
4.3
3.9
3.5
3.1
3.3
With Barrier
Flood Ebb
4.6
3.1
4.1
3.0
3.7
1.2
0.1
4,5
2.9
5.1
3.7
3.5
1.7
0.1
Change (ft/sec)
Flood Ebb
•tO. 6
-0.2
40.9
40.7
-0.1
-0.8
-2.5
40.2
-0.5
40.8
-0.2
0.0
-1.4
-3.2
TABLE 7.10
Effects of Chipps Island Barrier on Salinities
(Average Salinities Throughout Depth in ppt)
Station
E Golden Gate
C West of Treasure
Island
D East of Treasure
Island
G San Pablo Strait
H San Pablo Bay
I Carquinez Strait
J Suisun Bay
R At Barrier
Tides
21-22 September 1956
Without Barrier With Barrier
Max. Hin.
32.7
32.3
31.5
30.0
26.7
21.8
9.8
3.5
31.2
29.8
28.9
22.7
20.8
11.7
1.5
0.5
Max. Min.
33.0
31.8
30.9
31.0
29.7
25.3
16.0
13.7
32.3
30.2
29.1
26.5
24.2
18.7
12.8
13.0
13 September 1954
Without Barrier With Barrier
Max. Min.
32.7
32.0
30.6
26.7
21.2
15.3
5.5
5.5
30.5
28.2
27.3
19.0
14.5
6.2
4.5
4.5
Max. Min.
33.5
32.3
31.9
31.0
30.0
27.3
20.5
18.7
32.0
30.0
30.1
28.2
26.7
21.5
17.8
17.7
375
-------
TABLE 7.11
Effects of Che Dillon Point Barrier on Salinities
(Average Salinities Throughout Depth in ppt)
Station
E Golden Gate
C West of Treasure
Island
G San Pablo Strait
H San Pablo Bay
P San Pablo Bay
I Carquinez Strait
Tides
21-22 September 1956
Without Barrier With Barrier
Max. Min.
32.7
32.3
30.0
26.7
24.7
21.8
31.2
29.8
22.7
20.8
16.8
11.7
Max. Min.
33.2
33.5
31.2
29.2
28.0
28.2
32.2
31.3
28.0
26.8
26.0
26.3
13 September 1954
Without Barrier With Barrier
Max. Min.
32.7
32.0
26.5
21.2
-
15.2
30.5
28.2
19.0
14.5
-
5.7
Max. Min.
32.0
31.5
30.7
29.3
-
27.7
31.5
30.7
27.8
27.5
-
26.7
The Point San Pablo Barrier was tested with the spring tide of 11-12 November 1958
and was found to cause a small increase in current velocities of less than 1.0 ft/sec in South
Bay, and a reduction in Central Bay from the Golden Gate to the barrier. The maximum reduction
occurred at the barrier, where flood and ebb were reduced to 0.6 ft/sec (reductions in flood
and ebb of 2.6 and 3.3 ft/sec, respectively). Salinity measurements were not made.
The 1956 tide was used in the tests with the Sierra Point to Mulford Landing Barrier.
As in the case of the barriers in the upper Bay system, the Sierra Point to Mulford Landing
Barrier created a relatively stagnant pool adjacent to the barrier and for some distance down-
stream therefrom. At the barrier, velocities (averaged over the depth) were reduced to about
0.1 ft/sec (flood reduced 2.3 ft/sec, and ebb, 2.1 ft/sec). At Station C, west of Treasure
Island, flood and ebb velocities were reduced to 0.7 and 1.0 ft/sec, respectively (reductions
of 2.6 and 2.4 ft/sec). At Station E in the Golden Gate, flood and ebb velocities were reduced
to 3.0 and 2.4 ft/sec, respectively (reductions of 1.0 and 1.9 ft/sec). Elsewhere throughout
the Bay system, velocities were reduced or increased from 0.1 to 0.7 ft/sec. The effect of the
barrier on the salinities in the Bay was slight with the greatest change in Suisun Bay and
Carquinez Strait. The effects of this barrier are summarized in Table 7.12.
Dumbarton Barrier, the southernmost of the barriers examined, was tested with the
1956 tide. As might be anticipated, the barrier had little effect on either the currents or
the salinities throughout the Bay system. The barrier created a stagnant pool which extended
some distance from the barrier. At the barrier, the velocities were reduced to zero (reduc-
tions in flood and ebb of 3.0 and 2.3 ft/sec, respectively); at Station M, approximately midway
between Dumbarton and San Mateo Bridges, flood and ebb were reduced to 0.3 and 0.2 ft/sec,
respectively (reductions in flood and ebb of 1.9 and 1.5 ft/sec); and from Station M to Station
C, west of Treasure Island, reductions amounted to 0.5 to 1.0 ft/sec. Between the Golden Gate
and Station I in Carquinez Strait, the maximum reduction was 0.6 ft/sec; and in a few areas,
flood or ebb velocities were increased from 0.1 to 0.2 ft/sec. The greatest change in salinity
376
-------
TABLE 7.12
Effects of the Sierra Point to Milford Landing Barrier em Salinities
Station
E
C
G
H
I
J
K
Maximum Salinities
(Parts Per Thousand)
Base
Test
32.7
32.3
30.0
26.7
21.8
9.8
2.0
Plan
Test
33.5
31.8
30.0
26>.8
21.5
12.2
3.2
Change
40.8
-0.5
0.0
+0.1
-0.3
+2.4
+1.2
Minimum Salinities
(Parts Per Thousand)
Base
Test
31.2
29.8
22.7
20.8
11.7
1.5
0.5
Plan
Test
32.3
30.0
23.7
22.0
14.8
3.3
1.0
Change
+1.1
+0.2
+1.0
+1.2
+3.1
+1.8
+0.5
occurred at Station I in Carquinez Strait, where the average maximum salinity throughout the
depth was reduced to 2.8 ppt. At all other stations, the maximum and minimum salinities were
generally reduced from 0.4 to 1.7 ppt.
An interesting aspect of the investigation of the solid barriers in the physical
model is that it presented an opportunity to compare the results of the physical model with
those of an analog model. An earlier report by Harder (1957) described the results of cal-
culating tidal range with and without the Chipps Island Barrier using an analog computer. In
this study, a sinusoidal tide of period 12.4 hours was employed which corresponded very closely
to the tide of 13 September 1954. The hydraulic model results for the 1954 tide with and with-
out the Chipps Island Barrier are compared with analog results in Table 7.13. Hie differences
in these results are attributed to a number of factors, for example, the use in the electric
analog of a pure sinewsve tide with a 12.4-hour period, whereas in the hydraulic model the
actual tide with period of 24.8 hours was used (with some differences in the ranges of the first
and second 12.4-hour periods). Another suggested source of the disagreement is the extreme
difficulty of accurately determining the roughness of the prototype channel and areas outside
of the channel for use in the analog model.
4.2.3 Application of Dye Releases
The physical model has been employed on a number of occasions to provide information
on the dispersive characteristics of the Bay using flourescent dye as a tracer. The Corps, of
course, has conducted extensive studies of dye releases in the model (USCE 1963), and addi-
tional studies are reported in O'Connell and Walter (1963), Bailey et al. (1966), Lager and
Tchobanoglous (1968), and McCullough and Vayder (1968). The principal objective of the dye
releases performed by the Corps (1963) was to provide Information on the dispersion patterns
of different wastes released into the San Francisco Bay System, both under existing conditions
and with the implementation of the proposed barriers. All tests were run for an inflow of
16,000 cfs from the Sacramento and San Joaquin Rivers, except for the instances where barriers
377
-------
TABLE 7.13
Comparison of Results of Electric Analog and Hydraulic Model
(Effect of the Chipps Island Barrier
on the Tide of September 13, 1954)
Station
Presidio
Selby(a>
Eckley
BeniciaO>>
Roe Island
Nichols
Chipps Island
Electric Analog
Tidal Range in Feet
No
Barrier
5.35
5.20
5.10
5.10
4.70
4.50
3.90
With
Barrier
6.70
7.30
8.20
8.90
9.20
9.30
Increase
In Feet
1.50
2.20
3.10
4.20
4.70
5.40
Hydraulic Model
Tidal Range in Feet
No
Barrier
5.50
5.80
5.70
5.40
4.80
4.55
4.10
With
Barrier
5.40
6.35
6.95
7.30
8.00
8.35
8.50
Increase
In Feet
-0.10
0.55
1.25
1.90
3.20
3.80
4.40
(a) Hydraulic model data were taken from HHW and LLW profiles from nearby gages,
since there are no fixed tide gages at these locations.
(b) Station 20 at Martinez (across the Strait from Benlcia) used in hydraulic
model.
restricted flow from these two rivers. Dye release points and the barriers tested are shown in
Figure 7.52. Within this series of tests, 18 dye release points were employed and all base
tests (i.e., no barriers) were conducted using the approximate mean tide of 21-22 September
1956. Each test run was carried for 20 or 40 tidal cycles per release depending on the par-
ticular case. Dye concentrations were obtained for contour plots after 1, 3, 5, 10, 15 and
20 cycles of the 20-cycle tests, and also after 30 and 40 cycles for the 40-cycle tests. The
quantities of dye released depended upon the expected dispersal characteristics in the vicinity
of release points and the particular test duration; that is, shorter tests and areas with
slower flushing characteristics required smaller amounts of dye. In most cases, the dye was
released over one tidal cycle (14.9 minutes in the model), approximately a slug release. Both
continuous flourometer readings and discrete samples were obtained during each test run.
The results for the Sierra Point to Roberts Landing Barrier is indicative of this
work. In Figure 7.53 are shown the dye contours resulting from an injection at point "G" with-
out the barrier after 3, 5 and 10 cycles.- Under these (base) conditions, that part of South
San Francisco Bay north'of the barrier site exhibits better circulation of tidal currents, and
consequently more rapid dispersion and flushing action, than that part of the Bay south of the
barrier site. In Figure 7.54 are shown the corresponding plots with the solid barrier from
Sierra Point to Roberts Landing. Dye releases at Point "G" with the barrier showed action in
South Bay north of the barrier to be considerably more sluggish than without the barrier, dis-
persion and flushing action occuring more slowly with high concentrations of dye persisting
378
-------
NAPAi
f
REBER PLAN
BARRlEt
PITTSBURG
CHIRPS ISLAND
BARRIER
SIERRA PT. TO ROBERTS
LOG. BARRIER
A-O DYE TRACER RELEASE POINTS
DUMBARTON
BARRIER*
ALVISO
rtororrfi
SCALE IN FEET
10000 0 JO000 »0000
7.52 ^ye tracer release points In San
Francisco Bay physical model.
379
-------
U)
g
CYCLE 10
MODEL LIMITS
A INJtCI ION STAI ION
««UIOM»IIC FIOOROMEHR I
RECORDER
I. CONTOURS REPRESENT CONTAMINATION
CONCENTRATIONS OH UHII OR WEIGH! »
10 Kl LMCR »S FRAC1IONS OF !Ht
INITIAL QU»NIITT OF CONTAMINANT RELEASED.)
2. FRESH-WATER INFLOW AT CHIPPS ISLAND .16.000 CFS
3. DYE 1RACER CONCENTRATION 500 PC" QUANTITY 2.Of
Fig. 7.53 Distribution of dye in physical model (HHW Slack) without barrier.
-------
CYCLE IO
— MODEL LIMITS
Fig. 7.54 Distribution of dye In physical model (HHW Slack) with Sierra Point
to Roberts Landing Barrier.
-------
CYCLE 10
Fig. 7.55 Distribution of dye In physical model (HHW Slack) with Sierra Point
to Roberts Landing Barrier with partial opening.
-------
for longer periods of time. The results were quite different with a 10,000 foot navigation
opening in the Barrier at the ship channel. These results are displayed in Figure 7.55. The
addition of the Sierra Point Tidal Barrier with a 10,000 foot gate caused dramatic changes in
the dispersion patterns in South Bay and improved the dispersion and flushing characteristics
of this bay, as is evident by the faster dispersals of dye in all directions from the points of
release of the dye. The improvement was due to the increase in current velocities and tur-
bulence created by the navigation openings especially in deep water both upstream and downstream
from the barrier. In general, there was a spread of dye in all directions at least equal to or
greater than that in the base test. The movement of the dye was predominantly toward the west
parallel to the barrier (rather than alonj the shore in the north-south direction). Once the
dye reached and entered the navigation opening in the barrier, the increased current velocities
moved the dye southward in deep water faster than in the base test and the overall progression
in South Bay was more to the south than to the north.
The studies reported in O'Coimell and Walter (1963), Bailey et al. (1966) and Lager
and Tchobanoglous (1968) were conducted in a similar manner, except that the results were
extended to be interpreted in terms of a nonconservative tracer by applying a decay factor to
the station history curves. In Bailey et al. (1966), comparisons were made between dye meas-
urements in the prototype and the model to verify the latter's ability to represent mixing
phenomena in the Bay. Release and sampling points'in the prototype are shown in Figure 7.56.
The model was operated with 16,000 cfs inflow, approximately that of the prototype. The dye
concentration histories at three stations for model and prototype are displayed in Figure 7.57.
The prototype values have been multiplied by 10&, the mass scale factor, and have been corrected
for dye loss using an empirical exponential fit to the observed time histories. In general,
the dye concentrations in the model varied from about twice the prototype values near Chipps
Island to almost equal the prototype values in Suisun Bay. The discrepancies are suggested to
be due to boundary effects in the model or variations between the constant model tide and the
varying prototype tide. In Figure 7.58 are shown steady-state dye dispersion curves, corrected
to represent a conservative tracer, obtained from four different injection points at 16,000 cfs.
It can be seen that the concentrations are sensitive to the location of the discharge point in
Suisan Bay but not nearly so much in San Pablo Bay. Model tests were also performed to evaluate
the influence of flow on the steady-state concentrations, as is exemplified by Figure 7.59.
4.3 DIGITAL COMPUTER MODEL
The numerical hydraulic-water quality model for San Francisco Bay described in this
section was developed principally by Water Resources Engineers, Inc. (WRE), through a series
of contract agreements with the PHS and later FWPCA. The model effectively represents two
spatial dimensions in the horizontal, and is general in that it can be applied to any estuarine
system sufficiently well mixed in the vertical. Indeed, it has been applied to Sydney Harbor
(Australia), San Diego Bay and the Columbia River, among others. However, it was originally
developed for San Francisco Bay and most of the computational experience with the model has
been obtained in the Bay system, so it is appropriate to undertake the discussion of the model
in this section.
4.3.1 Theory and Computational Framework
The computation of currents and of the distribution of substance (i.e., the water
quality) are undertaken separately. Fundamentally, the computations are independent, but
383
-------
) A DISCHARGE POINTS
• SAMPLING STATIONS
Fig. 7.56 Location of sampling stations and discharge points for dye
dispersion tests In the San Francisco Bay System. From
T. E. Bailey, C. A. McCullough and C. G. Cunnerson, Mixing
and Dispersion Studies In San Francisco Bay, Proc. ASCE.
92, No. SA5 (Oct., 1966), pp. 23-45. Used with permission
of the American Society of Civil Engineers.
STATION t (CHIPPS ISLAND)
16 18
STATION & (PORT CHICAGO)
0 2
-------
100
I
SAN PABLO BAY
CARQUINEZ STRAIT
SUISUN BAY
I I I-
SACRAMENTO _
RIVER
1
10
ct
UJ
a.
a.
<
H 1.0
DISCHARGE POINT
WESTERN SUISUN BAY DISCHARGE POINT
EASTERN CARQUINEZ STRAIT DISCHARGE POINT
WESTERN CARQUINEZ STRAIT DISCHARGE POINT
SAN PABLO BAY DISCHARGE POINT
\
o
_i
m
•a
0.1
100
10
I .0
0. 1
Fig. 7.58
Longitudinal distribution of conservative steady-state concentrations
at HHW Slack resulting from fo.ur separate discharges at 16,000 cfs
inflow. From T. E. Bailey, C. A. McCullough and C. G. Gunnerson,
Mixing and Dispersion Studies in San Francisco Bay, Proc. ASCE. 92,
No. SA5 (Oct., 1966), pp. 23-45. Used with permission of the
American Society of Civil Engineers .
100
loo
— 1 .0
0. 1
Fig. 7.59 Comparison of concentration ratios of constituents discharged at
Martinez into 1,000 and 16,000 cfs inflows. From T. E. Bailey,
C. A. McCullough and C. G. Gunnerson, Mixing and Dispersion Studies
in San Francisco Bay, Proc. ASCE. 92, No. SA5 (Oct., 1966), pp. 23-45.
Used with permission of the American Society of Civil Engineers.
385
-------
usually the result of the former is used as an Input to the latter. The basic approach is to
represent the estuary as a network of uniform channels interconnected at junctions or nodes.
This permits a one-dimensional treatment of a two-dimensional system, the degree of approxima-
tion governed by the spatial refinement of the channel network.
The equations of motions employed in the hydraulic model are
aH _ _ 1 3(vA) (7i5)
dt B 3x
»v__v av _Kv (v| _g an (7 6)
at ax ax
where H - height of water surface above a reference datum
v - velocity along longitudinal axis of channel
B - channel width
A - cross-sectional area of channel
g » acceleration of gravity
K - frictional resistance coefficient
By using the continuity equation and the assumption that B is independent of t and x , one
can transform the momentum equation to the form
(7.7)
This expression is computationally more convenient than the former. In these equations K
denotes a frictional resistance .coefficient given by
4/3
2.208 R
where n is Manning's "n" and R is the hydraulic radius.
The momentum equation is applied to the channels, so that for a given channel
length L connecting junctions 1 and j , the finite-difference expression is
(7.8)
The notation A/ At denotes the divided difference from one time level to the next. The con-
tinuity equation is applied at the junctions in the finite form
(7.10)
At A8
where A8 is the surface area associated with the junction (see below). £„(}„ is the
(algebraic) sum of the flows into the junction, not only from the channels, but also from
386
-------
discharges, withdrawals and other external effects. The variables v, B, A and K are
associated with the channels, and H and AS are associated with Junctions.
Figure 7.60 is an example of how a channel-junction network might be established
for both one-dimensional and two-dimensional systems. Mean depths are obtained from standard
bathymetric charts or equivalent sources. If the natural system is essentially one-dimensional,
e.g. a system of channels, the mean width and surface area can be estimated from maps. For a
two-dimensional body, the system of centroids (the points of intersection of the perpendicular
bisectors of the channels) determines both the widths and surface areas as indicated in
Figure 7.60. The cross-sectional area of every channel is the product of Its width and depth.
Two networks for the same system, Suisun Bay, are shown in Figures 7.61 and 7.62, the former
from WRE (1968) and the latter from Orlob et al. (1967).
SURFACE AREA
IN BAY
CHANNEL LENGTH
CHANNEL WIDTH
SURFACE AREA
IN A CHANNEL
Fig. 7.60 Calculation of Surface Area.
387
-------
Fig. 7.61 Computational Network for Suisun Bay from WRE (1968).
Fig. 7.62 Computational Network for Suisun Bay fron Orlob et al (1967).
388
-------
The solution of these equations is performed in an explicit manner using a modified
Runge-Kutta method, which calculates the variables at the temporal half step t + At/2 then
uses these half-step values in computing the variables at t + At . Details of the program
operation may be found in Feigner and Harris (1970) and WRE (1968); mathematical aspects of
general Runge-Kutta methods are summarized in Todd (1962), Sections 9.2-9.5.
The transport per unit area per unit time of a conservative constituent in a channel
is assumed to be given by
T - M . QC - AE 2± (7.11)
fit 3x
where Q Is the flow in the channel, c the concentration and E the dispersion coefficient.
The corresponding finite-difference expression is
(7.12)
L
the quantity c is a representation of the concentration advected in the channel and is chosen
to provide the "best" numerical behavior (e.g. numerical mixing, stability, and accuracy). In
Orlob et al. (1967) the most satisfactory methods were found to be
ci > cj
ci
-------
4.3.2 Model Application and Verification
The model described in the preceding section has been operated extensively in many
forms for the San Francisco Bay system. A version used extensively by WRE for the Bay-Delta
Water Quality Control Program employed a coarse network of 250 junctions and 325 channels for
the entire Bay system, in comparison to the preceding representation of Suisun Bay (Figure 7.61)
which alone utilized 135 junctions and around 200 channels.
Figure 7.63 exemplifies the ability of the hydraulic model to produce the tidal stages
when the boundary tide is input at Benecia (see Figure 7.68). An example of the calculation of
salinity distribution with this model is given in Figure 7.64. In this case there is a dynamic
change in the chloride distribution during a one-month period, the following inputs were used:
1. Average July stream inflows and qualities.
2. Average July export values for the USER pumping plants at Tracy and the Contra
Costa Canal.
3. Initial chloride conditions (mean tidal values).
4. Average municipal and industrial flows.
5. An average July 1963 tide for the hydraulic model boundary at the Golden Gate.
6. A constant chloride value of 18,000 ppm for the quality boundary at the Golden
Gate.
The hydraulic model was operated for three 25-hour tidal cycles in order to obtain a dynamic
equilibrium. The computed net Delta outflow was 5,190 cfs, which represents a moderately high
Delta summer outflow. The dynamic water quality model was operated for thirty-one 25-hour
tidal cycles using the output from the hydraulic model. In Figure 7.64 are plotted salinities
for the 31st tidal cycle along with the historical values of July 1, 1963, and July 31, 1963.
The empirical data for July 1 were used as initial conditions in the model calculation. The
curves represent the salinity gradient from the Golden Gate into the Delta. The model results
compare favorably with the observed concentrations, particularly with regard to the gradients,
and represent well the effect of salinity intrusion into the Delta.
Examples of the results of the fine grid model for the San Pablo-Suisun-Delta system
are presented in Figures 7.65 through 7.69. Chlorinity predictions were made for two periods,
July and September 1955, the former representing a period of salinity intrusion in the
prototype and the latter, a period of salinity repulsion. Inputs to the model were daily
river inflows, tide stage at Point Orient, exportations, mean monthly agricultural consumption,
mean monthly evaporation, and municipal and industrial withdrawals and discharges. The net
Delta outflow at Chipps Island was ca. 1,570 cfs in July and 5,540 cfs in September. Boundary
conditions at Point Orient are shown in Figure 7.65. The hydraulic model results were in
excellent agreement with the predictions in the tide tables for these periods. A comparison
between the net flows calculated by the model and those predicted by the California Department
of Water Resources for the two periods is given in Table 7.14. Again the agreement is good.
The chlorinity boundary conditions for July (Figure 7.65) were Increased by 2,250 mg/1 after
27 days of operation to approximate the increase of salinity at the boundary. Similarly the
chlorinity boundary values for September were decreased by 890 mg/1 after 15 days of operation.
The results of the chlorinity calculations for July and September for selected stations are
displayed in Figure 7.66 and 7.67 along with empirical data for the corresponding periods. It
is apparent that the agreement is quite good.
390
-------
5.0
4.0
2.0
1.0
SACRAMENTO RIVER AT SACRAMENTO
STATION 1
4.0
3.0
2.0
1.0
-1.0
I i I I I I I I ' i
STOCKTON SHIP CHANNEL
i i l i
4.0
3.0
2.0
I I
COLLINSVILLE
STATION 2
4.0
3.0 ,
- 2.0 -
- 1.0 -
-1.0
-1.0
SOUTH FORK
MOKELUMNE RIVER AT NEW HOPE
4.0
3.0
2.0
1.0
i I I l I \ T l
OLD RIVER AT
CLIFTON COURT FERRY
STATION 3
-1.0
0000
0600 1200
4.0
3.0
2.0
1.0
-1.0
THREEMILE SLOUGH
AT SAN JOAQUIN RIVER
STATION 6
I l i
I I I
1800 2400 0000
TIME IN HOURS
— HISTORICAL TIDE
0600
1200
1800
2400
•COMPUTED TIDE
Fig. 7.63 Comparison of computed and observed tidal stages at
selected locations in the Delta, 13 July 1959. After
Kaiser Engineers (1969).
391
-------
LOCATION MAP
OF STATIONS
00 IDEM
GATE
89 101 lit
20k
LSAN P»BLOj L SUtSUMT.
MY •" BAY *
DELTA
LEGEND:
• • OBSERVED 1 JULY 1963
o—-o OBSERVED 31 JULY 1963
Ar«—A COMPUTED 31 JULY 1963
101
• MODEL NODE
COMPUTER BOUNDARY CONDITION
18 P.P.T. CL AT GOLDEN GATE
10 20 30 kO 50 60 70
DISTANCE FROM GOLDEN GATE IN MILES
80
Fig. 7.64 Simulation of salinity intrusion into the
Bay-Delta system dynamic water quality model
Period 1 July to 31 July 1963. After Kaiser
Engineers (1969).
392
-------
9 12 15 18
TIME, HOURS
Fig. 7.65 Specified Boundary Conditions - July and
September 1955 Chloride in San Francisco
Bay-Delta. After Feigner and Karris (1970).
TABLE 7.14
Comparison of Predictions by the California Department of Water Resources
and by FWPCA Numerical Model of Net Flows in Delta Channels
From Feigner and Harris (1970)
Channel
Sac. River @ Sac.
Sutter Slough
Steamboat Slough
Delta Cross-Channel
Georgians Slough
July 1955
DWR
Prediction*
(cfs)
8990**
1550
820
2950
1850
FWPCA
Model
(cfs)
8990**
1539
670
2916
1561
September
DWR
Prediction*
(cfs)
9841**
1750
1000
3100
1950
1955
FWPCA
Model
(cfs)
9841**
1811
795
3177
1755
* Empirical relationship
** Specified
393
-------
20
^
a.
-
Sio
gg
O
_J
X
«j
0
10
•- 8
a.
• 6
taJ
O
i k
5 2
0
11)1
A
A A«-r A * ^
-
CROCKETT
i i i i
i i i A A
A A A
A A A A -
— ^^ A ^^^^
A
"
PORT CHICAGO ~
I I 1 1
20
0.
a.
w
oto
-
a.
Q.
W
O
ae
o
X
u
I l I 1
A A j_ A A A
A
BENICIA
1 l l i
1 II 1
A A A-
^n^f"^^^^^^ —
^
«^*^*^A
A A
A A
? 0 & A FERRY -
1 1 1 1
0 10 20 30 0 10 20 30
TIME, DAYS A PROTOTYPE TIME. DAYS
MODEL
Fig. 7.66 July 1955 Chloride concentration
histories—San Pablo and Sulsun
Bay Stations. Feigner and Harris
(1970) .
^
0.
a.
£10
et
o
_i
X
0
10
»—
a.
a.
fai
S 5
oc
o
X1
u
l l i 1
CROCKETT
1 l i i
^"•"•N^J 1 1
^***^"^>>*^ A
^"""•••••N,,^
~ *"
PORT CHICAGO
i 1 i i
AU
1—
a.
Q.
Sio
K
O
_J
X
U
8
a.
a.
.
UJ ,
O If
K
O
^
X
u
n
_"*+±U** + *.
BENICIA
till
1 1 1 1
A 0 & A FERRY
A AV>A A
^S*|A**^*«^ ^
^^^^^•"^•^^
A
1 1 1 1
° 0 10 20 30 0 10 20 30
TIME. DAYS A PROTOTYPE TIME, DAYS
— MODEL
Fig. 7.67 September 1955 Chloride concentration
histories--San Pablo and Suison Bay
Station*. Feigner and Harris (1970).
394
-------
CONTRA COSTA
CANAL
Fig. 7.68 Study Area with Tracer Sampling Stations. Feigner and Harris (1970).
25
20
15
a.
<
§10
S
^-
z
20TH
19TH
OBSERVED AT LLWS , 19TH AND 20TH -
OBSERVED AT HHWS, 19TH DAY DAYS
• -• COMPUTED FOR LLWS, 20TH DAY
A-A COMPUTED FOR HHWS, 20TH DAY
I I I I
20
IS
10
12 16 20 24
DISTANCE, 1000 YARDS
28
32
}6
Fig. 7.69 Tracer Concentrations in Ship Channel--Benicia to Collinsville.
Feigner and Harris (1970).
395
-------
FWPCA performed an extensive dye release experiment in the Suisun Bay-Delta area in
1966, releasing Rhodamine WT continuously for 21 days from the Antioch Bridge and performing
fluorometric measurements through, the duration and for five weeks following the release. The
point of discharge and the measuring stations are shown in Figure 7.68. The FWPCA model was
operated to predict the tracer distribution using the following inputs:
1. Mean September hydraulics for the first 10 days with constant dye injection.
2. Mean October hydraulics for the next 11 days with constant dye injection.
3. Mean October hydraulics for 21 more days without dye injection.
An example of the measured data and the computed concentrations is presented in Figure 7.69.
This is the longitudinal distribution of dye in the ship channel after roughly twenty days.
A loss rate of 3.4% was assumed for these calculations.
Extensive application of this model (particularly the WRE coarse network) has been
made in the Bay-Delta Water Quality Program (Program Staff 1969) in evaluation of problem
situations for management of the Bay system. Investigations have'been made using this model
to assess the Impact on water quality of the proposed peripheral canal, deepening of the
Stockton Ship Channel and the restriction or closure of the entrance to Old River in the Delta,
as well as proposed barriers. Studies have been made concerning the required Delta outflow to
maintain acceptable salinities, the effects of various BOD loads and locations, the effects of
relative toxicity of treated municipal and industrial wastes, and the efficacy of various local
and regional systems of waste transport and disposal. Figure 7.70 Is the result of an example
calculation illustrating the effect of location on the concentrations resulting from a
106 Ib/day load of a conservative constituent.
4.3.3 Comparison of Physical Model and Digital Computer Model
During the Bay-Delta Water Quality Control Program, an opportunity was presented to
compare the results of the mathematical and physical models of the Bay system. The condition
simulated was the flushing of the Bay by increased flow and the subsequent reintrusion of
salinity with a decrease in flow. The physical model was operated under the following
conditions:
1. Initial run with 2,000 cfs net Delta outflow; 120 tidal cycles required to bring
system to steady state.
2. Flushing run with increased net flow of 30,000 cfs from Delta; 60 tidal cycles
required to bring system to steady state.
3. Reintrusion run with 2,000 cfs net outflow from Delta; 114 tidal cycles required
to reacquire steady state.
The computer model was operated under the following conditions:
1. Initial run of 10 tidal cycles under a 3,000 cfs net Delta outflow.
2. Flushing run of 30 tidal cycles at 30,000 cfs outflow.
3. Reintrusion run of 60 tidal cycles at net Delta outflow of 3,000 cfs.
396
-------
100.0
50.0
10.0
5.0
vO
-J
1.0
0.5
0.1
NODE NO.
WATER
QUALITY
ZONES
WATER
QUALITY
ZONES
NODE NO.
SAN JOAQUIN RIVER
9 _
200
202
205
STOCKTON-)
208 209
DISCHARGE AT ANTIOCH
(NODE 200)
DISCHARGE AT RICHMOND
(NODE A)
I
39
[DUMBARTON
BRIDGE
2 -
28 18
SAN MATED
BRIDGE
6
BAY
56 71 89
SAN RAFAEL CARQJ
100
BRIDGE
BRIDGE BRIDGE
113 HI*
133
152
157
NOTE: NET DELTA OUTFLOW - 2,000 CFS
SACRAMENTO RIVER
*f 9 -
Fig. 7.70 Computed concentrations resulting from 10
After Program Staff (1969).
Ibs/day discharge at two different locations.
-------
Two other simulations were also made except that run 2 was replaced with 10,000 and 20,000 cfs net
Delta outflows. The transient responses of the two models are compared in Figure 7.71, with
the computer results properly shifted to correspond to the changes in inflow in the physical
model.
19
18
17
16
15
14
13
12
11
I I I
POINT SAN PABLO (NODE 56)
I I I I
1 I I
Ul
o
o
o
a.
o
I I I I I
CARQU1NEZ STRAIT (NODE 89)
-------
5. GALVESTON BAY
Galveston Bay is the largest bay on the Texas Coast and is considered to be the most
important, both economically and ecologically. The Bay is approximately 520 square miles in
surface area (Figure 7.72) and includes the Port of Houston, Texas' most important commercial
port. Moreover, the Bay provides receiving water for the wastes of the vast urban-industrial
complex of the Houston area, and in this capacity has played an important part in the economy
and continued growth of the region. The Bay system receives waste discharges from domestic and
industrial sources, natural runoff, the Trinity River, San Jacinto River, Buffalo Bayou and the
Houston Ship Channel, and various other small tributaries. The major source of industrial
waste is the industrial complex located along the Houston Ship Channel above Baytown, estimated
(Gloyna and Malina 1964) to average in excess of 200 million gallons per day. The commercial
fishing industry in Galveston Bay is the most extensive and prolific along the Texas Gulf Coast,
both by poundage and value. Furthermore, Galveston Bay hosts a diversity of aquatic life, and
it is estimated that the Bay is the nursery grounds for approximately 807. of the total fishery
product taken from the Gulf of Mexico along the Texas Coast (Curington et at. 1966).
Galveston Bay is typical of many of the estuaries along the Gulf Coast, in that the
bay is shallow (average depth of 8 feet) and is nearly separated from the Gulf by barrier
islands. The tidal range in the bay is small, one to two feet, and both water levels and cur-
rents are- greatly influenced by wind. The Houston Ship Channel is maintained from Bolivar
Roads to Morgan Point at a project depth of 40 feet, and continues up Buffalo Bayou some
25 miles to the Houston Turning Basin. Several other dredged channels traverse the Bay.
Galveston Bay has been characterized (Carter 1970) as an estuary in crisis, whose
productivity and ecological functions are in immediate danger due to the combined effects of
area development, e.g. waste disposal, shell dredging, freshwater diversion and upstream
damming. Concern with this deterioration of the Bay's quality has brought about the strength-
ening of regulatory agencies to cope with these problems and the'initiation of extensive
research programs to provide a basis for regulation.
Within the last five years models have come into extensive use as predictive tools
for water quality of Galveston Bay. (Prior to this physical models had been applied for
problems of shoaling, harbor design and dredging, and limited mathematical models had been
developed, such as the work of Urban and Masch 1966 based on an extension of the tidal prism
approach of Ketchum 1951.) The models principally employed for water quality studies are a
physical model operated by the Corps of Engineers Waterways Experiment Station and the mathe-
matical models developed (or currently under development) by the Galveston Bay Project.
5.1 GALVESTON BAY PROJECT MATHEMATICAL MODELS
The Galveston Bay Project has been initiated by the Texas Water Quality Board and is
basically directed toward the development of a comprehensive management program by which a
suitable water quality level in the Bay can be determined and maintained with consideration
for the many demands on the Bay resources. The Galveston Bay Project (GBP) has been the focal
399
-------
SAN JiCINTO RIVER
TRlNlTV RIVER
ANAHUAC
HOUSTON
ids
SOUTH JETTY
GULF OF
MEXICO
SCALE
0 5 N.M.
»SAN LUIS R8SS
Fig.7.72 The Galveston Bay System
400
-------
point for the development of mathematical water quality models for the Galveston Bay System
(Espey et al. 1968). Mathematical models for water quality parameters developed or under
development for the Galveston Bay Project include both steady-state and time-varying models.
The time-varying models are employed to represent long-term (i.e., tidal-averaged) varia-
tions as well as intratidal variations which are of particular importance in considering
the local effects of individual outfalls. Due to the shallowness of the Bay and hence the
augmented mixing effects of waves, Galveston Bay is nearly vertically homogeneous, except,
of course, in the ship channels and in the neighborhood of large discharges. Therefore the
distribution of parameters is represented by vertically averaged models, i.e. with two
horizontal spatial dimensions. The Houston Ship Channel above Morgan Point is represented
as a one-dimensional system for detailed studies of the transport and reactions of polluting
substances in this area.
A time-varying two-dimensional hydrodynamic model is used to calculate the current
velocities, which are retained on magnetic tape and used as basic input, appropriately time-
averaged, for the advection terms in the various water quality models. Parameters for which
extensive model development and verification have been undertaken are salinity, temperature,
BOD and DO. However, application to other parameters such as coliforms, nutrients and a
toxicity parameter is an important adjunct to this work. Computational techniques used in
the time-varying and steady-state water quality models are straightforward: in the former,
explicit time-differencing is employed, and in the latter, the concentration field is
determined by successive overrelaxation using the finite-difference approximation to the
steady-state mass balance equation.
The Galveston Bay Project activities include an extensive data collection program
which is subdivided as follows:
1. a large-scale sampling program involving monthly sampling of the entire bay
system;
2. a high-frequency sampling program in which numerous samples are collected at
short-time intervals over a tidal cycle at selected points in the bay.
The large-scale program* maintains thirty-nine sampling locations throughout the Bay and Ship
Channel as shown in Figure 7.73. Samples are taken at two points in the vertical at each
station in the bay proper, and Ship Channel stations are sampled at four locations in the
vertical. All are sampled within roughly a twelve-hour period. Water quality parameters
measured in this data program include dissolved oxygen, temperature, pH, conductivity,
chlorinity, BOD, total and fecal coliform, nitrogen series and total phosphates. The high-
frequency program has concentrated on evaluating the short-term variations of certain water
quality parameters including dissolved oxygen, conductivity, temperature, pH, and some data
on free CC>2- Some limited current measurements and meteorological observations are made
during both the large-scale and the high-frequency programs.
5.1.1 Hydrodynamic Model
5.1.1.1 Model Formulation and Application
The hydrodynamic model for Galveston Bay is based principally upon the work of
Reid and Bodine (1968), which is a numerical solution to the equations of motion appropri-
ately simplified for the computation of storm surges in Galveston Bay. The model represents
401
-------
• MONTHLY SAKfLIKG STATIONS
OHICH FMQUtNCT STATIONS y
GALVESTON BAY SAMPLING STATIONS
HOUSTON SMP CHANNEL SAMPLING STATIONS
Til. '.73 CtlTMtoo l<7 rrojtct i^pl
-------
the Bay system in two horizontal spatial dimensions, and employs the time-dependent vertically
integrated equations of motion and continuity, allowing for rainfall, wind stress, and bottom
friction, but neglecting transport of momentum and the Coriolis force. Allowances are made in
this model for submerged barrier reefs and the overflow of low-lying barrier islands. The
dynamic equations of motion and of continuity are written
Hi + gD ^S - X - fQUTJ"2
at 8x
*X + go £3 - Y - fQVD"2
3t - 3y
. 9H , 9U , 9V „
and — + — + — • R
3t 3x ay
where U, V - vertically Integrated x and y components of velocity (dimensions L2/T)
H - water level elevation relative to MSL
D - water depth
Q - yU^+V2 , the magnitude of the vertically integrated velocity
f - nondimensional bottom coefficient of friction
R - rainfall rate
X, Y = the x, y components of wind stress divided by the water density
The wind stress is represented by
X - KW2 cos 8
Y - KW2 sin 6
in which W - wind speed
6 - angle between the wind velocity vector and the x axis
K « an empirical function of wind speed, proportional to the drag coefficient
(see, e.g., Wu 1969)
The numerical solution to these equations is accomplished by representing the
Galveston Bay area by a two-dimensional lattice with a constant grid spacing and solving the
finite-difference form of the equations by advancing the solution explicitly through finite
time increments over the entire grid system, the storm surge predictions of Reid and Bodine
(1968) were obtained using a grid spacing of two nautical miles. A depth and frictional
coefficient are assigned to each of the nodes. The parameter values for each node represent
averages over the area represented by that particular cell. With each node is also associated
the necessary information to determine that cell's orientation with respect to surrounding
land barriers and internal boundary constraints. A set of initial condition and boundary
conditions is required including, among other factors, tidal stage at the passes, freshwater
403
-------
inflow, and topography. The initial conditions used in the model also affect the duration and
behavior of the starting transients.
The storm surge model (Reid and Bodine 1968) was calibrated by adjusting the fric-
tional coefficient f to reproduce measured tides at selected points in Galveston Bay. The
friction factor was varied between 0.01 and 0.001; the resulting tides are shown in Figure 7.74
for Kemah Docks and the railroad causeway. It can be seen that if the bottom friction factor
is too large, the surge wave amplitudes are reduced, and the higher harmonics are filtered out
of the tidal oscillations. For a friction factor too small, the computed amplitudes of the
tidal waves are too large. The friction factor that was finally selected was a constant 0.0025
throughout the bay. Shown in Figure 7.75 are the final computed tides compared with the
observed tides at Morgan Point, South Texas City Dike and the South Jetty.
Although the storm surge model is probably inadequate for the computation of currents
under less singular conditions, it was thought to form a suitable point-of-departure for the
development of a more general model to be used in the Galveston Bay Project. The equations of
motion were augmented by inclusion.of the Coriolis deflection and.the nonlinear field accelera-
tions (when important, as in the locality of large discharges, see Section- 5.1.3). The bottom
stress expressions were reformulated in terms of Manning's n . A grid spacing of one nautical
mile proved to be the best compromise between spatial refinement and constraints of computer
time. The initial verification of the GBP one-nautical-mile model consisted of comparison
of predicted tidal variations with recorded tidal elevations. The model was operated using the
driving tide displayed in Figure 7.76 along with comparisons between measured and computed
tides for various points in the Galveston Bay system. In general, good agreement was obtained.
It should be noted that a slight transient is in evidence during the f: .-'. one-and-a-half tidal
cycles (i.e., the measured and computed tides are slightly out of phase), but by the third
cycle reasonable phase correlation is obtained.
The primary output of the hydrodynamic model is a time history of the velocity pattern
over the bay, which is required for the advection terms of the water quality models. As an
example of the velocity information provided by the hydrodynamic model, calculations have been
made using the input tide of Figure 7.76 and the ten-year average freshwater inflows to the bay
given in Table 7.15. For use in the steady-state mass conservation model, the results of the
hydrodynamic model were time-averaged over a tidal cycle to remove the oscillatory components
of tidal period and thus providing the "net" nontidal velocities. The results are displayed
in Figure 7.77 as a vector pattern of the net velocities superimposed on a map of the system.
The outline of the boundary of the model grid structure is also displayed. As an example of
the use of the model to predict the effects on currents resulting from physiographic modifica-
tions, the model was operated using identical input conditions with the addition of a proposed
diversion of 3,200 cfs across Umbrella Point. The model results in Figure 7.78 depict the
alteration in the net flow pattern as a result of the diversion.
5.1.1.2 Comparison of Mathematical Hydrodynamic Model with Physical Model
In April 1969, experiments were performed in the Ship Channel physical model (see
Section 5.2) at Vicksburg as a part of the GBP. The mathematical model was operated using the
input tide and freshwater inflow as specified in the physical model test, and velocities obtained
from the mathematical model were compared to those measured in the physical model. The compari-
sons were made for three ranges in the bay, namely (1) one station between Atkinson Island and
Mesquite Knoll, (2) two stations between Red Fish Island and Smith Point, and (3) two stations
between Red Fish Island and Eagle Point (Ranges 1 and 6 in Figure 7.92). Current velocities
measured in the physical model were reported to the nearest 0.1 ft/sec at hourly intervals
during the tidal cycle. Time histories of velocity from the mathematical model calculations
404
-------
Ill : I I I
z
o
i '-5
UJ
i '-o
oc
£ 0.5
I
MSL
-0.5
-1 .0
OBSERVED
COMPUTED
- F - 0.0100
RAILROAD CAUSEWAY -
•••• F - 6.0025 (FINAL VALUE)
,F
,0.001,
I
I 200 2000
2 SEPT 1964
01+00 1200 2000
3 SEPT 1964
T IME IN HOURS
0400 1 200 2000
4 SEPT 1964
Fig. 7.74 Computed Astronomical Tide for two locations in Galveston
Bay for three different friction factors: Observed value
also shown. After Reid and Bodine (1968).
1 .5
i .0
0.5
_ MSL
C-0.5
UJ
£ i.o
g °-5
- MSL
5-0.5
S-1.0
LU
o: 1.5
UJ »
•- 1.0
<
* 0.5
MSL
-0.5
-1 .0
1.5
MORGAN POINT
i
SOUTH
TEXAS CITY DIKE
l i i i
i i i ii iii
SOUTH JETTY
j i i i i
i i i i i i
1200 2000
2 SEPT
0<*00 1200 2000
3 SEPT 1964
TIME IN HOURS
0400 1 200
4 SEPT 1964
2000
Fig. 7.75 Comparison of observed and computed Astronomical Tide for
three locations in Galveston Bay using final friction
factor. After Reid and Bodine (1968).
405
-------
BOLIVAR ROADS
AND SAN LUIS
PASS
1200 2000 01(00 1200 2000 O'tOO 1200 2000
TIME IN HOURS
1.0
0.5
MSL
-0.5
-1.0
-1.5
MORGAN POINT
— OBSERVED
— MODEL
i i I I
1.5
1.0
0.5
MSL
-0.5
-1.0
-1.5
KEMAH DOCKS
I I I
— OBSERVED
— MODEL
1200 2000 O'tOO 1200 2000 0400 1200 2000
TIME IN HOURS
1200 2000 . O'tOO 1200 2000 O'tOO 1200 2000
TIME IN HOURS
Fig. 7.76
Comparison of measured and computed tidal elevations In Calves ton Bay
(Stover ejt al. 1971).
-------
Fig. 7.77 Calculated tidal-averaged velocities for
ten-year-mean inflow.
Fig. 7.78 Calculated velocities for conditions of
Fig. 7.77 with added diversion across
Umbrella Point (top right).
407
-------
were obtained at each node in the grid system, and for comparative purposes these model results
were spatially averaged over the cells which corresponded to each range. Comparison of the
results from the two models is presented in Figures 7.79-7.81. Velocity is plotted as a
TABLE 7.15
Ten-Year Average Freshwater Inflow (cfs)
to Galveston Bay System
From Smith and Kaminski (1965).
Dickinson Bayou 100
Double Bayou 75
Clear Creek 200
Trinity River 7,900
Cedar Bayou 150
Goose Creek 25
San Jacinto River 2,100
Carpenters Bayou 18
Greens Bayou 150
Hunting Bayou 35
Vince Bayou 15
Sims Bayou 90
Brays Bayou 125
Buffalo Bayou 340
Total Inflow 11,323
function of time over one tidal cycle, with the positive ordinate representing flood. With the
exception of high water slack period on the range between Red Fish Island to Eagle Point, the
results compare quite well.
Point currents in the Galveston Bay System are highly influenced by the local effects
of wind, waves and physiography, so even with the correct boundary conditions, comparison of
the mathematical hydrodynamic model with currents measured in the field is difficult. For this
reason the above type of comparison of computed velocities with physical model measurments is
of some value as it allows a degree of verification of the hydrodynamic model velocities inde-
pendent of their use in the water quality models. This provides, for example, some protection
from incorporating errors in velocity prediction into the dispersion coefficients used in the
water quality models.
5.1.2 Salinity Model
Salinity is the most important hydrobiological parameter in the Galveston Bay estuary.
Preservation of the natural ecosystem is a major concern to the management of Galveston Bay,
and this requires capability for the prediction of the salinity distribution in the Bay,
particularly with regard to effects of the decrease in freshwater inflow due to urban consump-
tion and upstream diversion, e.g., the Texas Water Plan (Texas Water Development Board 1968).
Figure 7.82 displays a typical summer salinity distribution in Galveston Bay. The
marked influence of the Houston Ship Channel on the isohalines should be noted. As stated
previously, the predominant sources of inflow into Galveston Bay are the San Jacinto River -
408
-------
1.0
0.8
0.6
$0.2
5 -o.z
3
2-o.k
-0.6
-0.8
-1.0
ooo
PHYSICAL MODEL:
STATION 2
-MATHEMATICAL
MODEL
10 15
TIME (HOURS)
20
25
Fig. 7.79 Comparison of velocities from physical
model and mathematical model. Range 1,
Atkinson Island to Mesquite Knoll (see
Fig. 7.92).
PHYSICAL MODEL
STATION k
O STATION 5
-1.6
to 15
TIME (HOURS)
25
Fig. 7.80 Comparison of velocities from physical
model and mathematical model. Range 6,
Red Fish Island to Eagle Point (see
Fig. 7.92).
409
-------
10 15
TIME (HOURS)
Fig. 7.81 Comparison of velocities from physical
model and mathematical model. Range 6,
Red Fish Island to Smith Point (see
Fig. 7.92).
Fig. 7.82 Salinity contours from me«iureinenti of 17 September 1968
in Calves ton Bay.
410
-------
Houston Ship Channel and the Trinity River (although the lesser tributaries discharging into
Galveston Bay are occasionally significant). These inflows tend to be highly erratic, as is
indicated in Figure 7.83. Variations in inflow of two orders of magnitude within a month span
are not uncommon. Due to the irregularity of these inflows, as well as wind conditions, evapo-
ration and tidal currents, the salinity at a given point in the bay tends to fluctuate markedly
over a wide range. Evaluation of the model operation has been difficult because of this natural
fluctuation in the measured data together with the lack of precise input information (e.g., com-
plete inflow records, evaporation measurements, and records of saline discharges in the Bay).
Host of the verification to date has been directed toward the "steady-state" calculations.
Steady-state distributions appear to be acquired in the Bay when the inflows are sustained for
several months (which usually occurs at low flows). Moreover, the effects of even large fluc-
tuations in the inflows can be removed by time-averaging over sufficiently long periods so that
quasi-equilibrium profiles are obtained.
The calculation of equilibrium profiles in the Bay system is effected by successive
overrelaxation using the finite-difference expression of the steady-state two-dimensional mass
transport equation with boundary conditions appropriate for salinity. Most of the work per-
formed on salinity in the Galveston Bay Study has employed a spatial increment of one nautical
mile. The net current (i.e., averaged over a tidal cycle) at each point is obtained from the
hydrodynamic model and used in the advective terms.
Comparisons of model predictions with mean measured chlorinities for two different
"averaging times" are given in Tables 7.16 and 7.17. The empirical chlorinities of Table 7.16
are the means over the three-month period November 1968 through January 1969. During this
period the inflow was generally quite low but with intermittent peaks ranging to as much as
13,000 cfs in the Ship Channel. The model velocities were obtained assuming relatively low
inflows of 230 cfs in the Ship Channel and 550 cfs in the Trinity. Agreement between model
predictions and the averaged field data is satisfactory with the exception of Stations 24 and
25 in Upper Trinity Bay. These stations, however, are near a low, marshy area, where they are
subject to random overland runoff from the Trinity. Table 7.17 tabulates the predicted concen-
trations using the ten-year mean inflows of Table 7.15. For comparison, the average of a year
of monthly salinity measurments (July 1968 through June 1969) are given. Dispersion coeffi-
cients were the same for both computations and were obtained from measurements of chlorinity
made in the summer of 1968.
5.1.3 Thermal Discharge Model
As a part of the Galveston Bay Project and with the joint support of Houston Lighting &
Power Co. (Houston), Central Power & Light Co. (Corpus Christ!), and Gulf States Utilities Co.
(Beaumont), an intensive study was performed of the behavior and distribution of temperature
in selected estuaries of the Texas coast. A principal objective of this study was the develop-
ment of a model for predicting the temperature profile in the vicinity (viz. within the "inter-
mediate" or "transitional" mixing region, see Chapter IV) of the thermal discharges into shallow
areas. The basis of this model is the time-dependent vertically integrated expression of the
conservation of energy, employing formulations of the various heat fluxes through the surface
(Stover et al. 1971). As with the other mathematical models developed by the Galveston Bay
Project, the solution is accomplished by finite-difference techniques, in this case using
explicit tine-differencing for stepping the solution forward in time.
For application to the Immediate vicinity of an outfall, both the temperature model
and the associated hydrodynamic model were implemented using a (horizontal) spatial grid of
relatively fine resolution, i.e., with a spatial increment of a few hundred feet. Boundary
411
-------
"48000
40000
32000,
24000
16000.
8000
-ft. -
OCT ' NOV ' DEC ' JPN ' FEB ' «* ' RPR ' MflT ' JUN ' JUL ' HUG ' SEP
OISCHflRGE FOR 1968 HflTER TERR. HOUSTON SHIP CHflNNEL
48000.
40000.
„ 32000.
24000
16000
8000 .
OCT NOV DEC JflN FEB MSR flPR MRT JUN JUL ' flUG ' SEP
DISCHRRGE FOR 1968 HflTER TEflfi. TRINITr RIVER BT HOMflTOR
Fig. 7.83 Inflows in Trinity River and Houston Ship Channel
(at Morgan Point) for 1968 water year.
412
-------
TABLE 7.16
Comparison of Steady-State Model Predictions and
Measured (Nov. 1968 - Jan. 1969 Average) Chlorinities (ppt)
GBP
Station
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Predicted
17.0
15.0
13.4
10.6
10.2
10.0
10.0
7.1
9.1
8.4
7.2
12.5
15.4
14.4
13.5
15.3
15.4
Measured
14.6
13.1
13.2
12.1
11. 8
10.4
9.2
6.4
8.8
8.2
7.2
13.3
15,0
12, a
13.0
14,6
14.0
GBP
Station
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Predicted
12.3
10.5
9.6
10.1
10.4
8.7
7.3
6.7
7.5
7.9
10. L
11.4
10.8
15.4
15.3
10.0
9.0
7.9
Measured
11.3
10.3
9.8
10.0
9.9
7.9
3.8
3.0
7.2
7.2
7.8
9.4
8.5
14.9
14.5
11.4
8.9
7.8
TABLE 7.17
Congwrison of Steady-State Model Predictions and
Measured (July 1968 - June 1969 Average) Chlorinities (ppt)
GBP
Station
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Predicted
15.3
13.3
9.5
5.9
5.5
5.3
5.3
2.2
4.3
3.7
2.9
11.8
14.6
13.1
9.5
13.4
13.4
Measured
14.6
11.5
12.4
9.6
8.7
6.4
s.e
3.4
5.3
4.9
3.8
10.3
13.7
11.4
10.6
11.8
11.9
GBP
Station
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Predicted
8.0
5.9
5.2
5.4
5.7
3.6
2.4
1.9
2.5
2.8
4.9
6.2
5.6
13.5
13.4
5.3
4.2
3.3
Measured
8.7
6.3
5.8
6.7
6.7
5.2
2.8
2.0
3.6
3.4
5.0
6.6
5.8
12.6
12.6
7.9
5.7
4.5
413
-------
conditions for the hydrodynamic model were obtained by Interpolation from the results of the
coarse, one-nautical mile grid hydrodynamic model described in Section 5.1.1. The momentum
equations used in this hydrodynamic model are the same as those of Section 5.1.1 with the addi-
tion of the nonlinear field acceleration terms, as in such proximity to the outfall these terms
become significant. Buoyancy effects are neglected.
This model was applied to several steam power plant discharges along the Texas coast.
One such application is to the Houston Lighting & Power Company P. H. Robinson Generating Sta-
tion which is located on the western shore of Calves ton Bay and discharges into upper Galveston
Bay in water of nominally five foot depth. The plant has a maximum generating capacity of
1,475 megawatts and utilizes 667,000 gallons per minute (approximately 1,500 cfs) of once-
through cooling water with a 17.6°F rise across the condensers. The outfall area is shown in
Figure 7.84. The outfall structure consists of low weir, boat barrier, and a pair of parallel
sheet-metal groins extending out about 1,000 feet perpendicular to the shore. The computational
grid system for both the hydrodynamic and temperature models as applied to the outfall site is
based on a 500 by 500 foot lattice covering an area of approximately seven square miles near
the discharge. The grid spacing provides a fairly fine representation of the physical system,
including the shoreline, groins and variations in the depth. The orientation of the grid was
dictated by the groins and shoreline, as these features required precise representation within
the computational framework.
As an example of the thermal discharge model operation, a comparison is made between
some of the field data obtained at the P. H. Robinson site and the associated model results.
The data were collected during the period 2-3 October 1969. Water temperature surveys were
performed by traversing the plume while towing multiple-depth temperature sensors. The outputs
of the sensors were continuously recorded on a chart recorder inside the boat. On-slte meteoro-
logical data were also collected (displayed in Figure 7.85), and the plant operating conditions
were obtained. The tidal range was larger than normal, so that tidal currents were more pro-
nounced than usual. The general direction of the wind was from the shore and the plume was
only slightly affected as it was•sheltered from the wind by bluffs along the shoreline. Later
in this period, however, the wind shifted and the movement of the plume became dominated by
wind-driven currents. Water temperature surveys were made at approximately 2:00 p.m., 5:00 p.m.,
and midnight on 2 October, and at 9:00 a.m. and noon on 3 October. In Figure 7.86 are shown
temperature contour plots obtained from the model at times corresponding to the water tempera-
ture surveys, with the boat traverses and recorded temperatures superimposed upon the predicted
contours. The disagreement between the model predictions and the measured temperatures was
greater than expected for the midnight survey; it was suggested to be due to difficulties in
determining positions at night (Stover et al. 1971). The computer time required for these
computations, using a time step of three minutes, was approximately four minutes per tidal
cycle on the UNIVAC 1108 computer, including the time required for generating the temperature
contour plots.
Another example of the operation of the thermal discharge model, but not in Galveston
Bay, is an application to the Central Power and Light Company Nueces Bay Generating Station,
which is located on the south shore of Nueces Bay. Nueces Bay, a small secondary embayment off
Corpus Christ! Bay, has a surface area of approximately 20,000 acres and an average water depth
of approximately three feet. It receives limited freshwater inflow from the Nueces River during
most of the year. The Nueces Bay plant has a maximum generating capacity of 240 megawatts and
utilizes 120,000 gallons per minute (270 cfs) of once-through cooling water which is pumped
from the Corpus Christ! Ship Channel at a depth of thirty feet and is discharged into Nueces
Bay from a dredged channel. The plant imparts a 15°F temperature rise across the condensers.
414
-------
500 X 500 FT
COMPUTATIONAL GRID
Fig. 7.84 Location map and computational grid for thermal discharge model.
TIME -HOURS
AIR TEMPERATURE
15 It 11 24
O 1 * » 1
TIME — HOURS
RELATIVE HUMIDtTV
11 1* 21 02
TIME -HOURS
OUTFALL TEMPCHATURE
TIME — HOURS
SHORTWAVE RADIATION
N
TIME — HOURS
WIND SPEED AND DIRECTION
Fig. 7.85 Environmental conditions at P. H. Robinson Plant on 2 - 3 October.
415
-------
2000
2000 HOOO 6000
DISTANCE FROM OUFFflLL IFEETI
9000
: ' . K)
(a) 2 October, 2:00 P.M.
2000
2000 KOOO 6000
DISTflNCE FF10H OUTFflLL (FEET)
8000
loon
(b) 2 October, 5:00 P.M.
Fig, 7.86 Calculated temperature ( F) contours with measurements from boat traverses.
P. H. Robinson discharge In Galveston Bay. From Stover et al. (1970).
416
-------
jooo uooo fu
i1I5t«NCF. FBOM JUTFBU iFEE II
(c) 2 October, Midnight.
2000
2000
14000 6000 8000
OISTBNCE FWDH OUTFRLL IFEETI
10000
I ?OOU
(d) 3 October, 9:00 A.M.
Fig. 7.86 Continued.
417
-------
Due to the plant size and the small area influenced by the discharge, the grid size chosen for
this computation was 200 by 200 feet, which proved adequate for the representation of the
physiographic features of the discharge area. Because of the proximity of the Corpus Christ!
weather station, meteorological data from that station were used in the temperature model cal-
culations. Computed contours together with point measurements of water temperature are dis-
played In Figure 7.87 for three different dates in the summer of 1969. Agreement between
computed and observed temperatures is evidently quite satisfactory. The effect of the southeast
wind of 10 ca. 15 knots on the isotherms, as shown in Figure 7.87(a), should be expecially noted.
The thermal discharge model has also been applied to the prediction of the plume sire
for expanded Nueces Plant operations of 569 MW (TRACOR 1970a) and for a new plant, Houston
Lighting and Powers Cedar Bayou Generating Station, located on the north shore of Galveston
Bay taking Its cooling water from Cedar Bayou and discharging into Trinity Bay (TRACOR 1970b) .
NUECES BAY
27 JUNE 1969, 9:00 AH (COT)
COMPUTED TEMPERATURE
• MEASURED TEMPERATURE
02468 lOOOft
(a) JUNE DATA, SOUTHEAST WIND AVERAGING 15 MPH , DISCHARGE FLOW 270 eft
Fig. 7.87 Comparison of calculated temperatures from thermal discharge model
and measured temperatures for three environmental conditions.
418
-------
If •
NUECES BAT
It AUG 19*9. 9:10 AM (COT)
COMPUTED TEMPERATURE
• MEASURED TEMPERATURE
(bl AUGUSl DMA CALM. DISCHARGE flOW 270 cf«
NUCCES MY
17 SEPT 19*9. »:00 AM (C01)
COMPUTED TEMPERATURE
• MEASURED TEMPERATURE
(Cl SEPTEMSER DATA, CALM. DISCHARGE FLOW 270 eft
Fig. 7.87 Continued
419
-------
Although this computational model has been discussed only insofar as its application
to thermal discharges, it is apparent that it nay be applied to the distribution of any sub-
stance discharged in a similar manner into a broad, shallow area, provided the sources and
sinks of the substance in the receiving water can be represented mathematically.
5.1.4 Application of the GBP Models to Ecological Parameters
Odum and Copeland (1969) classified the Calveston Bay estuary as an emerging new
system characterized by close association with Industrial Man, and whose ecosystem is subjected
thereby to multiple environmental stresses. The evident effects of area development on
Galveston Bay and the extreme ecological importance of the Bay motivated as a part of the
Galveston Bay Project a series of biological investigations performed by the University of
Texas (Copeland and Fruh 1970). The GBP mathematical models proved to be of value in inter-
preting some of the results of these studies, and it is this aspect that is discussed here.
Due to the large and varied waste discharges into the Bay system, the possibility of
the introduction of toxic compounds inhibitory to phytoplankton growth was given careful atten-
tion. Specifically, objectives were to ascertain the existence and importance of toxicity,
the relative toxicity at different points through the Bay, and the toxicity of variously
treated effluents in the Bay. In order to carry out these investigations, a phytoplankton
bioassay parameter was devised: the growth-rate parameter k . A blue-green alga
(Coccochloris elebans) was selected as the assay organism because it possesses the following
properties:
(1) ubiquitous to the marine environment;
(2) was shown to respond to environmental parameters in a way similar to the
resident phytoplankton;
(3) stable organism, that does not readily mutate;
(4) rapid growth rate of roughly one generation per 2.5 hours;
(5) sensitive to nutrient and toxic components;
(6) ordinarily stays in suspension with no clumping;
(7) considerable information available concerning its physiology and growth
potential.
The Coccochloris was maintained in pure cultures until needed for experimentation. The test
organism and a nutrient medium were added to diluted, 'pasteurized water samples collected in
Galveston Bay and the resulting growth followed colorimetrically. The growth, represented by
the optical density D which is proportional to the number of cells, followed an exponential
form with (base 10) rate constant k . Precisely,
log1Q (100 x D) - kt
The effect of toxicity was found to be a depression of the growth rate k from the
optimal rate of 2.0 (or 6.6 generations per 24 hours). Through the use of careful laboratory
procedures and sufficient controls, k proved to be a sensitive and utilitarian indicator of
the toxicity of the sample water. The following comments from Copeland and Fruh (1970) amplify
the interpretation of the k parameter.
420
-------
To assess the effects of toxicity on the test organism, its growth rate is measured
in liquid culture tubes. This growth rate is expressed as a k value (the increase
of log optical density x 100 per 24 hours). ... In this manner, an average k can
be obtained for a growth curve; thus giving a realistic growth evaluation. Any lags
in growth or erratic growth will be incorporated in this calculation.
In order to put the k values in perspective with growth rates and productivity,
we endeavor to discuss the meaning of the k value. This will allow us to inter-
pret the full meaning of k and how it relates to the evaluation of toxicity in
the natural setting.
k is the slope of a line on a graph and is defined as the log of the change in
optical density during a 24-hour period. This simply represents the ability of the
cells to reproduce—or the number of generations of the test organism per 24-hour
period. ... A k of 0.301 represents one generation per day (or a doubling of the
original number of cells in the test culture) and a k of 1.9 represents 6.3 genera-
tions per day (or a doubling of the cells in the original culture 6.3 times). This
means that the k value in relation to the number of generations is linear and
that a doubling of k means a doubling of growth rate.
Optical density is used as the measurement criteria and a change in optical density
of 0.1 means a change in the number of cells of approximately 1.05' x 107 cells
per ml. . . .To relate optical density changes, cells numbers and k values,
consider the change in optical density from 0.1 to 1.0 in 24 hours. This repre-
sents a k value of 1.0 (change in log of 0.1 to log of 1.0 is equal to a k of
1) and a change in the number of cells per ml from 1.05 x 107 to 10.5 x 107--or a
doubling of the cell population 3.3 times (thus relating k and optical density
in terms of growth of the population).
Values of k were obtained for six stations in the Bay system from February through
December 1969. These six stations were Trinity River, Houston Ship Channel, Bolivar Roads,
Dickinson Bay, Hannah's Reef, and the Texas City Dike. Experimental model computations were
made to predict the distribution of k through the Bay system assuming a steady-state distri-
bution. The period August through October represented a sustained low inflow period and
appeared to approximate an equilibrium state. It was assumed that k was a conservative
parameter transported by advection and diffusion through the Bay, with the important boundaries
being the Trinity River, Houston Ship Channel at Morgan Point, and Bolivar Roads. Thus the
solution technique employed was the same as the steady-state salinity calculations of Section
5.1.2 (i.e., successive overrelaxation on the finite-difference approximation of the mass con-
servation equation) with boundary conditions appropriate for toxicity. Dispersion coefficients
were the same as employed in the salinity work. Model predictions were compared with the
measured k values for Jthe period August through October. To eliminate the effect of experi-
mental fluctuations the measured data was averaged for the period August-October, and the
averaged values for the boundary points were used in the model computations. Additional model
calculations were made averaging the data over the periods March-October and July-October but
the agreement between predicted and observed k values was somewhat poorer; this is attributed
to the transient effects of large freshwater inflows during spring. The inflows, boundary
conditions, and comparisons of model results with measured values of k are presented in
Tables 7.18, 7.19 and 7.20, respectively. The agreement between measured k and model pre-
dictions is quite satisfactory.
Copeland and Fruh (1970) also investigated the relation of the phytoplankton toxicity
parameter k to the species diversity indices of various populations. Species diversity was
estimated by
where n^ is the number (or weight) of individuals in the i th species and N is the total
number (or weight) of individuals in the collection. Species diversity indices were determined
421
-------
TABLE 7.18
Inflows for Computations of Steady-State k Distribution
River or Tributary
Trinity River
Houston Ship Channel
(at Morgan Point)
Clear Creek
Dickinson Bayou
Double Bayou
Cedar Bayou
Goose Creek
Inflow (cfs)
564
230
126
110
78
44
40
TABLE 7.19
Averaged k Used for Boundary Conditions
Boundary
Averaging Period
August-October July-October March-October
Houston Ship Channel 0.99 1.06 1.08
Trinity River 0.49 0.54 0.76
Bolivar Roads 1.98 1.91 1.91
TABLE 7.20
Computed and Observed (mean) k Values for Galveston Bay Stations
Station
Dickinson Bay
Hannah's Reef
Texas City Dike
Averaging Period
August-October
pred.
1.34
1.21
1.66
obs.
1.45
1.35
1.70
July-October
pred.
1.33
1.20
1.65
obs.
1.35
1.47
1.61
March-October
pred.
1.33
1.20
1.65
obs.
1.51
1.40
1.59
for fin fish (by number and by weight) , nekton, zooplankton, benthos and phytoplankton from
seasonal samples at each of the GBP stations (Figure 7.73) over a year. The model calculations
of the distribution of k in the Bay system were compared with the mean annual diversity
indices observed in the system. Quite good correlation was obtained between the phytoplankton
diversity and the growth parameter, as is evident in Figure 7.88. A degree of correlation
might be expected a priori between empirical measurements as both parameters are a measure of
the response of the phytoplankton community to the presence of inhibitory constituents, in
which case the correlation between measured diversity and a predicted k constitutes an addi-
tional degree of verification that k can be modeled as a transported parameter. More impor-
tantly however, this correlation attests to the possibility of using models as predictors of
the ecological "health" of the Bay. An example of the correlation between the predicted k
and another population diversity is shown in Figure 7.89. Correlation between nekton weight
diversity and predicted k values is somewhat poorer, but this is not especially surprising
422
-------
(Y)
2.0
l.5
1.0
0.5
PHYTOPLANKTON
r - 0.88
o!2
0.8
1.0
1.2
1.4
1.6
1.8
2.0 (X)
PREDICTED TOXICITY k
Fig. 7.88 Relationship between annual mean species
diversity and model predictions of toxicity
parameter (Copeland and Fruh 1970). Numbers
are GBS sampling stations, Fig. 7.73.
2.0
1.0
1 I
NEKTON WEIGHTS
023
i r i
031
ol5 !7_
o3
o37
032
if
05
026
o25
022
028
021
036
o27
J
1
1
0.6 0.8 1.0 1.2 1.4 1.6 1.8
PREDICTED TOXICITY k
Fig. 7.89 Relationship between predicted k and
annual mean nekton weight diversity.
Copeland and Fruh (1970).
2.0
423
-------
since one would not expect a phytoplankton bioassay Co be a complete measure of the response
of the nekton community. Indeed, from this standpoint, the degree of correlation obtained is
encouraging.
It must be emphasized that this work is preliminary. Much remains to be learned
about the sources and nature of toxic materials in the system and the dynamic behavior of their
distribution. Moreover, the relation of k to species diversity and other ecological indica-
tors requires additional study with regard to seasonal effects, the higher trophic levels, and
so on. Nevertheless, the value of the growth rate k as an index to toxic substances and to
the ecological conditions in Galveston Bay was established by the extensive work of Copeland
and Fruh (1970). Furthermore, the evident feasibility of predicting its spatial distribution
from knowledge of its sources and of the transport characteristics of the system indicates that
it may possess utility as a management parameter.
5.2 PHYSICAL MODELS
The U. S. Corps of Engineers Waterways Experiment Station at Vicksburg, Mississippi,
operates three physical models of Galveston Bay or parts thereof. These three models are
designated Galveston Bay Entrance Model, Galveston Bay Surge Model, and the Houston Ship
Channel Model. The spatial extent of these models is illustrated in the location maps,
Figures 7.90-7.92. The scale ratios are given in Table 7.21.
TABLE 7.21
Scale Ratios in Galveston Bay Physical Models
Horizontal 1:500
Vertical
Tidal Day
The Entrance Channel Model, which can be operated as either a fixed-bed or a movable-
bed model, was constructed for studies related to the relocation of the entrance channel to
Galveston Bay and the rehabilitation of the jetties. The Surge Model is a highly distorted
model which was constructed primarily for comparison with the numerical surge model developed
by Reid and Bodine (1968). This model is a "gross phenomena" model and includes all of
Galveston Bay, East Bay to Rollover Pass, West Bay to Freeport, and the entire surrounding
flood plain up to elevation 20 feet. The Surge Model has been verified for water surface ele-
vations throughout the Bay with data from a known hurricane surge. (Hurricane Carla was used
for this purpose.)
The Houston Ship Channel Model is the most important physical model from the stand-
point of water quality studies of the Galveston Bay system. This model is a very large fixed-
bed model designed principally to study sedimentation In the Ship Channel, and related
maintenance dredging. The model includes most of the Bay system, all the Houston Ship Channel
424
Entrance
Model
,:500
.:100
19.91 min.
Houston Ship
Channel Model
1:600
1:60
19.25 min.
Surge Model
1:3,000
1:100
4.97 min.
-------
to above the Turning Basin, the important freshwater inflows and the most significant physio-
graphic features throughout the Bay. The model does not Include West Bay, but rather is
terminated on the west near the Galveston Causeway. Similarly, only a portion of East Bay is
included. The Ship Channel portion of this model has been verified with tidal amplitude,
salinity, and velocity data collected within the Channel during selected portions of a one-
year period. Verification within the shallower parts of the Bay proper has not been as
thoroughly carried out.
The Alpha and Gamma barriers (Figure 7.92) have been proposed to protect the developed
areas of the Bay from hurricane surge. They are provided with wide navigation gates that are
to be kept open in normal circumstances but closed to form a solid barrier whenever a hurricane
threatens. Concern has been expressed that the presence of the barriers, even when open, may
reduce the exchange of Bay waters with the Gulf so as to alter salinity profiles and to reduce
the assimilative capacity and hence the general quality of the system. The Ship Channel model
has been employed to study the distribution of contaminants (using dye releases) and the
•CALC III WLCI
Fig. 7.90 Entrance Channel Model—Waterways
Experiment Station.
425
-------
vv,
0 <
/
^
^
^^
/
/'
/
y
Fig. 7.91 Surge Model—Waterways Experiment Station.
426
-------
LOCATION MAP
SCALC5 IN FEET
Fig. 7,92 Ship channel model (Waterways Experiment Station)
showing proposed Gamma and Alpha hurricane barriers.
salinity distribution In the Bay with and without the proposed hurricane barriers. It has also
been used to study the Influence on currents and salinity patterns of natural and regulated
inflows and the proposed Cedar Bayou Plant diversion across Umbrella Point. Some documentation
of this work has appeared (Bobb and Boland 1970) and the remainder Is presently in preparation.
The results of the model operation will be briefly discussed, insofar as salinity is concerned,
in this section.
Two sets of Inflows were used In the model tests (Figure 7.93), the natural hydro-
graph for 1965 and the same hydrograph with projected 1980 regulation and a 3,200 cfs diversion
across Umbrella Point. Additional boundary inputs employed for the model studies were the
input tide shown In Figure 7.94 and a Gulf salinity of 32.5 ppt. This Input tide is the
average prototype tide at Pleasure Pier which was determined from several years record.
The effects of flow regulation alone on salinities as predicted by the model are
illustrated In Figure 7.95 which are annual time histories at various stations. Station
427
-------
24
I—
J32
o
?4o
o
V)
£48
I I li
INVERTED HYDROGRAPHS
TOTAL FRESHWATER INFLOW
INTO GALVESTON BAY
I 1 I I I I I I I I
-I960 REGULATED
HYDROGRAPH
-I96S NATURAL
HYDROGRAPH
I I I
u. 0
o
(A •%
0 *•
4 -
6 -
I8
^'"
1 1 1
.^
—
T-
-f
l_T>-1980 REGULATED U
•••• HYDROGRAPH
• 1965 NATURAL
HYDROGRAPH
INVERTED HYDROGRAPHS
TOTAL FRESHWATER INFLOW
J INTO HOUSTON SHIP CHANNEL
" 1 1 1 1 1 1 1 1 1 ' 1 1 1 1 1
b o
u.
0 8
24
S'32
= 48
1980 REGUUTED
HYDROGRAPH
I I I I
INVERTED HYDROGRAPHS
TRINITY RIVER PLUS
DOUBLE BAYOU
I I I I I I
I I I I I I I
Fig. 7.93 Inflow conditions—Galveston Bay Physical Model.
STATION T-A - PLEASURE PIER
-1
I
I
I
I
I
6 8 10 12 lit 16 18
TIME IN HOURS AFTER MOON'S TRANSIT OF 96TH MERIDIAN
20
22
Fig. 7.94 Input tide at Bolivar Roads used in physical
model experiments.
428
-------
OCT | NOV I DEC | JAN I FEB | MAR I APR I MAY | JUN | JUL I AUG I SEP
_ i i i Ti 1 fi i r I 1 T I 1
STATION TBO-4
20
15
10
5
I I I I I
I I I I I I I
I 1 I I I I I I I
5 -
30
25
20
15
I i i i i i i i i i
I I I I I I I I I I I I I
25
20
15
10
5
I I i I I I
J I I I I I T I I
. I STATION
I I I I I I I
J I I L 1
i i i i
STATION UGO-J*
I I I II 1*1 I I I I
I I I I I
0 20 40 6080 100 120 UO 160180200 220 2W 260 280 300320 3W 360
TIDAL CYCLE
1965 CONDITIONS
SURFACE
BOTTOM -
1980 CONDITIONS
SURFACE
BOTTOM
'U.S. BUREAU COMM. FISHERIES
SALINITY MEASUREMENTS 1965.
ENCIRCLED POINT INDICATES
INVERSION.
Fig. 7.95 Salinity measurements in physical model for natural
conditions (1965) and regulated inflows (1980).
After Bobb and Boland (1970). 1965 USBCF salinity
data from Pullen and Trent (1969).
429
-------
OCT 1 NOV I DEC t JAN IFEB j MAR | APR | MAY | JUN | JUL | AUG | SEP
I I I I I I I I I I I I I I I I
30
25
20
!5
10
30
j I
J I
I I I
I I I \ I I I I I I 1 i
I I I I I I I 1
i i i
20
15
10
5
25
20
15
10
5
0
"^S
STATION lit
t I I I I I I I I
I I I I
I I 1 I I I I I I I I
20
15
10
5
0
Till
I I I I
I I I I I I
STATION TBO-3
I I I I 1
I I *«U«d till
Fig. 7.95 Continued
430
-------
locations are shown in Figure 7.92. The effects may be summarized as slight, being most
manifest in the upper reaches of the Ship Channel.
During the 1965 water year, salinity data was collected by the Bureau of Commercial
Fisheries (Galveston) throughout the Bay (Pullen and Trent 1969). A number of USBCF stations
correspond to those monitored in the model and it is of interest to compare the salinity his-
tories from the prototype with the 1965 model predictions. Accordingly, the empirical data for
representative stations in Upper Galveston Bay, Trinity Bay and East Bay are plotted along with
the physical model results in Figure 7.95. Both surface and bottom measurements were obtained
but in most cases the difference did not exceed .5 o/oo. When a larger surface-to-bottom differ-
once existed, both measurements are plotted. (At Station UGO-4, the circled points indicate a
salinity inversion.) It is evident that the model results generally are in poor agreement
with empirical data. It is the opinion of the staff of the Waterways Experiment Station that
the 1965 inflow hydrographs used for the model tests did not account for all of the freshwater
inflow to the bay complex, especially to Trinity Bay during the February-March period. They
note that this same problem has been encountered with other estuary models, and in every case
where the inflow hydrographs have been refined to the point of accounting for all of the fresh-
water Inflow, what first appeared to be a discrepancy between model and prototype salinities
was resolved. They did not recommend such a refinement in the case of the Galveston Bay
barrier studies, since the results desired from the model were the changes in salinity caused
by the barriers, rather than the absolute salinities with barriers installed for specific
inflow conditions (Simmons 1970).
The effects of the planned Alpha and Gamma hurricane barriers on salinity were
determined for the 1980 flow conditions. Examples of these model results are given in
Figures 7.96 and 7.97, respectively. The effect of either barrier appears from these results
to be slight compared to the natural fluctuation in salinity throughout the Bay.
431
-------
i i i r r i i i i i i t i
STATION TM-k
I I I I I I 1 I I
I I I I 1
IS
10
is
10
1 I I I I I I I I I I I I I I
STATION TW-«
J5
so
zs
to
IS
10
I I I I I i I I I I I I I I I I
20
IS
10
5
JO
IS
20
IS
10
I I I
i i i r
STATION MSO-J
I I I I I
I I I I I I I
I I I
I I I I I I J I III
0 JO W 60 SO 100 110 1*0 160 1(0 2002202*0 2*0 2(O 300 320 1*0 MO
TIOAL CYCLE
WIIHOUT BAM 1C*
SIMFACE
tOTTOH
WITH URI> I CD
SURFACE™"••
BOTTOM
Fig. 7.96 Salinity measurements from physical model for
1980 inflows, showing effects of Alpha Barrier.
After Bobb and Boland (1970).
432
-------
oci , MOV i etc i jut ,nt , m» . «n> , mt, jmi , jui , tut ,str
i i i i i i i
}S
JO
IS
20
IS
10
t 5
I I 1 I
I I I I I I I I I I I I 1:1!
1 I I I I I I I I I I I I I I I I
SI»tlOM 7
PNJ
0 20 *0 to SO 100 110 1*0 1*0 1(0 200 220 2W 1*0 280 500 320 J*0 350
110*1. CYCLE
WITHOUT IMKICIt
SIMFACC
ntTOM
WITH B1RKICR
SURFACE
Fig. 7.97 Salinity measurements from physical model for 1980
Inflows, showing effects of Gamma Barrier. After
Bobb and Boland (1970).
433
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REFERENCES
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studies In San Francisco Bay. Proc. ASCE. 92. No. SA 5 (October), pp. 23-45.
Bobb, W. H., and R. A. Boland, Jr., 1970: Galveston Bay hurricane surge study; Report 2:
Effects of proposed barriers on tides, currents, salinities, and dye dispersion for
normal tide conditions; Report.3: Effects of Plan 2 Alpha and Plan 2 Gamma barriers
on tides, currents, salinities, and dye dispersion for normal tide conditions.
Tech. Rep. H-69-12, U. S. Army Waterways Experiment Station, Vicksburg, Miss.
Dunce, R. E., 1967: A discussion and tabulation of diffusion coefficients for tidal waters
computed as a function of velocity. Tech. Paper No. 9, Chesapeake Technical Support
Lab., FWQA, Annapolis, Md.
Carter, Luther J., 1970: Galveston Bay: Test case of an estuary in crisis. Science. 167.
20 February, pp. 1102-1108.
Consoer, Townsend and Associates, 1968: A comprehensive study of the waste treatment require-
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San Jose, Calif.
Copeland, B. J., and E. G. Fruh, 1970: Ecological Studies of Galveston Bay. Final Report to
Texas Water Quality Board, Galveston Bay Study Program, Austin, Texas.
Curington, H. W., D. M. Wells, F. D. Hasch, B. J. Copeland, and E. F. Gloyna, 1966: Return
flows—Impact on the Texas Bay Systems. Report to the Texas Water Development Board,
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DiToro, Dominic M., 1969: Maximum entropy mixing in estuaries. Proc. ASCE. £5, No. HY 4 (July),
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Durum, W. H., and W. B. Langbein, 1966: Water quality of the Potomac River Estuary at
Washington, D. C. Potomac River Studies, Geological Survey Circular 529-A.
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Feigner, Kenneth D. and Howard Harris, 1970: Documentation report, FWQA dynamic estuary model.
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Gameson, A. L. H., and I. C. Hart, 1966: A study of pollution in the Thames Estuary.
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Gloyna, E. F., and J. F. Malina, Jr., 1964: Galveston Bay water quality study—Historical and
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Harder, J. A., 1957: An electric analog model study of tides in the Delta Region of California.
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Hetling, L. J., 1968: Simulation of chloride concentrations in the Potomac Estuary. CB-SRBP
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Kaiser Engineers, 1969: San Francisco Bay Delta water quality control program, final report.
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Ketchum, B. H., 1951: The exchange of fresh and salt water in tidal estuaries. J. Marine
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Kneese, Allen V., and Blain T. Bower, 1968: Managing Water Quality; Economics. Technology,
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Lager, John A., and George Tchobanoglous, 1968: Effluent disposal in South San Francisco Bay.
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study. Proc. ASCE. 94, No. SA 5 (October), pp. 809-827.
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Odum, H. T., and B. J. Copeland, 1969: Coastal ecological systems of the United States.
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Or lob, G. T., R. P. Shublnski, and K. D. Feigner, 1967: Mathematical modeling of water quality
in estuarial systems. National Symposium on Estuarine Pollution, Stanford Univ.
Pence, George D., John M. Jeglic, and Robert V. Thomann, 1968: Time-varying dissolved-oxygen
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Program Staff, 1969: San Francisco Bay-Delta Water Quality Control Program. Final Report,
Abridged Preliminary Edition.
Pullen, E. J. and Lee Trent, 1969: Hydrographic Observations from the Galveston Bay System,
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Reid, R. 0., and B. R. Bodine, 1968: Numerical model for storm surges in Galveston Bay.
Proc. ASCE. 94, No. WW 1 (February), pp. 33-57.
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Smith, Ethan T., and Alvin R. Morris, 1969: Systems analysis for optimal water quality
management. Jnl. WPCF. 41. 9 (September), pp. 1635-1646.
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Ship Channel, Houston, Texas. Geological Survey, Water Resources Division, Open
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Stommel, Henry, and Harlow G. Farmer, 1952: On the nature cl estuarine circulation.
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Stover, J. E., W. D. Bergman, and R. J. Huston, 1971: Mathematical Modeling of Thermal Discharges
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Texas Water Development Board, 1968: The Texas Water Plan Summary. Austin, Texas.
Thomann, R. V., 1962: The use of systems analysis to describe the time variation of dissolved
oxygen in a tidal stream. Doctoral Thesis, New York University.
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TRACOR, Inc., 1970b: Water quality analysis of the Cedar Bayou Generating Station. Document
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Water Pollution Research Laboratory, 1964: Effects of polluting discharges on the Thames
Estuary. Water Pollution Research Technical Paper No. 11, Her Majesty's Stationery
Office, London.
Water Resources Engineers, Inc., 1968: Hydrologic-water quality model development and testing,
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437
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CHAPTER VIII
BIOLOGICAL MODELING IN ESTUARIES: A NOTE
G. J. Paulik
The preceding material in this report documents considerable progress in modeling the
dynamic behavior of physical and chemical parameters in estuaries. In this section we examine
the feasibility of extending these numerical modeling techniques to simulate the ecological
processes which are so Important in determining man's commercial, recreational and aesthetic
use of estuaries.
1. CHARACTERISTICS OF ANIMAL COMMUNITIES IN ESTUARIES
Fish and birds are the most visible members of the animal communities associated with
estuaries and have received the most attention from both the general public and the management
agencies. Few nations of the world have major marine fisheries so closely associated with a
system of estuaries as does the United States. About two-thirds of the economic value of the
commercial catch and most of the marine sport catch is composed of species which spend at least
part of their lives in estuaries (McHugh 1966). Except for the tuna fleet, the United States
has no true high-seas distant-water fishing fleets.
Estuaries play a variety of critical roles in the life histories of important species
of fish and shellfish. Some sessile animals are permanent residents of the estuaries, while
the anadromous species pass through the estuaries during migrations to freshwater spawning,
rearing and maturing areas. Other species use the estuaries as spawning and rearing areas.
Table 8.1 (from McHugh 1966) lists commercially exploited species which are estuarine-dependent.
Four of the top five commercial species by value (shrimp, salmon, tuna, oysters and menhaden)
landed by the U. S. flag fleet are closely associated with estuaries. The role of the estuarine
environment in the life history of each of these four species is described below In order of
species Importance.
Penaeid shrimp breed in the sea at varying distances from land. A series of early
life history stages, nauplil, protozoea and my sis, are completed as the larval shrimp move from
the sea toward the estuary. The estuarlal stage of life begins about three to five weeks after
hatching and lasts from two to four months. The juvenile shrimp then return to sea where they
are recruited into the fishery and complete their life cycle.
Salmon deposit their eggs in the gravel of freshwater streams, rivers or lakes with
upwelling areas. After hatching, the young fry may spend anywhere from a few weeks to several
years in fresh water. The downstream migrants, either as fry or as smolt, pass through the
estuaries where they often linger for extended periods of time. High biological productivity
438
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TABLE 8.1
ESTUARINE-DEPENDENT SPECIES TAKEN IN
UNITED STATES FISHERIES (After McHugh 1966)
Coupon Names
Alewlves
Catfish
Croaker
Drums
Eel
Flounders
Garfish
Gizzard shad
Hickory shad
Hogchoker
Menhaden
Salmon
Scup or porgy
Sea bass (Atlantic)
Sea robin
Sea trout or weakfish
Shad
Silversides
Smelt
Spot
Striped bass
Sturgeon
Swellflsh
Tautog
Tenpounder
White perch
Yellow perch
Crabs
Horseshoe crab
Shrimp
Clams
Mussels
Oysters
Periwinkles
Bay scallop
Terrapin
Turtles
Bloodworms
Sandworms
Scientific Names
Alosa pseudoharengus. A. aestivalts
Ictalurus spp.
Mieropogon undulatus
Pogonias cromls. Sciaenops ocellata
Anguilla rostrata
Psuedopleuronectes americanus. Paralichthys spp.,
Platlchthys steJLlatus
Lcpisosteus spp.
Dorosoma cepedlanum
Alosa mediocris
Trinectes maculatus
Brevoortia spp.
Oncorhynchus spp., Salmo salar. Salmo gairdneri
Calamus spp., Stenotomus spp.
Centropristes strlatus
Prtonotus spp.
Cynoscion spp.
Alosa sapidisslma
Menldia spp.
Families Atherinidae and Osmeridae
Leiostomus xanthurus
Roccus saxatilis
Actpenser spp., Scaphirhynchus platorhynchus
Sphaeroides maculatus
Tautoga onitis
Elops saurus
Roccus americanus
Perca flavescens
Callinectes sapidus. Cancer magister. Carclnus maenas
Limulus spp.
Penaeus spp., Xlphopenaeus spp.
Cardlum corfals, Saxidomus nuttalli. Protothaca stamtnea,
Mercenarla mercenarla. Mya arenarla
Mytilus spp.
Crassostrea spp., Qstrea lurida
Littorina spp.
Aequlpecten irradians
Malaclemys spp.
Various species
Family Glyceridae
Nereis spp.
439
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in estuaries promotes rapid growth of the juvenile salmon during this period. While some stocks
of salmon spend their entire life in an estuary or adjacent waters, it is more common for salmon
to undertake extensive ocean migrations during which they may travel several thousand miles.
The adult spawners must traverse the estuaries to return to their natal streams.
Oysters are estuarine residents and oyster farming is practiced extensively in U.S.
estuaries. Except for the early free-swimming stages of their life which are usually completed
in two or three weeks or less, oysters are sedentary. The larval stages of the oyster are
specially sensitive to estuarine water quality. Oyster biologists believe that poor water
quality has already caused sharp declines in certain species of oysters. One species so
affected is the native oyster Ostrea lurida in the State of Washington, which is highly suscep-
tible to pulp mill waste matter.
Menhaden of the genus Breyoorita use the extensive estuarine systems along the
Atlantic and Gulf states for nursery areas. Juvenile menhaden spend the major part of the
first year of life in estuaries. Larval and juvenile menhaden generally undergo a series of
metamorphoses in the estuaries. Their feeding habits change from-relying heavily on zooplank-
ton as a primary food source to direct utilization of diatoms and dinoflagellates. The relative
abundance of these two types of phytoplankton varies considerably from estuary to estuary, and
from season to season within a single estuary.
The foregoing indicates the extent to which estuaries are related to the recruitment
process in fisheries. Conditions in an estuary can be critical in determining the abundance
of entering year classes. Because of the requirements for the type of environment provided by
an estuary of the early life stages of some highly mobile species, transient conditions in an
estuary one year may determine commercial catches several years later, hundreds or even thou-
sands of miles away. Long range migrations complicate greatly the use of such tools as benefit-
cost functions in the economic evaluation of either enhancement or deterioration of water
quality in a specific estuary. The opportunity for biological modeling is perhaps greatest
for these Important or "key" species since a large backlog of biological information on the
details of their life histories does exist.
The biological communities in estuaries depend critically on the timing of such
factors as the nutrient and salinity cycles, and their life histories are synchronized with
a particular temporal sequence of a series of events. Some of these events may be biological,
others physical. Hedgpeth (1966) says
From this sequence of events in several estuaries it is obvious that analysis of a
factor such as gross photosynthesis by itself cannot indicate what may really be
going on in an estuary; indeed the same values for different times of the year may
represent different phases of a complex system. There is obviously no shortcut to
understanding an estuary; the spring and early summer diatom crop may depend on
the summer dinoflagellate crop, and vice versa, and the copepods may depend on
this alteration of various dominant types of phytoplankton throughout the year,
while the fish may depend on the interrelations of the phytoplankton and copepods
or influence the phytoplankton directly if they are herbivores, and consequently
have some effect on the next cycle of phytoplankton.
The high natural productivity of estuaries is often augmented by various types of
fertilizing agents introduced by man. The ultimate effect of different concentrations of
nutrient material on the productivity of fish stocks is not known except in the most general
sense. Existing knowledge of biological processes in estuaries is far from adequate for many
trophic levels. Most of the available information is concentrated either at the level of the
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phytoplankton which form the base of the food chain, or at the level of top predators, such
as fish or birds.
Much of the phytoplankton data is concerned with the relation between nutrients and
productivity, and much of the fish data with the demographic characteristics of particular
stocks. Both types of information are available for certain populations of oysters. The func-
tion and structure of intermediate links in the food chain are poorly understood. Quantitative
information is almost nonexistent on such crucial factors as the basic population dynamics of
herbivorous and carnivorous zooplankton in estuaries, the kinetics of estuarine bacteria,
and trophic interactions involving the organisms of benthic communities. Even for such birds
as ducks and geese which have been studied extensively, it is not possible to guess the popu-
lation consequences of eliminating the estuarine environment.
From the biological point of view the essential characteristics of the estuarine
ecosystem are:
(1) high organic productivity;
(2) extreme fluctuations in both species composition and biomass with season;
(3) animal communities adapted to, and dependent upon, the precise sequencing of events in
major chemical, physical and biological cycles;
(4) complex food webs whose structure and controlling mechanisms are poorly understood;
(5) unique environmental factors essential for the spawning and rearing of juvenile stages
of most of the nation's important commercial and marine sport fish; and
(6) resident animal populations dominated by benthic mollusks.
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2. MATHEMATICAL MODELS OF BIOLOGICAL COMMUNITIES
Entomologists and fishery scientists have constructed the most realistic mathematical
models of animal populations (Watt 1966). Generally, these models have emphasized the life
history of single species and have not incorporated either interaction with other species or
spatial distributions.
Most published models of complete animal communities have consisted of a set of
simple differential equations which is used to represent energy transfers between grossly
defined trophic levels. The following system of differential equations is given as typical of
those employed to represent production of plants and animals per unit area of surface. Note
the linear food chain assumed (phytoplankton - zooplankton — forage fish - carnivorous fish) .
The dot indicates a time derivative.
P - P(ClITg(N) - C2J
Z - Z(c,.P - cfiFl - «
Fl ' F1(C7Z ' C8F2 '
J -
:7>
Cg)
F2 ' F2(C10F1 - ell>
N - c12f (t) - c13P + c14Z
where P is phytoplankton, Z zooplankton, Fj forage fish, and F£ carnivorous fish.
(Common units for these entities are calories, grams of carbon, or grams of dry or wet organic
tissue). Here N represents general nutrient abundance, e.g. average phosphate concentration
assuming some fixed ratio between phosphorus and other elements; g(N) is a positive, non-
decreasing and asymptotic function of N while f(t) is a function describing the changing
rate of nutrient import with season or exogeneous factors. The c^ for i - 1 ..... 14 repre-
sent numerical constants with appropriate dimensions. For example, c. is the grazing
coefficient or instantaneous rate of removal of phytoplankton per unit of zooplankton. I and
T are exogeneous variables representing incident radiation and temperature, respectively.
These are usually cyclic functions of time.
The validity of models of this type rests upon the following assumptions:
(1) The trophic levels specified can be defined and measured; matter or energy flows only
along the pathways indicated.
(2) All important organisms are included in the model.
(3) Growth and reproduction can be combined in general production coefficients and natural
mortality and respiration in general loss-rate terms.
(4) One nutrient acts as a controlling factor or meaningful index in the ecosystem.
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(5) The form of the functional relationship between productivity and the physical factors is
approximately correct.
(6) The ecosystem being studied can be treated as an isolated and homogeneous unit.
Methods of solving these equations range from use of analog computers to various
ad hoc procedures that arbitrarily assume existence of some sort of steady-state conditions.
We are not aware of any studies that include a truly rigorous statistical comparison between
results predicted by this type of model and field or laboratory observations. There have been
many crude comparisons between values calculated from ecological models and observational or
experimental data. However, because of serious deficiencies in sampling and data collection
and simplifications introduced to render the system of equations more tractable, these com-
parisons cannot distinguish between different, remotely plausible hypotheses of ecosystem
behavior.
Certainly, the past accomplishment of ecological models of this kind is to have
provided insight into the organization and functioning of ecosystems; it is doubtful whether
some of the present concepts could have been developed in the absence of these models. The
models have served also as a fertile source of ideas for laboratory experiments and the design
of field sampling programs. The models have further served the usual purpose of mathematical
models in synthesizing fragmentary studies and providing a focus for diverse experimental work.
One of the most successful practioners of this type of modeling is Riley (1963, 1965),
who examined steady-state solutions for models of biological productivity and applied these
solutions to field observations taken under quasi-steady-state conditions in different regions
of the ocean to solve for model parameters. Riley's model seems to provide useful esti-
mates of steady-state concentrations of phytoplankton, zooplankton and phosphate. Steele's
1958, 1961 and 1965 papers, as well as that of Dugdale and Goering (1967) should be consulted
for further variations and extensions of this general approach. Parker (1968) modeled the
response behavior of biological productivity in Kootenay Lake to a reduction in the amount of
inorganic phosphate pollution from a fertilizer plant located near the Kootenay River above
the lake. Parker integrated numerically on a digital computer simultaneous differential equa-
tions representing rates of change in phosphate concentration, algae concentration, cladoceran
density, and kokanee standing crop. Seasonal changes in river discharge and phosphate concen-
tration, lake temperature and phosphate concentration, and photoperiod are included in his
model. Parker does not claim a high degree of realism for the results he obtained.
A number of large-scale biological modeling projects have begun recently. Many of
these projects may involve concurrent field and laboratory studies, but at the heart of each
is a large computerized digital simulation model. The use of simulation is becoming so wide-
spread that there can be little doubt that it will be a standard ecological research method
in the future.
The.largest ecological simulation modeling efforts in the United States are currently
supported by the National Science Foundation as part of the U.S.'s participation in the Analy-
sis of Ecosystems Integrated Research Program of the International Biological Program. Each
of the main modeling studies is associated with a particular biome. Studies of the Grasslands
blome and Deciduous Forest biome are now receiving major funding. Four others, Coniferous
Forest, Desert, Tropical Forest and Tundra biome studies, are in various planning stages.
N.S.F. also is funding a modeling study on the biological productivity of upwellings in the
sea. No aquatic or estuarine biome studies are being supported on a scale comparable to the
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Grasslands or Deciduous•Forest biome studies. However, many of the current and proposed
modeling efforts are directed at developing basic conceptual approaches to ecosystem modeling
and modeling methodology, the N.S.F. programs are extremely broad and coordinate various
types of field work including experimental manipulation of ecosystem components with labora-
tory studies and computer modeling and simulation. Research in each of the biome studies is
classified into three main categories: major site studies, process studies and biome-wide
studies.
Currently, there are several numerical modeling studies of lakes in the United SI
and in Canada. One of the most ambitious of these, the total systems study of Lake Wingra
a multidisciplinary group at the University of Wisconsin (Watts and Loucks 1969), is being
incorporated Into the Deciduous Forest biome program as a major site study.
A particularly intriguing model-building exercise is that of the Environmental
Systems Group of the Institute of Ecology at the University of California at Davis. This
group is attempting to construct a simulation model of the human society and environment of
the State of California (Environmental Systems Group 1969).
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LAKE WINGRA AQUATIC ECOSYSTEM
OUTPUTS
INPUTS
ENERGY
RADIATION
(LIGHT)
THERMAL
MECHANICAL
(WIND)
FIXED ORG.
MATTER
PRIMARY
PRODUCERS
ZOOPLANKTON
FISH
MAN
BENTHOS
I. ALGAE
II. AQUATIC
MACROPHYTES
DISSOLVED
NUTRIENT
POOL
NUTRIENTS
NITROGEN
PHOSPHORUS
Fig. 8.1 A compartment model showing (1) the major inputs, (2) the pools of plants,
animals, dissolved nutrients and detritus, and (3) the major outputs of an
aquatic ecosystem. After Watts and Loucks (1969).
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3. . USE OF NUMERICAL MODELS TO CONSERVE, EXPLOIT
AND EXTERMINATE ANIMAL POPULATIONS
The most extensive and by far the most fruitful applications of numerical modeling
in applied ecology have been at the population level. Boiling (1963, 1964, 1966, 1968, 1969)
provides a comprehensive introduction to computer modeling of such basic biological processes
as predator-prey relations, and in addition gives key references to the development and the
application of strategies to control populations of insects harmful to man.
More relevant to the types of biological modeling problems associated with estuaries
are the modeling techniques developed to help provide rational guidelines for the harvesting
of marine mammals and fish. The more traditional models and approaches to determining the
"maximum sustained yield" of an exploited animal population are explained In detail in the
books by Beverton and Holt (1956), Rlcker (1958, 1966), Gulland (1969) and Watt (1968) and
by many contributors to the volumes edited by Watt (1966), Le Cren and Holdgate (1966) and
Van Dyne (1969). The usual analyses attempt to maximize the annual rate of ha rves table
biomass or value-adjusted biomass as a function of the mortality rate generated by harvesting
and of the earliest age exploited. More sophisticated analyses consider the possibility of
differential harvesting rates on various population components, e.g., the use of age-specific
or sex-specific exploitation techniques. The harvest of Pribilof Island seals is taken pri-
marily from three- and four-year-old males unable to acquire a harem and therefore no longer
essential to the population. The unqualified success of the internationally supervised manage-
ment of the seal herd is due in large part to the development of a mathematical model which
realistically mimics seal population dynamics.
Existing models usually employ historical time-series of demographic data on popula-
tion fecundity, growth, and mortality to establish steady-state management programs. These
models have been successful primarily when dealing with a single dominant species under envi-
ronmental circumstances that dampen excessive variability in recruitment.
There have been a number of large-scale and complicated simulation models constructed
to solve biological or economic problems in the management of exploited fish stocks. Up to
the present, virtually all of the realistic simulation models have been so directed at solving
specific problems that they have little general interest. Extensive bibliographies are given
by Newell and Newton (1968) and by Paulik (1970).
Royce, Bevan, Crutchfield, Paulik and Fletcher (1963) constructed a simulation model
to represent a comnon estuarine management problem, that of harvesting anadromous species as
they travel through an estuary. They investigated the economic and biological consequences
of various schemes for restricting the entry of gear into the net fisheries for salmon in
northern Washington State waters. Four species of salmon were represented in this model and
their rates of travel and routes of migration were determined by an analysis of catch data as
well as from tagging experiments. One of the basic features of the model was the management
decision algorithm which simulated the regulation of the fishery by the International Pacific
Salmon Fisheries Commission to control the rate of exploitation and also the division of the
catch of sockeye salmon and pink salmon between Canada and the United States. Cost functions
for three types of gear, purse seines, gill nets and reef nets, were developed and potential
economic benefits of various amounts of gear reduction were estimated using the model.
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4. RECOMMENDATIONS FOR BIOLOGICAL MODELING IN ESTUARIES
In the Immediate future the most useful numerical models of biological phenomena in
estuaries undoubtedly will be concerned with certain "key" species whose basic life charac-
teristics are fairly well known. While it is clear that models capable of representing the
dynamics of entire communities of animals in estuaries will evolve in time, major research
efforts comparable to the National Science Foundation's biome studies will be essential for
their development. It is almost impossible to estimate the length of time required for the
development of realistic and predictive general estuarine ecosystem models. Certainly, much
depends upon the possibility of "breakthroughs," especially in the area of methodology in the
Analysis of Ecosystem Integrated Research Program. The feasibility of a major Estuarine biome
modeling program should be investigated.
4.1 KEY SPECIES MODELS
Although the immediate feasibility of total ecosystem models is questionable,
extremely useful simulation models of certain important estuarine animals could be developed
using existing data and techniques. Considering their potential usefulness, the lack of
interest in such models is surprising. The following steps are necessary to develop a simu-
lation model of important estuarine species:
(1) The general characteristics of life histories have to be determined and the major predator
and prey species at different life histories established.
(2) Basic biological processes in the life cycles have to be modeled. Data from isolated
individual experiments should be synthesized by means of mathematical models. The types of
processes of special importance to estuary-dependent species include:
(i) synergistic physiological responses as a function of simultaneous exposure to
multiple stresses at different levels of critical parameters such as temperature,
dissolved oxygen, turbidity and salinity;
(ii) spawning requirements, and tolerance ranges and optimums for chemical and
physical variables;
(iii) larval behavioral responses to flow conditions and to such factors as dis-
solved oxygen and total dissolved solids;
(iv) migratory behavior and effect of environmental gradients on movements;
(v) growth response as a function of quantity and quality of nutrient supply under
different environmental conditions; and
(vi) normal mortality rates and mortality response to environmental pollution.
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Many of these processes would be expected to vary with intrinsic characteristics of the animal.
Thus, the model should include population structural characteristics such as age composition,
sex ratio, and spatial and temporal segregations of identifiable groups.
(3) Submodels of fundamental biological processes are combined in life history structures and
with sectors containing hydrodynamic submodels to determine values for physical and chemical
parameters. The biological sector of the model would be used to forecast responses of popula-
tions of key estuarine species to alternative management policies.
One of the most important uses of an estuarine simulation model is simply basic
bookkeeping on duration and intensity ot exposure to various stresses for transient populations
and in a highly dynamic environment. Even if behavioral information is not complete, expo-
sure boundaries can be established. Calculations involving multistage reactions, e.g. sequen-
tial decision-making as occurs in directed migrations in which movement is affected by local
conditions, are practical only when using a computer simulation model.
Modeling can effectively complement biological surveys and long term baseline studies
in unpolluted estuaries. The interaction between modeling and biological observation should
accelerate the model development.
Modeling of key species should include the effects of events occurring outside the
estuary on the abundance of transient animals. For certain important species, extensions of
the model to include economic factors such as the values of different spawning areas in an
estuary should be straightforward. In the area of forensic ecology, legal arguments involving
water quality standards are far easier to arbitrate directly in terms of economically important
animals than in terms of surrogate measures. Of course, the first objective of such a modeling
effort should be to summarize life history data in a form that makes them easily manipulatable
and accessible to estuarine managers.
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REFERENCES
Beverton, R. J. H., and S. J. Holt, 1957: On the dynamics of exploited fish populations.
Min. Agr. Fish and Food (U.K.) Fish Investlg. Ser. II. J.9, No. 1.
Dugdale, R. C., and J. J. GoerIng, 1967: Uptake of new and regenerated nitrogen in primary
productivity. Llmnol. Oceanog.. 12, pp. 196-206.
Environmental Systems Group, 1969: A Model of Society. Institute of Ecology, Univ. of
California, Davis.
Gulland, J. A., 1969: Manual of Methods for Fish Stock Assessment; Part I - Fish Population
Analysis. F.A.O.
Hedgpeth, Joel W., 1966: Aspects of the estuarlne ecosystem. A Symposium on Estuarine
Fisheries. A.F.S. No. 3, pp. 3-11.
Rolling, C. S., 1963: An experimental component analysis of population processes. Mem. Ent.
Soc. Can., 32, pp. 22-32.
Rolling, C. S., 1964: The analysis of complex population processes. Can. Ent.. 96, pp. 335-347.
Rolling, C. S., 1966: The functional response of invertebrate predators to prey density.
Mem. Ent. Soc. Can.. 48, pp. 1-86.
Rolling, C. S., 1968: The tactics of a predator. Insect Abundance. Symposia of the Royal
Entomological Society of London, £, pp. 47-58.
Boiling, C. S., 1969: Stability in ecological and social systems. Symposium. 22. Brookhaven
National Laboratory, Upton, New York.
Le Cren, E. D., and M. W. Holdgate, 1966: The Exploitation of Natural Animal Populations.
Oxford, Blackwell Scientific Publication.
McHugh, J. L., 1966: Management of estuarine fisheries. A Symposium on Estuarine Fisheries.
A.F.S. No. 3, pp. 133-154.
Newell, W. T., and J. Newton, 1968: Annotated bibliography on simulation in ecology and
natural resource management. Working Paper, No. 1, Center for Quantitative Science
in Forestry, Fisheries and Wildlife, Univ. of Washington, Seattle.
Parker, R. A., 1968: Simulation of an aquatic ecosystem. Biometrics. 24. No. 4, pp. 803-821.
Paulik, 6. J., 1970: Digital simulation modeling in resource management and the training of
applied ecologists. Ecological Systems Research. B. C. Patten (Ed.). (In press.)
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Ricker, W. E., 1958: Handbook of Computations for Biological Statistics of Fish Populations.
Fish. Res. Bd., Canada, Bull. 119.
Ricker, W. E., 1966: Methods for Assessment of Fish Production in Fresh Waters. I.E.P. Handbook
No. 3, Blackwell Sc. Publ., Oxford and Edinburgh.
Riley, G. A., 1963: Theory of food-chain relations in the ocean. The Sea. Vol. II, M. N. Hill
(Ed.), New York, Interscience. pp. 438-463.
Riley, 6. A., 1965: A mathematical model of regional variations in plankton. Limnol. Oceanog..
Suppl. 10, pp. 2D2-215.
Royce, W. F., D. E. Bevan, J. A. Crutchfield, G. J. Paulik, and R. L. Fletcher, 1963: Salmon
Gear Limitation Jin Northern Washington Waters. Univ. of Washington Publ., Fisheries,
New Series 2(1).
Steele, J. H., 1958: Plant production in the northern North Sea. Marine Res.. Scot. Home
Department, _7» pp. 1-36.
Steele, J. H., 1961: Primary production. Oceanography. M. Sears (Ed.), Amer. Assoc. Adv.
Science Publ. 67, Washington, D. C., pp. 519-538.
Steele, J. H., 1965: Some Problems in the Study of Marine Resources. ICNAF Spec. Pub. 6,
pp. 463-476.
Van Dyne, G. M., 1969: The Ecosystem Concept in Natural Resource Management. New York,
Academic Press, pp. 325-367.
Watt, K. E. F., (Ed.), 1966: Systems Analysis in Ecology. New York, Academic Press.
Watt, K. E. F., 1968: Ecology and Resource Management: A Quantitative Approach. New York,
McGraw-Hill.
Watts, Donald, and Orie L. Loucks, 1969: Models for Describing Exchanges Within Ecosystems.
Institute for Environmental Studies, The Univ. of Wisconsin.
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CHAPTER IX
DISCUSSIONS
I. ESTUARINE HYDRODYNAMIC AND WATER QUALITY MODELS
THOMANN: I want to support very substantially Dr. Harleman's suggestion that we should get
on with looking at some of the sources and sinks. From my own point of view, I would say
the state-of-the-art of water quality modeling Is to the point where the hydrodynamics of
the system generally—now there are exceptions—are of less importance to us, in the sense
of our ability to understand what's happening, than the sources and sinks and the inter-
action between some of our variables.
I think we know very little about some of the interactions that are going on, the
order of the reaction rates, the linear versus nonlinear reaction systems. The distinction
between sources and sinks, and reactions here, I think, is well worth making. Feedback
effects, for example, we know very little about. Now they may be imbedded in some hydro-
dynamic system, but I think the state-of-the-art of describing the hydrodynamics of the
system is so much ahead of our ability to describe these interactions that we need much more
work in those areas. I think that when we get to the order of some of these reaction rates,
even in a first-order system, that we find, by and large in many applications in water
quality analysis, that the system is very responsive to the order of the reaction rate and
much less so to the details of the hydrodynamic system itself. Reaction rates occur, for
example, that are orders of magnitude greater than generally what was used in Dr. Harleman's
review here, and that may alter the way one looks at some of these profiles.
So I would certainly agree that we need to move on with some of these studies, to
take a look at sources and sinks, and interactions.
O'CONNOR: Are these hydrodynamic models really necessary to answer some of the quality
problems that we have?
CALLAWAY: In modeling the Columbia, what we're trying to do is solve this equation:
21 - -u H + *- (E 21) + S S
at 3x 3X x dx
You will notice that this velocity term is an input to the temperature model. We've gone
to a lot of trouble just to get the intertidal velocities.
O'CONNOR: Yes, that's what I mean. We've gone to a lot of trouble just to get one number.
Is one number that critical to the problem you have here?
CALLAWAY: Yes, I think so.
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O'CONNOR: We can use an equation like this to clarify a point, That E includes a lot of
garbage, a lot of things. You take the equations that you've worked with, and you lump
all of these effects together into the diffusion.
RATTRAY: The point being that those different things you lump together are lumped together
very differently for different river flows, or winds, and so on.
O'CONNOR: That's correct, and you can get an empirical correction of that E with the Q
plus the width, and so forth.
Seeing the relative importance of these things is one of the more important things
we have to do in an estuary, the relative importance of the hydrodynamics versus the
kinetics of the reactions, and 1, right now, am undecided as to how to pursue research on
this point. This is one of the big questions we face. 1 realize that these both are going
to go ahead simultaneously, but I would like to get some feeling for the relative weight
of the kinetics versus the hydrodynamics.
HARLEMAN: I think the point really boils down to the fact that in most of these water quality
models there are variable coefficients or knobs that one can turn, most of them related to
reaction rates. It's always possible with any model, even the most crude one, if you have
enough knobs to turn, that you can match a set of data. 1 think it's very important to
remove as many of the knobs as we can, so that the final ones which are left to turn are
really related to the ones that we have no other way of finding.by analysis. Really the
whole point of this discussion, as 1 see it, in my portion of Chapter II, is related to
removing one of the knobs, and that is dispersion. I think we should get on and twist the
knobs relating to everything else, but not twist that one, because when you twist that one
you change all the others. And if you are ever going to find anything fundamental about
such basic things as even the rearatlon rate in an estuary, we've got to have less knobs
to turn. Otherwise, we're just finding an adjustment of a multitude of parameters which
fits a set of data. If you change anything in the system then to try to predict, all you
can do is keep all the knobs the same.
THOMANN: But it turns out when you turn some of those knobs the system behaves quite dif-
ferently than when you turn other knobs.
HARLEMAN: Well, I'm just saying we should reduce the number of knobs. Let's reduce the
hydrodynamic knob.
THOMANN: No disagreement with that.
HARLEMAN: In your discussion of the tidal advective models where you get down to talking
about whether they should have four or fifteen time steps per day, it seems that you have
to be precise of what it is you are talking about in the advective term. You can't just
write down the conservation of mass equation with a dispersion coefficient and by any means
assume that that dispersion coefficient is a parameter which you can pull up, because it
depends on how you define all the other terms, and, as I have already said, the importance
of it changes as you change the definition of time scales.
JAWORSKI: On the Potomac we've got the advantage of applying both the Orlob or "real time"
model and the Thoaatm or "tidal average" model. One thing about the tidal-average approach
is that one can perform sensitivity computations very cheaply. In some of- our work we can
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tell If we have to go to a real-time solution just by varying the dispersion and testing
the sensitivity. This is very simple. So I think using the tidal-averaged model for the
DO-BOD reaction system, which has high rates, is at times a very uncomplicated method of
getting a fast answer and order of magnitude.
The opposite appears to hold for nutrient transport systems. We are working trying
to model nutrient transport on an annual flow cycle. We are running into trouble using the
tidal-averaged model over a range of flows while keeping constant the dispersion coefficient.
I think the applicability of either the tidal-averaged or the real-time model comes in in
what kind of answer you want and what you're willing to pay for it.
HORNE: In this quick sensitivity test, you are saying to turn the dispersion knob?
JAWORSKI: Yes. We'll run various tests with the Thomann model just to see how sensitive it
is to dispersion. On the work we just finished doing, we found for low flow range the
predicted profile was the most sensitive to the decay rate. The same thing was observed
using the Orlob approach. So there it didn't matter what you were using, a tidal-average
or a real-time model, you've got to know the reaction rates.
THOMANN: To reinforce that, in some of the algal modeling we've been doing, plankton model-
ing, we have growth and death rates of 2 per day, so we're up another order of magnitude
from the .2 or .3. The system tends to be almost completely dominated by what's going on
in those kinds of terms.
JAWORSKI: At high decay rates, we've varied the longitudinal dispersion in the average tidal
models from 1.0 to 6.0 sq. miles/day and obtained similar profiles, depending on the
hydrologic conditions. We get approximately the same answer in the real-time model. I
think in the BOD-DO sag relationship it's more important to define the reaction rate than
the hydrodynamics.
WASTLER: Yes, but, depending on your model (and I don't want to get into the merits of
various models), if you' don't know what the relationships are about the BOD decay rate and
the hydrodynamics, if you are just going to the curve-fitting and the knob-twisting routine,
what you don't know about your dispersion coefficient will appear in your BOD decay rate.
THOMANN: Not in the same way. Absolutely not.
WASTLER: Yes, but in an entirely unpredictable way.
THOMANN: You can't arbitrarily turn those knobs and expect to get exactly the same responses.
As I pointed out, the decay rate acts as a multiplier in these systems, not in the same way
as dispersion. I think the point is that the equations show there isn't an equal trade-off
in the solution of the equations with E as there is in K . I think they behave quite
differently with responses to changes in K as to changes in E . One interesting way to
look at that is to start from the conservative salt balance equation where K is 0, and
just put in a little bit and you get a fair amount of reaction response. Generally, the
responses are quite nonlinear with respect to that reaction coefficient. Quite nonlinear.
HARLEMAN: I think if you get into salinity regions it's always going to depend on dispersion
quite heavily. I don't think that on a one-dimensional basis you can really ever discount
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that. Only If you're in certain regions, I would agree with you. But, in general, you
cannot discount that.
THOMANN: No, I'm not discounting it. I'm saying that the solution response is relatively
insensitive to it, over a wide range of our choice of that coefficient. No, it definitely
has to be there, but I'm saying that a solution response in highly reactive systems is
relatively insensitive to that coefficient, and it may be worthwhile to trade-off that
time-dependent model for that coefficient.
HARLEMAN: I said in the conclusions of my chapter that there was a place in the future for
both of these models. Really, I think all we really need to be concerned with is that we
know what we are doing, and I can pee a very good place for calibration of the non-tidal
advective model using the real time model. In other words, you do a brief study using the
real tine models to determine some of the grosser effects of nontidal models. You do a
numerical study instead of a field study, which ought to be a gain in time and cost. So
there is a place for both.
BAUMGARTNER: Dr. Pritchard brought up the problem of disregarding the vertical eddy diffusivity
on the basis of what looked to be a uniform distribution vertically of salinity, and there
was some agreement that this was important. And what I want to do is ask, if possible, to
show some comparison of what kind of difference this might make in the Mersey or the James
River, if you have any examples, where the concentration would be an order of magnitude,
or 10%, or how much different if that assumption was embraced and if it was not.
PRITCHARD: Well, I can give you some examples which draw on biology rather than chemistry of
the environment. I don't think a model has been run for the Mersey or the James in which
one neglected the shear terms, or did not neglect them. But if you take, for example,
small organisms that are essentially carried by the water and consider this as a pollutant
or an indicator, that is, some component which naturally stays in one place or moves and
its movements are controlled to a good extent by the same processes that control the dis-
tribution of properties, and realize that if you did not have the shear term, the distribu-
tion that you observed couldn't possibly exist. An example of this is, and here I'll go
to a larger body, the Chesapeake Bay, which is in many ways just an enlarged James as far
as the terns that are important in it go, the hard-headed croaker, which is a major winter
food for striped bass. The croaker spawns off-shore, in the fall, just about the time when
we begin to get overturn in the temperature structure, and it seeks the warmer water, which
is now not at the surface but near the bottom, and as a consequence is brought up the bay.
These are little fingerlings, so long. They can't direct their migration up the bay. But
in two months after they are born, we find them in the deep holes right up where the
striped bass are wintering, and they're food. Now again it so happens If you compute
transport time of a particle which stays in that part of the structure that has' an upsteam
motion it gets there in two months. So these organisms are using this flow pattern to its
fullest in moving into a nursery area, besides their being food. In a one-dimensional
model you couldn't predict this.
BAUMGARTNER: I think that's an example of what Dr. Harleman said, that if it isn't one-
dimensional, don't try to use a one-dimensional model.
PRITCHARD: He might look at perhaps some experiments with tracer materials, remembering that
in Don's examples here essentially the upsteam extension from the source is about a tidal
excursion in some of the cases, where we'll find the upstream excursion of a pollutant to
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be many tidal excursions. Now I don't know how much of this will affect the local
concentration near the point of highest concentration, but it certainly affects a lot of
trends.
HARLEMAN: Those examples were in the nonsaline portion, and if you get into the larger
dispersive zones then you will find it more than one tidal excursion.
BAUMGARTNER: This is the kind of comparison I'm looking for, if we can provide something for
the record to show what the magnitude of our concern is, if we happen to be trying to
model concentration of some pollutant or some water quality parameter, whether it is an
order of magnitude or ten percent or whatever it is.
PRITCHARD: Let me say this then. I think that certainly the effects of the vertical shear
can be modeled by some sort of dispersion coefficient. The shape won't be exactly the
same as occurs, but it grossly can be modeled by some sort of dispersion coefficient. And
the question 1 would make is, just as in the case of the two-dimensional systems or one-
dimensional systems which are sufficiently uniform over a cross section or in the vertical,
where you find if you look at more detail in the time-dependent horizontal velocity field
you don't have to worry about the dispersion coefficient, if in the same way you look at
more details in the vertical structure, maybe you don't have to worry so much about some
of these diffusion terms.
RATTRAY: For a number of classes of estuaries, the longitudinal salinity distribution is
basically governed by the one-dimensional model, although in the local salinity change the
horizontal diffusion term is really a small part of the local salt balance. This means
that whereas you can do a reasonably good job of cross-sectional mean salinity determina-
tions with the one-dimensional model, other kinds of distributions that are not put in in
the same way, essentially over a cross section at the seaward end, but more local sources,
are not going to be nearly as well described in the one-dimensional model. This is a real
problem if you're talking about something other than salinity, or other than simple sources.
WASTLER: Would it be possible to get some criteria about under what conditions you would
use the one-dimensional versus the two-dimensional versus the three-dimensional, or at least
some guidelines as to how such criteria should be set up?
HARLEMAN: 1 think it's a question of availability at the moment. It isn't really ore of
just making a choice.
PRITCHARD: Right.
WASTLER: Well, certainly there is some criterion which makes you recognize that it's a
two-dimensional problem versus a one-dimensional problem.
HARLEMAN: If you have an embayment then obviously this is not one-dimensional, no matter
how you cut it. Just by looking at the geometry you can see that the major directions of
flow are simply not going to be defined by one dimension.
WASTLER: Certainly, but what we need is an explicit statement of that. This is a very
real and very critical problem, at least from my standpoint. When we are being asked to
evaluate certain types of models in certain types of estuarlne systems, it would be very
nice to have some kinds of guidelines to make a judgment as to whether there was a
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reasonable chance of a particular model actually being useful. It should be possible to
at least lay it out to such an extent that the additional information could be gathered to
make it quantitative.
RATTRAY: Well, you're asking really a very multipronged sort of question. In terms of
salt flux, I can give you a pretty good answer, but In terms of something that has a half
life of a few days, that's a much more complicated kind of a problem. For the kind of a
biological organism that has the behavior that has just been described, you'd have one set
of answers, and another set if you're talking about something with a different time frame
and spatial frame and scale distribution. An estuary you might have treated one-
dimensionally would have to be treated two-dimensionally, just because you are looking
at, in the same estuary, a different phenomenon. It is fairly complicated.
THOMANN: There are some situations where, even though there is an embayment involved, the
extent of tidal mixing is so great that for a particular water quality variable one is
studying it may be useful to consider the entire bay completely mixed, and look at the
dynamics of what is happening as far as growth-death. That is., physically it may have all
the features of a two-dimensional system, but for the particular problem you are looking
at, you might just get away with a simpler representation.
PRITCHARD: You don't really care about the horizontal-vertical variation in the water quality
structure. You care about what that does to the system. So you treat this as • mixed
bucket with inputs and outputs and what happens in the middle.
THOMANN: That's right, and what happens in the middle is where you bring to bear all the
knowledge you have on the dynamics of fluctuations, prey-predator relationships, and all
that.
PRITCHARD: And they are probably much more important than the differences you get out of
the hydraulic system.
THOMANN: Right.
LEENDERTSE: Actually, In a three-dimensional model, you need to know what are the largest
contributions of terms that go Into it, then you can make your selection. So if you have
very rapid decay rates, you need to develop them more. Hydrodynamically, in the horizon-
tal you can make a distinction in two ways, I would say. As soon as the flows are nut:
predominantly unidirectional, but the terms start turning around, then you can say you
need definitely a two-dimensional model. Also, you very likely need a two-dimensional
model when you have tidal flats. But to give you a meaningful list, we'd have to go, at
least I'd have to go, to a certain number of estuaries and see what their main character-
istics are.
WASTLER: What you Just said is perfectly satisfactory from the standpoint of what I was
trying to get at.
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PRITCHARD: I would like to make one other conment with respect to things that ought to be
done, and that is, again depending on where you are, your awareness of the concerns and
certain aspects of the problem vary. I've been working in a very large estuary, where, if
we look at its total assimilative capacity, the total capacity is quite large. I don't
find the problems for which any of these models would really be very pertinent. I can on
a very simple basis determine how much diluting water is available and we could make use of.
And the real pollution problems that we face are generally local in character, that is,
peripheral tributaries, small embayments, that are badly polluted because they don't flush
with the less-contaminated water in the larger bay. Where discharges go into the larger
bay, the problems have been to develop ways to make use of the diluting water. So we face
engineering design problems of how to most effectively use the capacity of the system, both
the diluting capacity and in some cases the capacity of the system from standpoint of non-
conservative losses of some of the material.
This is very true, say, in the case of some of the thermal inputs. The problem
with a big power plant like a nuclear plant is not this large-scale problem in which one
can show how the pollution distributes up and down the estuary in a one-dimensional sense
or even in a two-dimensional sense (where we are now looking at two dimensions vertically,
not horizontally). this is a relatively minor problem.. The heat is spread a hell of a
distance, but at very, very low temperatures like a few hundredths of a degree, up and
down the estuary. The problem is that in the tidal segment adjacent to the plant site
you've got potentially a pretty steady supply of about 90,000 cubic feet a second. If you
could make use of that, if you could design the input to take advantage of this, you can
get temperatures down to 4/10ths of a degree or 6/10ths of a degree due to dilution very
fast. For many of the problems that we've faced, our problems of design of intake and dis-
charge structures make use of the diluting water.
From the standpoint of site location for such systems, the desire is to find sites
which have potentially large diluting capacity. This means that narrow streams or the upper
reaches of estuaries aren't very good sites, but for large estuaries, well down in the
estuary may be a good site. An open coastline may be a good site if you make use of the
diluting water by proper design. In many cases the work of Don and other people on dis-
persion for horizontal jets becomes very important. The effect of longshore currents upon
the entrainment of momentum jets is a very important term which potentially has large dilu-
ting capacity, if you could make use of that. Also large tributaries from which you can
dissipate the heat from regions of very low temperature excess. In the thermal modeling
that I've been doing, I have found that these are the pertinent things and not the gross
or large-scale distributions.
THOMANN: Yes, but the point there is that the physics of heat are quite well understood.
I think here, while there may be a design problem associated with dilution effects to dis-
perse the effects of heat, it's that link between heat and what happens in the higher-order
ecological system that really is where we're hung up badly.
PRITCHARD: Yes, but the link is through temperature, not through heat. What I'm getting at
is that you want to dilute the heat.
THOMANN: But we don't know that. I think that's the problem. We don't know that
hundredths of a degree may not make a difference.
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PAULIK: It seems that one of the problems is that matter of basic objectives.
PRITCHARD: But it's still in terms of degrees, not BTU's.
HARDER: That hundredth of a degree might be the difference between a cloudy and a sunny
day. I'd like to suggest that in terms of the natural variations that this hundredth of
a degree may not be important.
THOMANN: In terms of this temperature excess problem, we ran a whole time series analysis
on temperature. Background variance that is generally unexplainable runs on the order of
at least a degree. All tests indicate somewhere on the order of a degree.
EDINGER: I'm surprised it's that Ipw.
PRITCHARD: Yes, I am too.
THOMANN: That's absolutely the last little bit of variability, that is completely
unexplainable.
EDINGER: You mean in terms of periodicities?
THOMANN: Yes. In terms of any periodicities. Just background noise.
PRITCHARD: We did a field study of the Chalk Point plant for something like a nine-day
period. Within that nine-day period, our best estimate of the background temperature
showed diurnal and tidal fluctuations for a total range of 9°. And we're looking at 12°
temperature.
HARLEMAN: And you're limited to 1%°?
THOMANN: That's why I brought up the mention of degrees. I understand some of the require-
ments are on the order of one degree, and background variability, strictly background noise,
is on the order of a degree.
PAULIK: I'm disappointed we haven't mentioned any of the biological criteria, because I
think that's the thing that generated all the current interest in temperature. If, as
Dr. Edinger mentioned, salinity is the most important factor determining the density
structure of an estuary, certainly the temperature is a critical thing in determining its
total biological characteristics. And the local temperature distribution is just exceed-
ingly important. Even though we get a lot of variability, especially on the east coast,
I think some of the west coast estuaries are very stable in terms of temperature structure.
Many of the organisms, especially the economically important organisms like oysters and
salmon, are keyed to the specific temperature, and their whole life cycle depends on this
temperature being realized within a certain season. Temperature controls the spawning of
oysters. Just a few degrees will trigger the oyster spawning. It certainly controls the
return of salmon to certain streams. I feel that this should be given some emphasis in
the report, especially the local effects of temperature below some discharge point, because
the importance of the local spatial distribution of temperature cannot be overemphasized.
EDINGER: You know, when we began in this heat business—sometimes I wonder why—about
eight years ago, we were pretty naive looking back now, but one thing was obvious at the
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time, and that was the fact that we might be able to come up with the best temperature
prediction methods In the world If we put enough money into it, 'but the one big job was
to have some method of getting this interpreted. Can the biologist give us some idea of
how much detail he needs In terms of a temperature distribution so that he can make that
interpretation? Then we can go ahead and make our predictions fit them. This, to me, would
have been the criterion. And after eight years of working on a number of studies in which
very competent biologists have been involved, I don't think we've met this criterion yet.
I'm not about, myself, to undertake a discussion of the effects of temperature on organisms.
PRITCHARD: Here again, it depends on where you are. It seems to me that this is not part of
the modeling. It's what you need to kLow In order to say what the effects of any discharge
are, and in order to get the criteria on what this discharge means. I don't think it's
part of the state-of-the art of modeling.
PAULIK: Well, I think this is where our modeling should be going.
PRITCHARD: Well, I agree, and I think the state-of-the-art is pretty damn poor.
PAULIK: It's pretty poor, but I don't think we're using the information that we have
available.
EDINGER: Let me ask the question this way. I heard this morning some estimate that some-
one wanted to know temperatures within .01 degrees, I hope centigrade. This is to do with
the data question we were discussing. My feeling is that we can hit temperatures within a
couple-of degrees depending on what you want for spatial detail and depending on what you
want for temporal detail. On temporal detail it becomes a function of your input data.
How good is your meteorological data, and how much experience have you had working with
this particular set of data? Now you take a guy and sit him down and let him run through
a set of data, once even making a forecast, then let him go back and make a hindcast. His
second time through on his forecast he'll be pretty darn clqse to those records. How good
is your boundary condition data? You see, the question here is what do you feel you need
for detail in terms of temperature distribution? I think you're going to have to admit
that you're going to have to answer this almost case by case, knowing the existing water-
body, knowing habitat areas, and things like this.
PAULIK: I think you have to look at the particular organism that's going to respond. You
might do things like make on-site tests of the water under different temperature conditions,
and possibly set up some sort of a physiological lab at the site. But then your model
should be able to incorporate that Information and use it to forecast.
PRITCHARD: Well, It seems to me that you don't have to Incorporate a biological model into
the temperature model. You predict the distribution of temperature, and then you take that
and predict the consequence of that with a biological model. Now, to give you an example
of a very simple way of approaching this, if we look at, say, the Chesapeake Bay, one can
look at Maryland and say that there are certain plants that raise the sectional mean
temperature or the temperature in the tidal segment opposite the plant site by four-tenths
of a degree. Now you want to know what that means. It displaces the time curve by that
amount, really. And one way to say what it means is to say, where do I have temperatures
like that? I've just got to move to the south end of the bay, the Virginia segment of the
bay, where the mean annual temperature is eight-tenths of a degree higher than the Maryland
section of the bay. And I've got a model.
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PAULIK: It's nice to have control. Right.
PRITCHARD: But the thing that we've imposed on those relatively small differences is the
fact that the annual temperature varies at one place during any month of the year, from
one year to another, by a range of six-and-a-half degrees.
PAULIK: Well, that's in this particular area. Maybe you have eggs, say, of some particu-
lar organism that are going to be down there for two months. So they integrate the
temperature essentially, and have a so many day-degree requirement to hatch. That will
integrate out a great deal of the variability that you're talking about. You might have
some sort of biological stabilizing thing.
PRITCHARD: Well, a warm year would be warmer somewhat, and a cold year would be colder some-
what. I've seen fourteen years out of a fifty year record where the average temperature
was a degree-and-a-half higher than the fifty year mean.
PAULIK: Well, maybe in that case you need the variability of temperature itself.
PRITCHARD: No, you don't. You don't change the time fluctuation whatsoever. You just
change the mean. But I think there are models that you could look at in nature.
PAULIK: Well, maybe we should set up some as control.
PRITCHARD: These kinds of studies have been going on. And at the end, you say "What did
you find out?" They say, "We found the environment is variable." And that the organisms
vary and the response varies, and that's about what they found out. It's been a very
disappointing experience over the years.
PAULIK: I'll agree with that, but I think we are beginning to accumulate the type of
physiological background that we need to begin to incorporate some of these factors into
the model. I see no reason why the model shouldn't include, or at least plan to include,
biological responses.
EDINGER: there is no reason why it can't. Because here's your temperature prediction and
it's not going to be Influenced by the organisms unless you put a whale in there or some-
thing. In other words, the temperature is independent in a sense that I don't think we
have to worry too much about it influencing the dynamics, except locally near a discharge
where it will stratify.
PAULIK: But, how do you know how much detail you'd like to put in until you try to put
some of this into a biological model and see what the response is, to help you determine
your criteria? This is what you need. Otherwise, it's going to be determined on a politi-
cal basis.
PRITCHARD: It doesn't have to be this particular model. It needs to be a model that takes
the temperature distribution and predicts what it means to the biota.
The Important thing in temperature is the time response. We can show for instance
that some of the organisms drawn through the condensers at Chalk Point are killed. There
is a certain fractional mortality of passing through that plant, which discharges to a
7,000 foot canal and takes two-and-a-half hours to discharge with virtually no temperature
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loss. So they're raised 11.4 degrees above ambient for two-and-a-half hours. Now that's
quite different than if they're raised 10 degrees above for four minutes. Then the dis-
charge undergoes very rapid dilution and temperature decrease. Where oyster larva are
killed going through the Chalk Point plant, in a plant that is designed to minimize the
time of contact, the oyster isn't killed.
PAULIK: Yes. I agree wholeheartedly.
PRITCHARD: These kinds of experiments are being run for that kind of simple model. It's a
fairly simple question you're asking. You're not asking what happens to the migration of
shore fish past a plant. That's a very much more difficult one.
PAULIK: You can, say, put an envelope of the minimum-maximum exposure, and put a time
factor in for a migratory fish. If he has to migrate by this plant within a certain period
of time, he takes the route that will minimize his temperature increase exposure, or he
takes the route that will maximize it. At least you can involve things like that, which
are really difficult to compute without models or without incorporating hydrodynamic factors
or temperature.
PRITCHARD: Well, we haven't had the expertise on modeling in the biological field that we
need.
ESPEY: Do you think the sources and sinks in the temperature model such as longwave and
shortwave radiation are fairly well-defined for estuaries? These empirical formulas that
have been studied on freshwater reservoirs, how about their application to estuaries?
EDINGER: I haven't gone into that problem explicitly, so I really can't tell you, except
we tm-.id be hiding ignorance on evaporation in the same empiricism in an estuary that we
do cu fresh water. I don't know, offhand, of any reason to believe that our rates should
be any different, except for the fact that we probably get longer wind fetches and higher
winds on a more open waterbody.
PRITCHARD: Gi- n a wind and an ambient temperature, including the second order effects,
humidity, etc you can predict the equilibrium temperature about as well there as anywhere
and you can prt- let the driving function there about as well as anywhere, and certainly the
surface cooling coefficient. I think now the uncertainties don't exist in that term,
because the distribution within the area that are concerned is so dominated by dilution.
If you're going to have a power plant that fits, that is, if it's designed so that it can
go there, you've got to dilute. You can't depend on cooling. So you find that you could
vary the cooling coefficient by fourfold and not get very significant differences in the
distribution of temperature. In most real cases we find that 80% of the heat is lost at
temperature excesses of less than one degree. So it doesn't matter much what size cooling
coefficient you take in the part of the plume that you are concerned with.
I'm agreeing with what John has said, that we're now not concerned with hundreds-
of-fold dilutions. We're being concerned with dilutions of the order of ten-fold. In
that range, the temperature distribution is dominated by mixing, not by cooling. You get
reasonable answers by neglecting cooling completely. And this is something that other
people are coming very rapidly to the same conclusion. For instance, Parker's report of
studies that have been done down at Vanderbilt, tells about systems in rivers where they
got their prediction almost as good as what they observed in the river by neglecting
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cooling completely. Mow, cooling has got to be the end product. It's sort of like
molecular diffusion. It's got to be in there, but it's in there at some place where it
doesn't matter as far as the predicting technique goes.
ESPEY: Well, in some of these shallow estuaries on the Gulf Coast, would cooling possibly
be more of a constraint?
PRITCHARD: My point is that when cooling becomes a constraint, the plant is too damned big
for that waterway. All you have to do is to compute the size required for an effective
cooling pond, and you'll find that the areas contained within high excess temperatures
are too large.
EDINGER: Well, though, there is another point in here. The way to double your rate of
cooling is to double your temperature excess. And cooling pond design depends upon what
temperature excess you want to operate your pond at above equilibrium. One way to get to
small areas, if you're working in an area where you have, say, a mean weekly equilibrium
temperature of 85 or 90°, is to operate your pond at 20° above, that, with a 15° condenser
rise, so that what's coming back Into the condenser is 5° above equilibrium. In this case,
you've doubled your temperature excess over that, which means you've confined your heat to
a very small area, and you're recirculating it. The question is: do you want to take a
shallow estuary and design it to be a cooling pond? If you're going to use it for that,
design it to be that; the plant will be much happier.
PRITCHARD: Yes, but the organisms won't. I think one of the biggest misconceptions in power
plant design that exists today is the concept that you gain by having regions of high
temperature in a natural environment because those are the regions that you lose heat
fastest. Because those regions are just going to turn out, with thousand megawatt plants,
to be too large for the regulatory authorities to accept. I can give you case after case
of the situation where you're talking about having temperature excesses, say, above 10°
or above 5°, of thousands of acres, as compared to, if you design the plant right, of
fifteen acres. And that-fifteen acres is because of dilution, not cooling.
PRITCHARD: I'm reminded of a study where we were concerned with a distribution around the
discharge, and we were concerned with predicting the distribution within one or two tidal
excursions, what the shape of the distribution was as it flipped around during the changing
tide. But this concern was limited to something like one or two tidal excursions in a
system that was maybe sixteen or twenty tidal excursions long. We weren't concerned with
tidal times when we were looking at scales of 0 - 180 miles, only a sort of central mean.
But we were concerned with tidal times when we were looking within plus or minus twelve
miles or so of the system.
WASTLER: I think part of it goes back to the same scaling problem that Dr. Harleman pointed
out this morning, with regard to the mathematical models. If you're using a broad-scale
model, where you are talking about annual changes, you can'1: use it to look at the small
scale variations, at least not with the same prototype data base. And the same thing with
hydraulic models. If you build a hydraulic model at such-and-such a physical scale that
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you can look at the broad scale phenomena, the scale simply Isn't right to look at the
outfall locations. But you could do that with the physical model if you built a model, say,
on the 100 scale.
HARLEMAN: The trouble is you have to build the whole estuary usually.
WASTLER: Right. So it lacks the flexibility that the mathematical modeling technique has.
RATTRAY: Well, you could do something. Some even fairly small-scale distributions may not
be affected by horizontal turbulence processes. In that case, if you scale the vertical
ones correctly, you don't really care what the horizontal ones aren't scaled correctly.
They aren't important.
PRITCHARD: Yes, that's the point I was trying to make.
PAULIK: I think biologists kind of like physical models, especially if you model the
local phenomena. You can apply your experience and observations to the model and get some
ideas of how an animal might respond to a particular set of conditions. I worked on the
Columbia River and several of the dam problems, and looking at the models there and the
locations of the fish ladders has always resulted, 1 would say, in much better design.
In fact, a couple of times we've gone to a model, after the ladder did not rate properly
and were able to diagnose the problems. Occasionally the problems had been such that when
the original design, for example, had spill gates located right next to a ladder and yet
the ladder did not rate properly, going back to the model you could see what was happening
.and by changing the spill pattern you could get the fish to respond and use the ladder very
nicely. The physical model seems to work with certain types of, say, local problems which
mathematical models perhaps can't handle. It's one way of simulating what's happened in
the prototype.
WASTLER: I think major structural changes, too, are something that mathematical models at
the present state-of-the-art can't handle as well as physical models. Again, it's right
back to the empiricism. You have to go on faith for every bit of it.
RATTRAY: Well, there's a difference between getting a qualitative answer and a quantitative
answer, too. You can get a very good qualitative answer even if it's distorted.
PRITCHARD: There's a lot of difference in being able to take the city fathers out and stand
beside the model and say, "Look, you put your outfall there, see the way the dye is going."
There is that kind of demonstration advantage, which probably isn't worth the cost unless
there is one of the models in existence.
LEENDERTSE: Don, I do not completely agree with you. You can do exactly the same thing with
the mathematical model. I have made movies of mathematical models; and they are very
instructive.
PRITCHARD: I would understand them. I'm not sure the mayor would.
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PRITCHARD: I have a comment about the terminology when we talk about a digital model or an
analog model. The model is the math. The model is the symbolism which you've developed,
the functional relationships you are going to consider to be important, in parametric form,
and then the computers are simply means of solving this. In one case you simulate the
functions by RC networks, and the other case you take the difference forms.
THOMANN: The distinction is really arbitrary, you solve the equation whether you use either
an analog or a digital.
RATTRAY: It would be much more reasonable to talk about one-dimensional models or linear
models.
CALLAWAY: Let me say one thing. • I think the Bureau of the Budget has indicated that they
would like to know what are the advantages of analog versus digital versus hydraulic models,
which one costs more and so on.
PRITCHARD: Tell them that that's the wrong question.
O'CONNOR: Absolutely, I think there's unanimity among us on that.
PRITCHARD: I've served on n committees which dealt with that same problem, and it's just not
the right question.
THOMANN: It seems to me that again we've heard several times this morning and this after-
noon the notion that there are a number of models around, and in fact there are really
relatively few, both from the hydrodynamic point of view and from the water quality modeling
point of view. Again, there are different methods of solution, but we really only have, in
the sense of a water quality model, one model that we can write down without too much diffi-
culty, and that's a multidimensional partial differential equation with advection and
dispersion, and that is about it. Now whether the solution of that equation feeds forward
into another equation, such as in the BOD and DO, that's conceptually a relatively easy
extension to make.
PRITCHARD: A few general comments. I would want to suggest that perhaps the requirement
for the development of conceptual models adequate to treat the problems requiring answers
is more important now than how one solves the mathematical formulation of the conceptual
models after they're obtained. No degree of sophistication in the use of the digital or
analog computers will correct deficiencies in the conceptual model. And further, I suggest
that the conceptual model to be employed fit the problem requiring solution.
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2. COMPUTATIONAL ASPECTS
LEENDERTSE: On the steady-state feedback thing, I wonder why you do a steady-state analysis.
You can go much faster through the variable time-dependent model than you can go through
the steady-state modeling. This is not the case when you have a small number, say thirty
sections of the river. If you have many more than that, like if you have a two-dimensional
model, you can approach the steady-state condition much faster with the time-dependent
solution than inverting all the matrices. In effect, inside the matrices you generally
have an iteration procedure to find the solutions. You should be very careful not to go
along this line, in my opinion. You could save a lot of money on computer time. Besides,
that is the way to also handle the time-dependent problem simultaneously.
THOMANN: I find that kind of hard to really grasp. What you are saying is that one can run
the time-dependent general feedback model faster? To reach equilibrium?
LEENDERTSE: Yes, you can run it faster where you have more ways of arranging it. I have
worked very extensively with multioperational schemes by which you do not have one repre-
sentation of the finite differences, but two. I have always worked with two, but you do
not necessarily have to use two. That approaches the steady-state condition much faster
than just inverting matrices.
THOMANN: Yes, but we don't invert the matrices.
LEENDERTSE: Well, a relaxation technique is, in fact, a time-dependent solution technique.
THOMANN: I see.
LEENDERTSE: Also you can include the nonlinear processes. The decay operation in, let's say,
BOD-DO reactions is not really linear. Even if you assume it is linear there are stops in
it. You can get negative concentrations in BOD as well as in DO, so you have to stop those
reactions. And that prevents you from using those types of models except only in those
cases where you are also assured that you don't have anaerobic conditions.
JAWORSKI: One comment concerning feedback. I think we are going to see more parameters in
water quality than BOD and DO. I think we are going to need a model that will run simul-
taneously, at least seven parameters. I can envision the seven as being DO, carbon, organic
nitrogen, ammonia, nitrite-nitrate, phosphorus, and salinity. We will need both a one-
dimensional and a two-dimensional hydrodynamic (real time) model, an expanded capability in
multi-parameters with appropriate intake and feedback. I think this is the direction in
which we are going, especially in the Potomac estuary. I think the capability of running
at least seven parameters simultaneously, even with crude first-order systems, is something
that we've got to start designing for today.
BAUMGARTNER: I don't think there's any question about that, and there doesn't seem to be any-
thing special about, say, the Potomac. I think we are all concerned with the same thing in
Puget Sound.
JAWORSKI: My point is we don't have it. If you wanted to do that today, you couldn't do
it. There isn't a model that can handle it. The Orlob model can handle five.
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The feedback is very Important, as in the nitrogen cycle, and I want to reaffirm
what Dr. Thomann stated.
PAULIK: Biologists heartily approve of the attempt to model several parameters simul-
taneously, because if an animal is responding, for example, to a decrease in available
dissolved oxygen, the water temperature almost completely controls the response, or it may
modify this by a factor of several-fold. Then you can add the other parameters in, and
what you are really working with is a multi-variate response surface, and the animal is
responding in a synergistic way to a combination of effects. It seems that one of the
major contributions of modeling to biological interpretation of pollution is the fact that
you can synthesize the animal's exposure to a variety of stresses, and you can actually
integrate over, say, some migratory path for example, through different stresses in an
estuary, and then interpret his total response to this combination of factors. The book-
keeping just makes this out of hand for anything but some kind of computer model. Of
course, the other approach is to almost ignore the model and expose the animal directly and
try to interpret the animal's response. But 1 think we have to work from both ends until
we build some type of effective model. Perhaps we need a scheme for evolutionary modeling,
so that the data we accumulate can be incorporated in the model, and monitored data be
incorporated in the model. So we get a feedback here to eventually improve the model and
the monitoring system.
LEENDERTSE: The results, of course, are not finished, but at the moment we are running six
constituents simultaneously. These can have all the feedbacks between any one of them, and
we found the numerical solutions for that. It can also be done in nonlinear processes,
that is, have feedbacks between all of the constituents. So the solution methods are
available. This is one of the basic questions which can be resolved right now. Then, in
connection with what just was said, there have been studies made on modeling biological
systems. I have here a report in front of me applied to the blood stream, physiological
models, the oxygen content in the blood. Many of these programs seem to be applicable to
the problems that we are,faced with here in water quality studies, and we haven't touched
upon them. There has been a lot of development on this in medicine, and I have a feeling
that we can really step forward by using the approach which has been made in those areas
and try to translate them into the water quality area.
THOMANN: Yes. I would second the notion, in answer to this tracking of multiple constitu-
ents, that I don't see any conceptual bog right now in computation. We also have a
multivartable time-dependent feedback with arbitrary type of kinetics that's operating, and
the problem is that we don't really know what to put into it. We don't have data yet for a
lot of these areas. So I don't see any computational bog to constructing any kind of system
that we want to model. I think that the problem is that sometimes we don't know what
systems we want to model, we don't know which interactions are important, and we don't have
data to put into them. I think the answer to Norb's question about what he doesn't have,
and I believe this is your point, is that you don't have a model. The computational
procedures exist.
JAWORSKI: If I could have the model tomorrow, I'd put it to use in designing next summer's
program for the Annapolis Laboratory.
THOMANN: Well, let me give you an example. Jan could give you this, too. One could give
you tomorrow a computational framework that had reaction kinetics in it. The instructions
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to the user are: you have to write what you think those reaction kinetics are; you have to
write what you think the nonlinear interaction is (and in some cases we don't even know what
that is yet). But the spot is there. When we all learn what to put in there then we can
run some.
JAWORSKI: When I talk of seven parameters, I'm referring to running the nitrogen cycle
simultaneously with the carbon and DO system, and also the salinity up to the salt water
intrusion point. We could use it tomorrow. We are using the Orlob model which has five
constituents running right now.
FEIGNER: To set the record straight, there is no real limitation on the number of constitu-
ents the Orlob model can handle. It happens that five are incorporated in the model, but I
don't see it as a major programming effort to increase that to a larger number.
THOMANN: Oh, it very definitely can be. Yes. Where you are talking about a constituent,
you are talking about consecutive reactions, but not with general nonlinear feedback. That
is a different order-of-magnitude problem.
FEIGNER: [Added in proof] The basic program logic certainly does not preclude a large
number of constituents being run simultaneously. The biggest problem will be to define the
functional relationships between the constituents, and that can be a significant problem.
PAULIK: I was just reacting a little bit to these respiratory models. There must be at
least ten of these, and there are a couple of good articles summarizing the evolution of
respiratory models. Still there is tremendous disagreement between different theoreticians
on the appropriate model of human respiration. And to think of the homogeneity of blood
compared to the heterogeneity of water in an estuary, I think we are facing a very difficult
problem, and I don1t know how it can be solved by proceeding in that direction.
HARDER: I would like to point out the importance of immediate communication between the
computing component of solving a problem and the interpretation of results, together with
what is sometimes an intuitive readjustment of parameters, which constitute a component
best supplied by the human mind. Perhaps in line with Don Harleman's suggestion of knob
turning, we might imagine a situation in which one is trying to adjust a large number of
parameters associated with a model so that its behavior duplicates that of a prototype for
which we have field-determined data. This procedure is called verification of the model.
Even if the model takes the form of a digital computer program, it might be well to have
laid out in front of an operator a series of knobs that would control the value of such
things as diffusion coefficients through potentiometer settings. These potentiometers
would be connected through a scanning mechanism and ah analog-digital converter to the
digital computer program. This would bypass having to punch new holes into new cards each
time there is a change and at the same time show, through the pattern of knobs, the geome-
try of the prototype. This would be a help to the operator, who would be trying to get
solutions back within his memory span so that he would not be trying something more-than
once, and could enable him to more quickly achieve verification by allowing him to make his
adjustments while watching the changing behavior of his model through, say, an oscilloscope
display of some dependent parameter.
I have included in my chapter a description of a technique which is not yet a part
of the state-of-the-art, but which I hope will soon be included in that category. This is
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a method I and ay students at the University of California at Berkeley have called nonlinear
systems identification. It's essentially a black box approach in which one attempts to
define the system without knowing anything about the details. I shouldn't suggest that if
you do know something about the details that you should ignore it. If you know the equations
for example, you should by all means put the equations into your analysis.
The technique will relate a series of input variables to an output variable and
enable one to use the past to predict the future by a method one might call multi-
dimensional Interpolation. It has proved to be very powerful, since it does not assume that
the inputs are linearly independent; and we know that most, if not all, significant problems
involve nonlinear relationships. We have applied it to systems that have one or two time-
dependent inputs. Each input interacts within parts of itself so that yesterday's value of
an independent variable interacts with that of the day before, and input one interacts with
input two at all different times in the past, and these combine in their influence on the
present value of the output variable.
In those instances in which we know very little about the equations, this would be
the appropriate technique because all it requires is a sufficiently long series of measure-
ments of the system.
JAWORSKI: On this feedback system—if we're going to go feedback in our quality models (a
four- or five-parameter related system), one needs a lot of time and field data to verify
the various reactions. I can see a distinct advantage in a hybrid system. If we could use
the cathode ray scope, and one wanted to control four reactions simultaneously, you have
the capability of a direct communication with the hybrid system. This would be a tremendous
advantage.
HARDER: Well, I'm not suggesting that direct communication is the monopoly of hybrid or
analog systems. This can be developed for digital systems.
JAWORSKI: Say,, you have a complete nitrogen cycle model in which there is linkage in the
organic nitrogen to ammonia to nitrite to nitrate and back to organic. You have about four
or five reaction rates to verify or adjust which govern the concentration of a given
species at a given time.
HARDER: What you mean is that a particular reaction rate depends upon three or four other
dependent variables and the reaction rate of each of these variables depends upon it and
the others, so that they are all coupled, and they cycle and in a sense there is a feedback.
But when you simulate such a system, you do it by carrying out an integration in time of
all dependent variables at the same time. And at each step you pick up the value of the
others, insert that into an equation, and find the derivative of each one of them and carry
it forward one step. I think that's the modeling technique, and that, I believe, is what
you mean by feedback, and it's certainly a very powerful method. In particular, it brings
out your ignorance of some of the terms.
JAWORSKI: Many terms are unknown to us. Moreover, there is often quite some time elapsed
between data collection and model verification. This time could be anywhere from weeks to
a year until you have adjusted and readjusted the coefficients to your satisfaction. I'd
like to reduce some of that time and we may be able to do this with the use of the hybrid
system.
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HARDER: Well, that's where the immediate back-and-forth communication between the computer
and the operator is very, very important. And if you can effect that with some digital
computer oriented equipment, it would do just as well as if it were an analog computer
oriented equipment. I think the important thing is the rapid communication. The particular
machinery, whether it's analog or digital, is not so important.
I'd repeat though the advantage that you have with analog equipment, and that is
essentially you don't have any continuing operating costs. And that gives you the kind of
comfort of knowing that you can waste a little time without going over the budget. The
digital computers are dreadfully expensive if you have to go through many, many, many steps
to achieve verification.
PAUL1K: There are some systems like DYNAMO and others that are used by economists. I
don't know if you've looked at any of these and have any opinions on them. They're simple
first-order linear difference equations, but they can handle thousands of equations.
HARDER: Yes. They're coupled first-order differential equations, I'm pretty sure. The
trouble is that we don't know the coefficients that go into the terms. Solving the equa-
tions is not the problem.
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3. DATA COLLECTION AND VERIFICATION
JAWORSKI: BOD-DO modeling is something chat we can usually do quickly, but I chink when you
start looking at nutrients, phytoplankton populations, zooplankton populations, you're
talking about quantum jumps in numbers of field data. One thing we have to impress upon our
administrators is that we are talking large populations of data required to verify our more
complex modeling.
WASTLER: It's all very well to say this from a philosophical standpoint, but when you get
down to it, where do you sample, when, with what degree of accuracy and how are you going
to use it? There are the questions that come flying right back at us every time we start
talking about sampling programs, data collection, and mathematical models. But this is not
the kind of thing that you give to an administrator. You give him some guidelines by which
he can decide what level of resources have to be committed to produce the results.
CALLAWAY: The verification process again. Doesn't that still boil down or come round-circle
to verification not being so much a problem of not having the right data at the right time,
but not having the right information on coefficients?
THOMANN: Yes. We're always interested in returning to some kind of first principles. In
this context of reducing numbers of knobs, certainly.
WASTLER: Would there be any way to explicitly state how much data would be needed for
reasonable verification?
THOMANN: Well, I think that bears heavily on the problem context, what kind of problem are
we actually attempting to solve in a given context. If, for example, we are concerned about
hour-to-hour fluctuations where some of these real time models are concerned, then that's
one type of sampling program. If, however, we're interested in mean value type programs,
such as the overall effect of a treatment plant on a particular estuary, that's a different
kind of sampling program.
WASTLER: I would like to see some comment possibly on the specific kinds of studies and
information needed to fill in some of these holes. One of the things we're working toward
right now is development of essentially a coastal monitoring system, which hopefully is not
going to be just a means of going out and collecting numbers. But we would like to develop
it within the concept of the kinds of information that are actually needed to solve some of
these problems. So any specific comments any of you gentlemen would have regarding this
kind of information would be greatly appreciated, particularly if we can use your names in
vain as having recommended them.
DOBBINS: In the estuaries themselves there's no question that nitrogen effects are
extremely important. We know that every gram of nitrogen requires maybe four grams of
oxygen to oxidize it. And I know that this is true in some of the young freshwater streams
away from the estuaries themselves where there are cases that the quality of the stream is
actually deteriorated by building a secondary treatment plant. There are small tributaries
of larger rivers with small primary treatment plants located on them which, say, has two
days hold time to reach a safe haven in the big river. The stream will be somewhat degraded.
But then you build, let's say, an activated sludge plant on this stream and you turn out an
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effluent ripe for nitrification, then the nitrogenous oxygen demand is going to be exerted
in those two days, and the stream will actually be probably worse off then it was before
you put in the plant.
I think the whole biological interaction between bacteria and algae is very com-
plicated and deserves a lot of study. This falls back to biologists. These things are
complicated by the fact that we have many fundamental periods involved. First, there is
the tidal period, twenty-four-and-a-half hours, then you have the diurnal period which is
the period that most of the inputs, the algae effects, follow, and over a long period of
time you have the seasonal effects. There's no such thing as a true equilibrium ever being
reached.
WASTLER: Well, one of the apparent things was that on a routine monitoring basis it would
of course be impossible to measure all possible parameters. One of the things that we're
concerned about, recognizing that to pin down a number of these interactions will require
intensive study in specific areas, is what kinds of information could be collected on a
routine basis which would support this kind of investigation?
PRITCHARD: I'm not sure you're at the stage of saying just "routine" sampling. We need to
know more fundamental facts about the response of the ecological system to water quality.
For instance, one finds quite a difference between the effects of nitrogen in systems of
fresh water than in the estuary proper where there's salt water. We can find great
differences in the character of the primary and secondary food chains in the upper Chesapeake
Bay over periods of time. Every spring the major nitrogen source is the input from the
river Susquehanna, and during the spring runoff the nitrate concentration runs around
70 mlcrogram atoms per liter or higher. By the first of June, the concentration of nitrogen
throughout the upper sixty miles of the Bay and all the tributaries is about 40 microgram
atoms per liter. By the end of summer and beginning of autumn that's down to four-tenths
of a microgram atom per liter. That difference is all going into the bottom in the form of
unoxidized organic material which does not return. We never go anaerobic except in very
local regions. And the supply of nitrogen is renewed every year and chewed up every year,
and there's not any great difference between a hundred-fold variation of nitrogen concen-
tration. But we do see significant differences in the estuary with respect to the phosphate.
Now, maybe it's not the phosphate but something that goes along with it, other organic
stimulants or something else. But it's quite a different thing than in the freshwater system
where the nitrification appears to be an important part of the system. My point being that
in some systems you might say that ten microgram atoms per liter of nitrogen is too much,
but here we've got a system that responds not essentially any differently between when the
nitrates (and in this case this is most of the nitrogen) in dissolved form is 40 microgram
atoms per liter in one case and down to .4 in the other, and still functions in essentially
the same way.
One very fundamental question, for instance. We have been trying to run the
nutrient balance in the upper Chesapeake Bay. There is in fact a sizable amount of nitrogen
and phosphorous produced by the Baltimore treatment plant. We don't see it in the upper Bay.
Why? Well, we think we know why we don't see the phosphorous. But we have no simple
explanation for nitrogen which we also don't see, and we should. Now there's some fundamental
thing happening to the nitrogen that has been put out of the sewage treatment plant.
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THOMANN: Did you try running a nitrogen model? You know, trying to track the nitrogen and
see where it is going?
PRITCHARD: We are trying, but it is very difficult.
RATTRAY: Another thought about monitoring, a partial answer. If you can get some measure-
ments that give a measure of rates. I've been jogged again by the people worrying about
nonconservative reactions or properties which have rates of reaction or exchange associated
with them. I keep coming back and saying that the equilibrium salinity distribution is not
a measure of the rate of salt flux and it is the rate of movement of salt through this cross
section, which is the rate of movement of water through it at different levels, that will
have a much more marked effect on the time-varying distribution where a rate is involved,
which has a time clock built into it. Something goes by and is decaying as it moves by.
Its distribution would be very much affected by the rates that things are going on, the
rates of movement of water, whereas the salinity, which doesn't have that time clock built
in, is not very hard to determine. There would be a big difference here. Not only do you
have to worry about what the rates are from biological and chemical factors, but the hydro-
dynamic models which will have to go along with it will, I think, be necessarily more
complicated, as you have to worry about the rates of things as opposed to what's in
equilibrium.
PAULIK: In a way it seems the total objective of the monitoring system should be con-
sidered. In solving a number of agriculture problems when we set up the land grant colleges
to work on these problems, we certainly didn't have the advanced physiological and chemical
knowledge that we have now, and we did this empirically. The monitoring systems in particu-
lar areas are going to be tied into key species, which are desired species in that area and
have to be protected, and these will be simple bioassay systems or continuous bioassays.
Here you might be concerned with sampling problems to make sure that you have a sort of
consumer, or vendor, and purchaser risk as in selling any type of material, but you are
concerned with the basic problem of sampling some key species to make sure that this species
is protected. And until you define all your objectives of a particular monitoring system,
it seems very difficult to set any general set of rules which will establish the basic
criteria for a monitoring system you can employ in any estuary in the country.
WASTLER: The basic purpose behind any monitoring system run by FWQA is to maintain
surveillance of water quality in order to determine where water quality degradation exists
so the proper mechanisms can be taken to combat it.
PAULIK: Well, this brings in the problem of defining what is degradation of water quality.
You might be concerned with the striped bass in San Francisco Harbor, which have a critical
time in their life when they are dependent upon adequate spawning grounds and the eggs have
to be protected during a certain, very specific period, whereas in another estuary you might
have a very different set of problems. It seems these problems are the ones that define
what you mean by high pollution and degradation.
WASTLER: Each system is in some degree different from all others, and the scope of the
monitoring system, including the timing and parameters measured, has to be related to the
problems and characteristics of that system. But what I am driving at here is, in the
process of carrying out this general objective, we have to take certain measurements on a
certain time scale. I am trying to get an idea if in carrying out this objective we can,
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say, modify it slightly to contribute toward the basic effort of developing predictive
models and understanding the interactions. I would not like to give the idea that such a
monitoring system would be set up purely for developing information for mathematical
modeling or any other kind of modeling. But I have a feeling that it can contribute
significantly to the data base which would be useful in developing such models.
PAULIK: It seems like it should go hand-in-hand with modeling so that these short-term
forecasts can continually be made on the basis of the data you're collecting. You could
somehow integrate this with the type of fragmentary results which are available in biology
now. All types of physiological studies are being put together in modeling efforts, or at
least in estuarine problems. In trying to make forecasts, your data could very nicely pull
into models based on synthesis of some of the physiological work that is being done.
RATTRAY: At this stage it seems like a few, easily identifiable quantities which don't
interact very much, only one or two interactions so that the system which you consider is
simple, might be very useful. They may not be the ones important to water quality but they
might be the ones very useful in measuring and determining what is going on in the system.
WASTLER: That's quite true. And once you're out there collecting samples or you have
established an automatic monitoring set-up to do it, it's sometimes quite simple just to
add another parameter which would be useful for a specific purpose. We would certainly be
happy to hear of any specific parameters that might be useful. Not necessarily right now,
but any time they might occur. I think Bob, in particular, understands the problems.
THOMANN: I think the problem there is that some of the variables that we would eventually
ask you to take a look at would be more difficult to get than you may care to incorporate
into a routine monitoring program.
WASTLER: This is certainly true, but...
THOMANN: But you would like to know them anyway.
WASTLER: We would like to know them anyway. As you very well know, sometimes if you can
do some advance planning, you can take care of such things over a period of two or three
years.
PAULIK: Well, it seems important, just as a general rule, to somehow simultaneously
observe the reactions of the higher organisms, zooplanktons for example and some of the
fishes too, at the same time you are taking the chemical and physical measurements. This
is the one problem that continually plagues biologists trying to build models, that we have
grab samples or samples taken at infrequent intervals or when the weather is favorable, and
we try to find physical and chemical measurements taken at the same time. It's very tough
to get an integrated series of biological and other measurements to interpret the biological
measurements.
I have a couple of comments. One on Chapter III where some of the philosophy of
model verification_is put forth. I wondered if some of this could be tightened up a little,
and perhaps some sampling strategies suggested. There seem to be a number of problems here.
How do you lay out a sampling grid? What's the frequency at which you collect samples?
What are the problems of verifying the model using grab samples versus some sort of continuous
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monitoring system? What are the precision requirements on the individual parameters that
you're measuring? You can develop a theory of propagation of errors in which you can look
at the precision with which you measure seme elementary variables and then combine these in
various functions to build up your final random variable that you'd like to plug into the
model. And I think that Dr. Thomann has laid out some of the problems here, but I wonder
if there is any theory or any references that would be of help to people building models on
how they can set up sampling programs to verify the model or to test model validity. I
don't think there are any general statistical techniques. Perhaps some of the time series
techniques give you some idea of the model validity over time. But I think that the problem,
that at least occurs to me, is how do you begin to sample in terms of the statistical theory
of sampling, and
-------
WASTLER: Well, it would have a basic relationship to how you would go about setting up a
sampling program.
THOMANN: My own experience has been that it helps just very, very generally. But what
usually constrains it? What kind of program you can mount within a given time period. You
know there are so many other extraneous variables that come to play in mounting a sampling
program that rigorous statistical frameworks may not give you the kind of information you
need. Budgets, sampling programs, lab facilities, how many people you have to put up.
Those tend to be almost overriding considerations in most of these sampling programs. Now
if you are posing the question: suppose there were absolutely no constraints. Then that's
a different situation.
WASTLER: Well, agreed that the budgetary-personnel factors are overriding. But there is
little point in, say, spending a half million dollars to develop a highly sophisticated
mathematical model when you don't have the data base to verify it. And it's this kind of
criterion that I'm looking for some guidance on. As you know, this is one of the things
we have to face everyday.
BAUMGARTNER: I think those things you mentioned, Bob, as constraints are not things which are
foreign to people who have to design sampling programs. And I think there are pretty well-
defined and perhaps rigorous, in their mind, techniques for how you get the best sampling
program for these constraints. So I don't think it's unrigorous, or I don't think we can
say that kind of rigorous analysis is not available.
THOMANN: Well, by rigorous analysis, I mean some kind of analytical framework.
BAUMGARTNER: I do, too.
THOMANN: No. I haven't seen it. Seriously, I don't know where it is. Isn't that exactly
what you're asking?
BAUMGARTNER: No. I'm talking about sampling programs as designed by statisticians who are
working with surveys. Books like Cochran and Cox for one, I recall. Maybe you know of
this one, Jerry. And I think the framework is pretty rigorous.
PAULIK: Well, the framework there is very rigorous. You can define a frame, a sampling
frame, and you also know what your population is. You certainly have certain clear objec-
tives. Here, a lot of the problems to me seem to be very fuzzy. Approaching a problem in
an estuary, I as a biologist would start looking for the key species, looking for the life
histories and their interactions with other animals. I would eventually get down to how I
thought the environment affected these species, instead of starting with the very detailed
modeling of the environment and building up to the species. I'm not at all sure how you
begin to sample in this context. That's one of the reasons I'm partly asking for informa-
tion and addressing the question to Bob. I think there is very good statistical sampling
theory in certain areas, for certain objectives, and I'm not clear, I guess, as to what the
total objectives of estuarine modeling are. That's my problem.
THOMANN: If you ask questions like: how many samples would I have to take to be 95% con-
fident that a particular value is within plus or minus five percent of something. That we
can do without any sweat at all. And that's done very often in collecting waste effluent
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samples, to some extent water quality samples. My own experience is that once you do that,
it's a nice number, one looks at it, and eventually what happens is that an overriding con-
sideration is what you can do within a specific time frame. But if you phrase the question
rigorously then there's a rigorous framework. It turns out some of those sampling require-
ments go up nonlinearly, as, for example, the variance of the process. There's another
thing. We don't know very much about the variances of some of these water quality
constituents, and very little about their statistical behaviors. We can use some non-
parametric statistics, but that doesn't bring us very far either. We've got a lot more, no
doubt, to do. And because those sampling requirements go up nonlinearly, it becomes
important. But 1 don't think we have any rigorous approach. We don't. There may be other
fields that have it.
BAUMGARTNER: 1 have the feeling that many of the people who are working in this area don't even
appreciate that this kind of statistical test is available.
WASTLER: This relates back to Dr. Harleman's comments this morning, with regard to the
different time scales. Obviously if you are going to talk about tidal fluctuations, samples
taken every week are not going to help you very much. Likewise, if you're talking about
annual trends, there's hardly any point in going out and measuring every thirty seconds. I
was looking for some general guidelines or directions.
DOBBINS: I think you put your finger on something there. I think if you have so many
dollars to spend on sampling, and you have so many estuaries to sample, it's more efficient
to concentrate, studying each estuary over a relatively short period of time, very intense
sampling patterns with short time intervals, then move to the next one, rather than study
them all at the same time on a very coarse network.
RATTRAY: I would certainly second that. I've never seen, and maybe nobody has seen, an
over-sampled estuary.
DOBBINS: If you would sometime just move in, maybe one month, and really concentrate
sampling and get the pulse of this one, and then move to the next one. But taking a sample
once a week in all ten of them isn't going to give you anywhere near as much information as
taking ten samples in one.
HARDER: I might bring out something which is fairly obvious, and pardon me if I am
elucidating the obvious, and that is if you know the rate at which a certain variable is
changing in nature, that automatically gives you the time scale at which you ought to
sample. Certainly you should sample no less frequently than two times within the period of
the highest frequency at which something changes or the frequency at which you think it is
important to record things. And the same thing relates to distance. You can define a
spatial frequency, if you like, and you ought to sample at least twice every time you go
through a cycle of spatial variability. These things then would give you a guide. Now
certainly if you are going to construct a model the sampling ought to be reasonably well
related to the nodes at which you expect to make the computation, which is a way of feeding
back from the model to the sampling program.
FITZGERALD: We continually get this question. We're going out and sample this estuary for two
weeks. How many samples should' we take? How should we take them? How often should we
sample? This is continually asked on an operational level. Normally we just say once every
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half hour every hundred feet, since the majority of them aren't going to do it anyway. But
I don't really know if there is a clear answer to deal with this point.
DOBBINS: I think you have to define the objectives of your central program. If it's merely
a monitoring program to determine what the conditions are over long periods of time, then
use a coarse network over a large area. But if the objective is to determine parameters
which will enable us to improve the mathematical models, then you've got to do intensive
sampling.
THOMANN: I think in the final analysis a lot of these questions, at least in my own mind,
can be answered by actually playing the forecasting game itself, and we haven't done this
very often at all in any of this water quality work, or in fact not really in any of the
hydrodynamic modeling either. By forecasting I mean that you know everything up to time t.
Given that information, what happens at t + At?
DOBBINS: This is okay if the inputs to a system change after time t, but if the inputs
remain the same then we're still fitting the same curve. This is what bothers me about
these long-time-averaged models. We look at DO in an estuary. It is a fairly simple sag
curve, oxygen declines then it rises again. The old Streeter-Phelps equation gives you a
functional shape like that, and you can fit the Streeter-Phelps equation with these very
simple inputs to many of these sags. Then we go back and we become aware of more and more
of the inputs. We include those in the equations. We still get a good fit. We now have
to go back to adjust the first values of the coefficients we had. As we keep adding more
and more inputs and feedbacks into the system, we're still getting a good fit. We're
getting a good fit with all these models. So therefore it leaves you with the agonizing
doubt, does the latest one still have all the inputs of concern?
THOMANN: And the answer is probably: no.
DOBBINS: And the answer's probably no. And verification, in my mind, is really curve-
fitting. It's a model to fit a curve, to put a functional description to a curve that you
can measure in the field. But the only true verification it seems to me to be to determine
if, when you remove some of the inputs, will the effects of this removal will be borne out
by your equations, the predictive equations. I don't know of any case where this has been
really done. The test of all these is to come.
THOMANN: Well, I think I disagree. I generally agree with you on several points, but I
think I disagree that verification for the kinds of models we're talking about here is
nothing more than passing functional relationships through a set of data. After all, one
could take a plot of data as a function of x and do some regression analysis on it and then
fit some function. The point of departure here of these models is some kind of fundamental
physical phenomenon.
DOBBINS: Oh, I agree.
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iHOMANN: Now, at any point In time, we may not appreciate the entire depth of what that
physical phenomenon involves, hence the old Streeter-Phelps sat around forty years and
really wasn't dealt with much beyond that. So I don't see it just as passing functional
relationships through data, because we could do that with regression.
DOBBINS: All I'm saying is that we should be very cautious about believing that we've got
the final model.
PAULIK: Well, you don't kid yourself if you're just estimating, say, the coefficients in
a nonlinear model. If you have a simple food chain model, you've got data from radioactive
tracers moving through the food chain, and then try to generate the same data from a simu-
lation model and where you know the underlying structure put some error terms on it, and
then try to employ statistical theory to estimate the constants, you loose total faith in
your original model, your original concept of how the system worked. I have the feeling
that some of these models may be of the same type, that we have a whole system of constants,
and the constants are very difficult to estimate statistically. I know of no standard
theory for how to generate an adequate sample to verify the model. If you have a number of
different segments for example in a stream, and you are using your model, you might ask the
question: how many of these segments have to be sampled to give some sort of reliable
estimate of the coefficients in the model? Is it possible to make that type of judgment?
DOBBINS: The objective of all this, as I see it, is to help us come up with engineering
judgments as to what we have to do, and what inputs to the system do we have to remove in
order to restore these estuaries to an acceptable form or level.
THOMANN: I think that's true, and, to me anyway, these models have been very useful. I
certainly agree that we can't be overconfident. Now the state of knowledge of what's
happening is increasing very rapidly and we can't really think at any one stage that we've
got the whole thing wrapped up. But I found these particularly useful for several reasons.
It helps to set a kind of framework for decision makers that didn't exist in the past.
Order of magnitude responses, for questions like the New York Harbor situation where after
secondary treatment is installed, what is the general level of improvement you could expect,
even given these crude models. Well, it turns out a couple of tenths of milligram per liter
dissolved oxygen. Now that, in and of itself, is the kind of information that didn't exist
five or ten years ago, and is very useful toward the decision-making process. Now, if you
are in fact going to go out now and try to sample the system to detect that two-tenths of a
milligram per liter of DO increase in the face of background noise, that requires some
rigorous design in how many samples you have to take, and so on.
WASTLER: The question is, do you believe that result?
THOMANN: Well, I believe it's not two. I believe it's not off by more than an order of
magnitude. That's the point.
EDINGER: Isn't there also a data requirement for observations we need to determine how to
design the model? For instance, I don't think we're at the stage yet where we can say that
we can predict for n interactions every possible combination of thing that could go on, and
go out in the field and apply it and come up with anything. Take, for instance, your work
a couple of years ago on phosphate cycling where you suddenly discovered just from rough
field data that you were turning over quantities of phosphate every two or three days. I
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don't Chink it would show up in the model until you explicitly designed it in. And yet it
is things like this that we are suddenly discovering that makes a water quality problem
what it is. The same thing I think is true with the problem of the nitrogen cycle in a
body of water. For instance, as I understand it now, and this is out of my field, apparently
there is some concern over nitrogen in the Delaware being very temperature sensitive, that
below certain temperatures there is very little oxygen response to it. Would this have
been built in a priori, or are we still in the stage yet that we have to intuit these things
out and get the data to discover that they are going on?
THOMANN: Well, in a lot of cases, yes, it is very temperature dependent. The question is:
five years ago, ten years ago, it probably wouldn't have been there because it simply didn't
occur to us. But 1 think the point here is that I don't see us building models in a vacuum.
The data that we get from the field shapes the models, and the models help shape how we
collect data, a continual type of interaction between data collection and modeling and data
collection again. We can't hope to model something we've never even observed.
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4. PREPARED REMARKS
4.1 DISCUSSION by George H. Ward
Dr. Pritchard has rightly pointed out that the kinds of estuarine water quality
problems that you face, the decisions to be made, and hence the kinds of models you require
depend to an enormous extent upon where you are. This observation can hardly be overemphasized.
From a practical standpoint, there is no universal estuarine model: in an estuary, the perti-
nent water quality problems must be determined and the relevant processes identified in order
that meaningful and utilitarian models b*e employed. Conversely, the limitations of existing
estuarine models should be similarly considered, their underlying assumptions and the approxi-
mations made in their development, so that they are not applied to a situation for which they
are not strictly valid.
Now this merely restates some sentiments that have already been expressed by several
of the people in this study. But it does serve to underscore, however, the fact that the bulk
of the state-of-the-art of estuarine modeling, at least as represented in this report, is con-
cerned with coastal plain estuaries of the type extant on the north Atlantic coastline of the
United States. While it is true that models have been developed for systems such as Jamaica
Bay on the east coast and San Francisco Bay on the west, which require a two-dimensional repre-
sentation, and while it is true that many of the model formulations are of more general appli-
cability, nonetheless most of the work in developing and applying estuarine models has been
directed toward our northeast coast estuaries, as this is where the earliest pollution problems
have occurred. But in the context of this report, we must point out that there are other kinds
of estuaries and other kinds of problems which also require study. I am thinking in particular
of the Gulf Coast estuaries which have been, undeservingly, somewhat slipped over in this
report.
By "Gulf Coast estuaries" I mean to include those bays and sounds contiguous to the
Gulf of Mexico and lying along the arc of coastline extending from behind the Florida Keys
around to Padre Island on the Texas coast, excluding, however, the estuary of the Mississippi.
Typically, these estuaries are broad and shallow with an average surface area of some 500
square miles and a mean depth of ten feet, and are fed by multiple inflows around their periph-
ery. Many of these estuaries are nearly isolated from the Gulf by barrier island paralleling
the shore. These estuaries are highly productive and are extremely important to the commer-
cial fishing industries in the Gulf. (See Gunter 1967 and Odum 1967 for general features of
these estuaries.)
Hydrodynamically, these estuaries present something of a different problem than such
geomorphological cousins as San Francisco Bay. The influence of tides, for example, is much
less important. The tidal range in the Gulf of Mexico varies from a maximum of just over two
feet to a minimum of a couple of inches. The influence of this feeble tide is further dampened
by the semienclosed, shallow physiography of the estuary. Winds, on the other hand, are a
dominating factor in these estuaries, governing both the advective and diffusive transports
through driven currents and wind waves (Collier and Hedgpeth 1950, Gunter 1967); models capable
of predicting these phenomena range from primitive to nonexistent. The dynamic influence of
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the horizontal salinity gradient (see Chapter II) may be especially significant in many of
these estuaries as well. The intrinsic transience of these systems is a difficulty in itself,
their response to changing meteorological conditions, surges of inflow alternated with periods
of relative drought, and so on.
As a general rule, dissolved oxygen is not yet a problem in Gulf Coast estuaries —
there are local exceptions of course, the Houston Ship Channel for one, nor can we be sure how
long this situation will last. However, there are other quality problems, many of which are
becoming critical. These include the Introduction of types of wastes which although consti-
tuting a "low-level stress" exert profound influences upon the ecology, bacterial contamination
of shellfish areas, the effects of dredging and channeling, accumulation of nutrients, and the
upstream impoundment or diversion of freshwater inflow. (See Wohlschlag and Copeland 1970,
Chapman 1968, and Copeland 1966 for a discussion of these and related problems.) Thermal dis-
charges can be a problem, but here I must disagree with the statement (Chapter IX, Section 1)
that when cooling becomes a constraint the discharge is "too big" for the estuary. Due to the
extreme shallowness of these estuaries, the effective dilution is considerably less than would
be expected in, say, an east coast estuary; consequently the plume ares is larger and loss to
the atmosphere becomes a significant part of the heat balance requiring explicit incorporation.
But this in itself does not mean the discharge is too big for the estuary, for both the surface
area of the estuary and the discharge location must be considered in assessing the effects of
the discharge. For example, in Galveston Bay a typical discharge into five-foot water produces
an area 1°F above ambient on the order of 500 acres. This, however, is only slightly over one-
tenth of one percent of the total surface area of the Bay. The critical question then becomes
whether that area affects the habitat, breeding or nursery areas of any temperature-sensitive
species, or encroaches on the migratory paths of such species.
In sum, the Gulf Coast estuaries present somewhat different problems, hydrodynamicslly
and quality-wise, than are encountered on the east coast, many of which require fundamental re-
search. Because of the singular fertility and complexity of these estuaries, a considerable
effort may be necessary to clarify the links between the activities of man, water quality
parameters, and the natural ecosystems. Furthermore, because of their hydrographic character-
istics, they exhibit a fa«ter response to man's activities than other estuaries. This aspect,
particularly with regard to the industrialization of the Texas coast, has motivated the sug-
gestion that they be considered a paradigm for the management of man-nature systems (Odum 1967).
Clearly, the use of models is an essential part of this.
REFERENCES
Chapman, Charles, 1968: Channelization and spoiling in Gulf Coast and South Atlantic estuaries.
Proceedings Marsh and Estuary Management Symposium, J. D. Newsom (Ed.), Baton Rouge,
La., Thos. J. Moron's Sons, Inc.
Collier, Albert and Joel W. Hedgpeth, 1950: An introduction to the hydrography of tidal waters
of Texas. Publ. Inst. Marine Science. Univ. of Texas, !_ (2), pp. 121-194.
Copeland, B. J., 1966:* Effects of decreased river flow on estuarine ecology. J. WPCF. 3£, 11
(Nov.), pp. 1831-1839.
481
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Gunter, Gordon, 1967: Some relationships of estuaries to the fisheries of the Gulf of Mexico.
In Estuaries. George Lauff (Ed.), Publ. No. 83, AAAS, Washington, D. C.
Odum, Howard T., 1967: Biological circuits and the marine systems of Texas. In Pollution and
Marine Ecology. T. A. Olson and F. J. Burgess (Ed.), Hew York, Interscience.
Wohlschlag, D. E. and B. J. Copeland, 1970: Fragile estuarine systems—ecological considera-
tions. Water Resources Bulletin. _6, January-February.
4.2 DISCUSSION by Richard J. Callaway
Some reports don't need summarizing—this one does. Since no one else has volunteered
to put things right I'd like to take at least a superficial whack at putting it in perspective.
(In addition, I don't think the authors or reviewers are detached enough to do it themselves.)
Not too many years ago it wasn't unusual for oceanographers to be highly proficient
and productive in many aspects of their science. For instance, G. A. Riley (admittedly, not
your typical oceanographer) probed the physical, chemical and biological processes of the ocean
as well as being the first to employ Lotka-Volterra prey-predator relationships cum nutrients
to model these processes. In the past few years the oceanography camp has gone the way of most
"new" sciences from the era of the generalise to that of the specialist and, for better or
worse, rather esoteric niches are being created at a great rate. It seems to me that estuary-
type sanitary or civil engineers have also arrived at a departure point, except instead of
becoming narrow specialists they can become more diverse, at least for a while before things
get too complicated, sophisticated and unmanageable for one man as it did for the oceanogra-
phers. Until that time arrives (and some would insist it already has) there are opportunities
for the versatile to do many interesting and useful things covering a broad range of subjects.
In this pioneering sense, the material covered in this report is still manageable although per-
haps some pages are harder to read than they really need to be.
What this state-of-the-art report is all about is solving the advection-diffusion
equation with source and sink terms,
Is+ull + vll + wM-ra-fK *2.)+i_(K *£.)+!_( K *S\] - is
at ax ay az Lax^ x ax' ^ y ay' ^ z az'J
The equation might be viewed as consisting of hydrodynamic terms on the left-hand side and
pollution terms on the right. In the past it has been the practice to consider that the heart
of the problem lay on the right-hand side of the equation. For a long time in sanitary engi-
neering the right-hand side has usually consisted of BOD and DO terms only; actually sanitary
engineers have been making an honest living for about 45 years by employing first-order reaction
rates for BOD as a carbonaceous sink. More and more, but somewhat reluctantly, they have begun
to consider (and use) other sources and sinks and to employ Michaelis-Menten kinetics. Much
of the discussion revolves around how to get at the velocity and diffusion terms, especially
when nonsteady-state conditions prevail. There is a tendency to try and get around the u and
K terms by fair means or foul, such as by averaging them out of existence or expressing them
functionally. While the right-hand-side people are bedevilled by rate constants, the left-
hand-siders agonize over isotropy, and von Neumann conditions (after all, if he was interested
482
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in error analysis then the problem isn't likely to be trivial and, plain to see in Leendertse's
paper, it isn't). The position of the practicing engineer is that he must be concerned with
both sides and really can't devote his life to fretting about any one of the K's or S's in
particular.
In a way, the discussions and papers presented here seem to accentuate Levins' (1966)
ideas that mathematical models sacrifice either generality or realism or precision at the ex-
pense of the remaining two quantities. For instance, box models on the one hand are abhorred
by some because one might unknowingly leave out something; in effect they may be too general.
On the other hand the box model has been, and still is, quite successful in investigating sys-
tems that cannot be quasi-realistically described any other way either because of the size of
the system studied (see, e.g., Broecker and Li 1970), or because the hydrodynamics are largely
unknown. It also seems a bit silly to argue the precision of one's guess on the magnitude of
exchange processes. The one-dimensional estuary model, whether employing tidally averaged or
real-time concepts, also has its reluctant admirers and has been highly useful in spite of the
fact that no real one-dimensional systems exist. It is hoped that the newer, hydrodynamicslly
founded, two- and three-dimensional models will be tractable enough to encourage their routine
use while still being developed.
In the case of (electronic) analog and hybrid modeling, the state-of-the-art appraisal
seems to be that the analog peaked out in usefulness a long time ago. The almost immediate
visual display of results that was once the sole property of analog machines can be handled,
although not routinely, by CRT devices which are now standard peripheral equipment on time-
sharing digital computer systems. (There is always a danger that those who get good enough to
use these devices routinely will become more enamoured of programming and the computer system
than the engineering problem.)
The feeling on physical hydraulic models ranges from viewing them as gigantic toys to
seeing them as having great potential in carefully selected pollution problems. As in all
things, there is a danger in misuse or misapplication. For instance, obtaining diffusion coef-
ficients from the hydraulic model in order to insert them in a mathematical model is not the
best of ideas if only because the hydraulic model is not scaled to reproduce eddy diffusion.
Likewise, proper scaling of the processes governing the distribution of nonconservative sub-
stances, such as those controlled by biological phenomena, doesn't appear to be all that easy.
Since there are better ways to estimate rate coefficients such efforts would be better spent
elsewhere. The hydraulic model is here to stay, however, because it still is the best method
in which to study channel modifications (for which it is chiefly intended, at least by the
Corps of Engineers).
The hydrodynamicists are in the almost enviable position of knowing a good deal about
what they don't know: i.e., how to relate something measureable to something that has long
defied routine field measurement and correlation, namely vertical and lateral viscosity and
diffusion terms. Functional relationships for vertical eddy terms and velocity profiles can
be expressed for a variety of laboratory plume conditions; when exposed to the cruel light of
even the well-mixed estuary they are sometimes useful as a guide as to how nice things could
be. (One is reminded of the probably apocryphal story of Einstein's advice to his son, a dis-
tinguished fluid dynamicist, to avoid hydrodynamics on pain of horrible boundary conditions.)
When it becomes necessary to include the effect of feedback on the computation of
motion or concentration, additional problems arise. Such events are handled routinely through
coupled differential equations. We are bound to see more and more of these with the systems of
483
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equations necessary to relate the nutrients-phytoplankton-zooplankton on up the scale to the
highest user. One should not lose sight of the problem because of imagined complexities in
solutions. For instance, the five equations Dr. Paulik showed can be solved with about two
dozen program statements if the coefficients are known. The real problem is again getting at
the coefficients, only this time there are fourteen of them and they are all maddeningly elu-
sive and uncertain. (Add to this that what Paulik has shown is the simple case.)
the report has accentuated a recurring theme in the history of the science and engi-
neering game: what were once thought to be solved problems are becoming unsolved with frus-
trating rapidity. It behooves the engineer to make use of the best tools available and to make
sure he trades in the old ones, an old saw to be sure but especially relevant when he is re-
sponsible for making recommendations on million-dollar investments in treatment plants and
disposal schemes. So now let us minimize our uncertainties and press forward into the Age of
the Great Coefficient Hunt.
REFERENCES
Broecker, Wallace S. and Y. H. Li, 1970: Interchange of water between the major oceans. Jnl.
Geophy. Res.. 75 (18), pp. 3545-3557.
Levins, R., 1966: The strategy of model building In population ecology. Amer. Sci. 54,
pp. 421-431.
4.3 SMC PROJECTIONS FOR THE IMMEDIATE FUTURE by Jan J. rendertse
The contributions in this volume show evidence of historical developments and
approaches used by different investigators in the engineering and the scientific disciplines,
who are'now tackling the problem of establishing cause and effect relations in estuaries. From
these developments certain projections can be made for the use and power of mathematical models
in the near future.
The development of models for estuarlne processes is difficult, complex, and, in
addition, costly. Generally speaking, models are made for the investigation of results of
certain actions, or for the evaluation of alternate approaches concerning fluid waste disposal
or engineering construction in the estuary or the adjacent watershed. The purpose of the in-
vestigation provides the analyst with tentative answers to two basic questions always raised
in modeling: what problem will the model help to solve and what is the accuracy required. The
answers to these questions will guide the investigator in the selection of the important pro-
cesses which have to be modeled.
The contributions in this report give evidence that even with modest accuracy, complex
models are required. Modeling of estuarine phenomena is a major undertaking which is generally
underestimated. Therefore, it is warranted to analyze the model development effort itself.
The major steps are shown in the figure with the main stream of the processes indicated. The
data is placed at the top for a good reason. No modeling effort can be started without data
as such data give the indication of the major processes involved. The figure is self-
explanatory, except possibly the connection between conception and finite-difference models.
484
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DATA
CONFIRMATION BY
SIMULATION
CONCEPT ION OF
PHYSICAL AND
BIOLOGICAL PROCESS
COMPUTATIONAL FORMULATION
IN FINITE DIFFERENCE
EQUATIONS (MODEL)
MATHEMATICAL FORMULATI ON
IN PARTIAL DIFFERENTIAL
EQUATIONS
Fig. 9.1 Construction of estuary models.
Conception and the resulting mathematical formulation are often inherited from the pre-computer
era. In those days, the only hope for solution of the derived equation was an expansion of
products of the dependent variables followed by a simplification so that analytical methods
could be used. At present other formulations are required which are more suitable for the
subsequently derived finite-difference equations. As indicated, not only approximations to the
partial differential equations can be made but also conservation of properties with a physical
meaning can be achieved.
The loop in* the steps of the development shown in this figure is closed by verification
of the model using actual data. It may be necessary to iterate through this development several
times. In the verification of the model, we interface with the technology of data collection
and with the aspects of data representation, both of vital importance for a successful investi-
gation. Digital records from field explorations can often directly be used for control of the
more sophisticated models, while the subsequent model results can be presented simultaneously
with the field observations in graphical form by computer-controlled plotters for an immediate
evaluation by the investigator.
Mathematical modeling has an advantage which has not yet been used extensively. It
enables representation of results in the form of an animated movie by plotting results of
successive steps of a computation in graphic form on 35- or 16-rora film. This permits presen-
tation of research results in an understandable manner to a large audience. It is also a tool
for the investigator in the understanding of the relationships of transients which do occur.
For example, water level gradients, which are not perceptible in nature or in the physical
model, can be shown in relation to currents by plotting the water surface in an isometric pro-
jection above the plan view with current vectors.
485
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From this presentation, it is clear that the organization of an investigation using
mathematical models is difficult, particularly as experience with these models and with related
fluid flow models has shown that effective and productive research can be done only with a small
group of investigators, where the principal investigator is intimately familiar with all aspects
of the investigation. Research "by committee" is highly inefficient and has led to complete
failure of certain projects in the computer industry. It is apparent from large model develop-
ments in related disciplines that many critical requirements have to be satisfied for the
organization and support of an effective and productive research group developing mathematical
models. It may be expected that the existing pattern of research and also the relations between
academic institutions, governmental research departments, and the field exploration groups will
be drastically altered to adjust to the extensive use of this new tool.
It was indicated that modeling of certain physical processes is possible. Conse-
quently, many physical processes c*an now be studied effectively with mathematical models in
a manner such as used in the modeling of the fluid dynamics of the atmosphere and the oceans.
The tremendous power of digestion and representation of data by computer methods opens the way
for investigations of mass transport in estuaries, circulations and generation of surges and
seiches by wind and tides. All these are important in water quality studies.
Up to this part of this discussion, emphasis has been on the physical processes,
which have been more accessible for formulation and subsequent modeling. Biological systems
are notoriously complex, highly nonlinear, and their interactions not well understood. An
important tool has now appeared, namely BIOMDD, which is an interactive computer-graphics
system for biological modeling. This system operates on an interactive graphics console com-
prising a cathode ray tube screen, a tablet, and a keyboard. It allows a user to draw block
diagrams of the biological system, each component of which can be defined by another block
diagram. The relationships which are presumed by the investigator in the model component can
be inserted as differential equations, chemical equations, or data curves. The system allows
use of handprinted text or the use of a keyboard for the mathematical expressions and param-
eter values, etc. The system operates in two phases, a construction phase, in which the user
draws and specifies models, and a simulation phase, which simulates the models and outputs
results. In the simulation phase the computational programs are written by the computer and
subsequently used in the simulation. Graphics results are depicted on the screen during simu-
lation. The investigator controls nearly all action with a pen-like stylus on the tablet. The
mathematical models of biological communities of the type described in this report by Dr. Faulik
can become operational In a single session. Since the modeling system is interactive, new path-
ways envisioned by the Investigator can be explored vejry rapidly; also, the Influence of coef-
ficients can rapidly be evaluated.
From this discussion it will be clear that mathematical models will become a more
powerful tool as time progresses, a tool whose usage we have to learn, but also a tool which
we urgently need for solving one of the most threatening problems of our civilization.
REFERENCES
The Rand Corporation, 1971: Water quality simulation in Jamaica Bay (16-mm computer graphics
output, 3 minutes duration).
Groner, G. F., R. A. Berman, R. L. Clark and E. C. DeLand, 1970: BIOMOD: A user's view of an
interactive computer system for biological modeling (a preliminary report). RM-6327-
NIH, The Rand Corporation, August 1970.
486
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4.4 DISCUSSION by W. H. Espey
Estuaries constitute one of our most valuable and vulnerable natural resources. As
the economic value of the estuarine zone rises, and population pressures increase, the conflict
between competing uses of the estuaries becomes a vexatious problem. Approximately 30% of the
total population live within fifty miles of the coastline of the continental United States, and
of this coastline, nearly 857. of the Atlantic and Gulf Coast, and 15% of the Pacific Coast are
estuarine (Singer 1969). Yet this same coastal belt represents only 8% of the total land area
of the United States (Pritchard 1966). As the entire population call upon and derive some bene-
fit from the estuarine zone, the nation has been forced to recognize that what it has in surplus
is now in jeopardy. Increasing conflict over the use of our valuable estuaries has dictated the
necessity for comprehensive water quality management of this valuable natural resource.
Water quality models play an important role in this management, as they provide a
tool by which quantitative assessment can be made of various water quality control programs and
their effect on the marine environment. Numerous management alternatives, such as low flow
augmentation, relocation of outfalls, various waste treatment schemes, and even physical modi-
fication of the estuary, can be examined with respect to meeting specified water quality stan-
dards within the estuary.
All too often, however, the role which the models are to play in the management process
is hampered from the outset by a lack of clear definition of the water quality problems which the
models are expected to address. Specification of the requirements of the water quality models,
that is what the models will be expected to do, is essential in the initial selection process.
These requirements will undoubtedly influence the modeling approach. In many cases a slide rule
computation will suffice to answer certain questions; however, detailed questions with regard to
such things as waste distribution around an outfall, or the effect of a barrier on the water
quality conditions in an area of the bay require a more detailed model. Existing models or
modeling techniques may be sufficient, however, the development of new techniques might be
required. These are the questions which should be resolved early in the course of any modeling
endeavor.
A particular aspect of the selection process concerns spatial resolution. In general,
models which describe the distribution of a constituent within an estuarine system, while capable
of evaluating the effect of a single discharge on the total system, cannot yield detailed infor-
mation in the immediate vicinity of the discharge. A good example is the evaluation of a thermal
discharge into the Calveston Bay system as described in the Case Studies, Chapter VII, Section 5.
The models developed as a part of the Galveston Bay Project, utilizing a one square nautical
mile (850 acres) spatial resolution, can hardly be expected to describe the mixing zone of the
discharge which averages some 650 acres. In order to evaluate the thermal effects near the out-
fall, a much finer resolution is required. These fine grid models, as they are sometimes re-
ferred to, are based on the same physical principles; however, new problems are often encountered.
Experience has shown that boundary conditions, stability criteria, source and sink definition,
and in a more general sense, solution techniques may well differ in this fine grid environment.
In summary I wish to emphasize that a clear definition of the kinds of problems which
the models are expected to address is of paramount importance in the design or selection of the
appropriate model. A't times users of water quality models have the impression that any model
can do everything. Simply stated, a lack of communication often exists between the modeler and
the user. If we are to successfully withstand the challenge of resolving the conflict of demands
on our estuarine waters, given all the technical and financial limitations, enough cannot be
487
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said for a sound foundation of discussion between the primary participants in the modeling
effort. In the future we can expect new problems in our estuaries as a result of new waste
materials from industry. Increased concern for ecological considerations will probably require
the modeling of other constituents such as toxicity and certain heavy metals in the marine
environment. This will require investigation of the various sources and sinks that effect the
fate of that constituent in the estuary.
•REFERENCES
Pritchard, D. W., 1966: Fisheries vs. the exploitation of the nonextractive resources in
estuaries. Exploiting the Ocean. Transactions of the Second Annual Marine Technology
Society Conference and Exhibit, Washington, D. C., pp. 173-179.
Singer, S. Fred, 1969: Federal interest in estuarine zones builds. Environmental Science and
Technology. 3, 2 (February), pp. 124-131.
488
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5. REVIEW AND COMMENTS
Arthur T. Ippen
The state-of-the-art of our knowledge on estuarine water movement and quality with
its transient features has been in rapid development in the past ten years. While much has
been accomplished many more gaps in our knowledge and in our methodology of dealing with estua-
rine management problems have also been laid bare. With our Increased capabilities in the
analytical, experimental and computer-based methodologies, progress is substantial and pro-
ceeds unabated as more sophisticated demands are placed before engineers and managers by the
national community concerned with problems in shoreline and estuarine environments.
The more intensive research activities started more than twenty years ago when the
U. S. Corps of Engineers found its design capabilities for proper maintenance of navigation
channels in tidal regions severely limited. This was true particularly with regard to the
unknown interaction of the saltwater-freshwater flows in estuaries to which the shoaling pat-
terns had been traced. It proceeded through its Committee on Tidal Hydraulics towards the
systematic investigation of these phenomena through basic analysis and experimentation on phys-
ical models assisted by interested members of the academic community. The fortuitous advent of
the computers supplied a vastly increased range of opportunities for extensive analytical
studies to these investigators. More detailed solutions of the basic differential equations
for tidal and water quality conditions in real estuaries became possible with physical informa-
tion supplied by hydraulic models and field explorations. Concern with the dispersion of con-
servative and nonconservative pollutants and with their effects on water quality had developed
greatly the range of interest in the governing hydrodynamlc aspects of the complex flows in
estuaries.
A detailed summary of the state-of-the-art at this time as attempted in this volume
seems therefore highly appropriate. The array of contributions by the leading investigators is
most impressive not only with regard to their individual summaries but also with regard to the
effort made to express alternative viewpoints and to reconcile "interdisciplinary" problems.
The wide range of topics covering hydrodynamics, chemical and biological processes and the
various modes of the research technologies place the subject of the estuarine water quality
analysis beyond the expertise of individuals. Hence, this collective effort by carefully
selected professionals was the only feasible way towards a comprehensive review of present-day
knowledge. In the opinion of this reviewer, who had the privilege to follow its development,
it is a most successful effort and should be most useful to all those active in the field who
must be aware of the wider aspects and implications of their particular tasks in the context
of estuarine water control.
Limiting the following remarks to the area of this reviewer's interests, a few more
specific comments are. made on the chapters pertaining to the primary dynamics of the carrier
fluids in estuaries, the saline and fresh waters involved in the tidal processes in complex
geometric boundaries.
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It is fair to say at this point that ouch of our knowledge in the past in the area of
prediction of tidal phenomena in estuaries including the saltwater-freshwater mixing processes
was primarily guided by investigations on "Physical Hydraulic Models." The chapter under this
title written by my colleague, Dr. D. R. F. Harleman, gives an excellent summary of the analyti-
cal premises and of the physical limitations of this methodology of problem-solving in estu-
aries. Hydraulic models have been built all over the world and have acquired a reputation of
general reliability for the solution of specific engineering problems. As long as the predic-
tive capacity pertained to tidal stages and velocities in complex estuaries this confidence
level was justified even to the extent of analyzing shoaling trends under verified salinity
intrusion patterns. As Dr. Harleman points out, however, severe restrictions exist with regard
to the utilization of hydraulic models for pollutant dispersion studies in view of the usual
distorted scales employed. The analytical conditions for dispersion in the nonsaline portion
of the estuary are different from those in the salinity intrusion length. It is most important
to realize this limitation when conducting pollutant dispersion experiments; nevertheless com-
parative studies of dispersion in three-dimensional models can still serve useful purposes
since, except for expensive field studies, no other means are at hand to predict dispersion
coefficients which are also necessary for numerical models. However, it is clear that mathe-
matical modeling with its more flexible possibilities of changing boundary conditions will re-
place the physical models in estuaries of simple geometry where one-dimensional approaches are
deemed adequate. The value of Dr. Harleman's Chapter V on "Physical Hydraulic Models" lies in
its complete and sound examination of the analytical principles for such models and of the
limitations inherent in the impossibility of meeting all similarity criteria within the eco-
nomic restraints applicable to model construction and operation. This chapter must be con-
sulted when hydraulic models are being planned so that their probable usefulness for the
solution of the respective problems is apparent from the outset.
Chapter II on "Hydrodynamic Models" is divided into three parts successively proceed-
ing from the complex cases of three-dimensional and two-dimensional models treated by
Dr. Pritchard to the one-dimensional models dealt with by Dr. Harleman. This sequence corre-
sponds to the progressive ability to deal with mathematical modeling as the number of dimensions
considered is decreased, but also to the increasing inability to represent mathematically all
the physical factors adequately which usually govern the tidal and dispersion phenomena in
estuaries. Research in the area of hydrodynamic modeling for practical reasons is proceeding
therefore in the opposite direction replacing bulk properties of the one-dimensional models
with their more sophisticated components which represent more realistically the details of the
physical system in the two- and three-dimensional models. This research is presently most
active and, while difficult, will yield increasingly useful returns as computational techniques
and capacities are developed and more competently exploited in addition to more exhaustive field
studies to define necessary Inputs to the two- and three-dimensional schemes. Dr. Pritchard
sets forth very clearly the various procedures and assumptions needed to make the three-
dimensional set of equations more amenable to solution. In particular, attention is given to
the definition of the eddy-diffusivities and of the eddy-viscosities, which have been deter-
mined numerically only for a few isolated instances. Analytical determination of these
coefficients in salinity Intrusion regions with various degrees of mixing are not available.
Numerical solutions for three-dimensional mathematical models of real estuaries are concluded
to be still beyond practical attainment.
The various approaches to two-dimensional models are also discussed in detail by
Dr. Pritchard based on the introduction of vertically averaged "effective" diffusion coeffi-
cients for the two horizontal directions. These are different in value from the three-
dimensional coefficients and again are required to be known for a numerical solution. This
490
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dilemma is shared to a lesser degree by the scheme proposed by J. J. Leendertse, but also this
scheme requires prior knowledge of alternate local diffusivities as vertically averaged coeffi-
cients. Specific applications of two-dimensional models covering tidal and salinity character-
istics are not given. It should be feasible, however, to cope numerically with two-dimensional
models at least in cases where the estuary is vertically nearly homogeneous in velocity and
salinity.
Dr. Fritchard appropriately calls special attention in both of his sections to the
presence in the dynamic equations of the isobaric gradients in the salinity intrusion zones.
Their numerical values are shown to be most significant as illustrated by sample calculations
and comparisons with surface gradients.
The "One-Dimensional Mathematical Models" which so far have had considerable practi-
cal use are described in careful detail by Dr. Harleman in Chapter II. Since many schemes have
been developed and are used, specific attention is focused on the definitions of velocity and
salinity as averaged in various schemes over time and space. Three-dimensional mixing pro-
cesses are not neglected but are incorporated in the redefined longitudinal dispersion coeffi-
cient. The independent variables are reduced to the tidal velocity, the cross-sectional area
and the dispersion coefficient, all of which may be functions of longitudinal distance and time.
The determination of the redefined dispersion coefficients for the one-dimensional case has
yielded practical values from theory for uniform density sections and from salinity measure-
ments for the intrusion regions. Tidal velocities are readily computed from numerical schemes
while the necessary cross-sectional information is available from surveys and is used to divide
the estuary into appropriate sections for numerical computation. A number of specific solu-
tions are reviewed from the simple case of the uniform estuary for which extensive experimental
material exists to the more complex cases.
Emphasis is rightfully given by Dr. Harleman to the solution of the real time mass
transfer equations as being most accessible through theoretical and practical determinations
of the dispersion coefficients. The difficulties encountered with suitable dispersion coeffi-
cients in mathematical models under nontidal advective conditions are effectively illustrated
by examples. It is concluded that the real time mass transfer equation offers the best
approach to future analysis when finite-difference techniques are used in computer solutions
and results in most realistic representations of dispersion phenomena. Since the major prob-
lems of pollution center on the dispersion of nonconservative substances and on chemical-
biological interaction in estuaries, this aspect seems most promising for future progress in
research.
In conclusion it may be stated that this entire estuarine state-of-the-art study as
represented by this timely collection of reviews by different authors fills an important func-
tion in our still continuing search for better predictive as well as monitoring methods.
Undoubtedly major mantnade changes in the estuarine environments of this country will be forth-
coming in the future. Their effects on the ecology can hopefully be assessed with greater
confidence. Research as reviewed in this volume will contribute to our awareness of the con-
sequences of such changes and will help to prevent unexpected and possibly irreparable damage.
May this volume also stimulate the urgently needed support for expanded research from all
agencies concerned so that present momentum in interest and knowledge may not be dissipated.
We have a long road ahead.
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CHAPTER X
CONCLUSIONS AND RECOMMENDATIONS
Throughout this report, conclusions have been drawn and discussed concerning the
various aspects of estuarine water quality modeling. Recommendations have been made concerning
the research required to remedy present(deficiencies in the models and to develop more advanced
models capable of treating the complexities of estuarine water quality. The background develop-
ment and discussion context are basic to their interpretation, and therefore the reader is
referred to the individual chapters for the recommendations and conclusions appropriate to that
topic. Nevertheless, it is of value to attempt a compilation and summary in this section.
As in the preceding sections of this report, we continue to discuss separately the
hydrodynamic processes and the reaction processes. As emphasized below, however, in a specific
estuary, the relative importance of hydrodynamics and kinetics must be ascertained in applying
or evaluating the adequacy of a model.
1. CONCLUSIONS
1.1 ESTUARINE HYDRODYNAMIC MODELS
(1) The adequacy of present hydrodynamic models is dictated to a large extent by
their dimensionality. The present state-of-the-art of one-dimensional hydrodynamic models is
well advanced, and good results may be obtained if one employs "real time" computational tech-
niques especially in the constant density region. The existing theory and computational tech-
niques for vertically averaged two-dimensional hydrodynamic models (particularly appropriate
for broad, shallow estuaries) appear to be adequate although in many instances the application
may have been less than state-of-the-art. The theoretical principles underlying the vertical
plane (laterally averaged) two-dimensional models as well as the general three-dimensional
models are generally well-in-hand. However, the solution techniques are not advanced at all
due to the necessity for representing gravitational circulation, which requires solution of
the coupled equations of motion and salt transport. Some similarity solutions have been ob-
tained for the steady-state two-dimensional vertical plane model.
In each case, the relative importance of the various terms in the momentum balance
must be ascertained. One term in particular, the horizontal pressure gradient due to varying
density, is often omitted, yet appears to be important in many estuaries. Inclusion of this
term may also present computational difficulties as it requires simultaneous solution for at
least the salinity distribution.
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(2) The greatest weakness of present hydrodynamic models is the treatment of the
eddy diffusivities and dispersion coefficients, particularly within the salinity intrusion
region of estuaries. Present methods rely on measured data (usually salinity), adjusting the
dispersion coefficients so that the model reproduces the measured data. The effects of the
estuarine environment and physiography on the dispersion are virtually unknown, so the predic-
tive capabilities of the model are severely limited. A theoretical or quasi-theoretical founda-
tion is needed for the calculation of the dispersion coefficients on the basis of external
influences (wind stress, waves, etc.) and the gross hydrographic parameters of the estuary.
The meaning and magnitude of the diffusion-dispersion coefficient is highly dependent
upon the spatial and temporal averaging employed in deriving the fundamental equations. For
the one-dimensional case, dispersion coefficients for a "real time" model (which considers the
tidal motion) may be estimated sufficiently well in the uniform density region by an extension
of the Taylor-Elder formula, although within the salinity intrusion region dispersion coeffi-
cients must be determined empirically. For a one-dimensional "non-tidal advective" model (which
considers only the freshwater through-flow), dispersion coefficients can only be determined
empirically. In the application of one-dimensional non-tidal models to some estuaries, the
constituent profiles may not be especially sensitive to the dispersion coefficients, so that
accuracy in their determination is not critical. It has been suggested that for long-term
estuary studies, both real time and non-tidal advective models be developed, using the former
to calibrate the latter.
(3) Although one-dimensional and vertically averaged two-dimensional models have been
employed in numerous estuaries, there is a sparsity of thorough verification. The usual prac-
tice is to verify the model by comparing the tidal height and phase at various points in the
estuary. It appears, however, that this is an inadequate measure of how well the model pre-
dicts currents in the estuary. Although the question of field programs and data collection
lies somewhat outside an assessment of estuarine modeling, nevertheless this assessment requires
adequate observations in real estuaries in order to identify the relative importance of the
hydrodynamic influences. For many types of estuaries, such observational data are lacking.
1.2 ESTUARINE WATER QUALITY MODELS
(1) It is concluded that as a general rule, the modeling of the hydrodynamic trans-
port of a constituent in an estuary is much further advanced than the modeling of its reaction
kinetics. This must be qualified according to the particular constituent and circumstance:
there are important problems which are hydrodynamically dominated, just as there are consti-
tuents whose reaction kinetics are well established. However, the most commonly unsatisfactory
aspect of present water quality models is the specification of the source and sink terms.
(2) Many of the physico-chemical processes affecting the concentration of parameters
lack adequate formulation, in that not only are the coefficients unknown, but the forms of the
reactions are poorly established. These include sedimentation and deposition of particulate
matter, nonlinear reaction kinetics, surface exchange of gaseous constituents, and chemical
and biological reactions.
(3) Many of the important water quality parameters behave in a highly coupled manner,
e.g., the species of nitrogen. Specification of the reactions involved as well as the formula-
tion of computational techniques for treating general nonlinear interdependencies ("feedforward"
and "feedback" reactions) are inadequate at the present.
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(4) Small-scale problems such as distributions in the locality of a discharge, ther-
mal discharges in the initial and intermediate scales, nearshore disposal, peripheral embayments,
etc., pose modeling problems for which grosser models as applied to the entire estuary are not
sufficient. These problems require refinement in both the transport and reaction kinetics, and,
due to the fine temporal and spatial detail, also may impose extreme demands on the solution
procedures.
(5) Modeling of the relation of water quality and the estuarine biota is not well-
advanced at all. Models of phytoplankton production, of nitrogen cycling, and of gross ecologi-
cal parameters (species diversity, bioassays of toxicity) have been attempted, but presently
these are at best experimental. Only recently have Michaells-Menten type kinetics been utilized
in estuarine and oceanic models. Models of finfish population dynamics, on the other hand, have
a longer history and, hence, a somewhat more advanced development.
1.3 UTILITY OF PHYSICAL MODELS
(1) The greatest potential value of the physical model is its ability to represent
three-dimensional systems, whose complexity precludes conventional analytic techniques. Despite
the lack of complete similitude, due to vertical distortion, useful results can be obtained by
comparative tests of alternative schemes.
(2) It must be borne in mind that estuarine physical models are Froude-scaled and
therefore, if properly verified, are capable of representing Froude-type phenomena such as
tidal currents, momentum entrainment and gravitational circulation. On the other hand, local
currents and turbulent eddies which are not Froude-type phenomena are not necessarily well-
represented in physical models. For this reason, there is considerable distortion of diffusive
processes in the physical model that makes its utility in quantitative concentration distribu-
tion studies dubious.
The validity of extrapolating dye releases In the physical model to the real estuary
is presently in doubt, due not only to questions of similtude, but to questions of measurement
in the physical model. Inadequate verification of such releases prevent a thorough appraisal
of the model's utility in this case.
(3) Modeling of complex reactive constitutents and coupled reactants at present is
not possible with the physical model.
(4) From a qualitative standpoint, the physical model possesses an excellent demon-
stration capability for the visualization of flow patterns and resultant concentration distri-
butions. With the formidable problems of communicating the results of water quality investi-
gations to a nontechnical audience, this capability should not be underrated.
1.4 SOLUTION TECHNIQUES
(1) In general, analog computers are falling into disuse relative to their digital
counterparts. This appears to be due to the unwieldiness and lack of versatility of the analog
computer as well as the increased availability and ease of programming of digital computers.
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Both have advantages and disadvantages, and the choice of a particular technique in the final
analysis is contingent upon the characteristics of the equations, the intended use of the model
results, required accuracy, available resources, experience of the personnel, and similar cri-
teria which are really ancillary to the problems of estuarine water quality modeling.
(2) The technique of finite-difference approximations is the most popular and promis-
ing method of solution to the complex partial differential equations which constitute estuarine
water quality models. In applying this technique, consideration should be given to the problems
associated with the approximations, for instance, accuracy and stability of the computation,
convergence questions (when appropriate), numerical distortion of the transport and reactive
processes, etc., in order to ensure the validity of the results. In many past numerical studies,
these considerations have been disregarded.
1.5 APPLICATIONS
(1) For any particular estuarine quality problem, a preliminary effort should be
devoted to determining the model appropriate to that problem. The features of an estuarine
system should be considered in determining the spatial dimensionality, temporal and spatial
refinement, and so on, of the model. All too often, the physical integrity of the model is
sacrificed for simplicity or expedience. On the other hand, models have been constructed which
are far too detailed for the problem under investigation. Similarly the relative influence of
the hydrodynamic and the kinetic terms should be investigated, so that the model employed is
neither overly complex nor superficial.
(2) Utility of models in evaluating both the effects of streamflow modification
(including augmentation, regulation, and upstream diversion) and the effects of physiographic
modification is basically limited by the predictive capability of the hydrodynamic models. In
the case of the former, many estuaries amenable to a one-dimensional treatment have been modeled
with various magnitudes of inflow and the results employed in management studies. Very little
work using more complex mathematical models has been performed. Physiographic alterations have
been studied almost exclusively with physical models.
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2. RECOMMENDATIONS FOR FUTURE RESEARCH
The distinction should be recognized between a conceptual problem and a computational
problem. Much work is required in strengthening the conceptual aspects of estuarine models,
whereas available computational techniques, employed rigorously, are in many cases sufficient.
Emphasis in future research should be placed upon the source and sink processes of
specific water quality parameters. There are, of course, important aspects of estuarine hydro-
dynamics which require continued study, but an increased effort is needed in the study of the
various reactions in order to produce models adequate for treating estuarine water quality
problems. Such work will not necessarily be strictly of a modeling nature. Rather, laboratory
and field studies of the processes will also be required. The results of these studies, how-
ever, should be capable of incorporation in estuarine models.
2.1 HYDRODYNAMIC MODELS
(1) Research is required in establishing methods whereby the dispersion coefficients
(or eddy diffusivities, as the case may be) may be predicted from hydrographic and environmen-
tal factors.
(2) Development should be undertaken of models capable of representing the vertical
dimension, viz. laterally averaged two-dimensional models and three-dimensional models. These
models must represent the hydrodynamic effects of density gradients, and may therefore require
the development of appropriate computational procedures.
(3) In all hydrodynamic models, the importance of the pressure gradient due to den-
sity (i.e., due to the internal slope of the pressure surfaces) must be considered and, if
necessary, included in the model.
2.2 WATER QUALITY MODELS
(1) Work is needed in the study and mathematical delineation of the reactions to
which estuarine constituents are subjected. Clarification of the processes affecting dis-
solved oxygen is required, as well as modeling of other parameters, e.g., sulfates and sul-
fites, heavy metals, carbon dioxide, partlculate matter, and so on. The phenomenon of gaseous
diffusion through the surface and its dependence upon physico-chemical features of the estuary
require study.
(2) Research is needed into the behavior and modeling of coupled reactions. In
particular, the problem of general nonlinear coupled reactions (with feedforward and feedback
dependencies) requires Investigation from both a physical and a computational standpoint.
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(3) Research should be pursued in the modeling of biota and' their interaction with
constituents of estuarine water. This ranges from models of phytoplankton populations through
models of the higher trophic levels up to fishery resource models.
(A) Models are required that are capable of representing the local effects of a
discharge in the nearshore environment. These will encompass both the hydrodynamics and kine-
tics of such a discharge. Considerable effort may be required in developing suitable computa-
tional techniques for this kind of problem. Physical models may also prove to be useful here,
If accompanied by a rigorous analysis of similtude requirements.
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