&EPA
United States
Environmental Protection
Agency
Environmental Research
Laboratory
Duluth MN 55804
EPA-600/9-80-033
July 1980
Research and Development
Proceedings of the
Second American-
Soviet Symposium
on the Use of
Mathematical
Models to Optimize
Water Quality
Mangement
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7 Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/9-80-033
July 1980
PROCEEDINGS OF THE SECOND AMERICAN-SOVIET SYMPOSIUM
ON THE USE OF MATHEMATICAL MODELS TO OPTIMIZE WATER QUALITY
MANAGEMENT
Bloom-field Hills, Michigan, USA
August 27-30, 1979
Edited by
Way!and R. Swain
and
Virginia R. Shannon
ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
DULUTH, MINNESOTA 55804
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DISCLAIMER
This report has been reviewed by the Large Lakes Research Station,
Environmental Research Laboratory-Duluth, Grosse lie, Michigan, U.S.
Environmental Protection Agency, and approved for publication. Mention of
trade names or commercial products does not constitute endorsement or recom-
mendation for use.
n
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FOREWORD
The Environmental Research Laboratory-Duluth is concerned with the
effects of pollutants on freshwater ecosystems, particularly the Laurentian
Great Lakes. The development and verification of mathematical models de-
scribing the transport, fate and effects of pollutants in freshwater eco-
systems are carried out at the Large Lakes Research Station at Grosse He,
Michigan.
A part of this development, calibration and verification activity has
been carried forward in cooperation with scientific personnel from the
Soviet Union under the section entitled, Prevention of Pollution of Lakes
and Estuaries under the Water Pollution Portion of the US-USSR Joint
Agreement on Cooperation in the Field of Environmental Protection. The
contributions to new knowledge contained in this volume clearly demonstrates
the utility of joint scientific collaborations on an international basis.
Norbert Jaworski, Ph.D
Director
Environmental Research Laboratory
Duluth
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PREFACE
This volume contains the proceedings of the papers presented at the
Second US-USSR Symposium on the Use of Mathematical Models to Optimize Water
Quality Management. All of the papers were presented either in English or
in Russian with simultaneous interpretation into the corresponding language
at the Cranbrook Institute of Science in Bloomfield Hills, Michigan, USA
during August 27-30, 1979.
Identical copies of this volume are being simultaneously published in
the Russian language under the direction of Dr. A.M. Nikavorov, Director of
the Institute for Hydrochemistry at Rostov-on-Don in the USSR. This joint
bilingual publication represents a reaffirmation of the continuing commit-
ment pledged by both countries to cooperative environmental activities.
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INTRODUCTION
The Joint US-USSR Agreement on Cooperation in the Field of Environmental
Protection was established in May of 1972. These proceedings result from
one of the projects, Project 02.02-12, Effects of Pollutants on Lakes and
Estuaries.
As knowledge related to fate and transport of pollutants has grown, it
has become increasingly apparent that local and even national approaches to
solving pollution problems are insufficient. Not only are the problems
themselves frequently international, but an understanding of alternate
methodological approaches to the problem can avoid needless duplication of
efforts. This expansion of interest from a local and national framework
represents a logical and natural maturation from the provincial to a global
concern for the environment.
In general, mankind is faced with very similar environmental problems
regardless of the national or political boundaries which we have erected.
While the problems may vary slightly in type or degree, the fundamental and
underlying factors are remarkably similar. It is not surprising, therefore,
that the interests and concerns of environmental scientists the world over
are also quite similar. In this larger sense, we are our brother's brother,
and have the ability to understand our fellowman and his dilemma, if we but
take the trouble to do so. It is this singular idea of concerned scientists
exchanging views with colleagues that provides the basic strength for this
project. While our methods may vary, our goals are identical, and therein
lies the value of such a cooperative effort.
Wayland R. Swain, Ph.D.
Project Officer, U.S. Side
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CONTENTS
Foreword iii
Preface iv
Introduction v
Figures ix
Tables xviii
Acknowledgment xx
1. Basic Trends in the Study of Hydrochemical Fields
and the Structure of Their Space-Time Heterogeneities
V.L. Pavelko, B.M. Vladimirskiy, I.T. Gavrilov and
G.V. Tsytsarin 1
2. Modeling the Great Lakes - A History of Achievement
William C. Sonzogni and Thomas M. Heidtke 9
3. Data Management Requirements for Great Lakes Water
Quality Modeling
William L. Richardson 37
4. Optimal Sampling for Long Term Trends in Lake Huron
David M. Dolan 58
5. A Model Approach to Estimating the Effects of Anthro-
pogenic Influences on the Ecosystem of Lake Baikal
A.B. Gorstko, Yu.A. Dombrovskiy, V.V. Selyutin,
F.A. Surkov, A.M. Nikanorov and A.A. Matveev 71
6. Species Dependent Mass Transport and Chemical Equilibria:
Application to Chesapeake Bay Sediments
Dominic M. DiToro 85
7. Simulation of the Distribution of Polluted Water in
Reservoirs From Concentrated Emissions
A.V. Karaushev and V.V. Romanovskiy 122
8. Results of a Joint USA/USSR Hydrodynamic Modeling Project
for Lake Baikal
John F. Paul, Alexandr B. Gorstko and Anton A. Matveyev . . 129
vn
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9. The Structure of Hydrochemical Fields and Short-Term
Prediction
A.M. Nikanorov, B.M. Vladimirskiy, V.L. Pavelko, Ye.V.
Melnikov and K.L. Botsenyuk 156
10. Transport of Mining Waste in Lake Superior
M. Sydor, G.E. Glass and W.R. Swain 180
11. Principle of Organization of an Automated Information
System
'V.L. Pavelko 197
12. A Multi-Layered Nested Grid Model of Lake Superior
G.J. Oman and M. Sydor 207
13. A Review of Some Methods and Parameters Used in Assessing
Effects of Water Intakes on Fish Populations
Richard L. Patterson 237
14. Control of the Water Resources of the Azov Sea Using the
"Azov Problem" Family of Simulation Systems
A.B. Gorstko, F.A. Surkov, L.V. Epshteyn and A.A.
Matveyev '. 252
15. The Transport of Contaminants in Lake Erie
Wilbert Lick 261
16. Self-Organization of Three-Dimensional Models of Water
Pollution
A.G. Ivakhnenko and G.I. Krotov 306
17. A Spatially-Segmented Multi-Class Phytoplankton Model
for Saginaw Bay, Lake Huron
Victor J. Bierman, Jr. and David M. Dolan .' 343
18. A Segmented Model of the Dynamics of the Ecosystem of
Lake Baikal Considering Three-Dimensional Circulation
of the Water
V.V. Menshutkin, O.M. Kozhova, L.Ya. Ashchepkova and
V.A. Krotova 366
19. The Task of Optimal Planning of Discharge of Pollutants
in a Dynamic Model of Self-Purification of a Body of
Water
O.M. Kozhova and L.T. Aschepkova 379
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FIGURES
Section Page
1 Variation of level of study of a component with
interval of sampling 5
3 Great Lakes modeling-management process 39
3 Large Lakes Research Station mini-Storet system 45
3 Summary of Lake Superior data from Storet water quality
file 48
3 Summary of Lake Michigan data from Storet water quality
file 49
3 Summary of Lake Huron data from Storet water quality
file . . . 50
3 Summary of Lake Erie data from Storet water quality
file 51
3 Summary of Lake Ontario data from Storet water quality
file 52
3 Summary of Connecting Channels data from Storet water
quality file 53
4 The Lake Huron system 60
4 Simplified mass balance model 62
5 Diagram of surface currents in Baikal during the ice-
free season 72
5 Time-series plots of concentrations of an arbitrary
pollutant in the various regions of Lake Baikal 75
5 Concentration field of a pollutant in the 0-20 m layer
near BCPC 77
5 Diagram of cycle of matter and energy in pelagic Lake
Baikal 79
ix
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Section Page
6 The vertical distribution of sediment organic carbon
and nitrogen as percent dry weight 93
6 Chesapeake Bay Station 856: Observed and computed
vertical distribution of total aqueous carbon dioxide,
aqueous ammonia nitrogen and their ratio 98
6 Chesapeake Bay Station 856: Observed and computed
vertical distribution of interstitial water 100
6 The effects of species dependent transport and solid
phase reactions on net alkalinity 103
6 Characteristic shapes for net alkalinity 105
6 Chesapeake Bay Station 858C: Observed and computed
vertical distributions of dissolved methane, mg
carbon/1; total aqueous carbon dioxide, mg carbon/1;
and total gas phase concentration, moles of gas/1 of
gas plus aqueous phase volume 108
6 Chesapeake Bay Station 858C: Observed and computed
vertical distributions of dissolved nitrogen gas, mg
nitrogen/1; dissolved argon, mg argon/1; and mole
ratio 109
6 The effect of an upward gas phase velocity Ill
6 The effect of an upward gas phase velocity with zero
boundary conditions 112
6 The effect of an upward gas phase velocity with zero
lower boundary conditions 113
7 Diagram for calculating the diffusion of pollutants in
an aquatic ecosystem 124
8 Lake Baikal 131
8 Frequency of winds over Lake Baikal'in the summer and
autumn 137
8 Hydrodynamic model calculation for Lake Baikal with
southwest wind 139
8 Hydrodynamic model calculation for Lake Baikal with
southwest wind 140
8 Hydrodynamic model calculation for Lake Baikal with
northwest wind 141
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Section Page
8 Hydrodynamic model calculation for Lake Baikal with
northwest wind 142
8 Observed surface currents in the Selenga River Region
of Lake Baikal 143
8 Lake Baikal whole lake dominant currents 145
8 Dispersion model calculation for Lake Baikal with
southwest wind 146
8 Dispersion model calculation for Lake Baikal with
southwest wind 147
8 Dispersion model calculation for Lake Baikal with
southwest wind 148
8 Dispersion model calculation for Lake Baikal with
southwest wind 149
8 Sample results from Hydromet cruise in Selenga Shallows
on 28-29 May 1976 150
8 Sample results from Hydromet cruise in Selenga Shallows
on 28-29 May 1976 151
8 Landsat satellite image of Lake Baikal 153
9 Berezina River at Gorval, 1951-1956 165
9 Berezina River at Gorval, 1951-1956 168
9 Berezina River at Bobruysk 170
9 Sula River at Zelenkova 171
10 Study area showing the outline of a deep trough where
tailings are discharged and the discharge source loca-
tion at Silver Bay 181
10 Comparison at Duluth of the measured winds and wind
function used for modeling of Lake Superior for the
November 1975 storm 184
10 Comparison at Duluth of the measured water level fluctu-
ations with the fluctuations derived from the numeri-
cal model of Lake Superior for the November 1975
storm wind 185
10 Water transport due to westerly wind stress 187
xi
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Section Page
10 Calculated transports for November 13, 1975 showing
water movement for high fetch northeasterly winds .... 188
10 Numerical simulation of the plume of mining waste from
point source at Silver Bay, Minnesota 190
10 Turbidity vs. Landsat Band 4 intensity above background . . 191
10 Turbidity plume derived from Landsat data for November
14, 1975 . 192
10 Discrete patches of upwelled tailings observed in
Landsat data for July 2 and 3, 1973 195
11 Variation in number of errors v with time of continuous
work t 203
12 Grids used in various models 208
12 Map of Lake Superior showing regions deeper than 200 m . . 215
12 Surface currents after 3 hours o' constant easterly
winds 216
12 Surface currents after 9 hours of constant easterly
winds . .' 217
12 Surface currents after 15 hours of constant easterly
winds ' . . . . 218
12 Surface currents after 24 hours of constant easterly
winds 219
12 Currents in layer 2 after 24 hours of constant easterly
winds . 220
12 Currents in layer 3 after 24 hours of constant easterly
winds . 221
12 Currents in layer 4 after 24 hours of constant easterly
winds 222
12 Layer 2 downdwelling in western Lake Superior after 24
hours of constant easterly winds ............ 223
12 Layer 2 upwelling in western Lake Superior after 24
hours of constant easterly winds 224
12 Surface currents in nested subgrid after 24 hours of
constant easterly winds . 225
xii
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Section Page
12 Layer 2 currents in nested subgrid after 24 hours of
constant easterly winds 226
12 Surface currents in nested subgrid after 24 hours of
constant westerly winds 227
12 Layer 2 currents in nested subgrid after 24 hours of
constant westerly winds 228
13 Box and arrow diagram summarizing transfers of fish
biomass 240
14 Diagram of regions of the Azov Sea 256
14 Program structure of the "Azov Sea" SS 259
15 Lake Erie bottom topography 262
15 Settling velocity versus percent of suspended sediment
for the Western Basin sediment 265
15 Side view of the flume 270
15 Example of the concentration time history data for the
shallow-based sediment with TW = 0.92 dynes/cm^ and
a water content of 6.7% 272
15 The entrainment rate as a function of the average bound-
ary shear stress for the shale-based, Western Basin,
and Central Basin sediments 273
15 The entrainment rate as a function of shear stress of a
linear scale for the Western Basin sediment 274
15 The reflectivity parameter as a function of shear stress
for the shale-based, Western Basin, and Central Basin
sediments 275
15 Friction coefficient as a function of Reynolds number
for an oscillatory flow over a smooth bottom 277
15 Significant wave height in meters for a wind speed of
11.2 m/sec (25 mi/hr) and a southwest wind 278
15 Significant wave period in seconds for a wind stress of
11.2 m/sec (25 mi/hr) and a southwest wind 279
15 Bottom stress in dynes/cm2 for a winds speed of 11.2
m/sec (25 mi/hr) and a southwest wind 280
x i i i
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Section
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
Near-surface total suspended solids map for the Western
Basin of Lake Erie on March 8, 1976
Observed surface sediment concentrations for the Western
Basin of Lake Erie on March 11, 1976
Calculated surface sediment concentrations on the Western
Basin of Lake Erie on March 11, 1976
Temperature distribution at 30 days for constant depth
basin
Velocities (u and w) at 30 days for constant depth
basin
Velocities perpendicular to cross-section at 30 days for
constant depth basin
Temperature distribution at 90 days for constant depth
basin
Velocities (u and w) at 90 days for constant depth
basin
Velocities perpendicular to cross-section at 90 days for
constant depth basin
Temperature distribution at 50 days for variable depth
basin
Velocities (u and w) at 50 days for variable depth
basin
Velocities perpendicular to cross-section at 50 days for
variable depth basin
Temperature distribution at 80 days for variable depth
basin
Temperature distribution at 120 days for variable depth
basin
Temperature distribution at 150 days for variable depth
basin
Temperature distribution at 180 days for variable depth
basin
Contaminant concentration at 30 days after release at
surface and 53 km from left side
Page
282
283
284
287
288
289
290
291
292
294
295
296
297
298
299
300
302
XIV
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Section Page
16 Characteristic curves of change of external, mixed and
internal criterion with increasing complexity of model
structure 308
16 Use of simple models for extrapolation of the area of
pollution between three measurement stations 1, 2 and
3 in the direction of the x-axis 313
16 Run-through of simple models and production of data
tables for the prediction problem 316
16 Run-through of simple models of diagonal shape and pro-
duction of data tables for the prediction problem .... 317
16 Selection of a model of optimal complexity based on the
combined criterion 320
16 Change of emission and concentration of pollutants with
time 331
16 Model 1-1 332
16 Model II-l 333
16 Model II-2 335
16 Model II-3 336
16 Model III-l 337
16 Model III-2 339
16 Model III-3 340
17 Saginaw Bay and the spatial segmentation scheme used for
the phytoplankton model 344
17 Sampling station network in Saginaw Bay 346
17 Gradient among spatial segments in annual average
chloride concentration 348
17 Gradient among spatial segments in annual average
phosphorus concentration 349
17 Gradient among spatial segments in summer average
chlorophyll jj concentration 350
17 Schematic diagram of principal model compartments and
interaction pathways 351
xv
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Section
17
17
17
17
17
17
17
17
17
17
18
18
18
18
18
19
19
Comparison between model output and field data for
Comparison between model output and field data for
Comparison between model output and field data for
biomass of non-Nz-fixing blue-greens in segment two ...
Comparison between model output and field data for
Comparison between model output and field data for
Comparison between model output and biomass of green
Comparison between model output and field data for
biomass of non-N2-fixing blue-greens in segment four . .
Comparison between model output and field data for
biomass of N2-fixing blue-greens in segment four ....
Comparison between model output and field data for
dissolved ortho-phosphorus in segment two
Comparison between model output and field data for
dissolved ortho-phosphorus in segment four
Regional ization of Lake Baikal in the model and
direction of horizontal transfer
Geostrophic circulation on the surface of Baikal and
depth of circulation average for August-September
1925-1965
Diagram of horizontal exchange between segment in
model
Horizontal distribution of phytoplankton throughout
the year in the 0-50 m layer (generated by model) ....
Distribution of phytoplankton in Lake Baikal in the
0-25 m layer in 1964
i . = ( i i . i „ . i ~ . i , )
Temperature curve
Page
354
355
356
357
358
359
360
361
363
364
368
369
373
375
376
383
383
XVI
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Section Page
19 Subdivision of body of water into chambers 383
19 Diagram of water exchange 383
19 Picture of propagation of a pollutant through a body of
water corresponding to optimal discharge levels 389
xvii
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TABLES
Section Page
2 Summary of Water Quality Model Relevant to the Great
Lakes 11
2 Summary of Circulation/Transport Models Relevant to the
Great Lakes 14
2 Summary of Available Heat Dispersion, Nonpoint Source,
Toxic Substances and Miscellaneous Models Relevant
to the Great Lakes 16
3 Summary of Principal Model Characteristics 42
3 Summary of Great Lakes Data in Storet 47
3 Storet Water Quality File Summary of Great Lakes
Eutrophication Data 54
4 Lake Huron Major Inflows and Outflows 59
4 Chemical and Biological Indicators of Trophic Status -
Main Lake Huron 61
4 Lake Huron Surveillance Plan Trend Detection Capa-
bilities 65
4 Sensitivity Analysis for Standard Error in Southern
Lake Huron for Total Phosphorus 67
4 Comparison of Stations Required by Different Strategies . . 68
5 Morphometric Characteristics of Lake Baikal 71
5 Biotic Balance of the Pelagic Ecosystem of Lake Baikal . . 78
6 Chemical Fast Reactants Structure and Aqueous Diffusion
Coefficients RQ
6 Chesapeake Bay Sediment Parameters 92
6 Sediment Electron and Nitrogen Stoichiometry 95
xv i i i
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Section Page
6 Station Sediment Parameters 96
8 Annual Average Water Balance for Lake Baikal During the
Period 1901-1970 130
8 Parameters for Lake Baikal Hydrodynamic Model 136
9 Factor Solution for the Results of Chemical Analysis of
Water Samples of the Gorky Reservoir in 1954-1958 .... 158
9 Factor Solution for Results of Chemical Analysis of
Water Samples of Tributaries of the Gorky Reservoir,
1954-1958 159
9 Factor Solution for Results of Chemical Analysis of
Water Samples from Kuybyshev Reservoir, 1954-1961 .... 160
9 Factor Solution for Results of Chemical Analysis of
Water Samples from Kuybyshev Reservoir Taken Near
Vyazovyye (Volga River) and Sokoli Gory (Kama River)
and Near Komsomolskiy (Tailwater), 1958-1959 . 161
9 Factor Solution for Results of Chemical Analysis of
Water Samples of Tributaries of the Volga River and
Kuybyshev Reservoir, 1954-1961 162
9 Factor Solution for Results of Chemical Analysis of
Water Samples from Volgograd Reservoir, 1954-1961 .... 163
9 Factor Solution for Results of Chemical Analysis of
Water Samples from Tributaries of Volga River Below
Kuybyshev Reservoir, 1954-1961 . . 164
9 Factor Solution for Results of Analysis of Water Samples
of Gorval'Tributary, Berezina River, 1951-1956 166
9 Factor Solution for Results of Analysis of Water Samples
from Bobrusk Tributary of Berezina River, 1957-1973 ... 167
10 Comparison of Currents Near Duluth 186
10 Comparison of Currents Near Silver Bay 186
10 Band Combinations for Signature Classification of
Contaminants 193
17 Statistical Comparison Between Model Output and Field
Data 353
xix
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ACKNOWLEDGMENTS
In any project of the scope and complexity of this effort, the Project
Officers become increasingly indebted to a large number of individuals who
contribute their time and effort with no thought of personal gain. Un-
fortunately, the list of persons who materially aided the effort is too ex-
tensive to allow a complete discussion. However, while those persons who
made outstanding contributions to the success of this project are acknow-
ledged below, the editors also wish to thank all those others, both Soviet
and American, whose efforts and assistance smoothed the way to a satisfac-
tory completion of this phase of the project.
Sincere thanks are extended to Mr. Igor Kozak, Mr. Igor Korobovsky, and
Ms. Nina Ivanikiw whose assistance with translations and interpretation at
the time of presentation have made possible the publication of this volume.
The substantial contributions and tireless efforts of Ms. Debra Caudill to
the preparation of the proceedings are acknowledged with deep appreciation.
It is also impossible to ignore the sustained interest and deep commit-
ment to this effort made by Edward Lerchen, Acting President, and his staff
at Cranbrook Institute of Science. Particularly, the planning of Dr. V.
Elliott Smith and the implementation of efforts provided by Millicent
Worrell were far above and beyond the call of duty. It was the contribu-
tions of these, and a host of others at the Cranbrook Institute which made
the symposium so successful.
xx
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SECTION 1
BASIC TRENDS IN THE STUDY OF HYDROCHEMICAL FIELDS AND THE STRUCTURE OF
THEIR SPACE-TIME HETEROGENEITIES
V.L. Pavelko1, B.M. Vladimirskiyl, I.T. Gavrilov2
and G.V. Tsytsarin2
Currently the basic task of hydrochemical research is the development of
prognostic and functional models of water quality. These models must corre-
spond to the requirements not only of operational evaluation, but also of
quality control, i.e., providing the necessary information for making the
corresponding administrative and scientific-technical decisions. To do
this, one must have full and representative data on the quality of water,
and the nature of its changes in time and space to understand the processes
which determine water quality. These data, naturally, should be obtained at
minimum cost. This statement, while obvious, is still improper, due to the
following factors:
a) The increasing difficulty of obtaining objective data describing
mean and extreme values of water quality indices;
b) The set of tasks which must be performed, for which the concept
of "water quality" is ambiguous, and in relationship to which
the "completeness and representativeness" of the information
must be evaluated, with an ever-increasing list of substances
which are limited in terms of maximum allowable concentrations;
c) The lack of any scientifically well-founded criteria or methods
for estimation of the "minimum cost".
Each of these factors is worthy of extensive discussion in both the
theoretical and the practical aspects. In the present article, however,
discussion is limited to the following considerations:
iHydrochemical Institute, 192/3 Stachky Prospect; 344090 Rostov-on-Don,
USSR.
^Moscow State University, Lennin Hills, Moscow, USSR.
-------
1. Under natural conditions, water quality parameters usually vary
in diurnal, seasonal or annual cycles, which may be disrupted
by hydrochemical factors. The existing system of observations
produces more or less reliable information. As the influence
of man becomes stronger, the quality of water undergoes great
changes, the amplitude of which increases greatly. In reser-
voirs and streams, flows of varying density with small degrees
of mixing are seen. Thus, concentrations may vary by orders of
magnitude when measured at points not greatly separated. Many
pollutants do not move in the same way through bodies of water.
The concept of "lag time" for water quality may differ signifi-
cantly, since the speed of a stream varies greatly with depth.
The hydrochemical lag time may also include a component of re-
laxation. Naturally, under these conditions the reliability of
the information produced is determined in the first stage—as a
sample is taken. Depending on the task at hand, the sampling
system must assure either the maximum possible averaging over
time and space coordinates, or the selection of samples in ex-
treme situations. The development of such a system requires
the accumulation of data on structures, and space-time hetero-
geneity of hydrochemical fields for typical aquatic systems.
2. The use of the concept of "minimum costs" involves the solution
of optimization problems, but leaves unanswered a primary ques-
tion: "Can new costs be correlated with old costs, or must the
cost of losses prevented be considered as well?" A comparison
with future costs may also be made, if mankind will be forced
to rehabilitate or restore certain ecologic niches.
It follows then that optimization of the system of observations cannot
be absolute and final, and that suboptimal solutions should be sought for
specific problems which are pressing for the present period of development
of society. These solutions should be systematically reviewed, following
development and application of new achievements in science and technology.
At the present time, obviously, it would be most expedient to:
a) limit consideration to the present level of costs, or plan in-
creased costs within limits of the same scale;
b) consider that the functional weights of the observed components
of chemical composition within the limits of the vector of
states are characterized by a definite hierarchy, e.g., used in
producing a generalized water quality index (Vainer 1975).
Under these conditions, the task of equalization of the degree of hydro-
chemical study of components, locally homogeneous regions, and the signifi-
cance of periods and phases of processes is formulated (equalization of
residual entropy in space and time and with respect to weights of the compo-
nents observed).
To perform this task, it is necessary to consider the following:
-------
1) estimate the space and time study level of each component
given the existing parameters of the observation system;
2) find the mean level of study, and estimate its sufficiency
for the performance of the task at hand;
3) find the variation in the level of study with the space-time
scanning step; and
4) redistribute the equipment of the observation network, in order
to achieve a space-time level of study of each component agree-
ing with its weighted significance.
The practical achievement of these stages of the program requires mas-
sive detailed studies of typical hydraulic projects encompassing character-
istic periods and phases of the operating mode, and the use of water quality
indices. Bodies of water can be classified by means of pattern recognition
algorithms, using data files which have been collected, a priori concepts
from theoretical hydrochemistry, and the data used to classify sources of
pollution. The selection of typical components of chemical composition can
be based on the minimum mean-square deviation of the distance (X) of compo-
nent (K) from the center of the grouping. The search for quasi-steady sec-
tions of a body of water, incorporating periods and phases can be conducted
analogously (Rybnikov 1970; Borishanskiy 1970; Pavelko 1972).
These functions should be performed in the first stage of operation of
the program. After standardization and selection of representative objects,
their hierarchical ranking, evaluation of priorities, ordering of the in-
vestigation is planned in the second stage as a function of the significance
of the object. Finally, the plans for development of the national economy,
the degree of pollution of the water, the readiness of individual observa-
tion subdivisions, and the conduct of planned studies are incorporated as
the third stage in the program.
The fourth stage consists of determination of the variation of the
degree of study of a given component with the interval of sampling, and in-
formation in the coordinates of space and time, using the following arbi-
trary system:
x - distance along a stream or the longest axis of a body of water;
y - distance transverse or across the stream or body of water;
z - depth;
t - time.
The characteristics of the degree of study which have been suggested
(Popov 1953; Linnik 1958; Margolin 1962; Margolin 1965; Pavelko 1972;
Pavelko 1977) include the structural function, the spectral density of dis-
persion or an estimate such as:
°c
-------
which can be transformed to an estimate of the degree of study:
1 = 1-^. (2)
The graph of the variation of the degree of study (I) with the interval
of sampling of information (T) is shown in Figure 1.
Based on such functions for each of the components of a body of water
which is studied, for its quasi-steady regions, for phases and periods, and
considering the actual interval of sampling of information, it is possible
to estimate the degree of heterogeneity of the level of study of a body of
water for each component. This material can be used as a basis for develop-
ment of recommendations for improvement of the observation system. The
parameters of the observation systems can be adjusted to assure identical
levels of study, e.g., equal to the average of all existing levels of study,
or corresponding to some other comparative level, i.e., 0.7, 0.8 (see Figure
1), if this level is considered necessary and sufficient for the present
stage. These functions can be used to select representative sections and
times for sampling, and to provide a basis for the volume and degree of
averaging of samples. It would be quite desirable to supplement the graphs
of the level of study with the cost of observation, as a function of the
frequency of sampling (interval of information collection), and the accuracy
of measurements performed. This will allow a rather precise representation
of the "value of information", which is necessary to solve the problem of
cost minimization.
Obviously, these studies should be performed systematically in order to
improve the system of observation and measurement for preventing pollution
and for subsequent prediction. The most difficult and responsible stage
will obviously be the third stage — the conduct of studies.
The method of investigation must involve the collection of a large
quantity of observed data on an expanding list of components in various
bodies of water or parts of bodies of water during various periods and
phases. Thus, the observation equipment must be highly productive, with low
inertia and cost. The use of automated in situ, high time resolution equip-
ment is desirable. The following systems can be used:
a) The "cross"-placement of a series of units of sensors along
the x and y axes in a body of water at a fixed depth;
b) the "garland"-placement of sensors at various depths to per-
form synchronous observations along the z axis;
-------
1.0
CO
LU
>
UJ
TIME (r)
Figure 1. Variation of level of study of a component with interval of sampling.
-------
c) the "cross-garland"-placement of a number of clusters of
sensors at points along the x and y axes;
d) the "transverse-garland"-placement of sensor clusters across
of a body of water;
e) an on-board automated continuous recording instrument for the
conduct of observations in the x and y directions with fixed
depth of sensors from a vessel. Depth observations (in the z
direction) can also be made with the vessel on station by low-
ering and raising a group of sensors;
f) the mobile hydrochemical laboratory (truck and trailer with
light boat), which may be shifted to any desired observation
point. The boat is used to place the sensors, while the re-
cording units are located in the trailer, and the laboratory
is in the body of the truck.
The methods and techniques of in situ automated measurements, which form
the basis of such systems, have been previously reported (Gavrilov 1977). A
set of the hardware required has been tested in the Moscow River and in the
reservoirs of this system. The data produced have shown that automated mea-
surements in situ, even of a comparatively small number of parameters (5-7),
can greatly increase the effectiveness of a system of observation and mea-
surement of pollutants.
Calculations have shown that the area of spatial interpolation for
various components differs by an order of magnitude. With identical density
of the network and frequency of sampling, the spatial resolution of obser-
vations of oxygen and BOD may contain an error of up to 25 percent. When
petroleum products are considered, the error may be up to 100 percent, and
for other components, as high as 70 percent. The time resolution of obser-
vations of oxygen may sometimes contribute a significant degree of error.
Automated observations have recorded a case in which, as an internal wave
moved through a reservoir at a depth of 3 m, the oxygen content dropped in
a period of one hour, from 20 mg/liter (over 200 percent supersaturation) to
a life threatening 4 mg/liter (Tsytsarin 1975). These data confirm the need
for performance of the studies described. It is suggested that reference
points be created for the performance of studies in priority regions, from
which methodologic and technical guidance could be provided. The observa-
tions would be performed by the efforts of special mobile hydrochemical de-
tachments.
In the USSR, the planning and performance of observations, as well as
the processing and interpretation of the data produced, should be performed
with the participation of the Hydrochemical Institute, Moscow State Univer-
sity, and the Scientific Research Institute of Neurocybernetics of Rostov
State University, since all of these organizations share common scientific
views concerning the means and methods for performance of the task.
-------
REFERENCES
Borishanskiy, L.S. 1970. The problem of effective placement of water tem-
perature and sea level observation points in the littoral zone. Tr.
GOIN-Statisticheskaya obrabotka okeanograficheskikh dannykh, Moscow,
Gidrometeoizdat Press, No. 99, pp. 34-48.
Gavrilov, I.G. and G.G. Shinkar. 1977. Automated installation for in situ
recording of certain physical-chemical properties of natural waters.
Gidrokhimich. mat. vol. 70, Leningrad, pp. 75-83.
Linnik, Yu.V. and A.P. Khusu. 1958. Some considerations on statistical
analysis of nonuniformity of a polished profile. Sb. Vzaimozamenyae-
most1, tochnost1 i metody izmereniya v mashinostroyenii (Handbook on
interchangeable, accuracy and methods of measurement in machine build-
ing), Mashgiz Press, 47, pp. 144-146.
Margolin, A.M. 1962. An analytic method of determination of the accuracy
of mine-geometric plans. Vopr. marksheyderii i gornoy geometrii v
neftegazodobyvayuschchey promyshlennosti (Problems of mine surveying
and mine geometry in the oil and gas industry), Moscow, Gostoptekhizdat
Press, pp. 210-226.
Margolin, A.M. 1965. Variability of oil and gas deposits and estimates of
the error in results of their prospecting. Sb. Otsenka tochnosti
opredeleniya parametrov zalezhey nefti i gaza (Estimate of the accuracy
of determination of the parameters of oil and gas deposits), Moscow,
Nedra Press, pp. 178-190.
Pavelko, V.L., V.M. Kalinchenko, and V.S. Baronov. 1972. Sovershenstvo-
vaniye metodov marksheyderskikh rabot i geometrizatsii nedr (Improve-
ment of methods of mine surveying and geometric measurements beneath
the earth), Moscow, Nedra Press, pp. 269-279 and 238-246.
Pavelko, V.L. 1977. Variability and levels of study. Geometrizatsiya
mestorozhdeniy poleznykh iskopayemykh (Geometric measurements of de-
posits of useful minerals), Ed. V.A. Bukhrinskiy and Yu.V. Korobchenko,
Moscow, Nedra Press, pp. 95-113.
Popov, Ye.I. 1953. Estimating the accuracy of hipsometric plans of de-
posits. Zapiski Leningradskogo gornogo instituta, Vol. 37, No. 1.
Rybnikov, A.A., Ye.V. Markevich and N.V. Mertsalova. 1970. Methods of cal-
culation of the discreteness of observations in the ocean. Tr. GOIN-
Statisticheskaya obrabotka okeanograficheskikh dannykh, No. 99, pp. 5-
24.
Tsytsarin, G.V. 1975. The reliability of estimates of water quality based
on single samples. Vestn. MGU, ser. geografich., Moscow State Univer-
sity, No. 4, pp. 43-50.
-------
Velner, Kh.A., V.I. Gurarii and A.S. Shayn. 1975. Determination of water
quality criteria in streams for the control of water conservation
systems. Materials of Soviet-American Symposium "Use of mathematical
models for optimization of water quality control", Kharkov-Rostov n/D.
-------
SECTION 2
MODELING THE GREAT LAKES - A HISTORY OF ACHIEVEMENT
William C. Sonzogni and Thomas M. Heidtke1
INTRODUCTION
The North American Great Lakes, which border the United States and
Canada, are a unique system. The largest body of fresh water on earth, the
Great Lakes contain 20 percent of the world's fresh water. Lake Superior,
the largest of the North American Great Lakes, is the largest lake in the
world in terms of surface area, and only Lake Baikal and Lake Tanganyika are
larger in terms of total volume.
The watershed of the North American Great Lakes are heavily developed,
at least in the southern portions. For this reason the lakes have been sub-
ject to considerable pollution. Since the lakes are a valuable natural re-
source and provide drinking water to 70 percent of the region's population
(approximately 30 million), the governments of the United States and Canada
have been sensitive to pollution problems. As a result, considerable effort
has been devoted to the study of water quality/quantity phenomena in the
Great Lakes basin. Mathematical modeling has played an important role in
this research, as evidenced by the large number of different models which
have been used to investigate various aspects of the lakes. In fact, the
Great Lakes are among the most widely modeled bodies of water in the world.
Importantly, many of the modeling studies have been successful in terms
of the valuable information and insights they have provided for making plan-
ning and management decisions. These applications of mathematical models to
the Great Lakes system, which are likely to be relevant to the study of
other large bodies of water throughout the world, have also influenced sub-
sequent modeling research by revealing the limitations and shortcomings of
existing modeling technology.
^Great Lakes Basin Commission, 3475 Plymouth Road, P.O. Box 999, Ann Arbor,
Michigan 48106.
-------
REVIEW OF PAST MODELING EFFORTS
As a first step in evaluating the capabilities and potential utility of
mathematical models in the planning and management process for the Great
Lakes, a literature review was initiated. The review focused on models and/
or modeling activities which have been specifically designed for analysis of
Great Lakes water quality/quantity problems, as well as those models which
may be easily modified for application to the lakes. The primary objectives
of this review were: (1) to supplement and update other Great Lakes
modeling reviews completed several years ago (Hydroscience 1972; Tetra Tech
1974); and (2) to identify areas of relative strength and weakness in using
current modeling technology to assist in the study of specific Great Lakes
problems.
In order to isolate those models which may be appropriately applied to a
particular water quality/quantity phenomena within a given region of the
total basin, several broad categories were defined and used to group models
having similar characteristics. These categories are:
1. Water Quality Models
2. Circulation/Transport Models
3. Heated Effluent Dispersion Models
4. Toxic Substances Models
5. Nonpoint Source Models
6. Other Models
Using these six headings for delineating model types, the results of the
review indicated that well over a hundred mathematical models have been
developed during the past 15 years. The major thrust of this modeling re-
search has been in the areas of water quality (primarily eutrophication) and
lake circulation or hydrodynamics. Because of the relatively large number
of models which have been developed and applied in the study of Great Lakes
water quality conditions and circulation patterns, these models have been
further categorized in Table 1 through 3 on the basis of previous applica-
tions to specific lake basins. Tables 1 and 2 present summary information
specifically on water quality and circulation/transport models, respec-
tively. Table 3 incorporates information on heat dispersion, nonpoint
source and toxic chemical models, as well as a number of miscellaneous
models that are relevant to large lakes. It should be stressed that models
listed under a given lake heading (Tables 1 and 2) have been calibrated and/
or verified for this particular lake; this does not imply that the same
model cannot be calibrated, verified and applied in the study of other
lakes. A brief-discussion of the modeling review is given below for each of
the six different model'classifications. Detailed information on each model
covered by the review is contained in Heidtke (1979).
Water Quality Models
Examination of Table 1 reveals that water quality models have been most
frequently applied to study eutrophication in Lake Ontario, Lake Michigan
and Lake Huron. These models have been generally used to evaluate average,
whole-lake effects for individual basins or their major embayments, e.g.,
10
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TABLE 1. SUMMARY OF WATER QUALITY MODELS RELEVANT TO THE GREAT LAKES
Model Type"
Lake Superior
Lake Michigan
Eutrophication Models
Whole-Lake Effects
Great Lakes Total Phosphorus
Model (Chapra 1977)
Modified Great Lakes Total
Phosphorus Model
(Brandstetter et al. 1973)
Grand Traverse Bay
Phytoplankton Model
(Canale e_t al_. 1973)
Phytoplankton-Based
Food Web Model
(Canale et al_. 1976)
Phosphorus Residence
Time Model (Sonzogni,
et al_. 1976)
Green Bay Water
Quality Model - GBQUAL
(Patterson, et al.
1975)
MICH 01]
Integral Primary Pro-
duction Model (Fee,
1973)
Nearshore Effects
Grand Traverse Bay
Phytoplankton Model
(Canale et al_. 1973)
Green Bay Water
Quality Model - GBQUAL
(Patterson e_t al.
1975).
Conservative
Substances
Great Lakes Chloride Model
(O'Connor and Mueller 1970)
Lake Reactor Model
Other Water Quality
Organic Carbon Budget Model
(Maier and Swain 1978)
Grand Traverse Bay
Coliform Model
(Canale et al. 1973)
11
-------
TABLE 1 (CONTINUED)
Lake Huron/Saginaw Bay
Lake Erie
Lake Ontario
Great Lakes Total Phosphorus
Model (Chapra 1977)
Modified Great lakes Total
Phosphorus Model
(Brandstetter e_t a]_. 1973)
Vollenweider Nutrient Load-
ing Model (Vollenweider
1968)
Modified Vollenweider Nutri-
ent Loading Model (Vollen- -
weider 1976; Vollenweider
1974)
BAY 5 (Richardson and
Bierman 1976)
SMILE 1 (Bierman and
Dolan 1976)
SMILE 51
HURO 1 (Brandstetter
et al. 1973)
Saginaw Bay Phytoplankton
and Nutrient Cycling
Model (Limno-Tech, Inc.
1978)
Saginaw Bay Total Phosphorus
Model (Canale and Squire
1976)
ERIE 01
Phytoplankton Motel of
Western Lake Erie
(O'Connor e£ al. 1975)
Ri chardson/Klabbers
Eutrophication Model
(Richardson and Klabbers
1974)
Mesolimnion Exchange
Model (Burns and Ross
1972)
NOAA's Ecological
Lake Ontario Model
Scavia ejt al_. 1976)
Tetra-Tech Water
Water Quality Eco-
logical Model (Chen
et al. 1975)
LAKE 1 (Thomann et
al 1975; Thomann et
aT 1976)
LAKE 1-A1
LAKE 3l
Lake Reactor Model
Snodgrass/O'Melia
Phosphorus Model
(Snodgrass and
O'Melia 1975)
12
-------
TABLE 1 (CONTINUED)
Lake Huron/Sag1naw Bay
Lake Erie
Lake Ontario
Nutrient Accumula-
tion Rate Model
(Clark et al_. 1976)
Management Model for Saginaw
Bay (Limno-Tech, Inc. 1978)
AUTO-QUAL Estuary Model LAKE 31
(Delos 1976)
FLUSH 021
BAY 16 (Richardson 1974;
Richardson 1976)
Lake Erie Chloride Balance Rochester Embayment
Model (Rumer, et al. 1974) Model (Limno-Tech,
1976)
^Model developed at U.S. EPA's Large Lakes Research Station, Grosse lie,
Michigan. No published information currently available.
13
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TABLE 2. SUMMARY OF CIRCULATION/TRANSPORT MODELS RELEVANT TO THE GREAT LAKES
Lake Superior
Lake Michigan
Lake Superior Circulation Model
(Hoopes et al. 1973)
Lake Superior Transport Model
(Diehl 1977)
Lake Huron/Saginaw Bay
Estuary and Coastal Cir-
culation Model (Lorenzen
e_t ^1_. 1974; Leendertse
et al. 1973)
Alternati ng-Di recti on-Im-
plicit Model (Allender
1976)
Katz-Kizlauskas Model
(Allender 1976)
Modified Simons Model
(Allender 1976)
Bennett Model (Allender
1976)
Green Bay Diffusion Model
(Ahrnsbrak and Ragotzkie
1970)
Three-Dimensional, Vari-
able Density, Rigid Lid
Hydrodynamic and Heat
Dispersion Model^
Green Bay Hydrodynamic
Model (Patterson et^ al.
1975)
Nearshore Water Pollu-
tant Fate Model for Lake
Michigan (Wnek and
Fochtman 1972)
Lake Erie
Lake Ontario
Alternati ng-Di recti on-Imp 1i •
cit Model (Allender 1976)
Lake Erie Wind-Driven
Current Model
(Lorenzen et al. 1974)
Partial Ice-Cover,
Wind-Driven Current
Model for Lake Erie
(Lick 1976)
Steady-State Circul-
ation Model for
Shallow Lakes (Lor-
enzen et^ al. 1974)
Three-Dimensional
Lake Ontario Wind-
Driven Current Model
(Lorenzen et al.
1974) ~
14
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TABLE 2 (CONTINUED)
Lake Huron/Saginaw Bay
Lake Erie
Lake Ontario
Three-Dimensional, Variable
Density, Rigid Lid Hydrod
and Heat Dispersion Model
namic
Steady-State, Near-
shore, Wind-Driven
Current Model (Lick
1976)
Time-Dependent, Near-
shore, Wind-Driven
Current Model (Haq
and Lick 1975; Lick
1976)
Simons Free-Surface
Lake Circulation Model
(Lick 1976)
Constant-Depth, Finite
Element Circulation
Model (Lorenzen ejt
al_. 1974)
Lake Erie Wind Tide
Model (Lorenzen elt
al. 1974; Platzman
1963)
Lam/Simons Chloride
Transport Model (Lam
and Simons 1976)
Lam/Jaquet Transport
and Phosphorus Regene-
ration Model (Lam and
Jaquet 1976)
Markov Process Circulation
and Pollutant Distribution
Model (Howell et ^1_. 1970)
Great Lakes Diffusion Model
(Boyce and Hamblin 1975)
Lake Erie Hydraulic Model 1
Lake Ontario Winter
Circulation Model
(Paskausky 1971)
Bennett Model
(Allender 1976)
Lake Ontario Heat
Transport Model
(Simons 1975)
Lake Ontario Hydrau-
lic Model (Rumer et
al. 1974) ~~
General Circulation
Model for Lakes
(Huang 1977)
published information currently available.
15
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TABLE 3. SUMMARY OF AVAILABLE HEAT DISPERSION, NONPOINT SOURCE,
TOXIC SUBSTANCES AND MISCELLANEOUS MODELS RELEVANT TO THE GREAT LAKES
Heated-Effluent Dispersion Models
Nonpoint Source Models
Hoopes Heated Surface Jet-Steady Cross-
current Model (Policastro and Tokar
1972)
Motz/Benedict Heated Jet-Flowing Ambient
Stream Model (Policastro and Tokar 1972)
Stolzenbach/Harleman Heated Surface Jet
Model (Policastro and Tokar 1972)
Edinger-Polk Two-Dimensional Heated Ef-
fluent Model (Policastro and Tokar 1972)
Edinger-Polk Three-Dimensional Heated Ef-
fluent Model (Policastro and Tokar 1972
Csanady Offshore Outfall Model (Policastro
and Tokar 1972)
Csanady Surface-Discharge Model (Policastro
and Tokar 1972)
Kolesar/Sonnichsen Thermal Energy Transport
Model (Policastro and Tokar 1972)
Wnek Heat Dispersion Model (Policastro and
Tokar 1972)
Pritchard Thermal Plume Model (Policastro
and Tokar 1972)
Sundaram Thermal Plume Model (Policastro
and Tokar 1972)
Remote Sensing Thermal Plume Model
(Cataldo et aj_. 1976)
Universal Soil Loss Equation
(Forest Service 1977; Forest Ser-
vice 1977)
Storm Water Management Model
(Forest Service 1977; Forest Ser-
vice 1977)
Hydrocomp Simulation Program
(Forest Service 1977; Forest Ser-
vice 1977)
Pesticide Transport and Runoff
Model (Donigian and Crawford
1976)
Unified Transport Model (Forest
Service 1977; Forest Service
1977)
BaHelle Urban Wastewater Manage-
ment Model (Branstetter et al
1973)
Wisconsin Hydrologic Transport
Model (Forest Service 1977;
Forest Service 1977)
Storage, Treatment and Overflow
Model (U.S. Army Corps of En-
gineers 1975)
Agricultural Chemical Transport
Model (Frereet al_. 1975)
Nonpoint Source Pollutant Loading
Model (Donigian and Crawford
1976)
Agricultural Runoff Management
Model (Donigian and Davis 1978;
Donigian and Crawford 1976)
Water-Sediment-Chemical Effluent
Prediction Model (Bruce 1973)
16
-------
TABLE 3 (CONTINUED)
Heated-Effluent Dispersion Models
Nonpoint Source Models
Wu/Gallagher Thermal Plume Model (Wu and
Gallagher 1973)
Longshore Thermal Plume Model (Palmer
1969)
LANDRUN (Konrad et al. 1978)
Hydrological Land-Use Model
(Bedlent et al_. 1977)
Stormwater Overflow Pollution
Stream Model (Smith and Eilers
1978)
Stormwater Overflow Hydraulic
Stream Model (Smith and Eilers
1978)
Toxic Substances Mooels
Miscellaneous Models
PCB Model for the Great Lakes (Whitmore
1977)
Great Lakes Radionuclide Model (Sullivan
and Ellett 1976)
Strontium 90 Concentration/Time Model
(Lerman 1972)
Cadmium Food Chain Model for Lake Erie
(Thomann 1974)
Aquatic Food Chain Model for Lake Ontario
(Thomann 1978)
Mercury Model (Miller 1978)
Vinyl Chloride Model (Miller 1978)
Environmental Exposure Model (Miller 1978)
Pesticide Transport and Runoff Model
(Miller 1978)
17
Water and Related Land Resources
System Model (Haimes et al_. 1973)
Phosphorus Mass Flow Program
(Porcella and Bishop 1975)
Water Resource Allocation and
Pricing Optimization Model
(Narayanan e^t a1_. 1977)
Spatial Pollution Analysis and
Comparative Evaluation Model -
SPACE (Heilberg 1976)
Lake Erie Integrated Policy Model
(Mesarovic 1973)
Hydrologic Models of the Great
Lakes (Meredith 1975)
PLUARG Overview Model (Johnson
et a].. 1978)
Lake Erie Phosphorus Discharge
Simulation Model (Prober and
Melnyk 1974)
REMOVE (Drynan 1978)
CLEANER (Park et aj_. 1975)
-------
TABLE 3 (CONTINUED)
Toxic Substances Models ~~ Miscellaneous ModelT
Imboden Phosphorus Model (Imboden
1974)
Modified Imboden Phosphorus Model
(Imboden and Gachter 1978)
Oglesby/Schaffner Phosphorus
Model (Oglesby 1977; Oglesby and
Schaffner 1978)
Dillon/Rigler Nutrient Loading
Model (Dillon and Rigler 1974)
Lorenzen Phosphorus Model (Loren-
zen et al. 1976)
13
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Lake Huron's Saginaw Bay, Lake Michigan's Green Bay, etc. The water quality
variables most frequently simulated include phytoplankton, zooplankton,
nutrient concentrations and dissolved oxygen concentrations. Several of
these models have been used to evaluate the long-term response of receiving
waters to hypothetical management scenarios. For example, eutrophication
models have been applied to western Lake Erie for estimating reductions in
nutrient concentrations and phytoplankton levels which may be expected from
several alternative control strategies, including (1) 80 percent removal of
all incoming phosphorus, (2) a detergent phosphate ban, and (3) a 50 percent
decrease in the phosphorus load from agricultural runoff.
Circulation/Transport Models
Table 2 shows that several circulation models have been used to examine
water movements in Lakes Michigan, Erie and Ontario. On the other hand,
very little attention has been devoted to Lake Superior or Lake Huron. In
the past, the majority of circulation models were limited to the study of
average, two-dimensional (horizontal), central-lake current patterns for
fixed wind directions and magnitude. However, the state-of-the-art in this
area has advanced now to incorporate vertical variations in water movement,
as well as time-dependent circulation in both the central lake and nearshore
zones.
Hydrodynamic models have been used to identify dominant characteristics
of lake water movement, and are often used in conjunction with water quality
models to define the rates of water exchange among various segments of the
lake system. They can also be applied to assess the effects of man-made
structures on existing water movements in the nearshore zones of large
lakes. This was exemplified in a study of expected modifications to coastal
currents in the Cleveland area of Lake Erie as a result of constructing a
jetport island approximately six miles offshore.
Heated Effluent Dispersion Models
Heat dispersion models are used to predict temperature distributions in
waters receiving heated effluent discharges. They are designed for simu-
lating conditions in the near field (the lake region at or near the point of
discharge) and/or far field (the larger lake region where temperature ef-
fects of the thermal plume are still evident). Heat dispersion models have
been developed to accomodate either surface or submerged discharges, and can
be used to evaluate vertical as well as horizontal temperature distribu-
tions.
In reviewing several numerical models of heat dispersion in lakes (as
listed in Table 3 and described in detail in Heidtke 1979), it was found
that the state-of-the-art has progressed steadily over the past few years.
Models have been developed which are capable of providing reasonable esti-
mates of temperature fields in waters receiving heated effluents from
steam-electric generation. Although no single model or technique is avail-
able for providing a comprehensive description of both near and far field
temperature distributions, combinations of methods (e.g., physical models
and field tests in conjunction with mathematical models) can be used to
19
-------
generate reliable predictions under most conditions. A very informative
discussion of this area can be found in a review of hydrothermal predictive
techniques conducted by Jirka et_ aj_. (1976).
Nonpoint Source Models
Several models of land runoff quality and quantity have been developed
and used to assess nonpoint source pollution loadings from rural and urban
land. Their application to the Great Lakes has increased as recent studies,
such as the U.S. and Canadian joint study on pollution to the Great Lakes
from land drainage, have focused on the significance of nonpoint source
pollution in the Great Lakes basin (Pollution From Land Use Activities
Reference Group, 1978).
In generating output, mathematical models of land runoff rely upon de-
tailed information concerning the physical and chemical characteristics of a
given watershed. Input data generally includes predominant land use and
soil types, topography, rainfall, snowmelt, temperature and land management
practices. These models can be used to predict runoff quantity and quality
at very short time intervals (every 15 minutes) or over relatively long
period (average annual conditions). Several nonpoint source models incor-
porate a widely used relationship for estimating expected soil losses in
the region of interest (Wischmeier and Smith).
Models of Toxic Chemicals
As might be expected, very few models are presently available for in-
vestigating the fate of low-level toxic chemicals in the aquatic environ-
ment. Of the small number of models which are available, only a few have
been applied to study the effects of these contaminants in the Great Lakes
basin (Table 3). One of the primary reasons for this is a lack of quantita-
tive information on toxic chemical inputs to the lakes. A lack of empirical
data for model calibration and verification further compounds the problem.
There is an obvious need for increased research in developing modeling
techniques and data acquisition systems to assist in broadening our under-
standing of the effects of toxic inputs to the Great Lakes.
Other Models
In view of the frequent need to consider Great Lakes problems other than
those covered in the previous five categories, miscellaneous mathematical
models potentially applicable to analysis of other factors (e.g., socio-
economic considerations, waste loadings, phosphorus removal efficiencies,
etc.) were also reviewed. These models can be integrated with or used in
parallel with more traditional models to yield a more comprehensive informa-
tion base for decision-making. Because the general review revealed numerous
models which fit this description, only models which have been previously
used within the context of the lakes were considered in detail.- These are
listed in Table 3. A more complete description of each model is contained
in Heidtke (1979).
20
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The major emphasis in modeling the economic aspects of pollution control
in the Great Lakes basin is in the area of "optimization". Optimization in-
volves identifying an optimum policy under a given objective function (to
maximize benefits or minimize costs) and set of constraints (e.g., an upper
limit on costs, or a specification on the manner in which costs/benefits are
distributed within a region).
Johnson ejt aj_. (1978) employed a general model of pollutant loadings
(total phosphorus and suspended solids) to the Great Lakes to identify cost-
effective strategies for achieving reduced phosphorus.loads to the lakes.
Drynan (1978) used a model of phosphorus removal processes in municipal sew-
age treatment plants to generate the costs of achieving specified effluent
concentrations of total phosphorus. Each of these models can be used to
complement other water quality models in order to obtain a better perspec-
tive for evaluating the desirability of alternative pollution control
strategies.
MAJOR MODEL LIMITATIONS
Perhaps the most notable difficulty in attempting to model large bodies
of water such as the Great Lakes is the problem of model verification. The
relatively long response time of the lakes makes it difficult to test the
ability of many models to accurately predict the long-term lake effects at-
tributable to changes in different input variables (for example, reductions
in existing pollutant loadings). An extensive historical data base is also
necessary for quantifying the uncertainty associated with model projections.
Consequently, a lack of appropriate verification data often limits the ac-
ceptance of model predictions by water resources planners and managers.
Tables 1 and 2 indicate that only a few water quality and circulation/
transport models have been used to evaluate nearshore lake conditions. As
is the case for many large lakes, water quality conditions in the nearshore
zone of the Great Lakes is most critical from a human-use perspective. For
this reason, predictive models are extremely valuable in helping to identify
effective ways of managing this area. It is likely that the increased level
of spatial and temporal refinement necessary for evaluating nearshore condi-
tions can be readily accomodated using existing modeling techniques, al-
though more extensive field data must be available for "fine-tuning" the
models to localized lake conditions (shoreline features, bottom topography,
nearshore currents, pollutant loads from local tributaries, etc.). Cer-
tainly there exist limitations as to the practicality of applying models for
predicting water quality changes in nearshore areas. As finer and finer
scales of time and space are introduced in the model application, the com-
puter costs for simulating water quality profiles over a fixed period often
increase significantly. Additionally, there are constraints on detail which
are imposed by the numerical techniques often used in solving the quantita-
tive expressions common to many of these models (this is particularly true
for circulation/transport models shown in Table 2). Therefore, although
water quality models have not been extensively applied for evaluating near-
shore conditions, it appears that the technology is available if used within
practical limits.
21
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Although several heat dispersion models have been developed (Table 3),
very few have as yet been applied to analyze thermal discharges to the Great
Lakes. This is partially attributable to a lack of field data for calibra-
tion and verification. In order to better understand the limitations and
appropriate uses of heat dispersion models, more extensive testing of model
predictions against field measurements is required.
As noted in Table 3, several models are currently available to examine
nonpoint source pollution in the Great Lakes basin. One of the more recent
applications of a nonpoint source model within the context of the Great
Lakes was carried out by Konrad et aj_. (1978). In this case, the model
LANDRUN was used to estimate unit area loads of various contaminants from
several areas of the Menomonee River basin located in the State of Wisconsin
(tributary to Lake Michigan). However, because many models of land runoff
often require a rather detailed data base on the aforementioned input vari-
ables, their application for assessing nonpoint source pollution from a
large region such as the Great Lakes basin is limited. Several of the non-
point source models listed in Table 3 are better suited for predicting run-
off quality and quantity in small watersheds having well-documented physi-
cal and chemical characteristics.
PRACTICAL APPLICATION OF GREAT LAKES MODELS
While the many models developed to date have been extremely instructive
from a large lakes research point of view, the true mark of achievement has
been in the application of models in real planning situations. Back in the
late 1960's, the Great Lakes Basin Commission, a U.S. planning organization
with responsibility for the U.S. portion of the Great Lakes, began to view
each of the Great Lakes and their basins as an integrated system.
It was apparent to the Great Lakes Basin Commission at that time that
conventional planning techniques would not be sufficient for holistic evalu-
ations of the Great Lakes. Consequently, after consulting extensively with
scientific experts and other knowledgeable people from both the United
States and Canada, the Basin Commission sponsored a feasibility study to de-
monstrate the application of system models to existing or hypothetical
situations within the Great Lakes.
This feasibility study resulted in the comprehensive report entitled, "A
Limnological Systems Analysis of the Great Lakes, Phase I" (Hydroscience
1973). This report documented the practicality and desirability of uti-
lizing models in planning for the Great Lakes region, and made recommenda-
tions for specific modeling approaches which could be used in the near
future.
This early pioneering effort stimulated additional modeling activities,
further advancing the possibility of planning for the Great Lakes an an en-
tire system. Some of the models proposed in Hydroscience (1973), such as a
lakewide eutrophication model of Lake Erie and Lake Ontario, have already
been developed and applied to some extent (Tables 1 and 2). These and other
models have been used in several significant policy and resource management
22
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decisions, e.g., the Lake Erie Wastewater Management Study (U.S. Army Corps
of Engineers 1976) and the National Commission of Water Quality Regional
Assessment Study (Rolan 1975).
One of the best examples of how mathematical models can be used in water
resources decision-making was the formulation of desirable phosphorus loads
or inputs to the Great Lakes. Given different alternative conditions of
lake quality, a number of available models were used to determine the ap-
proximate total phosphorus loads necessary to achieve those conditions.
Without the modeling work that has been conducted over the last several
years, it would have been virtually impossible to determine the extent to
which current loads must be reduced to achieve the desired trophic condi-
tions.
One of the principal features of this particular application of models
was that several independently developed models (at least three models were
used in the analysis of a lake basin) were used to relate desired condi-
tions to target loads. Results and projections of each model were then com-
pared. The fact that different models, which varied in their complexity and
spatial and temporal scale, predicted similar results provided an additional
measure of confidence in the results. A detailed discussion of the models
used, as well as the method by which the loads were determined, can be found
in Bierman (1979) and Thomas ejb ^1_. (1979).
FUTURE GREAT LAKES MODELING - WATER QUALITY, CIRCULATION/TRANSPORT AND NON-
POINT SOURCES
To complement the recently completed review of past Great Lakes modeling
capabilities, a separate study (Heidtke and Sonzogni 1979) was undertaken to
provide some insight into ongoing or soon-to-begin modeling activities which
have applicability or potential applicability to the Great Lakes. Informa-
tion was obtained through a survey of several U.S. and Canadian agencies and
academic institutions involved in modeling aquatic ecosystems.
According to the survey, continued emphasis will be placed on modeling
the eutrophication process. Much of this work will be an extension and re-
finement of previous efforts to calibrate, verify and apply mathematical
models to eutrophication problems in the Great Lakes, particularly the near-
shore area of the lakes. Included in this work will be (a) refined spatial
and temporal detail, (b) distinction among various species of phytoplankton
and zooplankton present, (c) distinction among various forms of nutrients
and their potential for influencing algal growth, and (d) consideration of
the statistical reliability of model predictions. Additionally, water
quality models which have been previously verified in Great Lakes applica-
tions will be used more extensively as aids in the decision-making process,
i.e., in helping to identify and evaluate cost-effective strategies for
managing the lakes.
Specific water quality problems and issues which will be addressed using
mathematical modeling techniques include:
23
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1. Phytoplankton/zooplankton production in Lake Ontario, as
well as alternatives for achieving its control.
2. Statistical quantification of error bounds on output from
eutrophication models, including identification of the
major sources of error.
3. Development of waste load allocations for conventional
pollutants discharged to the Lower Fox River/Green Bay
system of Lake Michigan.
4. Evaluation of the impact of changes in nutrient loadings
on the growth and distribution of Cladophora in the
Great Lakes.
Continued emphasis will be placed on the refinement and application of
mathematical models for the study of Great Lakes hydrodynamics. It is anti-
cipated that previous developments in the state-of-the-art will facilitate
more practical applications and increased reliability in model predictions.
Although research will still be directed at improving the predictive capa-
bilities of circulation/transport models by revising and updating input data
and/or various model assumptions, an increased effort will be made to take
advantage of existing technology for planning and management purposes.
Specific developments in modeling the dynamics of water movements in the
Great Lakes will include:
1. Modification and application of previously verified three-
dimensional circulation models as tools for simulating
pollutant transport in the lakes.
2. Critical comparison of circulation models through tests of
model predictions against observed field data.
3. Assessment of modeling techniques for simulating wave
heights and directions in Lake Michigan, Lake Superior and
Lake Erie.
4. Design of methods for computing winds and associated surface
stress on the basis of weather observations and output from
large-scale weather prediction models.
5. Further integration of hydrodynamic models to provide more
comprehensive evaluations of lake conditions.
6. Combined remote sensing and mathematical modeling for the
study of sediment transport and thermal structure in Lake
Erie.
It would appear that modeling of nonpoint source pollution will receive
major emphasis over the next few years. Much of this work will be devoted
to refinement and application of previously developed nonpoint source
24
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models. A few of the anticipated activities
cussed below.
in this area are briefly dis-
1. Previous applications of nonpoint source models have been
limited to analysis of runoff quality and quantity from rela-
tively small watersheds, which represents a serious constraint
on their utility for generating basinwide estimates of non-
point source pollutant inputs to the lakes; therefore, an at-
tempt will be made to assess the predictive capabilities of
such models when applied to larger watersheds of varying
physiographic characteristics.
2. In recognition of the need to consider Great Lakes water
quality/quantity problems and their possible solutions from a
systems perspective, nonpoint source models will be linked
with water quality models in an attempt to evaluate the impact
of various land management practices on downstream receiving
waters.
3. Similar to work being carried out in the area of water quality
modeling, continued testing of certain nonpoint source models
will be conducted in order to (1) quantify the errors in
model predictions and (2) identify the source of those errors
and how they may be modified to improve model predictions.
4. One of the major obstacles to more widespread use of models
as planning and management tools is their failure to ade-
quately consider the social, economic and administrative fac-
tors involved in many decision problems; to alleviate this
shortcoming, research is ongoing which should provide valuable
information in the aforementioned areas.
FUTURE GREAT LAKES MODELING - CHEMICAL TOXIC SUBSTANCES
Until recently, toxic chemicals have received very little attention in
terms of attempts to quantify their pathways through, and effects on, the
aquatic environment. However, increased awareness and concern over the po-
tential health risks posed by these materials has now stimulated numerous
research activities designed to advance the state-of-the-art in toxic sub-
stances modeling (Heidtke and Sonzogni 1979).
Toxic substances modeling of the Great Lakes is particularly important
since the Great Lakes appear to be especially vulnerable to toxic contamina-
tion. Factors which may contribute to the sensitivity of the Great Lakes to
toxics, as suggested at a recent workshop sponsored by the Great Lakes Basin
Commission, include:
1. The Great Lakes are close to and often downwind of major
sources of pollution.
25
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2. Substances entering the Great Lakes are often subject to com-
paratively long retention times, resulting in high accumula-
tion of even low level inputs.
3. Atmospheric inputs to the Great Lakes are large and are not
"filtered" by soils.
4. The Great Lakes contain extensive oligotrophic areas with
particularly sensitive biota.
5. Particulate material in the deep Great Lakes is deposited at
a relatively slow rate, allowing more time for bioaccumula-
tion of toxics.
6. The relatively unproductive nature of the lakes (except Lake
Erie) may decrease the rate of removal of toxic substances
from the water column, thus increasing the amount taken up
by fish.
7. The low suspended sediment load per unit volume to each of the
Great Lakes (except Lake Erie) may contribute to their sensi-
tivity; higher volumetric sediment loads may provide more op-
portunity for sorption of toxics and their subsequent settling
out of the water column (higher solids loads may also serve to
"dilute" toxic concentrations in bottom sediments).
8. The active circulation and mixing which are characteristic of
the Great Lakes helps to rapidly distribute toxics throughout
the lakes.
These factors, which are also likely to be relevant to many other large
lakes of the world, need to be given special consideration in the develop-
ment of toxic substances models.
Included among the several toxic substances modeling studies which will
be directed at the Great Lakes in the future are:
1. A model of the accumulation of PCBs in Lake Michigan sediments,
as well as their subsequent uptake by pelagic fishes.
2. Development of a model to predict fluxes of toxic chemicals
to and from sediments in Saginaw Bay, Lake Huron.
3. Continuation of efforts to formulate a modeling framework for
the fate ,and effects of hazardous substances in the aquatic
food chains of the Great Lakes.
At present it is difficult to assess how successful each of these model-
ing efforts will, be in terms of increasing our understanding of the interac-
tions and effects of toxic materials on the Great Lakes ecosystem. However,
these activities likely represent only the beginning in terms of toxics
modeling developments within the next few years. As more becomes known
26
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about the behavior of toxic materials in the environment, priorities for re-
search may be expected to shift. For example, various studies will be using
models which have been previously applied in the analysis of other water
quality/quantity phenomena (eutrophication, nonpoint source runoff, etc.).
This work may reveal that the modeling technology already exists for evalu-
ating the effects of toxic inputs to the Great Lakes. On the other hand, it
may be necessary to pursue new and unique modeling avenues in order to ef-
fectively characterize the problem and answer critical management questions.
In either case, modeling research is progressing in this area and should
continue to do so over the next few years.
As a final note, perhaps one of the most critical needs for advancing
the state-of-the-art in toxics modeling is a carefully designed field
sampling program. This program must be coordinated with the spatial and
temporal detail required for reliable model calibration/verification. The
resulting historical data base should be regulary updated, easily acces-
sible, and maintained in a format which promotes its effective use for
evaluating models of toxic substances.
SUMMARY
During the past 15 years, over 100 mathematical models have been deve-
loped to assist in increasing our understanding of the Great Lakes system.
Many of these models have been successful in terms of the information they
have provided for making management and planning decisions, as well as the
insight they have provided for future research. Water quality models have
probably been given the most attention, with the majority designed to in-
vestigate conditions in the lower Great Lakes — Erie and Ontario. However,
recent modeling developments have more frequently looked at the Great Lakes
as an integrated system, an approach that is fundamental for comprehensive
resource management.
Perhaps the greatest constraint on the use of Great Lakes models is the
inherent difficulty associated with model verification. This difficulty is
due, at least in part, to the large size of the Great Lakes and their rela-
tively long water and chemical residence times. However, a measure of con-
fidence in model predictions has been achieved when different models, vary-
ing in complexity and temporal and spatial scale, have generated similar re-
sults.
Future Great Lakes modeling efforts should be directed toward toxic
chemicals. Contamination by toxic chemicals is currently the most pressing
problem affecting the Great Lakes. Modeling efforts should also concentrate
on the nearshore area, which is the most critical sector from a human use
perspective.
As a final comment, it should not be assumed that increased model com-
plexity is always a prerequisite for improving their utility as decision-
making tools. In the future it may be more effective, at least in terms of
long-range planning objectives, to place greater emphasis on models which
can be practically used to arrive at management decisions rather than on
27
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models which incorporate maximum detail for all system parameters and vari-
ables. Similarly, if models are forced to meet extremely rigid calibration/
verification tests as the criteria for judging their acceptability, much of
their potential may go unrealized. This notwithstanding, the role of models
in managing the lakes is now becoming better defined and their limitations
more clearly understood. Hopefully, this will generate added confidence in
mathematical models as a necessary and valuable asset in developing effec-
tive management plans for the Great Lakes and other large lakes throughout
the world.
ACKNOWLEDGEMENTS
This paper is a contribution of the Great Lakes Environmental Planning
Study. The assistance of Ms. Ann Davis in preparing the manuscript is
gratefully acknowledged.
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1-44.
Policastro, A.O. and J.V. Tokar. 1972. Heated-effluent dispersion in large
lakes: State-of-the-art analytical modeling. Part 1. Critique of
modeling formulations. Argonne National Laboratory, Argonne, Illinois,
Report No. ANL/ES-11.
Pollution from Land Use Activities Reference Group. 1978. Environmental
management strategy for the Great Lakes system. Final Report to the
International Joint Commission, Windsor, Ontario, Canada, 115 p.
33
-------
Porcella, D.B. and A.B. Bishop. 1975. Comprehensive management of phos-
phorus water pollution, Ann Arbor, Science Publ., Inc., Ann Arbor,
Michigan.
Prober, R. and D.B. Melnyk. 1974. A simulation model for phosphorus water
discharges in the Lake Erie basin. Proc. 17th Conf. on Great Lakes Re-
search, Part 1, Internat. Assoc. Great Lakes Res., pp. 514-519.
Richardson, J.M., Jr. and J.G. Klabbers. 1974. A policy oriented model of
the eutrophication problem in the Lake Erie ecosystem. Systems Research
Center, Case Western Reserve University, Cleveland, Ohio, Report No. SRC
74-1.
Richardson, W.L. 1974. Modeling chloride distributions in Saginaw Bay.
Proc. 17th Conf. on Great Lakes Res., Part 1, Internat. Assoc. Great
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Richardson, W.L. 1976. An evaluation of the transport characteristics of
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Richardson, W.L. and V.J. Bierman, Jr. 1976. A mathematical model of
pollutant cause and effect in Saginaw Bay, Lake Huron. Water Quality
Criteria Research of the U.S. EPA, U.S. Environmental Protection Agency,
EPA Report No. EPA-600/3-76-079, pp. 138-158.
Rolan, R.G. 1975. Lake Erie regional assessment study. Prepared for the
U.S. National Commission on Water Quality. Dalton, Dalton, Little, and
Newport. Cleveland, Ohio.
Rumer, R.R., Jr., K. Kiser and C.Y. Li. 1974. Lake Ontario hydraulic model
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Rumer, R.R., Jr., D.D. Meredith and C.C. Chien. 1974. Chloride build-up
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Scavia, D., B.J. Eadie and A. Robertson. 1976. An ecological model for
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34
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35
-------
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36
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SECTION 3
DATA MANAGEMENT REQUIREMENTS FOR GREAT LAKES
WATER QUALITY MODELING
William L. Richardson 1
INTRODUCTION
An important but often neglected aspect of water quality research and
management is data management. The time has past when individual investi-
gators, biologists, chemists, limnologists and geologists working coopera-
tively with other investigators and agencies on large, complex and inter-
acting systems such as the Great Lakes can record, process, and interpret
data without the benefit of digital computers and computerized data bases.
Data are being produced by numerous agencies and scientists for different
purposes and components of the eco-water system using analytical instruments
that produce analog and digital output at phenominal rates. These data must
be recorded, transformed, verified, reduced, stored, retrieved, and statis-
tically analyzed, before they become useful. This paper discusses the data
management requirements for Great Lakes water quality research, management
and mathematical modeling from a modeling perspective.
GREAT LAKES WATER QUALITY DATA
The need for data is perceived in various ways depending on one's view-
point. The planner requires information to develop plans for future devel-
opment of remedial programs while the regulator requires an assessment of
which water quality standards are being violated and the cause. The public
health administrator wants to know if the water is safe for drinking and
swimming and whether the fish are safe to eat. The waste water treatment
plant operator wants to know how much more effluent can be discharged with-
out violating standards and the water supply manager wants to know the char-
acteristic of the water so the plant can be operated in the most efficient
manner. The scientist (ecologist and limnologist) desires an understanding
of how and why the system behaves and the modeler quantifies hypotheses
describing this behavior.
'U.S. Environmental Protection Agency, Large Lakes Research Station, Grosse
He, Michigan 48138.
37
-------
committee and will not be discussed in detail here. However, any water
quality data base must provide for notation and qualification of the preces-
sion and accuracy of the data.
If data are to be transferred from one institution's computer to
another, however, certain basic criteria must be followed. To facilitate
efficient and accurate exchange of data the following guidelines should be
followed (International Joint Commission, in preparation):
a. That data be stored at a minimum resolution of "raw data" i.e.,
at the spacial and temporal resolution at which an individaul
sample or observation was collected.
b. That data be specially referenced and stored according to either
latitude-longtitude or Universal Transverse Mercater - 6 UTM
grid coordinate.
c. That measurements be directly or indirectly referenced to a uni-
form table of nomenclature for parameters in accepted units pre-
ferably SI Units.
d. That individual data elements be stored in reference to time
and date.
e. That the time of data turn-around from the field sample to
storage be reduced to as short a time as possible with a goal
of less than 90 calendar days.
f. That an index or catalog of data collected by participating
agencies be maintained as an on-line computerized query system.
g. That the participants accept the responsibility for comprehen-
sive verification of all data stored in their respective systems.
h. That the agencies maintain their data files indefintely.
i. That agencies process data requests from investigators as quickly
as possible.
DATA NEEDS FOR MODELING
The role of modeling in the water quality management process can be per-
ceived using Figure 1. Mathematical models provide the focus for the syn-
thesis of the surveillance data and experimental research. Model results
are used to translate data and tested theories into terms managers can use
for decision making. Decisions usually result which either alter the quan-
tities of residual materials allowed to enter the system or which alter the
environmental goals. Data provide the basis for this process and the effi-
cient, accurate, and timely management of these data is mandatory.
38
-------
DECISIONS
Treat
Plan
Enforce
Change goals
INPUT MONITORING
Waste loads
Tributary loads
Atmospheric loads
Meteorological conditions
LAKE SURVEILLANCE
Circulation
Morphology
Sediments
Biology
Chemistry
COMPARE
DATA
VERIFICATION )
LABORATORY
EXPERIMENTS
Figure 1. Great Lakes modeling-management process,
39
-------
Therefore, data originate from surveys to determine the temporal and
spatial characteristics of physical, chemical, and biological properties for
a multitude of purposes. For the Great Lakes, Canadian and U.S. Agencies
and laboratories are spending over $10 million per year on this effort, co-
ordinated through the International Great Lakes Surveillance Plan required
under the 1978 Water Quality Agreement (GLWQA 1978).
The general objectives of this plan include (International Joint Commis-
sion 1978):
1. To search for, monitor, and quantify violations of the existing
Agreement objectives (general and specific), the IJC recommended
objectives, and jurisdictional standards criteria and objectives.
2. To monitor local and whole lake response to abatement measures
and to identify emerging problems.
3. To provide data for determining the cause-effect relationship
between water quality and material inputs in order to develop
the appropriate remedial/preventative actions and predictions of
the rate and extent of local/whole lake response to alternate
abatement proposals.
BASIC REQUIREMENTS
The component of the International Great Lakes Surveillance Plan which
provides the primary means of data coordination is data management and
interpretation. A work group of the International Joint Commission (IJC)1
Surveillance Subcommittee has been established to develop a data management
plan and to provide general guidelines for data interpretation. Some basic
requirements of this plan include:
1. Accuracy and Timeliness: Timeliness is an essential factor in water
quality research and management. Delays in processing results to the
manager can result in inaccurate, erroneous decision-making and ineffi-
ciencies. However, accuracy should not be sacrificed for expediency. Ac-
curacy must be assurred through careful transcription, keying, and verifi-
cation. Accountability by the persons involved assures a higher degree of
accuracy.
2. Compatibility: Because Great Lakes data are collected by a multitude of
agencies and laboratories by various disciplines in two countries, one pro-
vince and eight states, data base compatibility is a necessity. Comparable
data first require uniform sampling and laboratory methodology and inter-
comparison studies and quality control must be incorporated into the sur-
veillance program. For the Great Lakes International Surveillance Plan this
is being coordinated by the Data Quality Work Group of the Surveillance Sub-
'A binational body which is responsible for the implementation of the U.S.-
Canada Great Lakes Agreement
40
-------
Generally, the data requirements for model calibration and verification
are different than those for detecting and quantifying objective violations
and water quality trends. The intensity of sampling depends on the time and
space scale for water quality variables to which the model is being devel-
oped and applied. Table 1 includes a list of eutrophication models that
were recently employed to establish the target phosphorus loadings for the
renegotiated Great Lakes Agreement with Canada (Bierman 1979). These models
are similar in that they relate loadings of phosphorus to phosphorus and
phytoplankton biomass (represented by dry weight or chlorophyll a_). They
are different in that each represents a unique hypothesis of cause and
effect. Each is an attempt to describe the eutrophication process for the
subject lake (or lakes) to a different level of complexity in terms of time
and space and biological and chemical resolution.
Input Requirements
Inputs represent the "knowns" in the mathematical expressions which re-
present the model. All of the models require basic morphometric data. In-
formation on depth, volume, hydraulic detention time are normally available
for U.S. lakes because of extensive work done in the past for other purposes
such as navigation charts, flood control, and fish management and this in-
formation remains almost constant in time relative to other inputs.
External Loads
The mass loads of the primary model variables represent the "forcing
functions" of the model equations. Loads are estimated by the product of
water inflow and concentration. Extensive work has been done to develop
loadings for the Great Lakes models (Corps of Engineers 1975; International
Joint Commission 1976) and an extensive and unique data base is required for
this purpose. The U.S. Geological Survey (USGS) measures flows of most of
the major Great Lakes tributaries and these data are available through both
USGS and EPA data bases. State and Provincial governments maintain monthly
or bi-monthly water quality monitoring sites near mouths of most significant
tributaries as required by the U.S.-Canadian Agreement and these data are
stored in computerized data bases.
The resolution and accuracy to which loads must be measured depends on
the system and the model. Vollenweider and Chapra's whole lake models need
loads for just phosphorus estimated on an annual average basis. Thomann and
DiToro's 2-layer models for Lakes Ontario (Thomann et^ _al_. 1975) and Huron
(DiToro and Matystik 1979) also operate with annual loadings because of the
long hydraulic detention times of these lakes, 23 and 8 years, respectively.
DiToro's model (DiToro and Connolly 1979) of Lake Erie includes three hori-
zontal segments with three layers with hydraulic detention times on the or-
der of three months for the Western Basin to about 1-1/2 years for the cen-
tral basin and seasonal loadings are needed for this finer resolution.
Bierman's five segment model of Saginaw Bay (Bierman ert aj_ 1979) requires
loadings resolved to the same degree as the daily hydrograph of Saginaw
River because of the short detention time of the smallest segment.
41
-------
TABLE 1. SUMMARY OF PRINCIPAL MODEL CHARACTERISTICS (Bierman 1979)
: —
Vollenweider1
Characteristic (All Basins)
Time Dependence
Dynamic
Steady-State X
Spatial Segmentation
None X
Horizontal
Vertical
Input Requirements
External loads for
primary variables X
Depth X
Volume X
Hydraulic detention
time X
Temperature
Light
Water circulation
rates
Sediment nutrient
release rates
Primary Variables
Phosphorus X
Nitrogen
Silicon
Total forms only X
Available and
unavailable forms
Secondary Variables
Chlorophyll X
Diatom/Non-diatom
chlorophyll
Multi-class biomass
Zooplankton
Dissolved Oxygen X
Direct Calculations
Empirical correla-
tion X
Thomann/3
DiToro4
(Lakes
Chapra^ Ontario
(All Basins) & Huron)
XV
A
X X
X
X X
X X
X X
X
X
X
X
X X
X
X
X
X X
X
X
X
X
i~ a
DiToro5 Bierman0
(Lake (Saginaw
Erie) Bay)
y V
A . A
x
A
X
X
x x.
XV
X
Xw
X
X X
X\l
X
X X
X
X X
X X
X X
X X
X
X
X X
X
X X
],From Vollenweider 1975.
^From Chapra 1977.
From Thomann _et al_. 1975.
,IFrom DiToro
^From DiToro
42 From Bierman
and Matystik 1979.
and Connolly 1979.
et al_. 1979.
-------
Primary and Secondary Variables
The models calculate concentrations of primary and secondary variables
as shown in Table 1. The simulations are compared to actual data during
calibration and model coefficents adjusted withi" acceptable ranges until
the simulated concentrations describe the data to a satisfactory degree.
Thomann (Thomann e_t aj_ 1979) has developed a rigorous statistical approach
to reduce arbitrary and qualitative judgement involved in this process.
To resolve seasonal variation in nutrient and biomass concentrations
enough data are required to estimate a mean and standard error for each of
the model segments at a sampling frequency sufficient to resolve important
fluctuations. Rigorous approaches for sampling design are lacking but in
general for each of the Great Lakes ten to twelve sampling cruises are re-
quired at a station density of one station per 100 square miles for Erie and
Ontario and three stations per 1000 square miles for Michigan, Huron, and
Superior. Vertical sampling density is related to defining the thermocline
and associated chemoclines but generally the Great Lakes surveillance plan
specifies depth samples at 1 meter, mid epilimn ion, lower epilimn ion, upper
hypolimnion, 1 meter above bottom (International Joint Commission, 1978).
Uniformity
An implied data requirement of a modeling project is that all data
should be obtained concurrently. Each lake and input sample must be ana-
lyzed for all model variables. For example, one cannot use ammonia data
from a previous study and combine it with nitrate data for the current study
and hope to model the nitrogen cycle. Also, one cannot use phosphorus load-
ings from one time period and combine it with concentration data from
another and hope to get reliable results. Therefore, this requirement ne-
cessitates large, coordinated surveillance programs for the Great Lakes in-
volving many agencies and laboratories. A uniformity of methods must be
maintained along with a common approach to data management.
In summary, enough data must be available to obtain statistically signi-
ficant estimates of mass loads and average concentrations for each variable
in each segment and each time period (cruise). The dynamic, multicompart-
ment, multi-segment (high resolution models) require more data than the
steady state, single segment (whole-lake) single compartment (one variable)
models. It is not the purpose of this discussion to judge the merits of
each of the models or model types. The point is that modeling projects
whose purpose is to obtain understanding of limnological processes at high
resolution require significantly more data and, therefore, data management
becomes much more critical.
GREAT LAKES DATA BASES
One means to effect data base uniformity is to require the surveillance
and research organizations to use the same computer and data base system.
This possibility was considered by the Data Management Work Group of the IOC
43
-------
Surveillance Subcommittee but it was determined to be too complex to imple-
ment between two countries. Rather it was decided to recommend maintenance
of existing data base systems with separate agreements between parties to
exchange data as needed. In the U.S., however, it has become EPA policy to
require all data collected by organizations under contracts, grants or
interagency agreements to be stored into its data base system, STORE!
(International Joint Commission, in preparation).
STORE! (STOrage-RETrieval) is a national, on-line data base system
developed by EPA (andHpredecessor agencies) for the archiving of water
quality data. Because of its utility, ease of use, and efficiencies, STORET
is used by most all U.S. State and Federal agencies involved with collection
of water quality information. The STORET water quality file contains about
60 million observations which are accessible on-line via a telecommunication
network.
For the Great Lakes, STORET is viewed by many as an essential tool for
meeting the mandates of federal laws and the Great Lakes Water Quality
Agreement. Not only does it provide for long-term storage of data but it
also provides for exchange of data as well as a means to statistically ana-
lyze data. Because the data base resides on on-line disks, access can be
made by any user at their convenience via a telephone communications net-
work. This eliminates much of the bureaucracy that has traditionally been
associated with accessing data from other agencies.
Although STORET and the computer system on which it resides provide for
some statistical application, many users would rather obtain copies of data
sets and transfer these to their own computer where they may have available
more specialized analysis programs. At the EPA, LLRS a system is being
developed whereby subsets of the STORET are retrieved and telecommunicated
to an in-house POP 11/45 minicomputer. This "Mini-STORET" system will have
various options which the user may specify. A schematic of this is shown in
Figure 2. The user will have the choice of various output formats, hard
copy listings, files for use by model programs, graphical output, or files
for statistical programs.
The key objective is to make data manipulation as efficient as possible
and easily accessible by non-computer personnel for interpretation. For
example using manual techniques it may take a man-year just to get the data
in the proper formats for modeling purposes. Using the Mini-STORET system
it is expected that this will be reduced by at least 75%.
EXISTING GREAT LAKES DATA IN STORET
The Great Lakes data base in STORET consists of data collected by the
U.S. participants in the Great Lakes Surveillance Plan and their grantees
and contractors. Also, all the open lake data collected by the Canadian
Center for Inland Waters (CCIW) has been stored through an agreement between
CCIW and EPA, LLRS. In exchange for these data, CCIW has been provided with
a STORET account for direct access to all U.S. data. Historical data has
44
-------
PRIMARY DATA BASES
r
STORE!
BIO STORE!
IN-HOUSE
DATA
0!HER
DA!A
I
!RANSFER
AND FORMA!
CONVERSION
en
IN-HOUSE
WORKING
DA!A BASE
OUTPUT AND ANALYSIS PROGRAMS
RAW DA!A
LIS!ING
INVEN!ORY
LIS!ING
SIMPLE
S!A!IS!ICS
COMPLEX
STATISTICS
GRAPHICAL
OUTPUT
1
MATHEMATICAL MODEL
DATA FORMATS
Figure 2. Large Lakes Research Station mini-storet system.
-------
been stored where it has been available, but this is a very tedious and ex-
pensive task.
A complete inventory of all Great Lakes Data stored in STORE! is con-
tained in Appendix A. A summary of this inventory is given in Table 2. As
this table indicates there are over 3 million observations for the Great
Lakes stored in STORET or about 5% of the U.S. water quality file. This is
not a significant proportion considering that the Great Lakes represents
about 95% of the surface freshwater in the U.S. Figures 3 through 8 show
the time distribution of data availability for each lake. Several interest-
ing observations can be made from these graphs that reveal the history of
Great Lakes surveillance. First it appears that prior to 1960 most data
collection efforts were done on the connecting channels. The period 1962
through 1966 represent the first Federal surveillance effort for the open
lakes. For Lake Ontario the peak in 1972 represents the data collected for
the International Field Year on the Great Lakes. The 1975 peaks represent
the surveillance effort of the IJC Upper Lakes Reference Study. Lake Erie
was the subject of intensive sampling from 1967 to 1975 by both U.S. and
Canadian agencies and the decline in 1976-1977 was due to the implementation
of the IJC Surveillance plan which diverted most of the U.S. effort for that
period to Lake Michigan and a new effort for Erie in 1978 and 1979.
The number of observations does not necessarily reflect the utility of
these data for modeling purposes, however; but Table 3 might give some indi-
cation of this for eutrophication models. Ideally, each sample collected
should have been analyzed for the basic state variables listed in Table 1.
It can be seen that Chlorophyll a^ data are generally lacking. For all the
lakes except Michigan Kjeldahl nitrogen data appears to be deficient. On
the other hand there appears to be an over abundance of silica data. For
Lake Michigan there is a higher proportion of phosphorus to nitrogen data.
Since this table includes the entire period of record, it may not represent
the more recent attempts by the Surveillance Subcommittee to obtain data for
modeling purposes. To determine exactly whether existing data in STORET are
suitable for modeling purposes, the interested reader is encouraged to re-
view "raw" data in detail. Appendix B presents examples of STORET retrieval
procedures.
BIOSTORET
Biological data have always presented difficulties for data management.
There exists such a variety of samples, components and characteristics,
that STORET has not been able to deal with the parameter code requirements.
This is particularly the case for toxonomic data. In response to the need
for a biological data base, EPA has developed a specialized system, BIO-
STORET. The system is similar to STORET in that it is user oriented and re-
ferences samples to sample location, depth, time, and concentration but ex-
tends the capability to deal with toxonomic information as well as other
descriptive data. This system and data base are in a pilot stage and some
of the Great Lakes data are being used to test the system. Appendix C con-
tains an example BIOSTORET retrieval, command file, and output.
46
-------
TABLE 2. SUMMARY OF GREAT LAKES DATA IN STORET
(As of June 1979)
Lake
Superior
Michigan
Huron
Erie
Ontario
Total Lakes
Connecting Channel
St. Mary's River
St. Clair River
Lake St. Clair
Detroit River
Niagara River
Total Conn. Channels
Total
Number of
Stations
414
2,249
1,089
2,426
1,653
7,831
162
159
275
335
66
997
8,828
Number of
Observations
115,621
663,144
306,717
1,050,639
592,294
2,728,415
51,795
46,863
31,472
144,017
45,260
319,407
3,047,822
Number of
Samples
8,944
70,463
26,582
128,445
78,610
313,044
4,653
5,532
2,973
21,376
4,246
38,780
351,824
47
-------
45,000
40,000
CO
2 35,000
1-
j£ 30,000
UJ
CO
0 25,000
LL
2 20,000
LLJ
^ 15,000
"2.
10,000
5,000
0
-
_
-
-
—
-
-
-
, |,|
1
ll..
^^ ^* CO IO ^^ O) ^~* CO LO ^^ O5
CO CO CO CO f,p CO ^^ ^^ ^^ ^^ ^^
*•" ^™* T^" T" " *™* 1^~ **" ^— >- T*1^ r~* T~
V
YEAR
Figure 3. Summary of Lake Superior data from Storet water quality file.
48
-------
oc
DO
O
LL
O
QC
CD
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
il
o ^
(O CO
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V
YEAR
Figure 4. Summary of Lake Michigan data from Storet water quality file.
49
-------
100,000
90,000
g 80,000
O
< 70,000
gj 60,000
CO
O
u-
50,000
40,000
30,000
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V
YEAR
Figure 5. Summary of Lake Huron data from Storet water quality file,
50
-------
CO
120,000
110,000
100,000
90,000
< 80,000
oc
gj 70,000
CO
O
u. 60,000
O
"J 50,000
40,000
30,000
20,000
10,000
0
.ll
ll
o>
V
YEAR
Figure 6. Summary of Lake Erie data from Storet water quality file
51
-------
180,000
160,000
CO
2 140,000
> 120,000
LU
CO
g 100,000
UL
° 80,000
60,000
40,000
20,000
0
cc
LU
CO
o
CD
V
YEAR
Figure 7. Surmiary of Lake Ontario data from Storet water quality file.
52
-------
60,000
55,000
50,000
45,000
CO
o
~ 40,000
c 35,000
LLJ
CO
QQ
0 30,000
LJ_
O
a: 25,000
LLI
m
§ 20,000
1 5,000
10,000
5,000
0
-
-
-
^
-
"
-
"
-
-
-
II
0 T-
CO CO
O) O)
CO
CO
O)
I
in r»» o>
CO CO
o> o>
CO
o>
IM
.
T- CO IT) px O5
O) G) O) O) O)
V
YEAR
Figure 8. Summary of Connecting Channels data from Storet water
quality file.
53
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TABLE 3. STORE! WATER QUALITY FILE SUMMARY OF GREAT LAKES
EUTROPHICATION DATA (As of June 1979)
Number of Observations '""'
Connecting
Superior Michigan Huron Erie Ontario Channels
Total Phosphorus 3,483 13,805 11,867 16,702 23,342 6,389
Phosphorus
Dissolved Ortho 3,729 6,531 8,638 13,064 19,427 331
Phosphorus Dissolved 2,610 5,891 10,531 11,889 16,272 1,761
Total Kjeldahl
Nitrogen 275 3,969 3,282 5,003 7,380 2,499
N02 + N03 Dissolved 3,793 2,680 11,799 9,761 20,534 275
NH3-N Dissolved 3,639 3,592 9,868 8,957 22,937 261
Silica Dissolved 3,853 6,717 12,404 16,508 24,787 1,004
Chlorophyll 'A'
(Corrected) - 388 3,168 6,830 9,200
Temperature 7,236 27,675 19,942 96,559 56,979 23,613
54
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RESOURCE REQUIREMENTS
Data handling is often done as an afterthought. As a result, deadlines
have been missed, scientific personnel have been misused, data have been
lost, more errors have been made, and project effectiveness reduced. To
remedy this managers must allocate adequate resources for data management.
Resources include a budget, trained personnel, and adequate computer hard-
ware and software.
The EPA, Large Lakes Research Station eutrophication studies on Saginaw
Bay devoted about 10% of total resources to data management. This covered
all effort required to get results from bench sheets to a data base.
Data base costs (STORET) include initial storage charges (computer time)
of about $.01 per observation and storage changes of about $.005 per obser-
vation per year. Based on these unit costs Great Lakes data base costs
about $15,000. per year to maintain on disk. (The entire STORET water
quality file costs about $300,000. per year to maintain). Another 10 to 15
percent of total surveillance resources are required for statistical ana-
lysis and modeling.
CONCLUSIONS
Efficient data management is essential to water quality management and
research programs. Program managers must recognize the role of data manage-
ment and make adequate staffing and budget provisions for its implementa-
tion. This is the case for whatever purpose to which the data are being
applied; however, it is an absolute requirement for timely development of
mathematical models.
Models require complete sets of data for all state variables for the
time and space resolution desired. Although the most critical need for
modeling is proper surveillance design in the first place, accurate and
timely data processing plays a key role in the modeling process. As models
become more complex the quantity and complexity of data increases and auto-
mated, computerized data base and data analysis tools become more important.
As a rule of thumb about 10% of total project costs should be devoted to
data management.
REFERENCES
Bierman, Victor J., Jr. 1979. A comparison of models developed for phos-
phorus management in the Great Lakes. Prepared for Conference on Phos-
phorus Management Strategies for the Great Lakes, Rochester, N.Y. April
17-19, 1979 (in press).
Bierman, V.J., Jr., D.M. Dolan, E.F. Stoermer, J.E. Gannon and V.E. Smith.
1979. The development and calibration of a multi-class, internal pool,
phytoplankton model for Saginaw Bay, Lake Huron. In press, EPA Ecolo-
gical Research Series.
55
-------
Chapra, S.C. 1977. Total Phosphorus model for the Great Lakes. Journal of
the Environmental Engineering Division, American Society of Civil
Engineers. 103 (EE2):147-161.
Corps of Engineers, Department of Army, Buffalo District. Lake Erie Waste-
water Management Study, Preliminary Feasibility Study, Volume 1 and 2,
December 1975.
DiToro, D.M. and W. Matystik, Jr. 1979. Mathematical models of water
quality in large lakes, Part I: Lake Huron and Saginaw Bay model devel-
opment, verification, and simulations. EPA Ecological Research In Pre-
paration, EPA Ecological Research Series.
DiToro, D.M. and J.F. Connolly. 1979. Mathematical models of water quality
in large lak^es, Part II: Lake Erie. In preparation, EPA, Ecological
Research Series.
Great Lakes Water Quality Agreement of 1978 Agreement with annexes and terms
of reference between the United States of America and Canada, signed at
Ottawa, November 22, 1978.
International Joint Commission. In preparation. Data Management and Inter-
pretation component of the Lake Erie Surveillance Plan prepared by the
Data Management and Interpretation Work Group for the Surveillance Sub-
committee, Great Lakes Water Quality Board, June 1979.
International Joint Commission. 1976. Water Quality Board. Great Lakes
Water Quality 1975, Appendix B, Surveillance Subcommittee Report. July
1976.
International Joint Commission. 1978. Lake Erie Surveillance Plan prepared
by the Lake Erie Work Group for the Surveillance Subcommittee, Great
Lakes Water Quality Board, Revised March 1978.
Thomann, R.V., D.M. DiToro, R.P. Winfield and D.I. O'Connor. 1975. Mathe-
matical modeling of phytoplankton in Lake Ontario, I. Model development
and verification, U.S. Environmental Protection Agency Ecological Re-
search Series. EPA-600/3-76-065.
Thomann, R.V., R.P. Winfield, J.I. Segma. 1979. Verification Analysis of
Lake Ontario and Rochester Embayment Three Dimensional Eutrophication
Model. In press, EPA Ecological Research Series.
Vollenweider, R.A. 1975. Input-output models with spatial reference to the
phosphorus loading concept in limnology. Schweigresche Zeitschrift fur
Hydrologie. 37:53-84.
56
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APPENDIX A
INVENTORY OF WATER QUALITY DATA FOR THE GREAT LAKES
CONTAINED ON THE STORET WATER QUALITY FILE
Available upon request from:
William L. Richardson
EPA, Large Lakes Research Station
9311 Groh Road
Grosse He, Michigan 48138
APPENDIX B
EXAMPLES OF STORET RETRIEVAL PROCEDURES
Available upon request from:
William L. Richardson
U.S. Environmental Protection Agency
Large Lakes Research Station
9311 Groh Road
Grosse He, Michigan 48138
APPENDIX C
EXAMPLES OF BIOSTORET RETRIEVALS AND PROCEDURES
Available upon request from:
William L. Richardson
U.S. Environmental Protection Agency
Large Lakes Research Station
9311 Groh Road
Grosse He, Michigan 48138
57
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SECTION 4
OPTIMAL SAMPLING FOR LONG TERM TRENDS IN LAKE HURON
David M. Dolan1
INTRODUCTION
A major goal of the International Surveillance Program for the Great
Lakes is to detect long-term changes in the concentrations of various para-
meters in each of the lakes. With this in mind, the Lake Huron Work Group
of the Surveillance Subcommittee of the International Joint Commission has
designed a Lake Huron Surveillance Plan. The plan calls for intensive sam-
pling of the main lake in 1980, and sampling of all the major tributaries
each year. The intensive main lake effort will be repeated in 1989, and
again in 1998. After each intensive sampling, the data will be examined and
trends evaluated.
In order for the Surveillance Program to be successful, data must be
provided of sufficient quality so that trends may be accurately assessed.
The required accuracy for trend detection can be expressed conveniently in
probablistic terms (DePalma 1977). For example, if it is desired to detect
changes in the lakewide average concentration of chloride of 1 mg/jl or
greater with 90% probability, it can be shown that the standard error of
chloride measurements must not exceed 0.158 mg/Ji. Since the measurement
procedure for chloride is well established, the usual way to decrease the
standard error is to increase the number of observations. However, each ad-
ditional sample collected results in higher surveillance costs. Thus, there
is a trade-off between accuracy and funds required. A method is needed to
optimize the sampling program to obtain the required accuracy at minimum
cost.
If a reliable mathematical model was available for the parameter of in-
terest for Lake Huron, sampling costs could be reduced drastically. It
would only be necessary to update the information on model forcing func-
tions, such as loading rates and water movements, to accurately identify
trends. Unfortunately, models of Lake Huron, even for the simplest conser-
vative substances, are imperfect because of inaccurate model parameters. It
is intuitively obvious that a procedure which would combine the best infor-
U.S. Environmental Protection Agency, Large Lakes Research Station, Grosse
lie, Michigan 48138.
58
-------
mation available from a model with the information from sampling a lake
would be highly desirable and could result in significant cost savings.
Such a procedure is available using the Kalman filtering technique.
The Kalman filter (Kalman 1960; Kalman and Bucy 1961) is an informa-
tion processing technique that has had much use in aerospace engineering.
It combines model calculations and sampling results to estimate a parameter
of interest. The estimate is, ideally, better than one obtained from either
a model or a series of measurements alone. The basic function of the Kalman
filter is the calculation of the Kalman gain. This gain provides the rela-
tive weighting of model and measurements to obtain a estimate at a desired
time. If the model is inaccurate compared to the data, the data will re-
ceive most of the weight. If the data is "noisy" compared to the model, the
model estimate will be given the most weight. The advantage of the line-
arized Kalman filter for lake concentration sample optimization is that the
estimation errors at future times can be calculated independently of the
actual concentration estimates. Thus, as a model increases in accuracy as
more data becomes available, the model output will receive higher weight in
the estimate, and fewer measurements will be necessary to meet accuracy re-
quirements.
The Kalman filtering technique has been applied to lake concentration
estimation for the case of Lake Michigan (DePalma et^ jil_. 1979). This paper
is an application of the procedure to Lake Huron.
THE LAKE HURON SYSTEM
The Lake Huron system has been described in detail elsewhere (Upper
Lakes Reference Group 1977). The system, the third largest of the Great
Lakes by volume, is composed of four interconnected water bodies: the main
lake, Saginaw Bay, North Channel and Georgian Bay (Figure 1). It is con-
nected to Lake Michigan at the Straits of Mackinac, to Lake Superior by the
St. Mary's River and to Lake St. Clair by the St. Clair River (Table 1). In
TABLE 1. LAKE HURON MAJOR INFLOWS AND OUTFLOWS
Lake Route
Michigan Straits of Mackinac
Superior St. Mary's River
St. Clair St. Clair River
Annual Average Flow
1920 m3/sec
2110 m3/sec
5050 m3/sec
general, water entering in the northern part of the lake flows south along
the Michigan coast. There are also strong northern currents along the
Ontario coast (Figure 1).
At present, main Lake Huron is considered to be an oligotrophic lake, as
is indicated by a number of chemical and biological parameters (Table 2).
59
-------
Midland Bay
Penetanguishene Bay
WASAGA BEACH
• Municipality
• Open Water Fith
Collection Suikxi
LOCATION MAP
OF
LAKE HURON
Figure 1. The Lake Huron system.
-------
TABLE 2. CHEMICAL AND BIOLOGICAL INDICATORS OF TROPHIC STATUS -
MAIN LAKE HURON
Nutrients
Spring Average Concentration
Total Phosphorus
Total Dissolved Phosphorus
Dissolved Reactive Phosphate
Dissolved Nitrate & Nitrite
Dissolved Ammonia
Dissolved Reactive Silicate
Dissolved Oxygen
5.4 yg/£
3.2 yg/£
0.9 yg/£
282.4 yg/£
6.1 yg/£
1.46 mg/£
13.2 mg/£
Biology
Chlorophyll a_
Phytoplankton Biomass
Zooplankton Biomass
1.4 yg/£
1000 mg/m3 (10% blue-green)
60 mg/m3 (60% calanoid copepods)
Georgian Bay is also classed as oligotrophic, and North Channel is con-
sidered mesotrophic. However, Saginaw Bay is highly enriched and is classed
as eutrophic. This bay receives wastes from the highly populated Saginaw
River Basin. These wastes include both industrial discharges and agricul-
tural runoff. The influence of this bay on trends in the main lake is sub-
stantial as will be shown.
As a result of the influence of Saginaw Bay, the assumption of a homo-
genous main lake cannot be made in the case of Lake Huron. In the previous
application of Kalman filtering to Lake Michigan, the open lake was treated
as one, well-mixed segment. Also, unlike Lake Michigan, Lake Huron receives
substantial net input of water from two other Great Lakes. Thus, the appli-
cation of this procedure to Lake Huron has some unique and interesting as-
pects.
Loadings of nutrients to Lake Huron are expected to increase in the next
three decades due to industrial and urban growth. Even slight increases in
the present levels of phosphorus and nitrogen concentrations threaten the
oligotrophic status of the main lake. The International Joint Commission
has set a goal of non-degradation for this lake. Therefore, any increase in
nutrient levels would not be in accordance with this goal. It is for this
reason, that the optimal sampling design procedure has been applied to the
nutrient parameters total phosphorus and nitrate-nitrite nitrogen.
MASS BALANCE MODEL
The model to be used with the Kalman filter to provide estimates of lake
concentration for trend detection is a dynamic mass balance model (Figure
2). The basic equation for each substance is:
61
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CONCEPTUAL WHOLE-SYSTEM MASS-BALANCE MODEL
TRIBUTARY AND
POINT SOURCES
LAKE INFLOW
ATMOSPHERE
1
LAKE
SYSTEM
I
PERMANENT
SEDIMENT
LAKE
OUTFLOW
Figure 2. Simplified mass balance model
-------
V j= = W + E QIN.*CINi - Z QOUT.*c - k*c*V (1)
where: V is the volume of the body of water
g£ is the rate of change in concentration with time.
W is the loading rate.
QIN. is the flow rate of the ith input
CIN.J is the concentration of the ith input
QOUT. is the flow rate of the ith output
•J
c is the concentration in the water body.
k is the first order removal coefficient which may be an apparent
settling rate or a degradation constant.
Included in the W term are atmospheric loading, direct loadings and small
tributary loadings. QIN and QOUT refer to major water bodies that interact
with the one of interest. An equation similar to (1) is written for each
body of water that can be considered well-mixed. In the case of main Lake
Huron, 2 segments must be formed, Northern Lake Huron and Southern Lake
Huron. The division of the lake follows that used by DiToro (in press). A
separate mass balance equation is written for each segment and the two are
coupled:
dc, 2
= W +
V g- = W] + I QIN. CII^- QNS c1 - k c]V] (2)
V2 3T = W2 + QNS cl " QOUT C2 " k C2 V2
where: QNS is the net flow between northern and southern Lake Huron.
Since c-] appears in both equations, they are considered to be coupled.
These equations are then linearized about nominal values, CNOM] and CNOM2
for use in the Kalman filter procedure.
Note that in (2) the QIN-j's refer to flows from Lake Michigan and Lake
Superior. The CINj's are concentrations of these lakes which are subject to
dynamics different from Lake Huron. To avoid including added complexity in
the model, these concentrations are treated as inputs. In other words, the
boundary concentrations for the lake are modeled separately and then stored
for later use with this model.
63
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MEASUREMENTS
There are three types of measurements that can be made on the Lake Huron
system that will improve the estimate of lakewide concentrations. The first
type is the actual sampling of the open lake. A second type puts the prin-
cipal of mass balance to use: tributary and point source sampling. Here
the assumption is that if inputs are changing, then lake concentrations
must, as a consequence, change. The third type is actually a miscellaneous
category and includes all field and laboratory measurements that improve the
accuracy of the mass balance model, and thus improve the estimate. Such
measurements include apparent settling velocity, boundary flow rates, atmos-
pheric loading rates, and intersegmental exchange rates for the case of more
than one segment. Each of these types of measurements can improve the ac-
curacy of concentration estimates for trend detection with the use of a
Kalman filter.
The actual sampling of the open lake is the most direct way to determine
the lakewide average concentration of a substance. If unlimited funds were
available, a model would not be necessary. However, open lake sampling
cruises are expensive and represent a major surveillance effort. Lake Huron
is to be sampled intensively only every nine years. This is, in part, a re-
flection of the enormous costs of such a project. However, if enough
samples are taken, open lake concentrations can be determined quite accu-
rately.
Tributary sampling is also extremely expensive. In this case each in-
dividual tributary must be sampled. There are 23 major tributaries to Lake
Huron. Since loading is often a seasonal phenomemon, accurate time his-
tories for each tributary are necessary. Fortunately, there are estimation
methods for those tributaries with complete flow records so that daily sam-
pling is not necessary (Dolan et_ aj_. in preparation; U.S. Army Corps of
Engineers 1975). Even if the tributary loads are known accurately, how-
ever, the resultant concentration estimate for the lake is still influenced
by the accuracy of the model, including parameters and other forcing func-
tions .
These various model parameters and forcing functions can be determined
to a considerable degree of accurancy by the proper experiments in both
field and laboratory. Many investigators have devised ways to estimate ap-
parent settling velocity. Atmospheric deposition of pollutants is the sub-
ject of much current research (Murphy and Doskey 1975; Murphy 1975; Delumyea
and Pete! 1979). There are several methods available for estimating bound-
ary flows and intersegmental exchanges (Quinn 1977; Chapra 1979; Upper Lakes
Reference Group 1977; Dolan et aj[. in preparation). If one of these com-
ponents is particularly limiting with regard to accuracy of concentration
estimates, specific research aimed at improving the knowledge of the com-
ponent can be initiated.
Obviously, there are costs and benefits associated with each of the
three types. The techniques described in this paper can choose the optimal
conbinations of these measurements to meet the accuracy constraints at mini-
mum costs.
64
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OPTIMAL SAMPLING STRATEGY
For purposes of comparison, the proposed Lake Huron work group surveil-
lance plan will be evaluated for trend detection in each of the two segments
of main Lake Huron. The optimal strategy will then be presented. It should
be noted that the surveillance plan includes sampling for many other para-
meters besides nitrate-nitrite and total phosphorus and thus, it is not
likely to be optimal with regard to these two parameters.
As stated previously, any increase in nutrient levels in Lake Huron
would not be in accordance with IJC goals for this lake. Therefore, the
criteria for trend detection is the following: The sampling plan must de-
tect changes of 1 yg/A of total phosphorus and 50 yg/A of nitrate-nitrite
with 90% certainty every 9 years. This criterion is met with ease in the
northern Lake Huron segment (Table 3). In the south, only the nitrate-
TABLE 3. LAKE HURON SURVEILLANCE PLAN TREND DETECTION CAPABILITIES
Northern Lake Southern Lake
w/o filter w filter w/o filter w filter
Parameter
Nitrate Nitrite
1980 18 yg/A 18 yg/A 37 yg/A 36 yg/A
1989 18 yg/A 18 yg/A 37 yg/A 35 yg/A
1998 18 yg/A 18 yg/A 37 yg/A 35 yg/A
Total Phosphorus
1980 .60 yg/A .60 yg/A 1.83 yg/A 1.67 yg/A
1989 .60 yg/A .59 yg/A 1.83 yg/A 1.64 yg/A
1998 .60 yg/A .54 yg/A 1.83 yg/A 1.63 yg/A
nitrite criterion is met. The use of the Kalman filter does improve the
accuracy of the estimate slightly, but the plan as it is presently con-
ceived, cannot detect changes in total phosphorus of 1 yg/A in southern Lake
Huron.
In order to meet this criterion, it can be shown that 65 stations must
be added to the open lake sampling plan in the southern segment of the lake.
This results in a total of 138 stations for the total main lake effort. If
the optimization procedure developed for Lake Michigan (DePalma 1977) is
utilized it can be shown that accuracy criteria in both lake segments can be
met with 105 stations if the only goal of sampling is trend detection. This
assumes that accuracy requirements are met exactly and that the Kalman fil-
ter is used in conjunction with a mass balance model to improve the esti-
mates where possible. This comparison emphasizes that uniform spatial
coverage of the open lake is not desirable for trend detection. Rather, it
is more important to locate extra stations in areas of high spatial vari-
ability.
65
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SENSITIVITY ANALYSIS
It can be seen from Table 3 that the use of the filter in conjunction
with the mass balance model previously described, only marginally improves
the accuracy of the concentration estimates. This is an indication that the
model itself is inaccurate, probably because some or all of the model para-
meters and forcing functions are imperfectly characterized. If some of
these quantities can be better determined, the model accuracy will improve,
the trend detection capability will increase and the associated sampling
costs will decrease.
The Kalman filtering technique has been used to estimate improvements in
trend detection accuracy attributable to research and experimentation in
each of five areas:
1. Atmospheric Loading - Of all the input to lakes of pollutants,
this component is usually the least accurately known. Research
is presently being conducted to improve measurement accuracy
of atmospheric loading (Murphy 1975; Delumyea and Pete! 1979).
Also, it is possible to model atmospheric fallout, greatly im-
proving estimates of loading in areas where no sampling occurs
(Sydor in press; Kabel 1975; Acres Consulting Services 1975).
2. Intralake Exchange - This is defined to be the net effect of
advection and diffusion at the assumed interface between northern
and southern Lake Huron. Estimates of this quantity can be
greatly improved using existing models. The Lake Huron surveil-
lance Plan includes a design for estimation of this parameter.
3. Interlake Exchange - This is defined to be the net effect of
advection and diffusion at interfaces with Lake Michigan, the
St. Mary's River, and the St. Clair River. The effects of diffu-
sion at the river interfaces is nil so that these two components
can be easily improved by measuring flow rates. Much work has
already been done on the Straits of Mackinac (Quinn 1977), and
these results with other measurements can greatly improve the
accuracy of this component.
4. Initial Settling Velocity Uncertainty - At present, the pro-
cedure assumes that very little is known about the settling rate
of nutrients, and that knowledge improves as measurements are
taken. In reality, much is already known, and more work is on-
going in this area (Chapra in press).
5. Tributary Load Estimation Error - This is always a large source
of error in any lake model. However, not only are methods of
sampling improving for tributaries, but also methods of cal-
culation to obtain better estimates with sparse data are be-
coming available (Dolan in preparation). In addition, 24% of
the tributary nutrient loads to Lake Huron come from the Saginaw
River alone. This tributary could be monitored intensively to
reduce estimation errors.
66
-------
The increases in accuracy with improvements in each of these five areas
are compared to a base case which is the optimal strategy with no improve-
ments (Table 4). The results are expressed in terms of standard errors of
TABLE 4. SENSITIVITY ANALYSIS FOR STANDARD ERROR (yg/£) IN SOUTHERN
LAKE HURON FOR TOTAL PHOSPHORUS
Run Description
Present Optimum
Improve Atmospheric Estimates
Improve Intralake Exchange Estimates
Improve All Exchange Estimates
Improve Initial Settling Rate Accuracy
Improve Tributary Load Estimate
Improve All of the Above
1980
.1571
.1569
.1570
.1562
.1556
.1535
.1467
Year
1989
.1575
.1572
.1573
.1559
.1570
.1541
.1465
1998
.1569
.1566
.1568
.1551
.1567
.1536
.1456
Improve All of the Above Plus Initial
Condition Estimates
Improve Open Lake Measurements
.1429
.0161
.1440
.0162
.1440
.0162
the estimates for total phosphorus in southern Lake Huron for each year in
which a determination of trends is desired. These estimates are most sensi-
tive to improvements in tributary load estimation accuracy. If all five
areas are improved, the accuracy of the estimates would increase by 7 per-
cent by 1998. If, in addition, the initial concentration of total phos-
ptiorus had been better estimated, further gains in accuracy would be rea-
lized. For comparison purposes, the effect of a 10-fold increase in the ac-
curacy of open lake measurements for total phosphorus has been included.
Naturally, the total phosphorus estimates are much more sensitive to this
kind of improvement; however, such increases in accuracy would probably be
impossible to achieve in practice.
DISCUSSION
Trend detection is an important part of any lake sampling plan. In the
case of oligotrophic lakes such as Lake Huron, the early observation of
trends towards mesotrophy is critical. Given this reasoning, the confidence
level of 90% used in this analysis may be too low. If high confidence is
required, the use of a model becomes even more important.
67
-------
Each of the five areas for improvement mentioned in the preceding sec-
tion are interesing for reasons beyond mass balance modeling and thus may be
the subject of limnological research in the future. If the optimal strategy
for trend detection is recomputed assuming significant improvement in all
these areas, substantial reductions in the number of stations necessary for
sampling would be realized (Table 5). Also, the model's capabilities would
increase as measurements are made,
TABLE 5. COMPARISON OF STATIONS REQUIRED BY DIFFERENT STRATEGIES
Strategy
Lake Huron Work Group Plan
Optimal Strategy
1980
138
105
Year
1989
138
104
1998
138
104
Optimal Strategy with Research on Model
Parameters 85 83 79
Another advantage would be that trends for other, less variable, para-
meters; such as chloride could be detected with the help of the model and
the tributary load estimates even during years when no lakewide measure-
ments were taken.
CONCLUSION
The techniques discussed in this paper are extremely useful for de-
signing and/or evaluating a plan for trend detection. Even if the model
associated with the Kalman filter is inaccurate, the procedure leads to
orderly acquisition and evaluation of data pertaining to trend detection.
If the model can be improved, significant increases in accuracy can be
realized or, alternatively, sampling costs can be decreased. In either
case, these techniques provide a systematic way of evaluating a proposed
measurement strategy as well as determining if another strategy could give
substantial gains in accuracy.
In the case of Lake Huron, application of this procedure has shown that
for nutrients, the northern segment of the lake is over-sampled, while the
southern segment is under-sampled. This situation can be improved by in-
creasing the number of stations in the southern segment, or by improving the
model to be used with the Kalman filter to obtain estimates of trends.
These results are probably generalizable to other parameters that could be
described well by a simple mass balance, such as conservative substances,
radionuclides, some heavy metals and suspended solids. Biological para-
meters and certain organic contaminants would require a more sophisticated
approach (Canale et al. in press; Chiu 1978).
68
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REFERENCES
Acres Consulting Services - Earth Science Consultants. 1975. Atmospheric
Loadings of the Upper Great Lakes. Phase Two Report. Canada Centre
for Inland Waters.
Canale, R.P., L.M. DePalma and W.F. Powers. In press. Sampling Strategies
for Water Quality in the Great Lakes. EPA Ecological Research Series.
U.S Environmental Protection Agency, Duluth, Minnesota. 102 pp.
Chapra, S.C. 1979. Applying Phosphorus Loading Models to Embayments.
Limnology and Oceanography, 24(1): 163-168.
Chapra, S.C. In press. Simulation of Recent and Projected Total Phosphorus
Trends in Lake Ontario. J. Great Lakes Research.
Chiu, C. (Ed.). 1978. Applications of Kalman Filtering to Hydrology,
Hydraulics, and Water Resources. Department of Civil Engineering,
University of Pittsburgh, Pennsylvania. 783 pp.
Delumyea, R. and R.L. Petel. 1979. Deposition Velocity of Phosphorus-
Containing Particles Over Southern Lake Huron, April-October, 1975.
Atmospheric Environment, 13: 287-294.
DePalma, L.M. 1977. A Class of Measurement Strategy Optimization Problems-
With an Application to Lake Michigan Surveillance. Ph.D. Thesis, The
University of Michigan, Ann Arbor, Michigan. 122 pp.
DePalma, L.M., R.P. Canale and W.F. Powers. 1979. A Minimum-Cost Sur-
veillance Plan for Water Quality Trend Detection in Lake Michigan. In
Perspectives on Lake Ecosystem Modeling, eds. D. Scavia and A.
Robertson, Ann Arbor: Ann Arbor Science, pp. 223-246.
DiToro, D.M. and W.F. Matystik, Jr. In press. Mathematical Models of Water
Quality in Large Lakes. Part I: Lake Huron and Saginaw Bay, Model
Development, Verification and Simulations. EPA Ecological Research
Series. U.S. Environmental Protection Agency, Duluth, Minnesota.
Dolan, D.M., A.K. Yui and R.D. Geist. In preparation. Evaluation of River
Load Estimation Methods for Total Phosphorus. U.S. Environmental Pro-
tection Agency, Large Lakes Research Station, Grosse lie, Michigan.
Dolan, D.M., R.D. Geist and K. Salisbury. In preparation. Applications of
a Dynamic Mass Balance to Water Quality Problems in the Great Lakes.
U.S. Environmental Protection Agency, Large Lakes Research Station,
Grosse lie, Michigan.
Kabel, R.L. 1975. Atmospheric Impact on Nutrient Budgets. In: Pro-
ceedings of the First Specialty Symposium on Atmospheric Contribution
to the Chemistry of Lake Waters. Internat.'Assoc. Great Lakes Res.
pp. 114-126.
69
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Kalman, R.'E. 1960. A New Approach to Linear Filtering and Prediction Pro-
blems. Trans. ASMF, J. Basic Engr. 828: 34-35.
Kalman, R.E. and R. Bucy. 1961. New Results in Linear Filtering and Pre-
diction Theory. Trans. ASMF, J. Basic Engr. 83D: 95-108.
Murphy, T.J. and P.V. Doskey. 1975. Inputs of Phosphorus from Precipita-
tion to Lake Michigan. EPA-600/3-75-005, U.S. Environmental Protection
Agency, Duluth, Minnesota. 27 pp.
Murphy, T.J. 1975. Concentrations of Phosphorus in Precipitation in the
Lake Michigan Basin. In: Proceedings of the First Specialty Symposium
on Atmospheric Contribution to the Chemistry of Lake Waters, Internat.
Assoc. Great Lakes Res. pp. 127-131.
Quinn, F.H. 1977. Annual and Seasonal Flow Variations through the Straits
of Mackinac. Water Resources Research, 13: 137-144.
Sydor, M. Particle Transport in Duluth-Superior Harbor. EPA Ecological
Research Series. U.S. Environmental Protection Agency, Duluth,
Minnesota.
Upper Lakes Reference Group. 1977. The Waters of Lake Huron and Lake
Superior, V. 2. International Joint Commission, Windsor, Ontario.
743 pp.
U.S. Army Corps of Engineers. 1975. Lake Erie Wastewater Management Study
Preliminary Feasibility Report. Buffalo District Army. Corps of
Engineers, Buffalo, New York.
Verhoff, F.H., S.M. Yaksich and D.A. Melfi. In press. Estimation of
Nutrient Transport in Rivers. J. Environmental Eng. Div. Amer. Soc.
Civil Engr.
70
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SECTION 5
A MODEL APPROACH TO ESTIMATING THE EFFECTS OF ANTHROPOGENIC
INFLUENCES ON THE ECOSYSTEM OF LAKE BAIKAL
A.B. Gorstko1, Yu.A. Dombrovskiy"1, V.V. Selyutin1,
F.A. Surkovl, A.M. Nikanorov2 and A.A. Matveev2
Lake Baikal is located in the southern portion of eastern Siberia
between 51°29' and 55°46'N (Figure 1). Ranking eighth in area and first in
depth among all continental bodies of water of the planet, it contains about
20 percent of the world reserves of fresh surface water. The basic morpho-
metric characteristics of the lake are presented in Table 1.
TABLE 1. MORPHOMETRIC CHARACTERISTICS OF LAKE BAIKAL
Number
Characteristic
Value
1
2
Length
a) of lake
b) of shoreline
Width
a) mean
b) maximum
c) minimum
Area
636 km
2000 km
48 km
79 km
26 km
a) total
b) of zone less than 50 m deep
c) of zone less than 250 m deep
4 Depth
a) mean
b) maximum
5 Volume
6 Altitude Above Sea Level
7 Area of Watershed Basin
31.5 103-km2
2.5 103-km2
6 103-km2
730 m
1620 m
23.6 103-km3
456 m
557 103'km2
1 Institute for Mechanics and Applied Mathematics, Rostov State University,
192/2 Stachky Prospect, 344090 Rostov-on-Don, USSR.
2Hydrochemical Institute, 192/3 Stachky Prospect, 344090 Rostov-on-Don,
USSR.
71
-------
50s
102°
104°
106°
110°
Figure 1. Diagram of surface currents in Baikal during the ice-free season.
72
-------
The annual surface runoff into the lake is about 59 km^, of which one-
half is provided by the Selenga River, while the annual outflow of the
Angara River is 61 km^.
The most important characteristic of Lake Baikal is the exceptionally
high quality of its water. It is distinguished by its low mineral content
(many ions in concentrations of less than 100 mg/liter), high transparency
(Secchi depth measurements of up to 40 m), an abundance of dissolved oxygen
(at least 9.5 mg/£, even at depths of 1300-1400 m), and low temperature
(mean annual surface temperature 4.5°C).
Given these conditions, a unique ecologic system exists in the lake, the
most important peculiarity of which is the existence of an endemic complex
of aquatic organisms populating the open regions of Baikal. The richness
and variety of the flora and fauna of the lake, numbering 2400 species and
subspecies, of which approximately three-fourths are endemic species; the
extensive branching of the trophic chain, ending in the Baikal seal; and the
nature and scale of the dynamic processes in the water enable Baikal to be
considered as a world ocean in minature. A large quantity of information on
Baikal has been systematized (Kozhov 1972), and the results of later studies
can be found in the works of the Limnologic Institute, Siberian Branch,
Acad. Sci. USSR (1977, 1978).
While it represents tremendous ecologic and scientific value, Lake
Baikal is also an important national economic resource. The economic util-
ization of Baikal consists of three basic uses:
a) water consumption,
b) culture and harvesting of fish, and
c) recreational uses.
The floating of timber and the construction of large industrial enter-
prises which might act as sources of pollution are forbidden in the imme-
diate vicinity of the lake, and navigation is limited. However, it is im-
possible to completely eliminate the entry of various impurities into the
lake. The most important sources are surface runoff and atmospheric preci-
pitation. Preliminary analysis shows that currently it is difficult to see
any evidence of changes in the ecosystem of the lake as a result of human
activity. On the one hand, this is a result of timely preventive measures
taken as a result of the resolution of the Central Committee and USSR
Council of Ministers of 16 June 1971, in which it was stated that lake pro-
tection should contain, "additional measures to assure rational utilization
and conservation of the natural riches in the basin of Lake Baikal"; while
on the other hand, it also results from the great self-purifying capacity of
the lake. Thus, it is of great importance to obtain well-founded long-term
predictions concerning the possibility of future effects of human activity.
The model approach to the prediction of the status of this ecosystem is in-
tended to provide the most objective estimate of such effects in the Baikal
region and provide an effective solution to the problem of its preservation
and efficient use. This paper considers some trends in the realization of
the model approach.
73
-------
LAKE DYNAMICS
Information on currents in the lake at various periods of time, repre-
sented in the form of a field of velocities or flows between various volume
elements of the lake, is necessary for modeling of the hydrochemical regime
and the dynamics of the plankton population. There are two possibilities in
this case: 1) to perform numerical hydrodynamic calculations, or 2) to use
the data from natural observations of speeds and directions of currents.
Both methods were used in the present work.
The starting point used was a diagram of convective-gradient currents
(Figure 1) obtained by processing observed data on water temperature by the
dynamic method (Krotova 1970). The surface of the lake was divided into 14
regions (Figure 2), and 4 depth layers were distinguished: 0-25 m, 25-50 m,
50-250 m and 250 m to the bottom. The extensive material obtained in field
observations, summarized in the collective monograph Techeniya v Baikale
(Currents in Baikal) (1977), allowed an estimation of movements through the
boundary regions selected for various seasons of the year.
In addition, a UNIVAC 1100/40 computer, utilizing a program developed by
J. Paul, was used to calculate the velocity fields corresponding to various
wind situations. These calculations yielded results similar to those ob-
tained by means of a baroclinic model (Tsvetova 1977).
WATER QUALITY
The basis of the models for the dynamics of the concentrations of sub-
stances entering the lake is the equation of turbulent diffusion of a non-
conservative impurity:
|£ = div (D grad c) + v grad c - kc + f (1)
grad c/S = 0 (2)
where: c(t; x, y, z) is the concentration of the substance;
D(t; z) is the coefficient of turbulent diffusion;
v(t; x, y, z) is the velocity vector;
k(t) is the decay rate;
f(t; x, y, z) is the loading of the substance into the
lake;
S(x, y, z) is the boundary surface.
In the calculations, Equation (1) was replaced by a finite-difference
system corresponding to the regionalization used in the model. The time-
series plots of concentrations of an arbitrary substance in the various
regions of the lake are shown in Figure 2. The basic sources are assumed
to be the Selenga River (Region 4), and the Baikal Cellulose-Paper Combine
(BCPC, Region 2).
74
-------
< cr
El
40
30
o
o
20
10
4
\ \ \\ \ II 1 1 1 Ml
z
o
5
-------
The concentration fields in the vicinities of "local" pollution sources
were calculated with a significantly smaller grid (0.5 km horizontally;
0.02 km in depth), again using Equation (1). Concentration of an arbitrary
substance in the upper (0-20 m) layer near the BCPC is shown in Figure 3.
Data on flow velocities were taken from field observations. Measurements
of velocities during 1977 were conducted by V.M. Sleptsova of the Baikal
Weather Observatory, using an especially detailed plan of observations in-
corporating the requirements of the mathematical model. The experiments
showed that the propagation of a patch of material in the lake depends es-
sentially on the velocity field, and changes significantly from season to
season. At the same time, concentrations were found to be relatively insen-
sitive to variations in diffusion coefficients. The mean area of calculated
patch of material was small, 4-5 km^.
BIOTIC CYCLE
The biotic cycle, i.e., the chain of transformation of living and dead
organic matter, binds together all of the biologic, biochemical, physical
and other processes occurring in the ecosystem. The most important index of
the biotic cycle is the biotic balance, reflecting the relationship between
the system of productive and destructive processes at all trophic levels.
The mean annual biotic balance of Baikal, calculated on the basis of results
of studies of the Limnologic Institute, Siberian Branch, USSR Acad. Sci.,
for 1964-1970, is presented in Table 2.
However, the practical usefulness of this balance is limited, due to its
static nature. For completeness, nonliving organic matter, which acts as a
nutrient medium for bacteria, and nutrients limiting primary production
should also be included.
The pelagic ecosystem of the lake, encompassing 80 percent of its area,
occupies the most important position in the cycle of organic matter in
Baikal. A simplified plan of the cycle of matter and energy in the pelagic
zone is presented in Figure 4.
Differentiation of dissolved organic matter (DOM) into easily oxidizable
(OOM) and non-oxidizable aquatic humus (IOM) is necessary to provide an ac-
curate estimate of the bacterial production. This differentiation was per-
formed arbitrarily, based on the relationships of permanganate and bichro-
mate oxidizability, and the ratio of organic carbon (C0rg) to organic nitro-
gen (N0rg). In particular, analysis of the vertical distribution of X =
Corq/Norq, reflecting the relative increase in the fraction of aquatic humus
in the DOM with depth, showed that for the complex of matter forming the
IOM, X = 5 can be assumed, while for the IOM9 X = 22. Then the fraction of
Corg represented by IOM (xc) can be calculated by the equation:
x
c
= 1-38 (1 - ) . (3)
The ratio of mineral nitrogen to mineral phosphorus in the waters of
Baikal lies between 7 and 10 (Votintsev et al_. 1975), approximately corre-
76
-------
E
^
i 4
LJJ
5 2
0.005
1
1
I
4567
DISCHARGE (x), km
8
10
Figure 3. Concentration field (mg/liter) of a pollutant in the 0-20 m layer near BCPC.
D = 104 cm2/s; K - 0.086 day-1.
-------
TABLE 2. BIOTIC BALANCE OF THE PELAGIC ECOSYSTEM OF LAKE BAIKAL
kcal/m2/year (from Botintsev et^ al_. 1975)
Component B P P/B K T T/B A R~
Phytoplankton 3.0 875 290 0.9 97 32 972
Bacterioplankton 9.4 315 34 0.5 262 28 602 602
Epischura
(Epischura
baicalensis) 6.0 80.5 13.5 0.25 245 41 326 407
Cyclops
(Cyclops
colensis) 0.3 3.4 11.3 0.27 9.2 30.6 12.6 15.7
Macrohectopus
(Macrohectopus
branizkii)3.1 4.7 1.5 0.26 13.3 4.3 18.0 22.5
Omul
(Coregonus
autumnolis
migration)' 1.9 0.38 0.2 0.33 0.76 0.4 1.14 1.4
Large and Small
Gblomyanka
(Comephorus
baicalensis
comephorus
dybowskii 3.69 2.95 0.8 0.45 3.52 0.9 6.47 8.1
Bullhead sac-fry
and finger lings
(Cottocomephorus
grewingki cotto-
comephorus
inermis)0.23 0.23 1.0 0.21 0.84 3.7 1.07 1.34
Nerpa (seals)
(phpca
sybirica) 1.26 0.22 0.18 0.17 1.08 0.8 1.30 1.6
B = biomass T = destruction (loss to metabolism)
A = assimilation of food R = ration
P = production A = 0.8 R; P = A - T; K£ = P/A
78
-------
VERTICAL AND HORIZONTAL WATER EXCHANGE
Figure 4. Diagram of cycle of matter and energy in pelagic Lake Baikal.
SOM = suspended organic matter (PM), detritus; OOM - oxidizable fraction
of dissolved OM; IOM = nonoxidizable fraction of dissolved OM, aquatic
humus, a) Direction of solid lines coincide with direction of flow of
matter and energy; b) Oriented dotted lines symbolize participation of
initial component in process indicated.
79
-------
sponding to their proportions upon photosynthesis. However, the more rapid
regeneration of phosphorus, relative to nitrogen, causes a decrease in this
ratio, frequently dropping to 0, during periods of maximum development of
phytoplankton. Since nitrite and ammonia nitrogen are practically absent in
the waters of Baikal (Votintsev 1961), the limiting biogenic element, com-
pleting the biotic cycle, is nitrate nitrogen.
To estimate the changes in the status of the lake's ecosystem consider-
ing the excess input of biogenic substances and other impurities, and an-
thropogenic action, distinct from natural background fluctuations, one can
use a model based on the dynamic balance method (Gorstko et_ a_L 1977). The
algorithm for this method is presented in the following section.
DYNAMIC BALANCE METHOD
Two groups of equations are used to describe the ecosystem. The first
group reflects the productive nature of the system, i.e., the fact that
matter in some components is produced by the consumption of others:
P(T) = AUHX* + P(T) + U(T)] (4)
where: X = (X], X2,..., Xn) is the state vector of the ecosystem,
the coordinates of which represent the instantaneous
values of biomass or concentration of the components;
P(T) = (PI(T), P2(i),..., PP(T)) is the production vector of
the ecosystem in period T;
U(T) = (UI(T), MT),..., Un(x)) is the control vector,
characterizing the arrival (or removal) of individual
components during period T;
A(T) = [a-jj(T)] is the production matrix of the ecosystem;
ajj(T) is the fraction of the matter of the jth component
consumed by the ith component in period T.
The superscript t represents discrete time. The other grouping reflects the
dynamic nature of the system. It is as follows:
Xt+T= B(T) [X*
P(T) + U(T)]
(5)
where: B(T) = diag(Bi(T), B2(x) ..... Bn(T)) is a diagonal matrix,
and B-J(T) is the fraction of the i component which
remains at the end of period (t, t + T).
The elements of matrices A and B are defined through the rates of mutual
consumption or transformation of components f-j-j (the flux of matter or
energy is directed from the jth component toward the ith component):
80
-------
£ y
^ '
ke Q
ke Q.
J
where: f^j is the rate of decrease of the jtn component as a result
of consumption by the ktn component;
is the conversion coefficient between the units of mea-
surement of the corresponding components;
QJ is the set of consumers of the jth component.
If all components of the model are expressed in units of one chemical
element (for Baikal, as noted, nitrogen can serve as such an element), ob-
viously, the system is balanced:
N t._ N t N
E Xt+T= £ X* + Z U. . (8)
1=1 1 i=l n 1=1 1
Equation (8) can be easily obtained by adding (4) and (5) term by term and
keeping in mind that
N , x
I a.. + B. = 1, Vj . (9)
To consider the flow of matter from the lake p0, which depends on internal
processes, the fictitious component XQ is introduced such that f-,-^ = °> vi-
t+Tn t
Then XQ xn + P0 and
N t._ N . N N t N t N
£ X. = E X|+ £ U. + E Xj = E r. + E U. - pQ . (10)
1=0 1 1=0 1=0 n 1=1 1=1 1=1
In particular, we can distinguish two such components in the model of
Baikal, reflecting irrecoverable losses from the lake: 1) the discharge of
organic, mineral substances, and plankton with the waters of the Angara, and
2) the partial sedimentation of QOM onto the deposits on the lake floor.
N
Since Z a.. 0,
81
-------
then equation (4) always has a positive solution (Bellman, 1969):
P = (I - A)"1 AU1 + U) (11)
where I is a unit maxtrix.
Consequently, this system is noncontradictory. The basic task of simu-
lation of the ecosystem is reduced to identification of the matrix F =
[fin-] as a function of the values of the state variables, and the environ-
mental factors (illumination, temperature, secondary impurities, etc.), with
subsequent allowance for various input perturbations and control actions.
Consideration of the three-dimensional structure of the ecosystem in the
model can be achieved by using the same approach in two modifications:
1. Combined Model. This model involves an increase in the dimension-
ality of the state vectors of production and control, the coordinates of
which are grouped according to the number of spatial units distinguished.
In the same manner, the dimensionality of matrices A and B increase, as they
are converted to block matrices, their elements decreasing due to addition
of the speeds of flow or migration of components between neighboring cells
in the denominator:
T S
rj
reQ.
( £ fU) Yf,
reQj ^ ^
(i)
7TT- i = J A k = I
1 + T ( z f(4) y - +
reQ. rJ ^
0 i 4 j A k = I
where the subscripts i, j, r indicate the numbers of the components,
k, £, s are the cell numbers;
g^1) is the rate of outflow (migration) from the kth cell into the
sk s^h cell by the i^h component;
f^' is the rate of loss of the jth component due to consumption by
the ith component in cell k.
82
-------
This modification allows simultaneous description of the biotic cycle
and transfer (migration) of components; however, the increase (proportional
to the square of the number of cells M) in dimensionality of matrix A, which
must be repeatedly transformed, places a limit on its use.
2. Method of Splitting. Sequential operation of two models is used:
1) a model of the dynamics of the water (migration of components), and 2) a
model of the biotic cycle.
If x|< is that portion of the state vector which relates to cell k:
x-j is that portion of the state vector which relates to component i:
R|< is the model operator of the biotic cycle in cell k:
Gi is the model operator of water exchange (migration of the itn com-
ponent, then:
= (*) k = 12...M
= 6. (7.)
= 1,2,. ..N
where: 7 is the intermediate vector;
M is the number of cells;
N is the number of components.
CONCLUSION
At present, anthropogenic
a minimum. Therefore, we can
adequate models for long-term
rather complete dynamic model
mary function as a predictive
link between various branches
provide mutual testing of the
tion for studies necessary for
support.
effects on the ecosystem of Lake Baikal are at
judge their possible effects only if we have
prediction. One such possible model is a
of the biotic cycle. In addition to its pri-
instrument, this model can act as a connecting
of limnologic study of Lake Baikal, helping to
materials produced, and indicating the direc-
elimination of restrictions in information
REFERENCES
Bellman, R. 1969. Vvedeniye v teoriy matrits (Introduction to matrix
theory), Moscow, Nauka Press, 367 pp.
Biologic productivity of the pelagic area of Baikal and its variability.
1977. Trudy LIN SO AN SSSR, Vol. 19(39), Novosibirsk, Nauka Press,
254 pp.
Botintsev, K.K., A.M. Meshcheryakova and G.I. Popovskaya. 1975. Krugovorot
organ icheskogo veshchestva v ozere Baikal (The cycle of organic matter
in Lake Baikal), Novosibirsk, Nauka Press, 189 pp.
83
-------
Botintsev, K.K. 1961. Gidrokhimiya ozera Baikal (The hydrochemistry of
Lake Baikal), Moscow, Acad. Sci. USSR Press, 311 pp.
Gorstko, A.B., Yu.A Dombrovskiy and V.V. Selyutin. 1977. Application of
the dynamic balance method to modeling of the biotic cycle under condi-
tions of anthropogenic eutrophication. Antropogennoye evtrofirovaniye
prirodnykh vod (Anthropogenic eutrification of natural waters), Part I,
Chernogolovka, pp. 75-79.
Kozhov, M.M. 1972. Ocherki po baykalovedeniyu (Essays on the science of
Baikal), Irkutsk, Vost.-Cib. kn. izd-vo press, 254 pp.
Krotova, V.A. 1970. Geostrophic circulation of the waters of Baikal during
the period of direct thermal stratification. Trudy LIN SO AN SSSR, Vol.
14(34), pp. 11-44.
Problems of Baikal. 1978. Trudy LIM SO AN SSSR, Vol. 16(36), Novosibirsk,
Nauka Press, 295 pp.
Techeniya v Baikal (Currents in Baikal). 1977. Novosibirsk, Nauka Press,
160 pp.
Tsvetova, Ye.A. 1977. Mathematical modeling of the circulation of the
water of a lake. Ibid., pp. 63-81.
84
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SECTION 6
SPECIES DEPENDENT MASS TRANSPORT AND CHEMICAL EQUILIBRIA:
APPLICATION TO CHESAPEAKE BAY SEDIMENTS
Dominic M. Di Toro^
INTRODUCTION
The analysis of the interactions between sediments and the interstitial
waters is a problem of substantial difficulty. The complex chemical and
biochemical reactions which affect the concentrations of the substances of
interest are coupled to each other and to the gas and solid phases of the
sediment. In addition, fluxes exchange mass across the sediment-water
interface and redistribute it within the sediment via interstitial water
diffusion and physical and biological mixing in the surface layer. Clearly
a comprehensive analysis is necessary in order to understand these interre-
lationships and to establish the primary controlling factors.
The calculation presented below is based on the mass balance models of
observed increases or decreases in interstitial water concentrations of
various substances (Goldberg and Koide 1963; Lerman and Taniguchi 1972;
Berner 1974). Typically these models are applied to a single constituent of
interest, e.g., ammonia. For multiple constituents a conceptual simplifica-
tion is available if the reactions are being driven by the decay of organic
matter of a fixed stoichiometry (Richards 1965). A further simplification
occurs if it is assumed that certain species are at chemical equilibrium.
For certain inorganic dissolved species and certain redox reactions this is
a well known approximation that has been tested by a number of investigators
(e.g., Garrells and Christ 1965; Kramer 1964; Thorstenson 1970). while it
has often been pointed out that overall and complete thermodynamic equili-
brium is never attained for all species in all settings it is also clear
that certain reactions occur so quickly that they are virtually in equili-
brium over the time scale of the analysis. Thus while not as universally
applicable as the principle of mass balance it is nonetheless a useful ap-
proximation in certain contexts. For the calculation of sediment behavior
presented below, the equations of mass balance and chemical equilibrium are
combined into a single structure for the analysis of sediment interstitial
water and gas phase concentrations.
^Environmental and Engineering and Science Program, Manhattan College,
Bronx, New York 10471.
85
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COMPUTATIONAL FRAMEWORK
The equations of mass conservation provide the starting point for the
analysis of the distribution of dissolved and particulate materials in sedi-
ments. For a steady state one dimensional analysis let Di be the diffusion
coefficient porosity product of species A-j and let wi be the corrected
(Imboden 1975) advective velocity of A-J, the velocity induced by the sedi-
mentation of mass relative to a coordinate system fixed with respect to the
sediment surface corrected for compaction, and assume they are constants in
depth. The one dimension mass transport equations for the Ns species, A-j,
are:
dtA.l
= f1 ([A^....,^]) i=l,---,Ns (1)
where f -j( [A-j],..., [ANsl) is the sum of all sources and sinks for species Ai.
For rapid chemical reactions such as acid-base reactions these equations are
very difficult to solve using straight-forward numerical methods (DiToro
1976).
The method developed for this situation depends on separating the fast
and slow reactions and requiring that the fast reactions be at equilibrium.
Consider the species (the fast reactants) that are involved in at least one
fast reaction and number them i=l,---,Nfs. For example HCOs~ would be in
this category. The remaining species (the. slow reactants) are numbered
i=Nfs+l,...,Ns. Particulate organic matter would be in this latter cate-
gory. For this division the mass balance equations can be written:
(2)
•pc 9 9 c \ /
where the rate at which Ai is produced by fast reaction j is vjjRj. Rj is
the reaction rate of the jth Of Nfr fast reactions and vji is the reaction
stoichiometric coefficient. S-j is the net source of Ai due to the slow re-
actions. This separation of the equations simplifies the solution procedure
since equations (2) for the fast reactants can be transformed into a smaller
set of equations for the Nc components, Bj, that form the fast reactants.
Using the stoichiometric coefficients, aij, of the species in terms of the
components equation (2) becomes (DiToro 1976):
Nfs 2 Nfs
f=1 (-Di^ + Widl)aik[Ai]^=1 ^ K-1.-.NC (4)
86
d2[A.]
-^^
d[A.]
lfz~
= Si
Nfr
+ Z V..R.
j=l J1 J
-------
Although the species concentrations, Aj, are nonlinear functions of the Nc
components B|< via the equilibrium relationships, the equations (3) and (4)
can be solved more directly than the species equations themselves. In terms
of number of equations, for example, the twenty-three fast reactants con-
sidered in the application below would each require an equation in each seg-
ment whereas seven component equations in each segment suffice. The solu-
tion procedure for equation (4) is detailed in the Appendix.
CLASSIFICATION OF THE REACTIONS
The initial choice for an application of these methods is the species to
be considered and whether they are fast reactants. For the calculations
presented below it appeared reasonable to restrict the computations to the
distributions of the species with components: carbon, nitrogen, sulfur,
hydrogen, oxygen, and, because of its availability in the data to be ana-
lyzed, argon. The species and their relevant properties are listed in Table
1.
Consider, first, the species involved in purely aqueous reactions. The
acid-base reactions are clearly rapid and at equilibrium over the time scale
of sediment diffusion. However it is not clear that the redox reactions are
either as rapid or at chemical equilibria. The principal aqueous redox re-
actions for the species in Table 1 are the reduction of sulfate and the pro-
duction and consumption of methane. Since bacterially mediated kinetics are
responsible for these redox reactions, it might seem at first glance that
the notion of chemical equilibrium is of little value 1n this context. How-
ever it has been pointed out that the thermodynamically predicted sequence
of oxidation-reductions is commonly observed in nature as oxidation of or-
ganic material occurs (Stumm 1966) so that this appears to be a reasonable
simplification of the complex reaction kinetics actually taking place if
they are, in fact, rapid. The assumption is also quite convenient since
equilibrium calculations are independent of the reaction pathways and no de-
tailed specification of the kinetics are necessary; only the thermodynamic
constants of the species of interest are required. One of the interesting
results of the application discussed below is that it appears reasonable to
assume that the reactions involving sulfate and methane are indeed rapid,
relative to other reactions and the transport, and at equilibrium.
For the redox reactions involving nitrogen, although it may be reason-
able to assume they are rapid, the equilibrium that is calculated from ther-
modynamics is unreasonable. For example, the thermodynamically favored oxi-
dation of ammonia to nitrogen gas within the zone of sulfate reduction does
not appear to occur, nor does the oxidation of nitrogen gas to nitrate in
oxic environments. As a consequence the only nitrogen species considered
in the fast reactant set is gaseous and dissolved N2- Ammonia is assigned
to the slow reactants; nitrite and nitrate are not considered as their con-
centrations are quite small in the application considered below.
Consider, next, the aqueous-gas phase reactions. As methane is produced
it is possible that a gas phase (bubbles) will form. Whether the gas phase
is in equilibrium with the interstitial water at that location depends on
87
-------
TABLE 1 CHEMICAL FAST REACTANTS STRUCTURE AND AQUEOUS DIFFUSION
COEFFICIENTS
Phase
Aqueous
H+
OH-
H?0
C,
02
C02
HCO;
cor
CH4
N2
H2S
HS"
S=
S07
Ar4
PK(1)
o.o(3)
-14 5
0.0(3)
-89.0
-0.77
-7.07
-17.6
22.18
-2.57
-1.12
-8.36
-22.74
-43.55
-2.23
Relative Diffusion
Coefficient (2)
4.582
2.597
1.0
0.989
0.842
0.583
0.488
0.733
0.984
0.748
0.852
0.406
0.524
0.990
Phase
Gas
HoO
02
CO 2
CH4
N2
H2S
Ar
pH
H+
pS
PK(I)
-1.56
-86.0, .
0.0(3)
24.3] .
0.0(3)
f f\\
0.0 3
/ *N \
0.0(3)
7.42
H2S
4.6
SLOW REACTANTS
Phase
Relative Diffusion
Coefficient(2)
Reaction
Source(+) Sink(-)
Aqueous •
NHj
Organic Sediment
CH302(NH3)Y
0.963
0.0
S = + KY [CH302(NH3)y]
S = - KY[CH602(NH3)y]
Aqueous phase concentrations in mole/£, Gas phase concentrations in mole
fractions. pK for the reaction with ths species concentration on the left
hand side and the component concentrations on the right hand side, e.g.,
[HCO^] = 1.0 [C02(g)] + 1.0 [H20(aq)] - 1.0 [H+(aq)f. For T = 15°C and gas
phase total pressure of 4 atm2. Aqueous C02 equilibria adjusted for C£ =
10%. (Wagman, et al., 1968; Stumm and Morgan, 1970; Atkinson and Richards,
1967; Yamamoto et aT., 1976; Harvey, 1966; Weiss, 1970).
Dspecies/DC£~ with DCJT = 2.032 • 10'5cm2/sec @ 25°C (Robinson and Stokes,
1959; Himmelblau, 1964; Chapman, 1967; Li and Gregory, 1974: Reid et al.,
1977).
3
Indicates a component as well as a species. e~ is the remaining component.
-------
the rapidity of the mass transfer across the liquid-gas interface. For
stationary or slowly moving bubbles, equilibrium is obviously reasonable.
For rapid bubble motion with mass transport across the interface, however,
the assumption is incorrect. Therefore assigning the aqueous-gas phase re-
actions to the fast set precludes the analysis of non-equilibrium mass
transfer within this framework.
The aqueous-sediment reactions are not mechanistically included in this
calculation. Rather the reactions controlling pH and pS are adjusted such
that the pH and pS are at their observed values. The relevant sediment buf-
fering and precipitation reactions are simulated rather than explicitly in-
cluded. In addition, it appears that ammonia adsorption can be neglected
based on the magnitudes of the transport and reaction parameters as shown
below, although its inclusion would not materially complicate the calcula-
tion.
The principal slow aqueous-sediment reaction considered is the decay of
sedimentary organic material. In fact the rate of this decay and the diffu-
sion and advective transport set the time scale within which the other reac-
tions must be rapid. The structure of the calculation which results from
these concentrations is shown in Table 1.
The aqueous diffusion coefficients are species dependent as shown and
these differences are included in the calculation. However, the coupling of
the diffusion fluxes due to the electrical potential generated by the dif-
fering diffusion coefficients appears to have only a small effect at sea
water concentrations (Ben-Yaakov 1972) and is neglected although its inclu-
sion presents no real difficulty.
SPECIES INDEPENDENT TRANSPORT
As can be seen in Table 1, the assumption that aqueous species diffuse
at the same rate is in error by a factor of two and, if H+ and OH" are con-
sidered, by a factor of twenty. The situation becomes more implausible if
the gas and stationary phases are considered to be diffusing as well. The
assumption of species independent transport also requires that the species
are advecting with the same velocity. This is reasonable for the aqueous
and solid phases in the absence of groundwater intrusion, but it is cer-
tainly possible that the gas phase is advecting at a different velocity and
in a different direction relative to the aqueous and solid phases.
Nevertheless the solution of the relevant differential equations are so
simplified by this assumption that it is useful to pursue the result. For
the decay of sedimentary organic material, with a sedimentation velocity, w,
a first order decay with rate constant, K, and zero diffusion the solution
to equation (3) is an exponentially decreasing concentration of sediment or-
ganic material in depth: c0 exp(-Kz/w), with surface boundary concentration
c0. The spatial distribution of the source due to this slow reaction is:
S£=KCO exp(-Kz/w), where £ is the species index of sediment organic matter.
89
-------
For the case of species-independent transport equations (4) each become
an equation in a component since ? a-jiJA-j] = [Bkl. The source term is just
the quantity of that component 1 being produced by the decay of sediment-
ary organic matter since a^ is its component stoichiometry. Further the
equations are not coupled to each other and they can be solved separately.
The result is:
[B.](z) = [B.] + ^SL-(1 - e-Kz/w) k=l,..-,N (5)
K K ° 1+KD/vr
where [B|<]o are the component boundary concentration at z = 0, the sediment-
water interface. For components that are not part of the stoichiometry of
the sedimentary organic matter, i.e., the Bk's such that a£k = 0, their con-
centrations are constant in depth.
Perhaps the most useful way of picturing the calculation implied by
equation (5) is to interpret £ = 1 - exp(-Kz/w) as a variable that reflects
the extent of a titration of sedimentary organic matter into the chemical
system defined by the boundary concentrations of the components. As depth
increases, £ increases and more sedimentary organic material is titrated
into the chemical system where it reacts with the boundary-defined component
composition to produce a new composition of species. Thus with species-
independent transport, the depth distribution in the sediment is simply the
result of an exponential titration of organic matter into the chemical
system defined by the surface boundary composition of the sediment.
It is interesting to note that the chemical equilibrium investigations
of reducing sediments by Thornstenson (1970) and Gardner (1973) are con-
ceived as titrations of organic matter into chemical systems comprised of
entirely dissolved species or with solid phases present. The above analysis
indicates that for species-independent transport the depth distribution of
species indeed reflects a simple titration and the results of such calcula-
tions are directly relevant.
It is useful to realize that this solution satisfies the conditions of
the mass balance equations with species-independent transport coefficients
so that, in fact it is a plausible solution for situations wherein that as-
sumption is reasonable. Conversely for species-dependent transport due to
gas phase motions, stationary solid phases, and the differing molecular dif-
fusion coefficients of aqueous species, the requirements of mass balance are
more complex and the distributions of components can no longer be calculated
from equation (5).
CHESAPEAKE BAY SEDIMENT PARAMETERS
In order to investigate the feasibility and utility of these computa-
tions for species dependent transport an initial application has been made
to observed distributions of carbon, nitrogen, sulfate and dissolved gases
in Chesapeake Bay sediments. The species chosen (Table 1) are dictated pri-
90
-------
marily by the available data. The other major cations and anions are not
considered as they were not consistently reported.
Ideally, the relevant sediment parameters: the diffusion and advection
coefficients, sedimentary organic matter stoichiometry, and its reaction
rate should be determined together with the observations of interstitial
water concentrations. While this is not the case for the historical data
considered in this application, a reasonably complete set of estimates can
be made from other investigations. These are listed in Table 2. The ratio
of sediment organic carbon to organic nitrogen is reasonably constant over
the 30 cm. of interest and both decay exponentially as shown in Figure 1
(Schubel et^ aj_. 1977). The boundary condition for sediment organic carbon
ranges between 2 and 3% (Biggs 1967). The sedimentation velocity has been
determined by Pb210 dating and other methods and ranges between 0.1 and 0.28
cm/yr. It is, therefore, possible to estimate the first order decay rate
constant, and knowing the water content and surface sediment concentrations,
the rates of organic carbon and nitrogen sources resulting from this slow
reaction. Essentially a three-fold uncertainty exists for both these
sources as shown. An independent estimate of the chloride diffusion coeffi-
cient is available, based on an analysis of the time variable chloride pro-
files in the sediment (Holdren ejb aj_. 1975).
The parameter values in Table 2 can be used to assess the importance of
ammonia adsorption and the need to include a solid phase ammonia species.
It has been shown (Berner 1977) that if DK/w2»l+K*, where K* is the linear
adsorption coefficient, K* = tads.NH4]/[NH4(aq)], the adsorption can be ne-
glected in the computation of the interstitial water ammonia concentration.
For the parameter ranges in Table 2, DK/w^ ranges from 5 to 130. Although
no ammonia adsorption coefficient is available for Chesapeake Bay sediments,
the value of 1.6 has been reported for a Long Island Sound sediment (Rosen-
feld and Berner 1976). Thus it is probable that ammonia adsorption can be
neglected in this calculation.
There is only one remaining uncertainty: the hydrogen (actually the
electron) stoichiometry of the sediment. Although it is conventional to
assume that Redfield's ratio (Ch^O) applies to sedimentary organic material
it is possible that other ratios might be appropriate. For the components
used in this calculation the required stoichiometry is
CHg+2 °2+6(NH3)Y = C02 + 6H+ + 3e~ + YNH3 + 6H2°
It is intersting to note that the oxygen stoichiometry is essentially irrel-
evant once the carbon and hydrogen stoichiometry are established since it is
related to the H20 content of the organic matter. The quantity of H20 in-
troduced by the organic material decay is negligible relative to the concen-
tration present and the convenient choice is 6 = 0 corresponding to the
"dry" stoichiometry of the organic material, i.e., ((X^jHgCNHsJy. Any addi-
tional water added to this material does not alter the results and is there-
fore irrelevant for this computation. For this convention the Redfield
stoichiometry is CH/^ corresponding to the electron to carbon stoichiomet-
ric ratio of glucose.
91
-------
TABLE 2. CHESAPEAKE BAY SEDIMENT PARAMETERS
ro
Description
Ratio of Sediment
Org C to Org N
Sediment Org C at
z = 0
Sedimentation
Velocity
Sediment First Order
Decay Coefficient
Water Content
Organic Carbon
Chloride Diffusion
Coefficient
Sediment Sulfide
Concentration
Gas Phase Volume
Symbol
[Org C]
[Org N]
[Org do
w
K
P
Oro-C* '
DQ£~
FeS(s)
FeS(s)
vgas
Value
10.9 + 1.1
2.7
2.0 - 3.0
0.09 - 0.12
0.1
0.28
0.005 - 0.0067
0.5 - 0.6
15.2 - 45.9
0.22 - 0.44
1.0 - 1.3
0.7 - 0.8
0.1 - 1.0
Units
mole/mole
% dry wt.
% dry wt.
cm/yr
cm/yr
cm/yr
yr-l
% wet wt.
yM/Vday
p
cm /day
% dry wt.
% dry wt.
% volume
Stationl
842D
842D
820
842D
842D
820
-
856C
61
61
Pier 22
Pier 31
Reference
Schubel et al. (1977)
Schubel et al . (1977)
Biggs (1"9"6"7T~
Schubel et al . (1977)
Carpenter (T975)
Powers (1954)
Figure (1)
Biggs (1967)
2
Computed
Holdren et _aj_. (1975)
Biggs (1963)
Biggs (1963)
Schubel (1974)
Schubel (1974)
See Bricker et a\_. (1977) for the relationship to latitude and longitude.
-S0rg_c(0) = K[0rg C]0(l-p)/p.
-------
CO
SEDIMENT COMPOSITION, percent dry wt
0.1 0.2 0.3 0.5 1.0
Figure 1. The vertical distribution of sediment organic carbon and nitrogen as percent dry weight.
Chesapeake Bay Station 842D (Schubel et al_., 1977). The solid lines represent exponential
decays: exp (-Kz/w), with w/K = 18 cm and a molar carbon to nitrogen ratio of 10.9.
-------
A few representative examples for other sediments and their organic
fractions are shown in Table 3. The electron to carbon ratio, 3, which is
computed from the reported elemental analysis (C,H,0,N) after conversion to
dry stoichiometry, ranges from 3.8 for fulvic acids to 5.3 for bitumen from
recent Bering Sea sediment. Thus the fraction of the sediment organic car-
bon that is decaying is likely to have an electron to carbon ratio in the
lower end of this range since the resistant fractions (bitumen and kerogen)
have the higher stoichiometries and the initially deposited material is pro-
bably close to the Redfield ratio of 4.0. The nitrogen to carbon ratio is
also listed which, except for bitumen, spans a range from 0.05 to 0.1.
CHESAPEAKE BAY STATION 856: CARBON AND AMMONIA
The parameter values in Table 2 set the probable ranges for Chesapeake
Bay sediments. For an application to any particular set of data it is
likely that the specific values for the parameters will differ somewhat,
however the ranges are known.
An extensive set of interstitial water chemistry data for Chesapeake
Bay sediments is available (Bricker et aj_. 1977) that can be analyzed within
the context presented above. A composite data set has been examined from
the winter of 1971-1972. All stations with the designation 856, that is at
the latitude 38°56' and for which measurements of pH, $04, alkalinity, and
NH4 are available simultaneously, have been grouped. The average and stand-
ard deviation of both the midpoint sampling depth and the concentrations
have been calculated for the overlying water, and for the midpoint depth in-
tervals: 0-5 cm., 5-10 cm., 10-25 cm., 45-85 cm., and 85-100 cm. In the
plots that follow the data are represented by the means +; the standard de-
viation of both the concentration and the midpoint sampling depths. If no
error bars appear they are smaller than the plotting symbol. The depth in-
tervals are chosen to minimize the overlap between the actual sampling in-
tervals used in the surveys. The total carbon dioxide concentration is cal-
culated from the reported pH and alkalinity data with the equilibrium con-
stants adjusted for the reported chloride concentration (Stumm and Morgan
1970).
The boundary condition concentrations are established from the overlying
water data except for the initial ammonia concentration which is chosen to
reproduce the total carbon dioxide to ammonia ratio at 5 cm. This refine-
ment affects only the computed carbon to ammonia ratio at the sediment-water
interface and is made in order to better reflect the initial ammonia concen-
tration just below the interface. These concentrations are listed in Table
4 together with the parameters chosen for the transport and sediment organic
matter stoichiometry. The diffusion, advection, and reaction rate coeffi-
cients are those reported (Table 2). The sediment organic carbon decay rate
and the electron and nitrogen to carbon ratios are adjusted within the re-
ported range to produce an acceptable fit. The resulting parameters are
listed in Table 4. The organic carbon decay rate is at lower limit of the
probable range as is the chloride diffusion coefficient. Since only the
ratio of these parameters affects the computed profiles the values chosen
appear to be quite reasonable. The electron to carbon ratio of 3.33 is sub-
94
-------
TABLE 3. SEDIMENT ELECTRON1 AND NITROGEN STOICHIOMETRY
CH602(NH3)
Y
Y
Reference
Fulvic Acid
Humic Acid
Marine Sediment
Marine Sediment
Lithified Marine
Sediment
Kerogen
Bitumen
3.80 + 0.36 0.0912 + 0.0247
4.24 + 0.19 0.0766 + 0.0129
4.04 - 4.23 0.089 - 0.091
4.64 0.098
4.99 0.054
4.64 0.062
5.29 - 5.33 0.01 - 0.012
Rashid & King (1970)
Rashid & King (1970)
Kemp (1973)
Trask (1938)
Trask (1938)
Philip & Calvin (1976)
Bordovskiy (1965)
- A + IM1 o [0] o [N]
[C] MCI " J [C]
95
-------
TABLE 4. STATION SEDIMENT PARAMETERS
Station
Parameter
Sampling Dates
Y"1 = [Org Cl/tOrg Nl
3 - [Org H]/[Org Cl
K
w
S0rq_c(0)
DCJT
[NH4]o
[ZC02]o
[A£k]0
[N2]o
pH
pS
Units
-
mole/mole
mole/mole
yr
cm/yr
yM/£/day
cm^/day
mgN/£
mgC/£
g CaCo3/£
mgN/£
-
-
856
1 Oct 71 to
31 Jan. 73
9.0
3.33
0.005
0.1
13.5^)
0.25^
!.55<4>
21.2(2)
0.073(2)
-
7.4
12.5
858-C
22-XI-66,
5- 1-67
10.6
2.67 - 3.0
0.005
0.1
30.0
0.25
—
18.4(2)
-
7.0(5)
7.4
12.5
Any value in the ranges in Table 2 will produce the same result so long as
S0rg_c(0)/DCjr = 13.5/0,25 - 27.0/0,50.
"Average of overlying water data.
Computed from reported C£ concentration 10.4°/00.
^
_Estimated from C/NHs-N ratio at 5 cm. Overlying water cone. = 0.3 mgNA-
^Estimated from 0-15 cm data.
96
-------
stantially lower than what would be expected (Table 3). Evidently the re-
ductions of iron and manganese, which are not considered in this calcula-
tion, are responsible for this difference. Those reactions are presumably
utilizing the electrons corresponding to the difference between 3 = 3.33
required for sulfate reduction and methane production and the expected
ratio of 3 '^ 4.
Figure (2) presents the comparisons to the observed ammonia and total
inorganic carbon data as well as their ratio. Solutions for both the
species dependent and the species, independent diffusion coefficients cases
are presented with the latter using the diffusion coefficient of chloride
ion for all species. No gas phase is computed to form in either case so
that its transport coefficients are not involved. For both cases the am-
monia profile is essentially the same since the ammonia diffusion coeffi-
cient is almost equal to that for chloride, but not for the total carbon
dioxide profile due to the substantially smaller bicarbonate diffusion co-
efficient.
The effect of species dependent diffusion is most markedly apparent in
the computed total carbon dioxide to ammonia ratio as shown in Figure (2).
The sediment organic matter carbon to nitrogen molar ratio is estimated to
be 9.0 whereas the interstitial water molar ratio is observed to be 16.0
near the sediment water interface and decreases to 12.5 at 100 cm. The
species independent case calculation begins at the specified molar ratio of
the boundary conditions and rapidly declines to the sediment organic matter
ratio. The behavior of the species dependent calculation is more complex.
The molar ratio declines to a plateau of 14.4 until a depth of 30 cm. after
which it abruptly declines to 13.5 at 40 cm. and further decreases to 12.5
at 100 cm.
The initial plateau value can be understood in terms of the differing
diffusion coefficients. The ammonia profile follows equation (5) since it
is a conservative slow reactant. Until the onset of methane production,
total carbon dioxide can also be described by a similar equation with a dif
fusion coefficient adjusted for the fraction of total C02 that is COz(aq),
M0%; HCOs, ^90%; and COs (negligible). The boundary concentrations are
quickly overwhelmed by the source due to the sediment decay and the ratio
approaches:
rnrn rl
LOrg CJ
[NH4]
Since the total C02 diffusion coefficient is substantially smaller than the
ammonia diffusion coefficient, and for the transport and reaction parameters
for this station: DNh^K/w2 = 44, the observed interstitial water ratio is
larger than the sediment organic matter ratio by the factor: DNH4/DZC02 =
1.6. The importance of this species dependent transport effect for
ammonia, sulfate, and phosphorus concentrations has been pointed out
(Berner 1977).
97
-------
CHESAPEAKE BAY STATION 856
500
_ 400
CO
E. 300
O
£ 200
100
75
-»c
CARBON/NITROGEN, g/g NITROGEN, mg/l
« o £ 8 ^ OK § S
TOTAL CARBON DIOXIDE (aq)
, 1 J
- iScL-L
- /'"
r
O4 | | | | |
10 30 50 70 90
DEPTH, cm
AMMONIA
(b)
- ^r^~\
j? \ i i i i
10 30 50 70 90
DEPTH, cm
OTAL CARBON DIOXIDE (aq)/ AMMONIA NITROGEN
(0
" r"-*Li i i -
- v. T r
_ 1 1 1 1 1
10 30 50 70 90
DEPTH rm
to
o
•i — "O
-P C CU i—
i. tO to i—
CU C 0 -P CU
CU .£= C -C •
< — s CD -P 4-* CU +-> IO
t/> o s- T- E c
CU S- O S ••- 4- O
C -P Q. TJ O «r-
r^ C C CU M C O
(0 C O CU
fO S- -i- CU -i- to
T3 -r- -P i— to -P
CU C CU (O -P
-P O -P T3 J= 'r- C
3 E c cu -p ;> cu
Q. E cu -c 1- T3 -i—
O C rB 0 T3
O to O) T3 4- "O CO
3 o. — S- in
13 O CU E fO
c~ cU "O CU O "O "O
(O 3 l/> T- C CU
o~ to ro to i —
-— ^ 03 O) O CU -P O.
in -i- -O 10 E
^- -— • O -P -i- to
O -Q CU S- S- O) to
JQ- — O- O O -C
E f> D.I— -P CU
to CU CU C O + I-P
~~" -r- -P S- f~ C 4-
"0 X -P O 10 O
CU O -C 4- CU
> .,- -i_> .(-> E to
s- -o o c cu -c:
CU JO CU 3 CU -P
WC T3 i— J= 0.
X> O S- C (O -P CU
O -O O CU > T3
S- 4- Q. -P
-» N)
o in o
MOLE C/ MOLE N
gure 2. Chesapeake Bay Station 8!
distribution of (a) total aqueous
(c) their ratio. The computat
(solid line), and the species •
diffusion coefficients set at
(Table 2). The symbols repr<
observations and the rnidpo'
SPECIES DEPENDENT SPECIES INDEPENDENT
TRANSPORT TRANSPORT
98
-------
The further decline of the total C02 to ammonia ratio after a depth of
30 cm is due to the onset of methane production. Since a fraction of sedi-
ment carbon is now being reduced to methane, the production of total C02 de-
clines somewhat while the production of ammonia continues. Thus the ratio
is calculated to decline as do the observations.
SULFATE, ALKALINITY, AND NET ALKALINITY
The increase of alkalinity and the decrease of sulfate are major fea-
tures of the interstitial water chemistry of sediments. The computed alka-
linity and sulfate profiles are compared to observations in Figure 3a and
3b. With all aqueous diffusion coefficients set to that of chloride
(species independent transport), a twofold increase in the sulfate diffusing
in and the bicarbonate diffusing out reduces the alkalinity and increases
the sulfate profiles as shown. If all aqueous diffusion coefficients are
set at the value for sulfate, the computed profiles are close to the species
dependent case.
The accumulated precipitated sulfide in the pS phase provides a check of
the calculation. The estimated quantity present in certain Chesapeake Bay
sediments is on the order of 0.07 - 0.23 mol/1 (Table 2) in comparison to
the calculated range of 0.08 - 0.37 mol/1 for station 856 sediments. As
pointed out by Berner (1974), this provides support for the values of the
sedimentation velocity used in the calculation.
The relationship between alkalinity, sulfate, ammonia, and the other
major cations in anoxic sediments has been investigated (Berner ejt efL 1970)
and applied to Chesapeake Bay interstitial waters for which the relevant
data were available (Bray 1973). The idea is to balance the charges of the
protolytic and nonprotolytic ions (Ben-Yaakov 1973). This charge balance
equation must be satisfied. The equivalent sum of the protolytic ions is
very nearly titration alkalinity (corrections for uncharged ions that con-
sume acidity should be made, e.g., NH3(aq) which is normally small at the
pH of interstitial water). As a consequence, the behavior of alkalinity can
be deduced from the behavior of the nonprotolytic ions in the charge balance
equation.
Following this reasoning the alkalinity is given by the relationship:
[Alk] = [HCOg] + 2[C03] + [HS~] + 2[S=] + [OH"] - [H+]
rri_, ,Alk . 9 S04 NH4 Ca ? Mg , (R]
= [Cl ] [- + 2 r- - e- - 2 - 2 - ...I (8)
] + [NH + ] + 2[Ca++] + 2[Mg++] + ...
where the ratios: Alk/Cl, S04/C1 etc. are the sea water ratios, and the
term multipled by [Cl~] expresses the contribution of the original sea
water. The remaining terms account for the effects of the interstitial
99
-------
CHESAPEAKE BAY STATION 856
AKALINITY
0.0
1.5
~5> 1.0
LJJ
I-
<
10
30 50 70
DEPTH, cm
SULFATE
0.5
CO
0.0
JL
-Q.
10 30 50 70
DEPTH, cm
NET ALKALINITY
90
I
Z^
I
o
to
30
25
^ 20
10 30 50 70
DEPTH, cm
90
0) •!-
'
cn 1/1
- SPECIES DEPENDENT
TRANSPORT
SPECIES INDEPENDENT
TRANSPORT
100
-------
water and sediment reactions on the major cations and anions. Note that if
all the cations and anions were included in this formula as implied by the
dots in the equation, then the sea water term would be zero (it is electri-
cally neutral) and the only relevant terms are the sediment interstitial
water concentrations themselves. Since the major variations occur for sul-
fate and ammonia, consider a quantity which might be termed net alkalinity:
[Net Alkl = [Alk] + 2[SO|] - [NH+1 = 2[Ca++] + 2[Mg++] + ... (9)
which is the equivalent sum of the sediment-derived cations and anions due
to precipitations, dissolutions, and cation exchanges. Strictly speaking,
the definition of net alkalinity should include nitrite and nitrate concen-
trations since their reduction to nitrogen gas increases alkalinity. How-
ever in anoxic marine sediments their effects are usually small relative to
other cations and anions. Net alkalinity should be more sensitive to
changes in solid phase-interstitial water reactions since the masking ef-
fects of the large sulfate and ammonia changes have been removed. In fact
net alkalinity is clearly independent of the aqueous reactions associated
with the decay of sediment organic matter (unlike alkalinity itself) and re-
sponds only to solid phase interactions and, as shown subsequently, species
dependent transport effects.
It is clear from this discussion that the calculation of alkalinity, and
certainly net alkalinity, requires a knowledge of the behavior of the sedi-
ment solid phases and the major cations. It is possible to simulate their
behavior by introducing two artificial phases, called the pH and pS phases
in Table 1, which keep the aqueous concentration of H+ and S~ at specified
values. The pH phase behavior is analogous to a reservoir of available
strong acid or base which is titrated into the aqueous phase as required.
The operation of the pS phase is based on the precipitation of ferrous
sulfides (Berner 1970). The net effect of this reaction on the species con-
sidered in this calculation is to remove H2S (Ben-Yaakov 1973) to the extent
determined by the activity of S=.
The computed net alkalinity, (Figure 3c) is quite interesting. For
species dependent transport the net alkalinity first decreases and then in-
creases, as do the observations. The species independent cases also show
decreases and then increases. Therefore both the species dependent trans-
port and simulated solid phase interactions are responsible for the shape
of the net alkalinity profile.
In order to understand the solid phase role consider the species inde-
pendent case. The sources and sinks that affect net alkalinity must come
from the solid phase-aqueous phase interactions. The production of carbon
dioxide by the sediment organic matter oxidation does not influence alka-
linity since it is uncharged. The same is true for the removal of H2S.
However both reactions affect pH. Therefore with the pH phase present the
removal of H2S tends to increase the pH and the pH phase responds by re-
moving alkalinity. This is the cause of the initial drop in net alkalinity.
Once sulfate reduction is over the pH tends to drop as CO^ production con-
101
-------
tinues. The pH phase responds by adding alkalinity. This is the cause of
the rise in net alkalinity after a depth of 30 cm.
For the species independent case with all species at the smaller sulfate
diffusion coefficient these variations in net alkalinity are intensified
since more C02 and H2S are retained within the sediment, relative to the
species independent case with all species at the chloride diffusion coeffi-
cient. Of course these changes are occurring in the alkalinity profile it-
self but they are masked by the large increase due to sulfate reduction.
The net alkalinity response is much more sensitive to these variations.
A more detailed investigation of the behavior of net alkalinity is given
in Figure 4. The first case (Figure 4a) considers the behavior with no sul-
fide precipitation (pS free) but with a fixed pH. The only significant ef-
fect is that due to carbon dioxide production at a fixed pH, which increases
net alkalinity. With all diffusion coefficients at the value of sulfate the
increase is larger than the chloride diffusion case due to the increased re-
tention of carbon dioxide within the sediment interstitial waters.
In order to understand the species dependent transport calculations con-
sider the principle ions that form net alkalinity in this calculation and
their diffusion coefficients:
[Net Alk] - [HCO~] + 2[SOp + IHS'l -
Diffusion (10)
Coefficients (0.583) (0.524) (0.852) (0.963)
As net alkalinity is produced or destroyed within the sediment by the solid
phases it tends to diffuse either out to or in from the boundary, since it
is driven by the gradient of net alkalinity. If the diffusion coefficients
of the species that make up net alkalinity are all lowered, more net alka-
linity remains in the sediment as can be ^een from the two species indepen-
dent cases in Figure 4a. The species dependent profile can be understood by
comparing it to the species independent profile at the diffusion coefficient
of sulfate. The differences are a slightly increased bicarbonate diffusion
coefficient relative to sulfate, equation (10), an increased bisulfide dif-
fusion coefficient and an almost twofold increase in ammonia diffusion. The
effect on the diffusion of net alkalinity toward the boundary is in propor-
tion to the quantity of each ion and the change in diffusion coefficients.
Computations show that the largest effect is due to the excess bicarbonate
diffusion, which causes a decline in the profile, followed by the bisulfide
diffusion, which is most effective in the region of sulfate reduction and
causes the initial drop in the profile.
The direction of the change to be expected from the excess diffusion re-
lative to sulfate can be understood in terms of the profiles of the indivi-
dual species that make up net alkalinity. Bicarbonate is increasing with
respect to depth so that the transport will therefore lower the net alka-
linity profile. The same is true for bisulfide. For ammonia, the diffusion
is also toward the boundary so that increased diffusion will lower the am-
monia profile. But this increases the net alkalinity since ammonia enters
102
-------
NET ALKALINITY
pH FIXED. pS VARYING
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pH VARYING, pS FIXED
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103
-------
the definition of net alkalinity with a minus sign, Equation (10). Thus the
decreases due to the excess bicarbonate and bisulfide diffusion are compen-
sated for to some extent by the excess ammonia diffusion. It is these ex-
cess diffusion effects that cause the species dependent profile in Figure
3c, the calculation with both pH and pS phases fixed, to be lower than the
species independent profile at the diffusion coefficient of sulfate.
These effects are even more apparent in Figures 4b and 4c which are cal-
culated without the pH phase present. Since the pH is now free to respond
to additions of carbon dioxide and the removal of H2S (Figure 4b) no net
alkalinity is generated and for the species independent transport case the
net alkalinity is constant. The decrease in the species dependent case is
due to the bicarbonate excess diffusion which is compensated for to some ex-
tent by the ammonia excess diffusion. With the pS also free in Figure 4c,
the only difference is the presence of HS" as a component of net alkalinity
and its excess diffusion further lowers the profile.
To summarize, the net alkalinity can initially decrease for two reasons:
either as a purely species dependent transport effect (Figure 4c), or in re-
sponse to the precipitation of ferrous sulfide at constant pH (Figure 3c),
where the decrease in this case is enhanced by species dependent effects.
The subsequent increase is in response to the production of carbon dioxide
at constant pH (Figures 3c and 4a) and is again modified by species depend-
ent transport effects.
An interesting question arises if one inquires into the generality of
these results. Since they depend so directly on the assumption of constant
pH controlled by a pH phase and since the pH of interstitial waters is sel-
dom constant, although for station 856 data it is very nearly constant: pH =
7.40 + 0.18, it might appear that their applicability to other situations is
restricted. However consider the following: The calculated responses of
net alkalinity to the C02 source and the H2S sink are the result of the re-
quirement of a constant pH, that is, an infinitely well-buffered system.
Suppose that in most sediments there are solid phase-aqueous phase reactions
which serve as buffers and tend to resist the pH changes induced by the C02
source and f^S sink. These reactions would titrate net alkalinity into or
out of the aqueous phase, much as the hypothetical pH phase does, but with-
out its infinite buffer capacity. Thus the pH would change somewhat in re-
sponse to the sources and sinks, but so also would the net alkalinity. As a
consequence one would expect to see profiles of net alkalinity that only in-
crease, if no sulfide precipitation occurs, or initially decrease and then
increase after sulfate depletion, or only decrease if sulfate is not de-
pleted.
A selection of net alkalinity profiles are shown in Figure 5. They have
been chosen to display the characteristic shapes that are calculated in Fi-
gures 3 and 4. Figure 5a shows an increase due, presumably, to the CO?
source but with sulfate present and approximately constant, indicating that
another electron acceptor was involved. Figure 5b from a station in the up-
stream reach of Chesapeake Bay (Cl < l°/°o) exhibits the expected increase.
The small quantity of available sulfate is depleted in the top five cm,
after which the net alkalinity increases in response to the C02 source. Fi-
104
-------
LONG ISLAND SOUND
CHESAPEAKE BAY. 922W
60
—
t-
§50
I-
UJ
Z
Af\
*»u
cc
DO
V
Z
< 55
_i
UJ
Z
45
•* +t
_ 32
1
ET ALKALINITY, m
8 8 8
MM
z
tA
£*v
w • 1 '
' '
1 '
^~* SO4" ~ CONSTANT
1 1 1 1 1
0 10 20 30 40 5(
DEPTH, cm
SOMES SOUND. MAINE
0
_ I— I
h- 1
_ SO4= = 0
I-HI-H
1 1 1 1 1
0 16 32 48 64 8(
DEPTH, cm
CHESAPEAKE BAY. 845 G
le)
- Q S04- > 0
0
00
0 ° 0
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1 1 1 1 1
0 20 40 60 80 10
DEPTH, cm
20
10
(b)
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20 40 60 80 100
DEPTH, cm
LAKE ERIE, CENTRAL BASIN
l.BU
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2.40
2.20
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1
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DEPTH,cm
CHESAPEAKE BAY, 8S6C
30
25
20
(f)
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4~ > 0
I
i
20
40 60
DEPTH, cm
80 IOC
Figure 5. Characteristic shapes for net alkalinity. The vertical dashed
line marks the depth of sulfate depletion. Increasing net alkalinity: (a)
Intertidal zone, Long Island Sound, Clinton, Conn. (Berner et aj_., 1970);
(b) Chesapeake Bay Station 922W, 18/XII/73 (Bricker et aj_., 1977). Net
alkalinity decreasing then increasing after sulfate depletion; (c) Somes
Sound, Maine (Berner et al., 1970). (d) Lake Erie, Central Basin. Mean
+ standard deviation of three stations: 41°49'03", 42°00'00", 42°27'14"9
sampled every month from May through August, 1971 (Weiller, 1972). Net
alkalinity decreasing; (e) Chesapeake Bay Station 845G, 21/VIII/72 (Bricker
et al., 1977). (f) Chesapeake Bay Station 856C, 30/VI/71 (Bricker et a]_-,
1977). Note the increase after the depletion of sulfate.
105
-------
gures 5c and 5d illustrate the initial decline due to sulfide precipitation
and the subsequent increase after the depth at which the sulfate is de-
pleted. The location is indicated on the figure. The example from Lake
Erie is included because of its similar behavior in all respects except that
the magnitudes of the changes in net alkalinity are an order of magnitude
smaller than the brackish and sea water examples. Figures 5e and 5f from
Chesapeake Bay stations with higher salinity (Cl =< 10°/00) have more exten-
sive regions of declining net alkalinity. In both cases the decline is
within the region of sulfate reduction as indicated. The majority of other
profiles examined from Chesapeake Bay sediments exhibit the decreasing and
increasing pattern in the saline regions, with the increase starting after
sulfate is depleted, and a purely increasing pattern in the fresher water
regions, as illustrated in Figure 5.
Similar patterns are present in the Santa Barbara Basin data of
Sholkovitz (1973). Two cores (B and Gl) exhibit a continual decline in net
alkalinity in the presence of decreasing sulfate concentration. For the
third core (G2), the net alkalinity profile is essentially constant to 30 cm
with an increase at the last measured depth (33 cm). The sulfate is essen-
tially constant as in the case of the Long Island Sound example (Figure 4a).
These observations tend to indicate that the results of the detailed
calculations of net alkalinity (Figures 3 and 4) and the interpretation of
the changes have a certain generality. The net alkalinity changes are due
to the solid phase buffering reactions only and rot aqueous phase reactions.
The buffereing reactions are responding to the precipitation of sulfide by
decreasing alkalinity and to the production of carbon dioxide by increasing
alkalinity after the depletion of sulfate. It is expected that the decline
is accentuated and the increase is somewhat mitigated by species dependent
transport effects.
DISSOLVED GASES AND GAS PHASE TRANSPORT
After the depletion of sulfate, methane forms and if the sum of the par-
tial pressures of the dissolved gases exceed the total static pressure at
that depth bubbles will form (Reeburgh 1969). The calculation presented be-
low is designed to reproduce this behavior and to examine the consequences
of various gas phase transport mechanisms on the distribution of dissolved
nitrogen, argon, and methane.
As in the case of sulfate reduction the processes by which methane is
formed are bacterial. The basic features of the reaction in sediments is
that methane is produced only if sulfate is essentially absent and it is
consumed within the zone of sulfate reduction (Martens and Berner 1977;
Reeburgh and Heggie 1977). This is precisely the behavior predicted by
thermodynamic equilibria calculations. In order to assign these reactions
to the fast reaction set it is necessary that the kinetics be rapid relative
to the principal slow reaction: the decomposition of sediment organic matter
(K = 0.005 yr-1) and the mass transport time scale (z2/2D = 200 - 1250 days
for z = 10 - 25 cm). No direct evidence exists that the kinetics have the
106
-------
required time scale so that the results of these calculations and their com-
parison to observations are indirect evidence.
The observations for Chesapeake Bay sediments come from station 858-C
which is at a sounding depth of 30 m (Reeburgh 1967, 1969) for the winter of
1967. The dissolved nitrogen, argon, and methane concentrations are re-
ported for two cores, the total carbon dioxide for one core. The reported
chloride concentration and temperature are used to set the solubility of the
dissolved gases at a pressure of four atmospheres (Table 1), and the bound-
ary concentration of sulfate. The nitrogen and argon boundary conditions
are set at the average of the reported 0-15 cm values in order to mitigate
the time variable effects of the seasonal variation of the overlying water
concentrations (Martens and Berner 1977). The depth of penetration for sea-
sonal time variable effects in on the order of /tfr/To = 5.5 cm (uj = 2ir/365
radians/day) so that the effect should not extend below 10 or 15 cm. Bio-
turbation effects are probably limited to this depth as well (Martens and
Berner 1977).
The purpose of this calculation is to investigate the effects of gas
phase transport. Consider the simpliest assumption: that the gas phase is
stationary relative to the solid phase and that it is not diffusing. Since
in these examples the location of gas phase formation is below 30 cm biotur-
bation need not be considered in the mixing of the gas phase. The results
are shown in Figure (6). The total C02 is higher than station 856 which
implies a larger decomposition rate. The remaining parameters are as be-
fore (Table 4). Increasing the electron to carbon stoichiometry increases
the concentration of methane and decreases the concentration of total C02
as more of the carbon in being reduced to methane. The availability of
more methane causes a larger gas phase which forms closer to the sediment
water interface. The dissolved methane profile flattens as the dissolved
gas equilibrates with its gas phase partial pressure. The calculated
quantity of gas is discussed below.
The computed dissolved nitrogen, argon, and their molar ratio, together
with the observations, are shown in Figure 7. The sharper decline of the
dissolved nitrogen relative to argon is due to its relative insolubility
(Reeburgh 1969). As gases equilibrate with the gas phase, nitrogen prefer-
entially migrates to that phase. An electron to carbon stoichiometry of
0=3 appears to be consistent with these data. The gas phase is computed
to be 85 - 88% Cfy which seems quite reasonable (Martens 1976).
The most interesting and initially unexpected feature of these profiles
is that they are linearly decreasing even within the zone of sulfate reduc-
tion where no gas phase is computed to exist. This is a direct consequence
of the mass balance requirements and the hypothesized gas phase transport.
The stationary gas phase is advecting at the sedimentation velocity which
causes a downward flux of nitrogen and argon. This downward flux must be
supplied from the sediment-water interface and in the presence of substan-
tail diffusion of the dissolved species a linear gradient is required.
Since relatively more nitrogen is advecting with the gas phase its gradient
is steeper than that for argon. The nitrogen to argon ratio reflects this
effect .
107
-------
CHESAPEAKE BAY STATION 858-C
METHANE (aq)
10 30 50 70
DEPTH, cm
TOTAL CO2 (aq)
800
10 30 50 70
DEPTH, cm
GAS PHASE
90
0.20
8 0.00
30 50 70
DEPTH, cm
90
•^ 10 >>
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108
-------
10
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z 6
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QC H
1.0
0.8
0.6
0.4
0.2
0.0
CHESAPEAKE BAY STATION 858-C
NITROGEN (aq)
0 = 2.3
0=2.7
0 = 3.0
10 30 50 70 90
DEPTH, cm
ARGON (aq)
O3 O
"•^nsm.—
0=2.7 0=2.3
0=3.0
10 30 50 70 90
DEPTH (cm)
NITROGEN(aq)/ARGON(aq)
10 30 50 70 90
DEPTH, cm
Figure 7 Chesapeake Bay Station 858C: Observed (symbols) and computed
(lines) vertical distributions of: (a) Dissolved nitrogen gas, mg
nitrogen/1; (b) Dissolved argon, mg argon/1; (c) Their mole ratio.
The conditions are as in Figure 6.
109
-------
The quantity of total gas formed is calculated to range from 10 - 70 mM
or 5 - 30% of the total volume at 4 atm. Although gas phase volume was not
measured for these sediments, observations by Schubel (1974) suggest that it
is on the order of 0.1 to 1.0% of the total volume when it is observed
(Table 2). A possibility is that the gas phase migrates to the sediment-
water interface via "bubble tubes" as observed by Martens (1977) without any
appreciable interaction with the pore waters. This would correspond to a
loss of the gas phase that is essentially independent of the transport me-
chanisms considered in these calculations. The sole effect of the bubble
tube transport mechanism would be to decrease the size of the gas phase but
not the mole fractions since the removal would be proportional to the gas
phase composition. Therefore the computed profiles of the dissolved species
would remain unchanged.
A second possibility is that the gas phase has an upward velocity rela-
tive to the solid phases and that this velocity is low enough so that the
aqueous gas phase equilibria is maintained everywhere. Figure 8 illustrates
the computed results for upward velocities of 0.5 and 5.0 cm/yr. As ex-
pected (Barnes et_ a]_. 1975) there is a peak of nitrogen, argon, and their
ratio at the location of the gas phase dissolution. As a consequence there
is an increasing gradient from the sediment-water interface to the location
of gas phase formation. The mass fluxes of nitrogen and argon associated
with this gradient are upward; that is, nitrogen and argon are being sup-
plied to the overlying water. Their source, in this calculation, is the
lower boundary of the sediment, which is set at the computed concentrations
of the previous stationary gas phase calculations. It is necessary to
specify the boundary concentrations at the lower boundary of the sediment
only if there is transport from this boundary into the sediment segments
within the calculation. For an upward gas phase velocity this situation
occurs, but for all previous calculations this boundary is irrelevant. It
is evident that the situation depicted in Figure 8 cannot persist inde-
finitely since eventually the nitrogen and argon at the bottom boundary of
the sediment will be depleted. As a consequence, although these peaks in
nitrogen, argon, and their ratio are consistent with mass balance, they must
be a temporary effect.
Eventually, for a constant upward gas phase velocity, the bottom bound-
ary concentrations will be zero and the computed profiles for this situa-
tion are shown in Figure (9) for total C02, methane and the gas phase con-
centration; and in Figure (10) for nitrogen, argon, and their ratio. The
upward gas phase velocity is chosen to be 5 cm/yr, which results in gas
phase volumes within the reported range (Schubel 1974). The gas phase con-
centration is highest at the point of formation since the upward gas phase
velocity is transporting gas that has been generated by the methane forma-
tion to this location. The transport of nitrogen and argon below the depth
of gas phase formation is downward via diffusion of the dissolved gases and
upward via the gas phase velocity. The net result is essentially a zero
gradient of concentration above the depth of gas phase formation.
The consequences of this transport regime is that no flux of gas is com-
puted at the sediment-water interface. If such fluxes are known to occur,
this would indicate that the results of the stationary gas phase with rapid
110
-------
CHESAPEAKE BAY STATION 858-C
NITROGEN (aq)
10
O)
Z 6
LLJ
(D
O 4
CE
-5.0 cm/yr
—0.5 cm/yr
1.0
_ 0.8
D)
E 0.6
| 0.4
or
** 0.2
0.0
10 30 50 70 90
DEPTH, cm
ARGON (aq)
—5.0 cm/yr
—0.5 cm/yr
0.1 cm/yr
I
I
I
10 30 50 70 90
DEPTH, cm
NITROGEN(aq)/ARGON(aq)
45.0
—5.0 cm/yr
—0.5 cm/yr
LU
20.0
30 50 70 90
DEPTH, cm
"O O) -r- CD
S- > -P 0)
ex o
to
cu
ens- -(-> O)
CD
CL)
O CXI
111
-------
CHESAPEAKE BAY STATION 858-C
TOTAL CO2 0) r—
C .c^ re
o +J re -P .
•r- O I
•P C 4->
C TD 10
O OJ 0)
O (/> C
3 -1-
-o
C
re
r- O
o
« 0)
a>
C
re re u_
JC
-M U1 C
O) 3 «r-
.C -o E Q. a)
-P C >>T3 to
•r- re to Q> to o
3 v— > re.
>> c -o
4-?0
O «/> S-
O 3 -r- O
0) 4-
«/) d) O
re 10
^ re c:
O..C O
8.
to O -P
-O r- T3
x—» to O
xi re Q.
(U
«/> •*->
re 10 3
o> re .a
cn-r-
-o s-
s- o -i->
re c 10
.
3 -M
c o "re
re Q. u
to «r-
«4- C •!->
o re s-
S- (U
•«->-«->>
u
(U -»->
c o
o E
.a T-
s- i—
re i—
o •!-
s-
o
o
10
OJ
to
re
E o
CD
E « O)
C (U
« O S-
OJ M- J=
•o +J -p
•r- re
x i- a>
O 4-> -C
•r- C I—
-o a»
u
c c
o o
_a u
re (u
o 10
re
t/i ^
3 Q.
O
-------
CHESAPEAKE BAY STATION 858-C
NITROGEN (aq)
10
8
UJ
CD
2 2
0
1.0
_ 0.8
E. 0.6
O
CJ 0.4
-------
bubbling would be more realistic. However without seasonal measurements of
the gas flux as well as complete interstitial and gas phase measurements, it
is not possible to decide if upward velocities with gas phase dissolution
does or does not occur at certain periods. The observations of Barnes ei_
al. (1975) suggest that it can occur. But the interstitial water measure-
ments themselves cannot be used to distinguish between the regimes of gas
phase transport discussed above.
CONCLUSIONS
The mass balance equations for species dependent transport and fast re-
actants at equilibrium can be solved using the methods presented above. It
appears to be a reasonable approximation to compute the distribution of the
species with components C,H,0, and S using the requirements of chemical
equilibrium, but not the nitrogen species. The gas phase - aqueous phase
reactions at equilibrium correspond to the case of slow gas phase motion.
For species dependent transport the computations reproduce the major
features of the observed data as well as the more detailed variations of
the carbon to nitrogen ration and the net alkalinity. The use of net alka-
linity as a variable that reflects only the effects of the solid phase -
aqueous phase reactions and species dependent transport simplifies the in-
terpretation of observed alkalinity variations. The initial decline appears
to be due to sulfide precipitation if it is occurring, and the subsequent
increase is due to C02 production. The net alkalinity change is due to the
response of the sediment solid phase - aqueous phase buffering reactions.
The computational framework provides the means for examining the relative
impact of the various mechanisms that affect net alkalinity and the other
species.
The distribution of the dissolved gases is affected by the transport of
the resulting gas phase. For the Chesapeake Bay station analyzed, a sta-
tionary gas phase is consistent with the dissolved gas observations but not
the gas phase volume, which is larger than that observed at other stations.
It is possible that the excess gas is vented sporadically via bubble tubes.
Alternately a slow upward motion of the gas phase is consistent with the
dissolved gas data and the probable gas phase volume but would not account
for observations of gas escaping from the sediment-water interface. It is
not possible to distinguish between these gas phase transport regimes with-
out simultaneous measurements of dissolved gas concentrations, gas flux and
sediment gas volume. For a complete analysis, seasonal data and a time
variable computation are necessary.
A completely comprehensive analysis would include the solid phase -
aqueous phase reactions explicitly; in particular, the reductions of iron
and manganese and the relevant sediment buffering reactions. It is clear
from this analysis that a very comprehensive data set is required that in-
cludes not only the complete interstitial water chemistry but also a de-
tailed analysis of the sediment solid phases. Both the organic material and
the inorganic phases must be characterized. The sediment buffering reac-
tions are central to a detailed understanding of the net alkalinity and
114
-------
their nature, e.g., precipitation or dissolution of calcium carbonate, clay
cation exchanges, authigenie mineral formations, etc., would have to be con-
sidered. An analysis within the framework presented above would then
clarify their relative importance.
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Atkinson, L.P. and F.A. Richards. 1967. The occurrence and distribution
of methane in the marine environment. Deep-Sea Res., 14. pp. 673-684.
Barnes, R.O., K.K. Bertine, and E.D. Goldberg. 1975. N2: Ar nitrification
and denitrification in Southern California borderland basin sediments.
Limnol. and Oceanogr. 20. pp. 962-970.
Ben-Yaakov, S. 1972. Diffusion of sea water ions. I. Diffusion of sea
water into a dilute solution. Geochim. et. Cosmochim. Acta. 36 p.
1395- 1406.
Ben-Yaakov, S. 1973. Buffering of pore water of recent anoxic marine
sediment. Limnol. and Oceanogr. 18(1). pp. 86-94.
Berner, R.A. 1970. Sedimentary pyrite formation. Am. J. Sci. 268. pp. 1-
23.
Berner, R.A. 1974. Kinetic models for the early diagenesis of nitrogen,
sulfur, phosphorus and silicon in anoxic marine sediments. In The Sea,
Vol. 5, E.D. Goldberg (Ed.). J. Wiley and Sons, N.Y. pp. 4?7-450.
Berner, R.A. 1977. Stoichiometric models for nutrient regeneration in
anoxic sediments. Limnol. and Oceanogr. 22(5). pp. 781-786.
Berner, R.A., M.R. Scott and C. Thomlison. 1970. Carbonate alkalinity in
the pore water of anoxic marine sediments. Limnol. and Oceanogr. 15.
pp. 544-549.
Biggs, R.B. 1963. Deposition and early diagenesis of modern Chesapeake Bay
muds. Ph.D. Thesis. Lehigh University, Pa.
Biggs, R.B. 1967. The sediments of Chesapeake Bay. J.n Estuaries. G.H.
Lauff (Ed.). Publ. No. 83. Am. Assoc. Adv. Sci., Washington, D.C.
pp. 239-260.
Bordovskiy, O.K. 1965. Accumulation and transformation of organic sub-
stances in marine sediments. Mar. Geol. 3. pp. 33-82.
Bray, J. 1973. The behavior of phosphate in the interstitial waters of
Chesapeake Bay sediments. Ph.D. Thesis. Johns Hopkins University,
Baltimore, MD. pp. 1-149.
115
-------
Bricker, O.P., G. Matisoff, and G.R. Holdren, Jr. 1977. Interstitial
water chemistry of Chesapeake Bay sediments. Dept. of Natural Re-
sources. Maryland Geological Survey. Basic Data Report No. 9. pp. 1-
67.
Carpenter, J.H. 1957. A study of some major cations in natural waters.
Chesapeake Bay Institute of the Johns Hopkins Univ., Tech. Report XV,
80 p. Quoted by Biggs (1963).
Chapman, T.W. 1967. The transport properties of concentrated electronic
solutions. Ph.D. Thesis. Univ. of Calif., Berkeley (UCRL-17768).
DiToro, D.M. 1976. Combining chemical equilibrium and phytoplankton
models - a general methodology. _In_ Modeling Biochemical Processes in
Aquatic Ecosystems. R. Canale (Ed.). Ann Arbor Science Press, Ann
Arbor, Mich. pp. 233-256.
Gardner, L.R. 1973. Chemical models for sulfate reduction in closed
anaerobic marine environments. Geochim. et. Cosmochim. Acta 37.
Garrells, R.M. and C.L. Christ. 1965. Solutions, minerals and equilibrium.
Harper, N.Y. 450 p.
Goldberg, E.D. and M. Koide. 1963. Rates of sediment accumulation in the
Indian Ocean. J_n Earth Science and Meteoritics. J. Geiss and E.D.
Goldberg (Eds.). North-Holland Pub. Co., Amsterdam, pp. 90-102.
Harvey, H.W. 1966. The chemistry and fertility of sea waters (2nd edi-
tion), Cambridge Univ. Press, London, p. 168.
Himmelbau, D. 1964. Diffusion of dissolved gases. Chem. Rev.
Holdren, G.R., Jr., O.P- Bricker, III, and G. Matisoff. 1975. A model for
the control of dissolved manganese in the interstitial waters of Chesa-
peake Bay. ]j^ Marine Chemistry in the Coastal Environment. T.M. Church
(Ed.). ACS Sym. Ser. 18, Am. Chem. Soc., Washington, D.C. pp. 364-381.
Imoboden, D.M. 1975. Interstitial transport of solutes in non-steady state
accumulations and compacting sediments. Earth . Planet . Sci. Letters
27. pp. 221-228, 1975.
Kemp, A.L.W. 1973. Preliminary information on the nature of organic matter
in the surface sediments of Lakes Huron, Erie and Ontario. Proc. Symp.
on Hydrogeochemistry and Biogerchemistry. E. Ingerson (Ed.). The
Clarke Co., Washington, D.C. Vol. II., pp. 40-48.
Kramer, J.R. 1964. Theoretical model of the chemical composition of fresh
water with application to the Great Lakes. Pub. No. 11, Great Lakes
Research Division, Univ. of Mich. p. 147.
Lerman, A. and H. Taniguchi. 1972. Strontium 90 - diffusional transport in
sediments of the Great Lakes. J. Geophic. Res. 77(3). p. 474.
116
-------
Li Y-H, Gregroy, S. 1974. Diffusion of ions in sea water and deep-sea
sediments. Geochim. et Cosmochim. Acta 38. pp. 703-714.
Martens, C.S. 1976. Control of methane sediment-water bubble transport by
macroinfaunal irrigation in Cape Lookout Bight, North Carolina. Science
192. pp. 998-1000.
Martens, C.S. and R.A. Berner. 1977. Interstitial water chemistry of
anoxic Long Island Sound sediments. I. Dissolved gases. Limnol. and
Oceanogr. 22(1). pp. 10-25.
Philp, R.P. and M. Calvin. 1976. Kerogen structures in recently deposited
algal mats at Laguna Mormona, Baja, California: A model system for the
determination of kerogen structures in ancient sediments. In Environ-
mental Biogeochemistry, vol. 1. J.O. Nriagu (Ed.). Ann ArlxJr Science,
Ann Arbor, Mich. pp. 131-148.
Powers, M.C. 1954. Clay diagenesis in the Chesapeake Bay area: Clays and
Clay minerals. Natl. Conf. 2nd Natl. Acad. Sci. Natl. Res. Council Pub.
327. pp. 68-80. Quoted by Biggs, 1963.
Rashid, M.A. and L.H. King. 1970. Major oxygen-containing functional
groups present in humic and fulvic acids fractions isolated from con-
trasting marine environments. Geochim. et. Cosmochim. Acta. 34. pp.
193-201.
Reeburgh, W.S. 1967. Measurements of gases in sediments. Ph.D. Thesis.
Johns Hopkins Univ., Baltimore, MD.
Reeburgh, W.S. 1969. Observations of gases in Chesapeake Bay sediments.
Limnol. and Oceanogr. p. 368.
Reid, R.C., J.M. Prausnitz and T.K. Sherwood. 1977. The properties of
gases and liquids. McGraw Hill, N.Y. p. 576.
Robinson, R.A. and R.H. Stokes. 1959. Electrolyte solutions. Butter-
worths, London, p. 463.
Rosenfeld, J.K. and R.A. Berner. 1976. Ammonia adsorption in nearshore
anoxic sediments. Abstr. Geol. Soc. Am. Annu. Meeting (Denver), p.
1076. Quoted by Berner, 1977.
Schubel, J.R. 1974. Gases bubbles and the acoustically imperetrable of
turbid character of some estuarine sediments. J[n Natural Gases in
Marine Sediments. I.R. Kaplan (Ed.). Plenum Press, N.Y. pp. 275-298.
Schubel, J.R. and D.J. Hirschberg. 1977. Pb210 - determined sedimentation
rate and accumulation of metals in sediments at a station in Chesapeake
Bay. Ches. Sci. 18(4), pp. 379-382.
Sholkovitz, E. 1973. Interstitial water chemistry of the Santa Barbara
Basin sediments. Geochim. et. Cosmochim. Acta. Vol. 37. pp. 2043-2073.
117
-------
Stumm, W. 1966. Redox potential as an environmental paramter: Conceptual
significance and operational limitation. 3rd Intl. Conf. on Water
Pollut. Res. Munich, Germany, Paper No. 13. pp. 283-308.
Stumm, W. and J.J. Morgan. 1970. Aquatic chemistry. J. Wiley, N.Y.
Thorstenson, D.C. 1970. Equilibrium distribution of small organic mole-
cules in natural waters. Geochim. et. Cosmochim. Acta. 34. pp. 745-
770.
Trask, P.O. 1939. Organic content of recent marine sediments. Jjn Recent
Marine Sediments. P.O. Trask (Ed.). Dover Publications, N.Y., 1968.
pp. 428-453.
Wagman, D.O., W.H. Evans, V.B. Parker, I. Halow, Bailey, and R.H. Schumann.
1968. Selected values of chemical thermodynamic properties. NBS Tech.
Note 270-3. National Bureau of Standards, Washington, D.C.
Weiller, R.R. 1972. The composition of the interstitial water of sediments
in the Central Basin of Lake Erie. Unpublished report. Canada Centre
for Inland Waters, Burlington, Ontario, Canada.
Weiss, R.F. 1970. The solubility of nitrogen, oxygen and argon in water
and seawater. Deep Sea Research 17. pp. 721-735.
Yamamoto, S., J.B. Alcauskas and T.E. Crozier. 1976. Solubility of methane
in distilled water and seawater. J. Chem. Engr. Data 21(1). pp. 78-80.
118
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APPENDIX A
FINITE DIFFERENCE EQUATIONS AND SOLUTION METHOD
Mass Transport Equation:
d ' -13 w. a., A. -
dA.
1-1
Ei aik
<=1,...,NC (AI)
where: A.J = i fast reactant species concentration, 1=1,...,Nfs
Si = i species source due to the slow reactants
ai|< = stoichiometric coefficient of kul component, B|<, in the i
species
wi = itn species sedimentation velocity
Ei = i species diffusion coefficient
z = depth, positive downward.
Finite Difference Equation: For modes located at z1, z2,...,zM+2 with
1
spacing hn = zn+l - zn:
i
H
1=1
n+1 .n+1 , .n n n-1 An-l I n
aik ci Ai + aik di Ai + aik ei Ai - sk
*• ' k
n
= 0
= 1,
= 2,
(A2)
u n+1
where: c.
Ei
n-
- Bn 2 WV
'i i i
119
-------
Nfs
" = I a, Sf (hn-] + hn)/2
k .=1 IK 1
, the finite difference weights for the velocity
"+2
, E2, the velocity and diffusion evaluated at the midpoint
1 ] between zn and zn+1.
Perturbation Equation: For estimates Bj, j=l,---,Nc; n=2,---,N+l, at the N
interior nodes and the boundary conditions at n=l
and n=N+2, evaluate equation (A2) yielding the
errors el"!. To compute correction 6B1?, to the esti-
n ^
mates Blj, solve:
J
NC [ Nfs Nfs
-------
Nf s ( = o k y j
z aik J?j
i=l 1K 1J I = 1 k = j
solve equation (A3) for B1? = - 6B1? as the initial estimate.
0 J
Newton-Raphson Iteration: Using B^U) as the £th approximation, and an
J
chemical equilibria calculation to obtain An im-
plied by the B"l(£), the residual errors are
evaluated using equation (A2). Equation (A3) is
solved, using the appropriate Jacobians, to ob-
tain SB1], the corrections. These are used to
\J
produce the (£+l)tn approximation:
n=2,...,N+l (A6)
where 0 < p < 1 is the relaxation parameter which is chosen small enough to
insure convergence and non-negative Bn(£) for the appropriate components.
Convergence achieved when: ^
|5B"U)/B"U)| <1Q-4 (A7)
J J
for approximately 0.01% error.
121
-------
SECTION 7
SIMULATION OF THE DISTRIBUTION OF POLLUTED WATER IN RESERVOIRS FROM
CONCENTRATED EMISSIONS
A.V- Karaushev and V.V. Romanovskiy1
Perceptible currents may be absent during certain periods of time in
areas of lakes and large reservoirs where waste waters have been emitted,
thus allowing pollutants to accumulate in the region of the emissions. As a
result of turbulence, the polluted water is moved by the surrounding waters.
Movement of patches (clouds) of the pollutant occurs both as a result of
emissions of new portions of waste waters and due to turbulent diffusion.
The process of diffusion of pollutants can be simulated on the basis of
an equation for turbulent diffusion in cylindrical coordinates (Karaushev,
1969). In essence, this equation is written for the case of a concentrated
emission, but it can also be used for dispersive emissions of relatively
small length scales as a first approximation, especially in the case of an
emission into a large, deep reservoir. It is arbitrarily assumed that
emission of the entire discharge of waste waters takes place through one
central opening located in the center of the coordinates. The equation de-
scribes the process of diffusion of a dissolved substance in the plane of
the surface of the reservoir; the concentration is assumed to be invariant
with respect to angular position.
The method of calculating the diffusion in reservoirs, developed in the
State Hydrological Institute (SHI) based on an equation for diffusion in
cylindrical coordinates, is reported as a first approach (Karaushev, 1969),
and as a more refined approach (Karaushev, 1979). The case of an emission
of a non-conservative substance into a reservoir, whose decomposition takes
place according to a first order reaction, is considered in the new, refined
approach.
The differential equation for turbulent diffusion in cylindrical coordi-
nates for a non-conservative substance is written as follows:
r§ £ * V= - ti . «>
''State Hydrological Institute, Vtoraya liniya, dom 23, 199053 Leningrad,
USSR.
122
-------
Here, as previously, c is the concentration of the pollutant in the water
(kg/m3); t is the time (s); D is the turbulent diffusion coefficient
(m^/s); r denotes the distance from the center (from the source of pollu-
tion); alternatively, this is the radius (m) of a circle or sector of pollu-
tion limited by an arc coinciding with the isopleth of concentration c; k^
is a coefficient (1/s) which accounts for the non-conservation of the sub-
stance; it is assumed that on decomposition of the substance kn < 0, and
when the amount of the substance increases due to internal processes in the
reservoir, k^ > 0. For conserved substance, kn=0, and the third term in
the left part of the equation is thus eliminated.
Parameter 3 is expressed by the equality:
R
3 = D -
cm
(2)
where Rcm is the discharge of waste waters (m3/s); H is the mean depth
of the reservoir (m) in the area of the discharge and diffusion of the waste
waters; is the angle of distribution of the waste waters from the focal
source expressed in radians: = TT for emission at the shore; $ = ZTT for
emission at a significant distance from the shore.
The method of finite differences is one of the simplest approaches for
applying equation (1) for practical calculations. This method is examined
below.
In calculating turbulent diffusion, radius r is transformed into calcu-
lated elements (rings or semi-rings) Ar. The ordinal number of element Ar,
calculated from the center of the coordinates, is designated by n. The
values of Ar, n and r are related by the correlation:
1
) Ar.
(3)
Adjacent to the center, the element (where r = 0) has the number n = 1.
Figure 1 shows the calculated grid in sector of the pollutant diffu-
sion zone. The values for the concentration of the pollutant in each sec-
tion of the grid at a given instant in time t = t-j and at the following
instant t = t-j+i = t-j + At are indicated in the figure. For a given
instant in time, we have the following values of the concentrations:
Cj n_-| (in section n-1), C-j>n+i (in section n+1). For the following
(calculated) interval of time in calculated section n, we have the
concentration C-j+i>n (see Figure 1).
The calculated equation is written as:
+MnCi,n+l '
(4)
123
-------
Figure 1. Diagram for calculating the diffusion of pollutants in
an aquatic ecosystem.
124
-------
Coefficients n, vn and Mn are first calculated according to the
formulas shown below. The first one is a constant, and the other two vary
with the length of radius r with respect to the increase in the section num-
ber n.
n = i . 2D ^ + kH At, (5)
Ar
vn ' 5- (6>
Mn '
Equation (1) describes a non-stationary process of diffusion, developing
in time t. The dilution is calculated according to formula (4) with re-
spect to the boundary and initial conditions. The absence of pollution
within the limits of the entire calculated area, or some constant for the
range of the concentration of the pollutant, e.g., corresponding to the
natural (basal) concentration, is used as the initial condition.
It is convenient to express the concentration in excess over the natural
concentration level in calculating the pollution. The boundary condition is
written for the first section of the calculated grid (i.e., for the first
element adjacent to the center) in the form of the following material
balance equation:
R 2 k
cm ir r \ - Ar 8C rW^c\ A». H ^ r
(Ccm - Cr) - T §1 ' ^ar'r Ar ' T Ar Cl
where Cr and (^)r are respectively the values of the concentration and the
derivative at distance r from the center of the coordinates; C] is the
mean concentration in the first calculated section.
The concentration in the first section for each interval of time At
calculated, including the initial one, is calculated according to the fol-
lowing formula derived from Equation (8):
CH1,1 =aCcm + bCi,l+dcl,2> (9)
where Cj \ and C-j 2 are respectively the mean values of the concentra-
tion in the first'and second sections in the interval of time preceding the
calculated interval.
The coefficients contained in formula (9) are calculated as follows:
125
-------
At
b = l . - 2D + kH At, (11)
r Ar
p
At cm At
. 9n
d = 2D — * - TIT- — ?•
2 *" 2
AT
Note that as the material decomposes, kn becomes negative; for this
reason, the term containing RH in formulas (5) and (11) is also negative.
When k^ is positive, this term is also positive.
The calculation begins with the fact that the area of possible diffusion
of pollutants from the source, which is set at the center of this area, is
estimated based on general considerations. Depth H is based on the data
from measurements, and the mean coefficient for the turbulent diffusion D
for the entire area is calculated. D is calculated according to a formula
which best fits the conditions in the reservoir considered. The arguments
contained in the formulas for diffusion coefficient D are assumed to be
average for the entire area of diffusion of pollutants. The size of the
calculated segment of radius Ar is indicated in consideration of the condi-
tion:
H±Ar=mR3,ex» <13>
where rn = 20 + 30, and R3>ex is tne expected value of the radius of the
area of diffusion of polluted water in the reservoir.
The method of calculation examined here has limited applications; it can
only be used for those cases where D _> - 3, which is equivalent to the in-
equality:
Rcml2
-------
these values. Correct calculation of the coefficients is verified according
to the conditions:
-kHAt =
(16)
a + b + d -
kR At
(17)
All coefficients should be calculated with high precision so that the
deviations from unity in control formulas (16) and (17) do not exceed 0.001.
Condition (16) should be fulfilled for each element n.
The sequence of calculating the field of the concentration is as fol-
lows: the concentration in the 1st section is first calculated according to
formula (9), then the concentration in the 2nd, 3rd, etc. sections is cal-
culated according to formula (4) up to the end of the area of diffusion of
the pollutants obtained by calculation, i.e., up to the section where the
calculation will produce C = 0. The next calculated interval is then con-
sidered, and the calculations are done in the same order. The area of dif-
fusion of pollutants is increased by one step of Ar for each interval of
time. The calculations for the case of diffusion of a conservative sub-
stance in the reservoir are done by the same method, taking into considera-
tion the fact that in this case, kn = 0.
The calculation for a conservative substance must be verified based on
the substance balance. The mass of the substance, ^pol^' entering the re-
servoir through the emission considered for the entire period calculated,
i
t. = I At, is first calculated.
Calculation of the diffusion allows obtaining the values of the concen-
tration in each nth section. After calculating the volume of each section
and multiplying it by the corresponding concentration, the mass of the sub-
stance in the elements is found.
In adding the mass of the substance in all elements, the total mass in
the zone of diffusion of the substance is determined. It is obvious that
the total mass of the substance at time tj should be equal to Tp0l,j-
If the sums do not coincide and errors are found, then the values of the
concentration obtained are correspondingly equal to the value of the error.
After the appropriate correction is made, the calculation can be continued
in the normal order. Monitoring and corrections can be performed re-
peatedly.
Note that the composition of these values in calculations for non-con-
servative substances should produce an inequality which expresses the
natural processes of the decomposition of the substance due to the physio-
logical or biological processes which take place in the reservoir. In this
case, the calculation is not verified by the method examined above.
127
-------
The calculations are made for the entire period when there are no uni-
directional currents capable of carrying the polluted water beyond the
limits of the area affected by the emissions, i.e., for periods of calm, or
a period of stable ice on the surface water.
REFERENCES
Karaushev, A.V. 1969. River hydraulics. Gidrometeoizdat, Leningrad. 416
P-
Karaushev, A.V. 1979. A model and a calculated solution to the problem of
diffusion in reservoirs. Proc. of the VI All-Union Symposium on Con-
temporary Problems in Self-Purification of Reservoirs and Water Quality
Control. Tallin, p. 45-47.
128
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SECTION 8
RESULTS OF A JOINT USA/USSR HYDRODYNAMIC MODELING PROJECT
FOR LAKE BAIKAL
John F. Paull, Alexandr B. Gorstko? and Anton A. Matveyev3
INTRODUCTION
The United States of America (U.S.A.) and Union of Soviet Socialist Re-
public (U.S.S.R.) are confronted with many environmental problems which can
affect the health and welfare of their respective societies. Expanding pop-
ulations and industries, and increasing urbanization and farming have re-
sulted in alterations to the hydrosphere and to changes in loads of waste
materials which effect the quality of the environment. The mutual concern
for the environment provided the impetus for the U.S.A./U.S.S.R. Agreement
on Cooperation in the Field of Environmental Protection signed in 1972.
As part of the Agreement, Project 02.02-12, Protection and Management of
Water Quality in Lakes and Estuaries, was initiated. Although the two na-
tions share no common boundaries on any lake or estuary, they do share a
common concern for water quality preservation, and the need to understand
the physical, chemical, and biological processes that effect and determine
water quality. To share scientific knowledge on limnological processes, a
joint modeling project was initiated during the exchange visit by Soviet
representatives to the U.S.A. in 1976.
In June 1977, Dr. Tudor T. Davies from U.S. Environmental Protection
Agency (EPA), Environmental Research Laboratory, Gulf Breeze, Florida, and
Dr. John F. Paul and Mr. William L. Richardson from EPA, Large Lakes
Research Station, Grosse lie, Michigan, visited the Institute of Mechanics
and Applied Mathematics, Rostov State University, Rostov-on-Don, U.S.S.R.
They met with Drs. A.A. Zenin, A.A. Matveyev, A.B. Gorstko and F.A. Surkov.
During this visit, the details of the joint modeling project were arranged,
and a project report outline prepared. As a first step, it was agreed to
U.S. Environmental Protection Agency, Large Lakes Research Station, Grosse
lie, Michigan 48138, USA.
Institute of Mechanics and Applied Mathematics, Rostov State University,
192/2 Stachky Prospect, 344090 Rostov-on-Don, USSR.
3Hydrochemical Institute, 192/3 Stachky Prospect, 344090 Rostov-on-Don,
USSR.
129
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compare formulations and results of hydrodynamic and transport models devel-
oped by the two groups. The objective was to provide a basis for further
verification of lakewide and nearshore hydrodynamic and transport models.
This paper describes the application of a hydrodynamic model, previously
developed for and applied to areas of the Great Lakes, to Lake Baikal in the
U.S.S.R. The model results are compared to available field and remotely
sensed data.
BACKGROUND ON LAKE BAIKAL
The basin of Lake Baikal is located almost centrally in Asia, in a very
rugged mountain province in south Siberia, the Baikal Region. The charac-
teristic geomorphological features of the region include medium and high
mountain ranges extending over 1500 km in the southwest to northeast direc-
tion, and an alternation of ridges and trenches, the largest of which is
filled with waters of the lake.
Lake Baikal is the oldest and deepest intracontinental body of water in
the world. The formation of the Baikal trench began about 30 million years
ago. The watershed area of the lake is 0.54 million km2, and the area of
the lake itself is 31.5 thousand km2. The length of the lake is 636 km;
maximum width, 79 km; minimum width, 25 km; maximum depth, 1620 m; and
volume of the water mass, about 23 thousand km3. The topography of the lake
is shown in Figure 1. The trench of Lake Baikal is divided into three
basins, of which the middle one is the deepest. It is separated from the
southern basin by the Selenga shallows, a delta formed by the lake's largest
tributary, the Selenga River. The contribution of the Selenga amounts to
about 50% of the total runoff into the lake. Table 1 provides an annual
average water balance for the lake (Vikulina and Kashinova 1973).
TABLE 1. ANNUAL AVERAGE WATER BALANCE FOR LAKE BAIKAL DURING THE
PERIOD 1901-1970 (km3/year)
Inflow
Outflow
Precipitation 9.29
Condensation on lake
surface .82
River inflow 58.75
Groundwater inflow 2.30
Total 71.16
Runoff to Angara River 60.39
Evaporation 10.77
Total 71.16
Lake Baikal contains approximately 4/5 of the total surface water re-
serves of the U.S.S.R. However, the importance of the lake does not end
there. During the past approximately 1 million years, when this body of
water was formed in its present boundaries, special characteristics were
developed: low solute content, high transparency, low temperature, and
high saturation with dissolved oxygen.
130
-------
UPPER ANGARA
NIZHNEANGARSK.,i/i'M
(Depth contours expressed in meters.)
Figure 1. Lake Baikal
131
-------
Lake Baikal's ecosystem is distinct and closely balanced. In the course
of its evolution, its organisms have adapted to conditions varying little
with time, and have reacted very sensitively to changes in these conditions.
Indicative of this sensitivity is the fact that organisms of the open deep-
water parts of the lake do not dwell in the shallow regions near the Selenga
delta, which are subjected to the action of the river runoff.
HYDRODYNAMIC MODEL
This model is composed of two components: the basic hydrodynamic cal-
culation and the transport of dissolved and/or suspended material. Each
component will be discussed separately.
Summary of Hydrodynamic Component
The equations for the hydrodynamic component are derived from the time-
dependent, three-dimensional equations for conservation of mass, momentum
and energy. The principal assumptions used are:
(a) The pressure is assumed to vary hydrostatically.
(b) The rigid-lid approximation is made, i.e., the vertical velo-
city at the undisturbed water surface is assumed to be zero.
This approximation is used to eliminate surface gravity
waves and their associated small time scales, greatly in-
creasing the maximum allowable time step in the numerical
computations.
(c) Eddy coefficients are used to account for the turbulent diffu-
sion effects. The horizontal coefficients are assumed to be
constant, but the vertical coefficients are assumed to be some
function of the local dependent variables.
The resulting equations are:
3u 9v 3w n
9x + 9y + 35 = °» 0)
+ + f _ 1 3P 3 . 8U
V " P 3x 3x IAH 3l
+ _§ (A ^) + — (A —)
+ ^ + 3V2+ 9vw _ f _ 1 3P 3 3v,
3x 3y + 3z Tou * - ~ ~y + 9x (AH 3x'
o
132
-------
$ (AH I' + 5 <\ 1
3T . 3uT 3vT 3wT . 3 ,. 3T, . 3 ,R 3T>
3t 3X 3y ~3Z " 3* l H jsx' sy lbH 3y'
* Tz z/h(x,y).
The boundary conditions used with the equations are the following: the
bottom and shore are taken as no-slip, impermeable, no-flux surfaces; in-
flows or outflows along with appropriate fluxes of heat are specified at
rivers; at the water surface, a wind-dependent stress and a specified heat
flux are specified. The initial conditions used are either some simple
133
-------
specification for the variables (e.g., zero) or specification of all vari-
ables from some previous calculation.
The equations are put into finite-difference form in both space and
time. The spatial discretizing is accomplished by integration of the dif-
ferential equations about appropriate grid cells. The finite-difference
equations are explicit in time, except with respect to the Coriolis terms
and the vertical diffusion terms which are written implicitly. This is done
to eliminate the small vertical diffusion time step restriction and the in-
stability associated with explicit form of the Coriolis terms.
The equations as written can not be solved directly if the rigid-lid
condition is to be satisfied. To develop a solution scheme, an additional
equation is derived directly from the other equations and the rigid-lid con-
dition. This equation is derived by vertically summing the two finite-dif-
ference equations for the horizontal velocities, then taking the finite-dif-
ference analog of the divergence. The rigid-lid condition is used along
with the vertically summed continuity and hydrostatic pressure equations.
The result is a Poisson-type finite-difference equation in the surface pres-
sure. The surface pressure is the "integration constant" resulting from the
vertical summation of the hydrostatic pressure equation. The surface pres-
sure is a function of the horizontal coordinates. This procedure for de-
riving the Poisson-type equation for the surface pressure, a modification of
the SMAC procedure by Amsden and Harlow (1970), is different than previous
models which first derived the pressure equation from the differential equa-
tion, then discretized it (Paul and Lick 1974, Paul 1976). The distinction
between these two procedures is that with the use of the latter, the finite-
difference equation that results for the pressure will not necessarily be
directly deriveable from the finite-difference forms of the horizontal
momentum equations. This is strictly a numerical error associated with ap-
proximating the differential equations; however, this error reflects in the
inability of the numerical solution to satisfy the rigid-lid condition to an
acceptable degree. Even if direct methods are used to solve the finite-dif-
ference pressure equation or if an iterative procedure is used with strin-
gent convergence criteria, errors are manifested in the vertical velocity at
the rigid lid. These errors are not always small, especially in problems
where there is significant differences in depths between the shallow and
deep areas of the water body. These errors might not appear to be of much
significance in some calculations, but they do create appreciable errors
when the resultant velocities are used to calculate dispersion of substances
in the body of water. Also, if advantage is taken of the time implicit na-
ture of the vertical diffusion terms (i.e., time steps are large with re-
spect to the explicit vertical diffusion time limit), these errors become
large and ultimately make the solutions meaningless. Complete details of
this numerical procedure appear in the report by Paul and Lick (1980).
Summary of Dispersion Component
The transport and dispersion of material in the turbulent flow will be
described in a manner similar to that used for the transport and turbulent
dispersion of heat and momentum. Refer to Sheng (1975) for a summary of
this procedure. The concentrations of the material to be dispersed are
134
-------
treated as continuous on the length scales considered. The concentrations
are sufficiently small so that they do not significantly alter the density
of the water (the momentum equations can be solved independently), and they
function as completely conservative substances in the water column convected
with the local fluid velocities. The only exception to the latter condition
will be when gravitational settling is important for the material con-
sidered. In this situation, the vertical convection of the material is en-
hanced by a settling velocity. The basic equation used to predict the dis-
persion of the material is:
3C + 3(Cu) + 3(Cv) ^ 3(Cw) _ 3 fn 3C,
O.4" <% »/
01. OA
3n./"*
/ n 0
-------
TABLE 2. PARAMETERS FOR LAKE BAIKAL HYDRODYNAMIC MODEL
Grid spacings
Horizontal extents
15 km y-direction
7.8 km x-direction
600 km y-direction
150 km x-direction
Number of grid points
Minimum depth
Maximum depth
Coriolis parameter (53° N)
Horizontal eddy viscosity
Vertical eddy viscosity
Surface wind stress
Wind directions:
Case 1
Case 2
Case 3
Case 4
River flows:
Selenga
Barguzin
Upper Angara
Angara
Horizontal eddy diffusivity
Vertical eddy diffusivity
Particle settling velocity
where:
h = local depth
h0 = reference depth
41 y-direction
21 x-direction
8 z-direction
10 m
1620 m
1.16xlO-4/sec
107 cm2/sec
3.85 (1 + 258.7 -^ cm2/sec
no
1 dyne/cm2
Southwest
Northeast
Northwest
Southeast
9.64x1O8 cm3/sec
4.10xl08 cnvVsec
5.78x1O8 cm3/sec
19.52xl03 cm3/sec
106 cm2/sec
.385 (1 + 258.7 Jl) cm3/sec
ho
10 m/day
136
-------
FREQUENCY OF FALL WINDS (PERCENT):
FROM LEFT SIDE (NW) FROM RIGHT
SIDE (SE)
UP LAKE (SW) DOWN LAKE (NE) FREQUENCY
OF CALM
FREQUENCY OF SUMMER WINDS (PERCENT):
FROM LEFT SIDE (NW) FROM RIGHT
SIDE (SE)
UP LAKE (SW) DOWN LAKE (NE) FREQUENCY
OF CALM
Figure 2. Frequency of winds over Lake Baikal in the summer and autumn.
137
-------
time were not obtainable during the course of this study. Since an insuffi-
cient amount of data was available on the thermal regime for the entire
lake, the calculations were performed assuming this aspect to be negligible.
The wind direction for Case 1, from the southwest, was also used in a cal-
culation with the northern basin of the lake ice covered. This type of cal-
culation is of interest to assess the significance of the ice cover on the
central and southern basins. The northern basin is generally ice covered
through the middle of May, and occasionally until the beginning of June
(Anonymous 1969).
Detailed results of the hydrodynamic calculations are presented in Paul,
Richardson, Gorstko, and Matveyev (1979). Representative results will be
discussed here. All of the calculations were performed for 12 days of real
time, after which essentially steady-state conditions were obtained. Fi-
gures 3 and 4 show the surface velocities and vertically integrated velo-
cities for the southwest wind (Case 1) and the northwest wind (Case 3). For
all of the calculations, the results are typical of what might be expected
from the simple theory of motion which balances foriolis force, vertical
friction, and horizontal pressure gradients. The deep areas of the lake are
generally characterized by geostrophic motion, while the shallower areas are
markedly influenced by vertical friction. The magnitude and directions of
the surface currents indicate the different balances in the motion as one
goes from shallow to deep waters. The general circulation of the lake is
composed of the basic circulation in the three basins. Because the northern
and southern basins are relatively flat in their long dimension compared
with the central basin, the magnitude of the vertically integrated velo-
cities for the southwest wind are smaller in these basins compared to the
central basin. This is in agreement with the steady-state solutions ob-
tained for simple basins by Gedney (1971). He solved the equations which
balance Coriolis force, pressure gradients, and vertical friction. For con-
stant depth basins, the steady-state vertically integrated velocities are
everywhere equal to zero, while for parabolic shaped basins, the vertically
integrated velocities form two rotating gyres, with the magnitudes dependent
on the degree of the bottom slopes. Thus, the gyres in the calculations for
the vertically integrated velocities are functions of the local topography.
Differences are apparent for the different wind direction calculations, but
this is because the basins are elongated and the topography is highly vari-
able.
Data on surface currents in the Selenga region are shown in Figure 5.
Three cases are shown: prevailing southwesterly winds, prevailing north-
westerly winds, and steady-last ing northwesterly winds. The first two are
separated by only a day, and it is apparent that the currents on the second
day show the effects of the previous day's winds. This can be seen by com-
paring the second and third plot. The currents in the third plot compare
quite well with the surface current calculation for the northwesterly wind
(Figure 4). The first plot compares reasonably well with the calculation
for the southwesterly wind (Figure 3), but discrepancies do exist, appar-
ently a function of the transitory behavior of the currents. The second
plot appears to be some combination of the two calculations, as would be ex-
pected.
138
-------
N
o
SCALE:
1
100
KILOMETERS
200 15 CM/SEC
CO
UD
WIND
SURFACE VELOCITIES
Figure 3a. Hydrodynamic model calculation for Lake Baikal with southwest wind.
-------
N
WIND
0
SCALE:
1 \
100 200
KILOMETERS
HiliV^' |
VERTICALLY INTEGRATED VELOCITIES
Figure 3b. Hydrodynamic model calucation for Lake Baikal with southwest wind.
-------
N
v
WIND
SCALE:
0 100 200 15 CM/SEC
KILOMETERS
SURFACE VELOCITIES
Figure 4a. Hydrodynamic model calculation for Lake Baikal with northwest wind
-------
N
v
WIND
SCALE:
1
100
KILOMETERS
200
ro
VERTICALL Y INTEGRA TED VELOCITIES
Figure 4b, Hydrodynamic model calculation for Lake Baikal with northwest wind.
-------
CO
WIND
WIND
(1) Prevailing southwesterly wind (30 Aug 1972).
(2) Prevailing northwesterly wind (31 Aug 1972).
(3) Steady lasting northwesterly wind (8 Sep 1972).
Figure 5. Observed surface currents in the Selenga River Region of Lake Baikal
-------
The only data for lake-wide currents is a plot which appears in the
Baikal Atlas (Figure 6). This is depicted as representative of the typical
currents that exist in the lake. With reference to the wind frequency plot
for the lake (Figure 2), it can be seen that for the most -part, the winds
from the southwest are the most typical. A comparison of the data plot (Fi-
gure 6) with the vertically integrated velocities calculated for a southwest
wind (Figure 3) indicates a very good agreement. The observed gyres are re-
plicated quite well in the calculation. The data plot does not indicate
magnitude of the currents so no comparisons can be made on this aspect.
Using the steady-state currents that were calculated, the transport and
dispersion of material in the lake were calculated. A series of 8 calcula-
tions were performed: for each of the four main wind directions, for
material which is neutrally buoyant and for material which has a gravita-
tional settling velocity of 10 meters per day. This settling velocity cor-
responds with the mean Stokes settling velocity for the predominant particle
sizes observed in the suspended material in the Selenga River runoff. The
two calculations for each current pattern indicate the difference in distri-
butions that can be achieved when one considers dissolved material (neu-
trally buoyant) and suspended material (positive settling velocity). The
parameters used for the calculations are listed in Table 2. The calcula-
tions were performed for 28 days of real time for each circumstance. Mate-
rial was entered continuously during the calculation through the Selenga,
Barguzin and Upper Angara Rivers. The material concentration in each river
was set equal to 1.0. Since the transport equations are linear in the mate-
rial concentration, the actual concentration level is not important in the
calculation. The results for all the calculations are presented in Paul,
Richardson, Gorstko, and Matveyev (1979). Representative results will be
discussed here. Figure 7 indicates surface and bottom concentrations with a
southwest wind for both neutrally buoyant and suspended material. The ef-
fect of the settling velocity on the concentration distributions is impor-
tant. The main reason for this is that the currents over the water column
are, in general, going in different directions. If the material remains es-
sentially uniform over the water column (i.e., for the neutrally buoyant
material), then the material is primarily transported by the currents over
the upper portion of the water column. These are the currents which have
the larger magnitude. When gravitational settling is introduced, the
material tends to concentrate near the bottom, and thus, is transported
horizontally by the currents that are near the bottom. Since these cur-
rents are in generally in a different direction than the near surface
currents, the concentration distribution appears different.
The data that is available for comparison with the calculations are
shown in Figure 8. These data are taken from the Hydromet cruise in the
Selenga River Region of the lake on 28-29 May 1976. The plots are for P04
(a dissolved substance) and total suspended solids. The winds over the re-
gion were highly variable during the cruise, ranging from southwest to
southeast during the first part of the cruise, and to northeast during the
final part of the cruise. A reasonable comparison can be made with the cal-
culations for the southwest wind (Figure 7). The currents in the vicinity
of the Selenga delta are easterly near the surface and over the most of the
vertical column near the river area of the lake. As one goes away from the
144
-------
SLYUDYANKA
UST' BARGUZIN
Figure 6. Lake Baikal whole lake dominant currents.
145
-------
SCALE:
CONCENTRA TION
(PER VOLUME)
A 1.0000
B 0.1000
C 0.0100
D 0.0010
E 0.0001
100
KILOMETERS
200
SURFACE CONCENTRATIONS (NO SETTLING VELOCITY)
Figure 7a. Dispersion model calculation for Lake Baikal with southwest wind,
-------
SCALE:
CONCENTRATION
(PER VOLUME)
A 1.0000
B 0.1000
C 0.0100
D 0.0010
E 0.0001
0
100
KILOMETERS
BOTTOM CONCENTRA TIONS (NO SETTLING VELOCITY)
200
Figure 7b. Dispersion model calculation for Lake Baikal with southwest wind.
-------
CO
SCALE:
CONCENTRATION
(PER VOLUME)
A T.OOOO
B 0.1000
C 0.0100
D 0.0010
E 0.0001
0
100
KILOMETERS
200
SURFACE CONCENTRATIONS (SETTLING VELOCITY, 10m/day)
Figure 7c. Dispersion model calculation for Lake Baikal with southwest wind,
-------
SCALE:
CONCENTRATION
(PER VOLUME)
A 1.0000
B 0.1000
C 0.0100
D 0.0010
E 0.0001
0
100
KILOMETERS
200
BOTTOM CONCENTRATIONS (SETTLING VELOCITY, 10m/day)
Figure 7d. Dispersion model calculation for Lake Baikal with southwest wind.
-------
en
O
NEAR SURFACE CONTOURS OF
P0~3, mg/l
IMPLIED FLOW OF MATERIAL
Figure 8a. Sample results from Hydromet cruise in Selenga Shallows on 28-29 May 1976.
-------
NEAR SURFACE CONTOURS OF
SUSPENDED SOLIDS, mg/l
IMPLIED FLOW OF MATERIAL
Figure 8b. Sample results from Hydromet cruise in Selenga Shallows on 28-29 May 1976,
-------
river area, the subsurface currents are generally southwesterly. The sus-
pended solids settle out as they enter the lake and are transported by the
near-bottom currents. The dissolved material remains nearly uniform verti-
cally, and is transported out into the lake by the subsurface currents. The
data generally agree with the calculated distributions.
Additional data available for the Selenga region is the Landsat satel-
lite image shown in Figure 9. This image records observations of 9 July
1975. Unfortunately, no wind information was available for this date. The
image does indicate the same sort of suspended solids pattern as was ob-
served during the 28-29 May 1976 Hydromet cruise (Figure 8).
DISCUSSION AND RECOMMENDATION
The work summarized in this paper represents just a preliminary step in
a possible comprehensive joint modeling and field study by scientists from
our two countries. It is hoped that the work discussed here will continue
and that this continued work will enable scientists from both countries to
expand their knowledge of the physical process in lakes.
It is recommended that a comprehensive joint USA/USSR modeling and field
survey program be initiated for a lake such as Lake Baikal. This program
would be used to both verify the models and increase the understanding of
the processes at work in the lake. Modeling work already underway, along
with information available from previous field surveys, would be used to
plan an extensive one year field survey. The information that would be
available from satellites (for example, LANDSAT and NIMBUS) would be
utilized. The usefulness of the satellite imagery to characterize suspended
solids in lakes, in conjunction with field data and modeling results, has
been demonstrated by Sydor and Oman (1977); Sydor, Stortz, and Swain (1978);
and Paul, Mielnik, and Shute (1979). The satellite provides synoptic infor-
mation on a whole lake basin, while the survey ships provide information at
different depths in the lake and can provide ground-truth for the satellite
information. Such a program would cover a period of three years: the first
year for preparation, the second year for the field surveys, and the third
year for the analysis.
ACKNOWLEDGEMENTS
This work could not have been undertaken without the sponsorship of the
former coordinators for the Lakes and Estuaries Project, Drs. T.T. Davies
and A.A. Zenin.
Dr. Michael Sydor, University of Minnesota-Duluth, provided the Landsat
satellite image.
152
-------
Ei86-0e Eie?-8ei N8S3-38;
C N5«-22'Eie?-27 N N5«-2l/E 107-3" nSS 5 R SUN EL52 R2I38 l95-233<>-«- : -N-? 2i
iEiee-ee
a9JUL75 C N52-57/ElB6-«3 N
iEie?-
c'r •" ^ a? ?:• If
5 R SUN EL53 flZl36 l*>-233"-B- 1 -N-0-2L oS 2^ 68-83 Jg-!
Figure 9. LANDSAT satellite image of Lake Baikal
153
-------
REFERENCES
Amsden, A.A. and F.H. Harlow. 1970. The SMAC method: A numerical techni-
que for calculating incompressible fluid flows. Los Alamos Scientific
Laboratory, Report No. LA-4370, Los Alamos, New Mexico.
Anonymous. 1969. Atlas of Baikal. Govt. Dept. Geodosy and Cartography.
Irkutsk and Moscow.
Crowley, W.P. 1968. A global numerical ocean model: Part 1. J. Computa-
tional Physics, 3: 111-147.
Gedney, R.T. 1971. Numerical calculations of the wind-driven currents in
Lake Erie. NASA TM X-52985.
Paul, J.F. and W.J. Lick. 1974. A numerical model for thermal plumes and
river discharges. Proc. 17th Conf. Great Lakes Res., I.A.G.L.R., pp.
445-455.
Paul, J.F. 1976. Modeling the hydrodynamic effects of large man-made modi-
fications to lakes. Proc. of the EPA Conf. on Environmental Modeling
and Simulation (W.R. Ott, ed.), EPA-600/9-76-016, pp. 171-175.
Paul, J.F- and R.L. Patterson. 1977. Hydrodynamic simulation of movement
of larval fishes in western Lake Erie and their vulnerability to power
plant entrainment. Proceedings of the 1977 Winter Simulation Conference
(H.J. Highland, R.G. Sargent and J.W. Schmidt, ed.), WSC Executive Com-
mittee, pp. 305-316.
Paul, J.F. and W.J. Lick. 1980. Numerical model for three-dimensional
variable-density, rigid-lid hydrodynamic flows: Vol. 1, details of the
numerical model. Report in preparation.
Paul, J.F., R.A. Mielnik, and P.A. Shute. 1979. Use of LANDSAT imagery to
characterize the suspended solids in Lake Baikal. Project report for
Remote Sensing of Earth Resources, Eastern Michigan University.
Paul, J.F., W.L. Richardson, A.B. Gorstko, and A.A. Matveyev. 1979. Re-
sults of a joint U.S.A./U.S.S.R. hydrodynamic and transport modeling
project. EPA-600/3-79-015.
Semenov, A.E., ed. 1972. Hydrometeorological investigations of the
southern seas and Atlanta Ocean. Collected Works of the Laboratory of
the Southern Seas, Volume 11. Moscow.
Sheng, Y.-Y. P. 1975. The wind-driven currents and contaminant dispersion
in the near-shore of large lakes. Lake Erie International Jetport
Model Feasibility Investigation Report 17-5, Contract Report H-75-1,
U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss.
154
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Sydor, M. and G.J. Oman. 1977. Effects on Nemadji runoff on Lake Superior,
effects of river inputs on the Great Lakes. Dept. of Physics, Univ. of
Minn., Duluth.
Sydor, M., K.R. Stortz, and W.R. Swain. 1978. Identification of contami-
nants in Lake Superior through LANDSAT 1 data. J. Great Lakes Res.,
4(2): 142-148.
Vikulina, Z.A. and T.D. Kashinova. 1973. Water balance of Lake Baikal..
Trudy GGI, Issue 203, pg. 268. Gidrometeoizdat, Leningrad.
155
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SECTION 9
THE STRUCTURE OF HYDROCHEMICAL FIELDS AND SHORT-TERM PREDICTION
A.M. Nikanorov, B.M. Vladimirskiy, V.L. Pavelko, Ye.V. Melnikov
and K.L. Botsenyuk^
INTRODUCTION
The concept of the field presumes the existence of coordinated indices
located in some predictable manner in space, and characterized by a set of
parameters which jointly fix the observed structure.
"Thus, ... the system of a field refers to a system having
the following properties. It is a spatial unity, at least
with respect to certain of its properties. It is also a
unity based on its interactions in the sense that it can
be deformed only as a unit and that in relationship to
certain significant ... peculiarities, a change in one por-
tion of the field has a significant influence on changes
of other parts. It is an organized whole with definite
integral manifestations and should be studied as a whole,
not as the summary result of the combination of its parts
and their activities" (Huxley 1935, cited by Waddington
1948).
The concept of the field structure of phenomena is supplemented with its
specific content in actual experiments, including observations and measure-
ments. Based on these data, models of the phenomena are constructed which
then serve as the basis for development of corresponding theories by the
method of trail and error. The predictive capabilities of models created on
the basis of accepted theories are obvious. The accuracy of predictions
using these models increases as the corresponding theories are refined and
expanded. The desire, frequently the need, to predict phenomena which do
not have corresponding theoretical interpretation or, consequently, models
based on them, makes it necessary to use empirical models for predictive
purposes.
^ydrochemical Institute, 192/3 Stachky Prospect, 344090 Rostov-on-Don,
USSR.
156
-------
The construction of empirical models, reflecting information concerning
the actual, real aspects of phenomena in a form convenient for use, is also
inseparable from observations and experiments. Thus, analysis of the struc-
ture of hydrochemical fields and resulting empirical descriptions are in-
separably related both to the points of measurement selected, and to the
mathematical apparatus used. Naturally, the mathematical model incorporates
certain subjective and contradictory aspects, related to the selection of
the specific model, to the particular goal of modeling, and to the set of
models with which the same result can be achieved. It is this contradiction
which must be resolved as a result of introduction of the concept of the
field in the creation of empirical models. Thus, the concept of the field
in the analysis of hydrochemical information is used not as a causal princi-
pal, but rather as a method of analysis.
HYPOTHESES CONCERNING THE STRUCTURE OF HYDROCHEMICAL FIELDS
The results of factor and regression analysis presented in this section
are based on earlier published data (Zenin 1961). From these results evolve
a number of statements which are of basic significance in both modeling the
structure of hydrochemical fields, and in empirical prediction.
The data used represent the results of many years of measurement of a
number of hydrochemical indices in reservoirs along the Volga River and its
tributaries. In all, some 30,000 measurements were used in the analysis.
The results of factor analysis are presented in Tables 1-7. Analysis of the
results demonstrate several facts. Both for the reservoirs and for the
tributaries, the factor solutions can be characterized by virtually the same
set of factors. Thus, it may be assumed that the same mechanisms lie at the
basis of formation of the assigned set of indices. It should be noted that
the significance of individual factors (see the rows of "all factors" in
Tables 1-7) changes, with one exception. The significance of this change is
identical for all factors without exception. Obviously, the remaining fac-
tors reflect the specific characteristics of the hydrochemical situation in
the reservoirs studied. It is important for the modeling of the structure
of a hydrochemical field based on the vectors of mean and covariation
matrices that the factor mapping remain constant, or that the internal fac-
tors of the covariance matrices and the variability of weights of the fac-
tors, the eigenvalues of the covariation matrices, remain constant.
The initial material used for analysis of the capabilities for empirical
prediction was made up of series of hydrochemical indices from three hydrau-
lic projects over a period extending from 1951 through 1973. It should be
noted that the frequency of all data available was rather arbitrary.
Examples of the time series of indices used for analysis are presented in
Figure 1. The results of factor analysis for these data are presented in
Tables 8 and 9, and enable confirmation of the hypothesis developed earlier
in our analysis of data on the Volga reservoirs.
Regression models were constructed in which the computational procedure
used was step-by-step regression. A linear model for 6 and 13 indices
analyzed produced a high degree of agreement (see Figure 2). Furthermore,
157
-------
TABLE 1. FACTOR SOLUTION FOR THE RESULTS OF CHEMICAL ANALYSIS OF WATER
SAMPLES OF THE GORKY RESERVOIR IN 1954-1958
1
Level
ci-
sof'
HCO^ mg/liter
Na+ + K +
zi
cr
so2-
HCOg % eq.
Mg2+
Na+ + K+
Weight of f actor 1
2
-0.05
0.70
0.79
0.95
0.93
0.89
0.23
0.99
-0.07
-0.25
0.29
0.16
0.22
-0.26
2.24
Factor
3
0.01
0.68
0.16
-0.03
-0.06
0.02
-0.01
-0.03
-0.99
0.17
0.28
-0.06
0.02
0.04
1.26
Mapping Coefficient
4
0.00
0.02
0.15
-0.06
-0.25
-0.12
0.82
-0.02
-0.06
0.33
-0.27
-0.95
-0.31
0.86
1.64
5
0.12
-0.03
-0.07
-0.12
-0.08
-0.41
0.38
-0.10
-0.01
-0.12
0.02
0.10
-0.90
0.38
1.16
6
0.08
0.10
-0.53
0.24
0.18
0.10
-0.24
0.12
-0.08
-0.78
0.84
0.19
0.00
-0.17
1.36
weight of the factor is calculated as the root mean square of the co-
efficient by columns.
158
-------
TABLE 2. FACTOR SOLUTION FOR RESULTS OF CHEMICAL ANALYSIS OF WATER
SAMPLES OF TRIBUTARIES OF THE GORKY RESERVOIR, 1954-1958
1
Level
cr
so|-
HCO~ nig/liter
Mg2+
Na+ + K+
zi
cr
so|-
HC03 % eq.
Na+ + K+
2
-0.00
0.24
0.19
0.17
0.04
0.10
0.60
0.19
0.04
-0.06
0.04
-0.75
-0.15
0.98
Factor
3
0.00
0.15
-0.37
0.26
0.20
0.13
0.01
0.17
-0.21
-0.99
0.93
-0.08
0.06
0.05
Mapping Coefficient
4
-0.00
0.10
0.01
0.12
-0.07
0.15
-0.02
0.08
-0.02
-0.02
0.02
-0.54
0.92
0.02
5
+0.00
0.74
0.83
0.94
0.94
0.92
0.74
0.96
0.34
-0.09
0.21
-0.37
0.35
0.18
6
0.00
0.49
-0.20
-0.08
-0.18
-0.20
0.11
-0.09
0.91
-0.04
-0.26
-0.07
0.10
0.01
Weight of factor 1.44 1.49 1.10 2.40 1.14
159
-------
TABLE 3. FACTOR SOLUTION FOR RESULTS OF CHEMICAL ANALYSIS OF WATER
SAMPLES FROM KUYBYSHEV RESERVOIR, 1954-1961
1
Level
ci-
s°r
HCQ- mg/liter
O
Ca2+
Mg2+
Na+ + K+
Ei
CT
s°r
HCOj % eq.
Ca2+
Mg2+
Na+ + K+
2
-0.05
0.45
0.79
0.96
0.92
0.80
0.45
0.92
-0.04
0.37
-0.19
-0.07
0.19
-0.05
Factor
3
-0.01
0.29
0.05
0.17
-0.06
0.01
0.77
0.18
0.26
-0.18
-0.08
-0.73
-0.32
0.89
Mapping Coefficient
4
-0.03
0.82
0.21
-0.06
0.22
0.12
0.35
0.21
0.93
0.05
-0.57
-0.13
-0.12
0.19
5
-0.10
0.07
0.19
0.16
0.13
0.55
-0.21
0.16
-0.00
0.21
-0.06
-0.12
0.87
-0.38
6
-0.03
0.13
0.52
-0.01
0.28
0.20
0.07
0.22
-0.07
0.82
-0.44
0.07
0.07
-0.08
Weight of factor 2.12 1.51 1.48 1.19 1.16
160
-------
TABLE 4. FACTOR SOLUTION FOR RESULTS OF CHEMICAL ANALYSIS OF WATER
SAMPLES FROM KUYBYSHEV RESERVOIR TAKEN NEAR VYAZOVYYE (VOLGA RIVER)
AND SOKOLI GORY (KAMA RIVER) AND NEAR KOMSOMOLSKIY (TAILWATER),
1958-1959
1
Level
CT
HCO^ mg/ liter
Mg2+
Na+ + K+
Zi
cr
HCOg % eq.
Ca2+
Mg2+
Na+ + 'K+
2
-0.02
0.84
0.45
0.14
0.36
0.23
-0.01
0.49
0^96
-0.10
-ii93
-0.72
-0.58
0.80
Factor
3
0.18
0.05
-0.10
-0.14
-0.09
-0.54
0.02
-0.06
-0.01
-0.07
0.05
0.11
-0.78
0.26
Mapping Coefficient
4
-0.01
-0.12
0.42
0.10
0.08
0.05
0.03
0.03
-0.23
0.98
-0.30
0.03
-0.12
-0.08
5
-0.00
-0.08
0.06
0.13
-0.04
0.07
-0.01
0.06
-0.02
-0.02
0.03
-0.46
-0.05
0.37
6
0.07
-0.51
-0.76
-0.95
-0.89
-0.77
-0.08
-0.86
-0.17
-0.04
0.20
0.43
0.10
-0.37
Weight of factor 2.16 1.03 1.16 0.62 2.06
161
-------
TABLE 5. FACTOR SOLUTION FOR RESULTS OF CHEMICAL ANALYSIS OF WATER
SAMPLES OF TRIBUTARIES OF THE VOLGA RIVER AND KUYBYSHEV RESERVOIR,
1954-1961
1
Level
cr
soj-
HCOg mg/ liter
Ca2+
Mg2+
Na+ + K +
zi
Cl"
SO2-
HCOg % eq.
Ca2+
Mg2+
Na+ + K+
2
-0.04
0.28
0.92
0.97
0.97
0.96
0.45
0.99
-0.15
0.62
-0.44
0.24
-0.05
-0.21
Factor
3
-0.08
0.33
-0.13
-0.04
-0.14
-0.04
0.59
-0.02
0.27
-0.26
0.03
-0.90
-0.05
0.89
Mapping Coefficient
4
-0.20
-0.86
-0.08
0.13
-0.06
-0.03
-0.47
-0.09
-0.94
-0.20
0.69
0.02
0.24
-0.13
5
0.05
-0.02
0.27
-0.02
0.12
0.08
0.03
0.08
-0.03
0.67
-0.52
0.13
-0.10
-0.05
6
-0.03
-0.10
-0.07
-0.06
-0.12
0.18
-0.26
-0.09
-0.09
-0.20
0.20
0.32
0.89
-0.28
Weight of factor 2.37 1.50 1.58 0.92 1.10
162
-------
TABLE 6. FACTOR SOLUTION FOR RESULTS OF CHEMICAL ANALYSIS OF WATER
SAMPLES FROM VOLGOGRAD RESERVOIR, 1954-1961
1
Level
CT
HCO^ mg/liter
Ca2+
Mg2+
Na+ + K+
Ei
cr
so*-
HCO^ % eq.
Ca2+
Mg2+
Na+ + K+
2
-0.09
0.68
0.84
0.95
SUM
0.06
0.61
0.92
0.01
0.42
-0.17
-0.11
0.03
0,04
Factor
3
0.14
0.71
0.18
-0.04
0.22
0.08
0.16
0.19
0.98
0.08
-0.46
-0.05
0.08
-0.00
Mapping Coefficient
4
0.15
-0.04
-0.09
-0.13
-0.00
-0.04
0.12
-0.09
0.05
-0.05
0.00
0.35
-0.98
0.20
5
-0.05
-0.02
-0.47
-0.06
-0.21
-0.03
-0.08
-0.20
• 0.15
-0.86
0.24
0.00
-0.09
0.05
6
-0.07
0.18
0.13
0.23
-0.06
0.02
0^75
0.21
0.07
-0.05
0.04
-0.88
-0.11
0.72
Weight of factor 2.10 1.36 1.08 1.07 1.43
163
-------
TABLE 7. FACTOR SOLUTION FOR RESULTS OF CHEMICAL ANALYSIS OF WATER
SAMPLES FROM TRIBUTARIES OF VOLGA RIVER BELOW KUYBYSHEV RESERVOIR,
1954-1961
1
Level
Cl"
so|-
HCO^ mg/ liter
Ca2+
Mg2+
Na+ + K+
zi
Cl
SOJ-
HCOq % eq.
Ca2+
Mg2+
Na+ + K+
2
-0.26
0.04
0.97
0.77
0.97
0.89
0.31
0.97
-0.31
0.85
-0.75
0.04
0.34
-0.21
Factor
3
0.06
0.31
-0.03
0.29
-0.13
0.05
0.82
0.17
0.26
-0.11
-0.16
-0.97
0.14
0.90
Mapping Coefficient
4
0.07
0.10
-0.08
-0.30
-0.05
-0.43
0.16
-0.10
0.14
-0.16
-0.05
0.19
-0.91
0.29
5
-0.02
0.93
-0.19
-0.20
-0.00
0.05
0.27
0.02
0.89
-0.46
-0.55
-0.16
-0.20
0.25
6
0.02
0.08
-0.06
0.40
0.09
0.04
0.16
0.14
-0.12
-0.09
0.23
0.02
0.02
-0.04
Weight of factor 2.44 1.66 1.15 1.57 0.56
164
-------
£^,300
i ^ 200
o c
5 10°
0
OC -e
2> 25
!/5 u E
^ ^y (j
< LU 15
oc
_ 50
Ol
(^
O
25
10
- 10
CO
+ 5
ra
Z
0
en
E 150
D
O
i 50
8
10
5
0
I I I I
i i
I I I I I
10
5
o
0
350
£ 250
8
<5
50
10
B5 5
0
200
150
"• 50
0
w _ 300
a ^ 200
< 100
0
i . 35
=! 25
3 ~
< en
I J
L II
il
i i i
III
1951
1953
1955
1951
1953
1955
Figure 1. Berezina River at Gorval, 1951-1956.
165
-------
TABLE 8. FACTOR SOLUTION FOR RESULTS OF ANALYSIS OF WATER SAMPLES
OF GORVAL TRIBUTARY, BEREZINA RIVER, 1951-1956
1
Discharge, m3
Ca2+, mg/liter
Mg2+
Na+ + K+
HCO§
so'-
Cl"
Hardness
Zi % Eq.
Si
Fe
Hardness
Perman. oxide
2
-0.35
0.69
0.56
-0.10
0.79
0.87
0.66
-0.12
0.88
0.02
0.03
0.09
0.00
Factor
3
-0.07
-0.28
0.08
0.92
-0.42
-0.02
0.32
-0.18
0.05
0.02
-0.17
-0.16
0.09
Mapping
4
0^40
-0.24
-0.08
-0.34
-0.39
0.17
-0.03
-0.04
-0.46
-0.16
-0.02
-0.42
0.06
Coefficient
5 6
0.40
0.11
0.00
-0.12
0.22
-0.01
-0.01
-0.42
0.10
-0.02
-0.08
-0.83
-0.35
166
-------
TABLE 9. FACTOR SOLUTION FOR RESULTS OF ANALYSIS OF WATER SAMPLES
FROM BOBRUSK TRIBUTARY OF BEREZINA RIVER, 1957-1973
1
Discharge, m3
Ca2+, mg/liter
Mg2+
Na+ + K+
HCO-
so2-
4
cr
Hardness
Z1 % Eq.
Si
Fe
2
-0.22
0.84
0.58
-0.05
0.88
0.75
0.19
0.87
0.06
-0.18
0.33
Factor
3
-0.20
0.12
0.16
0.16
0.11
0.35
O.J32
0.19
0.00
0.02
-0.02
Mapping Coefficient
4
0.09
0.03
-0.14
-0..87
0.08
-0.07
-0.11
-0.29
-0.05
0.18
-0.21
5
0.08
-0.11
0.31
0.03
-0.02
0.36
0.04
0.07
0.02
-0.00
0.00
6
0.35
-0.21
-0.36
-0.14
-0.40
0.10
-0.07
-0.34
-0.31
0.08
0.32
167
-------
50
O)
E 30
co"
O
10
O)
E 10
- 240
01
E 190
6° 140
o
X 90
40
_ 9
en K
E 6
G 3
0
Jsoo
Q- 180
60
10
< CO
SS
I- Q
QC
<
n:
1951
1953
1955
Figure 2. Berezina River at Gorval , 1951-1956.
163
-------
analysis of the regression equations indicates that by selecting the corre-
sponding concentrations of Ca2+ and HC03 ions as basic variables, one can
reliably predict four other variables, i.e., Mg2+, Cl~, the sum of ions, and
total hardness. In Figure 2, the broken line shows the predicted values for
these variables.
Attempts were made to use models generated for one of the points and the
interrelated variables utilizing linear regression for prediction of the be-
havior of variables at other points. As before, the values of the concen-
tration of Ca2+ and HC03 were used as the base variables. The results of
this prediction are shown in Figure 3 and 4. The high degree of accuracy
achieved remains constant throughout a 10-year period. Additional analysis
of these data has shown that the concept of multidimensional samples lacking
a natural mean, or a natural average of eigenvalues of the covariation ma-
trix is not suitable for statistical interpretation of series of hydrochemi-
cal indices. Correlation was found between measurements of the vector of
means, and the vector of eigenvalues in the time scale of reading. The in-
troduction of these hypotheses relative to the structure of the hydrochemi-
cal fields enables a modeling approach from the point of view outlined
below.
SELECTION OF PARAMETERS OF THE MODELING SYSTEM FOR DETERMINATION OF THE
SPATIAL STRUCTURE OF HYDROCHEMICAL FIELDS
Based on the above general considerations concerning the unity of all
indices forming the structure of a hydrochemical field, a method of place-
ment of a system of observations for determining the spatial organization of
the field naturally follows. The essence of this method is that parameters
of the empirical models are determined which can be used to make the transi-
tion from integral field characteristics to particular characteristics for a
predetermined point. The network of initial observations can be adjusted in
order to select empirical dependence adequate to the existing structure of
the hydrochemical field, considering the limited nature of the number of
measurements.
Sequential analysis of the structure of the hydrochemical field can be
represented by the placement of r measurements on a grid of k-£ possible
measurements (r «k-£). In this case, p and q are the ordinal numbers of
the points at which information is obtained, corresponding to the coordi-
nates Zp and Xp; p = 1,..., k; q = !,...£. The introduction of the vector
Bpq = BpqUp, Xq), of dimensionality n, characterizes a corresponding cell
of the field, obtained by the results of measurement of a list of n indices.
In each cell of the field, mpq measurements of the_ index ypq(s); s =
l,...,n are performed, and the vector of mean values ynq is calculated.
Since in the initial stage of analysis, mpq ty s, and the reliable matrices
of the indices cannot be calculated, we cair represent the field as follows:
169
-------
- 50
en
E
TO 30
o
10
14
- 10
en
2
170
«
O
90
50
15
- 10
O)
E
300
cr
cu
*-<
§ 200
W
d>
CL
100
c/>
. w
f%<
l§?
H <
I
0
8
II
J
II
L
JD
O
CO
•P
(O
s-
OJ
ra
c
•r-
N
HI
S-
OJ
QQ
00
0)
3
O)
1957 1958
1959
1965
1973
170
-------
- 100
en
E
«3 50
0
40
Dl
E
di 20
0
60
ro
E 40
n
O
o
I
20
- 30
O>
E 20
o
10
300
c
S
0)
^
200
100
0
30
LJJ
I 20
DC —
O
0
f A f f\ f
J -/ u vJ
i I I iv
o
^t
c.
to
s_
OJ
oo
QJ
1969 1970
-------
* * (ypq(i) ' *(1)) (ypq(i) •
p=l q=l
Suppose the mean value of the vector of observed indices, and the corre-
sponding covariance matrix at this point in the field can be restored by
means of a linear transform of the vector of mean values and covariance
matrix, characterizing the entire field as a whole, i.e.,
ypq = Lpqy;
where L is an n-n matrix of coefficients.
Suppose
Lpq = LJq
where a, apq and 3pq are the diagonals of an n n matrix of eigenvalues;
while H is an n-n matrix of eigenvectors.
In this case, the simplifications
(4)
can be applied.
The procedure of calculation of Bpq at each instant in time after a
group r of measurements includes the following sequence of operations:
172
-------
1. Refinement of vector ypq if measurements have been performed
at a given point in the field.
2. Refinement of the vector y.
3. Refinement of the matrices R, H~ , a.
4. Calculation according to (4) for each point in the field Bpq, apq.
In this case, the vector Bpq represents each cell of the field as inde-
pendent of all previous measurements in any of the cells of the field. The
expediency of this very procedure is demonstrated by the following consider-
ations.
The recursive nature of the estimation of parameters of a point in the
field, even with non-optimal placement of points of information measurement,
guarantees
where t is the next measurement time.
If the number of experiments is sufficient to guarantee (5), this plan
of estimation allows an explicit prediction to be given for an arbitrary
point in the network. The error of prediction in this case depends not on
the random errors of measurement at arbitrary points on the network, but
rather on the number of experiments performed and the resolution of the net-
work as a whole.
Adaptive estimation of parameters 3pq is possible, and is accomplished
by the introduction of the strategy of effective "omission". For example,
in the exponential form e = exp{y(t-j - t)}, where t is the current instant
in time of the measurement; t-j is the instant in time of the previous mea-
surement, then Y is the "omission" factor. This strategy accomplishes
experimental planning with respect to possible instabilities.
Finally, each cell of the network is characterized by the actual vector
of means and covariance matrix, which allows testing of rather complex
multidimensional statistical hypotheses concerning the network as a whole.
This, in turn, allows a tranformation of the network, increasing its re-
solving capacity.
Determination of optimal values of the frequencies of performance of ob-
servations , based on the spatial heterogeneity of the field of measured
statistical indexes , may be made. The regression equation for BDq(s),
where s = l,...,n (the number of measured characteristics at a point in the
field), in the case of two-dimensional extrapolation (Fedorov 1971; Krug et^
£]_. 1977) can be represented as follows:
173
-------
where L- (x ) is an analogous expression.
J H
The dispersion of this estimate is
dtcfZp.x^l-o2^ .^/?
-------
5. Discard in sequence one of the cells of the field and per-
form calculations in (2), (3) and (4).
6. Discard that field cell from analysis which by elimination
produces the greatest change in (9).
7. The procedure terminates with a transition to step 9 if the
discarding of any point leads to an increase in the value
of (9).
8. Return to step 1 .
9. Calculate cjj.- according to (8).
10. Recalculate the frequencies of measurements at the point
The calculations should be performed for all values of s = l,...,n; p =
l,...,k; q = !,...,£ (k-£-n plans in all). As a result of the calculations
performed, for each point the summary necessary number of measurements which
can be perfomed given the arbitrary distribution of measurement frequencies
at this point is obtained. This is converted to the assigned number of mea-
surements. Naturally, when the arbitrary distribution is achieved, priority
recommendations (cost or any other consideration) can be made for the con-
duct of measurements. This is achieved by modifying (6) and (9).
The requirements of (5) assured consistency of estimate , but the re-
sults of two-dimensional extrapolation, namely the calculation of the dis-
persion of the prediction for a number of points relative to the remaining
points, may differ significantly from the desired value. The heterogeneity
of the calculated statistical indexes Bpq results from the lack of homo-
geneity of the measured field. In this case, based on the information accu-
mulated, and by testing of the corresponding statistical hypotheses (Kulbak
1967), the entire space of observations is divided into the corresponding
subspaces.
CLASSIFICATION OF STRUCTURES OF HYDROCHEMICAL FIELDS
Sequential analysis of the structures of hydrochemical fields, which
forms the basis of solution of the extreme problem of placement of the net-
work of observations, utilizes the concept of smoothness of distribution of
the observed field parameters.
The lack of homogeneity of observed or calculated indices is interpreted
by the researcher as a lack of homogeneity in the observed field. This con-
clusion is correct within the framework of the conceptual or empirical model
selected. In this case, the model of the field description toward which the
researcher was oriented should be changed. Since the model uses not only
175
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the mathematical apparatus of empirical field description, but also the
selected configuration of the observation network, additional empirical in-
formation on the field is required.
The planning of measurements at new points on the field or the introduc-
tion of new indices is necessary, but the uncertainty of this planning is
great. In this case, the problem of classification arises, the solution to
which provides the necessary information for planning of new experiments.
The conceptual model of the field in this case is fixed by preceding stages
of research, and, within the framework of this model, the empirical material
must be classified so that the heterogeneous field is approximated by
classes of homogeneous fields. It is necessary to realize which of the two
tasks is being performed by the researcher: typization or separation
(Ayvazyan et^ ^1_. 1974). The task of typization (representation), the solu-
tion of which always exists, assumes separation of the set of observed in-
dices (in this case, vectors 3pq) into a comparatively small number of
grouping areas. The grouping areas are analogs of the grouping intervals
used in processing of one dimensional observations, which defines the nature
of the requirements: uniform coverage of the entire set of experimental
data by nonintersecting classes of samples, and assurance of the minimum
possible distance between these areas. The task of separation, whose solu-
tion may yield a negative result if the entire sample belongs to one clus-
ter, assumes natural stratification of the initial observations into a
clearly expressed cluster (in this case, vectors
Before indicating the relationship between the performance of the tasks
of typization and separation in the reconstruction of a model of the struc-
ture of hydrochemical fields, it is necessary to indicate the specific char-
acteristics of the use of the methods of pattern recognition in hydro-
chemisty. As we perform the task of recognition, the most interesting char-
acteristics are those for which the difference between the mean values of
classes is great in comparison with the standard deviations within classes,
and not those for which the standard deviations are great (Duda and Hart
1976).
The introduction of measures of difference between classes in the form
of Euclidian distances, transformation by means of the main components, or
factor mapping of the initial matrix of data on a standardized matrix (with
the introduction of Euclidian distances) presumes that the space of charac-
teristics is isotropic, i.e., the groups defined by these distances will be
invariant with respect to shifts, rotations, or dilations (Ayvazyan ei_ al .
1974; Duda and Hart 1976). However, as studies have shown, the assumption
that the fields of hydrochemical indices are isotropic is not valid. The
absence of isotropicity can be clearly seen from the tables of factor solu-
tions presented above. In addition to the constantly acting factors, and
practically constant dispersions of indices from object to object, the vari-
able parameters are the weights of the factors participating in the descrip-
tion of the dispersion of the measured indices.
A natural invariant for each of the measured vectors ypn(s) of the field
is represented by the calculated indices Bpn, where p, q are, as before, the
number of the measured point, while s is an ordinal number with respect to
176
-------
time. Actually, when the task of typization is performed for statistical
parameters (the number of vectors to be analyzed is only k*£, significantly
less than the number of all experiments in the network for measurement of
vectors ypq(s)), we obtain nontrivial information on the structure of the
hydrochemical field. The presence of 3(r), where r is the class number, en-
ables a fixed description for the entire set of characteristic measurement
points in the form of statistical characteristics, namely the covariance
matrices
R(r) = H3(r)a3(r)H"1; (11)
and the vectors of means
m(r) = H (r)H'1y; (12)
where H, a and y are defined and calculated according to (3), (4), while N
is, as before, the number of classes, r = 1,...,N.
The information measure of difference for classes i, j (Kulbak 1967) in
this case, if the hypothesis of mixed multidimensional normal distributions
is accepted, is as follows:
3(1,j) = 1 tr
n
Z {[B(i,a)ot(a)3(i,a) - 3(j,a)a(a)»(j,a)] x
a=l
(6(j,a)a(a)3(j,a))"1 - (3(i ,a)a(a)3(i .a))"1) +
yT HaH^y; (13)
where a = l,...,n is the ordinal number of the index
a(a) = [3(i,a) - 3(j,a)] [-* - }- - + -7 - 1 - ] x
B (j,a) a(a) 3 (i,a)a(a)
1.a) - p(J.a)] = [g'a -g.
3 (J»a)
Using the expression
177
-------
yT HaH'V = tr {H'V yT H) ; (15)
we can write
Analysis of the expressions (14), (16) indicates that both of the com-
ponents contain the terms
If the entire sample is normalized with respect to the mean, i.e., y = 0,
then (16) becomes
3M i) - ] y r32(i.a) - 32(j,a)2
8(1'J) ' 2 J, I BtUJ-eU.aJ 1 ' <]7)
It is seen from (17) that the Euclidian distance providing isotropicity
of the space of characteristic on the assumption of nonisotropicity of the
space of characteristics y has been achieved. This, in turn, opens the pos-
sibility of using factorial analysis for the solution of the problem of
dimensionality of the space of characteristics 3pq, which enables simplifi-
cation of the task of typization objects of observation.
However, the question of the minimum number of classes into which the
space of characteristics should be divided remains open. Following the com-
mon recommendations (Ayvazyan et al_. 1974; Duda and Hart 1976), it can be
indicated that the approach of~d"irected decision-making functions here. The
classification procedure leads to a decrease in (9), which is equivalent to
a contradiction between the statistical classification and spatial homo-
geneity of classes.
Thus, the methods of analysis suggested do not contradict the concept of
the field used in hydrochemistry, and can be used in the prediction of
space-time characteristics of hydrochemical indices, providing optimization
of the network, with respect to spatial parameters and with respect to time.
REFERENCES
Ayvazyan, S.A., Z.I. Bezhayeva, and O.V. Staroverov. 1974. Classification
of multidimensional observations, Moscow, Statistika Press, 240 pp.
178
-------
Duda, R. and P. Hart. 1976. Pattern recognition and cost analysis, Moscow,
Mir Press, 511 pp.
Fedorov, V.V. 1971. Optimal experiment theory. Moscow, Nauka Press, 312
pp.
Krug, 6.K. e* al_. 1977. Experimental planning in problems of identifica-
tion and extrapolation. Nauka Press, Moscow, 208 pp.
Kulbak, S. 1967. Information theory and statistics. Nauka Press, Moscow,
408 pp.
Waddington, K.H. 1948. Organizers and genes. Foreign Literature Press,
Moscow, 206 pp.
Zenin, A.A. 1961. Results of hydrochemical studies of the Volga reservoirs
in 1954-1961. Rostov-na-Donu, 209 pp.
179
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SECTION 10
TRANSPORT OF MINING WASTE IN LAKE SUPERIOR
M. Sydor, G.E. Glass1 and W.R. Swain2
INTRODUCTION
The western arm of Lake Superior has a 100 km trough which extends
along the Minnesota shoreline from Knife River, Minnesota where the trough
is 180 m deep past Silver Bay, Minnesota, where depths range around 250 m
(Figure 1). An average of 67,000 tons of fine iron ore tailings are dis-
charged at Silver Bay into the lake every day. The discharge is the largest
source of particulates along the Minnesota shore. Transport of tailings in
western Lake Superior have been the subject of considerable local interest
(Cook 1974) and general interest regarding dispersion of contaminants in the
Great Lakes. The transport of particulates can be investigated through use
of remote sensing and numerical modeling. Fine particulates in lakes are
identifiable through use of Landsat data (Sydor, Stortz, and Swain 1978;
Sydor 1978). Their remote sensing signatures provide valuable information
for studies of the dispersal processes and verification of numerical models
(Diehl ejt aj_. 1977). We consider here measurements and remote sensing data
which suggest transport of tailings by means of an upwelling. Two wind epi-
sodes are discussed in detail. In one instance transports and dispersal of
a plume is simulated numerically for the actual wind and pressure conditions
over the lake. The model results are subsequently compared with measure-
ments of suspended solids and the remote sensing data for the plume. The
results show that the direct transport of tailings cannot readily account
for the episodes of high concentration of tailings at the Duluth water in-
take. In the instance chosen as our second example, Landsat images for two
consecutive days show discrete high concentration patches of tailings which
appear to have upwelled during westerly winds. The successive images and
current measurements at two stations show that tailings patches are trans-
ported in accordance with the results from numerical models. The patches
appear to have originated from an upwelling near the Duluth water intake.
The data for the intake show that the concentration of tailings in the
'U.S. Environmental Protection Agency, Environmental Research Laboratory,
Duluth, Minnesota 55804.
2U.S. Environmental Protection Agency, Large Lakes Research Station, Grosse
lie, Michigan 48138.
180
-------
CO
N
Min nesota
Lake Superior
Duluth
km
W i s c o n s i n
Figure 1.
Study area showing the outline of a deep trough where tailings are discharged
and the discharge source location at Silver Bay.
-------
patches exceeded 3 mg/£, a level which is an order of magnitude higher than
the concentrations produced by the direct transport of turbidity to Duluth.
DISCUSSION
Information regarding direct transport of tailings in the form of a sur-
face turbidity plume along the Minnesota shore from Silver Bay to Duluth can
be obtained by using remote sensing data for a specific storm event. During
November 9-14, 1975, a severe storm produced an extensive turbidity plume
off Silver Bay. Remote sensing data for the event provides information on
the extent of the plume at one instant of time. To determine the dispersal
of turbidity throughout the event we can simulate the plume numerically for
the entire period and compare the results of the model with the remote
sensing data for November 14, 1975 at the instant of time when the image was
taken.
It was first necessary to calculate the transports in the lake through-
out the event. This was done using a depth integrated model for Lake
Superior performed on a 6 x 6 km grid, using quadratic bottom friction and
employing a numerical scheme due to Leendertse (1967). The results of the
model for a generalized steady wind function have been discussed in detail
by Diehl et_ _al_. (1977). However, for the case here a numerical method of
handling the actual pressure distribution term and the wind function which
occurred on November 9-14, 1975 was needed. The function and the pressure
term for the storm were devised from approximation of the actual winds mea-
sured at weather stations around Lake Superior (Maanum 1977). Weather re-
cords from nine Coast Guard stations around the lake provided information on
wind speed, wind direction, and the atmospheric pressure gradients every 2
hours. The problem of finding pressure gradients at every grid point was
handled by devising a fit to the atmospheric pressure data for a low pres-
sure cell which moved in the proximity of the lake during the storm.
The formula used to fit the atmospheric pressure data was
pa = f(a,r) (b+cx+dy)
where pa is the atmospheric pressure,
(x,y) is the position in cartesian coordinates relative to the center
of the storm,
r is the distance from the storm center,
a,b,c,d are empirical coefficients, and
f(a,r) is some function of r and a.
The coefficients b, c, and d were treated as quadratic functions of
time:
182
-------
pa = f(a,r) [(b0 + b^ + b2t2) + (CQ + C]t + c2t2)x + (dQ + d^ + d2t
The function f was taken as f(a,r) = r2 + ar3, which provided a good fit to
the radial dependence of the pressure term. The value of a. was varied while
the other 9 coefficients were found by the method of least squares until the
sum of the squared differences between the calculated and measured pressures
was minimized.
Atmospheric pressures calculated by this model agreed well with the mea-
sured pressures both in time and space. Once the coefficients were found,
the pressure gradients were also known by taking spatial derivatives of the
empirical formula. Subsequently, wind directions were obtained by rotating
the direction of the pressure gradient field 125° counterclockwise. A
form similar to the one used to fit pressures was fitted to the measured
winds, to obtain wind speeds everywhere.
Wind speeds were given by
W = f'(a',r) (b1 + c'x + d'y)
where W is the wind speed, the coefficients b1, c1, and d1 were treated as
4th-order functions of time, and f'(a',r) = r/(r2 + a12). Figure 2 shows
the fit to the wind data at Duluth. The calculated and measured water
levels are shown in Figure 3. Spectral analysis of the calculated water
levels produce free surface modes given by Mortimer and Fee (1976). To
check the direction and speed of the vertically averaged currents at Silver
Bay we considered measurements of currents for several wind episodes re-
ported by Baumgartner et^ aj_. (1973). The comparison was made for the well-
mixed isothermal conditions in March, April, October, and November and is
shown in Tables 1 and 2. The calculated transports agree reasonably well
with the expected currents. The transport pattern for the entire area is
shown in Figures 4 and 5. Generally, the patterns in Figures 4 and 5 show
typical results for westerly and easterly winds (Ruschmeyer 1956; Diehl et^
al. 1977). The transports for both winds move down the Minnesota shore from
Silver Bay towards Duluth. For westerly winds, Figure 4, the pattern shows
a broad turn around between Knife River and Duluth. Thus, the surface con-
taminants from Silver Bay would be normally transported away from the shore
in this region of the lake. Westerly winds produce a counterclockwise cir-
cultion cell in the extreme western Lake Superior west of a line from Sand
Island to Silver Bay (Ruschmeyer 1956). The transport pattern for easterly
winds, Figure 5, is quite different. For easterly winds the cell-like
structure disappears. Figure 5 shows transports following the Minnesota
shore all the way from Silver Bay to Duluth. For easterly winds the magni-
tude of currents along Minnesota shore ranges from 6-30 cm/sec, averaging
at 10 cm/sec. One would thus anticipate for the event modeled here, that
the tailings caught in the turbulence at the onset of the high winds on
November 10 would be transported from Silver Bay for a distance of about 30
km to the vicinity of Two Harbors by the time of the satellite overpass on
November 14. This was indeed the case, Figure 6.
183
-------
CO
-p-
J
a
30+
20+
10+
WIND FIT AT DULUTH
— calculated
O measured
11/8/75
10
11
12
13
Figure 2. Comparison at Duluth of the measured winds and wind function used for modeling of
Lake Superior for the November 1975 storm.
-------
CO
en
-30--
-45--
computed
measured
DULUTH, MM.
1
11/8/ 75
10
11
12
13
Figure 3. Comparison at Duluth of the measured water level fluctuations with the fluctuations
derived from the numerical model of Lake Superior for the November 1975 storm wind.
-------
TABLE 1. COMPARISON OF CURRENTS NEAR DULUTH
Wind
Conditions
Simulated North-
easterly
Measured North-
easterly
Simulated Westerly
Measured Westerly
Speed M/S Dir.
12
10-11.5
17
15
10-11.5
13
13
13
13
NE
NE
NE
NE
NE
NE
WNW
NW
WNW
Currents
Speed CM/S Dir.
13.2
13.6
20.2
19.2
21.1
17.5
1.5
2.7
3.1
224°
226°
230°
221°
243°
225°
50°
49°
30°
Date
11/10/75
4/3/76
11/13/75
Composite
10/24/73
11/13/73
Composite
11/11/75
10/26/73
Time
0400
1600
0200
0800
0400
0600
0000
Measured currents at 18 m depth at Duluth intake in Lake Superior.
Calculated current at grid 3 average depth 24 m at grid center 3 km from
Duluth intake.
TABLE 2. COMPARISON OF CURRENTS NEAR SILVER BAY
Conditions
Simulated North-
easterly
Measured North-
easterly
Simulated Westerly
Measured Westerly
Wind
Speed M/S
15
17.5
10
13
8
11
11
Currents
Dir.
NE
NE
NE
NW
NW
WNW
WNW
Speed CM/S
5
9.7
5.3
4-7.4 1
3.6
1.6
7.8
8
Dir.
243°
236°
210°
60-255°
234°
226°
243°
229°
Date
Composite
11/13/75
9/18/72
October
11/11/75
3/31/76
10/8/72
10/14/72
Time
40 hrs
0200
30 hrs
1800
0200
1400
1600
Measured currents 8 km from Silver Bay every 1/2 hour at 3 depths (15 m,
46 m, 191 m) time averaged over 4 hours were used to determine the depth
integrated flow speed.
Calculated current 12 km southwest from Silver Bay at grid center 14 average
depth 175 m.
186
-------
Date 11 Nov. 75 Time 0600
Vertically Averaged Currents
After 66 Hours
Scale ° 3°cm/sec. Silver
Bay;
Figure 4. Water transports due to westerly wind stress. A
circulation cell which forms for the westerly winds is
responsible for entrapment of pollutants and subsequent
increased biological productivity in this region of the
lake.
187
-------
00
CO
Date 13 Nov. 75 Time 0600
Vertically Averaged Currents
After J14 Hours
Scale
cm/sec.
Silver
Bay<
Superior
Figure 5. Calculated transports for November 13, 1975 showing water movement for high fetch north-
easterly winds. For northeasterly winds tailings discharged at Silver Bay are transported directly
to Duluth. Note that for easterly winds the transports along the Wisconsin shore also bring
turbidity to the Duluth area. (See the corresponding pattern for distribution of suspended
solids in Figure 8).
-------
To compare the dimensions and the concentrations of the observed plume
with predicted transport of tailings, the turbidity plume itself was
modeled. The discharge of tailings is a slurry which takes most of the
tailings down in a density current to the bottom of the lake. However, some
tailings are stripped off from the density current to form a turbidity
plume. To simulate the dispersal of the turbidity plume, transports had to
be approximated on much finer grid scale. The grid size used in the cal-
culation of transports had to be subdivided to the dimensions compatible
with representation of the mixing processes (Csanady 1973, Galloway 1977).
The original 6x6 kilometer grid used in calculation of transport was thus
subdivided in the dispersion model into 0.6 x 1 km subgrids, a size com-
parable to transport lengths per half hour integration time step used in the
turbidity dispersion model. The currents at the subgrids were obtained by
interpolation of currents found in the transport model. Near the shore the
tangential currents were held constant and the normal currents were termi-
nated linearly to zero.
In each subgrid of the dispersion model the residual suspended load
brought in by the currents and diffusion was completely mixed every half
hour. A diffusivity of 1 m2/sec was used (Orlob 1959). Since the currents
were largely responsible for the transport of the suspended load, the dif-
ferential scheme for turbidity dispersion was essentially first order in the
temporal and spatial coordinates and no problems with convergence or stabil-
ity occurred. Numerical mixing method accentuates the diffusion of the
source, and introduces distortion due to preferential spreading of the lead-
ing edge of the plume. Thus, the calculated plume overestimated the actual
plume length. To minimize this problem and to assure stability the upstream
suspended solids concentration gradients were used for calculation of new
concentrations at each grid, however, even with optimum time step slight
skewing of the concentrations was expected in the calculated plume. The
center of mass of the simulated plume was, however, transported properly.
The advantage of using the finite difference method for calculation of dis-
persal of turbidity comes from the ease of calculation and low machine time
cost.
In simulation of the November 14 plume at Silver Bay, the source of the
fine particulates was approximated by taking 3% of the 67,000 tons/day dis-
charge as the fraction of fines which would escape the density current under
storm conditions (Glass 1973). This approximation appeared reasonable upon
comparison of the actual and the simulated concentrations of the suspended
solids within the plume. An equilibrium concentration of tailings amounting
to 10,000 tons of fines distributed uniformly over a .6 km2 grid located at
the source was taken as the initial boundary condition. The resulting simu-
lated plume, Figure 6, compares well with the turbidities found from sam-
pling measurements near Two Harbors, Figure 7, and the distribution of tur-
bidity, Figure 8, derived from the relationship shown in Figure 7 and Land-
sat data for November 14, 1975. Many plumes observed in Landsat imagery for
Silver Bay show similar characteristics. The remote sensing data generally
verify the transport patterns exhibited by the numerical model and agree
with the distribution of suspended particulates predicted by the dispersion
model over a period of several days. One can thus expect the numerical
methods to yield reasonable results for longer term dispersal, say over a
189
-------
10
o
Dulut
Sus. Sol. mg/l
Lake Superior
20 kn
Figure 6. Numerical simulation of the plume of mining waste from point source at Silver Bay, Minnesota.
The plume represents the dispersal of 3% of suspended solids stripped off from the discharge slurry.
-------
TURB.-SUS SOL
TURB.-PAND4C R=086
NOV. 14 TURB - B4C
Turbidity (NTU)
Figure 7. Turbidity vs. Landsat Band 4 intensity above background. Measured
values of suspended solids for November 14 near Two Harbors are indicated
by the open triangles. This relationship used in conjunction with Landsat
data allows for derivation of the distribution of surface concentrations
of suspended solids in Lake Superior.
191
-------
Suspended Solids Nov. 14,1975
Silver
Bay
7R 7mg/| RED CLAY
2T 2mg/| TAILINGS
2M 2mg/| MIXED
N
MINNESOTA
Lake Superior
Duluth
WISCONSIN
Figure 8. Turbidity plume derived from Landsat data for November 14, 1975.
Particulate concentrations and type are marked by numbers, (mg/&) and
letters respectively; where R - red clay, T - tailings, M - mixed red clay
tailings. Note the red clay originating from erosion along the Wisconsin
shore is transported towards Duluth, and is subsequently taken out in a
narrow plume along the center axis of the lake. (See reference 4).
192
-------
couple of weeks. Starting with the turbidity distribution observed on
November 14, 1975 as the initial condition, plume dispersal calculation was
extended over a period of two weeks, a time sufficient for the transport of
the center of mass of the plume to Duluth. After two weeks of simulation
time the model produced concentrations of tailings near Duluth ranging
around 0.2 mg/£, a background concentration typically found at the Duluth
water intake. It is therefore unlikely that direct transport of large tail-
ings plumes from Silver Bay to Duluth could produce concentrations of tail-
ings near Duluth in excess of 1 mg/n, a value necessary to explain high mo-
mentary concentrations of amphibole fibers at the Duluth intake shortly
after a passage of high easterly winds (Cook 1974).
As mentioned before, satellite data can be used to identify the type of
particulates in the lake based on the spectral dependence of the volume re-
flectance of suspended solids (Sydor 1978). Using the signature given in
Table 3 for tailings and red clay, we obtain a classification of the tur-
TABLE 3. BAND COMBINATIONS FOR SIGNATURE CLASSIFICATION OF CONTAMINANTS
Contaminant Band Combinations
Red Clay (Landsat 1) (B4-B5) < 0 (B5-B6) > 1
(B4-B6)/(B5-B6) < 1
(Landsat 2) (64-65) < 1 (65-65) > 1
(B4-B6)/TB5-e6) < 1 2
Tailings (64-65) > 0 (84/85) >_ 1-5
(B4-B5)/(85-86) > 1.5
Tannin (64-65) <_ -2 (84-85) <_ -2
(84/85) < 0-6
64 - 6and 4 signal intensity above the clear water background.
bidities observed in the November 14, 1975 image. The results are shown in
Figure 8. It can be seen that the turbidity plume at Silver Bay and a tur-
bidity patch near Duluth are identified as tailings. The origin of the
plume has been already discussed. The area of tailings near Duluth appears
to have resulted from an upwelling produced by onset of westerly winds on
November 10 and 11. Unfortunately the ground truth data at Duluth intake
does not show a pronounced peak for tailings. This could be largely due to
the presence of high turbidity background due to intrusion of red clay.
To consider a clear possibility of upwelling of tailing we examine data
for June 26 - July 3, 1973 when high concentration of tailings was observed
in the Duluth area, but when winds were predominately northwesterly and no
turbidity plume was evident near Silver Bay. Suspended solids and currents
were monitored for those dates. Landsat images for July 2 and 3, 1973 show
that red clay turbidity was confined to the Wisconsin shore so that the
193
-------
identification of suspended solids would be largely confined to turbidity
due to single contaminant type rather than a mixture of contaminants as was
the case for November 14, 1975 when ambiguities in identification in ground
truth and satellite data could arise for the mixtures of suspended solids.
The July images, Figure 9, show distinct patches of tailings with concentra-
tions of 3 mg/&. The patches are well down stream from the Silver Bay dis-
charge source. The patches occurred after prolonged westerly-northwesterly
winds on June 26-30, 1973. Easterly winds over the lake were observed on
July 1, 1973, and the currents along Minnesota shore were directed on July 1
towards Duluth (Keillor 1976). The easterly winds on July 1 were insuffi-
cient to produce large plumes at Silver Bay. On July 2 and 3 the winds re-
turned to westerly directions. .The background concentration of suspended
solids along the Minnesota coast was 0.9 mg/1. The records at the Duluth
water intake (Cook et_ aj_. 1974) showed a pulse of tailings in excess of 3
mg/£, confirming the tailing patch detected in the satellite data. Compari-
son of the successive positions of the discrete tailings patches for the
consecutive July 2 and 3 images verify the circulating transports for
westerly winds. The currents observed for July 1-3, 1973 (Keillor 1976)
also support the transport patterns predicted by the numerical model.
Thus high concentration of tailings found at the Duluth water intake ap-
pear to arise from discrete patches of tailings upwelled from lake bottom
during the westerly winds. The material taken down with the discharge
slurry at Silver Bay remains unconsolidated at the bottom of the lake trough
and extends over a long stretch of the valley from Silver Bay to Two Har-
bors. This unconsolidated material forms at the rising slope of the trough
a secondary source of tailings close to the Duluth water intake. At times
of turbulence the material is readily resuspended and upwells with passage
of westerly winds. A following spell of easterly wind transports the mate-
rial to the Duluth water intake. The direct turbidity plume at densities
in excess of 1 mg/£ appear confined to the vicinity of Silver Bay. The
plumes generally disperse below 0.2 mg/£ level by the time they reach
Duluth.
REFERENCES
Baumgartner, D.J., W.F. Rittall, G.R. Ditsworth, and A.M. Teeter. 1973.
Investigation of pollution in western Lake Superior due to discharge of
, mine tailings, Data Report 1971. Pacific Northwest Environmental Re-
search Laboratory Working Paper No. 10. JJT^ Studies Regarding the Ef-
fect of the Reserve Mining Company Discharge on Lake Superior. USEPA,
Washington, D.C. p. 423.
Csanady, G.T. 1973. Turbulent Diffusion in the Environment. Boston: D.
Reidel.
Cook, P.M., G.E. Glass, and J.H. Tucker. 1974. Asbestiform amphibole mine-
rals: Detection and measurement of high concentrations in municipal
water supplies, Science, 185, 853.
194
-------
Silver
Bay
Tailngs signature
July 2, 1b73
JulyS, 1973
i r
t r
Minnesota
Knife
R.
o o
o o o
o o
Wisconsin
Figure 9. Discrete patches of upwelled tailings observed in Landsat data for
July 2 and 3, 1973. The material upwelled during westerly winds was
responsible for a high influx of tailings to the Duluth water supply. The
influx usually occurs when the winds shift from westerly to easterly
directions causing the transport of the upwelled tailings to Duluth.
195
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Diehl, S.R., W.E. Maanum, T.F. Jordan, and M. Sydor. 1977. Transports in
Lake Superior. J. Geophys. Res. 82(6): 977-79.
Galloway, P.M., Jr. and S.J. Vakil. 1977. Criteria for the use of vertical
averaging in Great Lakes dispersion models. J. Great Lakes Res.
3(1-2): 20-28.
Glass, G.E. 1973. A study of western Lake Superior: Surface sediments,
interstitial water and exchange of dissolved components across the
water-sediment interface. Jn_ Studies Regarding the Effect of the
Reserve Mining Company Discharge on Lake Superior. USEPA, Washington,
D.C. p. 1034.
Keillor, J.P., J. Young, and R.A. Ragotzkie. 1976. An assessment of the
environmental effects of dredged material disposal in Lake Superior,
Marine Studies Center, Univ. of Wisconsin, Madison Publication, 4, 79,
91.
Leendertse, J.J. 1967. Aspects of a computational model for long period
water wave propagation, Memo R.M. 5294-PR, Rand Corp., Santa Monica,
California.
Maanum, W.E. 1977. Numerical prediction of currents and transport in
western Lake Superior, Thesis, Univ. of Minnesota.
Mortimer, C.H. and E.J. Fee. 1976. Free surface oscillations and tides of
Lake Michigan and Superior. Phil. Trans. R. Soc., London A., 281, 1-
61.
Orlob, G.T. 1959. Eddy diffusion in homogenous turbulence. J. of
Hydraulics Division, ASEC 1149. pp. 75-101.
Ruschmeyer, O.R. and T.A. Olson. 1958. Water- movements and temperatures of
western Lake Superior. Univ. of Minnesota School of Public Health,
Minneapolis Publication.
Sydor, M., K.R. Stortz, and W.R. Swain. 1978. Identification of contami-
nant in Lake Superior through Landsat I data. J. Great Lakes Res.
4(2): 142-148.
Sydor, M. 1978. Analysis of suspended solids in lakes through use of
Landsat data. J. Canadian Spectroscopy. 2(3): 91.
196
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SECTION 11
PRINCIPLE OF ORGANIZATION OF AN AUTOMATED INFORMATION SYSTEM
V.L. Pavelko1
The tasks of control of environmental quality are both local and global
in nature. They range from the study of individual objects or even parts of
objects to planetary tasks, from evaluation of states in a narrow time sec-
tion to historical scale, from an individual process at some single trophic
level to an ecologic environmental system. This directly requires that
large volumes of both basic and supplementary information be processed, and
that large amounts of combinations of various data be processed. All of
this requires the use of computers. A step beyond this level is the con-
struction and study of ecologic mathematical models which is in and of it-
self a very cumbersome procedure, impossible without computers.
Therefore, a major problem of testing and control of the status of the
environment is the creation of an automated information system for water
quality (AIS) as a section of a more general ecologic information system.
The flow of information determines the external conditions of existence
of the processes which we are studying. These, in turn, define the func-
tion of the source of a pollutant and are a response to the condition of
water objects, and will be referred to as accompanying flows. These flows
of information are related to hydrochemical subjects as follows:
1. "External conditions" - hydrologic flow (discharge, speed
and direction of currents, water temperature); meteorology
(precipitation, solar radiation, direction and speed of the
wind); hydrobiologic data (the self-purifying capability of
rivers), etc.;
2. "Source function" - data on waste waters, pollution of the at-
mosphere and soil, hydrogeologic data, data on intentional and
natural sedimentation, burial of pollutants (in cases of second-
ary chemical pollution), hydrobiologic characteristics (in con-
nection with eutrophication, oxygen starvation, etc.);
1 Hydrochemical Institute, 192/3 Stachky Prospect, 344090 Rostov-on-Don,
USSR.
197
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3. "Response" - sanitary-hygienic data, hydrobiologic, ichthyo-
logic data, data on industrial water use, marine hydrochemistry,
etc.
Many of these flows were used in AIS which had been created before the
hydrochemistry AIS. Therefore, the results were analyzed and compilation of
the best achievements were made, in order to minimize the time required to
translate the data banks created from one system of coding and indexing to
another for purposes of merging the streams of information in various re-
search programs.
This is equally true of the realization of various programs based on the
hydrochemical branch in question. For example, in the USSR, the stream of
hydrochemical information includes a subsystem in the nationwide service of
observation and testing of the level of pollution (OGSNK), the State Water
Cadastre (GVK), the State Water Report (GUB) and the Global Monitoring
System (GSMOS).
The principles which were used in the creation of AlS-hydrochemistry
system are summarized below. Included are some of the results obtained,
which are recommended as principles for the creation of a global monitoring
AIS, which it is expected will ultimately include all nations, including,
the USSR and the USA.
1. According to estimates, there are 2-5 million polluting substances in
the water, of which over 500 are toxic. (Maximum allowable concentra-
tion values having been established for them). It is estimated that
polluting substances increase annually by about 20,000 compounds. Ap-
proximately 100 of these are determined by manual methods. However,
the annual increase in analytical capabilities for these substances is
only 4-10.
Some 10-15 components are able to be automatically determined, and the
annual increase in the number of such determinations is about one.
Therefore, as information is collected, for the immediate future, GSMOS
should be oriented to 100-300 water quality components, determined basi-
cally in stationary and mobile laboratories.
Result: 1.1 Hydrochemical codes containing the names of components,
regardless of taxonomy, should have a capacity of up to
300 units (V.V. Tarasov 1971, 1974).
Result: 1.2 Forms must be developed for recording of information ob-
tained primarily manually. The forms should be adapted
to manual and automatic punching.
2. Manual punching is a labor-consuming stage of automated data processing
systems, the source of various errors, and the reason for a severe de-
crease in the timeliness of automated information systems. Therefore,
the volume of punching should be greatly minimized without reducing the
information content of the data entered in the computer.
198
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Information is universally suitable for the solution of various esti-
mates, predictions and other tasks, if the initially processed data and
the results of analysis are stored in a computer. The storage of more
generalized characteristics prevents the performance of a number of
tasks. Minimization of the volumes of information to be punched and
stored should be achieved by classification and formal methods of reduc-
tion.
The classes used might be: results of analysis [x-j] and the correspond-
ing characteristics fd-jj]. The results of analysis x-j can be divided
into groups: principal ions, gas composition, heavy metals, oil and
petroleum products, fertilizers, toxic substances, etc. Preliminary
subdivision into nondegradable and degradable (self-purifying) sub-
stances may also be made. There are other classifications—as to type
of effect (organoleptic, persistent, toxic, etc.); type of biota on
which the substance acts, etc. However, these classifications are not
of particular interest for the planning of the AIS, since it is desir-
able to store all results of analysis of an individual sample together.
Combining storage, search, and processing, though it increases the cum-
bersomeness of the machine procedure involved, greatly increases the in-
formation content (both formal and inductive) of estimates of the status
(effect of completeness of results analyzed).
The characteristics td-jj] should be divided into constant and variable.
The corresponding class characteristics can be noted in service labels.
Permanent characteristics include the name of the body of water, region,
nearby town, etc. Variable characteristics include the list of observed
components, date and depth of sampling, etc. Permanent characteristics
should be recorded in computer memory once in the form of a catalogue of
permanent characteristics (CPC), significantly minimizing the amount of
punching which must be done.
Variable characteristics should be coded in the form of numbers and
stored together with the numerical results of analysis. The composition
and taxonomy of the characteristic depend on the type of task to be per-
formed which required their determination in the first place. For
example, if the output result from GSMOS is a prediction of the pollu-
tion of water through the atmosphere locally (within the limits of a
region) in a time cross section, the characteristics used include only
the names of chemical substances subject to atmospheric transfer and the
accompanying meteorologic characteristics. If the purpose of the model
of atmospheric transfer is the spatial distribution of pollutants, we
must also know the geographic coordinates of the points.
If the self-purifying capability of a body of water in a preserve is
studied, we must know the names of virtually all of the hydrochemical
and hydrobiologic characteristics, and such secondary characteristics as
water temperature, speed of current, etc. The characteristics in the
CPC might include landscape zones, characteristics of the climate,
underlying soils and rocks, altitude above sea level and other charac-
teristics which depend on the task to be performed using the data col-
lected.
199
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4. Characteristics have various weights (Gaussian distribution). An error
in storage or recording of one characteristic leads to an error in the
interpretation of an individual word or small group of words, while an
error in another characteristic might result in the introduction of er-
rors to a large data file. For example, an error in the code "petro-
leum products" might cause a given concentration for a given date to be
omitted not only from petroleum products, but also from all other head-
ings. An error in the entry indicating the year of observation would
cause the loss of all observations for the entire year (they could not
be retrieved according to the characteristic of the year). Therefore,
the data storage unit must include a code protection system, the reli-
ability of which should be directly dependent on the weight of each
characteristic.
5. Various hydrochemical components have different precision in terms of
methods of analysis and natural variability. Therefore, the same fre-
quency of observations leads to different errors in reflection of natu-
ral trends from the observations (a space-time resolution of an obser-
vation system which is uniform with respect to all observed components
becomes nonuniform). This requires that as the observation system is
planned, various data reading intervals be assigned, leading to obser-
vations of equal accuracy. This step simplifies their processing and
assures the highest reliability of output materials with the minimum
cost of observation. However, the varying frequencies of observations
of different components leads to the situation that each sampling date
generates a different list of observed components. This does not allow
a rigid model to be used to record the results of observations.
For diffusion systems, it is more economical to record the results of
observations [x-j] together with the variable characteristics [d-jj] in
the form of a polynomial:
, lit nun
which can be subjected to various convolutions.
6. The first convolution is that the characteristics [d-jj] are removed from
the brackets, e.g., as follows:
T = dddX +d + ... + d d + dm+1
dx2 + ...)...).
This means in practice that for a given sampling data (d]), it is pos-
sible to record all the results of analysis of samples taken on this
date, without repeating the date next to each determination. The set of
data for a certain observation year ([x-j]) may have a single notation da
in the observation file header for the year. The file of data for the
entire period of observation for a given biologic observation station
may have a single header—the station code d2, etc. Obviously, each
information branch and each observation program should have its own
code.
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These convolutions may not achieve the global minimum of the volume of
data stored, since one criterion used in data reduction is convenience
of retrieval of the data necessary for performance of the main tasks.
Result: 6.1 To optimize decision making in the creation of a monitor-
ing AIS, technical requirements must be placed on the
GSMOS materials in terms of reliability, completeness,
timeliness, need, etc.
In reduction of characteristics, the names of hydrochemical components
require that a semi-rigid entry standard is used. The form carries the
results of analysis of not one, but a large number of samples; the
names of the components determined are entered into the head of the
form so that each form has a rigidly defined list of components, but
the heads of two different forms are nearly always different. This al-
lows the entry of one row of characteristics d, d1, d", ..., for the
entire matrix (n x N) of data, where n is the number of components in a
sample, and N is the number of samples. Secondly, binary coding is used
to index the names of the components defined (Tarasov 1971). The es-
sence of binary coding is that a list of names of hydrochemical compo-
nents is developed. They are assigned codes in the following sequence.
1, 2, 4, 10, 20, 40, 100, 200, 400, 1000, 2000, 4000...
The results of analysis are entered into the forms in accordance with
this list, while the codes are summed and a so-called code row is thus
developed.
Example. Lists of components and their codes
Oxygen 1
BOD5 2
Petroleum products 4
Phenols 10
Surfactants 20
NH4 40
N03 100
NO2 200
DDT 400
HCCH 1000
Cl 2000
Cu 4000
Zn 10000
Ni 20000
If the form contains:
201
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Sample
02 BODs Phen NH4 NOs N02 DDT Cl
1
2
3
X
X
-
X
X
X
-
X
X
X
X
X
X
X
X
X
X
X
X
-
™
-
X
X
XX X
the code row of the form consists of these names.
02 J
BOD 5 2
Phenols 10
NH4 40
NO 2 200
DDT 400
Cl 4000
4753
which, after addition, yields the code 4753. This value unambiguously
decodes the names of the elements in the matrix.
This coding method allows all combinations of 8 x 37 = 296 different
hydrochemical components to be stored in eight 37-bit memory locations.
These eight words of the code line relate to virtually 100 elements of
the matrix. Therefore, where n = 30 and N = 30, the amount of memory
expended in recording of variable characteristics (names of components)
is a fraction of 1 percent of the volume of useful information, and thus
reduces the volume of punching to less than half.
8. Since modern computer central processers operate very rapidly, while
peripheral information input-output devices operate comparatively
slowly, it is desirable to record only the initial quantities, produc-
ing the intermediate values in the computer by computation.
Taken together, all of these suggestions allow the volume of informa-
tion punched and stored in the computer to be reduced by approximately
a factor of 12, without loss of information contents. This greatly de-
creases the cumbersomeness of manual operations, input-output functions,
data search and retrieval operations.
9. Preparation of data for punching, the punching operation, review and
correction of data may require one or two orders of magnitude more time
than the actual machine processing. This requires a cautious attitude
toward the organization of operations. It is suggested that the initial
data be punched onto paper tape, allowing the use of a non-rigid form
for recording of the initial data, and allowing transmission of data
through communications channels. Since operator fatigue depends on the
time of continuous work (Figure 1), it is suggested that operations be
202
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ro
o
CO
CO
CC
o
QC
DC
UU
U_
O
DC
LU
GO
10 11
12 13
TIME
14
a.
15 16 17
Figure 1. Variation in number of errors v with time of continuous work t. a) Work with one
large break; b) Work with numerous shorter breaks.
-------
organized according to plan "B" which reduces the number of errors
significantly, and generally increases the productivity of labor.
10. Since visual testing of the results of punching using a control tape or
computer printout requires intensive attention, and the reading of
large numbers of correct entries unavoidably reduces attention, it is
suggested that double punching be used (by two operators independent-
ly), with the punch tapes checked by the computer. If the relative
number of errors for individual punching is p = 0.5 percent, the pro-
bability of simultaneous appearance of an error on two punched tapes
in the same location is p-, x P2 = 0.000025, while the probability of
appearance of an error in the same digit of a given word and the pro-
bability that the error in this word will be identical is negligible.
After machine verification of the two punched tapes for agreement, the
technician checks only the words which disagree and gives the keypunch
operator instructions for corrections of both punched tapes, after
which they are tested once more.
11. Manual entry of information produced on forms, punching, and recording
from the temporary information form to computer memory should be per-
formed without preliminary sorting of data, as the results of analysis
come in. This freedom levels the work load on testers, and allows the
process of systematization to be automated. The short period of time
between receipt of data and entry and testing of the data in the com-
puter allows some of the erroneous results of analysis to be cor-
rected.
12. Machine testing of information recorded in a form should include the
performance of logic and calculation procedures. Logical testing
checks such relationships as the number of hours (not over 24) re-
corded, such as the time of taking a sample, the depth of the body of
water (no less than the sampling depth), equality of the sum of anions
and cations, etc.
Calculation procedures presume 1) that the results of analysis will
fall within certain tolerance limits determined by retrospective exami-
nation of information, 2) that relationships between certain components
will be maintained in multidimensional statistical models, and 3) that
deviations from the approximating surface will be statistically hetero-
geneous, etc. As a result of machine testing, data are classified as
false or doubtful. The flagged machine results can be conveniently
printed in a form identical to the input form. A final decision as to
whether data flagged by the computer are bad or good, as well as cor-
rection of some of the bad data, are the job of a hydrochemical spe-
cialist, in combination with the analytic technician who performed the
analysis.
13. If nonsystematic punching of tapes is used (contents of forms), a tem-
porary catalogue (TC) should be made, containing the label "sample" and
the address of the location of the sample in peripheral storage. The
TC has been used for rapid retrieval of the necessary information for
204
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systematization of the data (according to dates, depths and sampling
locations). The identifying code used to mate the information recorded
and the data of the TC (and, subsequently, with the CPC) could be the
sampling location, e.g., the geographic coordinates of the station, the
point of a vertical line unambiguously defining the location of the
point where the information was read.
14. The sampling location may have various types of tie-ins depending on
the type of task being performed. For example, the geographic coordi-
nates, the distance from the mouth of a river, the distance from local
landmarks (bridge, cliff, waterfall, dock, etc.) may be used. The form
should contain one type of tie-in which can be numerically expressed,
e.g., geographic coordinates. The other types should be stored in the
CPC (the TC and CPC should be included in the identifying code).
15. Systematization of data before long-term storage or before use can be
performed by various systems. The selection of a plan for systematiza-
tion depends on the type of tasks performed, which should also be in-
cluded in the technical assignment for the AIS.
The first (or main) systematization should provide for convenient per-
formance of priority tasks. It is possible to store initial data using
several models (systems), each of which is convenient for its own range
of tasks. The expediency of storage using several models can be easily
established by technical and economic calculations.
16. In addition to the storage of initial data, it is also possible to
store summary results, if they are used for systematic performance of
certain tasks. The desirability of storing files of summary data must
also be determined by engineering-economic calculations.
17. To solve each class of problems, retrieval files (RF) of characteris-
tics are formed from the CPC. Each class of tasks should correspond
to its own RF. The RF also includes the necessary list of characteris-
tics based on the model, which is oriented to the performance of a
given range of tasks. All of this facilitates retrieval and processing
of the necessary information.
18. Systematized data for long-term storage are standardized, equipped with
service labels and stored in the form of magnetic or binary microfilm,
and standardized by means of a special data description language.
Models are developed using information compression techniques, leading
to significant savings of machine time.
19. Processing algorithms may include:
- estimates of states based on various lists of components (independ-
ently for each component), made up for fixed periods of observation;
- combined estimates of states based on the entire list of observed
components;
- comparative estimates of states at various observation points;
- determination of trends at various frequency levels of variability;
205
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- determination of local time anomalies;
- prediction of hydrochemical states;
- a combined estimate of states for all media to be controlled;
- transfer between media;
- identification of anomalies with sources of pollution;
- combined prediction for all media tested;
- warnings of increasing pollution and the possibility of unfavorable
after-effects.
These algorithms, presumed to be the purpose of organization of moni-
toring, are in various stages of development in the USSR.
20. The creation of the monitoring AIS requires the selection of intelli-
gent proportions for standardization of codes, the organization of data
bases, internal and external software, the class of computers used and
algorithmic languages. This will determine both the success of indi-
vidual units of specific subsystems of the AIS, and the effectiveness
of functioning of these subsystems. Therefore, their performance is of
primary significance in the task of mass production of mathematical
models (both particular hydrochemical models and general ecologic
models), which is the technical and theoretical basis of the problem of
storage and effective utilization of the resources of the biosphere.
REFERENCES
Tarasov, V.V. 1971. One method of information storage. Tr. NIIAK, No. 77,
Moscow.
Tarasov, V.V., V.V. Pugolovkin and V.L. Pavelko. 1974. Coding of informa-
tion for computer storage. Ekspress-informatsiya, Obninsk, No. 4(24).
206
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SECTION 12
A MULTI-LAYERED NESTED GRID MODEL OF LAKE SUPERIOR
G.J. Oman and M. Sydor 1
INTRODUCTION
Lake Superior is the largest, cleanest, deepest, and coldest of the
Great Lakes. The size and complex bathymetry of Lake Superior pose serious
obstacles when attemping to understand lake dynamics. Numerical modeling
and remote sensing provide relatively easy means of studying large bodies of
water. The art of numerical modeling has developed to the point where it
has been able to correctly predict large scale circulation patterns in the
Great Lakes. An understanding of the physical processes occurring in Lake
Superior will probably best be achieved by means of numerical modeling in
conjunction with remote sensing data and direct measurements made on the
lake.
A variety of numerical techniques exist for modeling large bodies of
water. Several authors have attempted to assess the state of the art in
numerical modeling of the Great Lakes (Lick 1976; Katz and Schwab 1976;
Cheng et^ aj[. 1976). In addition, techniques used in atmospheric and oceano-
graphic models have potential applications to the Great Lakes (Paul and Lick
1979). A review of existing three-dimensional Great Lakes models is under-
taken to find an appropriate model for Lake Superior. The decision was made
to first develop and verify an isothermal model before dealing with the much
more difficult task of simulating stratified flows.
GREAT LAKES MODELS
Simons (1973) developed a free surface model which he used to simulate
storm surges on Lake Ontario (Simons 1974, 1975, 1976) and in the Baltic Sea
(Simons 1978). Katz and Schwab (1976, 1978) also used Simons' model to pre-
dict currents in Lake Michigan. The model employs grid A+B shown in Figure
1, which consists of two independent grids staggered in space and time.
This grid permits use of centered differences in space and time for terms in
the equations of motion which govern seiche oscillations. Since both compo-
nents of velocity are defined at the same point, terms for the Coriolis
force can be evaluated using centered semi-implicit time differences which
^Department of Physics, University of Minnesota, Duluth, Minnesota 55812.
207
-------
B
k+3/2
k+1 <
k+i/2
<
-1/2
k-i <
t . t
)(
j (
k *i/o — • — <
k-tl
+1/2 *
5 .... if
kl fn
-1/2 • •
1
II
II
II
II
j-1 j-1/2 J J+12 i + I
A + B
k+3/2
k+i o
k+i/2
k o
k-i2
k-i o
H j-1 2 j j+l/i j+1 j+3/->
k-i
j-1 j
j+l
Figure 1. Grids used in various models.
208
-------
assure both numerical accuracy and computational stability. Forward time
differences are used for diffusion. This model permits any number of fixed
permeable layers of variable thickness and may be altered to employ moveable
impermeable layers.
Free surface models require very short time steps because they take into
consideration fast-moving gravity waves. In Simons' model, the computation
time is reduced by separating internal and external modes of oscillation.
This separation is accomplished by first summing the equations of motion
over the vertical axis to separate out the external gravity waves. Then,
appropriate differences between layers are taken to filter out fast moving
gravity waves. In this way, external gravity waves are integrated using a
short time step, while internal waves which determine vertical current pro-
files are integrated using a much longer time step.
This model treats both inertial and seiche modes of oscillation well.
Its main drawback is grid dispersion. The two solutions on subgrids A and B
of Figure 1 are only weakly coupled. Because the two subgrids are staggered
in space, the two nearly independent solutions provided by them may begin to
differ appreciably as time progresses. This possibility appears particular-
ly ominous for complex bathymetry.
Lick and his associates at Case Western Reserve University have deve-
loped a number of different models of lake currents (Lick 1976). Included
are steady state models (Gedney and Lick 1970, 1971, 1972; Gedney et al.
1973), free surface constant density models (Sheng and Lick 1975, Sheng
1975), and rigid lid models (Paul and Prahl 1971; Paul and Lick 1973, 1974,
1975). Sheng ejt aj_. (1978) later developed an improved free surface model.
The earlier free surface model used a single horizontal grid like grid A
in Figure 1. Centered space differences were used for all terms. A forward
backward time differencing scheme was used in which all terms in the momen-
tum equations were stepped forward in time and water levels were calculated
using the new transports. Internal and external modes were not separated
and advection of momentum was ignored. The more recent free surface model
uses grid C, shown in Figure 1. Internal and external modes are separated
and advection of momentum is included. This later model represents in-
ertial and fundamental seiche oscillations well. Two point averaging is
needed to calculate terms describing gravity waves. This averaging reduces
the effective spatial resolution of the model and could give rise to false
two grid space gravity waves. The four point averaging needed to calculate
vertical velocities might also poorly represent vertical advection over com-
plex bottom topography.
The rigid lid variable density model also employs grid C shown in Figure
1. The rigid lid condition sets the vertical velocity at the surface to
zero. Although evaluation of the surface is not permitted, surface pressure
is defined. Surface pressure is obtained by solving a Poisson's equation
derived from cross differentiating the momentum equations and applying the
rigid lid condition. Surface pressure corresponds to water level oscilla-
tions in free surface models, except that gravity waves are eliminated.
The rigid lid condition, in effect, filters out gravity waves. Thus, rigid
209
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lid models may use much longer time steps and require considerably less com-
putation time. Sheng et_ aj_. (1978) have shown for both constant and vari-
able winds that currents from rigid lid and free surface models display very
similar long term trends.
Both the free surface and rigid lid models developed at Case Western
Reserve employ the stretched coordinate system (Freeman et_ a]_. 1972). The
vertical coordinate z is transformed to the sigma coordinate a by dividing
by depth. At each grid point currents are given at specified values of a.
In the transformed coordinate system, each grid point have the same depth
and the same number of layers. This greatly reduces programming complex-
ities and assures adequate vertical resolution of currents at all points.
The main problems are poor resolution of surface layers in deep water and
in shallow water excessively thin layers which require implicit integra-
tion of vertical diffusion.
Another rigid lid model was developed by Bennett (1977). Bennett's
model employs a single grid lattice like grid A in Figure 1. The momentum
equations are cross-differentiated using the rigid lid condition giving an
equation for the stream function. The momentum equations are stepped for-
ward in time, ignoring surface pressure. The new approximate currents are
used to calculate a new stream function. The currents are then corrected
for the neglection of surface pressure by using the new stream function. In
this way, only one array is needed for the stream function and each velocity
component. Since the two components of current are not defined at the same
point in Bennett's grid, four point averaging was necessary to calculate the
Coriolis force. This averaging gave rise to spurious inertial modes which
had to be controlled using excessively high horizontal diffusion.
Kizlauskas and Katz (1974) developed a two layer model of Lake Michigan.
The two layers which represent the epilimnion and hypolimnion are separated
by a moveable impermeable surface. This model would only be applicable
after lake stratification.
In addition to the models mentioned above, numerical techniques devel-
oped in simulating flows in estuaries, oceans, and atmospheres have poten-
tial application in the Great Lakes. In particular, semi-implicit time
integration techniques (Madala and Piacsek 1977; Kwizak and Robert 1971)
show promise for the Great Lakes (Paul and Lick 1979). Semi-implicit
schemes are much more complicated but allow much longer time steps and pro-
duce much smoother results. With a few modifications, a three-dimensional
estuary model developed by Leendertse (Leendertse et. _al_. 1973, 1975a, 1975b)
would also be applicable to the Great Lakes. This model employs two grids
like grid A in Figure 1 staggered in time. A pressure gradient averaging
technique used in meteorological models (Schoenstadt and Williams 1976;
Brown and Campana 1978) could also be used to almost double the maximum al-
lowable time step used in explicit, centered time integrations.
210
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THE MODEL
In developing a numerical model of Lake Superior, a number of points had
to be taken into consideration. Since model verification would be based
largely upon recorded water levels and remote sensing observations of major
and minor storms, a free surface model seemed more appropriate. Rigid lid
models are more useful for modeling extended periods of time, i.e., several
weeks or more. At the present time, such an effort would be hard to verify.
The complex bathymetry and highly variable thermal structure of Lake
Superior can be handled more conveniently with layers of fixed depth, let-
ting only the bottom layer vary in thickness. When using layers of fixed
depth, the surface layer also corresponds more closely to the layer seen by
remote sensing. Implicit and explicit time differencing should be con-
sidered. At the present stage of development, only explicit techniques have
been considered. Finally, in order to assure adequate spatial solution in
regions of interest, either a variable grid or a nested grid model should
be used. The latter is used in the present model.
A number of explicit numerical integration techniques exist for solving
the equations of motion for geophysical fluids. A number of authors have
analyzed these various methods (Fischer 1965; Grammeltvedt 1969; Matsuno
1966). These various techniques have desirable smoothing properties, but
none of them are more accurate or more efficient than centered time dif-
ferencing. Computational modes may develop with centered time differencing,
but these can be controlled using half time step starting procedures and
periodic explicit smoothing operators. In addition, a pressure gradient
averaging technique proposed by Shuman nearly doubles the maximum allowable
step for centered differences.
A lattice like grid A in Figure 1 permits centered differences in both
space and time for terms associated with gravity waves. If a single grid is
used, centered differences for the Coriolis force are not possible. Forward
differencing of this term can lead to computational instability (Fischer
1959). A second identical lattice staggered in time is often used to solve
this problem. In Simon's model, this second lattice was also staggered in
space. This method raises the possibility of grid dispersion. In the pre-
sent model, the second grid is staggered in time but not in space. This is
the grid structure used by Leendertse (1973). Simons also used tins struc-
ture in modeling the Baltic Sea (Simons 1978). As Simons pointed out, the
main problem with this grid is the development of spurious inertia! modes
associated with four point averaging used in calculating the Coriolis terms.
This can be controlled by increasing horizontal diffusion. Another possi-
bility would be to use grid C of Figure 1 which would permit centered semi-
implicit evaluation of the Coriolis terms. However, centered time differ-
ences for gravity waves seem to lead to computational difficulties on this
grid (Sheng, personal communication).
The Lake Superior model is similar to Leendertse's estuary model
(Leendertse e_t aj_. 1973, 1975a, 1975b). It ignores vertical accelerations
and assumes pFessure varies hydrostatically. It has a free surface and uses
constant eddy coefficients to represent vertical and horizontal diffusion.
Horizontal and vertical advection of momentum are represented using an
211
-------
energy conserving formulation. Centered space differences are used for all
terms. Centered time differences are used for all terms except horizontal
and vertical diffusion which are approximated using forward differences. A
quadratic bottom friction is used.
The equations are given by Leendertse and are listed in Appendix A.
Five important modifications were made to Leendertse's model. First, the
model was made isothermal. Leendertse's complex formulations for vertical
momentum exchange was replaced by simple vertical diffusion with constant
diffusivity. Second, the depth of the bottom layer was allowed to vary.
Following Simons (1973), the equations of motion were integrated from
either the bottom of the lake or the depth of a given layer to either the
surface or the bottom of the next layer. Third, a pressure gradient aver-
aging technique originally proposed by Shuman was added. Fourth, internal
and external modes of oscillations were separated. Fifth, provisions were
made for saving model parameters along open boundaries to be used later on
nested subgrids.
Shuman pressure gradient averaging is a very simple technique capable of
nearly doubling the maximum allowable time step for centered time differ-
encing of barotropic waves (Schoenstadt and Williams 1976; Brown and Campana
1978). In the equations of motion, time centered pressure gradient terms
associated with water level oscillations are replaced by a simple average
over three consecutive time s^ps. For example, at the nth time step the
pressure gradient in the x-direction Px is normally given by Equation (1) in
which £ is water level and y the acceleration of gravity.
Pn = g(
x yvgx
(1)
This is simply replaced by Eauation (2) below, where 8 is a constant which
may vary between 0 and .25 che case of no advection of momentum
3£\n
g[(1-2. x a)()
a
n-1
+ (i-^
(2)
The term (-r-) is available at time step n because in centered time dif-
ferencing water levels at step n+1 are derived from vertical velocities
at step n and water levels at step n-1.
The separation of internal and external modes was accomplished as fol-
lows. First, the equations of motion were integrated from bottom to surface
in^order to get an equation which included the effects of gravity waves.
This external oscillation mode was integrated using a very short time step.
Next, equations of motion at consecutive layers were subtracted to eliminate
the external gravity wave. For example, consider the x components of velo-
city Uk and Uk+-| at the k and k+1 layers. The equation of motion can be
written
= R,
(3)
212
-------
where t represents time, h layer thickness, g acceleration of gravity, C
water level, and R the remaining terms. Multiplying (3) by hk+1 and (4) by
hk and subtracting eliminates the pressure gradient term and hence the
gravity wave. Replacing the time derivatives by finite differences gives
- <; n
- Sk
where n refers to the time step. The form of S|< depends on the kind of
time differences used for the internal wave. In any case, it is evaluated
using values at time step n and earlier. If there are M layers, this pro-
cedure gives M-l equations in M unknowns. The equations used to derive the
velocity shears are stepped forward once using an internal time step which
is an integral number N times the time step used for the external depth-
integrated model. The external mode is then stepped forward N times to give
the total transport T at the next internal time n+1 . The provides an MiQ-
equation in Uk11"1"1 , namely.
= T (6)
Equations (5) and (6) can be readily solved by elimination for U|<. Bottom
friction is then calculated at the new internal time step and used in cal-
culating the external mode. In this manner internal and external modes are
treated separately and efficiently while still being allowed to interact
with one another.
The addition of open water boundaries and nested subgrids introduces a
completely new set of problems and modeling decisions. Upon reaching the
boundary between coarse and fine grid, waves can reflect and generate false
computational modes or false gravity waves. Proper formulation of open
water boundary condition can prevent this. Grids of differing size can be
coupled using either one-way or two-way interaction schemes. In one-way
interaction schemes (Chen and Miyakoda 1974; Miyakoda and Rosati 1977) the
entire lake is first modeled on a coarse grid ana model parameters on the
boundary between grids are saved. The saved parameters are interpolated in
space and time and used as open boundary values in the fine grid. In this
method, information is allowed to pass from the coarse grid to the fine grid
but not vice versa. In two-way interactions (Harrison and Elsberry 1972;
Browning et_ aj_. 1973) both fine and coarse grid models are integrated simul-
taneously. At the boundary between grids, information needed by the coarse
grid from the fine grid is obtained using values from the fine grid and vice
versa. In this way, full interaction between grids is possible.
In order to reduce computer memory requirements, the one-way interaction
scheme is used in the Lake Superior model. Two-way interaction schemes are
necessary only when processes occurring in the nested subgrid have profound
213
-------
influences on large scale circulation, as for instance in hurricane model-
ing. In the program the location of any number of open boundaries are read
and model parameters along these boundaries are saved for latter fine grid
calculations. These stored model parameters are then interpolated in time
using a four-point formula given by Wang and Halpern (1970). Space inter-
polation is achieved using a method which approximates linear interpolation
of currents but assures that the total normal and tangential transport and
the total surface elevation along the boundary is the same in both grids. A
local boundary smoothing procedure proposed by Chen and Miyakoda (1974) can
be used if necessary.
EXAMPLE: RESULTS FOR LAKE SUPERIOR
For purposes of illustration the model was run for a 24 hour periods for
easterly and westerly winds on Lake Superior, Figure 2. A 10 km square
grid and time steps of 12 minutes for the internal mode and 90 seconds for
the external mode were used. Vertical and horizontal diffusivities were .01
m2/s and 10 m2/s respectively. A constant wind stress of .675 nt/m2 and a
constant Chezy coefficient of 50 (m/s2)-l/2 were used. In these runs, four
layers were used with interfaces at depths of 20 m, 50 m, 110 m.
Figures 3-6 show surface currents for easterly winds after 3, 9, 15,
and 24 hours respectively. The currents in the central regions of the lake
are clearly rotating in a manner attributable to interaction of wind stress
and Coriolis force. As expected, currents in the lower layers show a simi-
lar rotation in the opposite direction. Figures 7-9 show currents in the
subsurface layers after 24 hours. Return currents are clearly evident in
deep trenches parallel to the wind. The results for westerly winds were
similar except current directions were reversed. Downwelling and upwelling
for easterly and westerly winds are shown in Figures 10 and 11 respectively.
Comparing Figures 8 and 9 with Figure 10 provides evidence that for constant
easterly winds, taconite tailings dumped into the lake at Silver Bay would
be downwelled and carried along the bottom toward the south shore where they
would be upwelled. Isolated patches of tailings have been detected in this
region using Landsat data (Oman and Sydor 1977).
The model was also run on a 2.5 kilometer nested subgrid in extreme
western Lake Superior. Surface and second layer currents for easterly winds
after 24 hours are shown in Figures 12 and 13 and for westerly winds in Fi-
gures 14 and 15. For both winds, surface currents in the extreme western
reach of the lake seem to form circulation cells. These cells seemed to
reach steady state quickly and are consistent with observations by
Ruschmeyer et_ a]_. (1961) which indicated that nutrients from the Duluth-
Superior Harbor often remain trapped within 15 to 30 km of the harbor for
extended periods of time. The return current in layer 2 for easterly winds,
Figure 13, corresponds exactly with resuspension plumes observed in Landsat
images. The hooking return current in layer 2 for westerly winds is also
consistent with Landsat observations of westerly wind events.
214
-------
IX)
(—»
en
N
KM
Figure 2. Map of Lake Superior showing regions deeper than 200 m.
-------
no
i—'
cr>
East Wind
Currents
One Grid Space = 50-0 cm/sec
Layer 1, 0 to 20.0 meters
Time = 0/0300:00
Figure 3. Surface currents after 3 hours of constant easterly winds.
-------
ro
East Wind
Currents
One Grid Space = 50.0 cm/sec
Layer 1, 0 to 20.0 meters
Time = 0/0900:00
Figure 4, Surface currents after 9 hours of constant easterly winds.
-------
ro
co
East Wind
Currents
One Grid Space = 50-0 cm/sec
Layer 1, 0 to 20-0 meters
Time = 0/1500:00
Figure 5. Surface currents after 15 hours of constant easterly winds
-------
ro
East Wind
Currents
One Grid Space = 50-0 cm/sec
Layer 1, 0 to 20-0 meters
Time = 1/0000:0
Figure 6. Surface currents after 24 hours of constant easterly winds.
-------
East Wind
Currents
One Grid Space =25.0 cm/sec
Layer 2, 20-0 to 50-0 meters
Time = 1/0000:00
Figure 7. Currents in layer 2 after 24 hours of constant easterly winds,
-------
(Ni
ro
East Wind
Currents
One Grid Space ~ 25-0 cm/sec
Layer 3, 50 to 110-0 meters
Time = 1/0000:0
Figure 8. Currents in layer 3 after 24 hours of constant easterly winds,
-------
Figure 9. Currents in layer 4 after 24 hours of constant easterly winds.
-------
ro
r\3
CO
Figure 10. Layer 2 downdwell ing in western Lake Superior after 24 hours of constant easterly
winds. One grid space corresponds to 1 mm/sec.
-------
ro
ro
Figure 11. Layer 2 upwelling in western Lake Superior after 24 hours of constant easterly
winds. One grid space corresponds to 1 mm/sec.
-------
ro
ro
en
EAST WIND
CURRENTS
ONE GRID SPACE = SO CM/5
LAYER 1, 0 -20 METERS
THE 1/QOOO:00
Figure 12. Surface currents in nested subgrid after 24 hours of constant easterly winds
-------
cr>
EAST WIND
CURRENTS
ONE GRID SPACE = 25 CM/S
LAYER 2, 20-50 METERS
TIME 1/0000--00
Figure 13. Layer 2 currents in nested subgrid after 24 hours of constant easterly winds,
-------
WEST WIMD
Figure 14. Surface currents in nested subgrid after 24 hours of constant westerly winds
-------
ro
ro
CO
ONE GRID s= 25
LAYER 2, 20-50
TIME 1/0000:
''..•..•:•: <<•>'>
Figure 15. Layer 2 currents in nested subgrid after 24 hours of constant westerly winds.
-------
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232
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APPENDIX A
THE FINITE DIFFERENCE EQUATIONS
The finite difference equations listed in this Appendix are based large-
ly upon the work of Leendertse (1973, 1975). The equations of motion for an
incompressible fluid are given in Equations (A-l)-(A-4). In these equa-
tions, vertical accelerations have been ignored and the hydrostatic assump-
tion imposed; Equation (A-3). The Coriolis parameter is also assumed con-
stant. As Simons (1973) points out, these equations can be integrated over
the kth layer to give Equations (A-5)-(A-7).
If layer thickness varies, terms involving products of the horizontal
velocity and partial derivatives of layer thickness are absorbed into the
vertical velocity term w. In these equations, the horizontal stress terms
are approximated by simple diffusion of layer transport. The vertical
stress term at the surface is given by a quadratic function of wind, Equa-
tion (A-8), and at the bottom by a quadratic function of current, Equations
(A-9)-(A-10). Stresses between layers are approximated by simple diffusion.
Before giving the finite difference equations, the finite difference
operators listed in Equations (A-ll)-(A-15) must be defined. First, all
model parameters must be defined on a grid lattice as in Equation (A-ll).
In the present model, a spatially staggered grid is used which is ideally
suited for solving the continuity equation. In this grid only normal compo-
nents of velocity are defined at the six sides of a rectangular box which
constitute a grid cell. Thus, the u, v and w components of velocity are de-
fined at half integral values of i, j and k respectively. Averaging and
finite differencing operators in the x-direction are given in Equations
(A-12) and (A-13). Similar operators are defined in the Y and Z directions
and in time. Future and past values are signified as shown in Equations
(A-14) and (A-15).
The finite difference equations which form the basis of the model are
listed in Equations (A-16)-(A-19). Stress terms between layers are given by
Equations (A-20) and (A-21). These equations were then modified as des-
cribed by the main text.
3u s(uu)
9t ~9x~~
233
-------
av + aw + ^ + ajvjl + fu + ] | . J (V + B + p, . „ (A.2)
2E + pg = 0 (A-3)
iM + 1 • » <*-<>
Where: t is time
x, y are coordinates in the horizontal plane
z is the vertical coordinate, positive upwards
u, v, w are velocity components in the x, y, z directions
f is the Coriolis parameter
p is the density of the fluid
p is pressure
T represents the stress tensor
g is the acceleration of gravity.
fhv
^
1X7 1X7 1 8(hAX S) 1
TX X2 ' X dX '
T p
+ (Hy)kJk . (wv) + fhu
'k+% ^p L >k-Jg p 9S p — ~ u vn'u'
~ (hv) + (w)k_% - (w)k44g = 0 (A-7)
where h is layer thickness, Ax, Ay are horizontal diffusion coefficients.
TQ = PCu"|u~ (A-8)
Where: _p is the density of the moving fluid,
u" is the mean wind speed, and
C is a drag coefficient.
234
-------
^^p
(A-9)
= pg r
r
(A-10)
F.. = F(iAx, jAy, h, , nAt)
(A-ll)
i+^k + Fi-k) at (^J.
(A-12)
(l.j.k)
(A-13)
F = F
n+1
ijk
n-1
ijk
(A-14)
(A-15)
at i, j, n
(A-16)
6t(hXu) = - 6x(hxu ux) - 6 (Fv
hx6z(1Jzwx)
- g
at i + k, j, k, n (A-17)
235
-------
at 1, j + %, k} n (A-18)
5x(hxu)
6y(Fyv)
at i, j, k, n + 1 (A-19)
xz-
at i
(A~20)
at i j
(A-21)
236
-------
SECTION 13
A REVIEW OF SOME METHODS AND PARAMETERS USED IN ASSESSING EFFECTS OF
WATER INTAKES ON FISH POPULATIONS
Richard L. Patterson1
INTRODUCTION
It may be difficult to imagine that a water intake could have any effect
whatsoever upon the status of a fish population. Unlike fishermen who move
from place to place to improve their catch wherever it may be found, or un-
like low temperatures or low dissolved oxygen concentrations which exert
their effects over wide areas on survival of all eggs and larvae, a water
intake remains in a fixed position and has the single function of removing
water from the river, lake, reservoir, or estuary. It does not pollute the
waters around it and the only organisms that are affected in any direct way
are those which happen to find themselves so close to the mouth of the in-
take that they are sucked into the intake canal. It would therefore appear
that the only possible direct effect of a water intake upon a population
might be measured in terms of total numbers of organisms removed by the in-
take or in terms of percentages of populations removed by the intake in a
given year. However, direct effects upon sections of the aquatic habitat
may conceivably be triggered by withdrawal of excessive quantities of water
which, in turn, may effect recruitment or survival of the species inhabiting
the area. If a water intake exhibits size or species selectively in its
cropping of individuals the possibility of imbalance in food chains might be
considered as an indirect effect. The losses in new recruits in subsequent
years due to cropping of individuals in the present year is another type of
indirect effect that may be important for a given population. Consequently,
the question of impact of a water intake upon fish populations becomes a
matter for investigation. Reviewed below are some population parameters and
methods which are used to assess direct and indirect effects of water in-
takes on fish populations. Certain of the techniques are not widely known
due to the recent emphasis in assessing point sources of losses of fishes
from their habitats. Although the natural assumption may be that impacts of
water intakes are "bad" there is nothing inherent in the methods or para-
meters reviewed below that requires possible impacts to be detrimental to
the populations.
^School of Natural Resources, University of Michigan, Ann Arbor, Michigan.
237
-------
NEED FOR HISTORICAL REVIEW
Aquatic populations of the Great Lakes have experienced the effects of
man's activities for the past one hundred twenty years. Wells and McLain
(1973) and Smith (1972) have described the historical trends in Great Lakes
fish stocks depressed by (a) exploitation due to commercial fishermen, (b)
introduction of exotic species, (c) accelerated eutrophication, and (d)
other forms of pollution due to a variety of causes related in one way or
another to industrialization and shoreland and shoreline development. When
evaluating the impacts of a particular factor of fish mortality (such as
power plant effects) on a fishery, this factor should be viewed within the
context of the larger set of factors whose cumulative net effects produced
the environmental conditions surrounding and mitigating the effect of the
factor to be evaluated.
Applying the above to the evaluation of effects of water intakes as, for
example, power plant cooling water intakes which cause entrainment and im-
pingement of yellow perch populations of western Lake Erie, commercial catch
data reported by Muth (1977) show the perch population to be in a depressed
condition in 1977 with production having declined each year from 1973-1976.
Moreover, analyses of catches showed that about 60 percent of the catch was
less than the 8.5 inch minimum size recommended by the Lake Erie Technical
Committee on Yellow Perch as a management strategy to protect the stocks.
The Ohio Division of Wildlife (1977) also reported that the majority of the
1973 year class in the Western Basin (due to slow.er growth) was sublegal.
Except for increases in 1971 and 1972 yellow perch production in Lake
Erie has declined steadily since 1968. Between 1952 and 1968 production
fluctuated considerably but the overall trend was upward. Jobes (1952) re-
viewed yellow perch production in Lake Erie prior to 1952 and found it to
be relatively low compared to the 1955-1968 period except for a few years
of high production in the 1930's. The effects and implications of power
plant induced entrainment and impingement mortality upon yellow perch in
western Lake Erie beginning in the early 1970's should therefore be inter-
preted in light of a set of factors whose net effect produces a declining
population (due most likely to a combination of overexploitation and ad-
verse habitat conditions resulting in poor year classes in recent years).
Such a trend suggests that the compensatory reserve of the population may be
exhausted. That is, if an additional amount of the population is removed
each year beginning in a given year, the population reaches a point at which
it is unable to recover any part of this added loss. A declining population
in which small sizes predominate in the catch suggests that compensation for
additional removals of stocks whether by fishing, natural mortality, or en-
trainment and impingement may be slight at best. Additional losses to a de-
clining population could project it into a prolonged or permanent state of
depression unless stresses are relaxed. Yellow perch in the Great Lakes ap-
pear to be capable of recovering from severe stress conditions when those
conditions are removed so that the possibility of its extinction is not
seriously considered.
238
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WATER INTAKES AS A FACTOR IN FISH POPULATION DYNAMICS
A box and arrow diagram, depicting flows of numbers or biomass of a fish
population (Figure 1), is a convenient tool for showing the standing crops
of subpopulations and their transfers which should be taken into account in
order to conduct analyses of direct (same year) and long run (subsequent
years) impacts of water-intake caused entrainment and impingement losses to
a fish population and its associated fisheries. Direct effects of entrain-
ment and impingement mortality refer to immediate (same year) losses sus-
tained by age or size classes both separately and collectively including the
egg state. Direct effects also include percentage loss of larval production
and percent reduction of the young-of-year class caused by entrainment
losses to the larval stage. An assessment of direct effects is complicated
by (a) the need to obtain larval production and natural mortality estimates
as well as estimates of standing crops of other size classes and (b) the
differential impact that a particular water intake may exert if it selec-
tively crops size classes and the implications of this differential cropping
of size classes upon the percent reduction in the population.
Longer range or subsequent year-to-year effects are caused by losses of
fishes in various size classes in earlier years. Such subsequent year ef-
fects may occur in the form of reduced numbers in different size classes,
reduced yields, changes in production rate of larvae per individual adult,
and changes in the natural mortality rate of larvae. Longer run effects may
be termed indirect effects "to emphasize their separation in time from im-
mediate or direct effects. Long run predicted effects of water intake
caused mortality may or may not be verifiable from direct measurements of
standing crops in subsequent years, being a particularly difficult because
of (a) the confounding of possible multiple causes of reduced population,
and (b) the high statistical variability of estimates of standing crops.
Methods of predicting long run impacts of water intake caused mortality re-
quire the application of numerical simulation models of a fish population of
which there are a number of different types in use. They can be lumped into
two groups depending upon whether they simulate the differences in a popula-
tion (with and without the presence of water intake caused mortality)
directly in which case they are called equivalent adult models or whether
they simulate the fish population without the presence of water intake
mortalities and again with additional losses caused by water intakes. The
use of numerical simulation models requires assumptions concerning fish be-
havior and estimates of parameters such as female fecundity and compensatory
reserve of the population. Therefore, the assessment of long run impacts
of water intake caused mortality upon a fish population has all the problems
of assessing direct effects plus the additional ones mentioned above.
REVIEW OF FACTORS NEEDED FOR ASSESSING DIRECT EFFECTS
Mortality Rates of Entrained Larvae
Physiological effects of chemical, thermal, abrasive, pressure and shear
stresses encountered by individuals during passage through the power plant
cooling cycle have been investigated and reported at various symposia.
239
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EGGS
ro
yjy
LARVAE
x_y
YOUNG-OF-YEAR
c
1-YEAR OLDS
J>
c
2-YEAR OLDS
*v^r
• ^—r
C
3-YEAR OLDS
•VJ<
C
M-YEAROLDS
v_y
NATURAL
MORTALITY
INTAKE
ENTRAPMENT
MORTALITY
INTAKE
IMPINGEMENT
MORTALITY
FISHING
MORTALITY
Figure 1. Box and arrow diagram summarizing transfers of fish biomass.
-------
(Saila et^aj_. 1975; Jensen et^ a_l_. 1977). These studies provide understand-
ing of causes of mortality of individuals and also provide a basis for esti-
mating the mortality rates of eggs and fishes of various sizes which are en-
trained or impinged. Laboratory studies have shown that mortality rates can
be highly variable, ranging from less than 20 percent to nearly 100 percent.
The percents mortality of entrained and impinged individuals of various size
classes become important when coupled with entrainment and impingement rates
ranging into millions of individuals per year. Mortality rates due to en-
trainment and impingement must be studied on a plant by plant basis unless
100 percent mortality of all individuals is assumed. Mortality rates of en-
trained individuals can be estimated by a number of formulae, perhaps the
most common being the difference between the percent alive at entry into and
discharge from the cooling cycle:
Percent mortality = pD - pE (1)
Where: PQ = percent alive at discharge
PE = percent alive at entry
For small sample sizes at either discharge or entry the standard error
of the estimate of the difference of two fractions is large. Other problems
arise in estimating pp and pp. Individuals are killed during capture and
identification and Equation (1) assumes that this effect is cancelled out by
virtue of the difference. Bias in the estimates of pn or p^ or both can oc-
cur due to stratification of larvae either horizontally or vertically in the
water column, by the use of sampling gear which tends to clog or operate
only intermittently or by sampling infrequently during a day or week. Each
of these possible causes of bias in the estimates of p^ and pp create prac-
tical problems if they are to be eliminated. Laboratory experimental re-
sults are not a sufficient substitute for field sampling to establish en-
trainment mortality rates at an individual power plant intake.
Size Distribution of Entrained Larvae
Equation (1) classifies larvae of all lengths into a single category.
That is, all individuals from approximately 5-35 mm are lumped into a single
class called "larvae" for purposes of computing pD and p£. While this pro-
cedure may be acceptable due to the difficulty of obtaining samples of suf-
ficient size, it is important to measure lengths of individual larvae in or-
der to estimate the size distribution of entrained larvae. It is easily de-
monstrated mathematically (Patterson 1979) that if larger (older) larvae
have a higher survival rate to the juvenile stage of development, their
losses have stronger implications for the population as a whole than does
the loss of newly hatched larvae. A second important reason for obtaining a
size distribution of entrained larvae is to compare it to the size distribu-
tion of larvae in source waters, so that it can be determined whether selec-
tive cropping of larvae by the water intake is taking place. If it is con-
cluded that selective cropping of larvae by size class is occurring then the
sample of larvae used to estimate total numbers entrained will produce a
biased estimate of the fraction of total larval production (or total stand-
ing crop) that is entrained.
241
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Total Numbers of Entrained Larvae
The numbers, E(t-j,t2), of larvae entrained in a water intake is equal to
E(tpt2) = £ Cp(t)-Qp(t)dt
(2)
Where: Cp(t) = concentration of larvae in water entering
the intake at time t.
and Qn(t) = water flow through the intake at time t.
Records of Qp(t) are usually maintained and so this factor may be determined
relatively precisely. Determination of Cp(t) is another matter and for
power plants with large intakes one may expect considerable error in esti-
mating the mean concentration of larvae in the water column over even a
short interval of time.
directly are:
Sources of error in the estimation of Cp(t)
1. Stratification of larvae both vertically and horizontally in the
intake canal the configuration of which may vary over time.
2. Random variation in larval densities within the water column.
3. Passage of large clumps of extremely dense larvae through the
intake without being detected.
4. Clogged or inoperable sampling gear.
Direct estimation occurs by measuring larval densities at the point at which
water enters or leaves the plant. Since water intakes can be very large in-
deed (10 m x 10 m cross sectional area) the impossibility of sampling more
than a fraction of one percent of the water becomes obvious.
Indirect estimation of Cp(t) occurs by measuring Cv(t), larval densities
in source waters (rivers, lakes, or estuaries) and assuming that
cp(t) =
Where:
•cv(t)
f = ratio of
water to
water.
(3)
mean concentration of larvae in intake
mean concentration of larvae in source
The parameter f is important when concentrations in the water intake must be
estimated using indirect methods and upon occasion has become the object of
much debate (Oak Ridge National Laboratory Report 1977). When source waters
are divided as, for example, in the case of the Monroe, Michigan power plant
Cv(t) is a weighted sum of mean concentrations in each source. Various
methods have been used to estimate Cv(t). Hubbell and Herdendorf (1977)
used four methods for estimating Cv(t) which differed in the way in which
larval concentrations in depth zones of the source waters were averaged and
242
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the lengths of time that larvae were assumed to be present. In each case
they assumed a value of 1 for the ratio f so that Cp(t) = Cv(t). Patterson
(1979) estimated total numbers of yellow perch larvae entrained at the
Monroe power plant in 1975 using three sets of data: (a) larval concentra-
tions in the Raisin River channel downstream from the water intake, (b)
larval concentrations at the point of discharge of cooling water from the
plant, and (c) larval concentrations in lake waters near the mouth of the
Raisin River. Applying Equations (2) and (3) using concentrations obtained
from data sets (a), (b), and (c) Patterson obtained two estimates of f:
0.60 and 1.36. An average estimated value of 0.98 was thus obtained for f.
Total Number and Size Distribution of Impinged Individuals
Impingement occurs when individuals too large to filter through a
screening device are caught and held against the device by the force of the
incoming water. Impingement is estimated directly by counting the number of
individuals accumulated in collecting pans or other collection hardware over
a fixed time interval. As in the case of entrained larvae it is important
to obtain the size distribution of impinged fishes for two reasons: (a) so
that a determination of whether the impinged individuals are representative
of the population as a whole, and (b) so that survival rates of fishes that
are impinged and later returned to the source wafers can be estimated by
size class. The latter reason is particularly important because much effort
is being devoted by power plant management to returning impinged fishes to
source waters in a viable condition. Therefore if mortality occurs, for
example, to selected size classes only, the end result amounts to selective
mortality among adults induced by the power plant.
Since a complete census of impinged fishes is not taken problems of
estimating total numbers impinged based upon sampling techniques arise.
Statistical sampling designs for estimating total impingement have been dis-
cussed by Murarka and Kumar (Jensen 1977, Pages 267-289).
Percentage of Annual Larval Production Lost Through Entrainment
The percentage of larval production lost through entrainment mortality
can be most directly and quickly estimated as:
Percent lost in week i = 100 ( ,- (4)
\v
Where: Ej = estimated number of larvae entrained in week i.
A.,- = estimated abundance of larvae in source waters in
week i.
If length data are not recorded on entrained and standing crops of larvae
then both E^ and A.,- are recorded in terms of total numbers only. A bias in
the estimate of the percentage will then occur if the length distributions
contained in E,- and A,- are significantly different. In addition, Equation
(4) will overestimate percent of production lost due to entrainment because
243
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abundance is less than production. This bias can be reduced or eliminated
by replacing Equation (4) by Equation (5):
I E.
Percent lost in week i = 100
stl-mated annua proucion
Where: E-j = estimated number killed due to entrainment in
week i. And estimated annual production is ob-
tained by some method such as Patterson's or
Pol gar's.
Used as an indicator of water intake effect the percentage of annual larval
production lost due to entrainment has the advantage of not requiring as-
sumptions of any kind about the parent population and can be calculated from
direct measurements obtained in the field. It has the disadvantages of sus-
ceptibility to bias and lack of interpretability in terms of implications to
the adult population in source waters.
Percent Reduction in Number of Young-of-Year Recruits Due to Larval Entrain-
ment Mortality
The percentage by which the number of young-of-year recruits is reduced
due to entrainment of larval fishes is estimated as:
Rl
Percent reduction in number of y-o-y recruits = 100 1 - A • • ' -ri (6)
Where: R] = number of young-of-year recruits in source waters
in the presence of water intake operations.
And R2 = number of larvae killed due to entrainment that
would have otherwise survived to reach young-of-
year age.
The estimate of R ] is:
/\
R] = (estimated number of larval produced but not entrained) x
(e-D-P)
Where: D = number of days in larval stage
And p = mean daily instantaneous natural mortality rate of
larvae.
An alternate estimate of R-j can be obtained by estimating young-of-year
standing crops directly and correcting for natural mortality of young-of-
year to produce an estimate of number of y-o-y recruits.
The estimate of $2
244
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R2 = (estimated number of larval killed due to entrainment) x
Q
/ ' (x) *6 "dx (8J
o
Where: x = age of larvae at entrainment
f(x) = relative frequency of larvae at age x when
entrained.
If length data are not obtained_ on entrained larvae (from which ages can
be estimated) than an average age d of larvae at entrainment must be used
and Equation (8) becomes:
/\
R2 = (estimated number of larvae killed due to entrainment) x
^(D-d)-p (9)
In contrast to Equation (5) the use of Equation (6) as an indicator of water
intake effect on the population projects larval mortality into a later stage
of physical development. If survival rates in stages subsequent to the lar-
val stage are known then it is theoretically possible to estimate percent-
ages by which later life stages are reduced as a result of larval entrain-
ment. However, the requirement of additional parameter estimates (survival
fractions for life stages) renders all such estimates more and more unreli-
able. Equation (6) only accounts for a projected equivalent loss of young-
of-year fishes due to entrainment. It does not consider number of young-of-
year fishes entrained or impinged directly and consequently does not
evaluate total loss to the young-of-year class. By projecting the number of
larvae lost due to entrainment through several subsequent stages of physical
development and applying survival factors at each stage to obtain estimates
of equivalent projected losses at later stages (in later years) it is ob-
vious that these estimates are no longer direct effects. Whether they can
be considered estimates of population impacts depends upon whether they are
projected to occur on a regular annual basis so that a long run change in
the population and fishery can be projected.
Percent Reduction in Size or Age Classes Due to Entrainment and Impingement
Losses
The number of individuals in each and every size or age class from
young-of-year through adult stages killed as a result of entrainment and im-
pingement can be estimated if the required data are obtained. Length dis-
tributions must be estimated together with total numbers impinged/entrained.
If some methods is used to return impinged fishes to the source waters in a
viable condition then impingement mortality must be adjusted. If estimates
of standing stocks of the population are available by comparable length or
age classes then estimates can be computed of percent reduction in standing
stock by size class due to impingement mortality.
Vulnerability of Larval Population to Entrainment
The vulnerability of a larva to entrainment is the chance that it will
be entrained during its life time (in the larval state). Larval vulner-
245
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ability depends upon the geomorphometry of the basin of the source waters,
advective and dispersive flow characteristics of the source waters, length
of life of the larvae, and flow of water into the intake. Hydrodynamic
models of river, lake, and estuary circulation have been used to estimate
larval vulnerability to entrainment (Paul and Patterson 1977; Lawler e_t a]_.
1975) and seem to be particularly relevant in situations in which water cir-
culation patterns (a) cause larvae to repeatedly pass near a water intake or
(b) transport high concentrations of larvae long distances to the mouth of a
water intake. Vulnerability is a difficult parameter to quantify because
there are many environmental factors which should be taken into account
quantitatively but which exhibit large variability (wind, water circulation,
density of spawners, larval swimming patterns) and are difficult to measure.
If a larval subpopulation occupies a certain location in source waters and
is never vulnerable to entrainment it may be argued that the subpopulation
sustains no direct impact but it does not necessarily follow that there is
no long run impact to that subpopulation. Since it is probable the vulner-
ability to entrainment varies across the source water basin it is of impor-
tance to determine the degree of variation in vulnerability throughout a
basin. In an extreme case it might be discovered that in only 25 percent of
the basin is there a positive probability of larval entrainment while larvae
which enter the remaining 75 percent of the basin become immune to entrain-
ment. If such a condition prevailed it would be important not only for pur-
poses of siting of future water intakes, but for purposes of managing flows
into existing water intakes to know the location of the zone of high vulner-
ability. It may be possible to adjust intake flow regimes so as to reduce
entrainment levels if a constant source of entrained larvae is identified.
If larvae are differentially vulnerable to entrainment depending upon their
location in the source water basin then the direct effect measured in terms
of percent of the population entrained will vary across the basin.
The ratio of total numbers entrained to total numbers produced in the
entire basin is simply an average percentage which may exhibit considerable
variation if the ratio were calculated on a sub-basin basis.
ESTIMATING POPULATION PARAMETERS IN SOURCE WATERS
Total Annual Larval Production
Total production of larvae must be carefully defined with reference to
the source waters as there can be numerous points of larval entry includ-
ing spawning and hatching of eggs directly in source waters, transport of
eggs and larvae into source waters by stream flow, and migration of larvae
from adjacent connecting lakes, embayments, and back waters. An obvious
definition of production is the number of eggs which hatch in the source
waters. This definition eliminates confusion of production with migration
at the theoretical level but does not eliminate the problem of separating
production from immigration in the estimation process when the number of
newly hatched eggs cannot be estimated directly (the usual case). A method
used by Patterson (1979) to estimate production of larval yellow perch in
western Lake Erie is based upon a differential equation of mass balance for
larval abundance in the source waters:
246
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N(t) = h(t)-v(t)-r(t)-m(t)-L(t)-E(t) (10)
Where: h(t) = daily input rate of larvae to source waters
v(t) = daily emigration rate of larvae from source waters
r(t) = daily rate of larvae recruitment into first
juvenile stage of development in reference volume
m(t) = daily rate of mortality of larvae in reference volume
L(t) = daily rate of withdrawl of larvae in reference
volume by water intakes other than power plants
E(t) = daily rate of withdrawl of larvae in reference
volume by power plants
N(t) = net daily rate of change of larval abundance in re-
ference volume.
By making assumptions about the forms of the various functions defined
above and by experimentally estimating each function wherever possible the
solved form of Equation (10) is fit to field based estimated of N(t) (lar-
val abundance on day t) by least squares. The by-products of this exercise
are estimates of total larval production and total natural mortality of lar-
vae in source waters. Patterson assumed that net daily migration was zero
and defined all larvae entering the reference volume to be part of produc-
tion.
A second method which has been used to estimate larval production in
source waters consists of (a) estimating total larval abundance on a
periodic basis (weekly in the case of Lake Erie) throughout the entire
period of larval abundance, and (b) summing the periodic estimates of abun-
dance. The total sum is the estimate of production. This method has the
advantage of requiring only estimates of abundance but requires the assump-
tions that (a) new crops of larvae are sampled each period, (b) all crops of
larvae are sampled, and (c) natural mortality does not significantly depress
each new crop prior to its being sampled.
A third method for estimating total larval production in source waters
was developed by Polgar (1977) and applied to the estimation of striped bass
larval production in the Potomac River. It is a refinement of the second
method mentioned above and incorporates a mortality function so that the
larval mortality rate is estimated together with production. In this res-
pect it is comparable to Patterson's method.
All three methods have been applied in situations in which larval den-
sities in source waters have been sampled at weekly intervals.
Natural Mortality Rate of Larvae
The natural mortality rate of larvae in source waters is difficult to
estimate and may be the least precisely known of all population parameters.
It is quite obviously an important factor in the determination of year class
strength and hence of successful propagation of the population and fishery.
The methods of Patterson and Polgar (preceding section) for estimating total
larval production both incorporate mortality parameters and, upon assuming
that mortality is proportional to abundance, yield (simultaneously with
247
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estimates of production) estimates of the instantaneous natural mortality
rate. Both estimates depend numerically upon periodic measurements of lar-
val abundances in the source waters but differ in another respect.
Patterson's method assumes a single instantaneous natural mortality rate
which is constant throughout the entire length of the larval stage. The
method of least squares is applied to fit the solution of Equation (10) to a
time series of larval abundance data (over the entire period of larval abun-
dance) over a two dimensional grid of mortality rate-production rate para-
meter combinations. Simultaneous pairs of production and mortality para-
meters are thus identified which minimize the mean square deviation of pre-
dicted abundance (from the solution to Equation 10) from estimated abun-
dance based upon field measurements of larval densities. Polgar's method
breaks the larval stage down into three sub-stages plus an additional stage
representing eggs. Three recursive difference equations of the following
form are solved:
Vi6 . fl.l±l!i±l („,
-p.t. -p.t. u '
1-e 1 n 1-e 1 n (i=0,l,2,3)
Where: Ai = total estimated abundance of stage i larvae in
source waters summed over all periods of observation.
Pi - mean instantaneous natural mortality rate of stage i
larvae.
ti = length of stage i development time interval.
One of the parameters pi must be estimated independently in order to se-
quentially solve Equations (11).
A third method for estimting the natural mortality rate of larvae was
presented by Hackney and Webb (1977). Total abundance of larvae for each 5
millimeter length class is estimated weekly based upon field sampling.
Total abundance, Ni, for the entire season of each length class i is then
estimated and a relationship between length class and age (days) of larvae
is then established. Finally, total abundance of each successive length
class i is plotted on semi-log paper against age for the respective length
class, and the slope is the estimate of the mean instantaneous natural
mortality rate pn-. The Hackney and Webb method is similar to Patterson's
method to the extent that it uses a special case of Equation (10) which is
then fit to abundance data. Instead of fitting a production function h(t)
of known form to abundance data (Patterson's method), Hackney and Webb
estimated a proportion of weekly production by measuring larval abundance
(weekly) by 5 mm length class.
REVIEW OF FACTORS NEEDED FOR ASSESSING LONG RUN EFFECTS
Assessment of effects upon a fish population caused by water intake
losses of previous years is difficult because of (a) large natural vari-
248
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ability of the population standing crop or fishery yield from one year to
the next, (b) confounding of effects of water intake caused losses with
other causes of population reduction, and (c) variability of the direct ef-
fects of the water intakes from one year to the next. Consequently, it is
very difficult to estimate the impact of a water intake by estimating dif-
ferences in the population standing crop directly in the field from one
year to the next. Special situations can be visualized such as a power
plant cropping fishes of all ages from a reservoir with other environmental
factors remaining fixed. If the fish population exhibits a downward trend
over a period of years it may be not plausible to assume that the losses
are due to the cumulative direct effects of the power plant. As plausible
as this assumption might appear it remains to be tested because fish popula-
tions have been known to go into states of decline for no apparent reason.
The accumulation of direct effects of annual entrainment and impingement
losses are very frequently analyzed by numerical models which fall into one
of two groups. A fish population is typically assumed to exist in a condi-
tion of statistical equilibrium implying that the full set of environmental
conditions surrounding the population are constant. Under these conditions
age distribution and fishery yields are constant. Let the population stand-
ing crop and fishery yields be denoted by N] and YI respectively. A water
intake caused loss is then introduced on a fixed annual basis. Let the new
equilibrium levels of population standing crop and fishery yields be denoted
by N2 and Y£ respectively. Then two long run impacts of the water intake
are measured in terms of N-|-N2 and Y]-Y2. One type of numerical model
attempts to mimic the fish population before and after the introduction of
the water intake and therefore estimates N] (or Y]) and N2 (or Y2) separate-
ly, after which the difference N-]-N2 (or Y-j-Yp) is calculated as a measure
of long term impact. Leslie-Matrix models and their modification (DeAngelis
1978) are examples of this type. A second type of model focuses upon the
annual losses into an equilibrium estimate of the amount by which the popu-
lation is reduced (N]-N2) or the amount by which the fishery yields are re-
duced (Y-j-Y?). This type of model is called an "equivalent adult" model be-
cause it only is concerned with converting annual water intake caused losses
into equivalent losses to standing crop or fisheries. In these models
standing crops are never estimated because it is only the differences N-]-N2
or Y]-Y2 that are of concern (Patterson 1978; Goodyear 1978). Each approach
has advantages and disadvantages involving assumptions and parameter re-
quirements, verifiability, and ease of development and application. It is
the opinion of some that numerical models of impacts represent a "last re-
sort" when analyzing effects of water intakes upon fish populations. In re-
cent years, however, it appears that this method of last resort is invari-
ably used as one means of evaluating power plant impacts upon fish popula-
tions. It is difficult to determine whether cropping caused by water in-
takes is sufficiently severe to project a population into an inferior com-
petitive position within an aquatic community. If such were the case then
it is an example of an ecological impact triggered by what would initially
appear to be a completely unrelated cause of fish mortality. Long run
ecological impacts of water intake cropping which go beyond a single popu-
lation may exist and may even be significant in some cases but are even more
difficult to assess.
249
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REFERENCES
DeAngelis, D.L. et &]_. 1978. A generalized fish life-cycle population
model and computer program. Environmental Sciences Division Publica-
tion No. 1128, Oak Ridge National Laboratory, Oak Ridge, Tenn.
Goodyear, C.P. 1978. Entrainment impact estimates using the equivalent
adult approach. Power Plant Project, Biological Services Program, Fish
and Wildlife Service, U.S. Department of the Interior.
Hackney, P.A. and J.C. Webb. 1977. A method for determining growth and
mortality rates of ichthyoplankton. Fourth National Workshop on En-
trainment and Impingement. EA Communications, A Division of Ecological
Analysts, Inc., Melville, N.Y. pp. 115-124.
Hubbell, R.M. and C.E. Herdendorf. 1977. Entrainment estimates for yellow
perch in western Lake Erie 1975-1976. CLEAR Technical Report No. 71,
The Ohio State University Center for Lake Erie Area Research, Columbus,
Ohio.
Jensen, L.D. 1977. Fourth National Workshop on Entrainment and Impinge-
ment. EA Communications, A Division of Ecological Analysts, Inc.
Melville, N.Y.
Jobes, F.W. 1952. Age, growth, and production of yellow perch in Lake
Erie. U.S. Department of the Interior, Fish and Wildlife Service.
Fishery Bulletin 70.
Lawler, Matusky and Skelly, Engineers. 1975. Report on development of a
real time, two dimensional model of the Hudson River striped bass popu-
lation (1975). LMS Project No. 115-49, Tappan, New York.
Muth, K.M. 1977. Status of major species in Lake Erie, 1976 commercial
catch statistics, current studies and future plans. U.S. Fish and
Wildlife Service, Sandusky, Ohio.
Ohio Department of Natural Resources Division of Wildlife. 1977. Status of
Ohio's Lake Erie fisheries.
Ohio Ridge National Laboratory. 1977. A selective analysis of power plant
operation on the Hudson River with emphasis on the Bowline Point
Generating Station. Oak Ridge National Laboratory Report No. TM-5877,
Volume 2. pp. 5-78.
Patterson, R.L. 1979. Production, mortality, and power plant entrainment
of larval yellow perch in western Lake Erie. U.S. Environmental Pro-
tection Agency Research and Development Report. EPA-600/3-79-087.
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Paul, J.F. and R.L. Patterson. 1977. Hydrodynamic simulation of movement
of larval fishes in western Lake Erie and their vulnerability to power
plant entrainment. Proceedings of the 1977 Winter Simulation Con-
ference. Vol. I. National Bureau of Standards. Gaitherburg, MD.
pp. 305-316.
Polgar, T.T. 1977. Population dynamics of ichthyoplankton. Proceedings of
the Conference on Assessing the Effects of Power Plant Induced Mortality
on Fish Populations, Pergamon Press, pp. 115-120.
Saila, S.B. 1975. Fisheries and energy production, A Symposium. Lexington
Books, D.C. Heath and Company, Lexington, Mass.
Smith, Stanford H. 1972. The future of salmonid communities in the
Laurentian Great Lakes. J. Fish Res. Bd. Canada 29: 951-957.
Wells, LaRue and A.L. McLain. 1973. Lake Michigan, man's effects on native
fish stocks and other biota. Great Lakes Fishery Commission Technical
Report No. 20.
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SECTION 14
CONTROL OF THE WATER RESOURCES OF THE AZOV SEA USING THE
"AZOV PROBLEM" FAMILY OF SIMULATION SYSTEMS
A.B. Gorstko1, F.A. Surkov"1, L.V. Epshteyn1
and A.A. Matveyev2
A well-developed industrial and agrarian complex has grown up in the
Azov Sea drainage basin. In a territory of 618,000 km2, with a population
of 33 million persons (13 percent of the entire population of the USSR),
some 15 percent of the industrial and 21 percent of the gross agricultural
products of the country are produced. Virtually the entire area of the re-
gion is utilized agriculturally, with crops covering 86 percent of the area
of the drainage basin. Industrial utilization of the deposits of minerals
in the area has reached a high level. Most of the deposits which have been
discovered are being utilized to some extent. The entire water area of the
Azov Sea (38,000 km2) is utilized for fishing, and the use of the sea
coast for recreation is growing. Thus, the most important natural resources
of the region have been involved in the process of economic activity. This
is referred to as the emerging regional natural-technical system (NTS) of
the Azov Sea basin (Borovich ejt al_. 1977).
The water resources represent an important structural element of the
natural-technical system of the Azov Sea basin. Significant volumes of
fresh water, which must be of high quality, are necessary for normal func-
tioning and further development of industry, agriculture, power engineering,
fishing, water transport, and for supporting normal living conditions of the
population. However, the region is poor in fresh water resources. Whereas
each resident in the European portion of the USSR has at his disposal some
6000 m3 of fresh water per year, in this area, only 1700 m3/yr per resi-
dent is available. Of all natural resources, it is water which presently
limits the development of agricultural production in this area.
The possibility of deterioration in water quality, due to its shortages
and the intensive agricultural utilization, plays a significant role in
Institute of Mechanics and Applied Mathematics, Rostov State University,
192/2 Stachky Prospect, 344090 Rostov-on-Don, USSR.
2Hydrochemical Institute, 192/3 Stachky Prospect, 344090 Rostov-on-Don,
USSR.
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limiting the rate of growth of agricultural production. An increase of
capital investment in agriculture and an increase in the sophistication of
land use can provide an increase in the productivity of agriculture, but
not rapidly enough and, more importantly, these measures cannot, in and of
themselves, guarantee stability of agricultural production. The impossi-
bility of broadening the agriculture base leaves the problem of intensifica-
tion of agricultural production in a region with no alternative solutions.
The decisive factor in the intensification and stabilization of agricul-
tural production in the region should be irrigated agriculture. The yield
from irrigated land is not only significantly higher, but is is also less
subject to fluctuations at the mercy of natural weather conditions than is
the yield from nonirrigated land. Improvement in the sophistication'of land
use could also be greatly aided by irrigated agriculture.
However, construction of irrigation systems in the region would lead to
a number of significant structural changes in the nature of transformation
and consumption of the water reserves in the area. First of all, we must
note the negative influence of the use of water for irrigation on the qual-
ity of water in rivers and streams, due to increased introduction of pollu-
tants (particularly if mineral fertilizer is intensively used), and due to
changes in the salt and nutrient composition of the runoff. Furthermore, an
increase occurs in the consumptive use of water for agriculture, as a result
of increased transpiration of water by plants in irrigated fields and the
evaporation of water from the surfaces of reservoirs and from irrigation
systems. Finally, regulation of runoff, in order to hold back flood waters
for the needs of agriculture, results in seasonal leveling of the discharge
of rivers and streams, which influence the hydrologic and water/salt regimes
in the Azov Sea.
By 1976, the total amount of irrigated land had reached 13,300 km^ in
this area (double the area of 1971). In order for irrigated agriculture to
become a truly decisive factor in the intensification and stabilization of
agricultural production in the region, this surface area must clearly be in-
creased. Plans for the development of irrigation call for continuation of
the high rates of growth of irrigated land areas through the end of the cen-
tury. By 1975, the total quantity of water diverted for irrigation was 9.22
km3, the consumption use was 8.69 km3 (Bronfman 1976).
The total consumption of water for industry, power engineering and by
the population of cities and towns was 10.6 knv3 in 1975, including nonre-
covered consumption on the order of 1 km3. The volume of water diverted
and consumed by these structural elements of the NTS is expected to grow
slowly in the future, as planned steps are taken to develop recycling
systems and introduce new technological systems which use less water.
The nature of the use of the water resources in the continental portion
of the Azov Sea basin will also determine the functioning of such structural
elements of the NTS as the ecosystem of the Azov Sea. This system is the
most productive marine body of water, in terms of fishing, in the world. In
the mid-1930's, the total catch in this sea reached 300,000 tons (8-9-103
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tons/km2), while the potential of the sea is estimated as 850,000 tons
(over 20-103 tons/km2).
Since the Azov Sea is the final link in the chain of transformation of
water resources in the region, it follows from general ecologic principles
that the effects of anthropogenic action throughout the basin will be accu-
mulated in the sea (Dyuvino and Tang 1973). Analysis of the results of ob-
servations of the status of the ecosystem confirms this belief (Vorovich et
al. 1977). The present level of consumptive use of water and of discharge
oT pollutants, the seasonal leveling of the discharge of rivers and streams,
the change in the biogenic and mineral runoff and other anthropogenic fac-
tors will cause a decrease in the biologic productivity of the sea.
Thus, the shortage of high-quality water will limit the rates of devel-
opment (and in some cases, will set absolute limits of development) of the
NTS of the region as a whole. To study this problem further, the region was
divided into territories, the boundaries of which do not agree with any ad-
ministrative-territorial boundaries. The selection of a region on the basis
of geography (more precisely, hydrogeology) meets the needs of this regional
study: analysis of water resources as a functional element in the natural-
technical system, and the development of the scientific principles for the
control of the water resources of the region.
The range of problems related to the use of water resources in the Azov
Sea basin, conservation or conscientious reconstruction of the ecosystem of
the Azov Sea, and the development of macroregional programs of supplementing
the continental runoff in the basin will be referred to as the Azov problem.
The Azov problem includes the problem of the Azov Sea in its entirety
(Vorovich et _al_. 1977).
Solution of the Azov problem requires systematic development of a multi-
tude of versions of regional ecologic policies and water management strate-
gies in the Azov Sea basin, and evaluation of the probable effects of these
versions in combination with macroregional projects. Tremendous human and
material resources, planning, project and scientific organizations have
been drawn into this process. The importance of the problem, unprecedented
in terms of the scale of macroregional water management steps to be taken
(Voropayev 1976; Dunin-Barkovskiy 1976), the complexity and nonintuitive na-
ture of the behavior of the NTS (Forrester 1974), requires organization of
the development and evaluation of plan versions designed to facilitate and
accelerate both the generation and the evaluation of alternative plans. It
seems that the basic principle of organization of all scientific plans and
plan development on the Azov problem should be the idea of the man-machine
system. This should include structural subsystems, decision making per-
sonnel (DMP) at various levels; experts on the Azov problem as a whole, and
experts on its individual aspects; plan developers; families of "Azov prob-
lem" simulation systems (SS); an information bank; and persons who plan and
conduct experiments with the SS.
The simulation system is a modular structure of significant complexity,
allowing a broad range of experiments to be performed. The elements of the
254
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structure, the modules, are essentially complete models of individual pro-
cesses.
It is this representation of the SS as an element in a man-machine
system which served as the basis for development of the "Azov problem"
family of simulation systems. Work on the creation of the entire family is
far from complete. However, the development and operation of SS should not
be regarded as consisting of sequential stages. They are actually phases
in a continuous process.
The "Azov problem" family of SS must include purely descriptive systems,
systems with partial optimization, and the more standard, unified and de-
tailed systems. At the present time, three simulation systems have been
developed: 1) the "rough prediction" system (ISGP), 2) the "Azov Sea"
system (ISAM), which is descriptive in nature, and 3) a detailed simulation
system, "the land of the Azov basin" (DISSAB), a descriptive system with
elements of optimization. The methodologic basis of the development of the
model has been the method of simulation modeling (Gorstko 1977).
In addition to these systems, the "Azov problem" family should include
the following SS, currently under development: 1) a "regional optimizer", a
standard economic model, developing the structure of branches of the econ-
omy, to be optimal from the standpoint of criteria related to water re-
sources; 2) "Aksakal", a simulation system which "allocates water" among
branches and territorial systems in case of shortages, zonal economic models
based on optimization of the utilization of water in agricultural produc-
tion; 3) a "local optimizer", a standard economic model answering the same
questions as the "regional optimizer", but developed for a significantly
smaller territorial breakdown of the region; 4) "water purity", a standards
SS, defining the necessary level of purification of water and the loss re-
sulting from failure to meet purification standards. The general properties
are shared by the SS of the family to various degrees.
In all SS, the object being modeled is considered to be divided into in-
dividual segments (regions). The status of each segment is described by a
set of ingredients, combined in the description of the segment (status vec-
tor). The territory (or water area) of the segment is considered homoge-
neous in all ingredients. The union of the status vectors of all segments
is called the status vector of the system as a whole.
The ingredients of the status vectors change discretely, the selection
of the time step representing a complex, informal process which is con-
ducted in combination with decomposition of the system into processes
modeled by individual modules and separation of the objects into segments.
The art and erudition of the creators of the SS, not only mathematicians,
but also scientists studying the system from the standpoint of natural sci-
ence and social-economic disciplines (hydrology, biochemistry, demography,
etc.) must be employed here.
In the ISAM and DISSAB, the model is limited to a single time step (of 5
and 10 days, respectively). In the ISGP, two time steps are studied, a
small step (1 month) to model rapid processes, and a larger step (1 year or
255
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5 years, depending on the purposes of the specific simulation experiment).
The latter is used to simulate long-term economic programs and versions of
possible climate-determined changes which are important for the status of
the water resources of the region.
The modular structure of the SS makes it possible to describe each pro-
cess modeled using the mathematical apparatus which is most adequate to the
nature of the process. Further, it allows independent identification of
each unit before it is included in the composition on the SS.
In the "Azov Sea" simulation system, the status of each of the 7 seg-
ments, into which the water area is divided (Figure 1), is described by a
Figure 1. Diagram of regions of the Azov Sea.
status vector including 120 ingredients. The processes modeled in ISAM are
distributed among 15 models (modules): WATER DYNAMICS, NUTRIENT ELEMENTS,
OXYGEN, WATER QUALITY, PHYTOPLANKTON, ZOOPLANKTON, BENTHOS, GOBY, PIKE-
PERCH, BREAM, ROACH, STURGEONS, HERRING, ANCHOVY, OTHER FISH. The module
entitled WATER DYNAMICS includes the volume of water exchanged among re-
gions of the sea during the five-day period in question. The module called
NUTRIENT ELEMENTS describes the cycle in the SS of the compounds of nitro-
gen, phosphorus and silicon considering the processes of transfer, break-
down, consumption, erosion of the shore and other processes. The module
termed OXYGEN models the enrichment of the water with oxygen during photo-
synthesis, and by atmospheric aeration; consumption of oxygen in biochemical
processes, and distribution of dissolved oxygen between surface and bottom
layers. The WATER QUALITY module, one of the most important in the system,
models the dynamics of the concentration of pollutants considering their in-
put and breakdown. The rate of self-purification of the bodies of water de-
pends on the temperature of the water, type of pollutant and condition of
the aquatic ecosystem.
The modules describing the dynamics of living organisms in the ecosystem
consider the processes of grazing, respiration, breeding, migration and
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death. Controlling influences such as removal of a fraction of the popula-
tion and stocking of the sea with young fish are considered for the fish
populations.
From the standpoint of program organization, the "Azov problem" simula-
tion system is a hierarchical structure, the elements of which are standard
program and information modules for this information system. The software
modules consist of banks of programs (BP) on information carriers, while the
information modules consist of banks of data (BD). The SS is a variable
composition system, i.e., without changing the overall structure of the
system, additional modules can be added to or removed from the BP and BD in
the system. The modeling, control and service programs in the BP are not
distinguished in the sense of rules of accessing them: Each module is as-
signed an arbitrary number, the first digit of which represents the hierar-
chical level of which the module belongs, while the second indicates its
functional purpose (modeling program, control program, service program, ser-
vice BD, etc.). Subsequently, several digits are provided for the class
number, and several more for modifications. Addition or removal of modules
in the BP includes a stage of updating the BP directory, which can be
printed out by a special service program. The BP catalogue contains basic
information on the module, the number, value, data of inclusion, a comment
section and the number of the last simulation experiment in which the module
of the BP and SS was used. The "Azov problem" family has from 3-5 hierar-
chical levels. Modules of higher levels hierarchy can be accessed only by
programs of lower hierarchical levels.
The first hierarchical level of the entire SS contains the control pro-
grams, which call the basic control modules, check the correctness of a call
and transfer control to it. The modules of the first hierarchical level
operate once each time the system is initialized (STARTER class).
Modules of the second hierarchical level (REGIME class) are the main
control programs. The functions of this group include actual organization
of operation of the SS in accordance with its program, assigned by one or
more information control files. These files are defined by the experimental
planner. Their specific structure is dictated by the structure of each SS
and differs greatly, e.g., between ISAM and DISSAB. We can say with suffi-
cient generality that the control files consist of macroinstructions which
are interpreted by REGIME.
The third hierarchical level of modules consists of the MODELS of vari-
ous classes (e.g., in the "SEA" SS - the class WATER DYNAMICS, NUTRIENTS,
etc.); the subordinate control programs (for ISAM - the classes REPLACEMENT,
CHECK, ADJUST); the service programs (the classes PRINT RESULT, NORMAL OUT-
PUT, EMERGENCY OUTPUT); programs for generation of external factors (STO-
CHASTIC PREDICTION, REGRESSION ANALYSIS, EXOGENOUS FACTORS); and the basic
programs for the systems servicing the BP and BD.
Modules of the fourth and fifth hierarchical levels include programs
which run algorithms common for several blocks (e.g., the modules FEED, MI-
GRATION, CATCH, and ISAM), for all programs of the complexes which service
the BP and BD, and which are accessed by the base modules of the third hier-
257
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archical level. The program structure of one simulation system - ISAM - is
shown schematically in Figure 2.
The data bank also has a hierarchical structure. All information mod-
ules have a standard format in accordance with one of the PROTOTYPES. The
PROTOTYPES developed for each SS are divided into two classes - INITIAL PRO-
TOTYPES and ANNUAL PROTOTYPES. INITIAL PROTOTYPES are designed for storage
of information entering the SS from the external world, and are developed
so that the information is easy and convenient to prepare. This SS is a
system of variable composition with respect to INITIAL PROTOTYPES. ANNUAL
PROTOTYPES describe the structure of the information modules used directly
in the operation of the SS program modules. Their composition is rigidly
defined, while the PROTOTYPES are developed either as unchanging, or as
having an unchanging portion (e.g., prototypes of files with floating bound-
aries such as the prototype CHAMBER DESCRIPTION in the model LAND).
The system of modules servicing the BD is designed for input and record-
ing of information modules with checking for the agreement of preparation of
the initial data with the PROTOTYPES, adjustment of files using CORRECTING
PROTOTYPES (a subclass of the class of INITIAL PROTOTYPES), sorting of in-
formation, elimination of modules, generation of new modules using FINISHED
PROTOTYPES, work with the BD CATALOGUE, and output of information concerning
the composition of the BD and printout of modules on the request of REGIME.
It should be noted that the FINISHED IS PROTOTYPE must include proto-
types of all control modules, while the INITIAL PROTOTYPES must include pro-
totypes of the correcting files, which can be used to generate a new control
file from those already present in the BD. Thus, the BD contains a set of
finished programs from the simulation experiments, and an experiment can be
repeated by simply inputting a single item of data, its number. Further-
more, due to the ease of generation of new information control files in the
BD, it is simple to perform new simulation experiments without preparing
large files of initial data.
The most important systems functions, among those defining the differ-
ence of the simulation system from a simple large model, are performed by
the modules which track the course of the simulation experiment and adjust
the calculation trajectory of the system. They can be considered as a sub-
system for automatic tuning of the SS in various functioning regimes during
the course of an experiment, depending on the results of calculations al-
ready performed. For example, the modules of the CONTROL class in ISAM
give the experimental planner the ability to see that the values of ingre-
dients or certain simple functions (with ingredients taken as their argu-
ments) do not go beyond established limits, or, if they do, to take certain
actions.
In order to solve the Azov problem to the extent which is objectively
required and possible, application of a combination of regional and macro-
regional, technical, economic and ecologic measures, using the "Azov prob-
lem" family of simulation systems as a tool for this purpose must be made.
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FIRST HI ERA RCHICA L LE VEL:
STARTER
SECOND HIERARCHICAL LEVEL:
REGIME
r-o
THIRD HIERARCHICAL LEVEL:
MODELS
Water d
Biogenic
ynamics
elements
c
SUBORDINATE
;ONTROL MODULES
Replacement
Checking
Adjustment
SERVICE MODULES
Print result
Scheduled output
Emergency output
G
EXTERNAL
ENERATION
Exogenous
FACTOR
MODULES
factors
Stochastic prediction
Regression analysis
BASIC MODULES
SERVICING BRAND BD
Module 1
Module 2
:\^ i ^^
FOURTH HIERARCHICAL
LEVEL:
^-^ * ^^
ISAM standard subunit
i
I
r
Standard statistical subunits
Standard BP and BD
servicing subunits
Figure 2. Program structure of the "Azov Sea" SS.
-------
REFERENCES
Bronfman, A.N. 1976. Experience in study and solution of ecologic-geo-
graphic problems of the Azov Sea basin. Chelovek i sreda. Materials
of the 23rd International Geographic Congress, Moscow, Nauka Press.
Dunin-Barkovskiy, L.V. and N.N. Moiseyev. 1976. A system of models of the
redistribution of the river runoff of the USSR. Vodnyye resursy, No. 3.
Dyuvino, M. and T. Tang. 1973. Biosfera i mesto v ney cheloveka (The bio-
sphere and man's place in it), Moscow, Mir Press.
Forrester, J. 1974. Dynamics of urban development. Translation edited by
Yu. Ivanilov, Moscow, Progress Press.
Gorstko, A.B. 1977. Simulation modeling. Izvestiye SKNTs VSh, Estestv.
nauki, No. 2.
Voropayev, G.V. 1976. Tasks and organization of scientific research in
connection with the problem of redistribution of water resources.
Vodnyye resursy, No. 3.
Vorovich, 1.1., A.M. Bronfman, S.P. Volovik, and E.V. Makarov. 1977. The
Azov Sea problem. IZvestiya SKNTs VSh, Estestv. nauki, No. 2.
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SECTION 15
THE TRANSPORT OF CONTAMINANTS IN LAKE ERIE
Wilbert Lick1
INTRODUCTION
The general purpose of our research is to understand more thoroughly
and be able to predict the transport and fate of contaminants in lakes.
This transport and fate is a complex matter which involves physical, chemi-
cal, and biological processes, all of which need to be considered before the
fate of contaminants can be predicted. In the present paper, only the
physical processes of transport are discussed in detail.
Much of our work has been concerned with contaminants in Lake Erie (see
Figure 1) and that work will be emphasized here. Lake Erie is a large shal-
low lake approximately 386 km long and 80 km wide with an average depth of
20m. Topographically, it can be separated into three basins: (1) a shal-
low Western Basin with an average depth of 7 m, (2) a large, relatively flat
Central Basin with an average depth of 18 m, and (3) a deeper, cone-shaped
Eastern Basin with an average depth of 24 m. The Western and Central Basins
are separated by a rocky chain of islands.
In the following section, a general discussion of the processes affect-
ing the transport of contaminants is given. The results of analyses of two
specific problems of contaminant transport are then presented. These are
(1) the resuspension, transport, and deposition of sediments in the Western
Basin of Lake Erie, and (2) the temperatures, currents, and the transport of
contaminants in the Central Basin of Lake Erie during summer stratification.
FACTORS AFFECTING THE TRANSPORT OF CONTAMINANTS IN LARGE LAKES
A brief discussion of the factors affecting the transport of contami-
nants in lakes is given here. The transport of sediments is discussed
first. Sediments are significant contaminants in themselves, since they in-
crease the turbidity of the water and, when heavy sedimentation occurs, may
require that large amounts of dredging be done. However, most importantly
'Department of Mechanical and Environmental Engineering, University of
California at Santa Barbara, Santa Barbara, California 93106.
261
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'" '»/>J/SW/JJS//'
LAKE ERIE LONGITUDINAL
CROSS SECTION
Figure 1. Lake Erie bottom topography.
262
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in the present context, sediments (especially the fine-grained, clay-sized
fraction) readily adsorb many other contaminants such as phosphates, heavy
metals, and toxic hydrocarbons. Hence, if we understand the transport of
sediments, we can then more readily understand the transport and fate of
other contaminants.
After this brief review of sediment transport, a few comments are made
on the transport and fate of other contaminants, specifically radioactive
materials, nutrients, and chlorinated hydrocarbons. The discussion is meant
to apply to large lakes in general. However, when specific values of para-
meters are needed for illustration, parameters appropriate to Lake Erie have
been chosen.
Sediment Transport
The primary sources of sediments in lakes are river inflows and shore
erosion while the primary sinks are river outflows and the deposition and
ultimate consolidation of sediments into the permanent sedimentary bottom
of the lake.
The process of the transport of sediments from the primary sources to
ultimate sinks occurs by frequent cycles of resuspension, transport, and de-
position. This cycle occurs on the order of a few days while the time from
input to final deposition and consolidation probably occurs on the scale of
months to years, perhaps tens of years. Transport may occur as suspended
load (for the finer sediments) or as bed load (for the coarser sediments).
The emphasis here is on the fine-grained sediments, and therefore suspended
load, since the fine-grained sediments are responsible (because of their
large surface area and hence adsorptive capacities) for the greater flux of
contaminants.
Mass Balance Equation--
Preliminary attempts have been made to develop predictive numerical
models of the transport of fine-grained sediments in large lakes (Lick e_t
£L 1976; Sheng and Lick 1979). These models are predicated on the solution
of the following mass balance equation,
8C j(uC) 8(vC)
1 9x 9y
where C is the concentration of the sediment, t is time, u, v, and w are
fluid velocities in the x, y, z directions respectively, ws is the settling
speed of the sediment, DH is the horizontal eddy diffusivity, and Dy is the
vertical eddy diffusivity.
In order to obtain solutions to this equation and hence to determine
sediment transport, various quantities such as the fluid velocities, the
settling velocities, the eddy diffusivities, and of course the entrainment
and deposition rates at the sediment-water interface must be known or cal-
culated. The determination of these quantities is discussed below.
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Settling Velocities-
It is known that settling velocities depend on the mineralogy and grain
size of the sediments, on the ionization strength of the water, and on the
sediment concentration (Owen 1978; Terwindt 1977; Fukuda 1978). Little work
has been done to characterize the settling velocities of fresh water se^-
ments and much more needs to be done. The distribution of settling /ce-
cities for a sediment from the Western Basin of Lake Erie is presented in
Figure 2. Shown is the percent of suspended sediment within various
settling velocity intervals. It is quite evident that the settling velo-
cities range over four orders of magnitude, a range that must be considered
in any quantitative description of sediment transport.
Entrainment and Deposition—
The net flux of sediment qs at the sediment-water interface is the dif-
ference between the entrainment rate E and the deposition rate D, or
qs = E - D (2)
As a first approximation, it can be assumed that the deposition is propor-
tional to concentration, and hence
qs = E - BC (3)
where 3 is the coefficient of proportionality, is known as the reflectivity
parameter, and has the units of velocity. It is convenient to rewrite the
above equation as
qs = 3(C - C) (4)
where Ceq = E/B and represents a steady-state, or dynamic equilibrium, con-
centration.
The above relations are strictly only valid for sediments of uniform
grain size. They are only approximations for naturally occurring sediments
which have effective grain sizes varying over several orders of magnitude
(indicated by the variation in settling velocities shown in Figure 2), but
nevertheless are useful for organizing experimental data. It is becoming
evident from our previous modeling attempts and from our experimental work
that this range of effective grain sizes must be considered in any quantita-
tive description of sediment transport.
The processes that govern the behavior of the parameters E, B, and Ceq
are not well understood (especially for fine-grained sediments in fresh
water). It is known that these parameters depend on the shear stress ap-
plied at the sediment-water interface, on the bulk sediment water content,
on the mineralogy of the solid sediment, and probably on the activity of the
benthic organisms present in the sediment. Experiments have been made
(Fukuda 1978; Fukuda and Lick 1979; Lee 1979) and are being made to quantify
these phenomena.
In experiments thus far, only the top few millimeters of sediment have
been entrained. Since properties of the sediment such as water content and
264
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6 30
•a
•o
s
20
10
median
I (deionized with
median
dispersant)
-4 -3 -2 -I
(deionized)
0 -4 -3 -2
-I
| 40 r r
I
"H 30
w
1 20
| 10
V)
55
( tap water)
median
1
"
M
MIMMH
__
—
mi
1 MIVVIIUI
( Is. 7 water)
_
"
-
.
r
— f|
mm
-4 -3 -2 -I 0 -4 -3 -2 -I
log(0(ws)
(cm/sec)
Figure 2. Settling velocity versus percent of suspended sediment
for the Western Basin sediment.
265
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grain size change rapidly with depth, it is expected that entrainment rates
will also change rapidly with depth. Experiments are now being made to
understand and quantify this variation, a variation that is essential to ac-
curately predict sediment transport.
It is also becoming evident (e.g., see Thomas et a/[. 1972, 1976; Sly
1978; Sheng and Lick 1979) that sediment properties vary greatly throughout
a lake and even throughout portions of a lake such as basins or bays. For
example, entrainment rates vary by at least two orders of magnitude for
sediments throughout the Western Basin of Lake Erie (Lee 1979). If this
variation is not considered, accurate predictions of sediment transport can
not be made.
A major influence on sediment properties that has not been quantita-
tively investigated as yet is the effect of benthic organisms on entrainment
and deposition rates. Benthic organisms are plentiful in lakes, especially
in the near-shore areas. They influence sediment properties by (1) rework-
ing the sediments (Fisher et^ al_. 1979; Fisher and Lick 1979) i.e., by bur-
rowing, passing particulate matter through their guts, and egesting fecal
pellets of different shape, size, and content than the original sedimentary
material, and (2) excreting mucus on the surface and on burrow walls which
may assist in binding the sediments. Preliminary work has been done on this
problem but adequate information on the effect of benthic organisms on en-
trainment and deposition rates is not available.
Wave Generation and Bottom Stress —
Our sediment flux experiments give the entrainment and deposition rates
as a function of the applied shear stress. In lakes, this stress is due to
wave action and currents, with wave action dominating in shallow water.
In order to predict this wave action, a wave generation analysis is
needed. In our computations (Kang e_^ ^1_. 1979), the standard SMB procedure
modified for shallow water (CERC 1973; Pore 1979) has been used and is be-
lieved to give satisfactory results. This procedure gives wave height and
period and water velocities throughout the water column as a function of the
wind velocity and fetch.
Additional theoretical work (Kajiura 1968) is then necessary to deter-
mine the bottom stress. Recent work has attempted to extend the analysis to
the case where waves and currents are considered simultaneously in their ef-
fect on bottom stress (Grant and Madsen 1979). However, no substantive
field work has been done to verify these analyses and this needs to be done.
Currents —
Of course, to solve Equation (1), one must also know the currents in the
lake. These may readily be obtained from present three-dimensional time-
dependent numerical analyses. For a non-stratified lake, this procedure is
relatively well understood, at least in principle, even though the practical
application of this knowledge leaves much to be desired. In contrast, the
temperatures, currents, and the dispersion of contaminants in a stratified
lake are not well understood, not even in principle, although progress is
being made in this respect.
266
-------
Thermal stratification significantly affects the dispersion of contami-
nants in a lake but the effects are not quantitatively understood. For
example, it is necessary to know how contaminants are transported from the
hypolimnion to the epilimnion. That is, after contaminants are released in
the hypolimnion, do they mainly diffuse vertically or are they convected
horizontally to a near-shore area and then diffused and convected vertically
into the epilimnion? Also, due to thermal stratification, strong internal
waves and hypolimnetic currents may be present in sufficient strength to
cause entrainment of bottom sediments. These processes are presently being
investigated by means of a two-dimensional model of a stratified lake
(Heinrich et al_. 1979).
Characteristic Times —
Particulate matter is transported vertically through the water column to
the sediment-water interface by a combination of convection, settling, and
turbulent diffusion. Vertical convection is generally only significant
where strong upwelling and downwelling occur, e.g., in near-shore regions,
while settling and turbulent diffusion are ubiquitous phenomena. A charac-
teristic time for settling is ts = h/ws while a characteristic time for tur-
bulent diffusion is tdiff = h2/2Dv, where h is the depth of the water
column. These times are equal when h = 2Dv/w$. For shallow waters where
h <_ 2Dy/ws, turbulent diffusion dominates and settling can be neglected to a
first approximation. As an example, for the Western Basin of Lake Erie, h =
7m, Dy = 25 cm2/sec, and ws = 2.5 x 10~2 cm/sec. Hence, ts = 3 x 104 sec
while t
-------
Although the above procedure will serve to verify the short-term resus-
pension, transport, and deposition processes, the major question is the ul-
timate fate of contaminants and this involves time scales of months and
years. Once the short-term processes are understood and verified, it is
then necessary to do a long-term calculation and verification. By a long-
term calculation, I mean either a one-year calculation under actual condi-
tions or a set of short-term calculations statistically-averaged over a
year. The latter may be more efficient and also statistically correct con-
sidering the short time scales associated with the resuspension-deposition
cycle as compared to a year or the long time scales for deposition and ulti-
mate consolidation. This long-term verification must be made by comparison
of model output with sediment accumulation rates and/or accumulation in the
sediments of other easily measured and easily modeled substances such as
radioactive materials.
Other Contaminants
Sediments are a relatively simple contaminant to model since no bio-
chemical transformations are involved. In attempting to predict the ulti-
mate fate of other contaminants, quite complex cnemical and biological
transformations must be considered as well as the physical processes dis-
cussed above. These transformations occur not only in the water column but
also in the bottom sediments and in aquatic organisms.
The basic mass balance equation for any substance is simply Equation (1)
with the addition of a source-sink term S which describes these biochemical
transformations. For radioactive materials, the next easiest contaminant to
model after sediments, this source-sink term is given by S = -AC, where I/A
is the radioactive decay time. Because of the simplicity of this reaction
and because of the observational data available (e.g., Robbins 1979; Robbins
and Edgington 1979a,b) it is possible to develop and verify a model of
transport of radioactive materials relatively simply compared to other con-
taminants. With radioactive materials and with other contaminants as well,
processes occurring in the sediments such as benthic activity, chemical
transformations and diffusion, and resuspension and deposition processes are
especially important and must be considered in the interpretation of verti-
cal sediment concentration profiles.
Nutrients are another important class of contaminants and their ultimate
disposition is an important problem. Water quality models (which describe
the interactions between nutrients, phytoplankton, and zooplankton) are
available (Bierman 1979; Thomann et_ _al_. 1976; DiToro 1979) and are con-
stantly being improved. These models are presently being coupled with
hydrodynamic models, at least in a crude sense. It is becoming more evident
that transfer of nutrients to and from the bottom sediments, particularly
when the lake is stratified, is a significant factor in the overall nutrient
mass balance equation. Experimental and theoretical work has been done
(Mortimer 1941, 1942, 1971; DiToro 1979) and is being done to understand
this process better. Detailed analyses coupling this sediment work and
hydrodynamic models with water quality models needs to be done.
268
-------
Probably the most important class of contaminants in lakes at the pre-
sent are the toxic chlorinated hydrocarbons. Although the physical pro-
cesses by which they are transported are generally understood, their bio-
chemical transformations are not generally quantitatively known. A descrip-
tion of their ultimate fate depends on obtaining this information.
With many hazardous substances, it is also imperative to consider the
accumulation of these substances throughout the aquatic food chain. A dis-
cussion of this problem and the appropriate equations for dispersal in the
aquatic system and accumulation in the aquatic food chain is given by
Thomann (1979).
THE RESUSPENSION, TRANSPORT, AND DEPOSITION OF SEDIMENTS IN THE WESTERN
BASIN OF LAKE ERIE
The Western Basin of Lake Erie is quite wide, with an average width of
approximately 50 km, but is extremely shallow, with an average depth of only
7 m. Because of this and because of large inputs of sediments from the
Maumee and Detroit Rivers and from shore erosion, large sediment concentra-
tions and especially large variations in sediment concentration are present
in the Western Basin.
These large variations in sediment concentration make the Western Basin
particularly amenable to sediment transport analyses. In addition, an ex-
cellent surface sediment concentration data base is available. This data
base consists of synoptic maps every few days of surface sediment concentra-
tions in March and April 1976 and was made by means of aircraft overflights.
For these reasons, it was decided to model the sediment transport in the
Western Basin of Lake Erie.
In order to have a valid, predictive analysis of sediment transport, it
is necessary to know (1) the resuspension and deposition rates as a function
of the applied stress for sediments throughout the area being investigated,
(2) the bottom stress as caused by the combined action of waves and cur-
rents, and (3) the currents as determined by winds, throughflow, and any
possible stratification. These components must then be coupled with the
sediment mass balance equation, Equation (1), to give the sediment concen-
tration as a function of position and time. A summary of our work on these
problems for the Western Basin of Lake Erie follows.
Resuspension and Deposition Rates
Experimental work has been done by us to determine the resuspension and
deposition rates for sediments from the Western Basin (Fukuda 1978; Fukuda
and Lick 1979; Lee 1979). The experiments were done in a circular flume
(see Figure 3) with an outer radius of 66 cm and an inner radius of 51 cm
for a channel width of 15 cm. The top rotates and produces a shear flow
which in turn exerts a shear stress on the sediment-water interface. Velo-
city profiles were carefully measured and were used to determine this shear
stress as a function of ring rotation rate.
269
-------
•66 cm
T
30.5 cm
i^_—^-*
51 cm —+]
in
\i\\\\i\\\\\\\\\mi\\m
HAT
Figure 3. Side view of the flume.
-------
The sediments were placed in the flume and allowed to settle for periods
of one to ten days until the desired water content was reached. The top was
then rotated at a prescribed rate corresponding to the desired shear stress.
Due to this stress, the suspended sediment concentration increased with time
until a steady state was reached.
A typical variation of concentration with time is shown in Figure 4.
The entrainment parameters (E, Ceq, and 6) were obtained from data of this
type, i.e., Ceq is steady state concentration, E can be determined from-the
initial slope of the concentration versus time curve, and $ can then be cal-
culated from 3 = E/Ceq.
Typical results for the entrainment rate for three different types of
sediments (a shale-based sediment with a high clay content of 68%, a sedi-
ment from the Western Basin with a clay content of 38%, and a sediment from
the near-shore of the Central Basin with a clay content of 34%) are shown in
Figure 5. It is quite clear that the entrainment rate varies rapidly with
the applied shear stress and bulk sediment water content and also depends
strongly on the sediment type. Results for Ceq show similar variations.
The entrainment rates on a linear scale for the Western Basin sediments are
shown in Figure 6. It can be seen that E increases rapidly near a critical
stress but then increases less rapidly at higher stresses. More work is
being done at these higher stresses to better delineate these parameter
variations.
The variation of the reflectivity parameter 3 with applied shear stress
is shown in Figure 7. It can be seen that the variation of B is rather
small compared to the variations in E and Ceq. The average value of 6 for
all three sediments is 8.4 x 10~3 cm/sec with a standard deviation of _+
5.9 x 10~3 cm/sec.
Additional work has been done on other sediments from the Western Basin
(Lee 1979). From these latter experiments, it can be shown that, although
the general functional dependences of E, Ceq, and B on shear stress and
water content are similar for all sediments tested, the values of these
parameters change from one sediment to another by as much as two orders of
magnitude. This obviously has a significant effect on sediment concentra-
tions and sediment transport.
Wave Action and Bottom Stress
Surface waves are generated by the winds through energy transfer from
the winds to the waves. Although recent theoretical developments are useful
in understanding the wind-wave energy transfer mechanisms, the most useful
procedures for predicting wave parameters are the semi-empirical methods
such as the PNJ method developed by Pierson, Neumann, and James (1955) and
the SMB method developed by Sverdrup, Munk, and Bretschneider (Sverdrup and
Munk 1947; Bretschneider 1958).
The SMB method as modified for shallow water (Bretschneider 1958; CERC
1973) has been used in our calculations. These relations give the signifi-
cant wave height H and significant wave period T as a function of wind speed
271
-------
c
(mg/l)
ro
•^i
ro
U
curve fit
-4 2
E =3.3x10 mg /cm -see
Ceq = 49
£ = 6-5* I0
"~
cm /sec
TW = 0.92 dynes /cm
% HgO = 61.7 %
1
20
40
60
80 100 120 140
TIME (min)
!60
180
200
Figure 4. Example of the concentration time history data for the shallow-based sediment with
TW = 0.92 dynes/cm2 and a water content of 6.7%.
-------
SHALE
• 61-62%
|63-68 %
• 69-70%
A 73-75%
WESTERN
BASIN
O 76-77%
Q 79-81 %
<> 81-82%
CENTRAL
BASIN
A 68.0%
O 71-73%
£ 74-78 %
0 77.9%
O 80.3 %
K>
( mg/cm -sec)
-2
.3
-4
J
0
234
rw (dynes/cm2)
Figure 5. The entrainment rate as a function of the average boundary
shear stress for the shale-based, Western Basin, and Central
Basin sediments.
273
-------
5r
E x 10'
(mg/cm -sec) 2
%H20
O 79 - 83 %
D 76- 79 %
rrr
2
i
4
s i
1
6
' /4v
-------
£x 10 *
( cm / sec )
3.5
3.0
2.5
2.0
1.5
1.0
0.5
SHALE
• 61-62 %
H 63-65 %
^69-70%
A 73-75%
% H20
WESTERN
BASIN
O76-78%
D 79-81 %
O 81 - 82 %
O
CENTRAL
BASIN
A 68.0 %
Q 71-73 %
0 74-75%
Q 77-81 %
4 6
_ 2
rw ( dynes / cm )
8
10
Figure 7. The reflectivity parameter as a function of shear
stress for the shale-based, Western Basin, and
Central Basin sediments.
275
-------
U, fetch length F, and mean depth D. From these relations and assuming in-
viscid flow/ the horizontal periodic flow at the sediment-water interface
can be determined. Near this interface, of course, a turbulent boundary
layer occurs and produces a shear stress on the sediment-water interface.
In general, this maximum shear stress TW can be written as
= CfwP U2 (5)
Tw
where P is the water density, U is the maximum velocity just outside the
boundary layer, and Cfw is a bottom friction coefficient which depends on
both surface roughness and flow characteristics in the wave boundary layer.
Kajiura (1968) has analyzed this problem and developed a relationship be-
tween Cf and R where R = U/(av)l/2, v is the kinematic viscosity, and a is
the wave frequency. This relationship is shown in Figure 8.
By use of the above equations and parameters, we have developed a
numerical routine to give wave amplitude, period, bottom velocity, and bot-
tom stress as a function of wind speed for any location on Lake Erie (Kang
et^ aJL 1979). Typical results are shown in Figures 9a, b, and c for the
Western Basin and for a South-West wind at a speed of 11.2 m/sec (25 mi/hr).
Of most interest for the present purposes is Figure 9c which shows the bot-
tom stress. It can be seen that in much of the Western Basin the bottom
stress is greater than one dyne/cm^ and therefore at least the fine surfi-
cial sediments can be readily resuspended. Along the northern shore, bottom
stresses of 5 to 10 dynes/cm^ are present leading to entrainment of all but
the coarsest particles. As can be seen in Figures 9a, b, c, a difficulty
with the present analysis is that it does not include effects of wave dif-
fraction and refraction, effects which are obviously important in the lee
of the islands.
Currents
The wind-driven currents were calculated by means of a two-mode, time-
dependent, free-surface model (Sheng and Lick 1979). Basic assumptions of
the model were that the pressure varied hydrostatically, eddy coefficients
were used to account for turbulent diffusion in both the horizontal and
vertical directions, and the density was constant. Vertically stretched
coordinates were used.
At the free surface, the wind stress was specified. At the sediment-
water interface, the shear stress TC due to currents was given by
Tc = PcfcuB I UB! (6)
where Cfc = .004 and is a dimensionless skin friction coefficient (Sternberg
1972) and ]UB| is the magnitude of the bottom flow velocity ug.
An Analysis of a Specific Event
The above investigations of sediment resuspension and deposition, wave
action, and currents were coupled with the sediment mass balance equation,
276
-------
10-2-
IO"3
Figure 8. Friction coefficient as a function of Reynolds number
for an oscillatory flow over a smooth bottom.
277
-------
N
10 Mi
I h—H
16.1 Km
WIND
DIRECTION
Figure 9a. Significant wave height in meters for a winds speed of
11.2 m/sec (25 mi/hr) and a southwest wind.
278
-------
N
WIND
DIRECTION
10 Mi
I 1 1
16.1 Km
Figure 9b. Significant wave period in seconds for a winds speed of
11.2 m/sec (25 mi/hr) and a southwest wind.
279
-------
N
WIND
DIRECTION
16.1 Km
Figure 9c. Bottom stress in dynes/cm for a wind speed of
11.2 m/sec (25 mi/hr) and a southwest wind.
280
-------
Equation (1), and used to calculate the sediment concentrations during a
specific, short-term event from 8 March to 11 March, 1976 in the Western
Basin (Sheng and Lick 1979).
In the numerical calculations, a one-mile grid was used in the Western
Basin and was coupled with an eight-mile grid throughout the rest of the
lake. Seven points in the vertical were used. Wind data at hourly inter-
vals from nine weather stations were used to generate the wind field over
the lake. A settling speed for the sediments of 0.05 cm/sec was assumed.
The results are not sensitive to the value of this parameter. To estimate
the bottom shear stresses due to the combined action of waves and currents,
it was assumed that the time-averaged shear stress Tg was given by
1 T ,
TB =T j{ |Tw+ Td dT (7)
where TW and TC are given by Equations (5) and (6) respectively and T is
the wave period.
The sediment concentration data at noon, 8 March, 1976 (obtained from
aircraft imagery and surface ships) were taken as initial data to the sedi-
ment transport model and are shown in Figure 10. The calculation was con-
tinued until noon, 11 March. The observations and the computed results at
this time are shown in Figures 11 and 12.
It can be seen that near the southern shore there is reasonable agree-
ment between the observations and the calculations. In other areas, signi-
ficant discrepancies exist. It is believed that the major reason for this
is an inadequate knowledge of the sediment resuspension and deposition rates
throughout the Basin. Recent work (Lee 1979) has indicated that significant
variations in sediment entrainment rates occur throughout the Basin. In
particular, near the mouth of the Detroit River, the entrainment rate has
been found to be less than the rate used in the calculation above by ap-
proximately a factor of ten. Other areas also showed much lower entrainment
rates than the one used above.
In addition, in the calculations, since no data was available, no ac-
count was taken of the change of entrainment rate with depth, a variation
that surely must be significant. It is also believed that the bottom
stresses are not adequately specified. More work is needed to analyze pro-
perly and especially to verify the stresses due to the combined actions of
waves and currents.
TRANSPORT IN THE CENTRAL BASIN OF LAKE ERIE DURING SUMMER STRATIFICATION:
A TWO-DIMENSIONAL MODEL
The Central Basin of Lake Erie is wide, flat, and shallow (see Figure
1). The Basin stratifies during the summer with a thermocline generally at
a depth of 16 to 20 m and only a few meters from the bottom. The epilimnion
temperatures in late summer are near 20°C while the hypolimnion temperatures
281
-------
I DETROIT R
SCALE
SUSPENDED SOLIDS, MG/L
A 10 F 100
B 20 6 125
C 30 H ISO
D 50 I 175
E 75
N
Figure 10. Near-surface total suspended solids map for the
Western Basin of Lake Erie on March 8, 1976.
282
-------
CONCENTRATION
( MG/L)
A 5
B 10
C 25
0 5O
E
F
6
H
75
100
150
200
Figure 11. Observed surface sediment concentrations for the
Western Basin of Lake Erie on March 11, 1976.
283
-------
ro
oo
CONCENTRATION
( MG/L)
A 5
B
C
0
E
F
G
H
I
10
25
50
75
100
150
200
250
Figure 12. Calculated surface sediment concentrations on the Western Basin of Lake Erie
on March 11, 1976.
-------
are near 10°C to 12°C. The stratification and the oxygen demand of the
sediments generally leads to anoxic conditions in the hypolimnion of the
Central Basin in late summer. This has a significant effect on water
quality and motivated our present work on stratified flow.
Flow in a stratified lake and the change with time of this stratifica-
tion are not well understood and much field work and numerical experiments
need to be done in order to understand and model the effects of different
parameters on this stratification and the associated currents. Three-dimen-
sional, time-dependent models of lake circulation including the effects of
thermal stratification have been developed and preliminary calculations of
stratified flow have been made. However, these three-dimensional models are
relatively difficult to use and consume considerable amounts of computer
time.
In order to do the numerical experiments efficiently and therefore be
able to investigate the effects of various parameters more readily, we have
developed a two-dimensional, time-dependent model of a stratified lake
(Heinrich, Lick and Paul 1979). In this model, a cross-section of an in-
finitely long lake is considered. It is assumed that properties of the flow
vary only in the vertical direction z and one horizontal direction x but not
in the other horizontal direction y. Coriolis forces are included. A pres-
sure gradient in the y-direction is assumed such that the net flux in the y-
direction at any time is zero.
The horizontal eddy diffusivity and conductivity have been assumed con-
stant. The vertical eddy conductivity Kv has been assumed to have the
general form
Kv =
Ks (C, •
* C2e-z/D)
* °
ZZ/D
(8)
w
where Ks is a function of the wind stress -TW and the depth h, where C], C2,
tf2, and D are empirical constants, g is the gravitational acceleration, and
a is the thermal expansion coefficient given by
a = 1.5 x 10'5 (T-4) - 2.0 x 1(T7 (T-4)2
(9)
where the temperature T is in degrees Centigrade. An analogous form for the
vertical eddy diffusivity is
A =
A
A. (C
s
,
'
C2e-z/D)
ego.
|Tw|
where As is a function of the wind stress and depth and a-j is an empirical
constant.
285
-------
Only representative calculations from this model are presented here. A
more complete set of calculations and a discussion of the model are given in
the report by Heinrich, Lick and Paul (1979).
Constant Depth Basin
To investigate the effects of various parameters without the complica-
tions due to a complex geometry, a series of calculations was made for a
constant depth basin 100 km wide and 25 m deep, dimensions which approxi-
mate the Central Basin of Lake Erie. In one such calculation, the following
was assumed: an initial, spatially uniform temperature of 6°C, a zero ini-
tial velocity, a constant heat flux of 0.003 cal/cm2-sec, a wind stress of
1.0 dynes/cm*, C-j = C2 = 1.0, 02 = .0045, a-j = 3a2 = .0135, D = 2000 cm,
As = Ks = 15.0 cm2/sec, and AH = KH = 10° cm2/sec.
In the calculations, after an initial transient, a quasi-steady state
results in which the velocities and temperatures are changing relatively
slowly. At 30 days, only a mild stratification is present as can be seen
in Figure 13a. The associated velocities are shown in Figures 13b and 13c.
At later times, the thermocline develops much more strongly and inhibits
the flow. At 90 days, the results for the temperature and velocity fields
are as shown in Figures 14a, b, and c. It can be seen that a strong thermo-
cline has formed at 15 to 16 m, the epilimnion temperatures are 15°C to 18°C
while the hypolimnion temperatures are 12°C to 14°C. The flow is largely
restricted to the epilimnion with a strong return flow near the top of the
thermocline.
Additional calculations were made to investigate the effects of varying
the heat flux, the level of turbulence, and the depth of the basin. In-
creasing the heat flux causes the thermocline to form higher and earlier,
the hypolimnion temperatures to be lower, the epilimnion temperatures to be
higher, and the convection to penetrate less deeply. A change in heat flux
from 0.003 cal/cm2-sec to 0.004 cal/cm2-sec caused a decrease in the thermo-
cline depth of 2 to 3 m, a significant variation when this change is com-
pared with an average hypolimnion thickness of only 5 m. Decreasing the
level of turbulence, i.e., decreasing the magnitude of the eddy coefficients
K and A for example from 15 cm2/sec to 10 cm2/sec, caused the depth of the
thermocline to decrease by 2 to 3 m, an effect similar to increasing the
heat flux.
Increasing the depth of the basin to 30 m, 35 m, or 40 m caused the
thermocline to form deeper. The thermocline was also more diffuse. In-
creasing the basin depth even further caused little further change in the
thermocline depth. Decreasing the basin depth below 25 m decreased the
depth of the thermocline and delayed its formation. From these calcula-
tions, it was quite evident that, for depths comparable to those found in
the Central Basin of Lake Erie, the basin depth significantly influenced the
thermocline depth and structure.
An interesting result which appeared in many of our calculations was the
formation of a second thermocline above the first thermocline some time
286
-------
WIND = 6 M/ SEC
no
00
Oi-
25
20
60
WIDTH ( KM )
100
Figure 13a. Temperature distribution at 30 days for constant depth basin.
-------
CO
00
WIND = 6 M/SEC
VELOCITY SCALES
HORIZONTAL *. -* 10 CM/SEC
VERTICAL : .025 MM/SEC
\J
5
2E 90
JE 15
Q,
UJ
Q 20
OK
C.9
(
r ^^.^, . ..y _ ^y. ^ _y _y_ .,,_ .y J J J ^ . ^
S* ^
t TTTT^TTTTTTT^*
- \ ------------ \
\ ;;::;;;:; ; : ; (
- \ I i . .,:::::. :7
\^^^^ ^^^j
\^^^..*^«.^^fc^,^
**.*.*--
) 20 40 60 80 l<
DO
WIDTH ( KM )
Figure 13b. Velocities (u and w) at 30 days for constant depth basin.
-------
WIND = 6 M/ SEC
INi
co
0
5
10
X ,5
Q.
UJ
0 20
25
20
10.0
7.5
5.0
2.5
0
-2.5
-5.0
40 60
WIDTH ( KM )
80
100
Figure 13c. Velocities perpendicular to cross-section at 30 days for constant depth basin.
-------
WIND « 6 M/ SEC
ro
UD
o
25
20
40 60
WIDTH ( KM )
80
100
Figure 14a. Temperature distribution at 90 days for constant depth basin.
-------
ro
WIND - 6 M/SEC
VELOCITY SCALES
HORIZONTAL ! -*• 10 CM / SEC
VERTICAL I
.025 MM /SEC
Q.
UJ
Q
10
15
20
25
5
f -
* <
* t
^
20
^
40
I
60
80
100
WIDTH ( KM )
Figure 14b. Velocities (u and w) at 90 days for constant depth basin.
-------
WIND = 6 M/SEC
ro
Q.
UJ
O
0
5
10
15
20
25
20
40 60
WIDTH ( KM )
80
100
Figure 14c. Velocities perpendicular to cross-section at 90 days for constant depth basin.
-------
after the formation of the first thermocline. This phenomena is mainly due
to the temperature dependence of the thermal expansion coefficient a.
Multiple thermoclines have been observed in lakes and it is interesting to
speculate whether the above mechanism may be partly responsible for the ob-
served phenomena.
Variable Depth Basin
A series of calculations was also made for a more realistic geometry
corresponding to a section across the Central Basin of Lake Erie from
Ashtabula, Ohio to Port Stanley, Ontario (see Figure 1). Only one set of
calculations, chosen to illustrate the formation, maintenance, and decay of
the thermocline, is presented here. Assumed values of the parameters are:
GI = 0.5, C2 = 1.5, a-| = a2 = .001875, D = 900 0112, As = Ks = 15 cm2/sec,
AH = 106cm2/sec, and KH = 3 x 105cm2/sec.
For the first 50 days, it was assumed that the heat flux was 0.003
cal/cm2-sec and the wind stress was 0.75 dynes/cm2 with the wind directed
from left to right. At 50 days, the temperature and velocity fields were as
shown in Figures 15a, b, and c. A thermocline has formed at approximately
16 m depth with the hypolimnion temperatures from 10°C to 12°C. Both of
these results are in agreement with observations.
Starting at 50 days, the heat flux was reduced in a linear manner such
that the heat flux was zero at 120 days and negative thereafter. At 50
days, the wind direction was reversed for a period of 25 days at which time
it was returned to its original direction, from left to right. The effect
of the wind reversal was to eliminate a second thermocline which tended to
form when the wind was not reversed. Little effect of this reversal on the
temperature structure other than this was noted.
The temperature fields after 80, 120, 150, and 180 days are shown in
Figures 16, 17, 18, and 19. At 120 days, the heat flux has decreased to
zero, the epilimnion temperatures are 18°C to 19°C while the hypolimnion
temperatures are 12°C to 14°C. At 180 days, the heat flux is negative and
the lake is rapidly cooling, the thermocline has disappeared and the temper-
atures throughout are between 13°C to 15°C.
These calculations are in reasonable agreement with observations. Addi-
tional improvements can be made to the model and the parameters can be re-
fined further so as to more closely approximate actual conditions. However,
at this point, three-dimensional effects tend to become important and
further improvements need to be made in conjunction with numerical results
from three-dimensional calculations.
Stratification obviously has a significant effect on the transport of
contaminants. Calculations are presently being made to investigate this
further. Results of two preliminary calculations are presented here. In
these calculations, a steady state flow field as assumed corresponding
approximately to that shown in Figure 16 (80 days).
293
-------
WIND = 6 M/ SEC
5.5
a.
UJ
QI6.5
22
40 60
WIDTH ( KM )
80
100
Figure 15a. Temperature distribution at 50 days for variable depth basin.
-------
ro
i-D
en
WIND = 4.5 M/SEC
0,-
VELOCITY SCALES
HORIZONTAL: -»• 10 CM/SEC
VERTICAL : | .025 MM / SEC
40 60
WIDTH ( KM )
80
100
Figure 15b. Velocities (u and w) at 50 days for variable depth basin.
-------
WIND =4.5 M/ SEC
ro
UD
cr>
Or
~ 5.5
Q.
UJ
'(6.5 -
20
40 60
WIDTH ( KM )
too
Figure 15c. Velocities perpendicular to cross-section at 50 days for variable depth basin,
-------
WIND =4.5 M / SEC
ho
UD
Oi-
19,0 18.0 17.0 16.5
0
40 60
WIDTH ( KM)
Figure 16. Temperature distribution at 80 days for variable depth basin.
-------
WIND * 4.5 M/SEC
19.0
ro
UD
CO
22
20
40 60
WIDTH ( KM )
Figure 17. Temperature distribution at 120 days for variable depth basin.
-------
WIND" 4.5 M/SEC
INS
-------
WIND * 4.5 M/SEC
U)
o
o
20
40 60
WIDTH ( KM )
80
100
Figure 19. Temperature distribution at 180 days for variable depth basin.
-------
In the first calculation, it was assumed that a contaminant was instan-
taneously released at time zero at the surface approximately in the middle
of the lake (actually at a point 53 km from the left side). The amount re-
leased was 6.11 x 103 gm/cm (corresponding to an initial concentration in
one cell volume of 100 x 10-6 gm/cm3). At 30 days, the concentrations are
shown in Figure 20. The concentrations are greatest near the surface but a
considerable amount of material is also present in the hypolimnion. At
earlier times, the contaminant is mainly restricted to the epilimnion. The
contaminant is then transported to the right by surface currents. Near the
shore, it is convected and mixed vertically into the hypolimnion and is then
convected to the left by currents in the hypolimnion and lower part of the
epilimnion.
In the second calculation, the same amount of contaminant was released
at the same horizontal location but at the bottom of the lake. Figure 21
shows the concentrations at 10 days. It can be seen that the contaminant is
restricted to the hypolimnion. This is because vertical mixing through the
thermocline in the middle of the lake is small and horizontal hypolimnetic
currents are also small. For later times, the contaminant does mix and con-
vect vertically near shore and it then appears in the epilimnion.
SUMMARY AND CONCLUSIONS
Along with a general discussion of the factors affecting the transport
of contaminants in a lake, two specific events have been analyzed: (1)
sediment transport in the Western Basin of Lake Erie, and (2) the tempera-
tures, currents, and the transport of contaminants in the Central Basin of
Lake Erie during summer stratification.
Reasonable and encouraging agreement between the results of the calcula-
tions and observations was obtained. Nevertheless, it is also obvious that
a great deal more needs to be done before realistic, quantitative, predic-
tive models of contaminant transport are developed and available for use.
For sediment transport, the research needs are: (1) more qualitative
and quantitative knowledge of the entrainment and deposition processes for
a wide variety of sediments in lakes, (2) further investigation and verifi-
cation of bottom stresses due to the combined action of waves and currents,
and (3) additional calculations of sediment transport and verification of
these calculations, especially for long-term events.
For stratified flow, it is necessary to further investigate, model and
verify: (1) short-term events such as upwelling and downwelling due to
strong winds, (2) the formation, maintenance, and decay of the thermocline
and associated currents, and (3) the dispersion of contaminants during both
short-term and long-term events, especially when there is coupling between
the stratification, sediment-water exchange, and biochemical processes in
the water column.
301
-------
WIND =4.5 M/SEC
Oi-
to
o
22
40 60
WIDTH ( KM )
80
100
Figure 20. Contaminant concentration at 30 days after release at surface and 53 km from left side.
-------
ACKNOWLEDGEMENTS
This research was funded by the U.S. Environmental Protection Agency.
Mr. David M. Dolan was the Project Officer.
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DiToro, D.M. and J.F. Connoly. 1979. Mathematical models of water quality
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DiToro, D.M. 1979. Species dependent mass transport and chemical equili-
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Fisher, J.B., P. McCall, W. Lick, J.A. Robbins. 1979. The mixing of lake
sediments by the deposit feeder, Tubifex tubifex. To be published.
Fisher, J.B. and W. Lick. 1979. Effects of tubificid oligochaetes on the
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water interface. To be published.
Fukuda, M.K. 1978. The entrainment of cohesive sediments in fresh water,
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Fukuda, M.K. and W. Lick. 1979. The entrainment of cohesive sediments in
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Grant, W.D. and O.S. Madsen. 1979. Combined wave and current interaction
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Heinrich, J.C., W. Lick, and J. Paul. 1979. The temperature, currents, and
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Kajiura, K. 1968. A model of the bottom boundary layer in water waves.
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Kang, S.W., P. Sheng, and W. Lick. 1979. Wave generation in Lake Erie. To
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Lee, D.Y. 1979. Resuspension and deposition of lake sediments. M.S.
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Lick, W., J. Paul, and P. Sheng. 1976. The dispersion of contaminants in
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Mortimer, C.H. 1971. Chemical exchanges between sediments and water in the
Great Lakes. Limnology Oceanography,, 16.
Owen, M.W. 1978. Problems in the modeling of transport, erosion, and de-
position of cohesive sediments. Vol. 6S The Sea, Ed. Goldberg.
Pierson, W.H., G. Neumann, and R.W. James. 1955. Practical methods for ob-
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Pore, N.E. 1979. Verification of automated Great Lakes wave forecasts of
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Robbins, J.A. 1979. Cesium-137 in the sediments of Lake Huron. Presented
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Robbins, J.A. and D.N. Edgington. 1979a. The distribution of Cesium-137 in
Lake Erie. Presented at the Twenty-Second Conference on Great Lakes
Research.
Robbins, J.A. and D.N. Edgington. 1979b. History of plutonium deposition
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Sheng, P. and W. Lick. 1979. The transport and resuspension of sediments
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Sly, P.G. 1978. Sedimentary processes in lakes: Chemistry, geology,
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Sternberg, R.W. 1972. Predicting initial motion and bedload transport of
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Sverdrup, H.U. and W.H. Munk. 1947. Wind, sea, and swell: Theory of rela-
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Terwindt, J.H.J.
Interactions
1977. Deposition, transportation, and erosion of mud.
Between Sediments and Freshwater, Ed. golterman.
In
Thomann, R.V. 1979. A trophic length model of the fate of hazardous sub-
stances in the aquatic food chain. To be published.
Thomann, R.V., R.P. Winfield, D.M. DiToro, D.J. O'Connor. 1976. Mathemati-
cal modeling of phytoplankton in Lake Ontario,
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I and II. U.S. Environ-
Thomas, R.L., A.L.W. Kemp, and C.F.M. Lewis.
tion and characteristics of the surficial
Sed. Petrol. 42.
1972. Distribution composi-
sediments of Lake Ontario. J.
Thomas, R.L., J.M. Jaquet, A.L.W. Kemp, and C.F.M. Lewis.
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1976. Surficial
305
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SECTION 16
SELF-ORGANIZATION OF THREE-DIMENSIONAL MODELS OF WATER POLLUTION
A.6. Ivakhnenko and G.I. Krotov^
SYNTHESIS OF MATHEMATICAL MODELS USING THE METHOD OF SELF-ORGANIZATION ON A
DIGITAL COMPUTER
Mathematics enhances the deductive approach to the solution of problems.
The construction of each solution is based on processing of an a priori
fixed system of input axioms and initial data. The basic shortcoming of the
deductive method is the need for a priori production of sufficiently com-
plete and accurate information on the object. Imprecise knowledge of even a
single characteristic can make the result of modeling incorrect (e.g., an
output quantity may increase when it should decrease, etc.). Furthermore,
the problem of the unique and optimal structure of the model is not solved.
A variety of reliable models may be composed for one and the same object.
In practice, this means: The number of models will be equal to the number
of modeling attempts, i.e., each model is a unique invention of its author.
According to the theorem of incompleteness of Godel, the so-called in-
ternal criteria, utilized by deductive mathematics, are not suitable for se-
lection of the structure of a model of optimal complexity. Any internal
criterion leads to the false rule: "The more complex the model, the more
precise it is." An example of an internal criterion might be the mean-
square error calculated for all points of experimental data as well as cer-
tain other statistical criteria, including significance, elasticity, and the
like (Ivakhnenko 1978).
In contrast, external criteria pass through a minimum as the model be-
comes increasingly complex, and this minimum defines a model of optimal com-
plexity. The new inductive method of modeling--self-organization of models
by computer—is directed toward comprehensively decreasing the volume of £
priori information required for modeling.
The experimental points assigned in the table of initial data contain
information on quite a variety of factors. For example, in modeling bodies
of water, extremely complex processes of self-purification of the water and
transformation of matter (which are still unclear) are reflected in the ex-
perimental data. This means that the mathematical model of such phenomena
Academy of Sciences, 252207 Kiev, USSR.
306
-------
can be found by trail and error, without deep knowledge of the mechanism of
self-purification and transformation.
In contrast to deductive mathematics, methods of self-organization are
based on the use of external criteria: a) regularity, b) minimum bias, c)
balance of variables, d) convergence of a multi-step prediction and e) com-
bined criteria. It is the use of external criteria which enables a unique
model of optimal complexity, corresponding to the minimum of the selection
criterion to be obtained (Figure 1). The internal criteria play a secondary
role in this case. They are used for ranking of models in a sequence for
successive testing on the basis of an external criterion.
The principle of self-organization can be formulated as follows: With
gradual complication of models, certain criteria (which have the property of
"external supplementation") pass through a minimum. The computer, by run-
ning through the models, finds this minimum and, consequently, indicates the
unique model of optimal complexity.
SELECTION CRITERIA
According to Godel's theorem of incompleteness, the task of identifica-
tion of the structure and parameters of a model is not properly stated:
Only with some external supplementation can a unique solution be achieved.
In all works on modeling in which, by one means or another, a unique model
has been achieved, certain supplementary external information has been used.
The basic shortcoming of contemporary modeling is that the external, supple-
mentary information selected is not adequate for the task at hand. The fol-
lowing criteria can be selected as expedient supplements for the theory of
self-organization of models:
1. The criterion of regularity: The mean square error calculated
at new points is not used to produce estimates of the coeffi-
cients of the model:
NB
* (qtabi,i -V2
A(B) = 1-! +min (1)
B
E qtabl,i
i=l
where:
NB is the number of points of an individual test sequence
of data;
Qtabl are the tabular values of the output variable;
q are values calculated using the model.
307
-------
OJ
o
CO
a:
LLI
H
DC
o
Selection of structure by
external criterion
Selection of structure by
mixed criterion
AU)
COMPLEXITY OF MODEL STRUCTURE (S)
Figure 1. Characteristic curves of change of external, mixed and internal criterion with increasing
complexity of model structure (No. of terms and power of polynomial).
-------
2. The minimum bias criterion: Requires maximum agreement of the
values of the output quantity of two models obtained using two
different parts of the table of initial data NA and NB:
a(NA+NB) 2
ncn = 1=1 NA+ND ' — " min (2)
\ 2
a 1^1 qtabl,i
where qA and qe are the values of the output quantity cal-
culated for all (NA - NB) points using the data of the
model obtained in parts of the sample NA and NB, respec-
tively; NA and No are, e.g., points with even and odd ordi-
nal indexes'. If the input data contain noise, it is re-
commended that the summation interval be increased: Coef-
ficient a should be selected proportional to the intensity
of the noise. It is taken within the limits a = 1.5 -^ 3.0.
3. Criterion of balance of variables: Used for simultaneous
long-term prediction of several variables. Frequently, a
certain interrelationship of variables is known a priori,
and the criterion requires that this relationship be ful-
filled for the future over the extrapolation interval.
Let us assume that it is known that q = f(x], X£, X3).
Then the balance equation is q - f(x,, X2> x3) = 0.
The criterion of balance of variables
following criterion is more resistant to interference
aN
.Zn (qA,i ' qAB,i)(qB,i
« qtabl,i
where q^, qg, q/\+g are the value of the output quantity calculated using
the model after training on sample N/\, NB and N/\ and NB together.
Ivakhnenko et aK (1978) compare the interference stability of various
versions of the minimum bias criterion.
309
-------
b =
N2
E [q. -
•j=|\|"| '
3 .
I , O, I
U2
rrnn,
(3)
i=Nl
where: N-| = Ty - At, N2 = Ty + At
and (NI - N2) is the time interval, which includes the prediction point.
The criterion of balance of variables frequently leads to ambiguous selec-
tion of several models; in order to assure unambiguity of selection, it is
used in combination with other criteria.
In problems in which the balance of variables is unknown, it is "dis-
covered" in advance by the use of the minimum bias criterion. In recent
studies, combined selection criteria have been widely used, combinations of
the individual criteria just described. The reason is that the selection of
a criterion depends on the purpose of solution of the problem and should be
entirely the privilege of the person who is designing the model. Usually,
the motivation of statement of the modeling task is in itself not completely
clear and precise. This leads to equal uncertainty in the recommendations
for selection of combined criteria. Still, the following rules can be
formulated:
1. For models used for short-term prediction, the combined cri-
terion "no bias plus balance of variables" is recommended:
K, =
mm;
(4)
For models
criterion '
intended for long-term prediction, the combined
no bias plus balance of variables" is recommended:
K2 =
mm;
(5)
For differential (finite difference) models, in which long-
term prediction is obtained by means of step-by-step inte-
gration, the combined criterion "no bias plus convergence of
the procedure of prediction" is recommended:
i2(N)
mm,
(6)
where: i'
i - qtabl,i)
qtabl,i
310
-------
is the sum of the squares of deviations of the curve of the process from the
curve of a repeated (multi-step) prediction, begun from the first point of
the interpolation interval. N is the number of points in the table of ini-
tial data. The criterion of the minimum bias participates in all combined
criteria, since bias is to be avoided in modeling.
Combined criteria not only provide a stable and unique prediction, but
also make its selection "sharper". Individual criteria frequently do not
provide a sufficiently sharp minimum.
Desiring to satisfy a number of requirements, other types of combined
criteria may be used, containing two or three particular requirements. For
example:
K = Vn+
2
4 ~ V "srn • u • > v; ' »"" . /j\
All of the component criteria should be reduced to the dimensionless form
with range of change from 0 to 1. This requirement is not obligatory if the
criteria are applied sequentially for selection of the model.
THE PROBLEM OF MODELING OF PHYSICAL FIELDS ON THE BASIS OF EXPERIMENTAL DATA
Modeling of physical fields has as its purpose identification (construc-
tion) of a picture of the field, and estimation of its parameters. Mathema-
tical models of the field are necessary for prediction of its subsequent
development, extrapolation to neighboring areas, and for synthesis of the
optimal control rule for the field. An example of a physical field to be
modeled might be the field of pollution in a body of water. This example is
used below to explain some very general methods of field modeling.
The number of control stations recording water pollution increases from
year to year. Nevertheless, there is not yet available sufficiently precise
data with a small number of measurements. This virtually eliminates the
possibility of using probabilistic models based on the construction of
multidimensional distributions. The possibility of modeling is related to
the use of inductive methods of self-organization of models by computer, '
based on running a large number of models (in the form of finite-difference
equations) on a computer. Actually, this method requires a relatively small
number of points of measurement, and when the proper external selection cri-
terion is used, it has significant interference resistance.
Obviously, a description of methods applicable to the problem is neces-
sary to clarify the conduct of subsequent measurements. Thus, the peculi-
arities of measurement determine the selection of modeling methods, while
the development of methods makes certain corrections to the process of in-
formation collection.
A field of pollution is constructed using the data from observations of
control stations, and data on the location, intensity, and time of discharge
of pollutant substances. The selection of the output quantity and the argu-
311
-------
ments defines the "statement of the task of modeling". This depends on the
purpose of modeling (interpolation, extrapolation, or prediction), and upon
the availability of initial information to be used to synthesize the model.
In this work, algorithms are presented for the solution of three prob-
lems of modeling of the field of a pollutant. In the following, statements
are 1) based on the data collected at control stations; 2) based on in-
formation on the discharge of pollutants, and 3) based on the two types of
data, taken together.
In the first problem, the equation of turbulent diffusion (in its
finite-difference form) is not assigned a_ priori, but rather is found from
the experimental data on the principle of self-organization, by processing
standard samples and their nonlinearity. The number of terms in the com-
plete equation is usally significantly greater than the number of experi-
mental data points. Estimates of the coefficients are found by the least-
squares method. The solution is found by a multi-step integration process
of the finite difference equation. It is not necessary to know its solution
in analytic form.
In the second problem, the arguments of the finite-difference model are
selected so that the form of the equation corresponds to the "input-output
matrix" (Tamura and Kando 1977; Yurachkovskiy 1977). It is possible to re-
fer to the area of interpolation as a portion of a space lying within a
polyhedron, the points of which are the outer control stations. The area of
extrapolation lies outside this three-dimensional polyhedron (Figure 2). In
prediction, the area of interpolation lies within the time interval of the
experimental data. The area of prediction lies in the future, outside the
area of interpolation of the process.
One characteristic feature of the area of interpolation is that, accord-
ing to the Weierstrass theorem, any curve described by a sufficiently com-
plex function fits the experimental points rather well. Only in the area of
extrapolation and prediction do the curves diverge rapidly, forming a "fan"
of predictions. The model should correspond to the function (or solution of
the differential equation) which corresponds to the future course of the
process in the area of extrapolation or prediction longer than the others.
Depending on the principle of selection of arguments, three different
tasks can be distinguished. The first is based on the "principle of con-
tinuity or near action"; the second on the opposite "principle of distant
action"; the third task is combined, using both principles simultaneously.
The model may be a point, one-dimensional, two-dimensional, three-dimen-
sional, multidimensional, algebraic harmonic or finite-difference (differen-
tial). If the model is constructed according to observed data in which the
spatial distribution of the sensors (or control stations) is not indicated,
it is called a point model, hence there is reason to act as if all data were
collected at one point in space.
The model is called spatial or a field model if the initial data in-
clude information on the location of the measurement stations in space.
312
-------
oo
CO
'*
1
-2,0
*
X
X
-1,0
X
X
0.+,
0,0
0,-1
&
1
+1,0
1
\
\
—
\
\
\
\
V X
3
Figure 2. Use of simple models for extrapolation of the area of pollution between three
measurement stations 1, 2 and 3 in the direction of the x axis.
-------
Spatial models require at least three stations on each axis. Models with
one argument are called one-dimensional, while models with several arguments
are called multi-dimensional.
The same table of experimental data can be used to construct both the
algebraic (or harmonic) model, e.g.,
q = f(t), (8)
and the finite-difference model, e.g.,
q-l» q-2' ••" q-T)'
The difference is only in the order of reading the data of the table. When
methods of self-organization of models are used, preference must be given to
the construction of finite-difference models, which are analogs of differen-
tial equations. The problem is that in problems of mathematical physics,
linear differential equations correspond to nonlinear solutions. In pro-
cessing models, it is easier to "guess" the linear nature of a finite-dif-
ference equation than the nonlinearity of its solution, which reduces the
time required to process the reference functions. Furthermore, the solution
of the finite-difference equation is more varied than the variation of
curves corresponding to the same algebraic function. Therefore, further
analysis will consider exclusively finite-difference models.
The symbols for the model arguments may be either the instantaneous
values of the coordinates t, x, y and z. or the values of measurements of
control stations q(i) and discharges Av~0 (instantaneous and delayed argu-
ments), where: J(T) .]'(T)
i is the control station number;
j is the number of the pollutant;
T is the delay in measurement of the argument in question.
(1)
Examples: ^2(-3) ""s the discharge of the second polluting com-
ponent of the first source, measured only a delay of
3 time cycles;
(2)
ql(0) is the measurement of the first pollutant at the
second station in the current moment of time.
If there is no subscript, this means that only one pollutant is being con-
sidered.
FIRST TASK: MODELING OF A FIELD USING FINITE-DIFFERENCE ANALOGS OF THE
EQUATIONS OF TURBULENT DIFFUSION, BASED ON THE PRINCIPLE OF CLOSE ACTION
The equations of turbulent diffusion are based on the principle of con-
tinuity or close effect of neighboring particles, forming a physical field.
The elementary model in the theory of finite-difference equations refers to
314
-------
a geometric figure showing just which neighboring points of the field are
used for construction of the equation structure. Changes in two neighboring
cells (cubes) of the field are sufficient to represent the first derivative
(on the axis of placement of the cells), while three cells are sufficient
for the analog of the second derivative (Ivakhnenko and Krotov 1977).
^Examples of the interrelations of linear differential equations, their
finite-difference analogs, and elementary models are shown in Figures 3 and
4.
For physical fields, certain deterministic field models are usually
known, mathematically expressed in the form of differential or integral-dif-
ferential equations. These equations of the deterministic theory can be
used as predictors for selection of the form of the following: a) argu-
ments, and b) functions participating in the full description of the model
(combined method). The deterministic equation of diffusion indicates a cer-
tain model which should be used as the basis of selection. The full model
is produced from the model corresponding to the deterministic equation of
diffusion by incrementing its dimensions by one or two cells on each axis.
This means that, in any case, the order of the equation is increased by one
(or two) in order to allow the algorithm to select a more general rule.
In modeling, two basic types of modeling algorithms are used:
a) a combinatorial algorithm which equates the various coeffi-
cients of the full polynomial to zero in turn, or
b) a multiple-row (threshold) algorithm.
SELECTION OF MODELS AND DEGREE OF NONLINEARITY (DOUBLE SELECTION)
When the number of components of the full polynomial is less than 20,
the combinatorial algorithm is used to select the model of optimal com-
plexity (setting all possible combinations of terms of the full polynomial
equal to zero), requiring that all possible partial polynomials be con-
sidered. In this case, it is sufficient to analyze only the one, most com-
plete, clearly sufficient model, the power of which is equal to the number
of arguments; selection of models is therefore not required. When the num-
ber of components of the full polynomial is greater than 20, the multi-row
(threshold) modeling algorithm is used. Testing of certain partial poly-
nomials may be omitted in this case.
In order to expand the selection, one should use:
a) processing of particular models (composition of arguments), be-
ginning with the simplest two-cell model (Figure 3);
b) processing of all possible polynomials for each of the simple models
using the multiple-row modeling algorithm with renaming of vari-
ables, ending with a polynomial in which the highest power is equal
to the number of arguments indicated in the simple model in question.
315
-------
FIELD OF INTERPOLATED DIGITAL DATA
n>
CO
r+ 73
3" C
(D 3
"O <-+•
-s ~=s-
rr> -s
Q. O
— i. c
n ta
-2,0
-1,0
0, -1
0,0
0,+ )
no
-»
Simple models
Data tables
Gradual complication of model and analysis
o o
33
O
.a a.
4- (D
o>
ci- -S
+ O
— ' Q.
c
o o
•• rt-
X -••
+ o
— ' =5
W
O O
•• -h
J3
_1. Q.
C-i. Q)
OJ
CT
(D
a
-5
0,0
+ 1,0
0.-1
0,0
+ 1,0
-1,0
0,0
0.-1
0,0
+ 1,0
-2,0
-1,0
0,0
0,+1
+1,0
1+lfl
^0,0
"+1,0
^0,0
"o,+i
Vi
^+1,0
\0
^0,+1
V.
^-1.0
9+,,o
^0,0
9o.+i
90,-i
9-1,0
9-2,0
dq
-?- + atq = f(tx)
at
— + a\—
' 9o,+p 90,-i. 9_,
-------
FIELD OF INTERPOLATED DIGITAL DATA
-1.-2
+1,0
Simple models
Data tables
Gradual complication of model and analysis
co
+1,0
-2,-3
+1,0
-1.-2
+1,0
9+1,0
V-i
dq
-
dt
dq
dx
9+1,0 = /I ('+1,0'
a2?
= f(tx)
> +
dt
a
-------
This double processing of simple models and nonlinear polynomials pro-
vides a more complete examination of the set of possible partial polynomials
in problems in which, due to the great number of arguments, it is impossible
to use the simpler combinatorial modeling algorithm.
Double selection produces three types of tables of data. It is neces-
sary to distinguish the table of initial data from the testing stations, the
table of interpolated initial data, and the table of simple models, obtained
as a result of shifting of the simple models over the digital field of the
table (Figures 3 and 4). Each position of a simple model on the number
field of interpolated data corresponds to one row (point) on the table. The
points are ranked according to the dispersion of the output value (in the
left part of the equation), and are used to determine the selection crite-
rion. To do this, the row of simple model tables is divided into parts A
and B.
The division into parts is performed not on the initial digital field of
data, but rather on the tables obtained in the course of shifting the
simple models over the field. For each simple model, a unique table of ex-
perimental data is produced. There will be as many tables as there are
simple models which are compared against each other. The simple model which
provides the greatest depth of the minimum of the combined criteria is best.
THE CONVERGENCE OF MULTI-STEP INTEGRATION OF FINITE-DIFFERENCE EQUATIONS
The convergence of integration may be defined as follows: Each finite-
difference equation is a discrete analog of a certain differential equation
with continuous derivatives. If the curve obtained as a result of step-by-
step integration of the finite difference equation coincides closely enough
with the curve of the analytic solution of the differential equation (or
with a decrease in the discretization step—coincides with it precisely),
the procedure for step-by-step integration converges. If there is no con-
vergence, the step-by-step solution frequently becomes infinite or falls to
zero.
Convergence depends on: a) the degree of bias of the estimates of the
coefficients obtained by the least-squares method; b) the relationship of
discretization steps of the axes; c) the form of the simple model; and d)
the accumulation of computational errors (e.g., in calculation with a small
number of significant digits). It is recommended that the discretization
step be decreased until the accuracy of the model increases noticeably. If
the steps are too small, the accuracy decreases once more.
It is known that simple models with a small number of points are more
stable than branched simple models. Ivakhnenko and Krotov (1977) have sug-
gested that the simple models be used with an "implicit" plan of step-by-
step integration, as a method of increasing convergence.
All partial simple models and the full simple model are evaluated on the
basis of the combined criterion "bias plus error of step-by-step integra-
tion":
318
-------
sm mm
'HNL^
where
is the index of bias:
"A+B
(qA -
sm
, where a = 1.5 •=• 3.0,
a ] qtabl
while i(N) is the index of stability of integration:
N
KM) =
A+B
Z (q
1
tabl
N
A+B
Z
1
The component of the combined criterion n assures selection of an equation
with minimum bias. Component i(N) is directed toward selection of a stable
step-by-step procedure for construction of predictions and extrapolations.
Frequently, the requirements nsm •* min and i -> min are not contradictory,
i.e., they lead to the selection of the same model. If there is divergence,
a compromise solution must be found (Figure 5). The use of modeling algo-
rithms does not require a priori knowledge of whether there is an algebraic
solution of the differential (finite-difference) equation. Neither is it
necessary to know the explicit algebraic form of this solution. The study
is conducted at the level of the differential equation, and does not reach
the level of its solution.
SUCCESSIVE APPLICATION OF CRITERIA WITH DOUBLE SELECTION
It has been experimentally established that models obtained for a given
composition of arguments, i.e., for one simple model, differ comparatively
little in the minimum value of the criterion of regularity and differ quite
sharply in terms of the minimum criterion of the minimum bias. Geometri-
cally, this is expressed in that points corresponding to the models compared
form lines on a plane, similar to spectrograms. The criteria which should
be used for selection of simple models are: regularity, stability of
balance of variables:
A(B) -^ min, i(N) -»- min, B -> min,
319
-------
o
cc
in
H
CC
o
DO
D
I o
I
I
I o
I
I o
\ s
\
I I
Figure
Selection of a model of optimal complexity based on the combined
criterion
Kl -
+ i'
i'2(N)
min.
0 = models with same composition of mean arguments.
320
-------
while for selection of nonlinearity of the polynomials, the criterion of
minimum bias should be used:
nsm + min.
This recommendation leads to a reduction in program length, and to more ac-
curate discrimination of models.
The examples show that successive application of criteria is not com-
pletely equivalent to the application of a combined criterion in all selec-
tion sequences. It yields somewhat simpler models with less variety of
arguments.
Example 1. Double Selection of Point Models
For a point problem with one variable q, selection of simple models is
equivalent to selection of the number of delaying arguments considered:
n
First simple model: q+i = f(qg) = ag + a-jq0 + a2qQ,
2 2
Second simple model: q+1 = f(qo>q_i) = ag + al% + a2cl-l + a3qo + a4q-l +
+ a5qOq-l>
Third simple model: q+] - f(q0,q_-|,q_2) = ao + alqO + a2q-l + a3q-2
?l + V-2 + a7qOq-l + a8V-2 +
a9q_lcl_2-
Selection of non linearities in this example refers to selection of par-
tial polynomials produced from the full polynomials using a combinational
algorithm, i.e., by setting various combinations of coefficients equal to
2 , where n is the index of the last coefficient (n = 0, 5, 9, . ..). As
soon as the number of components of the full polynomial exceeds 20, multi-
ple-row modeling algorithms must be used.
For a point model with two variables q and x, we have:
First simple model: q+-| = f(qQ,XQ),
Second simple model: q+, = f (qo'xn'q-Vx-l^s
Third simple model: q+1 = f (q0,x0,q_-|,x_-|,q_2,x_2), ... etc.
Here, the transition to a multiple row algorithm occurs more rapidly,
since the number of terms of the full polynomials is significantly greater.
Otherwise, the processing of simple models and nonlinearities have the same
form.
321
-------
Example 2. Double Processing for Three-Dimensional Models
Using the principle of continuity (or similar action) for spatial
models, several equations are composed (based on the number of coordinates
of time and space). Furthermore, the number of arguments includes variables
with different time delays and shifts in space. A two-dimensional spatial
model is analyzed in detail in the following example.
BASIC STATEMENTS OF THE TASK OF MODELING (SELECTION OF ARGUMENTS)
First Task: Modeling of the Field of a Pollutant Exclusively on the Basis
of Data Provided by a Few Measurement Stations
The first statement of the task of modeling of a field has as its pur-
pose the construction of the field of a pollutant, not only in the area of
interpolation (encompassed by the measurement stations), but also over a
significant distance outside this area, i.e., construction of an extrapola-
tion of the field and prediction of its development in time. It is assumed
here that the emission of the pollutant changes relatively little with time
and, therefore, information on the pollutant is not directly considered. It
is contained indirectly in the data of the measurement stations. Therefore,
in the first statement, the set of arguments contains only the data of the
measurement stations. If there are few stations, the data can be inter-
polated within the area of interpolation in order to produce the mean values
of the pollution index in all squares of the field t-x. Two-stage inter-
polation (Ivakhnenko and Krotov 1977) implies that first the data of the
stations are interpolated (between stations) by the least-squares method.
The "digital field" thus produced is used to synthesize finite-difference
models suitable for prediction and extrapolation outside the area of inter-
polation.
For brevity of presentation of the overall method, the equations are
shown below for the two-dimensional model, q = f(xt), only. It is easy, by
analogy, to compose similar equations for the three-dimensional (or four-
dimensional) field, q = f(x,y,z,t), as well.
The equation for the parameters of pollution with simple model 5 (Figure
3) can be written in vector form as follows:
a) for prediction;
0) = "jnp [qK(0,0)'qK(0,-l)'qK(0,+l)'qK(-l,0)'qK(-2,0)'"-'qK(-T,0)]
(15)
b) for extrapolation:
tqK(0,0)'qK(-l,0)'qK(+l,0)'qK(0,-l)'qK(0,-2)"'"qK(0,-T)]
322
-------
where otj is the vector of polynomial functions (linear, nonlinear without
covariation or with incomplete consideration of covariation) of dimensiona-
lity j.
In all of the equations presented, the vector of measured external vari
able contained in the right portion can contain both the variables them-
selves, their delaying values, and covariations (in pairs), which can be
considered individual variables, e.g.,
qi(o)q2(o) ••• qi(-
In the equations presented below, the covariation vector will not be
shown for simplicity.
q and q are vectors of the parameters of pollution with dimensionality
j, k = 1, 2, 3, . .. For example, to predict and extrapolate the first com-
ponent of pollution, considering equations (la) where j = 2, k = 2, T = 2,
we obtain:
a) for prediction
lnp [ql
q2(0,-l)»q2(0,+l)'q2(-l,0)'q2(-2,0)] (15a)
b) for extrapolation
ql(0,+l) = a!3K [ql
= a
q2(-l,0)'q2(+l,0)'q2(0,-l)'q2(0,-2)]
The "Source-Functions" in the First Task (Consideration of Settling of
Pollutant Particles and Lateral Influx of Pollutant, as well as ExteTnal In
fluences of the Environment
In the two-dimensional problem analyzed above, we did not consider the
process of diffusion of a pollutant from layer to layer, and vertical set-
tling of particles. In order to consider these phenomena, the "source func
tion" or "residue" is introduced, a function of the coordinates of the
square in which the output quantity is measured. Furthermore, the source
function also includes the perturbing effects P of the wind, V of current
on the x axis, etc.
For the parameters of the pollutant, in the general statement, we ob-
tain:
323
-------
the prediction equation
+ ajnp [qK(0,0)'qK(0,-l)'qK(0,+l)»qK(-2,0)'qK(-l,0) ..... qK(-T,0)]
(16)
and the extrapolation equation
+ aj3K [\(0,0)>qK(-l,0)>qK(0,-l)'qK(0,-2)'-'-'qK(0,-T)]
Simultaneous step-by-step integration of Equations (16, 16a) enables the
generation of a prediction (on the t axis) or an extrapolation (on the x
axis) of the field of the pollutant.
Second Task: Modeling of Fields Using Finite-Difference Equations Corre-
sponding to the Expanded "Input-Output" Matrix (Based on the Principle of
Long-Range Action)"
In addition to the analogs of the equations of turbulent diffusion, the
modeling of physical fields also involves equations corresponding to the
functional "input-output" matrix (second task). Performance of the second
task of field modeling is easy if a prediction of changes in the field of
the pollutant within the area of interpolation is required when there is a
change in the discharge with time. The solution of the second problem does
not provide an accurate extrapolation of the field far beyond the limits of
the area.
The second task has three statements:
1. Modeling by "input-output" matrix, using information of measurement
stations alone:
In the first statement, the following prediction equation is used for
the i'th station:
m
sjH
where qs = Eq,q^M },q*(_T) ,. . •
324
-------
qs are the parameters of the pollutant;
a is the operator of the polynomial functions;
m is the number of stations.
For example, where j = 2, T = 2, m = 3n (n is the number of components),
- a ra(2) _(2) (2) (2) (2) (2)
" a21 Lql(0)'ql(-l)'ql(-2)'q2(0)'q2(-l)'q2(-2)J +
ra(3) a(3) _(3) .(3) _(3) (3) ,
31 Iql(0)'ql(-l)q2(-2)>ql(-2)>q2(0),q2(-l)] (17a)
The pollution at the itn station (or point of the field) depends directly on
the value of the indexes measured from the closest neighboring stations,
with various time delays.
2. Modeling by "input-output" matrix, using information on discharge of
pollutants only:
In its second statement for the i^h station, the prediction equation is
as follows:
where: q are the pollution parameters,
X is the discharge vector,
>s _ r,s ,s ,s ,s ,s ,s ,
" L l(0)'"l(-l)>;vl(-T)' •"' An(0)'xn(-l)'xn(-T)]>
BSJ is the operator of the polynomial functions,
p is the number of sources.
For example, where n = 2, T = 2 and p = 2, we obtain
(2) (2) (2) (2) (2) (2)
(18a)
3. Modeling considering information on measurement stations and infor-
mation on the discharge of sources of pollution:
In the general (third) statement of the task, the "input-output" matrix
is expanded by introducing the data from the measurement of nearby stations
in addition to the discharges of pollution sources.
325
-------
In its general statement, the prediction equation for the i^ station
is:
The "Source Function" in the Second Task (Consideration of External Ef-
fects)
In order to consider external effects (the adjusted wind or current,
temperatue, relative humidity and other physical quantities), it is recom-
mended that Equations (17) and (18) or (19) be supplemented with the follow-
ing source function in the right part:
Q(V, P, ...).
As a result, we produce complete descriptions, in the form of sums of poly-
nomials:
First statement:
( i ) P c
qj(+1) - Q^V'.P,...) + z 3SJUS); (20)
Second statement:
m <;
£ asj(q ); (20a)
or in the general (third) statement:
(i)
The particular polynomials are produced as versions by setting various terms
of these full descriptions equal to zero.
The additive introduction of "source functions" to the full equation was
shown earlier. Physical fields are encountered in which this function is
included multiplicatively or is mixed.
For example,
m c P
qr+n = (UV'.P) + Q?(V',P) ( Z a (qs) + z R.
1 U ' * 5=1 SJ S=l SJ
(20b)
326
-------
It is necessary to select a multiplicative or mixed version of application
of the function depending on which will provide the deepest minimum of the
criterion of selection, or on the basis of a priori information.
Third Task: Modeling of Fields with Simultaneous Application of the Princi-
ples of "Near" and "Distant" ActTon
The third task has three statements:
1. Modeling using the principle of "near" action and the "input-
output" matrix, considering information from the measurement
stations. In this statement, a combination of the first two
problems is used.
2. Modeling using the principle of "near" action and the "input-
output" matrix, considering information on discharges, a com-
bination of the first two tasks.
3. Modeling using the principle of "near" action and the expanded
"input-output" matrix, considering both data from the measure-
ment stations and the discharges of pollutants.
Examples of Solution of Two-Dimensional Problem with Fixed Number and Posi-
tion of Measurement Stations
Three variables were measured at three points in a body of water at the
depth of 0.5 m: dissolved oxygen pk = q(t), biochemical oxygen demand BOD =
q(t) and temperature x(t). The measurements were performed from mid-May,
every two weeks, for a total of 8 times. The results are cited elsewhere
(Ivakhnenko and Krotov 1977).
Two-stage interpolation:
Using quadratic interpolation, the number of measurement points was
greatly increased from 3 x 8 = 24 to 16 x 16 = 256 (Ivakhnenko and Krotov
1977). The permissibility of this approach is based on the fact that the
smooth functions are well approximated within the area of interpolation.
However, for extrapolation (on the x axis) and prediction (on the t axis),
it is necessary to use finite-difference equations derived from the modeling
algorithm.
When the combined criterion k] is used as the external criterion, the
following types of interpolating polynomials are produced:
q+1.Q = 8.109 - 0.136x + 0.033T + 0.06U - 8'10~5xT +
+ 6.54-10~3x2 - 8-10~5t2;
K3min = °'8728' A(B)min = °-6500'
327
-------
Z+1 = - 13.017 + 1.725T + O.OUt + 1.8»10~7xt - 1.3-10"4x2 -
- 0.056T2 - 5.6«10~4t2;
K3min = °-985> A^B^min = Ot362;
The task of prediction is performed by identification of the finite-differ-
ence equation
q+i,o = f^xo,+i'to,+i'qo,o'q-i,o'q-2,o'qo,-i'qo,+r
Z0,0'Z-l,0'Z-2,0'Z0,-l>Z0,+l) '
In order to produce the table of initial data, simple model No. 5 (Figure 3)
is moved step-by-step over the digital field of the table derived from
quadratic interpolation in direction t. The task of extrapolation is per-
formed by means of identification of an equation such as
Z0,+l = f(x+l,0't+l,0'Z0,0'Z0,-l,Z0,-2'Z+l,0'Z-l,0'
qO,0'qO,-l'qO,-2'q+l,0'q-l,0T) '
To produce the data table, the simple model was shifted in the direction of
the x axis, using the combined criterion:
o o
ncm + A (c) '
A(c)—the error for an individual examination sequence. The following
equations were obtained for extrapolation.
qO,+l = 9'524 + °'094 Z0,0Z0,-2 + °-003 Z0,0T0,0 + °-230 Z0,-lZ0,-2
- °'140 Z0,-1Z+1,0 * °'027 Z0,-2 + °'039 Z0,-2 q-l,0
- °'140 Z0,-2Z0,-1 -°-029Z-l,0-°-105Z-l,0Z+l,0'
= 0.38.
= - 0.18 + 1.157 Z - 0.112 Z + 0.01
328
-------
A(c) = 4.586-10'4; A(B)min = 0.05.
for prediction:
q+M = 12.431 - 4.648 Z^ + 0.162 q_2f0Z^0 + 0.840
- °-003 zo,oz+i,o + -,
K = 1-182-10-3; n = 2.025 '10'4;
mn
A(c) = 1.165-10"3; A(B)min = 0.037.
= 1.115 + 0.020 ZT + 0.004
A(B) = 0.064; A(c) = 0.937-10"3.
EXAMPLE OF COMPARISON OF MODELS ON THE TEST PROBLEM OF CONSTRUCTION OF THE
FIELD OF A POLLUTANT IN A BODY OF WATER
The method of using test problems to compare models requires that the
field is calculated using a known, deterministic formula. The form of dif-
ferential equation used to produce the formula determines the type of re-
ference function used in the modeling algorithm.
In the present example, the field was calculated by a formula suggested
by Lapshin which presumes pure diffusion of particles in space. It allows
the change in pollution in space and time to be determined when there is no
current:
2r °° P~X
< = £ ' R f dx
x = Rt
where k is the coefficient of turbulent diffusion,
T is the distance of the measurement station from the source of
pollution,
t is the time from the beginning of emission to the moment of mea-
surement.
329
-------
Assuming there is one source, the rule of change of emission and concen-
tration of polluting substances with time is shown in Figure 6, according
to the initial data obtained from the equation. Finite-difference models
of seven types were synthesized, and compared for accuracy of step-by-step
integration both inside and outside the interpolation area.
As a result, the following improved models were obtaind.
First state of problem:
Model 1-1 (Figure 7)
= 1.06020 - 0.1779 qj] ^ + 0.01673 q_Q + 0.63904
(« - °-°1059
+ °-00233 *o-Q + °-00001 ^,! - °'1027
- 0.0894 q2 + 0.00947 q - 0.0912 qq(
+ 0.00844 qqg - 0.02334 q. + °-00188
- 0.00358 qn>0q(l| + 0,17850 ^l]^ - 0,25224 ^l]
0.00894 qQq - 0.02334 qoq+l + °-00188
0.07531 q _ 0>0064
K = 0.032; n = 0.017; A(c) = 0.027; i = 0.182.
Second statement of problem:
Model II-l (Figure 8a)
q[y = 2.73520 + 0.00031 q + 0.000001 q - 0.0004
+ 0.00001 q3) + 0.29626 q2V) + 0.46974
(o\ CON d\ /o\
- 0.66516 qg q0 ' + 4.86824 q^'q^ - 1.95950
330
-------
24
20
co 16
^) 12
8
4
0
0
0
0
l I 1 I i
i i
12
6 9
TIME, hr
12
I I
15
15
Figure 6. Change of emission and concentration of pollutants with time.
331
-------
CO
GO
ro
10 -
INTERPOLATION
0 1.5 3 4.5
PREDICTION
\
7.5 9 10.5 12 13.5 15
TIME, hr
Figure 7. Model 1-1.
-------
co
co
CO
INTERPOLATION
PREDICTION
6 7.5 9
TIME, hr
Figure 8a. Model II-l.
-------
+ 1.79479 q + 1-58739 Q - 1.41426 q_3
1.1132 q q_(3) + 1.513890 qV + 9-67937
_
- 0.10583 qf + 3.44364 q_(3)2 - 0.38696 q
- 3.87969
K = 0.061; ncm = 0.046; A(c) = 0.040; i = 0.188.
Model II-2 ^Figure 8b)
m ? , ,2
q+y = 2.73090 + 0.000035 X^ - 0.000006 XQX _ 0.00032 XQX_,
222
+ 0.000394 X_]X^ + 0.000548 X_?XQ - 0.000548 X_2X_-|
- 0.000002 xj - 0.000003 X2X23 + 0.000003 ^^
K = 0.089; ncm = 0.082; A(c) = 0.036; i = 0.169.
Model II-3 (Figure 8c)
(1) (9} (2} 2
q;, = 2.03610 - 2.18146 qV - 0.21021 qvl; + 0.00754 X'
+ 0.10985 qg q.g + °«39242 qvr + 0.00002
- 0.10895 q02V3^ - o.oooooi
K = 0.08; ncm = 0.054; A(c) = 0.059; i = 0.151.
Third statement of problem-:
Model III-l (Figure 9a)
qjJ^Q = 2.62225 - 0.25160 qo^q^o + 0.12961 qj1^2 + 0.00245
334
-------
OJ
co
en
INTERPOLA TION
PREDICTION
0
6 7.5 9
TIME, hr
Figure 8b. Model II-2,
10.5 12 13.5 15
-------
CO
CO
en
INTERPOLA TION
0 1.5
6 7.5 9
TIME, hr
10.5 12 13.5 15
Figure 8c. Model II-3.
-------
CO
CO
1.5 3 4.
.5 9 10.5 12 13.5 15
TIME, hr
Figure 9a. Model III-l.
-------
0.94186 qoQo + 0.32467 *[}}Q*[%Q - 0.13988
+ 0.01585 q + °'98315 <-lo
-------
to
co
CO
INTERPOLATION
PREDICTION
0 1.5 3 4.5 6 7.5 9 10.5 12 13.5 15
TIME, hr
Figure 9b. Model III-2.
-------
CO
J5»
o
6 7.5 9 10.5
TIME, hr
12 13.5 15
Figure 9c. Model II1-3.
-------
- °-14929
°-08369
°-22246
K = 0.115; n = 0.05; A(c) = 0.04; i = 0.426.
\f\\\
The solution of the first and third problems allows construction of a
picture of the field, prediction of its development with time and extrapola-
tion of it along the spatial coordinates.
The solution of the second problem yields the value of the parameters
of the pollutants at the points where the measurement stations are located.
Using the data obtained, predictions can be constructed and then, by means
of interpolation, the field in an area located within a polygon may be
found, the points of which are the points of the measurement stations, i.e.,
in the interpolation area (Figure 1).
Another version of the algorithm is also possible: It is possible to
first interpolate the data of the stations, and then later construct the
prediction for the points in the area of interpolation of interest. Predic-
tions can also be produced only for the area of interpolation by means of
step-by-step integration of the equations along the time axis. The solu-
tion to the second problem is not suitable for extrapolation of the field
into neighboring areas.
CONCLUSIONS - COMPARISON OF MODELS
The model using the principle of near action, 1-1, produces less accu-
rate predictions than improved models based on the principle of "distant"
action, II-3, and the combined model, III-2, if the rate of emission of the
sources of pollution change with time. The model of "distant" action, II-3,
which considers both measured data at the measurement stations and informa-
tion on changes in the rate of pollutant discharge, is the best of all the
models.
In the future, the authors intend to use the criterion of balance of
variables (Tamura and Kando 1977), which should be no worse than direct mea-
surement of emission, for organization of models.
REFERENCES
Dyachenko, V.F. 1977.
Press, Moscow.
Basic concepts in computational mathematics, Mir
Ivakhnenko, A.G. and G.I. Krotov. 1977. Modeling of pollution of the en-
vironment when there is no information available on the emissions of
sources of pollution. Avtomatika, No. 5, Kiev, Naukova dumka Press, pp.
14-30.
341
-------
Ivakhnenko, A.G. 1978. Inductive methods of self-organization of computer-
ized complex system models. Avtomatika, No.4, Kiev, Naukova dumka
Press, pp. 11-26.
Ivakhnenko, A.G., V.N. Vysotskiy and N.A. Ivakhnenko. 1978. Basic vari-
eties of the criterion of the minimum of displacement of a model and in-
vestigation of their interference stability. Avtomatika, No. 1, Kiev,
Naukova dumka Press, pp. 32-53.
Tamura H., and T. Kando. 1977. Large-spatial pattern identification of air
pollution by combined model of source-receptor matrix and revised paper
presented at IFAC Symposium of Environment System, Kyoto.
Yurachkovskiy, Yu.P. 1977. Improvement of the MGUA modeling algorithm for
prediction of processes (a review). Avtomatika, No. 5, Kiev, Naukova
dumka Press, pp. 76-86.
342
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SECTION 17
A SPATIALLY-SEGMENTED MULTI-CLASS PHYTOPLANKTON MODEL FOR
SAGINAW BAY, LAKE HURON
Victor J. Bierman, Jr. and David M. Dolan1
INTRODUCTION
Saginaw Bay is a broad, shallow extension of the western shore of Lake
Huron (Figure 1). The bay is oriented in a southwest^ard direction and is
82 km long and 42 km wide. For convenience, the bay has been divided into
five spatial segments on the basis of differences in observed water quality.
The average depth of the inner portion of the bay, represented by spatial
segments one, two, and three, is approximately 6 m. The average depth of
the outer portion of the bay, represented by spatial segments four and five,
is approximately 15 m. Seventy percent of the total water volume is con-
tained in the outer portion of the bay and the remaining 30 percent is con-
tained in the inner portion.
P
The total area of the Saginaw Bay watershed is approximately 21,000 km .
The Saginaw River is the major tributary and it accounts for over 90 percent
of the total tributary inflow to the bay. The principal land use categories
in the watershed are agriculture and forest. The total population of the
watershed is slightly over 1.2 million. Most of the population is concen-
trated into four major urban-industrial centers: Bay City, Midland, Sagi-
naw, and Flint. These centers are all situated along the Saginaw River or
its major tributaries.
The principal water uses in Saginaw Bay include municipal and industrial
water supply, waterborne transportation, recreation, commercial fishing, and
waste assimilation. These uses are severely impacted by the considerable
quantities of waste discharges and runoff to the bay as a result of human
activities in the watershed. In particular, the inner portion of the bay
suffers from highly degraded water quality.
The principal issue addressed in the developement of the present model
was cultural eutrophication, defined as overproduction of phytoplankton bio-
mass due to increased nutrient loadings. The purpose of the modeling effort
u.S. Environmental Protection Agency, Large Lakes Research Station, Grosse
lie, Michigan 48138.
343
-------
10 0 10 20 30 40 50
I I I I I I I KM
SCALE: 1:1,000,000
Figure 1. Saginaw Bay and the spatial segmentation scheme used
for the phytoplankton model.
344
-------
was to develop a deterministic phytoplankton simulation model that could
describe the cause-effect connection between external nutrient loading and
phytoplankton growth in Saginaw Bay. The objectives were twofold: first,
to gain insight into the relevant physical, chemical, and biological pro-
cesses affecting phytoplankton growth; and second, to use the model as a
tool for comparing the future effects of various wastewater management
strategies.
The plan of work for the study will include a calibration and a subse-
quent verification of the model to two independent sets of field data ac-
quired on Saginaw Bay during 1974 and 1975. The verified model will be used
to generate a set of predictions corresponding to expected reductions in ex-
ternal nutrient loads, principally phosphorus. These predictions will be
compared to the results of a follow-up field survey to be conducted in 1980
on the Lake Huron-Saginaw Bay system.
SCOPE
The purpose of this paper is to present results of the preliminary cali-
bration phase of the phytoplankton modeling effort. Calibration results
will be presented graphically, as well as in terms of statistical tests for
goodness of fit between model output and field data.
In addition to model calibration results, field data will be presented
which illustrate the large gradients in water quality that exist among the
five spatial segments in Saginaw Bay. These gradients have a strong impact
on the model development effort because it is difficult to obtain a unified
set of kinetics which can describe phytoplankton dynamics simultaneously in
all five segments.
FIELD DATA
As part of the Upper Lakes Reference Study sponsored by the Inter-
national Joint Commission, intensive field surveys were conducted on Saginaw
Bay during 1974 and 1975 (Bratzel et aj_. 1977). These surveys involved
several different institutions, coordinated by the Large Lakes Research Sta-
tion of the U.S. Environmental Protection Agency. In each of the two years,
13 sampling cruises were conducted. In 1974, samples were taken on a 59-
station grid at multiple depths (Figure 2). In 1975, samples were taken on
a 37-station subset of this grid. This paper is restricted primarily to the
results from 1974.
Analyses were conducted for 22 different physical-chemical variables,
including phosphorus, nitrogen, silicon, chlorophyll, chloride, temperature,
and Secchi depth. Biological measurements included species identification
and number concentrations for both phytoplankton and zooplankton. In addi-
tion, measurements were conducted for phytoplankton cell volumes and zoo-
plankton dry weights.
345
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SAG IN AW BAY
SAMPLING NETWORK
LEGEND: • BOAT STATION
A WATER INTAKE
SAG IN AW
RIVER
Figure 2. Sampling station network in Saginaw Bay.
346
-------
Horizontal spatial gradients for several principal water quality vari-
ables are shown in Figures 3-5. All data are presented as the mean plus or
minus one standard error. In general, the five spatial segments are ordered
in the following manner with respect to increasing water quality: one,
three, two, five, and four. The observed gradients range over a factor of
five for chloride, a factor of six for total phosphorus, and a factor of
ten for summer chlorophyll a_.
Large spatial gradients in water quality occur in Saginaw Bay primarily
because most of the external nutrient loading is from a single tributary,
the Saginaw River, and because of the predominant water circulation pattern
in the bay. Water circulation tends to be counterclockwise with relatively
clean Lake Huron water flowing into segment four, and relatively dirty
Saginaw River water flowing out through segments one, three, and five (Fi-
gure 2). Segment two is relatively dirty because of dispersion from seg-
ments one and three. Although segments one and three possess the poorest
water quality of the five spatial segments, these segments together consti-
tute less than 10 percent of the total volume of water in Saginaw Bay. The
remaining three segments contain approximately equal volumes of water.
MODEL DESCRIPTION
A schematic diagram of the phytoplankton model is shown in Figure 6.
The compartments represent the principal variables in the model, and the ar-
rows represent pathways for material transport among the compartments. The
reader is referred to Bierman (1976) and Bierman et_ aj_. (1979) for a de-
tailed description of model concepts and model equations.
The model includes phytoplankton biomass in terms of five functional
groups: diatoms, greens, non-N2~fixing blue-greens, N2-fixing blue-greens,
and "others". The latter category includes primarily dinoflagellates and
cryptomonads. Non-N2-fixing blue-greens consist of those species which have
an absolute requirement for dissolved combined nitrogen. N2-fixing blue-
greens consist of those species which have the capability for fixing atmos-
pheric nitrogen, as well as for using dissolved combined nitrogen. The
nutrients included in the model are phosphorus, nitrogen, and silicon.
The principal reason for this multi-class approach is that there are im-
portant physiological differences among the five groups. Diatoms are the
only group with a major absolute requirement for silicon. The N2-fixing
blue-greens are the only group which can grow independently of the supply of
dissolved available nitrogen. The relative maximum growth rates and temper-
ature optima among the groups are such that a typical successional pattern
during the growing season begins in the Spring with diatoms, progresses to
greens and "others", and finally leads to the development of blue-greens in
late Summer and Fall. An important characteristic shared by all of the
groups is an absolute requirement for phosphorus.
347
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MODEL CALIBRATION
The calibration process consists of adjusting model coefficients within
their reported ranges in the literature to obtain the best possible corre-
spondence between model output and field data. With a simple model, human
intuition is usually adequate to determine the goodness of fit. With
sophisticated models, it is usually necessary to resort to more systematic
methods.
For the present model, there were 12 independent variables in each of
five spatial segments for which field data were available. Consequently,
there were 60 segment-variable combinations for each of 13 sampling cruises.
It was impossible to rely completely on human intuition for model calibra-
tion.
Calibration of the model was facilitated by use of the Model Verifi-
cation Program (MVP) developed by Manhattan College (Thomann and Winfield
1976; Thomann et aj_. 1979). The MVP is a system of statistical programs
which can analyze the goodness of fit between model output and field data
for each model run. Analyses can include the Student's T-test, regression
tests, and tests for relative error. Only the Student's T-test has been
used to date for the Saginaw Bay phytoplankton model.
MODEL RESULTS
Model results for all 12 variables and all five segments have been re-
ported in terms of the MVP. Selected model results have been reported
graphically for segments two and four. Segment four represents the volume
of water in Saginaw Bay with the highest water quality, and segment two best
represents the lower water quality in the inner portion of the bay because
it contains the largest volume of the three inner bay segments.
Table I contains results of the MVP Student's T-test for the best cali-
bration run to date. The T-test has been applied on a cruise-by-cruise,
segment-by-segment basis for each of the 12 independent variables. Results
have been reported as percent of sampling cruises for which there is no
significant difference between model output and field data at the 95 percent
confidence level. The overall average for the five model segments was 83
percent. In a purely statistical context, this implies that the model out-
put for the indicated 12 variables describes the Saginaw Bay system approxi-
mately 83 percent as well as the data for these 12 variables describes the
Saginaw Bay system. It should be noted that variability in the field data,
as well as accuracy in the model output, will lead to favorable MVP re-
sults.
Graphical results for phytoplankton concentrations in segments two and
four are shown in Figures 7-14. All field data have been presented as the
mean plus or minus one standard error. A logarithmic scale has been used
because it was found that the phytoplankton data followed a log-normal
distribution. Such a scale causes difficulty in determining goodness of
352
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TABLE ], STATISTICAL COMPARISON BETWEEN MODEL
OUTPUT AND FIELD DATA
Model variable
Chloride
Diatoms
Greens
Others
Non-N2 blue greens
N2 blue greens
Total zooplankton
Total phosphorus
Total nitrogen
Dissolved phosphorus
Dissolved nitrogen
Dissolved silicon
Percent of sampling cruises
significantly different
not
Segment number
1
92
92
100
92
69
59
70
100
75
100
71
57
2
100
100
92
92
92
89
90
100
73
77
92
92
3
85
92
75
58
92
67
60
82
64
33
42
54
4
92
92
100
92
100
78
100
91
83
42
92
77
5
100
73
100
91
100
100
100
100
91
33
75
77
Average 81
91
67
87
87
353
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SAG IN AW BAY
1974
Diatoms
Segment 2
F M A M
' A ' S ' 0 ' N ' D
Figure 7. Comparison between model output and field data for diatom biomass in segment two.
-------
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SAG IN AW BAY
1974
Green Algae
Segment 2
A S 0 N D
Figure 8. Comparison between model output and field data for biomass of green algae in segment two.
-------
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SAGINAW BAY
1974
Non-N2 Blue-Greens
Segment 2
M ' A '
M
' 0 ' N ' D
Figure 9. Comparison between model output and field data for biomass of non-N2-fixing blue-greens
in segment two.
-------
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en
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1974
" N2 Blue-Greens
Segment 2
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Figure 10. Comparison between model output and field data for biomass of ^-fixing blue-greens
in segment two.
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1974
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Segment 4
M A M J J A
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Figure 11. Comparison between model output and field data for diatom biomass in segment four,
-------
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1974
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Segment 4
JFMAM'J'J'A'S' o ' N ' D
Figure 12. Comparison between model output and biomass of green algae in segment four,
-------
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1974
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Figure 13. Comparison between model output and field data for biomass of non-N2-fixing
blue-greens in segment four.
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1974
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Figure 14. Comparison between model output and field data for biomass of No-fixing
blue-greens in segment four.
-------
fit by strictly graphical means because of its non-linearity. This was an
additional reason for using the MVP.
In general, the concentrations for each phytoplankton group were ap-
proximately an order of magnitude higher in segment two than in segment
four. This trend was a reflection of higher nutrient concentrations and
the resulting lower water quality in the inner portion of the bay.
Model output generally followed the trends in the data for the various
phytoplankton groups. Difficulties were experienced in both segments two
and four with model output for diatoms during the third quarter of the year.
In addition, the trend of the diatom output was somewhat high during Spring
in segment four.
Results for dissolved ortho-phosphorus concentrations in segments two
and four are shown in Figures 15 and 16. Phosphorus is generally considered
to be the most important nutrient limiting the growth of phytoplankton in
freshwater systems. The trend of the model output was low, compared to the
field data, in both of the segments. The relatively poor fit for dissolved
ortho-phosphorus was consistent with below-average MVP results, especially
in segment four (Table I).
Initially, the poor fit for dissolved ortho-phosphorus was a matter of
serious concern because it was not possible to improve the fit by adjusting
any of the model coefficients. Model output for dissolved ortho-phosphorus
depended heavily on the external phosphorus loading and the water circula-
tion rates among the various segments in the bay. Both of these factors had
been determined independently and then used as input to the phytoplankton
model.
Subsequently, it was realized that the dissolved ortho-phosphorus data
for 1974 were probably biased higher than the true concentrations. In 1975,
the external phosphorus loading and the water circulation rates were essen-
tially the same as in 1974; however, the 1975 data for dissolved ortho-phos-
phorus were substantially lower than the corresponding data for 1974. In
fact, the trend of the model output for dissolved ortho-phosphorus in 1974
agreed closely with the trend of the dissolved ortho-phosphorus data for
1975.
The problem with the dissolved ortho-phosphorus data for 1974, es-
pecially at lower concentrations, was caused primarily by logistical diffi-
culties. In 1974, samples were filtered immediately on shipboard, but were
not analyzed until one to three days later. In additi-on, the analytical
method used had not yet been refined to achieve maximum sensitivity. These
problems were corrected in 1975 and the analyses for dissolved ortho-phos-
phorus were conducted on shipboard using a more refined technique. The con-
sequences of the different approaches used for 1974 and 1975 were not
readily apparent until after data had been acquired for the two years.
362
-------
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Figure 15. Comparison between model output and field data for dissolved ortho-phosphorus
in segment two.
-------
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Figure 16. Comparison between model output and field data for dissolved ortho-phosphorus
in segment four.
-------
DISCUSSION
The importance of systematic quantitative methods for analyzing the
correspondence between model output and field data has been emphasized. It
should be pointed out, however, that such methods are not substitutes for
human intuition and common sense. For example, the discrepancies in the
case of dissolved ortho-phosphorus could not have been resolved strictly
through the use of statistical techniques.
It has also been emphasized that phytoplankton models are not simply
arbitrary systems of equations which can be curve-fitted to any set of field
data. If a model is conceptually sound and internally consistent, discrep-
ancies should be expected to arise between model output and field data in
cases where anamolies exist in a given set of field data.
REFERENCES
Bierman, V.J., Jr. 1976. Mathematical model of the selective enhancement
of blue-green algae by nutrient enrichment. In: Modeling Biochemical
Processes in Aquatic Ecosystems ed. R.P. Canale, pp. 1-31. Ann Arbor,
Michigan: Ann Arbor Science Publishers.
Bierman, V.J., Jr., D.M. Do Ian, E.F. Stoermer, J.F. Gannon and V.E. Smith.
The Development and Calibration of a Multi-Class Phytoplankton Model
for Saginaw Bay, Lake Huron. In press in U.S. Environmental Protection
Agency Ecological Research Series, 1979.
Bratzel, M.P., M.E. Thompson and R.J. Bowden (Eds.). 1977. The Waters of
Lake Huron and Lake Superior. Vol. II (Part A). Lake Huron, Georgian
Bay, and the North Channel. Report to the International Joint Commis-
sion by the Upper Lakes Reference Group. Windsor, Ontario. 292 p.
Thomann, R.V. and R.P. Winfield. On the Verification of a Three-Dimensional
Phytoplankton Model of Lake Ontario. In: Proceedings of the Conference
on Environmental Modeling and Simulation, U.S. Environmental Protection
Agency, Cincinnati, Ohio, 1976. pp. 568-572.
Thomann, R.V., R.P. Winfield and J.J. Segna. Verification Analysis of Lake
Ontario and Rochester Embayment Three Dimensional Eutrophication Models.
In press in U.S. Environmental Protection Agency Ecological Research
Series, 1979.
365
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SECTION 18
A SEGMENTED MODEL OF THE DYNAMICS OF THE ECOSYSTEM OF LAKE BAIKAL
CONSIDERING THREE-DIMENSIONAL CIRCULATION OF THE WATER
V.V. Menshutkin1, O.M. Kozhova2, L.Ya. Ashchepkova2
and V.A. Krotova2
A model of the seasonal dynamics of the pelagic ecosystem of Lake Baikal
is described, the main components of which are the phytoplankton, zooplank-
ton, bacteria, nutrients and detritus. The surface of the lake is arbi-
trarily divided into 65 areas of equal size, and the volumes of water
divided into two zones: the photosynthetic and destructive zones. The in-
put functions of time include absorbed solar radiation, water temperature in
both zones of the various regions of the lake, and the inputs of organic
matter from the largest sources of the lake.
In calculating the status vector of an ecosystem, biologic and hydro-
logic processes are considered in each step. The redistribution of dis-
solved and suspended matter within the mass of water is performed in the
model by means of horizontal advection and vertical turbulent mixing. The
speed and direction of horizontal transfer in each region are constant
throughout the year and correspond to the overall system of circulation of
water in the lake. The turbulent flows are proportional to the vertical
gradients of matter. On the whole, the dynamics of spatial distribution of
plankton demonstrated by the model agrees well with the data of actual sur-
veys made in Lake Baikal.
The ecosystem of the lake is a unique object for modeling. Some fea-
tures are reminiscent of marine ecosystems in the temperate latitudes. The
great length of the lake along its long axis, its extreme depths, the
uniqueness of the structure and circulation of its water masses, and the in-
fluence of river runoff all create a broad range of conditions influencing
the seasonal dynamics of the ecosystem.
Models of the ecosystem of Lake Baikal must consider the regionalization
of the lake. In terms of horizontal distribution of plankton, a distinction
Institute of Evolutionary Physiology and Biology, Academy of Sciences of
the USSR, 44 Morisa Toryeza Prospect, 194223 Leningrad, USSR.
2Irkutsk State University, A/Y2H 24 Lenin Street, 664003 Irkutsk, USSR.
366
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is made between shallow areas (littoral zones, shoals at the mouths of
rivers, bays) and deep water regions (Kozhov 1962). In terms of the hydro-
meteorologic regime, there are three characteristic deep water zones in
southern, central and northern Baikal, specifically the central portion of
the lake and the two littoral areas adjacent to the eastern and western
shores, as well as embayments, the strait of Maloye More, and the Selenga
shallows.
The basis of the model consisted of two-layered segments. The water
area of the lake was divided into 65 fundamental areas of equal size, so
that each of them covered a region which was characteristic in its hydro-
meteorologic aspect (Figure 1). The size of each fundamental area was 484
km2, selected so that a single area could cover either of the large bays,
Chivyrkuyskiy and Barguzinskiy. Beneath these areas are segments consisting
of two zones. The upper zone, from 0-50 m depth, is the zone of photosyn-
thesis and the most mobile portion of the body of water. The lower, the de-
structive zone, extends from 50 m depth to the bottom. The shallow regions
contain only the upper zones in their segments.
Exchange between the segments in the horizontal and vertical directions
is achieved by advection and turbulent mixing. The system of currents in
Lake Baikal is complex, variable, and insufficiently studied. Therefore,
this paper seeks only to analyze the overall circulation of the waters of
Lake Baikal, specifically, the large-scale movements occurring over long
periods of time (Krotova 1969). The basis for the system of overall circul-
ation (Figure 1) was the result of calculations of geostrophic currents from
thermal survey data (July-October) covering the entire lake and its individ-
ual parts (Krotova 1970) and the mean multiannual dynamic topography of the
surface of Lake Baikal (Figure 2). This was obtained by dynamic processing
of 2256 hydro logic measurements performed for the months of August and
September from 1925 to 1965. The horizontal structure of the overall cir-
culation has been confirmed by the release of 1860 experimental drift bot-
tles, specially designed floats (Pomytkin, 1962), and instrumented measure-
ments from the surface to the bottom over a period of a number of days, both
during the ice season and during the navigation season (Verbolov 1977a,
1977b).
The available material indicates that a single cyclonic circulation en-
compasses the entire lake. Movement is most intensive and continuous from
south to north along the eastern shore. On the average, the summer move-
ment is traced from Slyudyanka to Davsha (Figure 2), and reaches depths of
50-75 m. The speed of this shoreline current in southern Lake Baikal
reaches 9-13 cm/sec, 5 km from the shore. In the central section, its
velocity is 4 cm/sec at 5-8 km from the shore, and in the vicinity of the
Svyatoy Nos Peninsula a velocity of 7 cm/sec is reached. North of Davsha,
the current weakens, and its vertical thickness decreases to 10-20 m.
The current in the opposite direction - from north to south along the
western shore - is not as strong. Its speed is 2-7 cm/sec, and it expands
down to 20-50 m depth. In a number of places, it is complicated by an up-
welling effect.
367
-------
UPPER
ANGARA RIVER #f
MALOYEMOYE.
STRAITS -.;
CHIVYRKUYSKIY
BAYJ:
{•SVYTOY NOS. PENINSULA
' /""
•BARGUZIN RIVER
SELENGA RIVER
ANGARA RIVER
S MISHIKHA RIVER
VYDRINAL RIVER
KHARA-MURIM RIVER
Figure 1. Regionalization of Lake Baikal in the model and
direction of horizontal transfer.
368
-------
CO
Ch
Figure 2. Geostrophic circulation on the surface of Baikal (a) and depth of circulation (b),
average for August-September 1925-1965. Speed in cm/sec, vertical thickness in m.
-------
The cyclonic formation, covering almost the entire lake, is divided into
quasi-steady circular systems moving in the same direction: one in the
southern extremity of the lake, another between Listvenichnyy Bay and the
Selenga River delta, a third outside of Olkhon Island, a fourth north of the
Ushkaniy Islands, and a fifth in the northern extremity of the lake (Figure
1). The cyclonic circulations are usually elliptical in form. Their trans-
verse horizontal side is comparable to the width of the lake, and their
longitudinal dimension reaches 7-106-9-106 cm. In the interior, the
currents extend much less deeply into the water than at the periphery; the
average depth for August-September being 10-15 m. In the region of
Listvenichnyy Bay, a quasi-steady anticyclonic circulation is formed, while
in Maloye More, Chivyrkuyskiy and Barguzinskiy Bays cyclonic vortices mea-
suring 1.5*106 - 4'106 cm are formed. Anticyclonic vortex formation is
observed during the summer down to 75 m.
On the long axes of the cyclonic circulations are divergence zones, in
which we see the highest values of the vertical components of flow speeds,
directed upward. We can estimate the order of magnitude of vertical speed
on the basis of the relationship of the horizontal (V) and vertical (W) com-
ponents of the speed and the horizontal (L) and vertical (H) dimensions of
the body of water (V/W = L/H). Since Lake Baikal is 636 km long and its
mean depth is 700 m, the horizontal speeds of 1-10 cm/sec correspond to
vertical speeds on the order of 10~3 and 10~2 cm/sec. Furthermore, we
know that the variation in depth of the zero flow surface at Lake Baikal
may be as great as 100 m (Krotova 1970). This surface may be wedged out in
the central regions of the cyclonic formations, located near the center of
the lake, or along its western shore, then drop to a depth of 100 m at the
periphery, e.g., along the eastern shore. In this case, W/V is equal to the
ratio of the change in thickness of the layer involved in the flow, AH, to
the distance AL over which this change occurs. Assuming V, as before, to be
equal to 1 and 10 cm/sec, and AL to be 20-40 km, where AH = 100 m we produce
vertical speeds of the same order of magnitude, 10~3 and 10~2 cm/sec.
Finally, the time of transfer of a mass of water by the vertical move-
ment from the surface to the bottom should be of the same order of magnitude
as the time required to transfer a mass through the entire body of water
(Chekotillo 1965). Given the values of horizontal speeds of 1 and 10
cm/sec, and the length of the horizontal path of about 1000 km, the duration
of vertical motion is approximately 3 years and 4 months.
In southern Lake Baikal, in addition to divergence zones, local areas of
upwelling and downwelling of water as a result of surge phenomena are well
known. Upwelling is most clearly expressed in two sections along the west
coast, from Kultuk to Cape Polovinny, and from the mouth of the Angara to
Cape Goloustny. In the regions of Peschanaya Bay to Cape Bolshoy Kadilny,
To'sty Bay to Cape Polovinny, the Khara-Murin River to the Vydrinaya River,
the Mishikha River to Istokskiy Sor, the water constantly descends (down-
welling). On maps of the dynamic topography, the dynamic horizontals inter-
sect the shoreline (Figure 2a).
In modeling Lake Baikal, it was assumed that the exchange between seg-
ments in the horizontal direction occurs as a result of advection, and in
370
-------
the vertical direction as a result of turbulent diffusion. Furthermore, it
was assumed that the direction and speed of the current and level of the
water at each point in the lake are constant throughout the entire year.
This means that in each unit of time, each segment gives its neighbors as
much water as it receives from them. The condition of balance also occurs
for any connected subsystem of segments.
The status of the ecosystem of Lake Baikal on each day of year t is
determined in the model by the concentrations of the five primary elements:
phytoplankton b] (k, i), zooplankton b2 (k, i), detritus 03 (k, i),
bacteria b4 (k, i) and nutrients b$ (k, i), where k = 1, 2 is the num-
ber of upper and lower zones, i = 1, 2, ..., 65 is the number of each seg-
ment. The concentrations are measured in kcal/m3. Variable b$ (k, i)
does not refer to a specific chemical element, but rather to certain hypo-
thetical nutrients; the energy equivalent of the potential material for the
creation of primary production. In the six shallow sections, mentioned
earlier, are segments with only a surface zone, so that the total number of
zones is 124. Thus, the status of the system is characterized by a vector
with 620 components. The time interval is equal to one day.
Calculation of the system status vectors for each step is performed in
two stages. In the first stage, biologic processes are simulated (growth,
grazing, metabolism, death). They are reflected by the same functions as in
the energy model of the Lake Baikal pelagic ecosystem constructed earlier
(Ashchepkova e_t aj_. 1977). Some of them are calculated with corrections for
temperature, which varies significantly in the first zone during the course
of the year. It is assumed, in particular, that the optimal temperature for
grazing of zooplankton is 10°C (Pomazkova 1970), for the growth of algae,
1.5°C. The daily changes in concentration Abj(k, i) in this stage are
determined from the equations:
Ab-
- C
12
-Q-, -
M
Ab2(k, i) = 0.8 x
C2) -
- Mk,
Ab3(k, i
= 0.2 x
Mk + Mk
1 2'
Ab4(k, i) =
Ab5(k, i
= - P
Qk + Q2
QJJ,
k = 1, 2.
where p is the rate of photosynthesis, Cj£ is the ratio of the a compon-
ent to the jth. Cn is the cost of metabolism of the jth component, M^ is the
death of the j*h component. In the upper zone P1 is determined in accord-
ance with the limitation principle of Libikh, in the lower zone P2 = 0.
371
-------
In the second stage of calculation of the status vector, hydrologic
transforms are applied to the vector produced earlier, simulating horizontal
and vertical transfer. The transfer coefficients (Figure 3), representing
the fraction of the volume of the upper zones of the segments which moves in
a given direction during a given day, are calculated from the condition of
balance, based on data on current speeds.
Detritus and the accompanying bacteria sink at a constant speed. Phyto-
plankton is held at the top surface of the water during the ice season.
After the ice thaws, most of the algae begin to descend into the deeper
layers of water, beyond the photosynthetic zone, due to the loss of floata-
tion by the cells (Skabichevskiy 1960). Thus, the quantity of algae, detri-
tus and bacteria sinking during the course of a day from the upper layer
into the lower layer is calculated by the equation
"J-
.-d, i), k = 1,
.d, i)/15, k = 2,
j = 1, 3, 4; i = 1, 2, ..., 65,
where V is the sinking coefficient:
0.05 for j = 3, 4 or j = 1 and t > 150,
Vj =
0 for j = 1 and other values of t
where t is the time in days.
Since zooplankton can move independently, the long-term sinking of this
group is not considered in the model. It is assumed that the distribution
of zooplankton through the mass of the water is in accordance with the feed-
ing and thermal conditions of the medium. The diffusion of water is inter-
preted in the model as exchange between vertical zones, proportional to the
difference in concentration of the components they contain:
W.j(K, i) =•
w..d) x (b..(2, i) - b..(l, i)), k = 1,
Wj(i) x (bjd, i) - bj(2, i)), k = 2,
i = 1, 2, ..., 65
where wj(i) is the mixing coefficient in segment i.
Calculations were performed on an BESM-222 computer. In all segments,
values of the input functions are assigned for each day of the year: ab-
sorbed solar radiation, influence of the photosynthetic processes in the
surface zones of segments, and water temperature. These quantities reflect
the mean indices of the sum of absorption of solar energy in the southern,
middle and northern trenches of the lake over a period of many years, as
well as the dynamics of effective temperature in the photosynthetic zone in
all parts of the lake (Rossolimo 1957; Bufal 1966; Verbolov 1969). In the
lower zones, the temperature is constant and is equal to 4°C. It is assumed
372
-------
0,1
<^ (34) •"s-Z
°-io5i^/yx |
(yh* * (S^« *V
205^^ » >/o
Figure 3, Diagram of horizontal exchange between segments in model.
Circles show number of section, arrows show direction of flow, numbers
arrows show fraction of volume of upper zone of segment moving in
corresponding directions.
373
-------
in the model that the annual runoff into Lake Baikal is through three
rivers, the Selenga, Barguzin and Upper Angara. The Selenga brings in half
of all the water, i.e,. 30 km3/yr. Since the exchange of each of the
upper segments is 242*10^ m3, the input rate of water from the Selenga
into Lake Baikal is equal to about 0.005 of the volume of an upper layer
during a day. The flow of water from the Barguzin and Upper Angara is 0.002
and 0.003 times the volume/day, respectively, so that the Upper Angara each
day brings in 1 percent of the volume of the upper segment adjacent to its
mouth. The rivers bring in more than 4-105 tons of organic matter each
year, about 2/3 of the total quantity coming in with the Selenga (Votintsev,
Popovskaya 1969). If we assume that the remaining quantity of organic mat-
ter is divided evenly between the Barguzin and Upper Angara Rivers, the
loading to the system corresponding to these rivers will contain the fol-
lowing concentrations of organic matter: Selenga River, 6 g/m3; Barguzin
River, 3.75 g/cm3; Upper Angara River, 2.5 g/m3.
At the initial moment in time, January 1, all components of the system
are evenly distributed among the segments. By the end of the first year,
the transient process is completed, and beginning on January 1 of the next
year, the spatial distribution of variables calculated is printed out. In-
formation on the dynamics of the system is generated in the form of maps of
the horizontal distribution of any of five components of the ecosystem in
the lower or upper layer of the segments, and in the form of graphs of their
change with time in the assigned layer for any selected section of the lake.
If the distribution of the concentration of phytoplankton obtained in
the model in the upper layer during the course of the year (Figure 4) is
examined, it may be compared with the results of observations (Figure 5).
Although 1964 characteristically has the longest series of observations on
the condition of the plankton, it differs significantly from an average
year, primarily in the very richness of the phytoplankton. This was a so-
called "mellosing" year, in which the ordinary dynamics of phytoplankton are
disrupted: The spring peak of biomass of algae was very great, but there
was practically no fall peak at all. Nevertheless, it is possible to com-
pare the qualitative picture of the dynamics of distribution of plankton in
the model and in Lke Baikal during the periods of formation and elimination
of the spring maximum of biomass.
A comparison of the dynamics of plankton throughout the year in the
model with the results of actual surveys leads to the conclusion that, on
the whole, the model reflects the actual picture rather well. One major ad-
vantage of the model is the fact that it enables the tracing of the path of
propagation of matter entering the lake, and considers its transformation in
the biologic system. Furthermore, it enables an estimation of how changes
in the status of the ecosystem at any point on the lake influence its state
in other regions. For scientific purposes, the model is also useful as a
research tool, revealing the key processes necessary for prediction of the
behavior of the system. It enables a determination of the direction of
further scientific research, outlining the contours of the problems for eco-
nomic model for planning and prediction of the economic development of
regions adjacent to Lake Baikal.
374
-------
Ill
IV
VI
VII
VIII
IX
CO
^J
en
Ftiytoplankton
kcal/m3
Figure 4. Horizontal distribution of phytoplankton throughout the year in the 0-50 m layer
(generated by model).
-------
MAY
JUNE
JULY
AUGUST
GJ
-»-J
CT>
Figure 5. Distribution of phytoplankton in Lake Baikal in the 0-25 m layer in 1964
(after Votintsev, e_t al_. 1975).
-------
REFERENCES
Ashchepkova, L.Ya., V.I. Gurman, and O.M. Kozhova. 1978. An energetic
model of the pelagic community of Lake Baikal. Modeli prirodnykh
sistem. Nauka Press, Novosibirsk, pp. 51-57.
Bufal, V.V. 1966. The radiation regime of the Baikal trench and its role
in the formation of climate. Klimat ozera Baykal i Pribaykalya. Nauka
Press, Moscow, pp. 34-71.
Chekotillo, K.A. 1965. The time of vertical transfer of water in the
ocean. Okeanologicheskiya issledovaniya, 13, pp. 24-29.
Kozhov, M.M. 1962. Biology of Lake Baikal. Acad. Sci. USSR Press, Moscow,
p. 315.
Krotova, V.A. 1969. The water balance. Level. Currents. Wave action.
Atlas Baykala. Main Administration for Geodesy and Cartography,
Irkutsk-Moscow, pp. 12-13.
Krotova, V.A. 1970. Geostrophic circulation of the waters of Baikal during
the period of direct thermal stratification. Techeniya i diffuziya vod
Baykala. Nauka Press, Leningrad, pp. 11-14.
Pomazkova, G.I. 1970. Zooplankton of Lake Baikal. Auth. abst. diss. for
degr. of Cand. Biol. Sci., Irkutsk, p. 22.
Pomytkin, B.A. 1962. Some information on currents in the southern portion
of Baikal. Meteorologiya i gidrologiya, 11, pp. 47-49.
Rossolimo, L.L. 1957. Temperature regime of Lake Baikal. Acad. Sci. USSR
Press, Moscow, p. 551.
Skavichevskiy, A.P. 1960. Planktonic diatoms of fresh waters in the USSR.
Moscow State University Press, Moscow, p. 352.
Verbolov, V.I. 1969. The temperature of the surface of a lake. The radia-
tion balance. Total absorbed radiation. Effective radiation. Atlas
of Baikal, Main Administration for Geodesy and Cartography, Irkutsk-
Moscow, p. 11.
Verbolov, V.I. 1977a. Horizontal currents in Baikal. The ice season.
Techeniya v Baykale. Nauka Press, Novosibirsk, pp. 10-16.
Verbolov, V.I. 1977b. General characteristics of the currents during the
navigation season. Techeniya v Bayale. Nauka Press, Novosibirsk, pp.
43-62.
Votintsev, K.K. and G.I. Popovskaya. 1969. Phytoplankton and water chemis-
try. Atlas of Baikal. Main Administration for Geodesy and Cartography,
Irkutsk-Moscow, p. 20.
377
-------
Votintsev, K.K., A.I. Meshcheryakova, and 6.1. Popovskaya. 1975. The cycle
of organic matter in Lake Baikal. Nauka Press, Novosibirsk, p. 190.
378
-------
SECTION 19
THE TASK OF OPTIMAL PLANNING OF DISCHARGE OF POLLUTANTS IN A DYNAMIC
MODEL OF SELF-PURIFICATION OF A BODY OF WATER
O.M. Kozhova and L.T. Aschepkova1
SUPPLEMENTARY RESULTS
We will be studying the vector differential equation
x = f(x,w), x(t ) = h(v) (1)
m Y* Y* n
with the parameters w e R , v e R . The function h:R -»• R is assumed con-
tinuously differentiable. The function f :Rn x Rm + Rn is continuous along
with its derivatives, with the exception of the smooth manifold
p^x.w) = 0, ..., ps(x,w) = 0 (2)
of dimensionality n + m - 1, on which finite discontinuities in its value
are permitted. The solution of Equation (1) with the discontinuous right
portion is taken from A.F. Filippov (I960). As was shown in his work, the
solution exists at least in a small region of the initial values.
Let us fix the number t-j > IQ and determine the mapping (v,w) -*- I(v,w),
assuming
where the solution x(t) of Equation (1), corresponding to v,w, is defined
for tg £ t £t-|, and I(v,w) = + m otherwise. Here : Rn ->• R is an as-
signed function of the class C-|(Rn). In what follows, the question of de-
rivatives of the function I is of interest. Thus, the corresponding re-
sults are presented (Aschepkov, Badam 1977).
Suppose the solution x(t) of Equation (1), corresponding to fixed v,w,
exists where tg <_ t <.t-|, and at instances t = T-| , .. ., TS it successively
intersects the surfaces of Equation (2) such that
1 Irkutsk State University, 1 K. Marks Avenue; 664003 Irkutsk, USSR,
379
-------
tg < T-| < ... < Ts < t-j,
PiW^Kw) = 0, p-jW-u^.w) ? 0, i?j,
Pi(x(Ti-),w) Pi(x(T.j + ),w) > 0, i,j = 1, .... s. (3)
Condition (3) implies that the phase point at moment t = T-J falls only on
one surface p-j(x,w) = 0, approaching and departing at non-zero angles;
Pi(x(T-j+), w) has the sense of the corresponding one-sided derivative of
function p-j(x(t), w) with respect to t, on th§ strength of Equation (1).
Let us relate to x(t), w the vector function $: [t0, t-\ 1 -* Rri, the solution
of the conjugate differential equation
(4)
with the condition of a jump at moment t = T-J:
- ^'(Ti + ) Ax(T.j) / pj(x(T.j-),w),
xtT.j) '= x(Tr) - x(Ti + ), i = 1, ..., S. (5)
where H(t|j,x,w) = ^'flxjw) is a Hamiltonian function, ' is the sign of trans-
position. Due to the linearity of Equation (4) and the unambiguity of the
conditions of the jump (5), the solution of iKt), tg <_ t <_ t-j , not only
exists but is unique.
Under the assumptions made above, function I has at point v,w the
partial gradients Iy, Iw, which are calculated by the equations
Iv(v,w) = h'y(v) iMt0),
s
IW(V,W) = I yi Piw(x(T.j),w) +
1=1
*1
+ / Hw(^(t), x(t),w) dt. (6)
*0
The procedure of calculation of the derivatives can be organized, e.g., as
follows:
1. Equation (1) is integrated forward in time, and the quantities
i, T.J, X(T.J-), X(T.J + ) are recorded;
380
-------
2. Equation (1), (4), (5) are integrated in reverse time with the
known initial values x(t-|), iKti) and with the additional equa-
tion
y = -HW(IM,W), y(*i) = °- (7)
In the process of integration, the sum z|< of vectors y-j p jw(x( TJ),W), i=s,
s-1, ..., k, is accumulated sequentially for k=s, s-1, ..., 1. The deriva-
tives of interest to us are calculated through the values of c •] (eg, c-i = const).
In areas with different levels of pollution, the composition of the aquatic
communities are also different. In strongly polluted areas, species pre-
dominate which are capable of breaking down the pollutant (decomposers).
Relatively pure areas are populated by organisms which previously inhabited
the pure body of water and have adapted to the new conditions. The activity
of the decomposers characterizes the self-purifying capability of the body
of water.
The concentration of organisms inhabiting the pure medium before the
pollutant was introduced may be represented by x-j, while x2 represents the
concentration of decomposers. The rate of change of x-|, x2 and X3 depends
on the concentration X3 at a given instant in time t. In the model, using
certain simplifying assumptions, we assume:
x-| = xi(a-|-a2x-|), x2 = 0, x3 = v, if x3 CQ;
381
-------
x., = - 3y XoXoj if co < X3 < cl»
x-j =0, x2 = ag x2x3, x3 = - a7 6 x2x3 + v, if x3 > c-|. (8)
Here a], ..., ay are non-negative constant coefficients; 6 = 0(t) is the
water temperature at time t; v is a term resulting from water exchange. The
values of v are defined as follows. The body of water is approximated by a
system of k chambers. Each of them is a parallelepiped with identical top
surface area, the height of which is equal to the depth of the body of
water. Each number j of the jth chamber relates to a set Jj of numbers of
neighboring chambers, with J j | = 4 elements, while AJ-J are the fractions of
the volume of water carried by the current from j into chamber i per unit
time. The order of numbering of the elements Jj is indicated in Figure 1.
If the quantity of neighboring chambers is less than 4, the corresponding
elements Jj and numbers Xjj are assumed equal to zero. The vajues. x-j % x2,
x3 and v relating to the jth chamber are represented by xJ, xJ, x
-------
b /i /
t
/,
Figure 1. j.. = (j^, J2> J3, J4)
10
8
6
4
2
0
I I I
0 1
45678
TEMPERATURE(D
9 10 11 12
Figure 2, Temperature curve.
u
Figure 3. Subdivision of body of water into chambers,
0.7
1
0.7
2
0.7
3
0.7
4
o.if jo.i o.iTJo.1 o.TJJo.1 o.ifTo.i
5
0.7
6
0.7
7
0.7
8
0.7 .
Figure 4. Diagram of water exchange.
Figures show fraction A-H of volumes of chamber
carried by current per unit time.
383
-------
T K 2
I(u) = / E [x^-c,], dt. (12)
0 j=l J '
The problem is to determine the rate of discharge of the pollutant in the
form of a function of t, piecewise continuous over 10, T], satisfying
limitations (11) and minimizing function (12).
With one additional assumption, this problem can be reduced to a problem
with parameters. The sector [0, t] may be broken into m pairs of adjacent
intervals T], ..., Tm of identical length h = T/m. In each of these inter-
vals, the control may be regarded as constant:
u(t) = w.j, t e T.J, i = 1, ..., m. (13)
The limitations (11) for Equation (13) take on the form
m
un < u < u-,, i = 1, ..., m; E w- = A/h. (14)
u - - I i=1 i
To the equations for x^, x^ and x^, may be added the additional differential
equations
X0 =
.
- c^ , x4 = 1 (xQ(0) = 0, x4(0) = 0)
for calculation on integral (12) and time (t = x4(t)). To Equation (10) for
the discontinuity surfaces, additional equations may be added,
x^ - ih = 0, i = 1 , . . ., m -1 ,
defining the moments of discontinuity of control (13). As a result, we ob-
tain a problem of the same type as specified in the first paragraph. Find
the vector of parameters w = (w-j, . . . , wm) satisfying limitations (14) and
achieving the minimum of the functional
I(w) = x0(T), (15)
assigned on the trajectory of the system
K
x^ = x^(a1-a2xJ), xjj = 0, x^ = vj, if x^ < CQ;
Xl = Xl(a3-Vl-a5 9 X2}> X2 = h 9 X2J4
if c.j < x^ < c,;
384
-------
= 0, x^ = ag
if X > c-; j=l, ..., k;
x4 = l, (16)
with the initial conditions
x0(0) = 0, x4(0) = 0,
xi(0) = xio» j = ]> 2' 3' j = U ••" k' {17)
The right parts of system (16) have finite discontinuities on the surfaces
PJ = x^ - c0 = 0, pk+J- = x^ - c-, = 0, j = 1, ..., k,
P2k+i = X4 " ih = °' ] = lf '••' m " ]'
The function vJ is defined by Equation (9).
METHODS OF SOLUTION OF THE CONTROL PROBLEM
In order to find the optimal parameters in Equations (13) - (18), the
gradient-type iterational minimization methods may be used. Let us the
equations used to calculate the derivatives of goal function (15). We com
pose the function
k ? k
H = 4^0 2 [*3 - c-,]^ + E Hi +
if
J < CQ;
^1x1(a3-a4x-,-a5 6 x2)
0 xa-a) + v, if CQ
H. = 6 x^x^ag^-ayi^) + ^3V, if x <
and use it to write the conjugate equations of (4)
= - Hx.j, i = 1, 2, 3, j = 1, ..., k (20)
385
-------
with the initial conditions
U,J(T) = 0, i = 1, 2, 3, j = 1, ..., k. (21)
Suppose the fixed w|, .... wm correspond to the solution xo(t), x-j(t),
X4(t) of system (16)-(17), intersecting at times TI, ..., TS, 0 < T-J < ...
< TS < T, the surfaces (18) with numbers i-j, ..., is {!» 2, ..., 2k + m - 1 }
such that conditions such as (3) are fulfilled. This occurs, e.g., for
those T^ for which ijj, > 2k. Since Equation (18) of the discontinuity sur-
faces is independent of XQ, it follows then from the conditions of the jump
(5) that function ^n(t) is discontinuous at times t = T£. Consequently, on
the strength of (19) and (21), 4>0(t) = 1. The conditions of the jump for
the remaining conjugate variables become
*i = *i + ^ api 4
1 — I , £, o, J I, >.., K, Jo I, *••, S,
where
£-), 1 <_ i£ £ k,
Zili^lT+lAY i T 1 / V (T I l^ <" 1 " '0 t. ix j
i=l ~~
3 2 -2
•j=1 i i
j 1, (i = 3, 1 < i£ < 2k)V(j = i£ V j = i£ - k),
'^ 0, otherwise
1, i£ > 2k,
3pi /3x4 =
£ 0, i < i££ 2k. (24)
We can see from Equations (23) and (24) that the function ^(t) is piecewise
continuous. At times of switching of control, i.e., at the ends of inter-
vals TJ, it .has first-order discontinuities. At these same times, the
functions ^(t), as Equations (22) and (24) show, remain continuous.
Writing the second equation of (6) for this problem, we produce the required
values of derivatives of the functions (15) in the form
Iw.(w) = / (t)dt, i = 1, ..., m. (25)
1 386
-------
The algorithm for calculation of derivatives described in the first para-
graph can be applied to this case without significant changes. Instead of
system (7) of supplementary differential equations, it is expedient to
introduce the scalar equation
» • -*§.
If we integrate this equation together with the initial and conjugate equa-
tions in sequence over sections T-j, i = m, m - 1, ..., 1 with the initia-1
conditions y(ih) = 0, at times t = (i - l)h we will have
Iw.(w) = y((i-1)h), i = m, m - 1, ..., 1.
Using the derivatives of (25), we can realize various gradient proce-
dures of minimization of goal function (15) considering the limitations of
(14). The theory and practice of solution of such problems is discussed in
detail elsewhere (Fiacco and McCormick 1968; Himmelblau 1972; Demyanova and
Rubinov 1968; Levitin and Polyak 1966).
RESULTS OF CALCULATION
Equations (13) - (18) were solved on a BESM-6 electronic computer with
the following initial data:
K = 8, m = 12, q = 4, T = 12,
CQ = 0.6, C1 = 1.9; U0 = 0, UT = oos A = 15;
a-, = a3 = 0.1, a2 = a4 = 0.001, a5 = 0.01, a6 = a7 = 0.001;
The graph of function 6(t), a diagram of the subdivision of the body of
water into chambers, and the values of water exchange are shown in Figures
2, 3, and 4.
.JL. JL, .A.
The research for the optimal parameters w =,(wl» ..., W]2) was Per-
formed by the method of the arbitrary gradient (Demyanov and Rubinov 1968)
by the iterational plan:
wv+1 = wv + o^ - wv),
I'(wv) w = min I'w w,
w weW
I(wv+1) = min I(wv + (v^ - wv)), v = 0, 1, ...,
0
-------
by Levitin -and Polyak (1966). As the initial approximation, we can select
the point w° with coordinates w° = 15/12, i =1, ..., 12. The temperature
curve was interpolated with respect to 6 shifting nodes by the method of
Lagrange. The integration of the differential equations was performed by
the explicit method of Euler with a step of 0.01. The optimal values of
the parameters were produced in the second iteration
w* = w* = ... = w* = 1.225, w* = 1.525,
I O I L. O
where I(w*) = 0. The dynamics of propagation of the pollutant over the body
of water, corresponding to the optimal level of discharge, are shown in Fig-
ure 5.
ACKNOWLEDGEMENTS
The authors express their gratitude to N.I. Baranchikova, who performed
the calculations.
REFERENCES
Ashchepkov, L.T. and U. Badam. 1977. Theoretical and computational aspects
of parametric optimization of systems with discontinuous right parts.
Abstracts of reports of 4th National Conference on Problems of Theoreti-
cal Cybernetics, Institute of Mathematics, Siberian Affiliate, USSR
Acad. Sci., Novosibirsk, pp. 86-87.
Demyanov, V.F. and A.M. Rubinov. 1968. Approximate methods of solution of
extreme problems, Leningrad State University Press, 180 pp.
Fiacco, A.V. and Y.P. McCormick. 1968. Nonlinear programming: sequential
unconstrained minimization techniques. J. Wiley & Sons, Inc., New York-
London-Sydney-Toronto, 240 pp.
Filippov, A.F. 1960. Differential equations with discontinuous right part.
Matematicheskii sbornik, 51(93), No. 1, pp. 99-128.
Himmelblau, D.M. 1972. Applied nonlinear programming. McGraw-Hill Book
Co., 536 pp.
Levitin, Ye.S. and B.T. Polyak. 1966. Methods of minimization with limita-
tions. Zh. vychislitel'noy matematiki i matematicheskoy fiziki, 6, No.
6, pp. 787-823.
Pshenichnyy, B.N. and Yu.M. Danilin. 1968. Differentiability of the solu-
tion of systems of differential equations with discontinuous right parts
on the basis of the initial value. Teoriya optimal'nykh reshenii No. 1,
Institute of Cybernetics, Ukrainian Acad. Sci., Kiev, pp. 64-68.
Rozenvasser, Ye.N. 1967. General equations of sensitivity of discontinuous
systems. Avtomatika i telemekhanika, No. 3, pp. 52-56.
388
-------
t = 0
1.5-1.9
1-0-1.5
0.6-1.0
] 0.4-0.6
0.2-0.4
J <0.2
Figure 5. Propagation of a pollutant through a body of water
corresponding to optimal discharge levels.
389
-------
TECHNICAL REPORT DATA .
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/9-80-033
3. RECIPIENT'S ACCESSION NO.
4. TIT' E AND SUBTITLE
PROCEEDINGS OF THE SECOND AMERICAN-SOVIET SYMPOSIUM ON
THE USE OF MATHEMATICAL MODELS TO OPTIMIZE WATER QUALITY
MANAGEMENT
5. REPORT DATE
July 1980 issuing date
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Environmental Protection Agency - USA
Institute of Mechanics and Applied Mathematics - USSR
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Large Lakes Research Station
Environmental Research Laboratory-Duluth
Grosse lie, Michigan 48138
10. PROGRAM ELEMENT NO.
A30B1A
11. CONTRACT/GRANT NO.
Joint US-USSR Project
02.02-12
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Research Laboratory - Duluth,
Office of Research and Development
U.S. Environmental Protection Agency
Duluth, Minnesota 55804
13. TYPE OF REPORT AND PERIOD COVERED
Inhouse
14. SPONSORING AGENCY CODE
EPA/600/03
15. SUPPLEMENTARY NOTES
Performed as part of Project 02.02-12 (Water Quality in Lakes and Estuaries)
of U.S.A./U.S.S.R. Environmental Agreement.
16. ABSTRACT
The Joint US-USSR Agreement on Cooperation in the Field of Environmental Pro-
tection was established in May of 1972. These proceedings result from one of the
projects, Project 02.02-12, Effects of Pollutants on Lakes and Estuaries.
As knowledge related to fate and transport of pollutants has grown, it has be-
come increasingly apparent that local and even national approaches to solving pollu-
tion problems are insufficient. Not only are the problems themselves frequently
international, but an understanding of alternate methodological approaches to the
problem can avoid needless duplication of efforts. This expansion of interest from
local and national represents a logical and natural maturation from the provincial
to a global concern for the environment.
In general, mankind is faced with very similar environmental problems regard-
less of the national of political boundaries which we have erected. While the
problems may vary slightly in type or degree, the fundamental and underlying factors
are remarkably similar. It is not surprising, therefore, that the interests and
concerns of environmental scientists the world over are also quite similar. In this
larger sense, we are our brother's brother, and have the ability to understand our
fellowman and his dilemma, if we but take the trouble to do so. It is this singular
idea of concerned scientists exchanging views with colleagues that provides the
basic strength for this project.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
COS AT I Field/Group
mathematical models
lakes
rivers
water quality
water pollution
water flow
US-USSR Agreement in the
Field of Environmental
Protection
Eutrophication
Ecosystems
06/D
08/H
12/A
13/B
18. DISTRIBUTION STATEMENT
Release to public
19. SECURITY CLASS (This Reportj
Unclassified
21. NO. OF PAGES
410
20. SECURITY CL.ASS (This page)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
U.S. GOVERNMENT PRINTING OFFICE: 1980—657-165/0072
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