EPA -660/3-73-009
August 1973
                                      Ecological Research  Series
       amic  Water Quality Forecasting

 And Management
                                   Office of Research and Development
                                   U.S. Environmental Protection
                                   Washington, D.C. 20460

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            RESEARCH REPORTING SERIES
Research reports of the  Office  of  Research  and
Monitoring,  Environmental Protection Agency, have
been grouped into five series.  These  five  bread
categories  were established to facilitate further
development  and  application   of   environmental
technology.   Elimination  of traditional grouping
was  consciously  planned  to  foster   technology
transfer   and  a  maximum  interface  in  related
fields.  The five series are:

   1.  Environmental Health Effects Research
   2.  Environmental Protection Technology
   3.  Ecological Research
   4.  Environmental Monitoring
   5.  Socioeconomic Environmental Studies

This report has been assigned  to  the  ECOLOGICAL
RESEARCH  series.   This series describes research
on the effects of pollution on humans,  plant  and
animal   species,  and  materials.   Problems  are
assessed   for   their   long-   and    short-term
influences.    Investigations  include  formation,
transport, and pathway studies  to  determine  the
fate  of  pollutants and their effects.  This work
provides the technical basis for setting standards
to  minimize   undesirable   changes   in   living
organisms   in   the   aquatic,   terrestrial  and
atmospheric environments.

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                                                    EPA-660/3-73-009
                                                    August  1973
                     DYNAMIC WATER QUALITY

                 FORECASTING AND MANAGEMENT
                              By

Donald J.  O'Connor,  Robert V. Thomann,  and Dominic M. Di  Toro
                     Project No. R800369
                   Program Element  1BA023
                        Project  Officer

                  Dr. Walter M.  Sanders III
       Southeast Environmental  Research Laboratory
     National  Environmental Research Center-Corvallis
           U.  S.  Environmental Protection Agency
                   Athens, Georgia  30601
                         Prepared  for
             OFFICE OF RESEARCH AND DEVELOPMENT
            U.  S.  ENVIRONMENTAL PROTECTION AGENCY
                   WASHINGTON, D.  C. 20460
 For sole by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 • Price $2.01

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                          EPA REVIEW NOTICE

This report has been reviewed by the Environmental Protection Agency
and approved for publication.  Approval does not signify that the
contents necessarily reflect the views and policies of the Environ-
mental Protection Agency, nor does mention of trade names or commercial
products constitute endorsement or recommendation of use.
                                 ii

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                       ABSTRACT

This report describes  the  formulation and initial verifica-
tion of two modeling frameworks.   The first is directed
toward an analysis of  the  impact  of the carbonaceous and
nitrogenous components and wastewater on the dissolved
oxygen resources of a  natural  water system.  The second
modeling framework concentrates on the interactions between
the discharge of nutrient;  both nitrogen and phosphorus, and
the biomass of the phytoplankton  'and zooplankton populations
which result, as well  as incorporating the overall impact on
dissolved oxygen.  The models  are formulated in terms of
coupled differential equations which incorporate both the
effect of transport due to tidal  motion and turbulence, and
the kinetics which describe the biological and chemical
transformations that can occur.  The modeling frameworks are
applied to the Delaware and Potomac estuaries in order to
estimate the ability of such models to describe the water
quality effects of carbon, nitrogen, and phosphorous dis-
charges .  The agreement achieved  between observation and
calculation indicate that  the  major features of the impact
of wastewater components on eutrophication phenomena can be
successfully analyzed  within the  context of the models pre-
sented herein.

This report was submitted  in fulfillment of Project Number R800369, by
Manhattan College, Bronx, New York, under the sponsorship of the
Environmental Protection Agency. Work  was completed as of December 31,
1972.
                            1.1 a.

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                       CONTENTS



Section                                                 Page


I         Conclusions                                     1


II        Recommendations                                 3


III       Introduction                                    5


IV        Nitrification and Its Effects on Dissolved     11
          Oxygen

V         A Dynamic Model of Phytoplankton Populations   73
          in Natural Waters

VI        A Preliminary Model of Phytoplankton Dynamics 143
          in the Upper Potomac Estuary

VII       Acknowledgments                               191


VIII      References                                    193


IX        Publications                                  201
                            v

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                        FIGURES
                                                         Page
1     Evolution of the Modeling Structures                   9

2     Major Features of the Nitrogen Cycle                  13

3     Two-State BOD Curve - Mohawk River at St.Johnsville,   24
      New York

4     Variation of Growth Rate with Substrate Concentration 26

5     Logistic and Exponential Growth of Microorganisms
      and Accompanying Nutrient Utilization                 32

6     Block Diagram of Nitrification and Dissolved Oxygen
      Utilization                                           39

7     Sequential Reactions in Nitrification - First Order
      Kinetics - Stream System                              42

8     Observed vs. Computed Nitrogen Profile, August 1964   51

9     Estimated Temperature Dependence of Nitrification
      Reaction Rates                                        54

10    Observed vs. Computed Nitrogen Profiles,              55
      November 1967

11    Observed vs. Computed Nitrogen Profiles, July 1967    56

12    Estimated DO Deficit Due to Nitrification             60

13    DO Deficits Under Different Nitrification Conditions   62

14    Verification of Kjeldahl Nitrogen for Potomac Estuary 68

15    Verification of Nitrite and Nitrate Nitrogen and
      Chlorophyll "a" for Potomac Estuary                   69

16    Computed Dissolved Oxygen Deficit Due to Nitrification
      in Potomac Estuary, July - August, 1968               71

17    Interactions of Environmental Variables and the
      Phytoplankton, Zooplankton, and Nutrient Systems      80
                          VI

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                         FIGURES

No.                                                      Page
18    Phytoplankton Saturated Growth Rate (Base e)
      as a Function of Temperature                        86

19    Normalized Rate of Photosynthesis vs  Incident
      Light Intensity                                     89

20    Nutrient Absorption Rate as a Function of Nutrient
      Concentration: Comparison of Michaelis Menton
      Theoretical Curve with Data from Ketchum            96

21    Measured Phosphate Absorption Rate, After Ketchum
      vs Phosphate Absorption Rate Estimated Using
      yN1N2/(Kml + N, )(Km2 + N2) Where N  and N  Are the
      Nitrate and Phosphate Concentrations,  Respectively  99

22    Comparison of Phytoplankton Growth  Rates as a
      Function of Incident Solar Radiation Intensity and 103
      Temperature

23    Endogenous Respiration Rate of Phytoplankton vs
      Temperature after Riley (1949)                     106

24    Grazing Rates of Zooplankton vs Temperature        110

25    Endogenous Respiration Rate of Zooplankton vs
      Temperature                                        117

26    Temperature, Flow, and Mean Daily Solar Radiation;
      San Joaquin River, Mossdale, 1966-1967             132

27    Phytoplankton, Zooplankton, and Total Inorganic
      Nitrogen; Comparison of Theoretical Calculations
      and Observed Data; San Joaquin River,  Mossdale,    137
      1966-1967

28    Theoretical Growth Rates for Phytoplankton and
      Zooplankton Populations                            141

29    Map of Potomac Estuary Showing Longitudinal and
      Lateral Segments                                   146

30    Interactions of Nine Systems Used in Preliminary
      Phytoplankton Model                                148

31    Temperature and Flow Regimes Used for 1968 and
      1969 Verification Runs                             155
                           VI1

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                         FIGURES


No.                                                     Page

32   Comparison of Range of Observed Data and Model
     Output - August 1968.  a) Chlorophyll a (yg/1)
     b) Total Kjheldahl Nitrogen (mg/1)                  157

33   a) Total Phosphorous - PO. (mg/1) Comparison
     b) Nitrate Nitrogen Comparison, August 1968         158

34   Effect of Zooplankton Grazing on Phytoplankton in
     Segment #9  a) No Zooplankton Grazing  b)  Zooplankton
     Grazing at 0.42 1/mg Carb-Day, 1968 Flow Regime     160

35   Effect of Zooplankton Grazing on Nitrate Nitrogen
     in Segment #9 and Ammonia Nitrogen in Segment #6,
     1968 Flow Regime                                    161

36   Sensitivity Run - No Tidal Bay Segments.
     August 1968 Profile and 1968 Conditions             163

37   Spatial Profile Comparison of Observed 1969 Data
     and Computed Values for Chlorophyll a and Total
     Kjeldahl Nitrogen                                   166

38   Spatial Profile Comparison of Observed 1969 Data
     and Computed Values for Total Phosphorous and
     Nitrate Nitrogen                                    167

39   Temporal Comparison of Observed 1969 Data and       170
     Computed Values for Stations at Miles 12.1 and 18.3

40   Temporal Comparison of Observed 1969 Data and Com-
     puted Values for Stations at Miles 26.9 and 38.0    171

41   1969 Simulation of Chlorophyll, June 30 and July 15.172

42   1969 Simulation Contour Plot                        173

43   1969 Simulation Contour Plot                        174

44   1969 Simulation Contour Plot                        175

45   1969 Simulation Contour Plot                        176

46   1969 Simulation Contour Plot                        177

47   1969 Simulation Contour Plot                        178
                            Vlll

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                         FIGURES


No.                                                      Page

48   1969 Simulation Contour Plot                         ._-

49   1969 Simulation Contour Plot                         .0_
                                                          1 o J
50   Temporal Variation in Chlorophyll a at Segments #9
     and #28, Median Flow Simulation, a) lyg/1 Chlorophyll
     Boundary  b)  25yg/l Chlorophyll Boundary             186

51   Median Flow Simulation Profile of Chlorophyll -      188
     July 15
                            IX

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                         TABLES


No.                                                     Page
1     Nitrogen Balance Sheet for the Harvested Crop       19
      Area of the United States, 1930

2     Estimated Municipal and Industrial Nitrogen         49
      Discharges to Delaware Estuary

3     Summary of Reaction Coefficients Determined in      58
      Verification Analysis of Nitrogen in Delaware
      Estuary

4     Estimated Significant Input Nitrogen Loads,         67
      Potomac Estuary

5     First Order Reaction Coefficients, Potomac Estuary  67

6     Maximum Growth Rates as a Function of Temperature   87

7     Michaelis Constants for Nitrogen and Phosphorus     97

8     Endogenous Respiration Rates of Phytoplankton      107

9     Grazing Rates of Zooplankton                       109

10    Endogenous Respiration Rate of Zooplankton         117

11    Dry Weight Percentage of Carbon, Nitrogen, and     121
      Phosphorus in Phytoplankton

12    Parameter Values for the Mossdale Model            136

13    Parameters Used in Verification of 1968 and 1969   164
      Potomac River Data - Preliminary Phytoplankton Model


14    Direct Discharge Waste Loads

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                       SECTION I

                      CONCLUSIONS

Based on the results of the mathematical model formulations
and verifications presented in this report it is concluded
and the impact on water quality of the carbonaceous, ni-
trogenous, and phosphoric fractions of wastewaters discharged
into an estuarine environment can be assessed, to guide the
preliminary planning of remedial actions to improve water
quality.  The water quality parameters that appear sus-
ceptible to such an analysis are the dissolved oxygen levels
and the phytoplankton biomass which result as a consequence
of natural and man-made inputs.  The analyses and verifica-
tions presented for the Delaware and Potomac estuaries are
viewed as the foundation upon which a rational investigation
of wastewater treatment alternatives can be based for the
management and control of estuarine water quality.

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                        SECTION II
                     RECOMMENDATIONS

The modeling formulations which have been developed in this
study should be applied to other estuarine water bodies in an
attempt to further strengthen the verification and refine the
kinetic structures employed.  In particular, the eutrophica-
tion analysis is an initial attempt to incorporate the varied
and complex interactions which characterize the growth and
decay of phytoplankton biomass in natural waters and the re-
lationships to nutrient concentrations.  This analysis should
be extended to include effects of other microorganisms as well
as the effect of chemical and biological parameters not in-
cluded in the preliminary formulations.

In principle the models developed herein can be extended to
apply to other settings such as lakes and coastal waters.
The modifications and adjustments for such an attempt would
be a fruitful continuation.  The need for further comparisons
between observed data and calculated concentrations cannot be
over emphasized.  Only in this way can progress be made in the
understanding of the complex phenomena which control water
quality in natural bodies of water.

Finally, it is recommended that the models developed in this
work be applied in the planning and evaluation of water quality
enhancement programs.  Although the models are in no way com-
pletely satisfactory from a scientific point of view, and may
in fact not incorporate certain effects which may prove to be
important, they have been shown to be realistic and capable
of reproducing the observed situation for the cases considered
in this report.  The models, therefore, are worthy of consid-
eration in any attempt to rationally plan and execute water
luality enhancement programs.

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                      SECTION III

                      INTRODUCTION
As the nation moves forward in its program for water pollution
control and water quality management,  the need for both the
short and long term operation and management of water resource
systems becomes ever more important.  Complex problems regard-
ing subtle interactions between waste treatment processes and
the environment must be considered.  These problems can be
conceptualized and formulated using the techniques of dynamic
systems analyses.  However the general theory must be specifi-
cally adapted to these problems.  A whole expertise must be
developed which enables a translation of the problems of water
quality maintenance and prediction and water resources manage-
ment into the abstract formulations of dynamic systems analyses.

As is well known, a sequence of profound biological changes
occurs in a natural body of water receiving untreated waste
water.  What is not so well known, however, is the chain of
events that is set into motion as a result of discharging
biologically treated, nutrient-rich, effluents.  When the
waste is untreated, the bacterial populations predominate.
These are of the form which oxidize organic carbon in their
metabolic processes.  When this material is removed in a bio-
logical treatment unit, the next component of the cycle, ni-
trification, becomes more significant.  If the treatment
processes are designed to allow the nitrifying bacteria to
develop in the plant, then an end product of relatively stable
nitrates results.  If nitrification does not occur during
treatment, or is only partially completed, the remaining nitro-
gen is discharged.  Both the residual carbonaceous material
and the nitrogenous material exert a separate and distinct
oxygen demand on the water bodies resources.

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Simultaneously, the photosynthetic and respiratory processes
of the algae are operative.  If nitrogen is not removed in
treatment, the environment is more conducive to the growth
of algae, which then proliferate.  When the algal blooms ex-
ceed the capacity of the system, mass mortality may occur.
Algal decay occurs and carbonaceous and    nitrogenous demand
is returned to the water body thereby initiating the cycle
again.  It is therefore necessary to proceed in a sequential
manner so that these important secondary ecological effects
are incorporated both in the analysis and planning phases.

Efficient dynamic regional management of these systems there-
fore requires a foundation of analytical tools and techniques
that will buttress the solution of the very complex problems
associated with short and long term water pollution control.

The basis for the methods presented herein is the principle
of conservation of mass.  It can be expressed in mathematical
terms as a partial differential equation which related the
concentration of a substance c(r,t) at a position r and time
t to the effects of mass transport, which are described by
a velocity vector field U(r,t) and a dispersion matrix E(r,t);
the effects of kinetic   transformations, which are described
in terms of sources and sinks of the substance S(c,r,t); and
the effects of direct discharges to the water body W(r,t).
The requirement that the mass changes are accountable in these
terms requires that:

  ff + ? 0 [-E(r,t)Vc + U(r,t)c]  =  S(c,r,t) + W(r,t)    (1)

where V = 9/9x i + 9/9y 3 + 9/9z k.  Given the mass transport
and kinetic descriptions and for specified direct discharges
and boundary conditions this equation is solvable, in principle,
for the resulting concentration distribution c(r,t) as a

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function of position and time.
For complex, interacting situations, it is necessary to
simultaneously characterize the concentrations c.(r,t) of
a number of substances.  In this case the kinetic interaction
terms are usually functions of all the concentrations,
S.(cn, c-,  ..., c , r,t) so that a set of n simultaneous
equations result of the form:
  3c
  -5± + V '- D .   =  S. (cn , . . . ,c ,r,t) + W. (r,t)  i = 1, . . ,n    (.2)
   o L.        i      i  j.      n         i

where ]".  =  - E(r,t)Vc. + U(r,t)ci/ the mass flux, due to
transport.

It is also common to simplify these equations by analyzing
less than the three spatial dimensions.  The simpliest situ-
ation occurs in a one-dimensional analysis for which the
set of equations  (2) becomes
                              > = Si(cl
The procedure followed in the applications described herein
has been to numerically integrate these equations in order
to characterize the distribution of substances of concern.

The water quality problems investigated center on the effects
of nitrogen and phosphorus discharges as well as the carbon-
aceous waste discharges.  The analyses are directed toward
the effects on the dissolved oxygen distribution on the one
hand and the effects on the first two trophic levels of the
food chain, the phytoplankton and zooplankton, on the other.
A series of models are presented and verified in the subsequ-

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ent chapters indicating the utility and power of these
methods.  The evolution of the modeling structure is il-
lustrated in Figure 1 which presents the development and
increasing scope of the modeling frameworks with the latter
models encompassing a relatively broad range of environmental
variables associated with the eutrophication phenomena.

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      CARBONACEOUS  BOD
      DISSOLVED OXYGEN
         Carbonaceous BOD
         Nitrogenous BOD
         Dissolved Oxygen
        Carbonaceous BOD
    Organic Nitrogen, Ammonia
     Nitrate, Dissolved Oxygen
       Phytoplankton Biomass
       Zooplankton Biomass
      Total Inorganic Nitrogen
       Phytoplankton Biomass
       Zooplankton Biomass
Organic Nitrogen, Ammonia, Nitrate
Organic Phosphorus, Orthophosphate
        Carbonaceous BOD
         Dissolved Oxygen
     Linear
     Steady-state
—   Dissolved Oxygen
     Models
     Non-linear
     Unsteady-state
     Eutrophication
     Models
Fiq.  1    Evolution of  the  Modelinq  Structures
                        9

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                    SECTION IV
           NITRIFICATION AND ITS EFFECTS
                ON DISSOLVED OXYGEN
The discharge of nitrogenous compounds into natural waters
produces a variety of changes in water quality.  Changes
occur not only in the various forms of nitrogen, but also
in those substances with which the nitrogen may react.  The
oxygen requirements for the ammonia and nitrite oxidation,
the utilization of ammonia and nitrate as a nutrient by phyto-
plankton, and the ammonia interference with chlorination, are
significant examples.  The nitrogenous compounds, which may
be either organic or inorganic in nature, are found  in urban
and agricultural runoff, domestic waste waters and industrial
effluents.  A "background" concentration of nitrogen is
present in most water systems and is the result of a dynamic
equilibrium of the natural sources of nitrogen in rainfall,
from the land and within the ground.  Although the man-made
effects are generally of greater significance in the pollu-
tion of natural waters, background nitrogen concentrations
may be present in amounts that must be considered in any
modeling effort.

The purpose of this section is to present several simplified
mathematical models of nitrification in rivers and estuaries
which are based on first order kinetics and the assumption
of temporal steady state.  Specific attention is directed to
the role of nitrification in the modeling of the dissolved
oxygen balance.  The models presented here progress from rela-
tively simple nitrogen equivalent biochemical oxygen demand
models to more complex models which incorporate feedback
effects.

The broad aspects of the nitrogen cycle are reviewed and the
importance of nitrogenous waste loads in the dissolved oxygen
                          11

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balance of natural wastes is stressed.  This review is follow-
ed by a critical review of the reaction kinetics that actually
prevail in nitrification.

Figure 2 schematically outlines the major features of the ni-
trogen cycle that are of importance.  It is appropriate to
initiate the cycle at the point where organic nitrogen
(amines, nitriles, proteins) and ammonia resulting from mu-
nicipal and industrial waters are discharged into a water
body.  The organic nitrogen undergoes an hydrolytic reaction
producing ammonia as one of the end products, which in addi-
tion to the ammonia present in the waste waters, provides a
food source for the nitrifying bacteria.  The oxidation pro-
cess proceeds sequentially from ammonia through nitrite to
nitrate.  The conditions under which these reactions proceed
are comparatively restrictive, but, if present, they provide
an appropriate environment which may result in large deple-
tions of dissolved oxygen.

The forward sequential reactions of the nitrification process
often are the dominating features of the nitrogen cycle in
bodies of water receiving large discharges of nitrogenous
waste material.  The reactions proceed in the forward di-
rection, provided the concentration of dissolved oxygen is
sufficiently high to meet the requirements of the nitrifying
bacteria.  However, under conditions of low concentration of
dissolved oxygen, bacterial reduction of nitrate to nitrite
can occur followed by the further reduction of nitrite pri-
marily to nitrogen gas, although a few species may reduce the
nitrite to ammonia.  These reactions provide a source of oxygen
for the microorganisms in the stabilization of organic matter
without utilizing whatever dissolved oxygen is present in the
water.
                             12

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NITRIFICATION
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In addition to the removal of nitrate by bacterial reduction
it may also be used by the phytoplankton as a nutrient source.
The nitrate must be converted to ammonia by enzymatic reaction
before it is assimilated.  The assimilated nitrogen becomes
part of the organic nitrogen in plants and subsequently in
animals.  Excretion and decay of this material releases or-
ganic nitrogen, thereby completing the cycle.  Ammonia, or
some form of nitrogen, is also required by the heterotrophic
bacteria in the oxidation of carbonaceous material; however,
this sink of nitrogen is relatively insignificant by contrast
to the oxidation process or algal usage.

In summary there are two broad areas of concern from the water
quality viewpoint with respect to the nitrogen cycle in na-
tural waters:  1. The oxidation and possible reduction of
various forms of nitrogen by bacteria and the associated
utilization of oxygen.  2. The assimilation of the inorganic
nitrogen and the release of organic nitrogen by phytoplankton
during growth and death respectively.  In some cases, either
one or the other of these conditions dominate, the former in
areas of large sources of waste water with little or partial
treatment and the latter in water bodies receiving biologically
treated effluents or agricultural drainage.

This section is concerned with the first of these broad areas.
Its specific purpose is to present a mathematical model of the
nitrification reactions in streams and estuaries.  The basic
theory of the nitrogen cycle is reviewed; several simplified
kinetic mathematical models of portions of the cycle are con-
structed and application of the simplified models to specific
situations is described.
                              14

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                     Nitrification

Nitrogenous matter in waste waters consists of proteins, urea,
ammonia and, in some cases, nitrate.  The intermediate decom-
position products of the proteins such as amino acids, amides
and amines are also present in varying degrees.  The proteins
are broken down by hydrolysis in a series of steps into a
variety of amino acids.  Both exocellular and endocellular
enzymes are involved in the process.  Aeration and alkaline
conditions favor the production of the exo-enzyme.  The amino
acids are very soluble in water and exert a strong buffering
action; the carboxyl group reacting with the hydrogen ion and
the amino group reacting with the hydroxyl.  The decomposition
of the amino acids which can occur in a number of different
ways, is endocellular.  Ammonia is released in this process
of deamination, which may be reductive, oxidative or hydro-
lytic reaction, depending on the nature and structure of the
amino acids.  In any case, the significant end product is
ammonia.

Ammonia is also released in the aerobic decomposition of pro-
teins by heterotrophic bacteria.  The ammonia, which is highly
soluble, combines with the hydrogen ion to form the ammonium
ion, thus tending to raise the pH.  In the neutral pH zone,
all of the ammonia is present in this form and at the higher
pH it is evolved as a gas.  The heterotrophic decomposition is
the typical reaction in the first stage of the biochemical de-
oxygenation of natural waters.  Ammonia is an end product of
both this reaction and the reaction associated with the hydro-
lytic breakdown of proteins.  The ammonia present in natural
waters is thus a result of either the direct discharge of the
material in waste waters or of the decomposition of organic
matter in various forms.
                           15

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The ammonia in turn is oxidized under aerobic conditions to
nitrite by bacteria of the genus Nitrosomonas as follows:
(Sawyer, 1960, Hutchinson, 1957, Stratton, 1967 and Delwiche,
1956)
(NH4) + OH  + 1.502          >• H  + N0~ + 2H2 + 59.4 K cal  (4)

This reaction requires 3.43 gins of oxygen utilization for one
gram of nitrogen oxidized to nitrite.  The nitrite thus formed
is subsequently oxidized to nitrate by bacteria of the genus
Nitrobacter as follows:  (Hutchinson, 1957)

        N0~ + 0.50, bacteria > N0~+18 K cal                 (5)
          ^       ^              J

This reaction requires 1.14 gms of oxygen utilization for one
gram of nitrite nitrogen oxidized to nitrate.  The total oxygen
utilization in the entire forward nitrification process is
therefore 4.47 gms of oxygen per gm of ammonia nitrogen oxidized
to nitrate.  The Nitrobacter bacteria process about three times
as much substrate as the Nitrosomonas bacteria to derive the
same amount of energy.  Nitrite is therefore converted quite
rapidly to nitrate.

Essential factors for nitrification are oxygen, phosphates and
an alkaline environment to neutralize the resulting acids.
Nitrifying bacteria are very susceptible to action of toxic
substances (e.g. manganese).  These bacteria are obligate auto-
trophs , which derive their energy from the oxidation of simple
inorganic compounds .  The energy obtained from these reactions
is utilized for the assimilation of carbon from either carbon
dioxide or bicarbonate ion, but not from organic carbon.
However, it has been indicated that Nitrobacter can be grown
heterotrophically on an organic substrate, without losing its
ability to oxidize nitrite.  Acetate was assimilated as cell
                            16

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carbon, the optimum pH range being 8.5 - 9.5.  The alkaline
environment is required to neutralize the acidic end-products.
Below pH of 6.0, which can occur in a poorly buffered system,
inhibition occurs.  The presence of organic matter, particu-
larly amino compounds in excessive concentrations, inhibits
growth and respiration of the nitrifying bacteria.  These
excessive concentrations in the order of thousands of mg/1
are seldom encountered in either waste water or natural waters.
However, even concentrations of organics in the order of hun-
dreds of mg/1 which may be found in practical cases, appear
to still retard the nitrification process.  At these concentra-
tions the heterotrophic bacteria probably predominate and
assimilate the ammonia in their metabolic processes.  After
the death and lysis of these bacteria, nitrification takes
place.  The experimental evidence in this regard is not con-
clusive and, in fact, is somewhat contradictory.  While the
majority of reported work indicates some degree of inhibiting
effects, other reports show minimal or no retardation.  At
lower concentrations of organic materials, as may be found
in natural water bodies, both heterotrophic and autotrophic
reactions may occur simultaneously.

Given the appropriate conditions, the concentration of the
organisms appears to be the significant factor controlling
the rate of nitrification, with the concentration of the re-
actant having a reduced effect.  However, if there is an ample
supply of organisms, the rate appears to be controlled by the
concentration of the reactant  (see also - Kinetic Models).
The number of nitrifiers is, of course, determined by the
generation time of organisms, which is in the order of one
day, by contrast to an order of a few hours for many hetero-
trophic bacteria.

The common source of the nitrifying organisms is rich soil,
                           17

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where they are usually found in high numbers.   In  rivers  re-
ceiving waste waters, nitrifying bacteria are probably  present
in varying degrees depending on the nature  and  the treatment
of waste waters.  Their number in sewage is  low (about  100/ml)
but increases through biological treatment  to an order  of 1000/
ml. Natural habitats are found on biological aggregates or  on sur-
faces where the environment appears appropriate for optimum
growth and acceleration metabolism.  Surfaces such as these
are found in trickling filters and activated sludge treatment
plants as well as the rocky beds of shallow rivers.

                     Denitrif ication

Under conditions of low concentration of dissolved oxygen the
bacterial reduction of nitrate can occur.   This reaction  is
to be distinguished from the utilization of nitrate and sub-
sequent reduction by aquatic plants.  A large variety of
facultative bacteria can reduce nitrate.  E Coli are cited
as a common bacteria capable of reducing nitrate.  (Hutchinson,
1957)   The reduction by bacteria of nitrate is  probably all
to N0_ and then to nitrogen gas although complete  reduction
to ammonia may also occur.  The primary reactions  seem  to be:
C6H12°6 + 12 N°3          > 12 N0  + 6 C02 +  6H20         (6)

which represents the reduction of nitrate to  nitrite and

ccHn°c + 8 N°o bacter:La > 4 N0 + 2 C00 + 4CO^ + 6 H00    (7)
for the reduction of nitrite to nitrogen gas.   It should  be
noted that these reactions serve  in place of an  oxygen  source
for micro-organisms in the metabolic oxidation  of organic
compounds in a water body, without drawing on the dissolved
oxygen resources of the stream.
                             18

-------
The dissolved oxygen conditions under which nitrate reduction
becomes significant are subject to some differences of opinion
resulting from the relatively small amount of work done in
this area.  Under completely  anaerobic conditions, nitrifi-
cation cannot occur since the nitrifying bacteria are strictly
aerobic.  There is some evidence to indicate, however, that
nitrate reduction goes on constantly and is greatly acceler-
ated under low (0-2 mg/1) dissolved oxygen conditions.

                   Sources of Nitrogen

Agricultural sources of nitrogen may be significant with
respect to concentrations found in natural waters.  The
nitrogen is present in the drainage water from these lands
and aside from that inherent in the soil itself, emanates
from the fertilizers, legumes and barnyard and silo effluents.
An attempt at a nitrogen balance for harvested crop areas of
the United States is given in Table 1.
                         Table 1
              Nitrogen Balance Sheet for the
      Harvested Crop Area of the United States, 1930*
                                                Nitrogen
                                              Ib/acre/y.ear
Additions
Rain and irrigation
Seeds
Fertilizers
Manures
Symbiotic nitrogen fixation
Nonsymbiotic nitrogen fixation
Losses
Harvested crops
Erosion
Leaching
4.7
1.0
1.7
5.2
9.2
6.0
25.1
24.2
23.0

27.8
72.3
Net Annual Loss                                       44.5
*After Allison, 1955, quoted by Feth, 1966.
                           19

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The value for erosion is equivalent to an average annual value
of almost 8 mg/1 of nitrate as nitrogen from a drainage area
of 1 square mile with a flow of 1 cubic foot per second.  This
value appears to be high for most natural waters but may be
present in rivers draining harvested lands during certain peri-
ods of the year.

Nitrogen leached from drained soil on which alfalfa and blue-
grass are grown in Kentucky can yield up to 10 pounds/acre
year and corn fields in Wisconsin have yielded in the range
of 20 and 40 pounds per acre year.  Irrigated lands with di-
versified forms drained between 2 and 25 pounds of nitrogen
per acre year.

Fertilizers are significant sources of nitrogen in harvested
lands.  The fertilizers are in forms such as ammonium sulfate
and nitrate, anhydrous ammonia and in various forms of nitrates
and phosphates, all of which are highly soluble.  During peri-
ods of excessive runoff or when irrigation is in excess of
plant requirements, nitrogen in various forms is carried off
to either the ground or surface water.  Barnyard and silo
drainage, as well as wash waters and effluents from feedlot
and dairy operations, undoubtedly add nitrogen to water courses,

A recent report described the nitrogen enrichment of surface
waters by absorption of ammonia which was volatilized from
cattle feedlots (Hutchinson, 1969) .  Laboratory studies in-
dicate that as  much as 90% of urine nitrogen may escape as
ammonia from feedyards.  A significant quantity of this ni-
trogen may be absorbed by water surfaces within a few kilo-
meters from the feedlots (between 10 - 20 kg/ha-yr).  A value
of 4 kg/ha-yr was measured at a site in which there were no
irrigated fields or feedlots within 3 km and no large lots or
cities within 15 km.  On the other hand, absorption rates of
                            20

-------
about 35 kg/ha-yr were reported for sites 2 km from a large
feedlot and 73 kg/ha-yr for sites 0.4 km from a large feedlot.
Based on these measurements the investigators conclude the in-
crease in nitrogen for a nearby lake could be as high as 0.6
mg/1.

Increases in nitrate concentration have been reported (Bormann,
1968) when forests of watershed systems are cut.  Over the
period of one year, during which the land was clear cut and
a herbicide applied to prevent regrowth, the concentration
of nitrate in the drainage increased from about 1 mg/1 to ap-
proximately 40 mg/1 the following year and 50 mg/1 the second
year.  The runoff from the cleared area also increased over
the period.  A mass balance of the nitrogen indicated an annual
loss of nitrogen from the drainage area of about 50 kg/ha.
The increase is accounted for by increase flow and drainage,
reduction of root surfaces and production of an environment
more favorable for bacterial mineralization.

The contribution of rainfall to the nitrogen balance may be
significant in larger bodies of water.  Assuming a range of
0.5 to 1.0 mg/1 in rainfall (Feth, 1966), the annual input
would be between 5 and 10 pounds per acre year for a rainfall
of about 50 inches per year.

The contribution of total nitrogen due to domestic wastes may
range from 5 to 50 pounds/capita year, depending on the eco-
nomic and social characteristics of the area.  In urban and
suburban areas population densities may vary from 5 to 25 or
more people per acre.  On an areal basis, therefore, nitrogen
from this source may run from 25 to 250 pounds/acre year.
This is obviously one of the most significant sources.  The
concentration of organic nitrogen in untreated municipal sew-
age ranges from 5 to 35 mg/1 with an average of about 20 mg/1
                             21

-------
and of ammonia ranges from 10 to 60 mg/1 with an average of
about 25 mg/1.

Industrial waste waters may contain appreciable quantities
of nitrogen in various forms.  Ammonia is one of the most
commonly used industrial chemicals.  Nitrogen in its various
forms is used in manufacture of dye, glass, explosives, many
chemicals and synthetic products, and may therefore be found
in significant quantities in these waste waters.  They are
not usually of importance in paper, tannery, textile, metal
and in some vegetable and fruit-processing wastes.  On the
other hand, they are present in meat-packing, brewery, dairy
and coke plant wastes.

The nitrogen in plant and animal proteins is measured by the
Kjeldahl method, which converts all of the organic nitrogen
to ammonia,  The suspended fraction includes the nitrogen
present in the living and dead plankton and in the animal
excreta, while the dissolved portion contains the nitrogen
from excreted materials, usually associated with the degrada-
tion of cells.  The inorganic forms of nitrogen, ammonia, ni-
trite and nitrate may be measured directly.  Thus the nitrogen
present in waste streams or natural waters is usually reported
in terms of the specific inorganic forms or as organic nitrogen,
which covers a range of compounds in various stages of degra-
dation.

The total oxidizable nitrogen may be measured indirectly by
the standard biochemical oxygen demand test.  The first stage
reflects primarily aerobic oxidation of the organic material,
in which an end product is ammonia.  Simultaneously the organic
nitrogen is hydrolyzed to release ammonia.  The ammonia thus
formed from these two sources, in combination with the ammonia
present in waste waters undergoes oxidation through nitrite to
                            22

-------
nitrate.  This process of nitrification is often referred to
as the second stage of biochemical oxygen demand.  In untreated
and heavily polluted water, the two stages do not usually occur
together, the first being substantially completed before the
second is significantly underway.  In treated effluents and
less polluted waters, the lag between the two is reduced. As
the first stage is reduced and with nitrifying organisms
present, the two stages may occur simultaneously.  An example
of this reaction in BOD samples is shown in Figure 3.
(O'Connor, 1968) .

                     Kinetic Models

The nitrification process, in which ammonia is converted
through nitrite to nitrate is an autotrophic biochemical
reaction.  The energy for the growth of the microorganisms
is obtained from the oxidation of ammonia or nitrite (see
Eqs. (4) and (5).   in reactions of this type the rate is
generally assumed to be proportional to the concentration
of the substrate and also of the microorganisms.  The sub-
strate concentration dependency, however, may be modified by
stipulating that the rate of the reaction is independent of
the concentration of substrate at high substrate concentra-
tion and becomes increasingly concentration dependent as the
concentration decreases.  Substrate limiting kinetics have
been used in the analysis of biological waste treatment phe-
nomena  (Lawrence and McCarty, 1970; Pearson, 1968).  A
general review of the literature is given in Lawrence and
McCarty  (1970) .  The growth equation for the microorganisms
may be written as :
                            =  KM                      (8)
where M  =  the microbial concentration
      K  =  microbial growth coefficient.
                            23

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       10



       9



       8
  UJ
  a
  X
  O
       0
                     K=0.19/(
                               O
day
 8    10    12   14    16
   INCUBATION TIME
      (DAYS)
18    20
                                       Q
                                       O
                                       oo
   Q
   O
   00
22
Fig. 3  Two-state BOD Curve - Mohawk River at St. Johnsville, N.Y.

-------
In unlimited growth, the solution of Eq. (.g)  indicates that
the microbial population would grow at an exponential rate.
If however, a single substrate Csay nitrogen) plays an im-
portant role in limiting the growth, one must also consider
the utilization of the substrate and consider that the growth
coefficient, K, in Eq. (8)  is not a constant but depends on
a substrate concentration.   Thus, let
                               K c
where K  is the maximum growth rate (I/day) ,  c is the concen-
tration of substrate (mg/1) and K  (the so-called Michaelis
                                 o
or half-saturation constant) is the concentration (mg/1) at
which the growth rate is one-half of the saturated rate.
Fig. 4  is a sketch of this functional form.
Substitution of Eq. (9) into (8) gives:

                         dM  =  KmMc                     (10)
                         dt     K +c
                                 s
This form of kinetic equation was first developed by Michaelis
and Menton to explain enzymatic reactions.  It was later ap-
plied by Monod (1942)  to systems involving growth of biological
organisms.  This equation can be written in terms of micro-
organisms only by substituting for the substrate concentration
its equivalence in terms of microorganism i^e a unit increase
of organisms is equal to a unit decrease in substrate multi-
plied by the appropriate stochiometric or yield coefficient.
This is simply an expression of the conservation law, in which
the total mass, M, is equal to the sum of the initial organisms
present plus the substrate utilized to produce microbes, i.e:

                         M  =  MQ + a(cQ-c)              (11)
                            25

-------
     o
     o:
                                                           K
                                                            m
            K
             s     SUBSTRATE  CONCENTRATION - C
Fig.  4  Variation of Growth Rate with.Substrate Concentrations
                              26

-------
in which a  = stochiometric or yield constant (nig bacterial
              mass per mg substrate)
         M  = initial bacterial mass
          o
         c  = initial substrate concentration
          o
Substitution of Eq .  (n)into Eq.(ioy gives upon integration

         Ks             Co          aK + M + ac     M
                ln (  --   )  +   *        ° ln
      M    ac       c -l/a(M-M )        VL  + ac      F
       o     o       o        o         o     o      o
A detailed dimensionless analysis of the behavior of this
type of equation is given below in terms of the utilization
of the substrate.

Equation  (10) indicates that at high concentrations of sub-
strate (c » K ), the rate is independent of c and the rate
              s
expression reduces to a first-order reaction in which the
growth rate is proportional to the concentration of micro-
organisms :

                         dM     _. ,,
                         j^r  =  K M
                         dt      m

This equation indicates an unlimited exponential growth of
organisms.  At low concentration of substrate  (K  » c), the
expression reduces to a second-order reaction  in which the
rate is proportional to the product of the concentration as in

                         dM  =  jn	                      (14)
                         dt      K
                                  s

This equation can also be written entirely in  terms of micro-
bial mass by considering the bacterial equivalent of the
initial substrate concentration and letting M  be the sum of
                            27

-------
the initial organisms and the substrate.  Thus
                     dM
                     dt
                      (MT - M)
(15)
where Mm =
M  + ac  and
 o     o
J]L_
aK
Equation (15) is referred to as an autocatalytic reaction in
which the rate is increased by the concentration of the end
products of  the reaction.  It is autocatalyzed by the micro-
organisms which increase as substrate is oxidized.

The interacting differential equations for the substrate c
and the microbial mass are given by
and
                       d£
                       dt
                       dM1
                       dt
                   -K M'c
                     m
                    K +c
                     s
                   K M'c
                    m
                   K +c
                    S
(16)
(17)
where all quantities are now expressed in terms of their
equivalent substrate concentrations.  Thus, M' = M/a, the
microbial mass in substrate equivalents.  Eq.  (.17) is the
same as Eq.  (14) except in terms of the substrate concentra-
tion.  In the interests of mathematical simplicity,  the
microbial loss due to endogenous respiration has been neg-
lected in these equations.  The solution to Eqs. (16) and (17) is
        1 + d
             M
               log
                     1+d., - c/c
                        M    ' o
            dM c/c
                      + log
                              1+dM * c/co
                                   M
(18)
where d  = K /c , a dimensionless "Michaelis Number" and
       S    SO
                            28

-------
      dM = M" /c ,  a dimensionless microbial-substrate
             o  o
           concentration ratio.
Equation (18) is simply a dimensionless reexpression of Eq. (12)
One can examine several special cases of Eqs.  (16) and  (17)
which will provide some understanding of the behavior of
these equations.

Case I - Small Michaelis constant, small initial microbial
         mass
The limiting value for this case is K  =0 which results in
                                     s
the exponential growth of microbes as given from the solution
to Eq. (13)   The substrate utilization is then

              c/CQ  =  1 - dM exp  (^t)                  (19)

In Eq. (18)  the second term dominates for this case.

Case II - Large Michaelis constant, small initial microbial
          mass
For this case, d  is considered large and one obtains the
                o
logistic growth equations (Eq.  (15). In terms of substrate
utilization, the solution is
                                                          (20)
   J exp(-(KMt) (l+dM)/ds)
*K + exp(-KMt(l+dM)/ds)
for small d  (initial microbial mass relative to initial sub-
strate concentration)
                 exp (-KMt/ds)
                     exp(-KMt/ds)   l+dM expCl-KMt/ds)    (21)
                           29

-------
Case ITI - Large initial microbial mass, large Michaelis
           constant
In this case, d  is considered large, i.e. there is a large
microbial mass relative to c  at t = 0.  Under this condition,
M' is relatively constant since the additional microbes pro-
duced by metabolism of c  is small and M1 - M' .  The appro-
                        o                     o
priate dimensionless differential equation is then
              d(c/Co)     -dM(c/co}
                                                         (21)
whose solution is
              ds log C/CQ +  (c/c0-l) = KMdMt             (23)

therefore, for large d , i.e small concentration of substrate
                      5
relative to the Michaelis constant, the substrate decays ex-
ponentially as
              c  =  CQ exp  (-KM  M t)                    (24)
                                d
                                 s
Case IV  - Large initial microbial mass, small Michaelis con
           stant
On the other hand, for small d , i.e K  « c  « M1 , the
                              s       s     o      o
solution (Eg. 20)  is linear:
              C  =  Co (1-KMdMt)
until log (c/c )ds ~ (c/c  - 1) where the total solution
(Eq. 20) indicates an exponential "tail".
Intermediate solutions can be thought of as a "linear com-
bination" of two separate models: (a) Case I, an exponential
                            30

-------
growth of microorganisms, as given by Eq.  (13)  where nutri-
ents are not limiting, tK  = 0 in Eq. (10))   and (b) Case II,
                         o
logistic growth of microorganisms, as given by Eq.   (14)
where K  » c.  In terms of the substrate utilization and
       s
utilizing the dimensionless notation, these two limits can be
summarized as shown in Fig. 5.  As indicated, therein, for
various d  = M1 /c ,  the substrate utilization varies from a
linear decrease to exponential to an autocatalytic form.  Under
certain conditions, therefore, an appropriate exponential de-
crease (first order kinetics) may be assumed in mathematical
modeling under full recognition that other forms may prevail
depending on the "mix" of microbial population, nutrient con-
centration and Michaelis constant.  In the logistic growth
region, for 0.1 < d  < 1.0 Fig. 5 - shows that for complex river
or estuarine systems  it would be very difficult to detect de-
partures from exponential substrate decay.  Some data are
available on laboratory studies to evaluate the range of M1  .
However, actual field estimates of microbial mass,  and sub-
strate concentrations are not available.  As a consequence,
laboratory studies generally use "large" initial substrate
concentrations (d  and d  small) leading to specific non-
exponential substrate utilization.  Stratton and McCarty  (1967)
for ammonia oxidation obtained values of M  @ 20 C of 0.0033 mg
bacterial mass/1.  At a yield coefficient, a, of 0.28 mg cells/
mg substrate, this is equivalent to M1  of about 0.01 mg/1 NH^-N.
For this case however, K  = 2.6 mg/1 and c  =5.5 mg/1 or
                        o                 \J
d  - 0.47.  Stratton thus obtained ammonia curves similar to
those shown in Fig. 5 for low values of d., intermediate between
                 ^                       M
exponential and logistic microbial growth.  This then repre-
sents an initial low nitrifying population and "high" initial
substrate concentration.  At 20°C, for K  * 1-3.0 mg/1 Knowles
                                        5
et al  (1965) for normally low initial ammonia concentrations
in natural waters, (0.5 mg/1), d  for ammonia could be "large".
If the ratio of initial cell mass, M1  to initial concentrations
is from 0.1 - 1.0 one should expect to see appropriate exponenti-
                            31

-------
        1.0-
        0.8-
        0-6-
   c/c
Co
Co
K)
        0-4-
        0.2-
                                                                   Logistic     KS»G, large ds


                                                                   Exponential   KS-H>O, small ds
                      i.O
2.0
3.0
4.0
 i
5.0
                                                                     6.0
 i
7.0
8-0
     Fig. 5  Logistic and-Exponential Growth of  Microorganisms  and Accompanying  Nutrient Utilization

-------
al behavior in the substrate.   Therefore,  for "dilute" systems
where substrate concentrations are "low" and for "well seeded"
systems where initial nitrifying peculations are high relative
to initial substrate concentrations,  first order decay of sub-
strate may be approximately justified.   The actual application
of these conditions must of course be made with care but as a
first approximation, first order kinetics  reactions for the
nitrification phenomenon is a  meaningful step and can aid in
understanding the behavior of  observed nitrogen forms in
natural water systems.
         First Order Steady State Mathematical Model
                  of Streams and Estuaries
With an assumption of first order kinetic reactions and steady
state conditions, several levels of mathematical models can
be constructed each of which are useful in predictions of
long-term effects of nitrification on water quality.  The
three levels of modeling are:
         a) BOD equivalent models
         b) Sequential reaction models
         c) General feedback models

Each of the levels increases somewhat in the complexity of
the equations and provides greater detail in understanding the
phenomenon.  The major reason for even considering several
problem levels of first order kinetic models is that all prob-
lem contexts do not necessarily require complex models for
their solution.  Much can be learned from the first order
models although it is important to stress again that the Actual
kinetic mechanisms are more complex than those which are con-
sidered herein.  If it is obvious that non-linear kinetics
apply then one should use the models presented below only as
a gross approximation, if at all.  For some problem contexts,
                           33

-------
the first order kinetics assumption may be justified on the
basis of the previous analysis.  In any case, first order
kinetics simplifies the structure of the models (linearity
is presumed) and provides for rapid numerical solutions.

BOD equivalent models are particularly appropriate when the
major portion of the nitrogenous demand is present as ammonia.
Either a direct measurement of the ammonia and its oxvgen
equivalence or the second stage BOD measurement mav be used.
Either is then inputted as a sink of dissolved oxvgen in a
DO model.  This approach becomes more approximate as the con-
centration of organic nitrogen becomes more significant.  The
organic nitrogen breaks down by hydrolysis to vield, among
other products, ammonia.  The oxidation of ammonia and the
associate use of dissolved oxygen may therefore be delayed
in accordance with the hydrolytic rate of reaction.  This ef-
fect, if appreciable, is taken into account in the oxygen
model by a reduction in the reaction coefficient of the ni-
trogenous BOD or by empirically introducing a lag in the
initiation of nitrification.  In spite of its simplicity, this
model is quite adequate in a number of practical cases.

As indicated above, the nitrogenous oxygen demand, NBOD is
given approximately by:

              NBOD  =  4.57 (Orc'-N + NH-j-N)

This demand can be considered as an oxygen sink in the oair of
equations describing the distribution of NBOD and DO deficit.
For one dimensional steady state systems with constant coef-
ficients, these equations are:
                      d2L      dL
              °  =  E -TT- U 3/~ KNLN + VX)         (26)
                           34

-------
                      d2D      dD
              0  =  E	 - u -=—^ - K DM + K-.L           (27)
                       dx2     dx     a N    N

where E  =  dispersion coefficient (usually included only for
estuaries), u = river velocity or net downstream velocity for
estuaries, x  = distance, L  = nitrogenous BOD, D  = DO
deficit  (due to nitrogenous BOD), K  = rate of oxidation of
NBOD, K   =  reaeration coefficient, W  = waste discharges
of NBOD.
Eqs. (26)  and (27)  can be recognized as the same set of equa-
tions that are used to describe the DO deficit due to the
discharge of carbonaceous BOD  (CBOD).  For the constant coef-
ficient case, the solutions to Eqs. (26)  and (27)  are:
              LN  •  LNO
                     KKTLN
              DN  =  irMr  [exD  (JNX) - exp^aX)]      C28b)
                      a  N
where for rivers where dispersive effects are small
               'N      u

                      K
                       a
                      IT                                (29)
                      W
                       N
              LNO  ~  ~Q~
and for estuaries where the tidal dispersion effect  is  im-
                            35

-------
oortant
                                    u
                              /    4K E
              •    _  _u_(i±/i+   a
               a     2E    ~        u2                  (30)

                           W
                            N
              LNO =
                              u2
where for JN and j  the positive sign of the radical is as-
sociated with the negative x direction and the negative sign
is associated with the positive x direction.  The DO deficit
profile due to the nitrogenous waste discharge is thus given
by Eq. (28)    The DO deficit due to the carbonaceous BOD
must be added to this profile to determine the total DO
deficit due to waste discharges.

The disadvantage of this simple model is that usually some
judgment must be exercised in determining the spatial dis-
tribution of the reaction rate, K .  In some instances, the
field data may  indicate that for various reasons {organic
nitrogen hydrolysis, insufficient nitrifiers) an apparent
lag exists in the exertion of the NBOD.  The judgment incorpo-
ration of this effect becomes particularly significant when
projections are made of expected water quality under different
treatment schemes.

However, it is still somewhat surprising that even this simple
model is often not considered in DO balance studies.  This ap-
pears to be especially true in the numerous analyses conducted
to estimate the effects of different levels of waste treatment
                           36

-------
where the nitrogen discharges are largely unaffected by
standard high rate biological treatment.

Sequential reaction models answer some of the disadvantages
described above and especially apply when organic as well
as ammonia nitrogen are present as inputs.  Furthermore,
when it is desirable to trace the individual components of
the downstream nitrification process; organic, ammonia,
nitrite and nitrate, this approach is useful.  The lag ef-
fect is incorporated into the kinetic expression by the
first step of a sequence of the coupled reactions.  By con-
trast to the previous simplified approach, modeling with
consecutive reactions is slightly more complicated analyti-
cally, requires more data, time and experience, but provides
greater understanding and increased confidence in prediction,

The sequential reaction models follow each of the nitrogen
components individually.  The ammonia and nitrite outputs
are then converted to oxygen demands and used as inputs into
the DO deficit equation.  The general constant coefficient
equations are:
                   d2N..      dN,
                   ,2       dN.,
                   a N3     _ ± - v  N  + K  N  +W (x) (33)
           0  =  E - - - u dx    *33N3 + *23N2 + ™3{X)
                   dx2

                    2
                                _         K0,N0 + W. (x)
                            _             0,0    .
           0  =  E - -udx     444    343    4
                    dx2
where N, , N~, N, ,  and N. are organic, ammonia, nitrite and
nitrate nitrogen respectively, K. .  represents the first order
decay of substance i, K. .  is the forward reaction coefficient
and W. is the discharge of substance . .

                            37

-------
Equations  (31) through  (34) permit the determination of the in-
dividual nitroqen components and therefore represent a more
realistic description of the nitrification process than
Eqs.(26J. and  (27).  The forcing functions for the DO deficit
are 3.43 K«-N« and 1.14 K_^N_; the first representing ammonia
oxidation and the second nitrite oxidation.  The entire
scheme is sketched in a block diagram form in Figure 6.
The equation for DO deficit due to the oxidation of nitro-
gen forms is:
        d*D      dD
0  =  E —T - u d/ - KaV 3'43K23N2(X) + 1'14K34N3(X)   (35)
         dx

where N2(x) and N., (x) are given from solutions of Eqs.  (31)
through (34).

Particular utilization of Eqs. (31) - (34) again depends on
the water system either a river system where E = 0 or an
estuarine system where tidal dispersion effects embodied in
the dispersion coefficient may be significant.

The nature of the solutions to Eqs. (31) - (34) can be de-
termined by considering the river case where advective forces
dominate and dispersion can be considered zero.  Further, one
can assume that K..  = K. ., E k.  i.e that all material is
conserved in each system and none of the forms of nitrogen
are lost,  for example, to the bottom sediments, (K..  > K.  . ,).
The rate equations for the advective stream are then:
           dN,
                =  ~ K1N1 + Wl(x)                        (36a)
                =  K1N1 ~ K2N2 + W2(x)                   (36b)
                          38

-------
           SYSTEM
NHj-N
SOURCE
    ORGANIC
    Nrt SOURCE
                                          DISTRIBUTION
                                          AND DECAY
                                          OF NH3-N
                                                               SOURCE
OJ
vo
                               DISTRIBUTION
                               AND DECAY
                               OF N02-N
                                                                      3A
                                                                       D.O.
                                                                    REAERATION
                                                    oo OEFcrr
                                             RGURE6
                       BLOCK  DIAGRAM OF NITRIFICATION  &  DISSOLVED OXYGEN
                                            l/nUZATlON

-------
           dN3
                =  K2N2 - K3N3 + W3(x)                  (36c)
           dN.
                =  K3N3 - K4N4 + W4(x)                  (36d)
where t* = x/u, the time of travel.


Eqs.  (36) represent a series of first order coupled  equations
The solution of Eq. (36a)  is:
           Nl  =  N01

where N~, is the initial value of N,  @ x = 0 given by mass
balance incorporating W, (x) .


Substitution of this solution into Eq. (36b)  and integration
gives the solution for ammonia, N2 as:

       K1N01
N2  =  K -K  (exp(K1t*)-exp(-K2t*)) + N^exp (-K2t*)     (37)
        ^  _L

Sequential substitution and integration gives the concentra-
tions of nitrite and nitrate.  For example, for nitrite,

     K-.K-   exp(-K, t*)-exp(-K-t*)   exp (-K0t*) -exp (-K,t*)
N  -  -1- z   r _ ± _ J    _ _ ± _ £ _ I N
 3   K2~Ki        K3~K1                   K3~K2

         K
     + JT-^-   [exp(-K2t*)-exp(-K3t*)l NQ2              (38)
Each of the preceding inputs is reflected in this equation.
In general, the total amount of end product formed by virtue
                           40

-------
of the initial sources, C.., 	>C_  (for n reaction steps)
is:
                 n
           c   = V   c
            nO   £   U0i
and due to the intermediate steps is
                   n  K._,Cn .  .
           Cni  =  I  / V
            nl    i=l Ki  Ki-l
the sequential nature of these solutions is shown in Figure 7
where the last nitrogen form, the nitrate nitrogen is assumed
conservative, i.e  K. = 0.

For estuarine situations, the solutions essentially follow
the same form except that a series of unknown coefficients
are introduced by virtue of mass balances that are required.
The number of coefficients is equal to the number of equations
that can be written down so that explicit numerical determi-
nation of the coefficients is possible.  For example, the
solution to Eq.  (31)  for point source loads is
           N
,  = B,  exp(S,x)  + C, exp (V, x)  + YEN           (39)
where S1 = 2f(l +   l + 4 l^E/u2)
      Vl = 2¥ (1 - 7l + 4 KiiE/u2>

B, and C, = constants to be evaluated from consideration of
            boundary conditions
and YBN  =  particular integral of sources and sinks of N,.
                           41

-------
   H-
  iQ

   ^l
 i cn
  ro
H-C
n ID
w 3
rt- rt
  p-
O B)
 (D 50
 n ro
   DJ
 « o
 H- rt
 3 H-
 0) O
 rt 3
 H- CO
 O
 CO H-
   3


 CO H-
 rt rt
 n n
 o> P-
 p hh

   O
 CO Q>
^ rt
 W H-
 rt o
 ffi 3
                                                       00
                                                     Deficit
                                                    CN2-N3)
  DO
Deficit
CN3-N4)

-------
An equation similar to (39)can also be written down for N~
with the input from N, appearing in the particular integral
Thus,
           N2  =  B2exp(S2x)

                       I\-1 A
                     K22~K11
                             [B..exptS_x)  + C-expOV-x)]
                                  "
Eq. (40)  can be compared to Eq. (37).  This procedure is
repeated for each nitrogen form resulting in four solution
equations (see also Eq. (38))  in eight coefficients.  Appli-
cation of boundary conditions, mass balance and concentration
equality at each section provides the necessary equations to
evaluate these coefficients.  Details are given in Anon., 1969
and the application of this model to the Delaware Estuary is
discussed below.

Another approach that can be used to model sequential reactions
is to replace the spatial derivatives with finite approxima-
tions.  This results in a series of algebraic equations which
can be solved simultaneously to obtain the spatial distribu-
tion of each nitrogen form.  A generalization of this approach
is explored in detail in the next section.

In summary,  models of sequential reactions are readily structur-
ed utilizing the steady state and first order kinetic assump-
tions.  For the nitrification effect, these models "track" the
spatial distribution of each of a series of forms allowing di-
rect computation of the DO deficit due to nitrogen oxidation.
For streams, the solutions can be obtained directly while for
estuaries some type of computer solution is often required.

Feedback models incorporate feedback loops between any of the
respective systems.  In principle, this third model can include
                            43

-------
many of the features of the complete nitrogen cycle if one
accepts the assumptions of linearity and first order kinetics.
Typical examples of feedback svstems include denitrif ication
and algal utilization.  The denitrif ication phenomenon may
involve two feedback loops; NO, to NO- (with attendant evo-
lution of N-, gas) and NO- to NH, .   Both of these feedback
reactions occur under conditions of "low" dissolved oxygen.
Actually, a more complete model of the system would include
a non-linear interaction between DO and the feedback reaction
rates .

The utilization of ammonia and nitrate by ohytoplankton also
introduces feedback loops.  Organic nitrogen is formed as a
part of the complex living matter in algae cells which upon
death release the organic nitrogen in dissolved form thereby
completing the cycle.  This is an obvious over-simolif ication
of the actual mechanism which is dynamic and includes non-
linear growth limiting terms.  However, the problem addressed
here is to introduce the feedback aspect into the steady state
sequential models discussed in the previous section.

The general feedback model makes use of a finite difference
approximation to Eqs . (31) - (34)  and incorporates the first
order feedback reaction coefficients.   The differential equa-
tions which incorporate feedback are:
      d2N,     dN,
      d2N      dN
      TT " U dx~ + K12N1 - K22N2 + K32N3 + K42N4 + W2
       dX
      d2N      dN
0 - E — ± - U   i - K  N4 + K  N  + ... + K  N4 + W (x)
                           44

-------
Equations (41)  indicate all possible feedforward reactions
K. .  (j>i) and all possible feedback reactions (i>j) .   The
separate inclusion of K . .  allows for possible loss of material
from the system due for example to bottom deposition.  In
general, to prevent creation of nitrogen it is true that
If a finite difference approximation is made to these equa-
tions (or equivalently a mass balance is constructed around
a finite section) a series of n algebraic equations results
where n is the number of finite sections (Thomann, 1972) .
It should be noted that this approach is not restricted to
one-dimensional systems.

A finite difference approximation to the first equation of
Eg. 41     is given for spatial segment k as:
0  =
     - Vll,kNl,k + VkK21,KN2,k +••• \K41,kN4,k
       Wl,k            (k = 1,2...n)                    (42)
where Q, .  is the net flow from section k to j (positive out-
        •^                        th
ward); V,  is the volume of the k   segment; a, .  is a finite
difference weight = max (1/2, l-E'/Q) chosen for solution
stability; BR. = 1
efficient given by

                      E'kj  =  L^TTT
stability; Bv• = 1 - a, .;  Ev.  is a bulk tidal dispersion co
            K}        K]    K.~)
                            45

-------
where A,  . is the interfacial cross sectional  area  and  L,
and L. are segment lengths of segments k  and  j;  and  W.
     D                                        •        1, K
is the direct discharge of waste material,  N, .   The  notation
N, ,  indicates the spatial distribution in  all k segments  of
 i, K
the water body of the first nitrogen  form.  If in  Eg.  (37)
all terms involving the dependent variables N, are grouped
on the left hand side and the incut forcing function,  W, ,
                                                        1 , K
and othe:
obtains:
and other nitrogen forms, N. on the right hand side, one
akkNl,k + I akjNl,j = Wl,k + VkK21,kN2,k +  •••VkK41,kN4,k   (43)

where           akk = I  (Qkjakj + E'kj) + VkKllfk          (43a)

                akj = Qkj 6kj ' E'kj                       (43b)

A total of n equations similar to Eq.  (43)  can  be written
down for each of the spatial segments.  The svstem of  equa-
tions is then
a,,N, ,+a,-N.. ,+...a, N1   = W  T+V-.K,,  ,N-  ,+...V  K.,  , N.  ,
 11 1,1  12 1,2     In l,n    1,1  1 2.L,L  i,L     1  41,1  4,1
a-,N, ,+a^-N, _+...a0 N,   = Wn _+V_K,)1  ,N_  9+. . . V-K,, N.  „
 21 1,1  22 1,2     2n l,n    1,2  2  21,2  2,2      2  41  4,2
anlNl+an2Nl,2+-" annNl ,n = Wl ,n+VnK21 ,nN2 ,n+  ' ' ' VnK41 ,nN4 ,n
In matrix form, Eqs.(44) are

                                                  ,HN.)     (44)
where the subscripts refer to the nitrogen  forms  and  [VK..]
is a diagonal matrix of the product of volumes  and  reaction
                            46

-------
rates.  The dimensionality of the matrices and vectors  repre-
sents the spatial distribution of the nitrogen species.   Thus,
[A] is a n X n matrix and  (N,) are n x  1 vectors.

The entire procedure used to obtain Eq.  (44)  for  the  first
nitrogen form is now repeated for the second  through  fourth
forms.  The four matrix equations are then given  by
tA1] (Nj_)

[A2](N2)

[A3] (N3)

[A4] (N4)
(W2)
(W3)

(W4)
(N2)
[VK12]

[VK13]
(N3)

(N3)
(N2)

(N2)
                          (N4)

                          (N4)

                          (N3)
(45)
It should be recalled that in Eqs.  (45) , the  [A.] matrices
differ only in the reaction coefficient K.. on  the main  di-
agonal (see Eg. 43a).   There are 4  n algebraic  equations in
Eqs. (45) which can be solved by a  variety of block  decompo-
sition and relaxation techniques.   Some additional insight
can be obtained by continuing the matrix analysis.   The  set
of matrix equations (45)  can be written as
                 -  [VK31] -
or
                   -  [VK34]  [A4]
                   [A]  (N)  =   (W)
                                                            C46)
                                 (47)
where  [A] is a 4n X 4n matrix and  (N) and  (W)  are  4n  X 1
vectors.  Of course, in Eqs.(46)    not all  feedback or feed-
forward loops need be included.  For a given  sequence of
                           47

-------
loops, the solution for the four variables in all spatial
segments is formally given by

                    (N)  =  [A]'1 (w)                     <48>

The DO deficit due to these reactions is now readily computed,
If (N_) represents ammonia and  (N.,)  nitrite then a similar
differencing procedure yields for the DO deficit

              [B](D)N
                           3.43  [VK231 (N2)+ 1.14  [VK34] (N3) (49)
where [B] is an n X n matrix identical to the form of the
[A.] matrices except that the reaeration rate, K , appears
  1                                             3
on the main diagonal instead of K..  and  (D)  represents the
spatial distribution vector of DO deficit due to the nitrifi-
cation effect.  Note that once Eas  (45) have been programmed
Eg. (49)  is a soecial case of that set of eauations and does
not require a separate computer program.

A general multi-dimensional first order kinetic steady state
water quality problem with any number of variables and system
configuration can thus be structured by solving an MN system
of equations  (Eqs. (45))  where M is the number of variabled
forms (for nitrogen, M = 4).

For one-dimensional estuaries, the form of the matrices  [A.]
is tridiagonal which simplifies the computations.  If no feed-
back loops are incorporated, Eqs. (45)  reduce to the sequential
reaction models discussed in the previous section and  [A] be-
comes upper triangular.  In anv case, multidimensional systems
with interacting  (first order) reactions can be readily and
quickly analyzed by simply solving a set of linear algebraic
equations.  This of course by-passes the entire issue of how
one determines the K.. coefficients.  This is discussed in the
succeeding sections on applications.
                           48

-------
           Application of First-Order Models
                to the Delaware Estuary
This estuary extends for about 86 miles from Trenton, N.J. to
Delaware Bay.  The river flows past the metropolitan Phila-
delphia area and enters Delaware Bay about 50 miles from the
Atlantic Ocean.  The estuary receives large amounts of carbo-
naceous and nitrogenous wastes from municipalities and indus-
tries and is characteristic of a system where bacterial
nitrification leading to oxygen depletion appears to be pre-
dominant (Anon., 1969; O'Connor et al, 1969).  The municipal
and industrial nitrogenous components are indicated in
Table 2.

                        Table 2
     Estimated Municipal and Industrial Nitrogen
           Discharges to Delaware Estuary *
                     (pounds/day)
Organic
Nitrogen
28,500
28,500
9,000
Ammonia
Nitrogen
48,500
32,500
81,000
1,000
Nitrate
Nitrogen
2,000
30,500
32,500
16,000
Municipal
Industrial

Delaware River
@ Trenton (3,000 cfs)
Total                   37,500      82,000      48,500

*Anon (1969)

A total direct load of 109,500 Ibs of oxidizable nitrogen
per day is discharged from the municipal and industrial
waste sources.

In order to incorporate the effect of local drainage along
the length of the estuary, a runoff load equivalent to about
                           49

-------
110 pounds of organic nitrogen/day/mile was inputted together
with an equal amount of ammonia nitrogen.

A continuous solution model was used to represent the organic,
ammonia, nitrite and nitrate forms.  Only sequential bacterial
nitrification was included specifically in the model.  Algal
and denitrification effects were included qualitatively.  The
continuous solution model  (see Eqs. 39-40) was applied to
seven reaches of the estuary each of which included its repre-
sentative geometry  (area, depth) and sequential reaction coef-
ficient (Anon, 1969).  This type of model in contrast to the
finite difference approximation models results in continuous
solutions.  Thus, although the first segment in the Delaware
Estuary nitrogen model under discussion here is thirty miles
long, a continuous solution (for constant spatial parameters)
is obtained throughout this length.  The cross-sectional area
changes in the estuary were therefore approximated by the seven
segments.   Major point discharges were grouped according to
this segment breakdown.  This is a simplification introduced
to avoid numerous segment junctions at each discharge location.

The verification procedure consisted of comparing calculated
profiles from the four system model to a set of observed data
representative of steady-state summer conditions.  The reaction
rates obtained for each of the nitrogen components were then
used in the examination and verification of other  profiles
under different flow and temperature conditions.  A consistent
set of first-order reaction coefficients was therefore obtained
which provides a reasonable representation of the observed
phenomena.

Figure 8  shows the results of the first verification analysis
of data collected by the Delaware Water Pollution Commission
during July - August, 1964.  The reaction coefficients shown
                           50

-------
  2.0
   00
   40
               0     a
          BOCXCROUND ASSUME
          cue TO PLANKTON
          	1	i
                                                    K-O.IO
130
   30
f
 to
   2O
   ID
   OO
120
no
100
90
60
70
60
50
                                                 o-juur jo. e&a
                                                 O-AUG 10, 1964
                                                 • -AUG 31. I96«

                                                 DWPC OWA
                                                 ALL LWS
                                                 FLOW^JOOO CFS
                                                 TEMP-26-27'0
     00
     I
        120       IK)      OO      90      80
                          I    OST&NCE-MLES
        	nil	J	001   I 001 I  OS I	
                                                   60
                                                   50
     Fig.  8        Observed vs.  Computed  (solid  line)
                Nitrogen Profile   August 1964  (after  Anon.1969)
  20
   1.0
   00
   5O
                                    i    :  JLI     I
                 0.30
                          |   002    \.OX I .110.1   0.3    '   03    I 03  CXf
O>
   30
 to
O
  2O
   IO
     130
         110       100      90      80
                  .    DISTANCE- MLES ,
        \   	     I   o/ft   I n.ftl nn I   rvn
                                                   70
                                                                 60
                                                                     50
                                    51

-------
for each of the species do not constitute a unique set of rate
coefficients but represent a trial and error fit of the ob-
served data.  The coefficients are consistent however with
other verification analyses.  This is explored further below.

Several other points should be noted in this verification
analysis.  The direct discharge of organic nitrogen waste
loads does not alone account for the total of about 1/mg/l
that was observed.  It was therefore hypothesized that 0.75
mg/1 of organic ^nitrogen was due to the presence of plankton
and does not enter into subsequent nitrification reactions.
This is equivalent to about 75 yg/1 to 150 yg/1 which is
within the range of observed chlorophyll measurements for
the Delaware.

It can also be noted that the ammonia profile was verified
by employing a reduced reaction rate from mile 100 to mile
85.  This was justified because of the low dissolved oxygen
(< 2 mg/1 and minimums of about 0.7 mg/1) in this reach.

The nitrate analysis indicates two areas of nitrate decay.
The first reach from mile 100-90 is attributed to denitri-
fication.  Because feedback loops were not used in this in-
vestigation, it was not possible to recycle this nitrate re-
duction into the system.  Nitrate decay at the lower end of
the estuary was assumed to be due to increased phytoplankton
utilization of nitrate.

Experimental information on the influence of temperature on
nitrification is meager.  Work on the Thames estuary (Anon,
1964) indicated for the temperature dependence of the oxida-
tion of ammonia:

                           _      T-20
                       KT  ~  K206
                            52

-------
with 0 = 1.017.  Laboratory work however indicated 9 = 1.10.
Others  (Stratton and McCarty, 1967; Buswell et al, 1959)
have estimated 9 at about 1.08 while for nitrite oxidation
a value of 1.06 has been determined (Stratton and McCarty,
1967).  After review of available results, the temperature
dependence of the reaction rates was hypothesized as shown
in Figure  9.

Data were available for the period November 1967 when water
temperatures were 7  - 10 C and river flows at Trenton were
about 8900 cfs.  1964 loads were used.  Figure 10 summarizes
the results of the application of the model to these lower
temperature data.  A reaction rate of 0.025 (@ temperature =
7.5 C) was used for the conversion of organic nitrogen, .01
for ammonia, .05 for nitrite and 0.0 for nitrate.  At these
low rates, almost all forms behave as conservative variables.
It was not necessary in this case to make the assumption that
plankton had synthesized nitrogen into living tissue.  At the
low estuary temperatures, most plankton  activity would be
minimal.  The ammonia plot in Figure .10 distinctly shows the
effect of reduced nitrification.  This can be especially seen
by comparing the Nov. 1967 data  (Fig. 10) to the August 1964
data  (Fig. 8).   The reduced nitrification effect at low
temperatures is also evident in the nitrite and nitrate pro-
files.  This effect will also be reflected in reduced oxygen
utilization and has a significant impact on treatment programs
that may use nitrification.

The final step in the verification analyses was to use the pre-
ceding reaction coefficients temperature dependence and support-
ing assumption to "independently" verify other profiles.
Figure 11  is an example.  As indicated, agreement is good sup-
porting the assumption of a consistent set of coefficients.
The only change in the coefficients was to distribute the area
                            53

-------
.25
20
.15
.10
.05
                                                                ORGANIC-NH,
                                                                       = LO8 TO 2O°C
                                                        NH3 OXIDATION
                       10
15        20        25
WATER TEMPERATURE-°C
30        35
4O
                                      Figure  9
    Estimated Temperature Dependence of Nitrification Reaction Rates (after Anora.1969)

-------
130     120
                                                    /(--aces
                     *+
_np_
loo-
                                 90      80      70     60
50
       + 11/16/67
       911/20/67
       . II/27/6T
       0 ' B9OO eft
       T=7-IO"C
       oftec DATA
130     120     110      ICO     90      80

                      DISTANCE IN MILES
                                                 70      60
                                               50
                           Figure  10
Observed  vs. Computed  (solid line)  -  Nitrogen Profiles
                 November 1967^Cafter  Anon,  1969)
1
z
* 0
1
3
2
z
c^
z 1
0
1

K'OOS
• —
-a 	 B-t-i. 	 »-.• 	 ins 	 ?-|-? 	 1 	 T*~~? *i i it t

30 120 110 100 90 80 70 60 50
DISTANCE M MILES
• •
• .*:. • •
JXT"-^"'
- e • -f t— — -^^^
:• » . ; .? ?t T • j^&s.
•i i*ri 1 1 1 1 1 1 i i i i i i

30 120 110 100 90 80 70 60 50
DISTANCE IN MILES
                              55

-------
2
1 V. |
CJ g
0
13
REACT. COEF
( I/DAY )
3
z= 2
' K>o>
°3
REACT. COEF
( I/DAY )
1.0
^^P ^"^
2 ~ QO
l<
REACT COEF
( I/DAY )
3
2
z cr
il '
0
i;
REACT COEF
( I/DAY )

ASSUMED PLANKTON
^BACKGROUND
r\^7#^
= 9® \ .
niiiiiin . i .
r^Sr^-
•
l
«« »

•

.



0 120 . 110 100 90 80 70 60 50 MILES

_

= /
&I1ILUII A A ^&
0.10
A 9
/' '
o
1


"\ .
^



^^v-
\^



^



O 120 110 100 90 80 70 60 50 MILES
Qll
0.001
0.05
Oil

-


50 120 110 100 90 80 70 60 50 MILES
03


^T""^ 	 «^~^
fliiiiini i
0002


N>v
0105
/

/*
031

4


\.
*



30 120 110 100 90 80 70 60 50 MILES
0.0

005

0.0


• -7/12/67
0-56OO cfs
                      Figure 11
Observed vs. Computed  (solid  line)  - Nitrogen  Profiles
                July 1967  (after 7Vnon.,i969)

                          56

-------
of nitrification inhibition in accordance with low DO reaches.
The additional verifications were obtained with ease, once
the order of the coefficients had been established from the
August 1964 and November 1967 analyses.  The consistency of
the reaction coefficients is illustrated in Table 3.  While
the set of coefficients is certainly not unique, Table 3 shows
that the general order of magnitude of the coefficients is
consistent providing allowance is made for variable spatial
distributions of the coefficients due to low dissolved oxygen
values.

The four system model can be used to estimate the effects of
the nitrification reaction on the dissolved oxygen deficit,
as indicated in Figure 6.

The organic and ammonia waste sources are inputted into the
initial two systems.  The output from the ammonia system is
then multiplied by the reaction rate K23 , which is given by
3.43 K».. .  This then represents the sink of dissolved oxygen
due to ammonia oxidation and is inputted into the third system
which now represents the dissolved oxygen system with its ac-
companying reaeration rate.  The total of three systems are
therefore used and the output from the third system represents
the dissolved oxygen deficit due to NH-. - NO,, oxidation.  A
similar procedure is followed for the N02 - N0~ oxidation,
where K34A = 1.14K34.

Figure 11 shows a typical result for the July - August 1964
condition.  The reaction rates for nitrification shown in
Figure 8 were used in the computation together with a constant
spatial reaeration rate of 0.18/day at 20°C.  The two components
of the nitrification are shown.  A slight shift downstream of
the N02 - NO, component relative to the NH~ - N02 component
can be noted.  Also, the peak in the total deficit occurs some
                            57

-------
                   Table  3
        SUMMARY OF REACTION COEFFICIENTS
DETERMINED IN VERIFICATION AMALiSIS OF NITROGEN
              IN DELAWARE ESTUARY

              First. Order Peaction Coefficient @  20°C

Survey
July-Aug.
1964
Q = 3000 cfs
T = 26 C
Reach Org.to NH-
No. NH3(K]2) N02'(JP
1 0.1 0,1
2
to N09-
123J N°
1
0.001
3 0.001
4 0.05
5-6 1 0.11
7 0.1 0.11
June- July
1965
Q = 2000 cfs
T = 20 C


1 0.1 O.li
2
3
4
5
6
7
0.001
0.001
0.5
0.110
t '
k
8 01 0.110
July 1967
Q = 5600 cfs
T = 21 C



1 0.1 0.11
2
3
4
5
6 4
0.001
0.001
0.001
0.05
I 0.11
7 0.1 0.11

Nov. 1967
Q = 8900 cfs
T = 7-10 C



First Order
Reaction
1 0.025 0.01
2
3
4
5
6 >l




f »




'
7 0.025 0.01
3TK34'
0.3
0.002
0.002
0.110
0.3
0.3
0.3
0.002
0.002
0.110
0.3
i
0.3
0.3
0.002
0.002
0.002
0.105
0.3
0.3
N03
(K44}
0.0
0.05
0.0
0.0
0.0
0.5
0.0
0.05
0.0


0 05
0.05
0.0
0.05
0.05
0.05
0.0
0.0
0.05
Coefficient @ 7-10°C
0.05




\ •
0.05
0.0




, ,
0.0
                       58

-------
10 - 15 miles downstream of the major waste sources reflect-
ing the inhibited nitrification upstream due to low dissolved
oxygen.  It is also quite interesting to note a general back-
ground of 0.7 - 1.0 mg/1 dissolved oxygen deficit due to the
nitrogenous discharges from tributaries and run-off as well
as municipal and industrial sources.  The oeak value of
2.5 mg/1 dissolved oxygen deficit is somewhat lower than
previous estimates which placed the peak at about 3.0 mg/1
dissolved oxygen deficit, but at approximately the same lo-
cation.  Part of this difference is attributable to the use
of the four system model rather than an approximation through
"nitrogenous BOD" with associated reduction in nitrogenous
BOD decay rate or simple translation of the input of nitroge-
nous BOD.  In general, then, Figure 12 confirms previous work
which recognized a downstream shift of the satisfaction of
the nitrogenous oxygen demand.  The results indicate that this
phenomenon is due to low upstream dissolved oxygen values which
together with the discharge of potentially toxic materials have
an inhibitory effect on the nitrifying bacteria.  As the dis-
solved oxygen recovers, the nitrifying flora begins to develop
and bacterial nitrification proceeds at a relatively rapid
pace.  This is then accompanied by an increasing utilization
of oxygen.

Projected effects of a nitrogen removal program can be esti-
mated using this model.  Nitrogen removal from waste effluents
can be accomplished in several ways including biological ni-
trification, air stripping and ion exchange.  Each of the
methods accomplishes varying degrees of removal of nitroge-
nous components and at widely varying costs.

Figure 12 showed the estimated dissolved oxygen deficit due
to nitrification for 1964 summer conditions.  It should be
recalled that the peak dissolved oxygen deficit occurred at
about mile 75 because of assumed nitrification inhibition
                            59

-------
                       JULY-AUG. 1964
a\
o
         O»
         I-

         O
         8
                                                                                     TOTAL
                  130      120
110
100      90
80
70
60
50
                                                   DISTANCE IN MILES
                                                       FIGURE  12

                                 ESTIMATED DO DEFICIT DUE TO NITRIFICATION (After Anon.,1969)

-------
from mile 100 - 85.  This inhibition was ascribed to low dis-
solved oxygen conditions in that reach.   If, following imple-
mentation of the waste control program,  dissolved oxygen
conditions were at a higher level (say greater than 2 mg/1
everywhere), it is informative to explore the resulting effect
of nitrification on dissolved oxygen.  Two possibilities exist
under improved dissolved oxygen:
     a) ammonia oxidation will take place throughout the entire
        length of the estuary at approximately a rate of
        O.I/day.  This will result in a shift of the maximum
        dissolved oxygen deficit upstream.
     b) because of generally improved water quality, algal
        utilization of ammonia may now increase throughout
        the length of the estuary.  Since many algal species
        utilize ammonia preferentially,  the ammonia would be
        tied up in organic form, and not contribute to the
        deficit until some time later in the year.  The rate
        of this phenomenon is unknown.

Both effects will probably proceed simultaneously.  However,
in order to provide a somewhat conservative estimate, it can
be assumed that all the ammonia will be oxidized and will
contribute to the dissolved oxygen deficit.  Under this as-
sumption, model runs were made using ammonia oxidation rates
of 0.11/day everywhere, and the dissolved oxygen deficit was
computed.  The results are shown in Figure 13.

As shown, under favorable nitrification conditions, with am-
monia oxidation proceeding uniformly, the maximum dissolved
oxygen deficit shifts upstream to about mile 90.  There is a
decrease of about 0.2 mg/1 in the maximum, and a general
spreading over a larger area.  At mile 90, the deficit in-
creases from about 0.5 mg/1 to about 2.2 mg/1 under favorable
nitrification conditions.  On the other hand, at mile 75, the
                           61

-------
o
                         FAVORABLE NITRIFICATION CONDfTIONS
                                  O% NIT REMOVAL
             I I I I I I I I I
         130
120
110
                                                      INHIBITED NrmfKATtON
                                                          MI. too-so
                                                        0% NIT. f&HOAL
                       FAVORABLE NITRIFICATION CONDITIONS
                              5O% NIT REMOVAL
                       \	I	
100       90        80


   DISTANCE IN  MILES
70
60
50
                                                  Figure  13

                   DO Deficits  Under Different Nitrification Conditions  (after Anon.,1969)

-------
dissolved oxygen deficit decreases from about 2.2 mg/1 to
about 1.4 mg/1.  The difference between the downstream dis-
solved oxygen deficit and upstream increase is due to the
increasing cross-sectional area of the estuary as one pro-
ceeds in the downstream direction.  If a 50% removal of
oxidizable nitrogen were accomplished, the estimated dis-
xolved oxygen deficit profile is as shown in Figure 13.  A
general decrease is noted with a maximum dissolved oxygen
deficit of about 1.3 mg/1 in the area of mile 90.  In order
to provide an overall estimate of the effect of this shift
in the dissolved oxygen deficit profile and projected water
quality goals, a preliminary analysis was made of the esti-
mated dissolved oxygen profile under existing waste removal
requirements.

The estuary proper has been divided by the Delaware River
Basin Commission into four (4) zones with ultimate carbon-
aceous BOD removal requirements ranging from 86% to 89%,
based on raw 1964 waste loads.  These requirements will gen-
erally be met by various types of secondary treatment, in-
cluding for municipalities, biological waste reduction.  It
is difficult at this stage to estimate the extent of nitrogen
reduction to be expected from this program.  The particular
design practices will govern this factor.  However, for esti-
mating purposes, a value of about 20% oxidizable nitrogen
reduction appears reasonable.

A dissolved oxygen analysis was therefore made, using the
same sectional breakdown as used in the nitrification model.
The analysis indicated that under a 20% nitrogen removal of
1964 loads the dissolved oxygen goal of the DRBC will proba-
bly be met.  The critical region is in the vicinity of mile
100-90.  Under 50% nitrogen removal of 1964 loads, the DRBC
dissolved oxygen goals will be met with a greater degree of
                           63

-------
assurance; the steady-state DO profile is estimated to be
everywhere above 4.0 mg/1.

The 20% removal program is equivalent to a discharge load
about 95,000 Ibs/day of oxidizable nitrogen which would be
allowable while a 50% removal program is equivalent to a
discharge load of about 60,000 Ibs/day of oxidizable nitrogen.
Ultimately therefore waste removal programs must assign both
carbonaceous BOD loads and nitrogen loads both on a oounds/
day basis.  General unquantifiable factors that will tend to
further enhance the attainment of dissolved oxygen goals in-
clude algal utilization of ammonia with subsequent reductions
in the dissolved oxygen deficit due to nitrification,  ammonia
oxidation at a slower rate than that assumed and specific en-
couragement of nitrogen removal.  Factors that will tend to
mitigate against achievement of the objective include a faster
rate of ammonia oxidation which will intensify and shorten the
area of minimum dissolved oxygen or carbonaceous removal de-
signs that result in oxidizable nitrogen removal of less than
20%.
           Application of First-Order Model
               to the Potomac Estuary
The Potomac River discharges into Chesapeake Bay and extends
over 100 miles upstream to the head of the estuary at Little
Falls.  The major waste source is the effluent of the Washington,
D.C. secondary waste treatment plant.  Water quality problems
include low dissolved oxygen in the vicinity of the District
of Columbia discharge and generally high algal concentrations.

Hetling and O'Connell (1968) and Jaworski, et al (March, 1969;
May, 1969) summarized data pertaining to dissolved oxygen,
nutrients and chlorophyll.  These results were obtained from
                           64

-------
a series of sampling stations in the Potomac  where  data were
collected over a period of months during  1967,  1968 and again
during 1969.  Waste load information from the point waste dis-
charge locations as well as from land run-off has also been
obtained.

A steady state feedback model which considers organic ammonia
and nitrate nitrogen form was constructed.  The assumption
of first order kinetics prevailed throughout.  A feedback
loop is incorporated which represents ammonia and nitrate
nitrogen utilization by algae with the  subsequent nitrogen
release upon death recycled to the organic  nitrogen form.
The nitrification phenomenon was also inputted  to the dis-
solved oxygen deficit system.  A total  of five  systems was
therefore modeled.

The equations are:

    i  rt  f    dNl 1   1
0 = A" dl   EA dlT   ' X d  K.. for  all i.  The feedback loop
appears as K.,N., a source term in the  first equation which
                           65

-------
 utilizes  the  solution of  the  fourth  equation.

 A finite  difference approximation was  employed  to  solve
 E<3-  (50).   The  spatial segmentation  and svstem  parameters  of
 other  work  (Hetling and O'Connell, 1968:  Jaworski  et  al,
 March  1969  and  May 1969)  was  used.  A  total  of  23  spatial
 segments  was  applied to the reach of the  Potomac from Little
 Falls  downstream,  a distance  of  about  100 miles to the ap-
 proximate entrance to Chesapeake Bay.   In matrix form, the
 equations to  be solved are (see  also Eq.  (46).
    [VK12]

      0

      0
      0    -  [VK25]
 where  the  [A.]  are 23  x 23  triadiagonal matrices  incorporating
 net  advective  flow and dispersion with the  K..  appearing  on
 the  main diagonal.   Note that [A-]  has the  reaeration  rate K
                                 o                           a
 on the main  diagonal,  [VK..]  are 23 x 23  diagonal matrices
 (N.) and  (D) and  (W) are 23 x 1  vectors of  the  nitrogen forms,
 DO deficit and input nitrogen loads, respectively.   The matrix
 Equation  (48)  is  therefore  composed of a  system of  115 algebraic
 equations, the simultaneous solution of which provides the
 steady state distribution of  the four nitrogen  forms and  the
 DO deficit in  all 23 segments.

 Data were available for the period July-August  1968 for veri-
 fication purposes.   These data included Kjeldahl  nitrogen
(representing the  sum of organic  nitrogen  from waste discharges,
 ammonia nitrogen  and algal  nitrogen), nitrite and nitrate ni-
 trogen and chlorophyll a measurement.  Major input  nitrogen
0
0
[A31
- [VK34]
0
[VK41]
0
0
[A41
0
0
o
0
0
[A5]
cv
(N2)
(N3) =
(N4)
(D )
(W^

-------
lt.Carb.BOD
Ibs/day
5,900
132,000
11,700
20,800
Org. N.
Ibs/day
1,300
20,000
1,300
1,400
NH3-N
Ibs/day
2,100
20,000
2,300
1,100
loads are summarized in Table 4 below.

                         Table 4
        Estimated Significant Input Nitrogen Loads
                      Potomac Estuary
                     July-August, 1968
Arlington, Va.
Washington D.C.
Alexandria
Ft. Westgate

Figures 14 and 15 show the results of a verification analysis
of data collected during July-August, 1968.  The first order
reaction coefficients for the verification analyses shown in
these figures are given in Table 5.

                         Table 5
            First Order Reaction Coefficients
                      Potomac Estuary
                     July-August, 1968
                       Temp. = 28°C
                                                 Reaction
Reaction Step                  Symbol          Coef.  (1/dav)
Decay of Organic - NH            K                 0.2
Organic Nit. -> NH -Nit.          K12               0.1
Decay of NH3 ~ Nit.              K_2              0.30
NH., Nit. ->• N03 - Nit.            K-.,              0.28
NH3 Nit. •+ Algal Nit.            K^              0.02
Decay of N03 - Nit.              K33              0.10
NO3 Nit. -> Algal Nit.            K34              0.10
Decay of Algal Nit.              K.,              0.12
Algal Nit. -> Organic Nit.        K.,              0.12
                            67

-------
CTl

OO
                                                                     COMPUTED WITH
                                                                     ALGAL FEEDBACK
                                                                     COMPUTED WITH
                                                                     NO FEED BACK
                        5          10

             I   I   1  2  I  3|4| 5   6
  30        35 MILES FROM CHAIN BR.
13 I  14  I    15    I
                                               SEGMENT NO.
                                                 Figure 14

                          Verification of Kjeldahl Nitrogen  for Potomac Estuary

-------
       o>
                                            JULY-AUG. 1968
                                            Q=2fXX) CFS


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                                                SEGMENT NO,

                                                 Figure 15

                       Verification of Nitrite  and Nitrate Nitrogen (upper plot)
                         and Chlorophyll "a"  (lower plot} for  Potomac Estuary

-------
These coefficients represent the end result of manv solutions
of Eq. (51) which tested the effects of various interactions
and levels of coefficients.  It can be noted that orqanic ni-
trogen is  "settled out" of the svstem because of the difference
between the decay coefficient of orqanic nitrogen and the
conversion of organic nitrogen to ammonia nitrogen.  This was
justified on the basis of bottom sampling which indicated
significant deposits in the vicinitv of the Washington D.C.
outfall.   Ammonia nitrogen followed two oaths: a)  utilization
in the algal nitrogen loop (K = 0.02 per day) and b) oxidation
to nitrate (K = 0.28 per day).  This split allowed a proper
spatial profile to be maintained.  Nitrate was recycled to
algal nitrogen all of which was allowed to decav to organic
nitrogen.

Figures 14 and 15 compare the observed data of the various
nitrogen forms to computed values generated by the model with
and without the feedback of ammonia and nitrate nitrogen to
organic nitrogen.  For Figure 14 onlv Kjeldahl nitrogen ob-
served data were available.  The effects of the feedback loop
are to increase all profiles in a non-linear spatial manner.
The relative downstream shift of the various nitrogen forms
is interesting and reflects the sequential nature of these
types of reactions.  Steady state analyses such as shown in
Figures 14 and 15 can provide a basis for estimating the ef-
fects of environmental changes on nitrogen distribution in
addition to the effects of nitrification on the oxvgen regimes.

This latter effect is shown in Figure 16 where the dissolved
oxygen deficit due to nitrification is given with and without
algal feedback of nitrogen.  Therefore, if the nitrogen is
completely stored in the algae, the dissolved oxygen deficit
is lower at greater distances downstream than if the algal
nitrogen is released and available for further nitrification.
                            70

-------
(J  C
^  .2
Q  "(5
   .y   2
II
 a
                                                                      without  feedback
                                                                      with algal feedback
                             10
15
                         20        25        30         35
                   Miles Below Chain Bridge
                          Figure 16
Computed Dissolved Oxygen Deficit Due to Nitrification
       in Potomac Estuary, July - August,  1968
                                                                                          40

-------
The peak value of 3 mq/1 DO deficit is significant from a
water quality management viewpoint and indicates the need
for nitrification of the principal waste discharges.

The nitrogen algal cycle in the Potomac estuary is obviously
more complex than given by this model.  Non-linear kinetics,
algal growth dynamics and environmental influences of
temperature and light all affect the observed data.  As a
planning tool however the simplified model using first order
kinetics provides a rapid means for estimating order of
magnitude responses and points the direction for more complex
modeling efforts.
                            72

-------
                      SECTION V
                  A DYNAMIC MODEL OF
      PHYTOPLANKTON POPULATIONS IN NATURAL WATERS

The quality of natural waters can be markedly influenced by
the growth and distribution of phytoplankton.  Utilizing
radiant energy, these microscopic plants assimilate inor-
ganic chemicals and convert them to cell material which., in
turn, is comsumed by the various animal species in the next
tropic level.  The phytoplankton, therefore, are the base
of the food chain in natural waters, and their existence is
essential to all aquatic life.

The quality of a body of water can be adversely affected if
the population of phytoplankton becomes so large as to inter-
fere with either water use or the higher forms of aquatic
life.  In particular, high concentrations of algal biomass
cause large diurnal variations in dissolved oxygen which
can be fatal to fish life.  Also, the growths can be nui-
sances in themselves, especially when they decay and either
settle to the bottom or accumulate in windrows on the shore-
line.  Phytoplankton can cause taste and odor problems in
water supplies and, in addition, contribute to filter clog-
ging in the water treatment plant.

The development of large populations of phytoplankton and, in
some cases, larger aquatic plants can be accelerated by the
addition of nutrients which result from man's activities or
natural processes.  The resulting fertilization provides more
than ample inorganic nutrients, with the resulting development
of excessive phytoplankton.  This sequence of events is com-
monly referred to as eutrophication.

Generally, the management of water systems subjected to ac-
celerated eutrophication because of waste discharges has been
                            73

-------
largely subjective.  Extensive programs of nutrient removal
have been called for, with little or no quantitative pre-
diction of the effects of such treatment programs.  A
quantitative methodology is required to estimate the effect
of proposed treatment programs that are planned to restore
water quality or to predict the effects of expected future
nutrient discharges.   This methodology should include a
model of the phytoplankton population which approximates
the behavior of the phytoplankton in the water body of inter-
est and, therefore, can be used to test the effects of the
various control procedures available.  In this way, rational
planning and water quality management can be instituted with
at least some degree of confidence that the planned results
actually will be achieved.

This chapter presents a phytoplankton population model in
natural waters, constructed on the basis of the principle
of conservation of mass.  This is an elementary physical law
which is satisfied by macroscopic natural systems.  The use
of this principle is dictated primarily by the lack of any
more specific physical laws which can be applied to these
biological systems.  An alternate conservation law, that of
conservation of energy, can also be used.  However, the de-
tails of how mass is transferred from species to species
are better understood than the corresponding energy trans-
formations*  The mass interactions are related, among other
factors, to the kinetics of the populations and it is this
that the bulk of this chapter is devoted to exploring.

                 Review of Previous Models

The initial attempts to model the dynamics of a phytoplankton
population were based on a version of the law of conservation
                            74

-------
 of mass  in which  the hydrodynamic  transport of mass  is  assumed
 to be  insignificant.   Let  P(t) be  the  concentration  of  phyto-
 plankton mass  at  time  t  in a  suitably  chosen  region  of  water.
 The  principle  of  conservation of mass  can be  expressed  as  a
 differential equation
                         dt
where  S  is  the  net  source  or  sink of phytoplankton mass within
the  region.   If hydrodynamic  transport  is  not  included, then
the  rate at which P increased or decreases depends only on  the
internal sources and sinks of phytoplankton  in the region of
interest.

The  form of the internal sources and sinks of  phytoplankton
is dictated by  the  mechanisms which are assumed to govern the
growth and  death of phytoplankton.  Fleming  (1939),  as de-
scribed by Riley (1963) , postulated that spring diatom  flower-
ing  in the  English  Channel is described by the equation,

                            =  [a -  (b  + ct)]P
where  P  is  the phytoplankton  concentration,  a  is  a constant
growth rate,  and  (b  +  ct)  is  a  death  rate  resulting  from  the
grazing  of  zooplankton.   The  zooplankton population, which
is  increasing owing  to its  grazing, results  in an increasing
death  rate  which  is  approximated by the linear increase of
the death rate as a  function  of time.

The less empirical model  has  been  proposed by  Riley  (1963)
based  on the  equation
                         ar  =  tph  -  R -  G]p
                            75

-------
where P,  is the photosynthetic growth rate, R is the endoge-
nous respiration rate of the phytoplankton, and G is the
death rate owing to zooplankton grazing. A major improvement
in Riley's equation is the attempt to relate the growth rate,
the respiration rate, and the grazing to more fundamental
environmental variables such as incident solar radiation,
temperature, extinction coefficient, and observed nutrient
and zooplankton concentration.  As a consequence, the coef-
ficients of the equations are time-variable since the
environmental parameters vary throughout the year.  This
precludes an analytical solution to the equation, and nu-
merical integration methods must be used.  Three separate
applications (Riley, 1946, 1947, 1949) of these equations
to the near-shore ocean environment have been made, and the
resulting agreement with observed data is quite encouraging.

A complex set of equations, proposed by Riley, Stommel and
Bumpus (1949) first introduced the spatial variation of the
phytoplankton v/ith respect to depth into the conservation
of mass equation.  In addition, a conservation of mass equa-
tion for a nutrient (phosphate) was also introduced, as well
as simplified equations for the herbivorous and carnivorous
zooplankton concentrations.  The phytoplankton and nutrient
equations were applied to 20 volume elements which extended
from the surface to well below the euphotic zone.  In order
to simplify the calculations, a temporal steady-state was
assumed to exist in each volume element.  Thus, the equations
apply to those periods of the year during which the dependent
variables are not changing significantly in time.  Such con-
ditions usually prevail during the summer months.  The results
of these calculations were compared with observed data, and
again the results were encouraging.

Steele (1956) found that the steady-state assumption did not
                           76

-------
apply to the seasonal variation of the phytoplankton popula-
tion.  Instead, he used two volume segments to represent the
upper and lower water levels and kept the time derivatives
in the equations.  Thus, both temporal and spatial variations
were considered.  In addition, the differential equations for
phytoplankton and zooplankton concentration were coupled so
that the interactions of the populations could be studied, as
well as the nutrient-phytoplankton dependence.  The coeffici-
ents of the equations were not functions of time, however, so
that the effects of time-varying solar radiation intensity and
temperature were not included.  The equations were numerically
integrated and the results compared with the observed distri-
bution.  Steele  (1964) applied similar equations to the verti-
cal distribution of chlorophyll in the Gulf of Mexico.

The models proposed by Riley et al and Steele are basically
similar.  Each consider the primary dependent variables to be
the phytoplankton, zooplankton, and nutrient concentration.
A conservation of mass equation is written for each species,
and the spatial variation is incorporated by considering finite
volume elements which interact because of vertical eddy dif-
fusion and downward advective transport of the phytoplankton.
Their equations differ in some details  (for example, the growth
coefficients that were used and the assumptions of steady
state) but the principle is the same.  In addition, these
equations were applied by the authors to actual marine situ-
ations and their solutions compared with observed data.  This
is a crucial part of any investigation discussion wherein the
assumptions that are made and the approximations that are used
are difficult to justify a priori.

The models of both Riley and Steele have been reviewed in
greater detail by Riley (.1963) in a discussion of their appli-
cability and possible future development.  The difficulties
                           77

-------
encountered in formulating simple theoretical models of phyto-
plankton-zooplankton population models were discussed by
Steele  (1965).

Other models have been proposed which follow the outlines of
the equations already discussed.  Equations with parameters
that vary as a function of temperature, sunlight, and nutrient
concentration have been presented by Davidson and Clymer  (1966)
and simulated by Cole  (1967).  A set of equations which model
the population of phytoplankton, zooplankton, and a species of
fish in a large lake have been presented by Parker  (1968) .
The application of the techniques of phytoplankton modeling
to the problem of eutrophication in rivers and estuaries has
been proposed by Chen  (1970).  The interrelations between the
nitrogen cycle and the phytoplankton population in the Potomac
Estuary has been investigated using a feed-forward-feed-back
model of the dependent variables, which interact linearly fol-
lowing first order kinetics (Thomann, 1970).

The formulations and equations presented in the subsequent
sections are modifications and extensions of previously pre-
sented equations which incorporate some additional physiologi-
cal information on the behavior of phytoplankton and zoo-
plankton populations.  In contrast to the majority of the
applications of phytoplankton models which have been made
previously, the equations presented in the subsequent sections
are applied to a relatively shallow reach of the San Joaquin
River and the estuary further downstream.  The motivation for
this application is an investigation of the possibility of ex-
cessive phytoplankton growths as environmental conditions and
nutrient loadings are changed in this area.  Thus, the primary
thrust of this investigation is to produce an engineering tool
which can be used in the solution of engineering problems to
protect the water quality of the region of interest.
                           78

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            Phytoplankton System Interactions

The major obstacle to a rigorous quantitative theory of
phytoplankton population dynamics is the enormous complexity
of the biological and physical phenomena which influence the
population.  It is necessary, therefore, to idealize and
simplify the conceptual model so that the result is a manage-
able set of dependent systems or variables and their inter-
relations.  The model considered in the following sections is
formulated on the basis of three primary dependent systems:
the phytoplankton population, whose behavior is the object of
concern; the herbivorous zooplankton population, which are
the predators of the phytoplankton, utilizing the available
phytoplankton as a food supply; and the nutrient system, which
represents the nutrients, primarily inorganic substances, that
are required by the phytoplankton during growth.  These three
systems are affected not only by their interactions, but also
by external environmental variables.  The three principal
variables considered in this analysis are temperature, which
influences all biological and chemical reactions, dispersion
and advective flow, which are the primary mass transport
mechanisms in a natural body of water, and solar radiation,
the energy source for the photosynthetic growth of the phyto-
plankton .

In addition to these external variables, the effect of man's
activities on the system is felt predominately in the nutrient
system.  Sources of the necessary nutrients may be the result
of, for example, inputs of wastewater from municipal and in-
dustrial discharges or agricultural runoff.  The man-made
waste loads are in most cases the primary control variables
which are available to affect changes in the phytoplankton
and zooplankton systems.  A schematic representation of these
systems and their interrelations is presented in Figure 17.
                            79

-------
         FLOW
      TEMPERATURE
         SOLAR
        RADIATION
                                        EOOPLANKTON
                                        PREY
      GRAZING
                                       PHYTOPLANKTON
                                    NUTRIENT
                                    LIMITATION
       NUTRIENT
         USE
NUTRIENTS
                                          MAN MADE
                                            INPUTS
Fig.  17  Interactions of Environmental Variables and the
        Phytoplankton, Zooplankton and Nutrient Systems

                              80

-------
In addition to the conceptual model which isolates the major
interacting systems, a further idealization is required which
sets the lower and upper limits of the temporal and spatial
scales being considered.  Within the context of the problem
of eutrophication and its control, the seasonal distribution
of the phytoplankton is of major importance, so that the lower
limit of the temporal scale is on the order of days.  The
spatial scale is set by the hydrodynamics of the water body
being considered.  For example, in a tidal estuary, the spatial
scale is on the order of miles whereas in a small lake it is
likely a good deal smaller.  The upper limits for the temporal
and spatial extent of the model are dictated primarily by
practical considerations such as the length of time for which
adequate information is available and the size of the computer
being used for the calculations.

These simplifying assumptions are made primarily on the basis
of an intuitive assessment of the important features of the
systems being oonsidered and the experience gained by pre-
vious attempts to address these and related problems in natural
bodies of water.  The basic principle to be applied to this
conceptual model, which can then be translated into mathe-
matical terms, is that of conservation of mass.

                  Conservation of Mass

The principle of conservation of mass is the basis upon which
the mathematical development is structured.  Alternate formu-
lations, such as those based on the conservation of energy,
have been proposed.  However, conservation of mass has proved
a useful starting point for many models of the natural en-
vironment.

The principle of conservation of mass simply states that the
                             81

-------
mass of the substances being considered within an arbitrarily
selected volume must be accounted for by either mass transport
into and out of the volume or as mass produced or removed
within the volume.  The transport of mass in a natural water
system arises primarily from two phenomena: dispersion, which
is caused by tidal action, density differences, turbulent dif-
fusion, wind action, etc.; and advection owing to a unidirec-
tional flow - for example, the fresh water flow in a river or
estuary or the prevailing currents in a bay or a near-shore
environment.  The distinction between the two phenomena is
that, over the time scale of interest, dispersive mass transport
mixes adjacent volumes of water so that a portion of the water
in adjacent volume elements is interchanged, and the mass trans-
port is proportional to the difference in concentrations of
mass in adjacent volumes.  Advective transport, however, is
transport in the direction of the advective flow only.  In ad-
dition to the mass transport phenomena, mass in the volume can
increase resulting from sources within the volume.  These sources
represent the rate of addition or removal of mass per unit time
per unit volume by chemical and biological processes.

A mathematical expression of conservation of mass which includes
the terms to describe the mass transport phenomena and the source
term is a partial differential equation of the following form
                 =  V • EVP - V • QP + S                  (52)

where P(x,y,z,t) is the concentration of the substance of in-
terest - e.g., phytoplankton biomass - as a function of position
and time; E is the diagonal matrix of dispersion coefficients;
Q is the advective flow rate vector; S  is the vector whose
terms are the rate of mass addition by the sources and sinks;
and V is the gradient operator.  This partial differential
                            82

-------
equation is too general to be solved analytically, and the
numerical techniques are used in its solution.

An effective approximation to Equation (52) is obtained by
segmenting the water body of interest into n volume elements
of volume V. and representing the derivatives in Equation (52)
by differences.  Let V be the n x n diagonal matrix of volumes
V.; A, the n x n matrix of dispersive and advective transport
terms; S , the n vector of source terms S  ., averaged over the
volume V.; and P, the n vector of concentrations P., which are
the concentrations in the volumes.  Then the finite difference
equations can be expressed as a vector differential equation
                      o
                     VP  =  AP + VS                       (53)

where the dot denotes a time derivative.   The details of the
application of this version of the dispersion advection equa-
tion to natural bodies of water has been presented by Thomann
(1963) and reviewed by O'Connor et al (1966).
The main interest in this report is centered on the source
terms S .  for the particular application of these equations
to the phytoplankton population in natural water bodies.  It
is convenient to express the source term of phytoplankton, S  . ,
as a difference between the growth rate, G  .,  of phytoplankton
and their death rate, D .,  in the volume V..  That is
                       PD                  I)

                     SP:  =  
-------
and death rates as functions of environmental parameters and
dependent variables is an important element in a successful
phytoplankton population model.

               Phytoplankton Growth Rate

The growth rate of a population of phytoplankton in a natural
environment is a complicated function of the species of phyto-
plankton present and their differing reactions to solar radi-
ation, temperature, and the balance between nutrient availability
and phytoplankton requirements.  The complex and often con-
flicting data pertinent to this problem have been reviewed
recently by Hutchinson (1967), Strickland (1965), Lund (1965)
and Raymont (1963).  The available information is not suffici-
ently detailed to specify the growth kinetics for individual
phytoplankton species in natural environments.  Hence, in order
to accomplish the task of constructing a growth rate function,
a simplified approach is followed.  The problem of different
species and their associated nutrient and environmental re-
quirements is not addressed.  Instead, the population is
characterized as a whole by a measurement of the biomass of
phytoplankton present.  Typical quantities used are the chloro-
phyll concentration of the population, the number of organisms
per unit volume, or the dry weight of the phytoplankton per
unit volume (Vollenweider, 1969).   With a choice of biomass
units established, the growth rate expresses the rate of pro-
duction of biomass as a function of the important environmental
variables.  The environmental variables to be considered below
are light, temperature, and the various nutrients which are
necessary for phytoplankton growth.

Consider a population of phytoplankton, either a natural as-
sociation or a single species culture, and assume that the
optimum or saturating light intensity for maximum growth rate
                            84

-------
of biomass is present and illuminates all the cells, and
further that all the necessary nutrients are present in suf-
ficient quantity so that no nutrient is in short supply.  For
this condition, the growth rate that is observed is called
the maximum or saturated growth rate, K1.  Measurements of K'
(base e) as a function of temperature are shown in Figure 18
and listed in Table 6.  The experimental conditions under
which these data were collected appear to meet the require-
ments of optimum light intensity and sufficient nutrient supply,
The data presented are selected from larger groups of reported
values, and they represent the maximum of these reported growth
rates.  The presumption is that these large values reflect the
maximum growth rates achievable.  From an ecological point of
view, it is necessary to consider the species most able to
compete, and, in terms of growth rate, it is the species with
the largest growth rate which will predominate.  A straight-
line fit to this data appears to be a crude but reasonable
approximation of the data relating saturated growth rate K1
to temperature, T

                      K1  =  ^T                          (55)

where K, has values in the range 0.10 ± 0.025 day"  °C~ .  This
coefficient indicates an approximate doubling of the saturated
growth rate for a temperature change from 10° to 20°C,  in ac-
cordance with the generally reported temperature-dependence of
biological growth rates.  The optimum temperature for algal
growth appears to be in the range between 20° and 25°C, al-
through thermophilic strains are known to exist  (Fogg,  1965).
At higher temperatures, there is usually a suppression  of the
saturated growth rate, and the straight-line approximation is
no longer valid.  It should also be noted that the  scatter in
the data in Figure 18 is sufficiently large so that the linear
                            85

-------
   4.0
<
Q
UJ


o:

i
o
o:
   3.0
    2.0
Q
UJ

I-   1.0
Z)

<
V)
                          10
15
20
25
30
                           TEMPERATURE  C
   Fig. 18  Phytoplankton Saturated Growth Rate  (Base e)

            as a Function of Temperature
                               86

-------
Table 6

Maximum Growth Rates


As a Function of Temperature
Reference
Tamiya et
al, 1964
Yentsch
1966
Spencer
1954
Riley
1949b
Myers
1964
H
Sorokin
1958
it
"
"
Sorokin
1962
Organism
Chlorella ellipsoidea
(green alga)
Nannochloris atomus
(marine flagellate)
Nitzschia closterium
(marine diatom)
Natural association
Chlorella pyrenoidosa
Scenedesmus quadricauda
Chlorella pyrenoidosa
Chlorella vulgaris
Scenedesmus obliquus
Chlamydomonas reinhardti
Chlorella pyrenoidosa
(synchronized culture)
(high-temperature strain)
Temper-
ature
25
15
20
10
27
19
15.5
10
4
2.6
25
25
25
25
25
25
10
15
20
25
Saturated
Growth Rate,
K'
-1
3.14
1.2
2.16
1.54
1.75
1.55
1.19
0.67
0.63
0.51
1.96
2.02
2.. 15
1.8
1.52
2.64
0.2
1.1
2.4
3.9
    87

-------
 dependence  on temperature  and  also  the  magnitude  of  K1  can
 vary  considerably  in  particular  situations.

 In the  natural environment,  the  light intensity to which  the
 phytoplankton are  exposed  is not uniformly  at  the optimum
 value but it  varies as  a function of depth  because of  the
 natural turbidity  present  and  as a  function of time  over  the
 day.  Thus, the phytoplankton  in the lower  layers are  exposed
 to intensities below  the optimum and those  at  the surface may
 be exposed  to intensities  above  the optimum so that  their
 growth  rate would  be  inhibited.   Figure 19b,c,d from Ryther
 (1956)  are  plots of the photosynthesis  rate normalized by the
 photosynthesis rate at  the optimum  or saturating  light in-
 tensity vs  the light intensity,  I, incident on the  popula-
(tions.   Figure 19a is a plot of  function
F(I) = Y~ exp
        s
                                                         (56)
 for  I   =  2000  ft-candles,  proposed  by  Steele  (1965)  to  des-
      o
 cribe  the light-dependence of  the growth  rate  of  phytoplankton.

 The  similarity between  this function and  data  from Ryther  is
 sufficient to  warrant the  use  of this  expression  to express
 the  influence  of nonoptimum light intensity on the growth  rate
 of phytoplankton.   Other workers have  suggested different  forms
 for  this  relationship (Shelef  et al, 1970; Vollenweider, 1965).
                             83

-------
             1.0 —
    (a)  P/Ps  0.5
                            I	I
                                                        10
CO

CO
LJ
I
CO
o
f-
o
X
Q.
u.
o
LU

<
CC
Q
UJ
N

-I
    CW
             1.0
(C)  P/ps  0.5
                                Chlorophyta
                                                    10
                                          Diatoms
           0
10
cc
o
•z.
(d)
         1.0 —
             0.5
                                                  Flagellates
                                                         10
                   LIGHT  INTENSITY  (FOOT CANDLES x I03)


       Pig.  19   Normalized Rate of Photosynthesis vs.  Incident
                 Light Intensity: (a) Theoretical Curve after
                 Steele (1965)  (b,c,d) Data after Ryther  (1956)
                                     89

-------
These variations approximately follow the shape of Equation
(56) for low light intensities but differ for the region of
high light intensities, usually by not decreasing after some
optimum intensity is reached.  In particular, Tamiya et al
(1964) have investigated the growth rate of Chlorella ellip-
soidea to various light and temperature regimes.  The saturated
growth rates as a function of temperature are included in
Figure 18.  The influence of varying light intensity fits the
function
                       F(I) = I   K'/a                  (57)

where K1 is the saturated growth rate and (a = 0.45 day
kilolux  ).  However, since K1 is a function of temperature,
the saturating light intensity for Equation (57) is also a
function of temperature.  Similar data obtained by Sorokin
et al (1962) using a high-temperature strain of Chlorella
pyrenoidosa support the temperature-dependence of the satu-
rating light intensity for chlorella.  Therefore, in using
Equation (56), a temperature-dependent light saturation in-
tensity may be warranted.

At this point in the analysis, the effect of the natural
environment on the light available to the phytoplankton must
be included.  Equation  (56) expresses the reduction in the
growth rate caused by nonoptimum light intensity.  This ex-
pression can therefore be used to calculate the reduction
in growth rate to be expected at any intensity.  However,
this is too detailed a description for conservation of mass
equations which deal with homogeneous volume elements, V.,
and the growth rate within these elements.  What is required
is averages of the growth rate over the volume elements.

In order to calculate the light intensity which is present
in the volume, V., the light penetration at the depth of
                           90

-------
water where  v- is located must be evaluated.  The rate at
which light is attenuated with respect to depth is given
by the extinction coefficient, k .   That is, at a depth z,
the intensity at that depth, I(z),  is related to the surface
intensity, I , by the formula

                   I(z)  = IQ exp (-kez)                 (58)

where z = 0 is the water surface and z is positive downward.
Thus, the reduction of the saturated growth rate at any depth
z resulting from the nonoptimum light intensity present is
given by Equation (58),  substituted into Equation  (56).
              -k z
           I e  e
F[I(z)]  =  -2—	 exp
                                   -I e
                                       -k z
                                         e
+ 1
                                                       (59)
To apply this equation to the finite volume elements, within
which it is assumed that the phytoplankton concentration is
uniform, it is necessary to average this reduction factor
throughout the depth of the volume element V..  Let H .  and
H, . be the depths of the surface and bottom, respectively,
of the volume element V..  For example, if the volume ele-
ment V. extends from the water surface to the bottom of the
water body, then H  . = 0 and H, .  is  the water depth at the
location of V..  For the sake of simplicity, it is assumed
that this is the case.  If H .  ^ 0, a straight-forward gen-
eralization of the following average is required.
In addition to an average over depth, it is also expedient
to average the phytoplankton growth rate over a time interval,
Since the time scale within which this analysis is addressed
is the week-to-week change in the population over a year, a
daily average is appropriate.  For simplicity, it is assumed
that the incident solar radiation as a function of time cv"
a day is given by the function
                          91

-------
               I0(t)  =  0
                                   0 < t < f
                                   f < t < 1
                                                        (60)
where f is the daylight fraction of the day  (i.e.  the  photo
period) and I  is the average im
             cl
tensity during the photo period.
period) and I  is the average incident solar radiation in-
             cl
Let r. be the reduction in growth rate attributed  to nonoptimum
light conditions in volume V., averaged over depth and  time.
Then r.  is given by

•v"
rj
J

1
H.
J J
>
H.
D

o
ff
1
T



O
-k .z

j exp
s
-k .z
-I e e:i
a
j
s
                                             + 1
                                                  dt dz   (61)
where T = 1 day, the time-averaging interval, H,. =  H.  =  the
depth of segment V., and k  . is the extinction  coefficient
in V. .  The result is
                    r .  =
                             ,.    -a, .    -a  .
                             1       li     01
                                 e    J -e    J
                           , - 7=
                           k  .H .
                                                          (62
where
                               -k  .H.
                                                          (63)
                    a
                            |a_
                            "s
The integral given by Equation (-61) is a  form of  an  integral
used by Steeman Nielson  (1952, Tailing  (1957),  and Ryther  (1952)
and Yentsch  (1957), as described by Vollenweider  (1958), and,
in particular, Steele  (1965), for  the purpose of  relating  an
instantaneous rate  (e.g., growth,  photosynthesis, etc  )  to
an average day rate and an average depth  rate.
                            92

-------
The reduction factor r. is a function of the extinction co-
efficient K  . of the volume V.  However, the extinction
coefficient is a function of the phytoplankton concentration
present if their concentration is large.  Thus, an important
feedback mechanism exists which can have a marked effect on
the growth rate of phytoplankton.  As the concentration of
phytoplankton in a volume element increases, the extinction
coefficient, particularly at the green wavelengths,  starts
to increase.  This mechanism is called self-shading.  The
most straightforward approach to including this effect into
the growth rate expression is to specify the extinction co-
efficient as a function of the phytoplankton concentration

              kej  =  k'ej + h(Pj}                    (64)
where k'  .is the extinction coefficient attributable to
other causes and k .  includes the phytoplankton1s contribu-
tion.  The function h (P .) has been investigated by Riley  (1956),
who found that it can be approximated by

              h(P.) = 0.0088 P. + 0.054 P.2/3         (65)

where P. has the units yg/liter chlorophyll  concentration
       3         -1                        a
and h has units m  .   A more recent investigation (Small,1968)
shows that this relationship applies to coastal waters of
Oregon for a range in chlorophyll  concentration of from 0
to 5.0 mg Chi /m3.
             a

A theoretical basis for this relationship is the Beer-Lambert
law, which expresses the extinction coefficient in terms of
the concentration of light-absorbing material.  For dense
algal cultures, this law has been experimentally verified
(Oswald, et al, 1953).   A similar relationship based on this
law has been proposed by Chen  (1970) from the data of Azad and
                           93

-------
Borchardt (1969)
                   h(P_.)  =  0.17 p                      (66)

for h in m   and P., the phytoplankton concentration is
mg/liter of dry weight.  This expression gives values com-
parable with Equation  (65) for a reasonable conversion factor
for the units involved.
To summarize the analysis to this point, the saturated growth
rate K1 has been estimated from available data and its tem-
perature dependence established.  The reduction to be ex-
pected from nonoptimum light intensities has been quantified
and used to calculate the reduction in growth rate, r., to
be expected in each volume element V. as a function of the
extinction coefficient and the depth of the segment.  The
mechanism of self-shading has been included by specifying
the chlorophyll dependence of the extinction coefficient. It
remains to evaluate the effect of nutrients on the growth
rate.

Th-s effects of various nutrient concentrations on the .jrowta
of phytoplankton has been investigated and the results are
quite complex.  As a first approximation to the effect of
nutrient concentration on the growth rate, it is assumed
that the phytoplankton population in question follow Monod
growth kinetics with respect to the important nutrients.
That is, at an adequate level of substrate concentration,
the growth rate proceeds at the saturated rate for the tem-
perature and light conditions present.  However, at low
substrate concentration, the growth rate becomes linearly
proportional to substrate concentration.  Thus, for a nu-
trient with concentration N. in the j   segment, the factor
                            94

-------
by which the saturated growth rate is in the j   segment re-
duced is: N./(K  + N.).  The constant, K ,  which is called
           j   m    ]                   m
the Michaelis or half saturation constant,  is the nutrient
concentration at which the growth rate is half the saturated
growth rate.  There exists an increasing body of experimental
evidence to support the use of this functional form for the
dependence of the growth rate on the concentration of either
phosphate (Dugdale, 1967), nitrate, or ammonia (Eppley, 1969)
if only one of these nutrients is in short supply.  An example
of this behavior, using the data from Ketchum (1939)  is shown
in Figure 20a for the nitrate uptake rate as a function of ni-
trate concentration and in Figure 20b for the phosphate uptake
as a function of phosphate concentration.  These results are
from batch experiments.  Similar results from chemostat experi-
ments, which seem to be more suitable but more lengthy for this
type of analysis, have also been obtained.   Table 7 is a list-
ing of measured and estimated Michaelis constants for ammonia,
nitrate, and phosphate.  The estimates are obtained by taking
one-third the reported saturation concentration of the nutrients
These measurements and estimates indicate that the Michaelis
constant for phosphorus is approximately 10 yg P/liter and for
inorganic nitrogen forms in the range from 1.0 to 100 yg N/
liter, depending on the species and its previous history.

The data on the effects of the concentration of other inorganic
nutrients on the growth rate is less complete.  Since algae
use carbon dioxide as their carbon source during photosynthesis,
this is clearly a nutrient which can reduce the growth rate at
low concentrations (Kuentzel, 1969).  Reported saturation con-
centration for Chlorella is < 0.1% atm  (Myers, 1964).

The silicate concentration is a factor in the growth rate of
diatoms for which it is an essential requirement.  The satu-
                            95

-------
UJ  1


8-8
CO
CD
O
       •20
       10
                   50
100        150        200
 v

   N04 (wg-N/l)
                                                           250
                                         300
Q
LU
CD
CH
O
CO
CD
O 03^
Q-  O
        10
               A =13.5

               K  = 30.4
                   10
20
          30
                                                 40
50
60
                               P04
  Fig.
       20   Nutrient Absorption Rate as a Function of Nutrient
           Concentration: Comparison of Michaelis Menten
           Theoretical Curve with Data from  Ketchum (1939)
                                  96

-------
Reference
                        Table 7
                  Michaelis Constants

              for Nitrogen and Phosphorus
Thomas
 1968

L.Tahoe
 A.C.1969

Riley
 1965
Gerloff
 1957

Eppley et
 al 1969
MacIsaac
et al 1969
Chaetoceros gracilis
  (marine diatom)

Scenedesmus gracile
Natural association
Microcystis aeruginosa
  (blue-green)

Phaeodactylum tricornutum
Oceanic species
Oceanic species

Neritic diatoms
Neritic diatoms

Neritic or littoral
  Flagellates

Natural association
  Oligotropic

Natural association
  Eutrophic
                                       25
                              total N 150
                              total P  10
>44
                              PO
                                       10'
        10
NO.,
T
NH,
3
NO,
"KTTT
NO,
NH^
3
NO,
NH,
3
NO.,
T
NH,
1.4-7.0

1.4-5.6

6.3-28
7.0-120
8.4-130
7.0-77

2.8
1.4-8.4

14

18
    Estimated.
                           97

-------
rated growth rate concentration is in the range of 50-100
yg Si/liter (Stickland 1965).

There are a large number of trace inorganic elements which
have been implicated in the growth processes of algae, among
which are iron [for which a Michaelis constant of 5 yg/liter
for reactive iron has been reported by Lake Tahoe Area Council
(Anon, May 1969)],  manganese, calcium, magnesium, and po-
tassium (Lund, 1965).  However, the significance of these
elements in the growth of phytoplankton in natural waters
is still unclear.  Trace organic nutrients have also been
shown to be necessary for most species of algae: 80% of the
strains studied require vitamin B12, 53% require thiamine,
and 10% require biotin (Droop, 1962).  Presumably, these nu-
trients are available in sufficient quantities in natural
waters so that their concentration does not appreciably af-
fect the growth rate.

In the preceding discussion of nutrient influences on the
growth rate, it is tacitly assumed that only one nutrient is
in short supply and all the other nutrients are plentiful.
This is sometimes the case in a natural body of water.  How-
ever, it is also possible that more than one nutrient is in
short supply.   The most straightforward approach to formulat-
ing the growth rate reduction caused by a shortage of more
than one nutrient is to multiply the saturated growth rate
by the reduction factor for each nutrient.  This approach has
also been suggested by Chen (1970).  As an example, the data
from Ketchum  (1939) for the rate of phosphate absorption as a
function of both phosphate and nitrate concentration can be
satisfactorily fit with a product of two Michaelis-Menton
expressions.  The resulting fit, obtained by a multiple non-
linear regression analysis, is shown in Figure 21.  The
Michaelis constants that result are 28.4 yg N03-N/liter and
                            98

-------
LJ
O
    12
o:
-p
 i   1C
CD
O

 X
O
CO

<

O
CO
LLl
     o k
      0
                                            0 ; '. i7 - ^
 Fig. 21  Measured  Phosphate  Absorption Rate,  after Ketchum (1939),
          vs. Phosphate Absorption Rate Estimated Using V^N,/ (Kml+
           (K  2+N2)  where  Ni  and N2 are the Nitrate and Phosphate
          Concentrations,  Respectively
                                      99

-------
30.3 yg PO.-P/liter, with a saturated absorption rate of
         _g
15.1 x 10   ug PO.-P/cell-hr.  This approximation to the
growth rate behavior as a function of more than one nutri-
ent must be regarded as only a first approximation, however,
since the complex interaction reported between the nutrients
is neglected.
The result of the above investigation is the following growth
rate expression.  For the case of one limiting nutrient, N,
with Michaelis constant K , the growth expression for the
             .th
                         m
rate in the j   segment is
          PD
„ „   / 2.718f ,
K, T..  /	r;	 (e
 1J
                                 -01,
                        k  .H.
- e
          V
                                                          (67)
in which Equations  (55) and  (62) have been combined.  This
is the functional form that is used subsequently in the ap-
plications of these equations to natural phytoplankton
populations.
       Comparison with Other Growth Rate Expressions

The most extensive investigation of the relationship between
the growth rate of natural phytoplankton populations and the
significant environment variables, within the context of
phytoplankton models, is that of Riley et al  (1949).  The
expression which results from their work is
   log
        K>Io - Gp
  = 22.884 + log v  - log I  -
           6573.8
             T1
(68)
                            100

-------
where G  is the growth rate (day  ), K1 = 7.6, i  = average
daily incident solar radiation  (ly/min), T' - temperature in
°K, and v  is the nutrient reduction factor for phosphate
concentration, N , defined as
v
v
= 1.0
= (0.55)
                         N
N  > 0.55 mg-at/nf
 P         '      -
N  < 0.55 mg-at/nf
                                                          (69)
where G  is the growth rate  (day  ), K1 = 7.6, I  = average
daily incident solar radiation  (ly/min), T1 = temperature
in °K, and v  is the nutrient reduction factor for phosphate
concentration, N ,  defined as
            v  = 1.0
             P
                       -1,
            v  =  (0.55)  N
             P            P
                 N  > 0.55 mg-at/m"
                 N  < 0.55 mg-at/nf
In order to compare this expression with that in the previous
section, let the nutrient reduction factor be replaced by a
Michaelis-Menton expression
                                  N
                              K   + N
                               mp    p
                                                          (70)
where K   is the Michaelis constant for phosphate.  To be
       mp
comparable with Equation  (67) , K   should equal approximately
            3             3      m^
0.20 mg-at/m   (6.2 mg P/m ).  Using Equation  (70) for v  , the
growth rate expression becomes
G— K ! T
P o
Y, (T)
Y (T)+I
.1 u

N
P
K
m
To ] + N
Y,(T)+I0J P
                                                          (71)
                            101

-------
where

               loo  v  (T) = 22.9CT) - 336.4
               J.og10Y, u;       T + 273                 (   '

and T is temperature in degrees centigrade.  To compare this
expression with that proposed in the previous section, con-
sider first the nutrient saturated growth rate as a function
of solar radiation intensity and temperature.  The equations
are compared in Figure 22a as a function of total daily solar
radiation for three temperatures.  The dotted line is Equa-
tion (71) , and the solid line is the product of Equations  (55)
and (56) .  The rate expressions are comparable, although two
differences are apparent.  In Riley's expression the effect of
temperature is less pronounced in the 15° to 25 °C range, and
the effect of higher daily average solar radiation intensities
is opposite (i.e., tends to increase the rate) to that of
Equation (56)  based on Steele's expression.  The growth rate
equations averaged over depth are compared in Figure 22.  The
depth average rate resulting from Riley's expression is
               Gp  =    fc'H   In  [1 + Io/r, (T)]           (73)
which is compared with Equation  (67) .  The differences are now
more pronounced.  In particular, the higher growth rates at
lower light intensities given by Equation  (67) are reflected
in the increased depth average growth rate.  This is not unex-
pected since the majority of the population is exposed to
lower light levels at the lower depths.  Also, the dependence
on temperature is quite different, being linear in the case
of Equation (67) but practically  suppressed in Equation (73) .

An interesting feature of Riley's Equation (71) is the multi-
plication of the Michaelis constant by an expression which
                           102

-------
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 Fig.  22   Comparison of Phytoplankton Growth Rates  as  a
          Function of  Incident Solar Radiation Intensity
          and  Temperature
                            103

-------
depends on temperature and light intensity.  The effect is to
lower the Michaelis constant at high temperatures and at high
light intensity levels, which seems to be a reasonable behavior
for a phytoplankton population.

More elementary growth rate formulations have been proposed
which do not span the range of conditions attempted in Equa-
tions (67) and (71).  In particular, a common proposal is to
make the growth rate linearly proportional to the various
environmental variables.  For example, Davidson and Clymer
(1966) assumed that the growth rate is proportional to phos-
phate concentration and photoperiod    and a temperature factor
                       2
given by exp [-(T - 18) /18].  This temperature factor is
quite different from all others proposed and greatly magnifies
the effect of temperature on the growth rate.  For example, at
T = 18°C, the factor equals 1.0, whereas at T = 9°C, the factor
drops to 0.01,  a 100-fold decrease, compared with approximately
a 2-fold decrease predicted by Equations (67) and (.71).  This
exaggerated effect seems to be unrealistic.

A complete investigation of the environmental influences on
the growth rate is still to be made.  In particular, it has
been emphasized that there is an interaction between nitrogen
and phosphorus limitations as well as other effects which in-
fluence the phytoplankton growth rate.  Also, these effects
are different for differing species.  The species-dependent
effects are important in the problem of the seasonal succession
of phytoplankton species.

For any particular application, it is advisable to investigate
the growth rate of the already-existing population, as the re-
sulting expression may differ significantly from the general
over-all behavior as described by Equations  (67) and  (71).
Also, in dealing with natural associations of species" of
                            104

-------
plankton, the various constants which result from such an
investigation can be considered to be averages over the
population, and so they represent in some average way the
population behavior as a whole.

                 Phytoplankton Death Rate

Numerous mechanisms have been proposed which contribute to
the death rate of phytoplankton: endogenous respiration rate,
grazing by herbivorous zooplankton, a sinking rate, and
parasitization (Fogg, 1965).   The first three mechanisms
have been included in previous models for phytoplankton dy-
namics, and they have been shown to be of general importance.

The endogenous respiration rate of phvtoplankton is the rate
at which the phytoplankton oxidize their organic carbon to
carbon dioxide per unit weight of phytoplankton organic car-
bon.  Respiration is the reverse of the photosynthesis process
and as such contributes to the death rate of the phytoplankton
population.  If the respiration rate of the population as a
whole is greater than the photosynthesis or growth rate, there
is a net loss phytoplankton carbon, and the population biomass
is reduced in size.  The respiration rate as a function of
temperature has been investigated, and some measurements are
presented in Figure 23 and Table 8.  A straight line seems to
give an adequate fit of the data; that is, Respiration Rate =
K2T.  For the respiration rate in days   and T in °C, the
value of K_ is in the range 0.005 ± 0.001.  The lack of any
more precise data precludes exploring the respiration rate's
dependence on other environmental variables.  However, an im-
portant interaction has been suggested by Lune (1965) .  During
nutrient-depleted conditions, "many algae pass into morphologi-
cal or physiological resting stages under such unfavorable
conditions.  Resting stages are absent in Asterionella formosa,
                            105

-------
    .15

-------
and this is why a mass death occurs in the nutrient-depleted
epilimnion after the vernal maximum."  In terms of the respira-
tion rate, the resting stage corresponds to a lowering of the
                           Table 8
       Endogenous Respiration Rates of Phytoplankton
                        (Riley,  1949)
                                               Endogenous
                                               Respiration
                               Temperature
                                  " °C
       Organism
Nitzschia closterium

Nitzschia closterium
Coscinodiscus excentricus

Natural Association
                                    6
                                   35
                                   20
                                   16
                                   16
                                    2
                                   18
                                    2,
                                      0
                                               Rate,  Day
(Basee)
                                   17.9
0.035
0.170
0.08
0.075
0.11
0.03
0.12
0.024±0.012
0.110±0.007
respiration rate as the nutrient concentrations decrease. Thus,
a Michaelis-Menton expression for the respiration rate nutrient
dependence may also be required, and this dependence should be
investigated experimentally.  This mechanism is quite signifi-
cant from a water quality point of view since the deaths of
algae after a bloom is of primary concern in protecting the
quality of natural bodies of water.  The resulting mass of dead
algal cells becomes a sink of dissolved oxygen which can danger-
ously lower the available oxygen for fish and other aquatic
animals.

The interaction between the phytoplankton population and the
                              107

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next trophic level, the herbivorous zooplankton, is a complex
process for which only a first approximation can be given.  A
basic mechanism by which zooplankters feed is by filtering the
surrounding water and clearing it of whatever phytoplankton
and detritus is present.  Thus, the presence of zooplankton
prey on phytoplankton as a food source.  The filtering or graz-
ing rate of some species of zooplankton have been measured and
are presented in Table 9.   The grazing rate is sometimes reported
as a volume of water filtered per unit time per individual.  In
order to be applicable to a natural zooplankton population con-
sisting of differing species, these grazing rates are converted
to a filtering rate per unit biomass of zooplankton and denoted
by C .   A convenient biomass unit for zooplankton concentration
is their dry weight.  As can be seen from Table 9, the resulting
values  of C  vary considerably.  This variation is not unex-
pected since the measurement of grazing rates of zooplankters
is a difficult procedure and subject to large variations in the
estimates.

Variations of the filtering rate with temperature change have
been reported (Anraku, 1963).  Examples of this variation are
presented in Figure 24 for four species of genus Daphnia, a
small crustacean(Burns, 1969); two species of Acartia  (Conover,
1956);  and two species of Centropages  (Anraku, 1963), both
copepods.  The copepods show a marked grazing rate temperature-
dependence while the grazing rates of the Daphnia do not vary
as markedly.  The filtering rate also varies as a function of
the size of the phytoplankton cell being ingested (Mullin,
1963),  the concentration of phytoplankton (McMahon, 1965), and
the amount of particulate matter present (Burns, 1967).  Se-
lective grazing of certain phytoplankton species has also been
reported (Burns, 1969).  The complexity of this aspect of phyto-
plankton mortality is such that the use of one grazing coeffi-
cient to represent the process must be viewed as a first
                             108

-------
                         Table 9
              Grazing Rates of Zooplankton
                                                 Grazing Rate
                                                    Liter/
                                        Reported    mg Dry
Reference        Organism             Grazing Rate  Wt.-Day
                 ROTIFER
     unson   Brae
 1967
Hutchinson   Brachionus calyciflorus   0.05-0.12a   0.6-1.5
                 COPEPOD
Riley 1949b  Calanus sp.               67-208b      0.67-2.0
Adams,       Calanus finmarchicus      27a          0.05
 Steele 1966
Mullin,      Rhincalamus nasutus       98-670a      0.3-2.2
 Brooke 1967
Anraku       Centropages hamatus                    0.67-1.6
 Omori,1963
                 CLADOCERA
Wright,1958  Daphnia sp.                            1.06
Burns,1969   Daphnia sp                             0.2-1.6
Ryther,1954  Daphnia magna             81a          0.74
McMahon,     Daphnia magna             57-82a       0.2-0.3
 1956
                 NATURAL ASSOC.
Riley,1949b  Georges Bank              80-110b      0.8-1.10

 Ml/animal-day
 Ml/mg wet wt-day
approximation.  However, since this mathematical expression
does represent a physiological mechanism that has been investi-
gated and for which reported values of C  are available, this
approximation is a realistic first step.  Also, it is difficult
to see, aside from refinements as to temperature and phyto
                            109

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Fig.  24  Grazing Rates of Zooplankton vs.  Temperature
                     110

-------
plankton concentration dependence, what further improvements
could be made in the formulation so long as the phytoplankton
and zooplankton population are represented by a biomass
measurement which ignores the species present and their in-
dividual characteristics.  For simplicity in this investiga-
tion, the grazing rate is assumed to be a constant.   The
death rate of phytoplankton resulting from the grazing of
zooplankton is given by the expression C Z . , where Z.  is the
                                        9 3          1     th
concentration of herbivorous zooplankton biomass in the j
volume element.

For models of the phytoplankton populations in coastal oceanic
waters and in lakes, the sinking rate of phytoplankton cells
is an important contribution to the mortality of the popula-
tion.  The cells have a net downward velocity, and they eventu-
ally sink out of the euphotic zone to the bottom of the water
body.  This mechanism has been investigated and included in
phytoplankton population models (Riley, 1949; Chen,  1970).
However, for the application of these equations to a relatively
shallow vertically well mixed river or estuary, the degree of
vertical turbulence is sufficient to eliminate the effect of
sinking on the vertical distribution of phytoplankton.

Therefore, considering only the phytoplankton respiration and
the predation of zooplankton, the death rate of phytoplankton
is given by the equation

                    D  .  =  K-T + C Z.                    (74)
                     PD      2     g }
and for a zooplankton biomass concentration Z.,  the mortality
rate can be calculated from this equation.

This completes the specification of the growth and death rates
of the phytoplankton population in terms of the  physical vari-
                          111

-------
ables:  light and temperature, the nutrient concentrations,
and the zooplankton present.  With these variables known as
a function of time, it is possible to calculate the phyto-
plankton population resulting throughout the year.  However,
the zooplankton population and the nutrient concentrations
are not known a priori since they depend on the phytoplankton
population which develops.  That is, these systems are inter-
dependent and cannot be analyzed separately.  It is therefore
necessary to characterize both the zooplankton population and
the nutrients in mathematical terms in order to predict the
phytoplankton population which would develop in a given set
of circumstances.
                The Zooplankton System

As indicated in the previous section, the zooplankton popu-
lation exerts a considerable influence on the phytoplankton
death rate by its feeding on the phytoplankton.  In some in-
stances, it has been suggested that this grazing is the
primary factor in the reduction of the concentration of phyto-
plankton after the spring bloom.  In the earlier attempts to
model the phytoplankton system, the measured concentration of
zooplankton biomass was used to evaluate the phytoplankton
death rate resulting from grazing.  However, it is clear that
the same arguments used to develop the equation for the con-
servation of phytoplankton biomass can be applied directly to
the zooplankton system.  In particular, the source of zoo-
plankton biomass S .  within a volume element V. can be given
                  ZD                           D
as the difference between a zooplankton growth rate G . and
a zooplankton death rate D  ..  Thus, the equation for the
                          ZD
source of zooplankton biomass, which is analogous to Equa-
tion  (54) is

                    S  .  =   (G  . - D  .) Z.                 (75)
                     ZD       ZD    ZD   D
                            112

-------
where G .  and D .  have units day   and Z.  is the concentration
       z}       zj                        D
of zooplankton carbon in the volume element V..  The magnitude
of the growth rate in comparison with the  death rate determines
whether the net rate of zooplankton biomass production in V.
is positive, indicating net growth rate, or negative, indicat-
ing a net death rate.

As in the case of the phytoplankton population, the growth and
death rates, and in fact the whole life cycle of individual
zooplankters, are complicated affairs with many individual pecu-
liarities.  The surveys by Hutchinson (1967) and Raymont (1963)
give detailed accounts of their complex biology.  It is, how-
ever, beyond the scope of this report to summarize all the
differences and species-dependent attributes of the many zoo-
plankton species.   The point of view adopted is macroscopic,
with the population characterized in units of biomass.  The
resulting growth and death rates can be thought of as averages
over the many species present.  These simplifications are made
in the interest of providing a model which is simple enough to
be manageable and yet representative of the over-all behavior
of the populations.

The grazing mechanism of the zooplankton provides the basis for
the growth rate of the herbivorous zooplankton, G ..  For a
                                                 ZD
filtering rate C , the quantity of phytoplankton biomass in-
gested is C P., where P. is the phytoplankton biomass concen-
tration in V..  To convert this rate to a  zooplankton growth
rate, a parameter which relates the phytoplankton biomass'in-
gested to zooplankton biomass produced, a  utilization effici-
ency, a   , is required.  However, this utilization efficiency
or yield coefficient is not a constant.  At high phytoplankton
concentrations, the zooplankton do not metabolize all the
phytoplankton that they graze, but rather  they excrete a portion
of the phytoplankton in undigested or semidigested form  (Riley,
                             113

-------
1947).  Thus, utilization efficiency is a function of the
phytoplankton concentration.  A convenient choice for this
functional relationship is a  K  /(K   + P.)  so that the
                      ^     zp mp   mp    -)
growth rate for the zooplankton population is

The resulting growth rate has the same form as that postulated
for the nutrient-phytoplankton relationship, namely, a Michaelis-
Menton expression with respect to phytoplankton bi.omass.  In
fact, the argument which is used to justify its use in Equation
(67) can be repeated in this context.  The difference is that
in this case the substrate or nutrient is phytoplankton biomass,
and the microbes are the zooplankton.  The Michaelis constant
K   is the phytoplankton biomass concentration at which the
 mp
growth rate G .  is one-half the maximum possible growth rate
a  C K  .  The fact that at high phytoplankton concentrations
 zp g mp                      ^    ••
the zooplankton growth rate saturates was incorporated by Riley
(1947) in the first model proposed for a zooplankton population.

The assimilation efficiency of the zooplankton at low phyto-
plankton concentrations, a  , which is the ratio of phyto-
                          zp
plankton organic carbon utilized to zooplankton organic carbon
produced has been estimated by Conover (1966) for a mixed zoo-
plankton population.  The results of 26 experiments gave an
average of 63% and a standard deviation of 20%.  Other reported
values are within this range.  Experimental values for K  ,
which in effect set the maximum growth rate of zooplankton,
are not available and would probably be highly species-
dependent.  Perhaps a more effective way of estimating K
is first to estimate the maximum growth rate at saturating
phytoplankton concentrations, a  C K  , and then calculate
                               zp g mp
K   .  Growth rates for copepods through their life cycle
                             114

-------
average 0.18 day   (Mullin, 1967).  For the Georges Bank
population, Riley used 0.08 day   (Riley, 1947)  for the maxi-
mum zooplankton growth rate.  For a value of the grazing
coefficient C  of 0.5 liter/mg-dry wt-day and an assimilation
coefficient of 65%, the Michaelis constant for zooplankton
assimilation, K  ,  ranges between 0.25 and 0.55 mg-dry wt/
liter of phytoplankton biomass.  However, these values should
only be taken as an indication of the order of magnitude of
K  .  It is probable that its value can vary substantiallv in
 mp
different situations.
The fact that the growth rate reaches a maximum or saturates
is an important feature of the formulation of the zooplankton
growth rate since in some cases the phytoplankton concentra-
tion during part of the year exceeds that which the zooplankton
can effectively metabolize.  If the zooplankton growth rate
is not limited in some way and, instead, is assumed simply to
be proportional to the phytoplankton concentration, as propos-
ed in simpler models, the resulting zooplankton growth rate
during phytoplankton blooms can be very much larger than is
physiologically possible for zooplankton, an unrealistic re-
sult.  The saturating growth rate also has implications in the
mathematical properties of the resulting equations.  In parti-
cular, the behavior differs significantly from the classical
Volterra Preditor-Prey equations  (Lotka, 1956).  This is dis-
cussed further in a subsequent section.

The growth of the zooplankton population as a whole, of which
the herbivorous zooplankton are a part, is complicated by the
fact that some zooplankters are carnivorous or omnivorous.
Thus, the nutrient for the total population should include
not only phytoplankton but also organic detritus as a food
source since this is also available to the grazing zooplankton.
However, for cases where the phytoplankton are abundant and
                            115

-------
the growth rate saturates for the significant growing periods,
the simplification introduced by ignoring the detritus is
probably acceptable.

The death rate of herbivorous zooplankton is thought to be
caused primarily by the same mechanisms that cause the death
of the phytoplankton, namely, endogeneous respiration and
predation by higher trophic levels.  The endogeneous respira-
tion rate of zooplankton populations has been measured and
the results of some of these experiments are presented in
Figure 25 and Table 10.

It is clear from these measurements that the respiration rate
of zooplankters is temperature-dependent.  It is also depend-
ent on the weight of the zooplankter in question and its life
cycle stage (Comita, 1968).  As a first approximation, a
straight line dependence is adequate, and the endogeneous
respiration rate is given by the equation: respiration rate =
K3T where K- = 0.1±0.005(dav °C)  .  The conversion from the
reported units to a death rate is made by assuming that 50%
of the zooplankton dry weight represents the carbon weight
and that carbohydrate  (CH-O) is being oxidized.  The data are
somewhat variable and serve only to establish a range of values
within which the respiration rate of a natural zooplankton as-
sociation might be expected.

The death rate attributed to predation by the higher trophic
levels, specifically the carnivorous zooplankton, has been
considered by previous models in a more or less empirical way.
The complication resulting from another equation and the un-
certainty as to the mechanisms involved are quite severe at
this trophic level.  In particular, it is probable that an
equation for organic detritus is necessary to describe ade-
quately the available food.  Hence, it is expedient to break
the causal chain at this point and assume that the herbivorous
                            116

-------
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 Fig. 25  Endogenous Respiration Rate of  Zooplankton
          vs.  Temperature
                                117

-------
                         Table 10
        Endogenous Respiration Rate of Zooplankton
Reference
Bishop
 1968
 Organism

Cladocerans


Copepods


Copepods
Riley
 1963
Comita
 1968
Calanus
finmarchicus
Diaptomus
leptopus
              D. clavipes
              D. siciloides
              Diaptomus sp.


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Respiration
Rate,
Ml OVMg Dry
Wt-Day
14.2
2.7
12.2
3.8
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6.5
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4.1
3.4
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                             118

-------
zooplankton death rate resulting from all other causes is
given by a constant, the magnitude of which is to be de-
termined empirically.  The severity of this assumption can
be tested by examining the sensitivity of the solutions of
the phytoplankton and zooplankton equations to the magnitude
of this constant.  Hence, the resulting zooplankton death
rate is given by
                       Dzj  =  K3T + K4               (77)

where K. is empirically determined.

With the growth and death rates given by Equations (76)  and
(77), respectively, the source term for herbivorous zooplankton
biomass is given by Equation (75).   The conservation of mass
equation which describes the behavior of Z. is given by Equa-
tion (53), with Z. as the dependent variables replacing P. and
S  . replacing S .  as the source terms.

This completes the formulation of the equations which describe
the zooplankton system.  The equations for the nutrient system
remain to be formulated.
                  The Nutrient System

The conservation of mass principle is applied to the nutrients
being considered in the same way as it has been previously ap-
plied to the phytoplankton and zooplankton biomass within a
volume segment.  The number of mass conservation equations re-
quired is equal to the number of nutrients that are explicitly
included in the growth rate formulation for the phytoplankton.
For the sake of simplicity, the formulation for only one nutri-
ent is discussed below.
                            Ill

-------
The source term S .  in the conservation of mass equation for
                 nD                           th
the concentration of the nutrient N. in the j   volume seg-
ment V. is the sum of all sources and sinks of the nutrient
within V..  The primary interaction between the nutrient
system and the phytoplankton system is the reduction or sink
of nutrient connected with phytoplankton growth.  The rate
of increase of phytoplankton biomass is G  .P..  To convert
                                         PD D
this assimilation rate to the rate of utilization of the nu-
trient, the ratio of bioraass production to net nutrient as-
similated is required.  Over a long time interval, this ratio
approximates the nutrient-to-biomass ratio of the phytoplankton
population.  For example, if the nutrient being considered is
total inorganic nitrogen and the phytoplankton biomass is
characterized in terms of dry weight, then this ratio is the
nitrogen-to-dry-weight ratio of the population.  For both
nitrogen and phosphorus, these ratios have been studied for
a number of phytoplankton species and natural associations.
An example of this information is presented in Table 11, con-
densed from Strickland  (1965)   If a   is the nutrient-to-
                                    np
phytoplankton biomass ratio of the population, then the sink
of the nutrient owing to phytoplankton growth is a  G  .P..

A secondary interaction between the biological system and the
nutrient systems is  the excretion of nutrients by the zoo-
plankton and the release of nutrients in an organic form by
the death of phytoplankton and zooplankton.  The excretion
mechanism has been considered by Riley (1965) in a generaliza-
tion of the equations of Steele.  The rate of phosphorus ex-
cretion has also been measured experimentally  (Martin, 1968).
Using the formulation for zooplankton growth rate proposed
herein, the rate of  nutrient excretion is the rate grazed,
a  C P.Z., minus the rate metabolized, a  G  . Z.; that is,
 np g D D                               np z;j 3
                            120

-------
the excretion rate is
                 anpcgzjpj  I i - K *VP  )           (78)
                                  mp    j .'
At high phytoplankton concentrations, almost all the grazed
phytoplankton is excreted since the bracketed term in Equa-
tion (78) approaches unity.
                       Table 11
     Dry Weight Percentage5 of Carbon, Nitrogen,
          and Phosphorus in Phytoplankton
               % Carbon       % Nitrogen       % Phosphorus
Phyto-
plankter
Myxo-
phyceae
Chloro-
phyceae
Dino-
phyceae
Chryso-
phyceae
Aver-
age Range
36 (28-45)
43 (35-48)
43 (37-47)
40 (35-45)
Aver-
age
4.9
7.8
4.4
8.4
Aver-
Range age Range
(4.5-5.8) 1.1 (0.8-1.4)
(6.6-9.1) 2.9 (2.4-3.3)
(3.3-5.0) 1.0 (0.6-1.1)
(7.8-9.0) 2.1 (1.2-3.0)
Bacillario   33   (19_50)   4.9     (2.7-5.9)   1.1    (0.4-2.0)
phyceae


 The units are (mg of carbon, nitrogen, or phosphorus)/(mg dry
 weight of phytoplankton) X 100%.
b
 Condensed from Strickland  (1965).
                           121

-------
There is a difficulty, however, in using this term directly
as a source of nutrient.  To illustrate this difficulty,
assume that the nutrient is inorganic nitrogen.  A part of
the excreted nitrogen, however, is in organic form, and a
bacterial decomposition into the inorganic forms must pre-
cede utilization by the phytoplankton.  The same is true for
the nutrient released by the death of phytoplankton, a  K0TP.
                                                      np c.  ~],
and that released by the death of zooplankton, a  K_TZ., where
                                                      J
a   is the nutrient-to-zooplankton biomass ratio.  Therefore,
strictly speaking, a conservation of mass equation for the
organic form of the nutrient is required.  The organic form
is then converted to the inorganic form.  For the case of ni-
trogen, the kinetics of this conversion have been investigated
and applied to stream and estuarine situations (Thomann, 1963).
If the conversion rate is large by comparison with the other
rates in the phytoplankton and zooplankton equations, then the
direct inclusion of these sources is approximately correct.

The sources of nutrients arising from man-made inputs, such
as wastewater discharges and agricultural runoff, are included
explicitly into the source term since these sources are usually
the major control variables available to influence the biologi-
cal systems.  An extensive review of the magnitude and relative
importance of these sources of nutrients, primarily nitrogen
and phosphorus, has recently been made  (Vollenweider, 1968).
A useful distinction is made between diffuse sources such as
agricultural runoff loads and ground water infiltration, which
are difficult to measure and control, and point sources such as
wastewater discharges from municipal and industrial sources,
for which more information is available.  The nitrogen and
phosphorus loads from agricultural runoff are quite variable
and depend on many variables such as soil type, fertilizer ap-
plication, rainfall, and irrigation practice.  The nutrient
                            122

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sources from point loads can be estimated more directly.  For
example, the nutrient load from biologically treated municipal
wastewater is in the order of 10 g/capita-day total nitrogen
and 2 g/capita-day total phosphorus.  The ratio of per capita
phosphorus to physiologically-required phosphorus is approxi-
mately 2 to 3, the excess being primarily the result of de-
tergent use.  Industrial loads can also be important, especially
effluents from food processing industries.  If the required
loading rates are available, their loads should be included
in the nutrient mass balance equations.  In particular, if the
investigation of the phytoplankton population is directed at
the probable effects of increasing or decreasing the nutrient
load, these loads must be explicitly identified and their
magnitude assessed.

Let W  . be the rate of addition of the nutrient to the j
     nD                                                J
volume element.  This source is then included as a component
in the nutrient source term in the mass balance equation.

An important additional source of inorganic nutrients which
may influence the availability of nutrients is the interaction
of the overlying water either with the underlying mineral
strata if exposed or with whatever sediment is present.  These
interactions can complicate the source term but they should
be included if they add significantly to the available nutrient.

The source term which results from the inclusion of the phyto-
plankton utilization sink, the zooplankton excretion and the
mortality sources, and the man-made additions is
         W  .                        /    a  K
   c
   S
    nj          nppjj    npgjj     Kmp + Pj
                                                           (79)
             + a  K0TP. + a  K..TZ .
                np 2  j    nz 3  j
                            123

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Any additional sources and sinks that contribute can be
added to the source term as needed.  With the source term
formulated, the conservation of nutrient mass equation is
given by Equation  (53) with N. as the dependent variable
replacing P . and S  . replacing S  . .

              The Equations of the Model

In the previous sections, the equations for phytoplankton
and zooplankton biomass and nutrient concentration within
one volume element have been formulated.  The resulting
equations are an attempt to describe the kinetics of the
growth and death of the phytoplankton and zooplankton popu-
lations and their interaction with the nutrients available.
The form of the equations for the volume V. are as follows:

    PJ = [GpjCP.,N.,t) - Dpj(Z.,t)] P. + Spj(P.,Z.,N.,t)   (80)

         Z, = [G ,(P.,t) - D  • (t)] Z. + S  .(P.,Z.,t)       (81)
          J     *• J  J       *• J      J    ^ J  J  J
                                                           (82)
where G  .and D  .are given by Equations  (67) and  (74) , G  . and
       PD     PD                                       ZD
D  . are given by Equations  (76) and  (77), and S  . by Equation
(79).  The dependence of the growth and death rates on the
concentration of the three dependent variables and time  is
made explicit in this notation.

These equations describe only the kinetics of the populations
in a single volume element V..  However, in a natural water
body there exists significant mass transport as well.  The
mass transport mechanisms can be conveniently represented by
the matrix A with elements a...  If for particular segments
                           124

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i and j the matrix element a..  is nonzero, then the volume
segments V. and V. interact, and mass is transported between
the two segments.  Letting P,  Z, and N be the vectors of
elements P., Z . ,  and N.  and letting S ,  S ,  S  be the vectors
          D   D       D               p   n   z
of elements S  .,  S .,  and S .,  the conservation of mass equa-
tions for the three systems including the mass transport and
kinetic interactions are

                  VP  =   AP + VS                          (83)
                  VZ  -   AZ + VS                          (84)
                   o            Z
                  VN  =   AN + VS                          (85)
                                n

where V is the diagonal  matrix of the volumes of the segments.
These are the equations  which form the basis for the phyto-
plankton population model.  The detailed formulation and
evaluation of the mass transport matrix has been discussed
elsewhere  (Thomann, 1963; O'Connor & Thomann, 1966; O'Connor
et al, 1971).
The form of Equations (83)-(85) makes explicit the linear and
nonlinear portions of the equations.  In the equation for P,
the phytoplankton biomass, the concentration P., in the volume
element V., is linearly coupled to the other P,  's through the
         D                                    K
matrix multiplication by A.  However, there is no nonlinear
interaction between P. and any other P, , k ^ j.  The reason
                     D                K
is that the transport processes are described by linear equa-
tions.  It is usually the case, however, that the A matrix is
a function of time, since at least the advective terms usually
vary in time.  The nonlinear terms in the vector S  involve P.
itself and the corresponding Z. and N..  Hence, the P equation
is coupled to the Z and N equations through this term.  Note,
however, that P. is not coupled to the Z, , k ^ j, in any other
               D                        *
segment, so that the coupling takes place only within each
volume segment.
                            125

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Therefore, the nonlinearities provide the coupling between
the phytoplankton, zooplankton, and nutrient systems.  This
coupling is accomplished within each volume and does not ex-
tend beyond the volume boundary.  The coupling among the
volumes is accomplished by the linear transport interaction
represented by the matrix A.  This matrix may be time-varying
but its elements are not functions of the phytoplankton, zoo-
plankton, or nutrient concentrations.  Hence, in many ways
these equations behave linearly.  In particular, their spatial
behavior is unaffected by the nonlinear source terms.  However,
the temporal behavior and the relationships between each P.,
Z.f and N. are distinctly nonlinear.
          Comparison with Lotka-Volterra Equations

The classical theory of predator-prey interaction as formulated
by Volterra involves two equations which express the growth
rate of the prey and the predator  (Lotka, 1956).  Within the
context of phytoplankton and zooplankton population, the prey
is the phytoplankton and the predator     zooplankton.  In
the notation of the previous sections, for a one-volume system,
the Lotka-Volterra equations are:
                 air  =   (GP - V* p - cgpz                 (86)

                 ar  =  - V + azPcgpz                     (87)
where all the coefficients, G , D' , C , D , and a   are assumed
                             P    P   9   z       ZP
to be constants and G  > D' .  This is a highly simplified
situation since, as indicated previously, the growth and death
rates are functions of time and, in the case of the phytoplank-
                            126

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ton growth rate, of the phytoplankton and nutrient concen-
trations as well.  However, for a situation with adequate
nutrients and low initial phytoplankton concentration, the
nonlinear interaction is small initially, and the time vari-
ation of G  can be small during the summer months.  In any
case, the analysis of this simplified situation is quite
instructive.

Although no analytical solution is available for these simpli-
fied equations, their properties are well understood  (Davis,
1962).   In particular, the equations have two sets of singular
points  corresponding to the solution of the righthand side of
Equations (86) and (87) equated to zero: the trivial solutions
P* =0, Z* = 0, and
                         D_            G._ - D'
                                         :g
\—,  Z*  =  -E-	K.          (88)
A perturbation analysis of Equations (86) and (87) about this
singular point shows that the solutions whose initial condi-
tions are close to P*, Z*, oscillate sinusoidally about this
singular point.  Hence, no constant solution is possible.
The prey and predator populations continually oscillate and
are out of phase with each other.  When the predator predomin-
ates, the prey is reduced, which in turn causes the predator
to die for lack of food, which allows the prey to proliferate
for lack of predator, which then causes the predator to grow
because of the prey available as a food supply, and so on.
The interesting feature is that these oscillations continue
indefinitely.

The classical Lotka-Volterra equations assume an isolated
population with no mass transport into or out of the volume
being considered.  To simulate the effect of mass transport
into the volume, assume that an additional source term of
                               127

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phytoplankton biomass exists and has constant magnitude P .
For this situation, the equations become
              g   =  (Gp - D'p)  P - CgPZ
                                    (89)
                -  =  D Z + a   CPZ
              dt      z     zp  g
The nontrival singular point for these equations is
                                    (90)
              P*  =
i4--  z* = ^§-
 zp g           z
                                  a__P_   (Gp - D'p)      (91)
The perturbation analysis about this singular point yields a
second order linear ordinary differential equation whose
characteristic equation has the roots A.  and A_ where
a P C -, /
zp o g
2 DZ * y
"a PC'
zp o g
2 °z J
2
- azPCgPo - (GP -
D'p)Dz
                                                          (92)
Since for P  > 0, these roots have negative real parts,  this
singular point is a stable focus,  and the steady state values
given by Equation (91)  are approached either by a damped sinu-
soid or an exponential (Davis, 1962).  Note that for P  =0,
the classical case,  the roots are  purely imaginary,  and the
oscillation persists indefinitely.

This analysis suggests that the effect of transport  into the
system stabilizes the behavior of  the equations and  in parti-
cular allows the solutions to achieve a constant solution.
This is in marked contrast to the  behavior of the classical
Lotka-Volterra equations.

Another modification, which has been introduced into the zoo-
plankton equations,  changes the behavior of the proposed
                           128

-------
equations in contrast to the Lotka-Volterra equations.   It
has been argued that the zooplankton growth rate resulting
from grazing must approach its maximum value when the phyto-
plankton population becomes large since the zooplankters
cannot metabolize the continually increasing food that is
available.  Thus, the growth rate a  C EZ is replaced by
a  C ZP K  /(P + K) where K   is a Michaelis constant for
the reaction.   The equations then become
         a!  -  WP - DVP - cgpz + po                 C93)
         ||  =  - D Z +  "g ?   "*-                      (94)
         dt        2      P + Kmp

The nonzero singular points are

                     D K
          P*  =       z mP
                 azp g mp    z
                  P^   (G_ - D' )
          Z*  =  c-^-H -Eg	E_                       (96)
                  g        g
   This solution reduces to the previous situation,  Equation
(91), for large K  .  This is expected since for K   large
             3   mp                               mp
with respect to P, the expression K  / (P + K  )  approaches
one.  However, an interesting modification from classical
predator-prey behavior occurs if the following condition is
met
          azpCgKmp  =  Dz + e                           (97)

where e is a small positive number.  For this condition,  P*
is large and positive.  What happens in this case is that the
zooplankton population, although it continues to grow expo-
                           129

-------
nentially, cannot grow quickly enough to terminate the phyto-
plankton growth by grazing, and the phytoplankton continue to
grow exponentially until P* is reached.  Of course, in the
actual situation, for which G  is not a constant, other
phenomena such as nutrient depletion and self-shading exert
their effect, and the growth may be stopped sooner.  However,
if the growth rate of zooplankton at a phytoplankton concentra-
tion equal to the Michaelis constant K   is only slightly larger
than their death rate D ,  then the zooplankton alone do not
                       z
rapidly terminate the bloom.

This condition is an important dividing line for the possible
behavior of the phytoplankton-zooplankton equations set forth
in the previous sections.   In particular, it indicates what
must be true for a system wherein the zooplankton are a signi-
ficant feature in the resulting phytoplankton solution.  How-
ever, if Equation (97) is satisfied, then the zooplankton are
not the dominant control of the phytoplankton population.
            Application - San Joaguin River

As an example of the application of the equations proposed
herein, consider the phytoplankton and zooplankton population
observed at Mossdale Bridge on the San Joaquin River in
California during the two years 1966-1967.  Mossdale is located
approximately 40 miles from the confluence of the San Joaquin
and the Sacramento Rivers.  The data presented below have been
supplied to the investigators by the Dept of Water Resources,
State of Calif (Anon,1966), as part of an ongoing project to
assess the effects of proposed nutrient loads and flow di-
versions on the water quality of the San Francisco Bay Delta
(O'Connor et al, 1972).
                            130

-------
In order to simplify the spatial segmentation and the calcu-
lations, a one-volume segment is chosen for the region of the
San Joaquin for which Mossdale is representative.  The volume
of this segment is, of course, somewhat arbitrary, and a more
representative spatial segmentation would remove this uncer-
tainty.  However, it is instructive to consider the behavior
of the solution of this simplified model.

The nutrient data available indicate that phosphate, biocarbo-
nate, silicate, calcium, and magnesium are available at con-
centrations well above the levels for which it has been sug-
gested that these nutrients limit growth.  Only the ammonia
and nitrate concentrations are low, and they both decrease
markedly during the 1966 spring bloom.  Hence, these nutrients
must be considered explicitly.  To simplify the computations,
the ammonia and nitrate nitrogen are combined, and the nutrient
considered is total inorganic nitrogen.

There is some uncertainty concerning the magnitude and the
temporal variation of the inorganic nitrogen load being dis-
charged to the system during the time interval of interest.
For lack of a better assumption, the inorganic nitrogen load
W  being discharged into the volume is assumed to be a constant,
the magnitude of which is determined by comparison with the ob-
served inorganic nitrogen concentration data at Mossdale.

The variation of the environmental variables being considered
- namely, temperature, solar radiation, and advective flow in
the San Joaquin during the two-year period of interest - and
the straight line approximations that are used directly in the
numerical computation are shown in Figure 26.  The influent
advective flow, which is assumed to have constant concentrations
of phytoplankton and zooplankton biomass and inorganic nitrogen,
                           131

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           30 -
   o
              0
         200
I   — "P*
fc   ^T" ^'














I   2 (i
                       !20

                       1966
240       360

    TIME -
50
         Fig. 26  Temperature,  Flow,  and Mean Daily Solar Radiation;

                  San Joaquin River,  Mossdale, 1966-1967



                                     132

-------
is routed through the volume.  Since Mossdale is located above
the saline portion of the San Joaquin, no significant dis-
persive mass transfer is assumed to exist by comparison with
the advective mass transfer.

The equations which represent this one-segment model are

          P =  (Gp - Dp)P + | (PQ - P)                  (98)

          Z  =  (Gz - Dz)Z + | (ZQ - Z)                 (99)

          N  =  - anPGPP + -V + I (No - N)              (100)

where Q = Q(t) is the advective flow entering and leaving the
volume; V is the volume of the segment; P , Z , and N  are
the phytoplankton, zooplankton, and inorganic nitrogen con-
centration of the flow entering the volume.  The remaining
terms have been defined previously by Equations (67),  (74),
(76), and (77).  In the nutrient equation, only the direct
source of inorganic nitrogen, W , has been included; the or-
ganic feedback terms representing excreted nitrogen, etc.,
Equation  (79) , have been dropped.  Since the magnitude of W
is uncertain and is assigned by comparison with observed data
and computed model output, these feedback terms can be thought
of as being incorporated in the value obtained for W .

The solution of Equations (98), (99), and  (100) requires
numerical techniques.  For such nonlinear equations, it is
usually wise to employ a simple numerical integration scheme
which is easily understood and pay the price of increased com-
putational time for execution rather than using a complex,
efficient, numerical integration scheme where unstable be-
havior is a distinct possibility.  A variety of simple methods
                            133

-------
are available for integrating a set of ordinary first order
differential equations.  In particular, the method of Huen,
described by Stiefel (1966),  is effective and stable.  It
is self-starting and consists of a predictor and a corrector
step.  Let y = f(t,y) be the vector differential equation and
let h be the step size.  The predictor is that of Euler:  with
y  the initial condition vector at t , the predictor value of
y at tQ + h = t-j^ is

                  YI*  =  yo + hf(to, yo)               (101)

the corrector value is simply

                  yl   =  yo + I [f(to'yo) + f^i^l**1 <102)

That is, the corrector uses the predictor value at t, to  esti-
mate the slope at t, which is averaged with the slope at  t
to provide the slope of the straight line approximation.   A
variation of this method is discussed at some length by
Hamming (1962).

Another simple two-step method is that of Runge, described by
Levy (1950) .  The Euler predictor is used with a half-step
integration.

                  y*  =  y0 +1 f(v V               (103)

This value of y is used to estimate the slope at the midpoint
of the interval, which is then used as the slope of the straight
line approximation

                  YI  =  yo + hf(to + £, y*)            (104)
                             134

-------
Both of these methods are second order methods, being accurate
                    2
to terms of order At  in a comparison of Taylor series ex-
pansions of the exact and approximate values, and both methods
require  two derivative evaluations per step.  The method of
Runge has been used in the calculations presented below.

The equations themselves are programmed for solution using
a continuous simulation language and a digital computer.  The
language, in this case CSMP/1130, is based on a block diagram,
analog computer, representation of the differential equations.
The flexibility of these languages which allow changes in the
equation structure to be made easily  is an asset in modeling
complex systems.

The biomass variables used in the calculations are total cell
counts for the phytoplankton and rotifer counts for the zoo-
plankton.  The rotifer population represented the large major-
ity of the zooplankton present on a weight basis as well.  In
order to relate these variables to comparable units, a series
of conversion factors have been used.  The phytoplankton count
-chlorophyll concentration ratio was measured.  However, the
carbon-chlorophyll or dry weight-chloroplyll conversions are
unknown.  Hence, the conversion to an organic carbon basis
is made rather arbitrarily.  However, the carbon-to-chlorophyll
ratio which results (see Table 12) is within the range reported
in the literature.  The same problem exists with the rotifer
counts to rotifer carbon conversion: the value used is given
in Table 12.

The comparison of the model output and the observed data for
the two-year period for which data are available is shown in
Figure 27.  The parameter values used in the equations are
listed in Table 12.
                           135

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                        Table 12

        Parameter Values for the Mossdale Model


Notation           Description            Parameter Value

K       Saturated growth rate        0.1 day   °C
        of phytoplankton

I       Light saturation intensity   300 ly/day
        for phytoplankton

k1      Extinction coefficient       4.0m
  e
H       Depth                        1.2m

K       Michaelis constant for total 0.025 mg N/liter
        inorganic nitrogen

f       Photoperiod             0.5+0.11 sin [0. 0172 (t-165) ] day

K?      Endogenous respiration rate  0.005 day   °C
        of phytoplankton

C       Zooplankton grazing rate     0.13 liter/mg - C - day

P       Influent phytoplankton       5.0 ug Chi/liter
 °      chlorophyll concentration

a       Zooplankton conversion       0.6 mg C/mg - C
  "     efficiency
        Phytoplankton Michaelis      60 ug Chi/liter

                                              -1
mp     constant
D       Zooplankton death rate       0.075 day
 z
Z       Influent zooplankton carbon  0.05 mg C/liter
 0      concentration

a       Phytoplankton nitrogen-      0.17 mg N/mg - C
  "     carbon ratio

C/Chl   Phytoplankton carbon to total 50 mg C/mg-Chl
        chlorophyll ratio

N       Influent total inorganic     0.1 mg N/liter
        nitrogen concentration

W       Direct discharge rate of     12500 Ibs/day
        nitrogen

V       Segment volume               9.7 x 108 ft3

        Phytoplankton total cell     100 cells/ml =
        count/phytoplankton              1.75 yg Chi/liter
                                        4
        Zooplankton count/zooplankton 10  No./liter =
        carbon ratio                     1.30 mg C liter
                            136

-------
    100,000 -

             i

     80,00 0[-


g            i
K _, 60,000 j-

25         i

3 -J 40,000 |-
Q. _J
o LJ

> ° 20,000 -

Q.

o
CL
O
o
rw
16,000



12,000 -



 8,000



 4,000
            0
                120
240
360
120
o
c:
O
  UJ
                                              .
      Fig.  27   Phytoplankton, Zooplankton,  and  Total Inorqanic Nitrogen;

               Comparison of Theoretical Calculations and Observed Data;

                       San Joaquin River,  Mossdale, 19666-1967

                                      137

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It is clear from both the data and the model results that a
classical predator-prey situation is observed in 1966: the
spring bloom of phytoplankton resulting from favorable tem-
perature and light intensity provides the food for zooplankton,
which then reduce the population during the summer.  The de-
crease of the zooplankton and the subsequent slight secondary
bloom of phytoplankton complete the cycle for the year.  It
is not clear, however,  from a casual inspection of the data,
whether the zooplankton population terminated the phytoplankton
growth, as in classical predator-prey situations, whether the
nutrient concentration dropped to a limiting value that reduced
the growth rate, or a combination of the two.  This point is
elaborated in the next section.

The situation in 1967 is quite different.  No significant
phytoplankton growth is observed until late in the year.  The
controlling variable in this case is the large advective flow
during the spring and summer of 1967 (see Figure 26) which
effectively washes out the population in the region.  Only
when the flow has sufficiently decreased so that a copulation
can develop do the phytoplankton show a slight increase.
However, the dropping temperature and light intensity level
terminate the growth for the year.

            Growth Rate - Death Rate Interactions

The behavior of the equations which represent the phytoplankton,
zooplankton, and nutrient systems in one volume can be inter-
preted in terms of the growth and death rates of the phytoplank-
ton and zooplankton.  The equations are as before
            air  =   (GP - Vp + I  (po - p)               (105)
                           138

-------
            H  =  (Gz - °z)Z + V- (Zo - Z)              (106)

where P  and Z  are the concentrations of phytoplankton and
zooplankton carbon in the influent flow, Q.  A more useful
form for these equations is
            ar  =  [GP - :(JDP + l)]  p + lpo            (107)
            aT  =  [Gz - (DZ + l)]  z + lzo            C108)

A complete analysis of the properties of these equations is
quite difficult since the coefficients of P and Z are time
variables and also functions of P and Z.  However, the be-
havior of the solution becomes more accessible if the
variation of these coefficients is  studied as a function of
time.  The expressions G  - (D  + Q/V) and G  - (D  + Q/V)
can be considered the net growth rates for phytoplankton and
zooplankton.  The advective or flushing rate, Q/V, is included
in these expressions since it acts  as a death rate in one
segment system.

The sign and magnitude of the net growth rate controls the
behavior of the solution.  For a linear equation,  for which
the net growth rate is not a function of the dependent varia-
ble  (i.e., P or Z), the type of solution obtained depends on
the sign and magnitude of the net growth rate.  That is,  for
the equation
             dT  =  ap + I po                            (109)
where a, Q, and V are constant, the solution is
                           139

-------
             P(t) = P(o) eat + P     (eat - 1)         (.110)
For a negative, that is, for a negative net growth rate, the
solution tends to the steady state value P  Q/|a|v.  However,
for a positive, the solution grows exponentially without limit.
Thus, for a negative but |a| small, or for a positive, the
solution becomes large; whereas for a negative but |a| large,
the solution stays small.  Hence, the behavior of the solution
can be inferred from the plots of the net growth rates.
Figure 28a is a plot of the following terms from the 1966
Mossdale calculation: G  without the Michaelis-Menton multi-
plicative factor included - i.e., the growth rate at nutrient
saturation denoted by G  (I,T); G  itself denoted by G   (N,I,T)
                       t~         P                    P
i.e., the growth rate considering the nutrient effects.  The
net growth rate G  - (D  4- Q/V) is also plotted.  Similarly,
in Figure 28b the growth rate of zooplankton G , the mortality
                                              z
rate D , the flushing rate Q(t)/V, and the net growth rate
      z
G  - (D  + Q/V) are plotted.
 Z     Z

The analysis of the 1966 model calculations can now be made
by inspecting these figures.  The net growth rate for the
phytoplankton G  - CD  + Q/V) becomes positive at t = 85 days
owing to an increase in G , the result of rising temperature
and light intensity, and a decrease in Q/V as the advective
flow decreases.  The positive net growth rate of the popula-
tion causes their numbers to increase exponentially fast until
the nutrient begins to be in short supply.  This is evidenced
by the departure of the G  curve from the G  at nutrient satura-
                         P                 P
tion curve.  At the same time, the D  curve is showing a marked
increase because of the increased zooplankton population and
their grazing.  The result is that the net growth rate becomes
zero and then negative as the zooplankton reduce the phyto-
plankton population by grazing.  The growth of the zooplankton
can be analyzed in a similar fashion using Figure 28b.  The net
                           140

-------
CO
UJ
a:
I-

I
cc
§5
I- Q
^
Q.
O
I-
>-

Q.
5
o
  CO
Q-
O
O
rw
       0.2
       O.I -
                 60
120    180   240

   TIME-DAYS
       1966
300    560
                 60     120    180   240   300   360
                           TIME-DAYS
        Fig. 28  Theoretical Growth  Rates for Phytoplankton
                and  Zooplankton Populations
                              141

-------
growth rate becomes positive when the phytoplankton popula-
tion is large enough to sustain the zooplankters.   Then the
zooplankton grow until they have reduced the phytoplankton
population to a level where they are no longer numerous enough
to sustain the zooplankton.  The net zooplankton growth rate
then becomes negative and the population diminishes in size.
This small zooplankton population no longer exerts a signifi-
cant effect on the death rate of the phytoplankton, D , and
its value decreases, causing the net phytoplankton growth rate
to become positive again, and the smaller autumn .bloom results,
The decreasing temperature and light intensity and the in-
creasing advective flow then effectively terminate the bloom
as the year ends.
                           142

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                        SECTION VI

      A PRELIMINARY MODEL OF PHYTOPLANKTON DYNAMICS
              IN THE UPPER POTOMAC ESTUARY

The Potomac Estuary has been the subject of investigation
and analysis over many decades and most intensively within
the past several years. This work has been motivated in large
measure by the location of Washington D.C. along its shores
and by a long history of poor water quality.  Also, if water
quality is to be improved nationally, the river flowing
through the Nation's capital must serve as a model situation.
Several Enforcement Conferences have been held on the water
quality of the Potomac River with the aim of establishing
required effluent controls to achieve specified water quality
objectives.  Important recommendations which evolved from
these efforts included waste load allocations which restrict
the mass discharge of oxygen demanding material, and nitrogen
and phosphorus residuals.  These recommendations relied to
some degree on the application of detailed mathematical
modeling of key water quality constituents such as dissolved
oxygen and various nitrogen and phosphorous forms.

The purpose of this section is to present a preliminary
model of the dynamic behavior of phytoplankton in the Upper
Potomac Estuary.  This research extends the previous model-
ing efforts on the Potomac to incorporate explicitly the
space-time variability of chlorophyll a as a water quality
parameter indicative of a eutrophied environment.  The
research also extends previous work in this report (see
Section  V  to an estuarine situation dominated presently
by municipal waste discharges.  This preliminary phyto-
plankton model is intended therefore to shed further light
on the water quality changes that can be expected when a
                              143

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waste reduction program is completed for the Potomac Estuary.
Primary emphasis here is on the efficacy of nutrient removal
programs especially in the Washington D. C. area and down-
stream.

The area of interest of the model is centered in the upper
forty-mile reach of the Potomac Estuary although as indicated
below,the model geographically extends over the entire 114-
mile length of the estuary from Little Falls to Chesapeake
Bay.  The upper reach of 30 - 40 miles is generally unaffected
by the incursion of salts from Chesapeake Bay.  The Potomac is
tidal in the vicinity of Washington B.C. and is several
hundred feet wide with a shipping channel of minimum depth
of 24 feet maintained to Washington.  The tidal portion
averages about 18 feet and is characterized by numerous
coves and embayments along its length with average depths
significantly less than the main estuary.  Presently, the
major waste source in the upper reach of the Potomac is the
effluent from the Washington D.C. secondary treatment plant.

Water quality problems include low values of dissolved oxygen
in the vicinity of the District of Columbia and high concen-
trations of phytoplankton especially of Anacystis, a blue
green form.  Detailed reviews are given bv Jaworski et al
(March 1969), Jaworski (Nov 1969), Aalto et al (1970) and
Jaworski et al (1971).

              Model Geometry and Kinetics

The basic model builds on previous efforts of phytoplankton
dynamics (Di Toro et al, 1971) and nitrification effects in
estuaries  (Thomann et al, 1970).

Spatially,  the estuary is divided into twenty-three longi-
                           144

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tudinal segments using data on depth,cross-sectional areas
and segment volume given by earlier modeling work in the
Potomac and summarized in Jaworski and Clark (undated).
Figure 29 shows the location of these main estuary segments.
As discussed more fully below, it was found necessary to in-
clude the effects of the tidal bays on main channel quality.
The tidal flat areas have been cited as an important phenome-
non in the Potomac River as far back as 1916 (Phelps).  Accord-
ingly, an additional fifteen spatial segments were incorporated
in the model to reflect lateral effects of the shallow water
areas.  These tidal bay segments are also shown in Figure 29.
A total of thirty-eight spatial segments were therefore used
to represent the Potomac with primary emphasis on segments
#1 - #15, the U.pper Estuary.  Each segment is assumed com-
pletely mixed and therefore represents a finite difference
approximation to a continuous medium.  In order to simplify
the analysis, waste loads were inputted into segments #5, 6
and 7 which accounts for the major portion of direct discharge
load.  No attempt was made in this preliminary model to input
urban or suburban runoff or overflows from combined sewers.

The model incorporates interactions between nine variables
which are space and time dependent.  The variables in the model
are
     1)  Phytoplankton chlorophyll "a" - P
     2)  Zooplankton carbon - Z
     3)  Organic nitrogen - N,
     4)  Ammonia nitrogen - N~
     5)  Nitrate nitrogen - N.,
     6)  Organic phosphorous - N ,
     7)  Inorganic phosphorous - N 2
     8)  Carbonaceous biochemical oxygen demand - L
     9)  Dissolved oxygen - C
                            145

-------
                                      N
                                                            SCALE OF MILES
                                             O = MAIN ESTUARY SEGMENTS
                                                = TIDAL BAY SEGMENTS
VIRGINIA
                                                                     10    15
                                                                  /f  CHESAPEAKE
                                                                  %*
                                                                   \  BAY
                                                                     N»^
        Fig. 29   Mao of Potomac Estuarv Shov/inq  Longitudinal
                              and Lateral Segments
                                   146

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Other variables, constructed from these primary variables
are also tracked through the estuary.  These secondary vari-
ables include total nitrogen and total phosphorous as the
most important.

Figure 30 shows the general interaction scheme of the nine
primary system variables.  The major interactions and phe-
nomena are a) predator-prey relationships between phyto-
plankton and zooplankton systems 1 and 2  (which is apparently
not an important phenomenon  at present in the upper Potomac) ;
b) nitrification of oxidizable nitrogen, given by systems #3,
4, and 5; c) conversion of organic phosphorous to inorganic
phosphorous and d)  dissolved oxygen depletions due to carbo-
naceous waste loads and effects of nitrification.  Nutrient
interactions are shown with phytoplankton chlorophyll.  The
basic kinetic equations are discussed below for each system.

The general matrix equation for the phytoplankton chlorophyll
system (Di Toro et al, 1971) under a finite difference ap-
proximation is
            [V]       =   [A] (P) +  [V] (sp)                  (111)

where (P) is an n x 1 phytoplankton chlorophyll "a" vector,
 [A] is an n x n matrix of transport and dispersion effects,
 [V] is an n x n diagonal matrix of segment volumes and  (S  )
is an n x 1 vector of source terms.  Specifically for
segment j
           S  .  =   (G  . - D  . )  P .
            PD       PD    PD   D
where G  . is the growth rate and D  . is the death rate.
       P:                         PD
Turning first to the growth rate, the light and temperature
effects are as given in Eq. (67).   The interaction with ni-
trogen and phosphorous is given by a product of Michaelis
                            147

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     WASTE LOAD-
WASTE LOAD-
                          SYSTEM.®
                         Carbonaceous
                          Biochemical
                            Oxygen
                            Demand
                           SYSTEM CD
                           Organic
                           Nitrogen
                  SYSTEM @
                  Zooplankton
                    Carbon
SYSTEM ©
 Organic
 Phosphorous
                          SYSTEM
                          Dissolved
                           Oxygen
                            Deficit
                          SYSTEM  @
                          Ammonia
                          Nitrogen
                 SYSTEM 0)
                Phytoplankion
                 Chlorophyll
SYSTEM  @
  Inorganic
  Phosphorous
                                     WASTE LOAD
                          SYSTEM©
                           Nitrate
                           Nitrogen
           Fig.  30   Interactions of  Nine Svstems Used  in Preliminarv
                                       Phytoplankton  Model
                                              148

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effects.  Therefore,
     Gpj  - Glt,i,T.f.e.«        -    a--          (112,
where NT  is the total inorganic nitrogen with K   as Michaelis
       In                  3          ^         mn
constant, N  is orthophosphate concentration with K   as the
Michaelis constant.  GT.  represents the functional relation-
ships of the growth rate and the solar radiation, I; photo-
period, f; water temperature, T; depth, H and light extinction
coefficient, K.  The latter coefficient is incorporated as a
non-linear relationship with chlorophyll as

     K  =  KE + .0088 P + .054 P'67                     (113)
where KE is the extinction coefficient (I/meter) estimated at
zero phytoplankton concentration.  The death rate, D  . is as
given in Eq. (74 )  and incorporates phytoplankton respiration
as a function of temperature and grazing by phytoplankton.

The governing equation for the zooplankton carbon system is

     [V] ^|1  =  [A] (Z)  + [V] [Gz - Dz](Z)               (114)

where Z is a n x 1 column vector of zooplankton carbon con-
centration.  The expressions for G .  and D  . are identical
                                  ZD       Z3
to those used in the earlier work.  Actually, this system
did not play a role in the verifications discussed below.
Some sensitivity runs were made however to show the effect
of different levels of zooplankton grazing.
The organic nitrogen system includes non-living forms of
organic nitrogen generated by the model and includes sources
due to death of zooplankton and phytoplankton and grazed but
unassimilated phytoplankton nitrogen.  The mass balance
                           149

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equation for organic nitrogen, N,, is

         d(N )
      [V] -^-  =   [A] (N.^ +  [V] (Snl) + Nnl              (115)

where  (N , ) is a vector of input  sources of N. .
        nl                   ••                1

The generalized source - sink term is given by

Snl   a^ DzZ + f(Z,P,Gz) + a]_G(P)N1 - K33 + K34(T)N1     (116)

where a, is the ratio of nitrogen to chlorophyll,  a   is  the
ratio of carbon to chlorophyll, K.,., is the settling rate of
organic nitrogen     K.,. is the hydrolysis rate of organic
nitrogen - a function of water temperature and f(Z,P,G  )  is
                                                       Z
the excretion of organic nitrogen (see Eg. 78  )•
Ammonia nitrogen  (System #4) is produced by the feed  forward
hydrolysis reaction of the organic nitrogen and any direct
sources of ammonia due to waste inputs or river runoff.   The
oxidation of the ammonia to nitrate forms an  important  sub-
system of the overall model.  This nitrification effect  is
used as a sink of dissolved oxygen.  The mass  balance equa-
tion for ammonia nitrogen N~, is
         d(N,)
      [V] -g£±-  =   [A] (N2) +  [V] (SN ) +  (WN )            (117)

where  (W  ) is a vector of input  sources of ammonia.  The
         2
source-sink term for the ammonia  is
     SN   =  K45N1 ~ K45  (T)N2 + alGPPa

where K._ is the rate of oxidation of ammonia due  to  nitrify
ing bacteria (a function of temperature) and a represents  an
                            150

-------
assumed ammonia preference by phytoplankton and is given by

                 a  =  N0/(N, + K  )                     (119)
                        2./  2.    mn

The first term in Eq.  (118) therefore represents the pro-
duction of ammonia due to hydrolysis of organic nitrogen.
The second term represents the utilization of ammonia by
phytoplankton and the oxidation of ammonia by nitrifying
bacteria.  Note that at concentrations of ammonia greater
than about 0.5 mg/1 Eq. (119) indicates a preference for
ammonia of about 95% at a K   = 0.025 mq/1.
                           mn

The primary source of nitrate nitrogen (System #5) is the
oxidation of ammonia by nitrifying bacteria.  The primary
sink of nitrate is the utilization of nitrate by phytoplankton.
The governing equation used is
     [V] --  =   [A] (N3) +  [V] (SN ) +  (WN )            (120)

where N., is the nitrate nitrogen concentration.  The nitrate
source-sink is
     SN3  =  K45 N2 - ^Pd-a)                         (121)

The first term in Eq. (121) represents the production of ni-
trate by nitrification while the second term represents the
utilization by phytoplankton.

The organic phosphorous system is assumed to be generated
from the death of phytoplankton and zooplankton plus any ad-
ditions of organic phosphorous due to the discharge of wastes
The organic phosphorous of this system therefore represents
the "non-living" phosphorous.  The sinks for System #6 are the
                             151

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conversion of organic phosphorous to inorganic dissolved
phosphorous forms and an assumed sink of organic phosphorous
out of the estuary, itself due to sorption of turbidity parti-
cles and subsequent settling.

The equation for System #6 is then
         d(N  )
     [V]    g1   =   [A] (N  ) +  [V]  (S   ) + W            (122 )
                         P            pi      pi

where N ,  is the organic phosphorous and the source sink term
is given by

     (SN  )  =  i^ DzZ + f (Z'P'Gz} + apDpP               (123)

                - K,,N ,
                   66 pi

where a  is the ratio of phosphorous to chlorophyll and K,,
is the overall decay of organic phosphorous.

The inorganic dissolved phosphorous system receives the output
from the conversion of organic nitrogen and any direct sources
of waste as input.  The primary sinks of the dissolved phospho-
rous are the uptake by the phytoplankton and sorption on sus-
pended material.  The System #7 equation is

         d(Nn2]
     [V]   ,p   =   [A] (N  ) +  [V](S   ) +  [V](S   ) +W     (124)
           at            P^          p2          p2      p2

where N „ is the dissolved inorganic phosphorous and

     SN _  =  K,-,N , - a- G P - K^_N _                   (125)
       p2      67 pi    2  p     77 p2
where K,., is the rate of production of dissolved inorganic
       b /
                            152

-------
phosphorous from the organic form and K__ is the overall loss
coefficient of inorganic phosphorous from the water column.
Total organic phosphorous and total phosphorous are computed
from Systems #1 ,  #2, #6 and #7 with appropriate conversion
factors .

In the carbonaceous BOD system only the direct sources of
biochemical oxygen demand (carbonaceous) are included.  A
traditional first-order decay is assumed.  The equation is
     [V]     -  =   [A] (L) +  [V] (SL) +  (WL)                (126)

where L is the carbonaceous BOD and ST is simply
                                     LI

                    SL  =  - K88
The DO deficit  System #9 is classical in concept except that
the effect of nitrification is computed internally using the
output from System #4.  The governing equation is
               =  CA] CD) +  tvl {                           (127)
where D = DO deficit.  The source-sink term for the deficit
is given by
     Sn  =  4.57  (K.-N,) + KQQL - K D + f, (P)            (128)
      D            QjZ.     o y     a     1
where KOQ is the deoxygenation coefficient due to carbonaceous
       o y
BOD, K  is the reaeration coefficient.  The first term in
      cl
Eq.  (128) represents the utilization of oxygen due to nitri-
fication while the last term represents the effect of phyto-
plankton on the dissolved oxygen.
                             153

-------
The nine system equations for each of 38 spatial segments re-
sults in a total of 342 simultaneous non-linear ordinary
differential equations.  The entire set was programmed for
solution by a CDC 6600 computer.  For a one-year simulation
at integration time steps of about O.I/day, normal central
processing times were about 6 minutes.

         Verification and Sensitivity Analysis
                       1968 Data

Two data periods were available for testing the validity and
degree of applicability of the preliminary model.  The pro-
cedure followed in the verification was to "tune" the model
to the first data set collected in 1968.  A forecasting
situation was then set up for 1969.  The validated model for
1968 was used to independently estimate the 1969 data.  The
only new input data used for 1969 was the river inflow with
associated water quality, water temperature and waste loads.
All coefficients determined from the 1968 validation were
used without modification for the 1969 verification.  This
latter verification then provides the basis for determining
the degree to which the model will simulate future conditions
under different waste removal policies.

Figure 31 shows the data periods and the flow and temperature
regimes used for the 1968 and 1969 runs.  During the 1968
data period, Figure 31 indicates a significant flow transient
in June 1968 with a lesser transient in September.  The piece-
wise linear approximations to the time variable flow were de-
termined  from the available data and do not account for the
day to day variability in inflow.   The emphasis was rather on
describing the seasonal variation rather than on shorter term
fluctuations.
                            154

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  WATER
   TEMP
    °C
30r


25


20





 10


 5


 0
             April
          30


          25
 FLOW
(lOOOcfs)  20
  INTO
  * I
          15


          10
         Moy   June  July   Aug   Sept   Oct

                         1968
                                                 Nov   Dec   Jan   Feb
           	—Data Period-1968
Mar   April   May   June  July   Aug   Sept  Oct   Nov  Dec

                1969
             -Data Period-1969-
             April  May  June  July   Aug  Sept  Oct    Nov   Dec

                              1968
                                                 Jan   Feb   Mar   April  Moy   June  July   Aug   Sept  Oct   Nov   Dec

                                                                                 1969
         Fiq.  31  Temoerature and  Flow  Regimes  Used for 1968  and  1969  Verification Runs

-------
Only summary plots of observed data were available for veri-
fication of the 1968 data.  These plots are given in Jaworski
et al  (1969).  Figures 32 and 33 show a variety of results
from the validation analyses of the May to November 1968 data.
As indicated previously, the system equations are all time
variable; the results in Figures 32 and 33 show the average
spatial profile during August 1968, a period of low flow just
prior to the late summer flow transient that year.

The three curves of model output in Figures 32 and 33 repre-
sent three conditions on the nutrients; nutrients are totally
conserved in the model and nutrients are decayed out of the
system at rates of .05/day and O.I/day.  As shown, the agree-
ment is good when the organic nitrogen and total phosphorous
are allowed to decay at rates of .05/day or O.I/day but there
is poor agreement when the nutrients are totally conserved
in the system.  This illustrates one of the steps in the
"tuning" of the model using the 1968 data.  It was obvious
from the early runs of the model where the nutrients were al-
lowed to recycle entirely within the model that the results
were not desirable below about Mile Point 15.  The observed
data on all variables shown in Figures 32 and 33 decreased
more rapidly than the computations.  Accordingly, the hypo-
thesis was adopted that incoming sediment load permitted
sorption of phosphorous and nitrogen with subsequent settling
to the bottom of the estuary and out of the domain of the
modeling framework.  This hypothesis is consistent with that
given by Jaworski et al (1971)  in determining phosphorous
balances in the estuary.

In addition, the determination of model parameters using the
1968 data indicated that zooplankton grazing of the phyto-
plankton was probably not a significant factor in the Potomac.
This is supported by the fact that the phytoplankton of concern
                           156

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   I60r
   140
   120
_i  100
_J

a.  80
O
IT
9  60
o
   40
   20
    0
      0
                                      (a)
         APPROX. RANGE
         OBSERVED DATA
                                                   NUTRIENTS
                                                    CONSERVED
   [; NUTRIENT DECAY = 0.0 5/doy

   D NUTRIENT DECAY= O.I /day
10        15       20      25

  MILES  BELOW CHAIN  BRIDGE
30
35
                            KNiT=0.05/day

                            KMT =0.1/day
                       10
         15       20       25       30

      MILES  BELOW CHAIN BRIDGE
    Fiq.  32  Comparison of Range  of Observed  Data and  Model Outnut
              August  1968  a) Chlorophvll a  (pg/1)
                            b) Total  K-jheldahl  Mitroqen  (mg/1)
                                  157

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                                 PHOSPHOROUS
                                      CONSERVED
                       Kphos. = 0.05/doy

                       Kphos. = 0.l/doy
  3.0
  2.5
o>
E2.0
                     10
                      15
20
25
30
35
  i Knit. =0.05/day

©Knit =0.1/day
                                               NITROGEN  CONSERVED
                                                         30
                                                         35
        Fiq. 33   a)  Total  Phosnhorous - PO.  (mq/1) Comnarison
                  b)  Nitrate Nitrogen Comparison, August 1968
                                  158

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in the Washington D. C. area down to about segment 15 are
Anacystis,  a blue green form that is toxic to zooplankton.
Several runs were made of the model to show the sensitivity
of the various systems to perturbations on the zooplankton
system.

Figure 34 (b) shows the effect of zooplankton grazing on the
phytoplankton chlorophyll "a" concentration in segment #9.
The time period extends from March to December 1968 with
an additional 60 days to show the general trend after
December 1968.  As indicated in Figure 34(b) with extensive
zooplankton grazing, ohytoplankton populations are almost
completely depleted in September and maximum concentrations
of only 60 yg/1 are computed as compared to observed values
of about 130 yg/1.  It is interesting to note that the sharp
drop in zooplankton at day 90 is the effect of flow transient
at that time and as a consequence phytoplankton populations
remain high since the predators are flushed past the segment.

Figure 34 (a) with no zooplankton grazing is considerably
closer to the observed data range as shown.  The transient
drop around September 1968 is due to a flow increase as shown
in Figure 31.

The effect of zooplankton predation on the nutrient forms is
interesting and is shown in Figure 35.  As shown, the ammonia
nitrogen is quite insensitive to the zooplankton effect under
the conditions run.  The drop in ammonia concentration at day
90 and day 180 is due to flow transients during that time
(see Figure 31).  With the cycling of organic nitrogen to
ammonia nitrogen there is a sufficient feed forward to the
ammonia system even when the phytoplankton population is high.

The effect on the nitrate concentration is marked however as
shown in the top of Figure 35.  Under extensive zooplankton
                          159

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                                        (a)
            140

            120


            100
MgCHLOROPHYLL
       I     60

            60

            40

            20
                                               RANGE OF OBSERVED DATA
                  30  60   90  120   150  180  210 240  270 300  330 360
              I  March April   May I June I July  I Aug
                                         Sept  Oct   Nov  Dec  Jan
                                                                Feb
                                       DAYS
                                       1968
                                      (b)
           100

            80

            60
ig CHLOROPHYLL
       I     40

            20

             0
                                           ZOOPLANKTON
1.0


.8


.6


.4


.2
                                                                      mg CARBON
                                                                           I
              0   30  60   90  120   150  180  210 240  270 300  330 360
               March I April I May I June I July I Aug I Sept  I Oct  I Nov I  Dec  I Jan I  Feb  I

                                       DAYS
                                       1968
         Fig.  34  Effect of  Zooplankton  Grazinq on Phvtoplankton
                   in  Segment  #9    a) No  Zooolankton Grazinq
                   b)  Zooplankton Grazinq at 0.42  1/roq Carb-Dav.
                   1968  Flow  Reqime
                                      160

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         3.0
         2.5

 NITRATE 20
    N
  (mg/l)   |5
SEGMENT
  **9     l.O
         0.5
         2.5

AMMONIA 2.0
    N
  (mg/l)  1.5
SEGMENT
         1.0

         1.5
                                     (a)
                                                  NO ZOOPLANKTON
                                                          EXTENSIVE ZOOPLANKTON
                                                                GRAZING
           0   30  60  90   120  150  180  210  240  270  300 330 360
                                  DAYS
                                     (b)
                                              EXTENSIVE ZOOPLANKTON GRAZING
                                              NO ZOOPLANKTON
           0   30  60  90   120  150  180  210  240  270 300 230  360
                                   DAYS
      Fig  35  Effect of  Zooplankton  Grazinq on Nitrate Nitroqen  (ton)
               in Segment 19 and Ammonia Nitrogen  (bottom) in Segment #6,
               1968 Flow  Regime
                                   161

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grazing, the utilization of nitrate is decreased and nitrate
approximately behaves as a conservative variable responding
only to changes in the flow regime while being fed by the ni-
trification system.  As a consequence, NO- values build up
to 3 mg/1 under extensive zooplankton grazing - considerably
greater than observed (see Figure 33b).   With no zooplankton
grazing, phytoplankton populations increase and therefore
utilize nitrate nitrogen and reduce the values to the 1 -
1.5 mg/1 level.  This is the range observed during 1968. All
of these runs indicate that even with minimal grazing rates
of the zooplankton on the phytoplankton, the phytoplankton
population is never computed to grow to the levels observed
in 1968.  It was concluded therefore that the predatory effect
of zooplankton on the phytoplankton was minimal at least for
the upper end of the estuary below Washington D.C.

A sensitivity run was also made to indicate the effects of
including the tidal bay segments.  The results are shown in
Figure 36.  Observed chlorophyll concentration in the main
channel appear to be significantly influenced by the growth
of phytoplankton in the shallower side-channel areas.  With
average depths of about 5 feet in some of the tidal bay seg-
ments, significant populations are computed to grow in these
areas.  These populations are then tidally exchanged with the
main channel flow and contribute to the population at that
location.  It is hypothesized then that the growth of phyto-
plankton in the shallow areas is significant  (even without
direct discharges to embayments) and can account for as much
as 40 yg/1 chlorophyll in the observed main channel data.

             Verification Analysis - 1969 Data

The parameters determined from the 1968 verification analysis
are listed in Table 13.  This set of parameters was then used
                           162

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  I60r
 140


,120


 100
Q.  80
O
IT
3  60
   40
   20

   0
         APPROX. RANGE
         OBSERVED DATA
                                NO TIDAL BAY SEGMENTS
                                MAIN CHANNEL ONLY
                     10        15       20       25
                       MILES  BELOW CHAIN BRIDGE
                                                     30
35
           Fiq. 36  Sensitivitv Run -  No  Tidal Bav Segments.
                    Aug.  1968 Profile  and 1968 Conditions
                                 163

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                       Table 13
           Parameters Used in Verifications
         of 1968 and 1969 Potomac River  Data
           Preliminary Phytoplankton Model
Program
 Name              Description
KIT   Saturated Growth Rate-Phyto.
IS    Saturating Light Intensity
KMN   Michaelis1Constant-Nitrogen
KMP   Michaelis1   "   Phosphorous
KMPL  Michaelis1   "   Phyto.Chlor
K2T   Phyto. Endogeneous
             Respiration Rate
CCHL  Carbon-Chlorophyll Ratio
NCHL  Nitrogen-Chlor. Ratio
PCHL  Phos.-Chlor. Ratio
K34T  Org.N-NH3 Hydrolysis Rate
K33   Decay of Organic Nit.
K45T
K67T
      NH3-N03 Nitrification
      Org P-Inorg.P Conversion
                            Rate
K77   Decay of Org. P.
K66   Decay of Inorg. P.
K11T  BOD Decay Coef.
    Unit         Value
I/day- °C         0.1
ly/day          300.0
mg-N/1             .025
mg-P/1             .005
yg-Chlor/1       60.0
Rate I/day-°C      .005

mg Carb/yg Chlor   .05
mg Nit/yg Chlor    .01
mg P/yg Chlor      .001
l/day-°C           .007
I/day             0.10
l/day-°C           .01
l/day-°C           .007

I/day             0.10
I/day             0.10
l/day-°C          0.01
                           164

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to verify independently the 1969 data under the flow and
temperature regime showed in Figure 31.  As shown in Figure
31, fresh water inflow conditions were generally unsteady
throughout 1969.  There was a gradual  decrease in the flow
from April to mid-July at which point the flow increased
markedly.

Figures 37 and 38 show the comparison between the computed
output and some observed data for the period at the end of
the gradual decline in river inflow, from June 30, 1969 to
July 15, 1969.  Three data surveys were conducted during
this period.  It should be recalled that the computed output
for 1969 was generated directly using the parameters from
the 1968 runs; the parameters were not adjusted for the 1969
runs.  Only flow, temperature and incoming concentrations
into segment #1 based on observed 1969 data were used in
the 1969 computations.  It should be noted, however, that
boundary conditions at segment #1 do influence results for
about the first 15 miles of the estuary.  Incoming chloro-
phyll concentrations were not measured during 1969; the
first station for which data were available was at Key
Bridge, Mile 3.3.  Boundary chlorophyll values were inputted
to approximately duplicate the order of magnitude of observed
data at that station.  The values ranged from 4 yg/1 in the
winter to a high of 55 yg/1 in September 1969.

Figure 37 shows the comparison between chlorophyll values
as observed and the computed values for the beginning and
end of the survey period.  The June 30 values varied mark-
edly especially in the vicinity of Mile 15 where a low value
of 30 yg/1 was observed, an apparently abnormal value for
this location at that particular time of year.  The general
shape of the spatial profile as computed is good and approxi-
mately reproduces the spatial behavior of the observed data.
                           165

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  160

  140




 1 100


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   60
   40
   20
               COMPUTED
                 JUNE 30
                 JULY I!
                      10       15        20      25       30       35
                               MILES BELOW CHAIN BRIDGE
                                                                            40
                                                    • = JUNE 30,1969
                                                    + = JULY 8
                                                    X= JULY 15
o>
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                                          COMPUTED
                                              UNE 30
                                             JULY 15
                      10        15       20       25        30
                              MILES BELOW CHAIN BRIDGE
                                                                   35
                                                                           40
     Fig.  37   Soatial  Profile  Comoarison  of Observed 1969  Data
               and Computed Values for Chloronhvll  a  (ton)  and
               Total  Kjeldahl Nitroqen  (bottom)
                                  166

-------
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                                    -JUNE ^COMPUTED
10        15       20       25
    MILES BELOW CHAIN BRIDGE  '
                                                        30
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                               • =JUNE 30,1969
                               +=JULY 8
                                     15
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30
35
                                                    40
       Fig.  38   Spatial  Profile Comnarison of Observed 1969  Data
                 and Computed Values  for Total  Phosnhorous  (top)
                 and Nitrate Nitroqen (bottom)
                                   167

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Peak values as computed during the period tend to be higher
than observed although the average of the two computed lines
of June 30 and July 15 is reasonably close to the average of
the observed data and differs in the peak region by about
20 yg/1 and by 5 - 10 yg/1 downstream of the peak region.

The total Kjeldahl nitrogen verification is also shown in
Figure 37 and agreement is very good with the exception of
the July 8 survey where there was no observed increase in
nitrogen in the vicinity of the major discharges.  There
is no readily apparent explanation for this discrepency.

Figure 38 shows the comparison between the spatial profile
of total phosphorous  (as PO,) as observed during the surveys
of June 30, July 8 and July 15, 1969 and the computed pro-
files for the time period bracketing those surveys.  As
shown the agreement is quite good although in the downstream
direction, computed values decline more rapidly than ob-
served values.  This is probably a consequence of the simple
first order adsorption kinetics that were used.  A second
order assumption as used by Jaworski et al, 1971, would give
better results.  Computed values in general are within 0.5 -
1.0 mg/1 of observed values of PO,.

The nitrate profiles are also shown in Figure 38.  The agree-
ment is good for the July 8 and July 15 surveys.  Agreement
was not obtained for the June 30 survey.  During that survey
the observed data showed a considerable downstream shift in
the observed nitrate data which was not duplicated by the
computed values.  This could possibly be due to low dissolved
oxygen values which would delay the onset of nitrification
but DO data were not available to confirm this hypothesis.
There is a more rapid decline in computed values of nitrate
nitrogen in the vicinity of Mile 20 than was observed.  This
                          168

-------
discrepancy may be due to the simple preference structure
used in the model.  A more detailed analysis of phytoplankton
preference for differing forms of nitrogen appears to be war-
ranted to provide better agreement.

In summary, the spatial agreement between observed and computed
data for 1969 conditions is good.  The general shapes of the
spatial profiles are obtained and approximate quantitative
agreement is obtained.  Several areas remain to be explored,
notably the model structure of the different nitrogen forms
where anomalous results were occasionally obtained.

Figures 39 and 40 show comparisons between the temporal vari-
ation in chlorophyll a at four stations.  As shown in Figure
39, the observed data are scattered but with a general peak
in July 1969.  The order and timing of this peak for both
Mile 12.1 and Mile 18.3 is properly duplicated by the model.
The rapid drop in chlorophyll concentration at the end of
July is attributed to an increase in fresh water inflow
(see Figure 31) at that time.  The decrease is also success-
fully duplicated by the model.  The model approximates the
subsequent fall bloom of phytoplankton at Mile 18.3 but at
Mile 12.1 the model calculations are somewhat higher than
the observed data.

Figure 40 shows the comparison at two stations further down-
stream.  At both stations a winter bloom of phytoplankton
occurred which was not modeled in this work.  At Hallowing
Point, Mile 26.9, the spring growth pattern and subsequent
decrease is adequately modeled.  In the fall of 1969 at
Hallowing Point, however, data indicated a significant in-
crease in phytoplankton  (maximum levels of 445 yg/1 chloro-
phyll) .  This fall bloom was not duplicated by the phyto-
plankton model as constructed.  It is not clear from the
                          169

-------
                                                  WOODROW WILSON BRIDGE
                                                      Ml. 12.1
                          [SEGMENT 7-
                  COMPUTED
                          'SEGMENT 6
   I60r


   140


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   100


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      	-  •     • —-^<
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g    I  Jon.  |  Feb.  | Mar. |   Apr. |   May |  June  |  July |  Aug. |   Sept. |   Oct. |  Nov. |  Dec. |
 -  I60r                                I969/^
                                                              •        •
                                                          i	i	i
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                                                    PISCATAWAY CREEK
                                                    Ml. 18.3
                          COMPUTED  •
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      0    30    60    90   120   150   180    210   240   270   300   330   360
      I  Jan.  I  Feb.  I  Mar.  I  Apr. I  May I  June  I  July  I  Aug.  I Sript.  I  Oct. I  Nov. I  Dec. I
                                       1969
   Fig.  39  Temporal  Comparison  of Observed  1969 Data and
             Computed  Values for  Stations at  Miles 12.1 and  18.3
                                  170

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   160
01
X
CL
o
rr
o
o
           30    60    90   120   150    180   210  240   270   300   330  360

                                   TIME OF YEAR
        Jon. I  Feb. | Mar.  | April |  May  |june | July   I Aug. | Sept.  | Oct.  | Nov.  | Dec.
   160
en
X
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                                                          POSSUM  POINT

                                                             MILE 38,0
            30    60    90   120   150    180   210   240   270   300   330  360
                                   TIME OF  YEAR
         Jan. I Feb. I  Mar.  I April I  May I.June I July  I Aug. I  Sept. I Oct.  I Nov.  I Dec.



    Fig.  40  Temporal Comparison of Observed 1969  Data and

              Computed Values for Stations  at Miles 26.9  and  38.0-'
                                     171

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data as presented whether the high values in the fall of
1969 at Mile 26.9 are a surface phenomenon and therefore
beyond the scope of the model formulation or whether the
high values extended throughout the water column.

For Mile Point 38, Possum Point, similar comments can be
made.  The apparent winter bloom of phytoplankton is not
captured by the model since the growth parameters for the
phytoplankton biomass reflect a warm water population.
The model does reasonably well until about August 1969
after which a wide scatter of chlorophyll values was ob-
served.  Peak values during August to October are not
obtained by the model although the model does approximate
conditions again during November and December.

In general, the phytoplankton model as formulated herein,
provided a reasonable approximation to 1969 conditions us-
ing a 1968 "tuning" of the model.  As an independent check,
recognizing that no changes were made to the model from the
results of the 1968 analysis, the verification for 1969 pro-
vides an added measure of credibility to the overall model
structure.  Overall spatial profiles during July 1969
verified well to about 40 miles downstream from Chain Bridge,
Dynamic variability in phytoplankton was verified well
throughout the first 20 miles and then only approximately
for the remaining 20 miles.  Transient blooms in the late
winter and early fall of 1969 were not duplicated by the
model.

                           Model Application

In order to illustrate the application of the preliminary
model, two simulations were prepared.  It should be stressed
that this 'model is largely explanatory in nature and the
                               172

-------
simulations discussed below are not to be considered as
quantitative projections.  At best, the simulations in-
dicate general qualitative trends.  Both simulations used
a 90% removal policy of present raw waste loads.  For the
first simulation the 1969 flow regime was used; untreated
nitrogen, phosphorous and carbonaceous waste loads were re-
duced by 90%; and incoming boundary values into segment #1
were the same as those used in the 1969 verification analysis.
For the second simulation, median flow conditions were used
with reduced waste inputs as in the first simulation.  In
addition several boundary conditions on chlorophyll were
explored in the median flow simulation.  For both simula-
tions it was assumed that all organic and ammonia nitrogen
was converted to nitrate nitrogen at the treatment plants
and all phosphorous residual load is    inorganic phosphorous.
The summary of the waste loads is given in Table 14.


                      Table 14
            Direct Discharge Waste Loads

                              Waste Input       Waste Input
                           for Verification    for Simulation
                                Ibs/day           Ibs/day
Total Nitrogen (N)               46,500             7,070
Total Phosphorous (P)            20,300             2,620
Carbonaceous BOD                151,200            10,000

As shown, the 90% removal of the untreated waste loads re-
sults in about a 15% residual discharge of the 1968-1969
discharged loads.  Although the residual loads used in the
simulations are in excess of the allocations established for
the Potomac  (Jaworski et al, 1971) the level is considered a
practical achievable level on a continued sustained basis.
                           173

-------
Figure 41 shows the spatial chlorophyll profile for June 30 -
July 15, 1969 and the 1969 simulation with 90% reduction of
nutrient waste discharge.  As indicated, even after waste
reduction, maximum concentration of phytoplankton chlorophyll
exceed 100 yg/1 or four times the objective of 25 yg/1.  The
effect of the removal program is significant however from
about Mile 25 to Mile 40.  It should be recalled that no dis-
tributed sources of nutrient were included which would increase
the nutrients available for growth.

Figures 42-49 show space-time computer generated contour
plots from the 1969 simulations.  As indicated in Figure 42
for almost two months, chlorophyll concentrations are greater
than 25 yg/1 due in part to the incoming concentration as
given at segment #1.  Figure 43, a plot of the computed
Michaelis expression N /(K   + N ) as in Eq. (112) indicates
                      p   mp    p
that phosphorous limits the growth and essentially results
in the decrease of the population from its peak value during
July.  The flow transients create the "waves" in the contour
plot.  As shown in Figure 43, the estuary from about segment
#12 on, is increasingly limited by phosphorous almost uni-
formly throughout the year.  Figure 44 shows the accompanying
inorganic phosphorous levels and indicates the large area of
the estuary at which phosphorous is at the Michaelis constant
concentration (see Table 13).  Note in Figure 44 the lack of
significant temporal dependence of the phosphorous.  Also,
concentrations of inorganic phosphorous in the vicinity of
Washington D.C. range from .05 to .1 mg/1 P significantly
above the Michaelis constant indicating that in that immediate
area, phosphorous is not limiting.  This is also indicated in
Figure 43.  It should also be recalled that other sources of
nutrients from urban and suburban runoff were not included.

Figure 45 is a contour plot of the saturated growth   rate of
                             174

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                      19.69  SIMULATION
                  90% REMOVAL  OF NITROGEN
                 90% REMOVAL OF PHOSPHOROUS
                                       COMPUTED BEFORE
                            COMPUTED
                              AFTER
                     OBJECTIVE
                       15      20     25      30
                   MILES BELOW  CHAIN BRIDGE
                                     	= June 30 Computed
                                     	= July 15 Computed
Fig.  41   1969 Simulation  of Chlorophyll.
         Solid Line,  July 15 Profile.
Dashed  Line, June 30:
Comnare to Fiq. 37
                          175

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-------
the phytoplankton with the light limitation.  At the beginning
and end of the year, growth rates are less than about .01/day
throughout the length of the estuary.  During the peak grow-
ing season, growth rates reached maximum levels of .2 - ,3/day.
The interaction of phytoplankton population and light penetra-
tion is evident in the vicinity of segments #7 - #10 during
the peak growth season.  Figure 46 shows the net growth rate -
growth versus death and predation.  As indicated, there is a
large region in space and time where the net growth rate is
positive but it should be recalled that the transport structure
will substantially modify the net growth rate due to "wash out"
effects.

Figure 47 displays the total phosphorous surface.  As shown,
total phosphorous downstream of segment #10 is temporally con-
stant and uniformly decreases spatially.  In the vicinity of
Washington D.C. total phosphorous concentrations range from
.3 - 0.5 mg/1 PO.-P even with 90% removal at the plant.

Ammonia nitrogen and nitrate nitrogen are shown in Figures 48
and 49.  The shape of these surfaces can be compared to the
phosphorous surfaces of Figures 44 and 47.  The comparison
indicates the more complex interactions and structure of the
ammonia and nitrate nitrogen.  Significant temporal and spatial
variations occur reflecting the recycling of the different ni-
trogen forms.  Nitrogen in all regions is above limiting con-
centrations .

Flows for the median flow simulation ranged from 13,000 cfs
during the winter to a high of 20,500 cfs in the spring, de-
creasing to 4,200 cfs in the later summer and then increasing
again to 9,500 cfs in December.  Incoming boundary concentra-
tions as noted previously have an important relative impact
on the estuary especially after waste loads are reduced by
                            184

-------
treatment.  For the nutrients, median concentrations as given
in Jaworski et al,  1971, were used.  Kjehldahl nitrogen of
0.95 mg/1 and nitrate nitrogen levels ranging from 0.1 to 0.9
mg/1 were inputted as boundary values.   Organic phosphorous
of 0.05 mg/1 and an inorganic phosphorous concentration of
0.05 mg/1 were also inputted as constant values throughout
the year into segment #1.  The effect of incoming phytoplankton
chlorophyll was examined under median flows for two cases:
a) chlorophyll a concentrations of 1 yg/1 entering the estuary
from up-river and b)  chlorophyll a boundary concentration of
25 yg/1, the promulgated EPA criterion.

Figure 50 shows the temporal variation in chlorophyll for the
main channel segment #9 and its associated tidal embayment,
segment #28, Piscataway embayment, using median flows.  For
the case of 1 yg/1 chlorophyll entering the estuary (Figure
50a), it is seen that the effect of the reduced waste discharge
on the main channel is minor and results in a late summer in-
crease to about 14 yg/1 chlorophyll.  In the embayment section,
however, chlorophyll concentrations rise to about 55 yg/1
(twice the objective) for a period of about 120 days.  It
should be noted however that the level of 50 yg/1 is consider-
ably less than the concentrations of 150 - 200 yg/1 before
any nutrient reduction.  The incoming concentration of 1 yg/1
chlorophyll is unrealistic however and one would normally ex-
pect higher incoming concentrations.

Figure 50b shows the effect of constraining the boundary
chlorophyll concentration at a level equal to the objective
of 25 yg/1.  As indicated, main channel chlorophyll levels
rise to over 50 yg/1 in August and in the embayment, concen-
trations rise to over 70 yg/1 again for a period of about
4 months.
                               185

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  160


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                       MEDIAN  FLOW SIMULATION



                       90% REMOVAL OF NITROGEN

                      90%  REMOVAL OF PHOSPHOROUS
                                    +

                 25  fJLg/\ CHLOROPHYLL ENTERING ESTUARY
                                          COMPUTED-JULY 15
                         -OBJECTIVE
     0
10      15      20      25      30

     MILES  BELOW CHAIN BRIDGE
                                                         35
40
               Fig. 50  Temporal  Variation in Chloroohvll a at

               Segments #9 and #28, Median Flow Simulation.

               a) luq/1 Chloronhvll Boundarv  b)  25ug/l  Chloroohvll

               Boundary

                                186

-------
Figure 51 is a longitudinal profile for July 15 for the median
flow simulation and a 25 yg/1 chlorophyll boundary condition.
For about a 25 mile region of the estuary, the concentration
of phytoplankton chlorophyll exceeds the objective and rises
to a maximum value of almost 60 yg/1.  Although this is above
the objective, it is nevertheless a reduction of about 60%
from maximum chlorophyll concentrations before any nutrient
removals.

These simulations permit the following    general observations,
Achievement of the objective of 25 yg/1 chlorophyll in the
estuary may not be possible, primarily because of the effect
of discharges from the Upper Potomac into the estuary.  Even
under a median flow regime  (4,000 cfs during the growing
season), maximum concentrations of 50 yg/1 chlorophyll are
calculated for the main channel and 75 yg/1 in some embayments,
It should also be noted that additional sources of nutrients
such as local drainage and urban runoff have not been included
in the model.  Therefore, even if reductions greater than the
90% used in these applications could be achieved, such reduc-
tions would be probably offset by other distributed nutrient
sources.  On the basis of the preliminary model, then, state-
of-the-art nutrient reduction (90% or better) in the Potomac
may provide reductions in average chlorophyll levels by about
60% under median flow conditions.  Again, the simulations
should be considered as indicative of general trends only and
not as a predictive certainty.

                             Summary

A non-linear dynamic model of phytoplankton chlorophyll in
the Upper Potomac estuary verifies the observed 1968 and 1969
phytoplankton and nutrient temporal and spatial trends and
approximately verifies the associated concentration levels.
                               187

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x 40
o

20





- (0)
l/ig/l CHLOROPHYLL ENTERING ESTUARY
—

—
Segment No. 28
(Piscataway \
Embayment) \. . 	 	 	
/ -•»
/ \
— / \
/ \
Objective / \
/ \
/ Segment No.9-\ \
/ ^ 	 ^^ \
---1 	 !. 1 	 ^ 1 ^4-^T 1 1 ^T 	 -4 	
i
p














0 30 60 90 120 150 180 210 240 270 300 330 360

Jan. Feb. 1 Mar. Apr. May | June July Aug. Sept. | Oct. | Nov. Dec.
TIME OF YEAR
120

^ 100
•*•
i
IJ 80
>
X
Q_
o 60
o
_j
5 40

20
Q
_ (b)
25ug/l CHLOROPHYLL ENTERING ESTUARY
— Segment No. 28
(Piscafaway V
Embayment) \
Y

^ ""*" ~~ ~~~ * "^ -^^
*•"** '^
s Segment No.9-i \
/ ^. 	 * 	 ^N^^ \
/ JT ^^^^^ \
- ^/ ./ ^^^^ ^
s, 	 •*" 	 ' Objective ^^v^
—
II 1 1 1 1 1















1
0 30 60 90 120 150 180 210 240 270 300 330 360

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. | Nov. Dec.
TIME OF YEAR
Fig. 51  Median Flow Simulation Profile of Chlorophyll
                        July 15
                         188

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The data for 1968 were used to tune the model and the 1969
data were then used for independent verifications using coef-
ficients from 1968 data and changing only flow, temperature
and incoming boundary conditions for the 1969 verification.
Late fall blooms of phytoplankton were not completely verified
and a late winter bloom was not verified.  Phytoplankton
species discrimination is not incorporated in the model.

The transport through the estuary assumes an important role
in the phytoplankton dynamics of the upper Potomac Estuary.
The nutrients and phytoplankton associated with the incoming
river flow are particularly important.  Unlike other problem
contexts such as dissolved oxygen,  critical conditions for
phytoplankton growth may occur during non-drought flow con-
ditions.  This is due primarily to the addition of nutrients
during high flows.  The verification analyses indicate that,
at present, phosphorous and nitrogen are probably sorbed
from the estuary and enter the estuarine sediments.  The
preliminary model does not incorporate any recycle of nutrients
from the sediments although the model does indicate that phos-
phorous and nitrogen are not conserved in the water column and
losses to the sediment can be considerable.  These losses could
alter the effect of incoming river concentrations.

Simulations with the preliminary model indicate that under non-
drought flows and a 90% reduction of untreated raw nutrient
loads, chlorophyll concentrations in the main channel may rise
to about 50 yg/1 with embayment values over 70 yg/1.  These
concentrations are considerably above the objective of 25 yg/1
but are about 60% less than present levels.  The simulations
are intended only to show possible trends under future loading
conditions and are not to be interpreted in any detailed quanti-
tative fashion.  Models of the type presented herein require
additional evaluation and testing on a continuing basis.
                             189

-------
Further refinement of the model is therefore necessary before
any definitive statements can be made of the effects of nu-
trient reduction on phytoplankton chlorophyll in the Upper
Potomac Estuary.
                          190

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                      SECTION VII

                    ACKNOWLEDGMENTS

The authors are pleased to acknowledge the guidance and
support of Walter Sanders of the Southeast Water Laboratory,
Environmental Protection Agency , in this research.

The Participation of John L. Mancini, Hydroscience, Inc.,
is also gratefully acknowledged.

The application of the nitrification model to the Delaware
Estuary was accomplished through a contractual arrangement
between the Delaware River Basin Commission and Hydroscience
Inc.

The assistance of Gerald Cox and Jack Hodges of the Department
of Water Resources and Harold Chadwick of the Department of
Fish and Game, State of California is also gratefully acknowl-
edges .

Richard Winfield, Research Engineer, Manhattan College, as-
sisted significantly in the verification and application of
the eutrophication model of the Potomac Estuary and his input
is specifically acknowledged.

Finally the authors are pleased to acknowledge the effect and
forebearance of Mrs. Berenice Maguire in the preparation of
this report.
                            191

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                     SECTION  VIII
                       REFERENCES
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    Quality Conditions and  Investigations  in the Upper Potomac
    River Tidal System" Tech.  Report No.  41, Ches.  Tech.
    Support Lab., FWQA, Annapolis,  Md., May 1970, V.  Chapters.

 2.  Adams,  J.A.,  Steele, J.H.  "Some Contemporary Studies  in
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    Calanus Finmarchlcus p. 19-35,  H. Barnes, Ed.,  G. Allen
    and Unwin Ltd., London, 1966.

 3.  Allison, F.E. "The Enigma  of Soil Nitrogen Balance Sheets"
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 4.  Anon, "Effects of Polluting  Discharges on the Thames  Estu-
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 5.  Anon, Department of Water  Resources,  State of California
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 6.  Anon, "Nitrification in the  Delaware Estuary" Prepared by
    Hydroscience Inc., Westwood, N.J. for  Delaware River Basin
    Comm.,  Trenton, N.J., June 1969, 43 pp + Append.

 7.  Anon, "Eutrophication of Surface Waters - Lake Tahoe"
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 9.  Azad, H.S., Borchardt,  J.A.  "A Method  for Predicting the
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10.  Bishop, J.W.  "Respiration  Rates of Migrating Zooplankton
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    58-62.
                           193

-------
11. Bormann, F.H. et al "Nutrient Loss Accelerated by Clear-
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12. Burns, C.W., Rigler, F.H., "Comparison of Filtering Rates
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14. Buswell et al, "Study of the Nitrification Phase of the
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16. Cole, C.R. "A Look at Simulation through a Study on
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18. Conover, R.J. "Oceanography of Long Island Sound, 1952-1954
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19. Conover, R.J. "Assimilation of Organic Matter by Zooplankton"
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20. Davidson, R.S., Clymer, A.B. "The Desirability and Appli-
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21. Davis, H.T. "Introduction to Nonlinear Differential and
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22. Delwiche, C.C. "Biological Transformations of Nitrogen
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23. Di Toro, O'Connor, D.J. and Thomann, R.V. "A Dynamic
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                            194

-------
24.  Droop, M.R., "Physiology and Biochemistry of Algae"
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25.  Dugdale, R.C. "Nutrient Limitation in the Sea: Dynamics,
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    12(4)  685-95.

26.  Eppley, R.W., Rogers, J.N.,  McCarthy, J.J. "Half Satura-
    tion Constants for Uptake of Nitrate and Ammonium by
    Marine Phytoplankton" Limnol. Oceanog.,  1969, 14(6),
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27.  Feth,  J.H. "Nitrogen Compounds in Natural Water - A
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    41-58.

28.  Fogg,  G.E. "Algal Cultures  and Phytoplankton Ecology"
    p. 20, University of Wisconsin i'Prcss, Madison, Wis. 1965.

29.  Gerloff, Skoog,  "Nitrogen as a Limiting  Factor for the
    Growth of Microcystis Aerugmosa in Southern Wisconsin
    Lakes" Ecology,  1957, 38,  556-61.

30.  Hamming, R.W. "Numerical Methods for Scientists and
    Engineers" p. 194-210, McGraw Hill, N.Y. 1962.

31.  Hetling, L.J. and O'Connell, R.L. "An 0~ Balance for the
    Potomac Estuary" Unpublished Manuscript, Chesapeake
    Field Sta., Ches. Bay-Sus.  River Basin Project, FWPCA,
    Feb. 1968.

32.  Hutchinson, G.E. "A Treatise on Limnology, Vol. I" J.
    Wiley & Sons, Inc., New York, N.Y. 1957, 1015 pp + xiv.

33.  Hutchinson, G.E. "A Treatise on Limnology Vol. II.
    Introduction to Lake Biology and the Limnoplankton"
    p. 306-54, Wiley, New York,  1967.

34.  Hutchinson, G.E. and Viets,  F.G. "Nitrogen Enrichment of
    Surface Water by Absorption of Ammonia Volatilized from
    Cattle Feedlots" Science,  Vol. 166, Oct. 24, 1969,
    pp. 514-515.

35.  Jaworski, N.A.,  Lear, D.W.  and Aalto, J.A. "A Technical
    Assessment of Current Water Quality Conditions and Factors
    Affecting Water Quality in the Upper Potomac Estuary" Tech.
    Report No. 5 Ches. Tech. Sup. Lab, FWQA, Annapolis, March,
        ~
                             195

-------
36. Jaworski, N.A. et al "Nutrients in the Potomac River
    Basin" Tech. Kept. No. 9, Ches. Tech. Sup. Lab., FWPCA,
    Dept. of Int., May 1969, 40 pp.

37. Jaworski, N.A. "Water Quality and Wastewater Loadings,
    Upper Potomac Estuary During 1969" Tech. Rept. No. 27,
    Ches. Tech. Sup. Lab., FWQA, Annapolis, Md., Nov. 1969,
    VII Chapters.

38. Jaworski, N.A., Clark, L.S. and Feigner, K.D. "A Water
    Resource Water Supply Study of the Potomac Estuary"
    Tech. Rept. No. 35, Ches. Tech. Sup. Lab, EPA, Annapolis,
    Md., April 1971, XII Chapters.

39. Jaworski, N.A. and Clark, L.J. "Physical Data, Potomac
    River Tidal System Including Mathematical Model Segmenta-
    tion" Tech. Rept. No. 43, Ches. Tech. Sup. Lab., FWQA,
    Annapolis,Md.undated,unpaginated.

40. Ketchum, B.H. "The Absorption of Phosphate and Nitrate
    by Illuminated Cultures  of Nitzschia Closterium" Am. J.
    Botany,  June 1939, 26.

41. Knowles, G., Downing, A.L. and Barrett, M.J. "Determina-
    tion of Kinetic Constants for Nitrifying Bacteria in Mixed
    Culture, with the Aid of an Electronic Computer" J. Gen.
    Microbiol., Great Britain, 1965, 38, 263-278.

42. Kuentzel, L.E. "Bacteria, Carbon Dioxide and Algal Blooms"
    J. Water Pol. Control Fed., Oct. 1969, 41 (10).

43. Lawrence, A.W. and McCarty, P.L. "Unified Basis for
    Biological Treatment Design and Operation" J. SEP, ASCE,
    Vol. 96, No. SA3, June 1970, pp. 757-778.

44. Levy, H., Baggott, E.A.  "Numerical Solutions of Differenti-
    al Equations" p. 91-110, Dover, N.Y. 1950.

45. Lotka, A.J. "Elements of Mathematical Biology" pp. 88-94,
    Dover, N.Y. 1956.

46. Lund, J.W.G. "The Ecology of the Freshwater Phytoplankton"
    Biol. Rev, 1965, 40, 231-93.

47. Maclsaac, J.J., Dugsdale, R.C. "The Kinetics of Nitrate
    and Ammonia Uptake by Natural Populations of Marine Phyto-
    plankton" Deep Sea Res., 1969, 16, 415-22.

48. Martin,  J.H. "Phytoplankton Zooplankton Relationships in
    Narragansett Bay III" Limnol. Oceanog, 1968, 13  (1).
                             196

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49. McMahon.. J.W. ,  Rigler, F.H. "Feeding Rate of Daphnia
    Magna Straus in Different Foods Labeled with Radioactive
    Phosphorus" Limnol. Oceanog.,  1965, 10 (1) 105-13.


50. Monod, J. "Recherches sur la Croissance des Cultures
    Bacteriennes" Hermann, Paris,  1942.

51. Mullin, M.M. "Some Factors Affecting the Feeding of Marine
    Copepods of the Genus Calanus" Limnol. Oceanog., 1963, 8
    (2),  239-50.

52. Myers, J. "Algal Culture from Laboratory to Pilot Plant"
    Growth Characteristics of Algae in Relation to the Problem
    of Moss Culture, p~. 37-54,J.S. Burlew, Ed. , Carnegie
    Inst. of Wash.  D.C, Publ. 600, 1964.

53. O'Connor, D.J., Thomann, R.V., "Stream Modeling for Pollu-
    tion  Control" Proc. IBM Sci. Computing Symp. Environ. Sci.,
    Nov.  14-16, 1966, p. 269.

54. O'Connor, D.J.  "Water Quality Analysis of the Mohawk River-
    Barge Canal" Prepared in coop, with Hydroscience, Inc, for
    N.Y.  State Dept. of Health, July 1968.

55. O'Connor, D.J., St. John, J.P. and Di Toro, D.M. "Water
    Quality Analysis of the Delaware River Estuary, J..SED, ASCE,
    No.  SA6, Dec. 1969, pp 1225-1252.

56. O'Connor, D.J., Di Toro, D.M., Thomann, R.V., Mancini, J.L,
    "Phytoplankton Population Model of the Sacramento - San
    Joaquin Delta Bay" Tech. Rept. to  Dept. of Water Resources,
    State of Calif, by Hydroscience Inc., Westwood, N.J. 1971.

57. Oswald, W.J., Gotaas, H.B., Ludwig, H.F., Lynch, V.
    "Photosynthetic Oxygenation" Sewage Ind. Wastes, 1953, 25
    (6),  692.

58. Parker R.A. "Simulation of an Aquatic Ecosystem" Biometrics
    1968, 24(4) 803-22.

59. Pearson, E.A. "Kinetics of Biological Treatment" Advances
    in Water Quality Improvement,  pp.  381-394, E. Gloyna and
    W.W.Eckenfelder, Ed.,Univ. of Texas Press, Austin, 1968.

60. Phelps, E.B., Stream Sanitation, pp. 148-49, J. Wiley & Sons,
    New York, N.Y.  1944.

61. Raymont, J.E.G. "Plankton and Productivity in the Oceans"
    pp 93-466, Pergamon, N.Y. 1963.
                              197

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62. Riley, G.A. "Factors Controlling Phytoplankton Populations
    on Georges Bank" J. Marine Res., 1946, 6(1) 54-73.

63.  Riley, G.A. "A Theoretical Analysis of the Zooplankton
    Population of Georges Bank" J. Marine Res., 1947, 6(2),
    104-25.

64. Riley, G.A., Von Arx, R. "Theoretical Analysis of Seasonal
    Changes in the Phytoplankton of Husan Harbor, Korea,
    J. Marine Res.,  1949, 8 (1) 60-72.

65. Riley, G.A., Stommel, H.,  Bumpus, D.F. "Quantitative
    Ecology of the Plankton of the Western North Atlantic"
    Bull. Bingham Oceanog. Coll, 1949, 12(3)  1-169.

66. Riley, G.A. "Oceanography of Long Island Sound 1952-1954
    II. Physical Oceanography" Bull. Bingham Oceanog. Coll.
    1956, 15 15-46.

67. Riley, G.A. "The Sea" Theory of Food-Chain Relations in
    the Ocean, p. 438-63, M~iN. Hill,Ed.,Interscience, N.Y.
    1963.

68. Riley, G.A. "Mathematical Model of Regional Variations in
    Plankton" Limnol. Oceanog, 1965, 10  (Suppl) R202-R215.

69. Ryther, J.H. "Inhibitory Effects of Phytoplankton upon the
    Feeding of Daphnia Magna with Reference to Growth, Repro-
    duction and Survival" Ecology, 1954, 35,  522-33.

70. Ryther, J.H. "Phytosynthesis in the Ocean as a Function of
    Light Intensity" Limnol. Oceanog, 1956, 1, 61-70.

71. Sawyer, C.N. "Chemistry for Sanitary Engineers" McGraw-
    Hill Bk. Co., New York, 1960, 367 pp + Viii.

72. Shelef, G., Oswald, W.J. ,  McGauhey, P..H.  "Algal Reactor
    for Life Support Systems" J. SEP, ASCE, No. SA1, Feb.1970.

73. Small, L.F., Curl, H. Jr.  "The Relative Contribution of
    Particulate Chlorophyll to the Extinction of Light off
    the Coast of Oregon" Limnol. Oceanog., 1968, 13(1), 84.

74. Sorokin, C., Krauss, R.W.  "The Effects of Light Intensity
    on the Growth Rates of Green Algae" Plant Physiol. 1958,
    33, 109-13.

75. Sorokin, C., Krauss,R.W. "Effects of Temperatures and Illu-
    minance on Chlorella Growth Uncoupled from Cell Division"
    Plant Physiol, 1962, 37, 37-42.
                             198

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76. Spencer, C.P. "Studies on the Culture of a Marine Diatom"
    J. Marine Biol.  Assoc. U.K., 1954; quoted by Harvey, H.W.
    "The Chemistry and Fertility of Sea Waters" p. 94,
    Cambridge Univ.  Press, 1966.

77. Steele, J.H. "Plant Production on Fladen Ground" J.  Marine
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78. Steele, J.H. "A Study of Production in the Gulf of Mexico"
    J. Marine Res.,  1964, 22, 211-22.

79. Steele, J.H. "Primary Production in Aquatic Environments"
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80. Stiefel, E.L. "An Introduction to Numerical Mathematics"
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81. Stratton, F.E. and McCarty, P.L. "Prediction of Nitrifi-
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82. Strickland, J.D.H., "Chemical Oceanography" Production of
    Organic Matter in the Primary Stages of the Marine Food"
    Chain,   Vol. 1,  p. 503, J.P.Riley and G. Skivow, Eds.,
    Academic, New York, 1965.

83. Tamiya, H., Hase, E., Shibata, K, et al "Algal Culture
    from Laboratory to Pilot Plant" Kinetics of Growth of
    Chlorella with Special Reference to Its Dependence on
    Quantity of Available Light and on Temperature, pp 204-
    234, J.S. Burlew, Ed., Publ. 600, Carnegie Inst. of
    Washington D.C.  1964.

84. Thomann, R.V. "Mathematical Model for Dissolved Oxygen"
    J. Sanit. Eng. Div., Proc.  ASCE, Oct. 1963, 83, SA5, 1-30.

85. Thomann, R.V. "Systems Analysis and Water Quality
    Management", ESSC Pub. Co.,  Stamford, Conn., 1972.

86. Thomas, W.H., Dodson, A.N.  "Effects of Phosphate Concen-
    tration on Cell  Division Rates and Yield of a Tropical
    Oceanic Diatom"  Biol. Bull., 1968, 134  (1) 199-208.

87. Vollenweider, R.A. "Primary Production in Aquatic Envi-
    ronments" Calculation Models of Photosynthesis - Depth
    Curves and Some Implications Regarding Day Rate Estimates
    in Primary Production Measurements, pp 425-57,C.R.
    Goldman,Ed., Mem.Inst.Idrobiol, 18 Suppl, Univ. of
    Calif.  Press, Berkeley, 1965.
                          199

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88. Vollenweider,  R.A. "Scientific Fundamentals of the Eu-
    trophication of Lakes and Flowing Waters, with Particular
    Reference to Nitrogen and Phosphorus as Factors in
    Eutrophication" p. 117, Organization for Economic Co-
    operation and  Development Directorate for Scientific
    Affairs, Paris, France, 1968.

89. Vollenweider,  R.A. Ed., "Manual on Methods for Measuring
    Primary Production in Aquatic Environments" Ch. 2, p.4-24,
    Blackwell Scientific Publications, Oxford, England, 1969.

90. Wright, J.C. "The Limnology of Canyon Ferry Reservoir,
    I.  Phytoplankton-Zooplankton Relationships" Limnol.
    Oceanog. ,  1958, 3(2) 150-9.            "

91. Yentsch, C.S., Lew, R.W. "A Study of Photosynthetic Light
    Reactions"  J.  Marine Res, 1966, 24(3).
                             200

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                      SECTION IX
                     PUBLICATIONS
1. "The Management of Time Variable Stream and Estuarine
   Systems" Robert V. Thomann, Donald J. O'Connor and
   Dominic M. Di Toro, Chemical Engineering Progress
   Symposium Series, Vol. 64, Nov. 1968.

2. "Photosynthesis and Oxygen Balance in Streams" Donald
   J. O'Connor and Dominic M. Di Toro, Journal of the
   Sanitary Engineering Division, ASCE, Vol. 96, No. SA2,
   Proc. Paper 7240, April 1970, pp 547-571.

3. "Modeling of the Nitrogen and Algae Cycles in Estuaries"
   Robert V. Thomann, Donald J. O'Connor and Dominic M.
   Di Toro, Proc. of Fifth International Water Pollution
   Conf., San Francisco, Cal., 1970.

4. "A Water Quality Model of Chlorides in Great Lakes",
   Donald J. O'Connor and John A. Mueller, Journal of the
   Sanitary Engineering Division, ASCE, Nol. 96, No. SA4,
   Proc. Paper 7470, Aug. 1970, pp. 955-975.

5. "Recurrence Relations for First Order Sequential Reactions
   in Natural Waters" Dominic M. Di Toro, Water Resources
   Research, Vol. 8, No. 1, Feb. 1972.

6. "A Linear Ecologic Model - Nitrification in Estuaries"
   Donald J. O'Connor, Robert V. Thomann, Dominic M.
   Di Toro, Systems Approach to Water Management, American
   Elsevier Co., New York, March 1972.

7. "Effect of Longitudinal Dispersion on Dynamic Water
   Quality Response of Streams and Rivers" Robert V. Thomann,
   Water Resources Research, April 1973.
                         201

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SELECTED WATER
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
                                             1. Ri,,ur» Vo. ?
                                          w
 DYNAMIC WATER QUALITY FORECASnNG AND MANAGEMENT
 O'Connor,  Donald J., Thomann, Robert V., and Di  Toro,  Dominic
 Manhattan College, Bronx, New York, Civil Engineering Dept.
                                                                 5. A .
                                                                 8.  ~ -form  - Orga  .-atioit
                                                                    Kepo-'No.
                                          M.
                                                                    R800369
                                                                  R800369
                                                                      v  Kept*  and
                                                                      Tioti Cr-'tred
'2.  S- isorir  ^rgar  tion
U. S. Environmental Protection Agency      Final Report
     Environmental  Protection Agency report number,
     EPA-660/3-73-009,  August 1973.
 This report describes the formulation and  initial  verification of two modeling frame-
 works.   The first is directed toward an analysis of the impact of the carbonaceous and
 nitrogenous components and wastewater on the  dissolved oxygen resources of a natural
 water system.  The second modeling framework  concentrates on the interactions between
 the discharge of nutrient, both nitrogen and  phosphorus, and the biomass of the phyto-
 plankton and zooplankton populations which result,  as well as incorporating the overall
 impact on dissolved oxygen.  The models are formulated in terms of coupled differential
 equations which incorporate both the effect of  transport due to tidal motion and
 turbulence, and the kinetics which describe the biological and chemical transformations
 that can occur.  The modeling frameworks are  applied to the Delaware and Potomac
 estuaries in order to estimate the ability of such models to describe the water quality
 effects of carbon, nitrogen, and phosphorous  discharges.  The agreement achieved
 between observation and calculation indicate  that  the major features of the impact of
 wastewater components on eutrophication phenomena  can be successfully analyzed within
 the context of the models presented herein. (O'Connor-Manhattan College)
i?a. Descriptors *water quality, *Mathematical  Models,  *Computer Models, Water pollution,
 Cycling nutrients, Eutrophication,  Dispersion,  Mass Transfer, Nutrients, Oxygen  Demand,
 Photosynthesis, Simulation Analysis
I'b. Identifiers
                      05C,  05G,  06G
                        19.  Security Class.
                           'Repo: i

                         1.  Se rityC'  s.
                           (Pave)
                                         21.
                       No. of
                       Pages

                       Pi: -
Send To:
WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR
WASHINGTON. D. C. 2O24O
         Donald J. O'Connor
                         Manhattan College. Bronx. New York
                                                     «U.S. GOVERNMENT PRINTING OFFICE: 1973 546-311/98 1-3

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