United States
Environmental Protection
Agency
Environmental Research
Laboratory
Duluth MN 55804
EPA-600 3-80-065
July 1980
f.l-
Research and Development
Mathematical
Models of Water
Quality in Large Lakes
Part 2
Lake Erie
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development. U S Environmental
Protection Agency, have been grouped into nine series These nine broad cate- <
gories were established to facilitate further de>/elopme~iT and aopl cation of en-
vironmental technology El mmation of trad t onal grouping was consciously
planned to foster technology transfe' and a maximum interface in related fields
The nine series are
1 Env ronmental Health Effects Research
2 Env ronmental Protection Technology
3 Eco'ogical Research
4 Environmental Monitoring \
5 SoC'Oeconomic Environmental Studies
6 Scientific and Technical Assessment Reports (STAR)
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8 ' Special ' Reports
9 Miscellaneous Reports
This report has been assigned to the ECOLOGICAL RESEARCH series This series
describes research on the effects of pollution en humans, plant and animal spe-
cies, and materials Problems are assessed 'or their long- and short-term influ-
ences Investigations include format on transoort, and pathway studies to deter-
mine the fate o" pollutants and their effects This work provides the technical basis
for setting standards to minimize undesirable changes in living organisms in the
aguatic terrestrial, and atmospheric environments
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161
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EPA-600/3-80-065
July 1980
MATHEMATICAL MODELS OF WATER QUALITY IN LARGE LAKES
PART 2: LAKE ERIE
"by
Dominic M. Di Toro
John P. Connolly
Manhattan College
Environmental Engineering and Science Division
Bronx, New York 10^71
Grant No. R803030
Project Officer
William L. Richardson
Large Lakes Research Station
Environmental Research Laboratory - Duluth
Grosse lie, Michigan U8138
ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
DULUTH, MINNESOTA 5580U
- Envron-rpr,*.,, r x
F> *.>,, i, v:,"'"'1-'' r-rotection flm
ll-;-u:i V. ?_;i---iii/ " x.-'
230 Sa-t;/.^:.?
rM-~r"".y, L-"'-"-'''c'.rn Street
L"'^go, ij|mG;s 60604
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DISCLAIMER
This report has been reviewed "by the Large Lakes Research Station,
Grosse lie Laboratory, U.S. Environmental Protection Agency, and approved
for publication. Approval does not signify that the contents necessarily
reflect the views and policies of the U.S. Environmental Protection Agency,
nor does mention of trade names or commercial products constitute endorse-
ment or recommendation for use.
U,S. Environmental Protection Agency
11
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FOREWORD
The Great Lakes represent a vast, complex system of competing water
uses including a delicate, and interacting ecosystem. They comprise 80% of
the surface freshwater in North America and proved 45 million people in the
basin with almost unlimited water for drinking, industrial processing, and
recreation. Lucrative sport and commercial fisheries rely on these waters
as do the transport of tremendous quantities of raw and refined commercial
products and the disposal of residual industrial and municipal materials.
Optimal use of these water resources demands that a balance be
maintained between the economic welfare of the region and the health of the
ecosystem. To arrive at this balance a rational and quantitative
understanding of the interacting and competing components must be developed.
This report documents the results of a three year research project to
develop a water quality model for Lake Erie. Sufficient detail is
presented so that much of the methodology could be applied to other water
bodies. Also, it is intended to present results and background for Great
Lakes researchers and managers who have already used this research in the
decision making process. Specifically, these results have been used in
renegotiating the U.S.-Canadian Great Lakes Agreement and to decide on the
allowable phosphorus load to Lake Erie.
Appreciation is extended to scientific reviewers at the University of
Michigan, NOAA-Great Lakes Environmental Lab, and the Canadian Center for
Inland Waters. The report has also received extensive review by several
state and Canadian agencies.
William L. Richardson, P.E.
Environmental Scientist
ERL-D, Large Lakes Research Station
Grosse He, Michigan 48138
m
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ABSTRACT
This research was undertaken to develop and apply a mathematical model
of the water quality in large lakes, particularly Lake Huron and Saginaw Bay
(Part 1) and Lake Erie (Part 2).
A mathematical model was developed for analysis of the interactions be-
tween nutrient discharges to Lake Erie, the response of phytoplankton to
these discharges, and the dissolved oxygen depletion that occurs as a conse-
quence. Dissolved oxygen, phytoplankton chlorophyll for diatoms and non-
diatoms, zooplankton biomass, nutrient concentrations in available and un-
available forms and inorganic carbon are considered in the model. Extensive
water quality data for Lake Erie was analyzed and statistically reduced.
Comparison of data from 1970 and 1973-7^- to model calculations served for
calibration of the model. A verification computation was also performed for
1975* a year when no anoxia was observed.
Recent developments in phytoplankton growth and uptake kinetics are
included in this analysis. The methods of sedimentary geochemistry are
expanded to include an analysis of sediment oxygen demand within the frame-
work of mass balances. Projected effects of varying degrees of phosphorus
removal on dissolved oxygen, anoxic area, chlorophyll, transparency and
phosphorus concentration are presented.
This report was submitted in fulfillment of Grant No. R803030 by
Manhattan College under the sponsorship of the U.S. Environmental Protection
Agency. This report covers the project period March 26, 197^ to March 25»
1977-
IV
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CONTENTS
Disclaimer ii
Foreword iii
Abstract iv
Figures vi
Tables xiv
Acknowledgement xvii
1. Introduction 1
2. Summary 3
3. Recommendations ih
U. Description of Study Area, Mass Discharge Rates, Segmentation 16
5. Data Compilation and Reduction 31
6. Kinetics 52
T. Sediment-Water Interactions: Mass Transport and Kinetics . . 90
8. Mass Transport Calibration 102
9. Structure and Computational Details Il6
10. Calibration 128
11. Verification 169
12. Estimated Effects of Phosphorus Loading Reductions 183
References 203
Appendix 210
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FIGURES
Number
1. Lake Erie Calibration Results, 1970. Epilimnion Western Basin
(lefthand side); Central Basin (center); Eastern Basin
(righthand side); chlorophyll a_, yg/£, (top); C1!| shipboard
primary production mg C/m3/hr (middle); total phosphorus
mg/£ (bottom). Symbols: mean +_ standard deviation; lines
are the computations
2. Lake Erie Calibration Results, 1970 Epilimnion. Western Basin
(lefthand side), Central Basin (center), Eastern Basin
(righthand side), orthophosphorus (top); nitrate nitrogen
(middle), reactive silica (bottom)
3. Schematic Diagram of the Dissolved Oxygen Mass Balance Computa-
tion. The Epilimnion, Hypolimnion (for Central and Eastern
Basins), and surface Sediment segments as indicated. Direct-
ly observed concentrations and reaction rates are indicated
(*). Transport rates estimated using tracers are indicated
(T) 7
h. Lake Erie Central Basin Calibration for Dissolved Oxygen and
related variables, 1970. Epilimnion (lefthand side); Hypo-
limnion (righthand side), Dissolved Oxygen, mg/& (top); pH
(middle); chlorophyll a_ ug/5, (bottom). The Hypolimnion
chlorophyll a_ data is 1973-1974, as no 1970 data are avail-
able 8
5. Lake Erie Central Basin Calibration. Epilimnion (lefthand
side), Hypolimnion (righthand side), shipboard C11* primary
production (top). No Hypolimnion data available, BODs
(middle). This is 1967 data as no 1970 data are available. . 9
6. Comparison of 1970 calibration (lefthand side) and 1975 verifi-
cation (righthand side) Dissolved Oxygen (top), chlorophyll
ja (upper middle), orthophosphorus (lower middle), nitrate
nitrogen (bottom) 11
7. Observed Hypolimnion Mean Dissolved Oxygen vs. Minimum Observed
Hypolimnion Dissolved Oxygen (top). Predicted Mean Hypolim-
nion Dissolved Oxygen just prior to overturn vs. Lake Total
Phosphorus Loading (bottom) 13
VI
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FIGURES (cont'd)
Number
8. Lake Erie Model Segmentation of Western, Central and Eastern
Basins: Water Segments 1-6, Epilimnion (top) and Hypolim-
nion (middle); Sediment Segments 7-10 (bottom) ........ 18
9. Total Phosphorus Loading (lO3 kg/day_ to Lake Erie, 1970-1975-
Western Basin (top); Central Basin (middle); Eastern Basin
(bottom) ........................... 25
10. Orthophosphorus Loading (lO3 kg/day) to Lake Erie, 1970-1975- 26
11. Organic Nitrogen Loading (lO3 kg/day) to Lake Erie, 1970-1975 . 27
12. Inorganic Nitrogen Loading (101* kg/day) to Lake Erie, 1970-1975 28
13. Dissolved Silica Loading (lO1* kg/day) to Lake Erie, 1970-1975 . 29
ih. Comparison of chlorophyll &_ concentrations (ug/£) for the West-
ern Basin and Eastern Basin Epilimnion. Segment averages +_
standard deviations for each cruise are shown for 1970, 1973-
7U and 1975 data sets .................... 38
15. Comparison of chlorophyll a_ concentrations (ug/£) for the Cen-
tral Basin Epilimnion and Hypolimnion. No chlorophyll data
available for the hypolimnion during 1970 .......... 39
16. Total phosphorus concentrations (mg/£) for the Western Basin
and Eastern Basin epilimnion ................. kO
17. Total phosphorus concentrations (mg/£) for the Central Basin
epilimnion and hypolimnion
18. Soluble reactive phosphorus concentrations (mg/£) for the West-
ern Basin and Eastern Basin epilimnion. No data available
for the Eastern Basin during 1973-7^ .............
19. Soluble reactive phosphorus concentrations (mg/£) for the Cen-
tral Basin epilimnion and hypolimnion ............
20. Comparison of ammonia concentrations (mg/£ as N) for the West-
ern Basin and Eastern Basin epilimnion. No data available
for the Eastern Basin during 1975 ..............
21. Comparison of ammonia concentrations (mg/£ as N) for the Cen-
tral Basin epilimnion and hypolimnion ............ 1*5
22. Comparison of nitrate concentrations (mg/£ as N) for the West-
ern Basin and Eastern Basin epilimnion ............ k6
vii
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FIGURES (cont'd)
Number Page
23. Comparison of nitrate concentrations (mg/£ as N) for the Cen-
tral Basin epilimnion and hypolimnion ............. 1*7
2k. Comparison of reactive silica concentrations (mg/£ as SiOa) for
the Western Basin and Eastern Basin elilimnion ........ 48
25. Comparison of reactive silica concentrations (mg/£ as SiOa) for
the Central Basin elilimnion and hypolimnion ......... 49
26. Comparison of dissolved oxygen concentrations (mg/£) for the
Central Basin epilimnion and hypolimnion ........... 50
2J. Lake Erie State Variable Interactions. Representations of dis-
solved oxygen (top) and phosphorus (bottom) nutrient cycles. . 55
28. Lake Erie State Variable Interactions. Representations of
nitrogen (top) and silica (bottom) nutrient cycles ...... 56
29. (a) The range of possible cell nutrient content Aq^ = (q^ - q )/
q , and (b) the ratio of the apparent half saturation constant
for growth, K1 , to the half saturation constant for uptake, K ,
s rn
as a function of 6, the ratio of maximum specific uptake, V /q
to maximum growth rate, u1, and K /q , the dimensionless half
m q o
saturation constant for the dependence of nutrient uptake on
internal nutrient concentration. The observed ranges (Table 9)
in B for silica, nitrigen, and phosphorus limited growth are
shown ............................. 6k
30. Normalized growth rate as a function of normalized extent nutri-
ent concentration: a comparison of the Michaelis-Menton ex-
pression to that derived from the internal cellular kinetics
with uptake as a function of both external and internal nutri-
ent concentrations ....................... 66-
31. A comparison of the solutions to the dynamic equations (25)- (27)
and the cellular equilibrium approximation equations (29)-(32).
The parameters are for Scenedesmus sp. (Table 9) and phosphorus-
limited growth. The retention time, t =2.0 days, and K /qp =
0.46. The initial conditions are the equilibrium solution for
an influent of S. = 10.0 ug P04-P/1. The influent is abruptly
increases to S. = 20.0 yg POU-P/1 at t = 0 ........... 70
viii
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FIGURES (cont'd)
Number Page
32. Percent error between the cellular equilibrium approximation
and the dynamic equations for various values of 3 and £ =
K /qn. The curves correspond to the errors in cell concen-
tration, S(t), and cell quota, q(t), as indicated 72
33. Calculated current patterns across the boundary of the Western
and Central Basins based on the prevailing direction and mag-
nitude of Lake Erie winds (direction of W 67 S and magnitude
of 5.2 m/sec) at a depth of 3 meters (top) and 9-1 meters
(bottom) 103
3^. Calculated heat flux (cal/cm2-day) to the Central Basin using
observed temperatures for each cruise. During the winter
period, the net hear flux shown is estimated 106
35- Average temperatures (°C) in two meter sections of the Central
Basin for 1970, Project Hypo Cruise 1 (left) and 1975, CCIW
Cruise h (right). This data was used to establish location
of the thermocline for each year 109
36. Lake Erie Model Transport Calibration, 1970: Temperature. The
calculated profiles are a result of the transport coeffi-
cients given in fig. 38. Symbols: means + standard devia-
tions; lines are the computation ~ Ill
37. Lake Erie Model Transport Calibration, 1970; Chlorides. Mass
input rates of chlorides to the lake are included in the cal-
culation 112
38. 1970 Model Transport Regine. Horizontal Transport and Vertical
Exchanges (m2/day) resulting from the calibration 113
39- Lake Erie Model Transport Calibration, 1975= Temperature.
The calculated profiles use the transport coefficients given
in fig. hO llU
ho. 1975 Model Transport Regime. Horizontal transport are assumed
to be the same as for 1970 while vertical exchanges differ
as shown 115
Ul. Comparison of one group and two group phytoplankton calcula-
tion (chlorophyll, mg/£) for Lake Erie Central Basin Epilim-
nion (left) and Western Basin (right) 118
h2. Mean dissolved oxygen (mg/Jl) for the Central Basin hypolimnion
segments vs. the minimum dissolved oxygen (mg/Jl) observed
for each cruise 119
IX
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FIGURES (cont'd)
Number Page
^3- Cruise average dissolved oxygen (mg/l) for the Central Besin
hypolimnion segments vs. the fraction of stations anoxic
for that cruise ....................... 120
kk. Average monthly wind velocities (mph) around Lake Erie (left)
and variation of non-algal extinction coefficient (m"1 ) with
wind velocity (right) .................... 122
1*5- Lake Erie model calibration for dissolved oxygen (mg/£), 1970.
Symbols: means + standard deviations, lines are the compu-
tations . . . . ~ ....................... 129
x
h6. Lake Erie model calibration for dissolved oxygen (mg/5), 1973-
'. . . . . 130
^7- Dynamics of Central Basin hypolimnion dissolved oxygen cali-
bration illustrating the relative importance of the indivi-
dual oxygen sinks ...................... 132
hQ, Lake Erie model calibration for phytoplankton chlorophyll
(ug/2,) 1970. The calculated curve is the sum of the diatom
chlorophyll and non-diatom chlorophyll ............
^9- Lake Erie model calibration for phytoplankton chlorophyll
(yg/£), 1973-197^. The calculated curve is the sum of the
diatom chlorophyll and non-diatom chlorophyll ........ 135
50. The distribution of chlorophyll (yg/£) between diatoms and
non-diatoms (left) and model calibration for % diatom bio-
mass (right) ......................... 136
51. Western Basin phytoplankton growth and death dynamics for dia-
toms (left) and non-diatoms (right). The curves are the cal-
culated rates from the appropriate kinetic expressions. . . . 137
52. Central Basin epilimnion phytoplankton growth and dea^h dynam-
ics for diatoms (left) and non-diatoms (right) ........ 138
53. Eastern Basin epilimnion phytoplankton growth and death dynam-
ics for diatoms (left) and non-diatoms (right) ........ 139
5^. Lake Erie model calibration, nutrient limitation multiplier,
1970. The curves are the growth rate reduction factors due
to silica, nitrogen and phosphorus limitation effects ....
55- Calculated nitrogen species distribution. Non-living organic
nitrogen, ammonia nitrogen, nitrat plus nitrite nitrogen,
phytoplankton nitrogen and the resultant total nitrogen . . .
x
-------
FIGURES (cont'd
Number Page
56. Lake Erie model calibration for organic nitrogen (mgN/£),
1970. Profiles are the sum of the computed phytoplankton
and the non-living organic nitrogen ............. ikk
57. Lake Erie model calibration for organic nitrogen (mgN/£),
1973-197^. Profiles are the sum of the computed phyto-
plankton and non-living organic nitrogen ..........
58. Lake Erie model calibration for ammonia nitrogen (mgN/£),
1970 ............................ Ik6
59- Lake Erie model calibration for ammonia nitrogen (mgN/£),
1973-197^ .......................... 1U7
60. Lake Erie model calibration for nitrate nitrogen (mgN/£),
1970 ............................ Ik8
6l. Lake Erie model calibration for nitrate nitrogen (mgN/£),
1973-197^ .......................... 1U9
62. Calculated phosphorus species distribution. Unavailable
phosphorus, orthophosphorus, phytoplankton phosphorus and
the resultant total phosphorus are shown .......... 150
63. Lake Erie model calibration for total phosphorus (mgPOi -P/£) ,
1970 ........................ .... 151
6k. Lake Erie model calibration for total phosphorus (mgPO,-P/£),
1973-197^ .......................... 152
65. Lake Erie model calibration for ortho-phosphorus (mg?0, -P/£) ,
1970 ........................ .... 15U
66. Lake Erie model calibration for ortho-phosphorus (mgPO, -P/&) ,
1973-197^ .......................... 155
67. Lake Erie model calibration for reactive silica (mg SiO /£),
1970 ........................ .... 156
68. Lake Erie model calibration for secchi depth (m), 1970 .... 158
69. Lake Erie model calibration for zooplankton carbon (mg/£),
1970 ............................ 159
70. Dynamics of zooplankton population for Western Basin (left)
and Eastern Basin epilimnion (right). The curves are the
calculated rates from the appropriate kinetic expressions. . l60
XI
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FIGURES (cont'd)
Number
71. Lake Erie model calibration for alkalinity, expressed as
mg/£ of calcium carbonate equivalents, 1970 .......... 162
72. Lake Erie model calibration for pH, 197C ............ 163
73. Lake Erie model calibration for the rate of primary production
expressed as mg/m3/hr of carbon, 1970. Observations are
shipboard C11* measurements
Ik. Lake Erie model calibration for BOD (mg/.02), 1967 ....... 165
75- Lake Erie model verification for phytoplankton chlorophyll
(yg/£), 1975. Symbols: means + standard deviations; lines
are the computations ..................... 170
76. Lake Erie model verification for organic nitrogen (mgN/£),
1975 ............................. 171
77- Lake Erie model verification for ammonia nitrogen (mgN/£),
1975 ............................. 172
78. Lake Erie model verification for nitrate nitrogen (mgN/i),
1975 ............................. 173
79- Lake Erie model verification for total phosphorus (mgPO, -P/£) ,
1975 ............................. Ilk
80. Lake Erie model verification for ortho-phosphorus (mgPO<-P/£),
1975 ............................. 175
81. Lake Erie model verification for reactive silica (mg SiC>2/£),
1975 ............................. 176
82. Lake Erie model verification for dissolved oxygen (mg/£) , 1975-
Note that 1975 oxygen conditions are radically different
than 1970 in Central Basin hypolimnion ............ 177
83. Dynamics of Central Basin hypolimnion dissolved oxygen. The
contributions of each of the oxygen sinks for the 1975 oxygen
profile are shown .......... , ............ 180
8k. Western Basin Loads (ibs/day) for orthophosphorus (top) and un-
available phosphorus (bottom) under tvo reduction schemes:
a) With point sources reduced to 0.5 nig/, and b) diffuse sources
reduced by 25? after point sources are reduced to 1 mg/j . . 186
85. Summer average epilimnion chlorophyll (pg/£) vs. lake total phos-
xii
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FIGURES (cont'd)
Numbers Page
phorus loading (mt/yr). There are separate responses for
point and diffuse source reductions and for each basin .... 188
86. Summer average epilimnion total phosphorus (ygPO, -P/&) vs.
Lake total phosphorus loading (mt/yr). The pattern is simi-
lar to the chlorophyll results ................ 190
87. Summer average secchi disk depth (m) vs. lake total phosphorus
loading (mt/yr). Load reductions will result in a signifi-
cant increase in water clarity in the Central and Eastern
Basins ............................ 191
88. Total phosphorus discharge to Lake Ontario vs. lake total
phosphorus loading (mt/yr). This loading is calculated based
on the Eastern Basin epilimnion concentration and the exist-
ing flow to the Niagara River ................. 192
89. Area of anoxia (km2) vs. lake total phosphorus loading (mt/yr)
showing both the short term and ultimate effects .......
90. Central Basin hypolimnion oxygen consumption rate (mg/£-day)
vs. lake total phosphorus loading (mt/yr). The consumption
rate is the sum of all water column and sediment oxygen sinks
in the hypolimnion ...................... 195
91. Minimum Central Basin hypolimnion dissolved oxygen (volume averaged
concentration just prior to overturn in mg/5,) vs. lake total
phosphorus loading (mt/yr) .................. 196
92. Summer average Central Basin hypolimnion dissolved oxygen (mg/£)
vs. lake total phosphorus loading (mt/yr) ........... 197
xin
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TABLES
Number
1. Sources and Sinks of Dissolved Oxygen Central Basin Hypolim-
nion - 1970 10
2. Dissolved Oxygen Calibration - Residual Analysis 1970 12
3- Lake Erie Flow Budget 17
k. Segment Parameters 20
5- Detroit River Phosphorus Loadings Rate (metric tons/,yr) .... 22
6. Nitrogen Fixation 30
7- Lake Erie Limnological Data Summary - Historical Data Sets . . 32
8. Lake Erie Limnological Data Summary - Data Sets Used in Model . 36
9. Growth and Nutrient Uptake Parameters 60
10. Comparison of Growth Rates 62
11. Phytoplankton Growth 77
12. Phytoplankton Respiration and Non-Predatory Mortality 79
13- Herbivorous Zooplankton Growth ,. 8l
14. Carnivorous Zooplankton Growth 82
15. Zooplankton Respiration 83
16. Nitrification 84
17. Denitrification 85
18. Mineralizations 86
19. Sediment Reactions 99
20. Heat Flux and Surface Temperatures 1970 108
xiv
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TABLES (cont'd)
Number Page
21. Sources and Sinks of Dissolved Oxygen Central Basin Hypolim-
nion - 1970 131
22. Annual Sedimentation Fluxes 166
23. Transport Parameters 167
2^. Computed and Observed Sediment Interstitial Water Concentra-
tions - Central Basin 168
25. Calculated Average Central Basin Concentrations and Oxygen
Consumption Rates for 1970 and 1975 178
26. Percentage Contribution of Oxygen Sinks to Dissolved Oxygen
Deficit at the Time of Minimum D.O. in the Central Basin
Hypolimnion 179
27. Calibration and Verification Residual Analysis 182
28. Results for Simulation of Phosphorus Point Source Reduction to
0.1 mg/£ 18U
29. Loading Condition vs. Annual Total Phosphorus Load 187
30. Simulation Results Summer Average 199>
200
31. Comparison of Results - This Work and the U.S. Army Corps of
Engineers Phosphorus Model 201
A-l. Morphometry and Hydrodynamic Regime for 1970 212-
2lU
A-2. 1975 Resegmentation 215
A-3. Water Temperature °C 2l6
A-U. Solar Radiation Data 217
A-5. Photo Period Data . . . 218
A-6. Loadings to Lake Erie Western Basin, Kg/day, Calibration Year-
1970, Verification Year - 1975 219
A-7. Loadings to Lake Erie Central Basin, Kg/day, Calibration Year-
1970, Verification Year - 1975 220
-------
TABLES (cont'd)
Number Page
A-8. Loadings to Lake Erie Eastern Basin, Kg/day, Calibration Year-
1970, Verification Year - 1975 221
A-9- Mass Loading of Chlorides to Lake Erie, MT/day, Calibration
Year - 1970, Verification Year - 1975 222
A-10. Phosphorus Loads Used for Projections - Western Basin 223
A-ll. Phosphorus Loads Used for Projections - Central Basin 22k
A-12. Phosphorus Loads Used for Projections - Eastern Basin 225
A-13. Boundary Conditions 226
A-lk. Lake Erie Model Initial Conditions 227
A-15. Settling Velocities 228
A-l6. Chemical Thermodynamic Parameters for Aqueous Phase Computa-
tion 229
A-17' Equations for 0_ and CO Aqueous Saturation 230
xvi
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ACKNOWLEDGEMENTS
This work would not have been possible without the assistance and
cooperation of many colleagues. The efforts of the members of the EPA Large
Lakes Research Station, Grosse lie, Michigan, were critical. In particular,
thanks are due to Nelson Thomas for his guidance, support, and knowledge of
Lake Erie, William Richardson, as an understanding and helpful project offi-
cer, and Victor Bierman, for his insightful and critical comments. The
efforts of the staff in transferring data to STORET is greatly appreciated.
Without that effort and the STORET system this analysis would have been much
more cumbersome.
The concurrent analysis of the mass discharges of nutrients to Lake Erie
by the members of the Lake Erie Wastewater Management Study group, U.S. Army
Corps of Engineers, Buffalo District, was another critical contribution. The
timely cooperation and responses to requests by Steven Yaksich is greatly
appreciated and his significant contribution is acknowledged.
The data sets used in this analysis were derived from three sources and
to each of these groups a special thanks is due:
The Canada Centre for Inland Waters group, for their cruise data
and to Noel Burns for the numerous discussions of Lake Erie
phenomena.
The Center for lake Erie Area Research (CLEAR) at Ohio State
University and to Charles Hendendorf for his timely cooperation.
The Great Lakes Laboratory at the State University College at
Buffalo and to Robert Sweeney for a useful set of sediment data.
Finally our own colleagues at Manhattan College who contribute in many
and important ways: Robert Thomann and Donald O'Connor and the members of
the staff: Walter Matystik, William Beach and Joanne Guerriero. The
friendly and timely assistance of Kathryn King and Eileen Lutomski who typed
this report is greatly appreciated.
xvi i
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SECTION 1
INTRODUCTION
This report presents an analysis of the interactions "between nutrient
discharges to Lake Erie, the response of phytoplankton to these discharges,
and the dissolved oxygen depletion that occurs as a consequence. The frame-
vork of the analysis is "based on the principle of conservation of mass which
is applied to the relevant variables. Section 2 presents a summary of the
results of the analysis and projections of the response of Lake Erie to re-
duced phosphorus mass discharges. Section 3 lists recommendations for im-
proving the utility and reliability of the computations.
Section k presents the details of the lake morphometry, hydrology, seg-
mentation, and nutrient mass discharges to the lake. Section 5 presents the
water column data used for the analysis. Section 6 discusses the kinetics
which represent the physical, chemical, and biological processes that affect
the variables of concern in the water column. Recent developments in phyto-
plankton growth and uptake kinetics are analyzed and a more tractable version
of these equations is presented. The complete kinetic structure is presented
in a series of tables which define the stoichiometry of the reaction, the
reaction rate expression, and the parameter values used in the computation.
In Section 7 the method for analyzing the interaction of the sediment
and the overlying water is developed. The methods of sedimentary geochem-
istry are expanded to include an analysis of sediment oxygen demand within
the framework of the mass balances. This novel approach related the sediment
oxygen demand to the flux of algae and detritus to the sediment, the decompo-
sition reactions in the sediment and the mass transport mechanisms that
affect the sediment solids and the interstitial water.
The seasonal vertical and horizontal mass transport for Lake Erie is
analyzed in Section 8. The method relies on the analysis of tracer sub-
stances, specifically temperature and chlorides. A simplified method for the
computation of interbasin transport based on heat balances is developed.
The overall structure of the computation and the rationales which lead
to the specific forms of the equations are presented in Section 9- The exo-
genous variables and the method for relating Central Basin hypolimnetic aver-
age dissolved oxygen to anoxic area are presented. The computational method
employed for the carbon dioxide-alkalinity systems are discussed and the
final forms of the mass balance equations are presented and related to the
specification of the water column kinetics in Section 6, the sediment reac-
tions and mass transport in Section 7, and the lake transport in Section 8.
Section 10 contains a detailed presentation of the results of the cali-
bration to the data presented in Section 5. Data from 1970 and 1973-7^ are
-------
compared to calculation for all the state variables and the successes and
failures of the computation are discussed. Dissolved oxygen, phytoplankton
chlorophyll for diatoms and non-diatoms, zooplankton biomass, nutrient con-
centration in available and unavailable forms, and inorganic carbon are con-
sidered. The sediment portion of the computation is also calibrated to
available observations although these are less complete.
Section 11 includes the results of a verification computation for 1975,
a year when no anoxia was observed. The computations are in reasonable
agreement with these observations as well. The principle cause of the abnor-
mal condition was a shallower thermocline but the consequences were quite
interesting and reflected the differing balances between areal and volumetric
sinks of oxygen. A statistical analysis of the calibration and verification
results is included.
Section 12 contains the projected effects of varying degrees of phos-
phorus removal. Considerable uncertainty results from the inability to pro-
ject long term deep sediment behavior, and the time scale of these effects.
Dissolved oxygen, anoxia area, chlorophyll, transparency, and phosphorus con-
centrations are all calculated as functions of total phosphorus discharge to
the lake. It is estimated that a discharge of 10,000 metric tonnes/yr will
essentially eliminate anoxia in the Central Basin. A crude estimate of the
uncertainty of this projection is 10-20% based on calibration and verifica-
tion uncertainty.
-------
SECTION 2
SUMMARY
This section is a summary of the principle results and conclusions of
this report as they relate to the relationship between phosphorus discharges
and dissolved oxygen depletion in Lake Erie. It summarizes the methods used
to analyze the present effects of phosphorus discharges on the algal and
dissolved oxygen distribution of Lake Erie and presents the results of calcu-
lations which attempt to predict the impact of phosphorus loading reductions.
The sections that follow present, in detail, the conceptual framework, the
data employed, the mathematical formulations and their justification, the
calibration and the verification calculations, and the complete results of
the projection calculations. This presentation is an overview of the
methods and results without detailed technical justifications.
FRAMEWORK
A direct causal chain links phosphorus inputs to the dissolved oxygen
concentrations in Lake Erie. Phosphorus entering the lake increases the in-
lake phosphorus concentration. If the other nutrients required for phyto-
plankton growth are in excess supply and the population is not light limited,
the plankton will respond and an increased concentration of biomass will
result. The organic carbon synthesized and the dissolved oxygen liberated
affect the dissolved oxygen concentration: the latter directly; the former
via subsequent oxidation. A quantitative assessment of these phenomena is
required if projections of the effects of phosphorus loading reductions are
to be attempted.
Two essentially separate phenomena must be understood: the interaction
of phytoplankton and nutrients; and the interactions of the sources and sinks
of dissolved oxygen. The methods employed for both these calculations are
"based on mass balance equations.
The kinetics employed are designed to simulate the annual cycle of
phytoplankton production, its relation to the supply of nutrients, and the
effect on dissolved oxygen. The calculation is based on formulating the
kinetics which govern the interactions between the biota and the forms of the
nutrients and applying them to the regions of Lake Erie within the context of
conservation of mass equations. The fifteen variables for which these calcu-
lations are performed are:
Phytoplankton
1. Diatom chlorophyll-a
2. Non-diatom chlorophyll-a
-------
Zooplankton
3. Herbivorous zooplankton carbon
k. Carnivorous zooplankton carbon
Nitrogen
5. Organic nitrogen
6. Ammonia nitrogen
7. Nitrate nitrogen
Phosphorus
Unavailable phosphorus
Soluble reactive phosphorus
Silica
10. Unavailable silica
11. Soluble reactive silica
Carbon, Hydrogen, Oxygen
12. Detrital Organic Carbon
13. Dissolved Inorganic Carbon
ill. Alkalinity
15. Dissolved Oxygen
CALIBRATION - PHYTOPLANKTON AND NUTRIENTS
A critical requirement for establishing the credibility of the analysis
is a complete calibration vhich compares the computation to the available
observations. A summary of these comparisons is shown in fig. 1. The aver-
age concentration of chlorophyll a_ is both computed and observed to decrease
from vest to east. The spring diatom bloom appears to be of comparable mag-
nitude in all basins, whereas the magnitude of the fall non-diatom bloom pro-
gressively decreases from west to east. Shipboard C1"* primary production
measurements are compared to the comparable kinetic expression in the calcu-
lation. The observed three-fold variation from west to east is correctly
reproduced. Total phosphorus concentrations are observed and calculated to
decrease from west to east. The higher observed concentrations in the late
fall and early spring are attributed to wind driven resuspension of sedimen-
tary phosphorus due to high winds during this period. These effects do not
appear to persist into the productive period of the year.
The calculated and observed orthophosphorus, nitrate nitrogen (the ammo-
nia concentrations are quite small), and silica concentrations are shown in
fig. 2. The depletion of silica terminates the spring diatom bloom, whereas
the exhaustion of nitrogen primarily in the Western Basin and phosphorus in
the Central and Eastern Basins terminates the non-diatom bloom. Again the
progressively decreasing concentrations from west to east are suggested but
not as pronounced as in the previous figure. The somewhat scattered behav-
ior of silica in the Western Basin is unexplained at present.
One of the principle purposes of calibration is to demonstrate that the
-------
LAKE ERIE CALIBRATION, 1970
EPIL1HNION
" 32.00
16.00
GC
o
x 0.00
o
WKTf/tN BASIN
J'F'M'A'M'J' J'A'S'O'N'D
CENTRAL BASIN EPtLIMNION
J'F'M'A'M'J' J'A'S'O'N'D
'. ivm^ : ^MTT
-s~_^ I i ^* * _
EASTERN iASIN [HLIUNION
J 'F'M'A'M'j1 J'A'S'O'N'D
JIFIMIAIMIJIJIAISI0INID|IJIF'M'ATMIJ1 J'A'S'O'N'D
EASTERN BASIN
j 'F'M'A'M'j1 J'A'S'O'N'D
f o.os
at
O 0.04
z
0.00
j 'F'M'A'M'J' J'A'S'O'N'D
CENTRAL BASIN EflLIMNION
_ Segment 2
jrFTMlATM1 j1 J'A'S'O'N'D
EASTERN BASIN EflLIUNION
13
J'F'M'A'M'j'J'A'S'O'N'D
Figure 1. Lake Erie Calibration Results, 1970. Epilimnion Western
Basin (lefthand side); Central Basin (center); Eastern
Basin (righthand side); chlorophyll a_, pg/£, (top); C14
shipboard primary production mg C/m3/hr (middle); total
phosphorus mg/5, (bottom). Symbols: mean +_ standard de-
viation; lines are the computations.
calculation can reproduce the major features of the seasonal distribution of
phytoplankton and nutrients over a range of observed concentrations. The
fact that the Western, Central, and Eastern Basin distributions are all
reasonably well reproduced using the same kinetic structure and coefficients
suggests that the calculation has a certain generality and can reproduce con-
ditions as distinct as those in the Western and Eastern Basin.
CALIBRATION - DISSOLVED OXYGEN
The second phenomena of concern are those that influence the dissolved
oxygen balance in the basins. Since the comparatively shallow Western Basin
is essentially completely mixed vertically and remains nearly saturated
throughout the year it is not of concern, although it is part of the oxygen
balance calculation. Three regions are represented in each basin: the epi-
limnion, the hypolimnion, and the surface sediment layer. The principle
sources and sinks of dissolved oxygen are associated with the phytoplankton
and detrital organic carbon in these segments. Although nitrification is
included in the calculation, its effect is negligible. The principle aerobic
reactions which affect dissolved oxygen are primary production and respira-
tion of phytoplankton, and the oxidation of detrital organic carbon via
-------
LAKE ERIE CALIBRATION, 1970
EPILIMNION
E 0.032
O 0.016
I 0.000
a.
WESTERN BASIN
- Segment 1
0
J'F'M'A'M'J'J'A'S'OND
CENTRAL BASIN EPILIMNION
_ Segment 2
j 'F'M'A'M'J'
EASTERN BASIN EPILIUNION
J'F'M'A'M'J'
z 0.00
3 1.20
_i
5 Q0.60
ta
§*
2 o.oo
y- * '
'F'M'A'M'J'J'A'S'O'N'D'
j'F'M'A'M'J'J'ATS'O'N
EASTERN BASIN EPILIMNION
. Segment 3
J'F'M'A'M'J ljTATs'70iNi
J'F'M'A'M^J'J'A'S'O'N'D
- Segment 2
CENTRAL BASIN EPILIMNIOH
m \n T -X .
j' F'M'/VM'j' J'A
jM-nyj
VrTi
EASTER* BASIN EPILIMNION
_ Segment 3
j'F'M'A'M'J'J'A'S'O'N'D
Figure 2. Lake Erie Calibration Results, 1970 Epilimnion.
Western Basin (lefthand side), Central Basin
(center), Eastern Basin (righthand side), ortho-
phosphorus (top); nitrate nitrogen (middle),
reactive silica ("bottom).
"bacterial synthesis and respiration. The primary production reaction liber-
ates dissolved oxygen and decreases total carbon dioxide. The aerobic res-
piration reactions consume dissolved oxygen and liberate carbon dioxide.
Thus both dissolved oxygen and total inorganic carbon mass balances are con-
sidered. This provides an additional variable for the calibration, giving
a total of four principle state variables of concern as shown in fig. 3. Of
these, three are directly observed. As discussed below, BOD measurements
provide an indirect measurement of the fourth variable, the detrital organic
carbon.
The concentrations of concern are affected by two classes of transport
phenomena: dispersion, which exchanges matter across the segment interfaces;
and advection, which, in the vertical direction, is associated with settling
of the particulates. These are shown in fig. 3 together with the principal
reactions of concern. If all the transport fluxes and reaction rates were
directly observed, then the mass balance calculation could be accomplished
by simple summations. These could then be compared to the observed concen-
trations as a check on the reaction and transport fluxes. Unfortunately,
-------
LAKE ERIE OXYGEN MASS BALANCE
(*) Directly Observed
(T) Estimated Using Tracers
- ADVECTION
« - DISPERSION
neither all the variables
nor all the fluxes are ob-
served, as indicated in
fig. 3. These must either
be estimated using tracers
or from the calibration it-
self.
Air-vater interface
exchange is assigned based
on wind velocity and empir-
ical relationships. Thermo-
cline exchange is estimated
using temperature as a
tracer. Hypolimnion-inter-
stitial water exchange is
estimated using chloride
as a tracer. Settling
velocities of phytoplankton
and detrital carbon are
assigned based on litera-
ture values and the cali-
bration. Sedimentation
settling velocity is esti-
mated using pollen tracers
and sedimentation trap re-
sults .
Certain reaction rates
are also directly observed;
for example primary produc-
tion. The aerobic respira-
tion rate of oxygen is
directly observed as bio-
chemical oxygen demand
(BOD). Since this rate is
the sum of both detrital
oxidation and phytoplankton
respiration, it provides an
indirect measurement of the
detrital carbon concentra-
tion.
For the sediment segment, direct observations of the dissolved oxygen
equivalents in the interstitial water are available. However due to the
complex chemistry associated with the anaerobic respirations the total C02
is not easily computed. The interstitial ammonia concentration provides a
tracer which is liberated as the anaerobic respirations proceed, so that this
reaction rate can be estimated.
As previously discussed, the purpose of the calibration is to estimate
the individual kinetic coefficients based on the observed state variable con-
centrations and composite reaction rates, and to demonstrate their applica-
AIR-WATER INTERFAC!
t
,
EPILIMNION
GROSS PRIMARY PRODUCTION * ,
CHLOROPHYLL AEROBIC RESPIRATION
T ^
1
i
THERMOCLINE '
1
1
I 1
DETRITAL
CARBON
i
(BOD)
AEROBIC
RESPIRATION
)
HYPOLIMNION
CHLOROPHYLL*
S, * <- '
SEDIMENT-WATER
INTERFACE
CHLOROPHYLL
DETRITAL
ORGANIC
CARBON
(BOD)
DISSOLVED*
OXYGEN
DISSOLVED *
INORGANIC
CARBON
,
T(T,
DISSOLVED
OXYGEN
DISSOLVED
INORGANIC
CARBON
1 (T) (SEDIMENTATION FLUX) * (SOD)
1
SEDIMENT
1
1
I ANAEROBIC RESPIRATION
k^l ^ '
DETRITAL
ORGANIC
CARBON
(AMMONIA) (T)
*
ANAEROBIC
RESPIRATION
(T)
i
DISSOLVED
OXYGEN
EQUIVALENTS
DISSOLVED
INORGANIC
CARBON
DEEP SEDIMENT DEMAND
Figure 3.
Schematic Diagram of the Dissolved
Oxygen Mass Balance Computation.
The Epilimnion, Hypolimnion (for
Central & Eastern Basins), and sur-
face Sediment segments as indicated.
Directly observed concentrations
& reaction rates are indicated (*).
Transport rates estimated using
tracers are indicated (T).
7
-------
LAKE ERIE CENTRAL BASIN CALIBRATION
1970
EPILIMNION HYPOLIMNION
12.00
3 6.00
J^
° 0.00
9.10
8.30
7.50
ISSOL VEO
OXYG£N
CENTRAL BASIN EPILIMNION
Segment 2
j 'F'M'A'M'J'J'A'S'O'N'D
CENTRAL BASIN EflLIMNION
S*gm*nt2
jlFlMlA'Mrj'J'A'S'O'N'D
5
» 16.00
oc
o
8.00
0.00
CENTRAL BASIN EfILlHNION
j'F'M'A'M'j1J'A'S'O'N'O
J'F'M'A'M'J" J'A'S'O'N'O'
8.40
7.20
6.00
CENTRAL BASIN HYfOLIMNION
j'F'M'A'M'j1J'A'S'O'N'o
CENTRAL 8ASIN HYPOLtHNION
_ Styncm 5
J'F'M'A'M'j'J'A'S'O'N'O
bility in differing re-
gions of the lake. The
results for the Central
Basin are shown in fig.
k. Dissolved oxygen,
while remaining essen-
tially saturated in the
epilimnion, declines
after stratification to
a minimum of 1.0 mg/£
before overturn. The
lack of fit after over-
turn is due to an inac-
curate estimate of the
vertical dispersion.
The pH comparison is a
direct calibration of
the total COa computa-
tion since the pH is
determined by changes
in total CO2, the alka-
linity being essentially
constant. There is a
dramatic difference be-
tween the epilimnion pH
reflecting the net total
COa loss which is ocur-
ring and the hypolimnion
pH reflecting the net
total CO2 production.
Figure h. Lake Erie Central Basin Calibration
for Dissolved Oxygen and related
variables, 1970. Epilimnion (left-
hand side); Hypolimnion (righthand
side), Dissolved Oxygen, mg/£ (top):
pH (middle); chlorophyll a_yg/&
(bottom). The Hypolimnion chloro-
phyll a data is 1973-197^, as no
1970 data are available.
The chlorophyll
concentrations also re-
flect the production in
the surface layer, the
settling into the hypo-
limnion and the respira-
tion and other losses.
Figure 5 gives the com-
parisons to epilimnion
primary production and
BOD in both layers. The
larger quantity of BOD in the epilimnion reflects the larger quantity of phy-
toplankton and detrital organic carbon. The table presents the observed and
computed sediment interstitial water concentrations. The ammonia and oxygen
equivalents are within the range of observations. The deep sediment flux of
oxygen equivalents is chosen to match both the observed oxygen equivalent
concentrations and also the observed oxygen equivalent flux which is observed
as the sediment oxygen demand at the sediment-water interface. Similar com-
parisons for the Eastern Basin have also been made with essentially compar-
able results.
8
-------
LAKE ERIE CENTRAL BASIN
£ 48.00
f
2 24.00
O
m
-------
TABLE 1. SOURCES AND SINKS OF DISSOLVED OXYGEN
CENTRAL BASIN HYPOLIMNION - 1970
Reaction Average Volumetric Dissolved Q
Depletion Rate Change
Source ( + ) Day 180-2^0
Sink (-) (mg 02/£-day) (mg 02/A)
Phytoplankton Respiration - 0.20 - 1.20
Detrital Organic Carbon - 0.06*16 - 3.88
Oxidation
Sediment Oxygen Demand - 0.0597 - 3-58
Thermocline Transport 0.0305 1.83
Total 02 Change - 6.83
fact what was observed and calculated is the result of the combination of
these two sinks of dissolved oxygen. The fact that the computations are able
to verify the 1975 dissolved oxygen distribution implies that the proportions
of volumetric and areal oxygen demand in the calculation are correct.
The chlorophyll, orthophosphorus, and nitrate nitrogen are also shown.
Note that slightly less chlorophyll is calculated in 1975 than in 1970 due to
a relatively small (12%) phosphorus loading reduction achieved between 1970
and 1975. Although the effect is slight in the chlorophyll it is calculated
and observed in the nitrate distribution. Slightly less nitrate uptake was
observed and calculated in 1975- Admittedly this effect is small but encour-
aging.
PROJECTIONS
A comprehensive calculation of this sort can be used to calculate the
effects of phosphorus loading reductions on the dissolved oxygen, in particu-
lar, in the Central Basin. An immediate question regarding these calcula-
tions is: how accurate are these projections likely to be. Although there is
no firm methodology available at present to answer this question, it is cer-
tainly the case that the projections are not likely to be any more accurate
than the calibration and verification. An analysis of the residuals, the
observed minus calculated dissolved oxygen concentration for the Central
Basin, is shown in Table 2.
Residual standard deviations of ^ 0.5 to 1.0 mg/£ imply that the predic-
tion error standard deviation is likely to be at least this large for any
specific time during the computation. In terms of median relative error
(^ 5-15%) the calibration and verification of the Central Basin is among the
more precisely calibrated of dissolved oxygen models (l).
10
-------
LAKE ERIE CENTRAL BASIN VERIFICATION
CALIBRATION 1970
VERIFICATION 1975
x
o
a
12.00
f 6.00
0.00
J'F
16.00
CENTRAL BASIN EPILIMNION
- S«gmnt2
8.00
0.00
j'F'M'A'M'J'J'A'S'O'N'D
CB EPILIMNION
J 'F'M'A'M'J ' JIATSTOTNTD
f 0.016
-------
TABLE 2. DISSOLVED OXYGEN CALIBRATION - RESIDUAL ANALYSIS 1970
RESIDUAL x*
STANDARD MEDIAN
CENTRAL BASIN MEAN DEVIATION RELATIVE ERROR
EPILIMNION
HYPOLIMNION
1970
1973-7^
1975
1970
1973-7^
1975
(mg/fc)
-0.13
-0.16
-0.2U
0.56
0.63
0.29
(mg/£)
0.60
1.26
0.95
0.77
1.08
0.93
(%}
h
1
6
12
15
8
*
Residual = Calculated-Observed
3£3f
Relative Error = |Calculated-Observed]/Observed.
tween loading and concentration (fig. 7)5 the standard deviation of the pro-
jected loading is linearly related to the prediction standard deviation.
For a predicted loading of 10,000 MT/yr and using a range of 0.5-1-0 mg/£
for the prediction standard deviation, the predicted loading standard devia-
tion is 1000 - 2000 MT/yr so that the prediction uncertainty due to calibra-
tion uncertainty is estimated to be in the range of 10-20$.
12
-------
HI
O
X.
o
Q
<
MEAN DISSOLVED OXYGEN vs. MINIMUM
DISSOLVED OXYGEN
i 16
- SEGMENTS
14
12
10
8
6
4
2f
1970 DATA
1973-74 DATA
* 1975 DATA
0 2 4 6 8 10 12 14 16
MINIMUM DISSOLVED OXYGEN, mg/l
MINIMUM CENTRAL BASIN HYPOLIMNION DISSOLVED OXYGEN
(VOLUME AVERAGED) vs. LAKE TOTAL PHOSPHORUS
O
o
UJ
' «?//vr SOURCE
REDUCTIONS
-«-- DIFFUSE SOURCE
REDUCTIONS
ULTIMATE
EFFECT
SHORT TERM
EFFECT
IS YEARSI
I
I
I
I
6000 8000 10000 12000 14000 16000
TOTAL PHOSPHORUS LOAD, MT/yr
18000 20000
Figure T. Observed Hypolimnion Mean Dissolved Oxygen
vs Minimum Observed Hypolimnion Dissolved
Oxygen (top). Predicted Mean Hypolimnion
Dissolved Oxygen just prior to overturn vs
Lake Total phosphorus Loading (bottom).
13
-------
SECTION 3
RECOMMENDATIONS
In the course of this analysis of the relationship between nutrient dis-
charges, phytoplankton growth, and dissolved oxygen depletion, a number of
shortcomings in "both theoretical understanding, observational data, and
experimental investigations have been uncovered. These recommendations are
based upon these considerations.
The most critical requirement is an assessment of the degree of uncer-
tainty to be associated with the projected response of Lake Erie to reduc-
tions in phosphorus loading. A number of different avenues of investiga-
tion should be pursued in order to develop a response to this question of
uncertainty.
The major source of uncertainty in the present state of the computa-
tions is the behavior of the sediment under present and projected conditions.
The computations in this report are based on a one-layer sediment segment.
This should be expanded to include a multiple layer vertical analysis which
can be calibrated in much more detail with both interstitial water and
sediment solids concentrations of the relevant variables. The expected time
to steady state under projected conditions should be calculated.
A more detailed vertical segmentation of the Central Basin is required
in order to assess the magnitudes of the areal sediment-water fluxes and
the volumetric water reactions. Advances in phytoplankton growth and uptake
kinetics should be included in the computation. This, when coupled to a
detailed sediment computation, should allow a more refined analysis of these
critical oxygen depleting and nutrient regenerating reactions.
Experiments designed to measure directly the flux of oxygen equivalents
from the sediments are required in order to confirm the theoretical struc-
ture used in the calculations presented in this report.
A comprehensive data collection program should be pursued which is
designed to measure directly all the areal and volumetric rates which either
produce or consume oxygen in the Central Basin hypolimnion. The data used
in this report are composited from several years and of particular impor-
tance is the volumetric depletion rate measurement (BOD) and direct sediment
flux measurements.
A comprehensive monitoring program for the measurement of nutrient load-
ing to the lakes should be continued. Methods of sampling and assumptions
regarding methods of calculating the loading from the observations should be
agreed upon. The U.S. Army Corps of Engineers efforts in this regard are to
Ik
-------
"be commended. The results of their analysis were a critical component of the
calculations presented in this report. For future reevaluations of the
response of the lake to changes in inputs, a comparable effort must be
expended in order to obtain reliable loading information.
A statistical methodology should be developed which is capable of
addressing the question of uncertainty of projections. The crude analysis
presented at the conclusion of this report is only a rough approximation
and should be taken only as an indication of the order of magnitude of the
uncertainty. Since this question of uncertainty is central to the use of
model computations for decision making it should be pursued with the same
effort as that expended upon the other facets of the analysis.
As phosphorus removal programs are implemented it is of utmost impor-
tance that the actual response of the lake be documented and compared to
projected responses. There is no substitute for observations of response
after large scale changes in loading are. accomplished. However a casual
inspection of routine monitoring data is not sufficient to test the predic-
tions since the lake is a complex and interactive system. Only integrated
calculations and complete observations can isolate the relevant effects,
check the predictions in detail, and determine the extent of agreement or
disagreement between observation and prediction. The design of the data
collection program should be compatible with the methods of analysis and
computation that are to be applied. For computations based on mass balance
principles of the sort used in this report the minimum requirements are
measurements of all the state variables and observable rates used in the
calibration, both in the water column and the sediment. The nutrient load-
ing rates to the lake are also required as discussed previously.
Such a comprehensive evaluation should contribute to the development
of more detailed knowledge and more reliable predictive methods for the
Great Lakes in general and Lake Erie in particular.
15
-------
SECTION k
DESCRIPTION OF STUDY AREA, MASS DISCHARGE RATES, SEGMENTATION
MORPHOMETRY
Situated "between Lake Huron and Lake Ontario, Lake Erie is the fourth
in the chain of Great Lakes. Although all the Great Lakes are similar in
their origin and history, Lake Erie exhibits distinct properties.
Volumetrically, Lake Erie is the smallest of the Great Lakes, ^70 km3
[2], The surface area of Lake Erie, 25,750 km2, is slightly larger than
that of Lake Ontario [2]. Lake Erie is the shallowest of the Great Lakes
with a maximum depth of 64 m, and an average depth of only 18 m [2], The
drainage area of Lake Erie is 86,762 km2 excluding Lake St. Clair [3].
Lake Erie is divided naturally into three basins: the Western Basin,
Central Basin and Eastern Basin. The Western and Central basins are sepa-
rated by a chain of bedrock islands. The Central and Eastern basins are
separated by a ridge of sand covered glacial clay.
The Western Basin is shallow and flat with a muddy bottom. It covers
approximately 3,110 km2 and has an average depth of 7.5 m [2], Conditions
in the basin are greatly influenced by the Detroit River.
The Central Basin is the largest of the three basins being approxi-
mately 16,300 km2 in area [2]. The maximum depth is about 25 m and the
average depth is 18.3 m [2], The bottom is relatively flat with a mud-sand
composition.
The Eastern Basin has a surface area of 6,220 km2, a. maximum depth of
6k m and an average depth of about UO m. In contrast to the other basins
the bottom is not flat but steeply sloped.
HYDROLOGY
Water inflow to Lake Erie is dominated by the Detroit River which
drains from Lake Huron-Lake St. Clair. The average flow of the Detroit
River between 1936 and 197*1 is 188,000 cfs. [k] which accounts for over 90
percent of the flow discharged via the Niagara River and Welland Canal.
However, since 1967 the flows have been consistently higher than the average,
reaching a maximum yearly average of 238,000 cfs in 1973 [4]. The flow from
the Detroit River controls the circulation in the Western Basin penetrating
far southward into the basin and then draining, primarily through the north-
ern island channels [2].
16
-------
Only four other tributaries to Lake Erie exceed an average flow of 1,000
cfs. They are, in order of decreasing contribution, the Maumee, the Grand
(Ontario), the Thames, and the Sandusky [2]. There are some twenty other
tributaries feeding Lake Erie. The flow in. each of these is small sometimes
reaching zero in summer.
Lake Erie has the greatest water loss due to evaporation of any of the
Great Lakes [5]. This loss has been estimated at about 85 cm (33.5 inches)
per year [6]. However, the loss is approximately equal to the water surface
precipitation and thus neither figure significantly in the water budget for
the lake [5,7].
Table 3 shows the flow budget for the Lake used in the subsequent cal-
culations.
TABLE 3. LAKE ERIE FLOW BUDGET
Source Flow (cfs) % of Discharge
Detroit River 209,000 95.0
Maumee River 4,000 1.8
Other Tributaries 7,000 3.2
Discharge
Niagara River & Welland Canal 220,600
Based on these flows, the hydraulic detention time of the lake is approxi-
mately 2.5 years, the shortest of the Great Lakes, approximately one-fourth
the Lake Ontario retention time and approximately one-tenth of that for Lake
Huron. The retention time is related to the lake's response time to changes
in mass input rates. However, as shown subsequently, the actual response
time for certain substances is significantly longer due to the sediment-water
interactions.
SEGMENTATION
For the purposes of the computations presented subsequently, lake Erie
is divided into ten completely mixed segments to represent the significant
geophysical, chemical, and biological variations exhibited throughout the
lake. Six of the segments are water column segments and four are sediment
segments. The lake is logically divided by basin. In the Central and East-
ern Basins where a thermocline develops, epilimnion and hypolimnion segments
are considered. The Central Basin hypolimnion is separated into two segments
representing the bulk of the hypolimnion and a thin bottom layer. Sediment
segments underlie each of the hypolimnion segments and the Western Basin
segment. Figure (8) shows this segmentation.
IT
-------
EPI LIMN ION SEGMENTS
0-BOTTOM
0-17 METERS
HYPOLIMNION SEGMENTS
. Jl
\.- 17 METERS-BOTTOM
-^^-
,--17-22 METERS
-22 METERS-BOTTOM
SEDIMENT SEGMENTS
ALL 5 CM. DEEP
Figure 8. Lake Erie Model Segmentation of Western, Central and.
Eastern Basins: Water Segments 1-6, Epilimnion (top)
and Hypolimnion (middle); Sediment Segments 7-10,
(bottom).
18
-------
Segment 1 represents the Western Basin. It extends from the surface to
the sediment interface which is assumed to be at the average depth of the
basin: 7.5 meters. The eastern boundary of the segment is a straight line
from Pelee Point to the edge of Sandusky Bay. Segment 1 receives flow and
nutrient loadings from the Detroit and Maumee Rivers which constitute the
majority of the mass inputs to the lake.
The Central Basin epilmnion is represented by segment 2. It is bounded
by segment 1 on the west and segment 3 on the east. The eastern boundary
extends from just west of Port Ravan, Ontario to a point about fifteen kilo-
meters west of Erie, Pennsylvania. This line follows the natural ridge sep-
arating the Central and Eastern Basins. Segment 2 extends from the surface
to a depth of 17 meters which is the approximate depth of the thermocline
during 1970 and 197^, the years chosen for calibration. For the 1975 veri-
fication, the depth of segment 2 is set at 13 meters to reflect the unusu-
ally high thermocline in that year, as described subsequently in section (8).
Segment 3 corresponds to the Eastern Basin epilimnion. As with segment
2, the depth is 17 meters. It is the outflow segment of the model dis-
charging to the Niagara River and Welland Canal.
Directly below segment 3 is segment k, representing the Eastern Basin
Hypolimnion. Although the actual maximum depth of the basin is 6k meters,
segment k extends from 17 meters to the average depth of the basin, ko
meters.
Segments 5 and 6 represent the Central Basin hypolimnion. The bulk of
the hypolimnion makes up segment 5. Segment 6 comprises a thin bottom layer.
This differentiation is established in order to reflect the dissolved oxygen
gradient in the hypolimnion. Segment 5 extends from 17 meters to 22 meters
and segment 6 from 22 meters to the bottom.
The well-mixed surface layer of the sediment which underlies the lake is
represented by segments 7 through 10. They are included in order to calcu-
late, dynamically, the nutrient exchange and oxygen demand at the water
column-sediment interface. A segment depth of 5 centimeters is chosen as the
depth of the well-mixed layer in the sediment as discussed subsequently.
The individual segment depths, areas and volumes are listed in Table h.
This information was compiled from bathymetric charts of Lake Erie [8,9].
Basin area versus depth profiles were constructed using a planimeter which
were then converted into segment volumes.
RATES OF MASS DISCHARGE
The first comprehensive documentation of the rates of mass discharge to
Lake Erie is reported in the 1969 I.J.C. water quality report [10]. Total
nitrogen, total phosphorus, chlorides, solids and BOD mass loadings are given
for 1967. These are subdivided into major municipal, industrial, and tribu-
tary components. The tributary component is further subdivided into the
municipal, industrial and other sources discharging to the tributary load.
19
-------
TABLE 4. SEGMENT PARAMETERS
DEPTH
SEGMENT METERS
1 7.5
2 IT
3 17
it 23
5 5
6 2.
7 0.05
8 0.05
9 0.05
10 0.05
AREA 2
kilometers
SURFACE BOTTOM
3010
15560 9400
6150 1+177
4177
9400 3750
3750
3010
5650
3750
^177
VOLUME
kilometers
22.6
234.6
91.9
82.7
43.5
7.5
0.15
0.28
0.19
0.27
Total Lake Surface Area = 24720 Km
Total Lake Volume = 482.8 Km3
20
-------
Since the 1969 report the IJC has published yearly reports [11,12,13,1^]
in which mass loading estimates to Lake Erie are made, separated into indus-
trial, municipal and tributary sources. However, the loadings are not sub-
divided by basins, which is a requirement for the present calculation. Burns
has reported basin specific mass loadings for 1970 [15]. For these estimates
tributary inputs were calculated from mean monthly flow and concentration
data. Direct discharges were estimated from the 19^9 IJC report. The mass
loadings are reported as annual averages for each basin.
The IJC report indicates that the Detroit River is by far the largest
contributor of mass loadings to Lake Erie comprising almost 60% of the total
phosphorus and 11% of the total nitrogen input. Since the Detroit River dis-
charges into the shallow Western Basin, the combination of the magnitude of
the loadings and the shallowness of the basin result in a strong coupling
between Detroit River and Western Basin conditions. It is necessary, there-
fore, that the Detroit River contribution be estimated as accurately as pos-
sible.
ESTIMATE OF DETROIT RIVER MASS DISCHARGE
The concentrations of various water quality constituents entering Lake
Erie from the Detroit River are monitored separately by two agencies: the
Michigan Department of Natural Resources (Michigan DNR) and the Ontario
Ministry of the Environment (OME). Because the Detroit River is character-
ized by lateral concentration gradients [e.g. ref. l6] measurements are taken
at several stations along a transect from Michigan to Ontario. The Michigan
DNR has eleven stations and OME monitors twelve.
Three different methods are used to determine the mass loading rate to
Lake Erie based on the transect data. Michigan DNR divides the river cross
section into panels each containing ten percent of the total flow and uses
the concentration at a station within the panel as the panel concentration.
OME uses the same ten percent panels with the average concentration for the
panel. The International Joint Commission (IJC) used station concentrations
in conjunction with the actual percentage of flow for an area of cross-sec-
tion one half the distance to the next station. All three agencies use flow
distribution measurements made by EPA in 1963 [17].
The estimates of the mass loadings based on the two data sets and three
calculation methods vary widely. Table 5, compiled from the Lake Erie
Wastewater Management Preliminary Feasibility Report [l8] , shows this varia-
tion in the loading estimates for total phosphorus. Differences between
the three calculation methods are to be expected. The variation in lateral
concentration averaging technique results in different estimates in direct
proportion to the steepness of the concentration gradient. Note that using
the OME data, the IJC method consistently yields smaller standard deviations
than the OME method. Comparing the two data sets using the IJC method, the
OME data yields lower loadings than the Michigan DNR data. For 1970, the
calibration year for the model, the OME estimate is 62% of the DNR estimate.
The standard deviation of the OME estimate is about one third that of the
DNR. The estimated difference is even more pronounced in 1972 when the
21
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Michigan DNR data yields a loading about twice that of the OME data. These
differences demonstrate a lack of concordance "between the data from Michigan
and the data from Ontario. The larger standard deviations of the Michigan
DNR data using the same calculation method imply that there is much more
loading variability in the river than that described by the OME data.
The strong influence of the Detroit River on the Western Basin and all
of Lake Erie makes the choice of data set and calculation method used for the
mass loading estimate an important decision. If the estimate is too large,
the vertical transport (settling) will be overestimated in order to match
observed concentrations. If the loading estimate is too small, sediment
sources will be overestimated. Therefore the particular estimate that is
chosen has substantial ramifications.
LAKE ERIE WASTEWATER MANAGEMENT STUDY
The most comprehensive study of mass loadings to Lake Erie has been per-
formed by the Army Corps of Engineers, Buffalo District. As part of their
Lake Erie Wastewater Management Study (LEWMS) the Corps have reviewed pre-
vious methods of estimating mass loadings and have proposed and implemented
a new method. The Corps analyzed the three main methods traditionally used
in mass loading estimation, comparing them to known loadings based on daily
measurements. The three methods are:
A) (Average Annual Flow) X (Average Annual Concentration)
B) (Flow at sampling time) X (Concentration at sampling time),
averaged over year
C) (Flow at sampling time) X (Concentration at sampling time) X
(ratio of sampling time flow to yearly average flow).
Loadings were calculated using each method with twelve data points taken
from the third, then the ninth, and then twenty-fifth day of the month, and
then twelve random dates. In addition, estimates were made using all forty-
eight measurements. In all cases, the estimates of the actual loads were
poor. The Corps concluded that the reason for the poor estimates was an in-
adequate accounting of high and low flow events due to an inappropriate
sampling frequency.
The calculation method developed by the Corp to overcome this difficulty
is known as the flow interval method. The flow is divided into equal inter-
vals up to the maximum flow for the period of record. The average mass
loading is calculated for each flow interval based on the concentration data
points within that interval. The average mass loading for a period of inter-
est is then the summation of the average for each interval multiplied by the
fraction of the period of interest during which the flow is within that
interval. When compared with the other loading calculation methods, the flow
interval method yielded lower error estimates although for the conditions
used (kQ measurements), its estimate was also not very accurate [l8]. How-
ever, when high flow measurements are included in the data set, the flow
interval method results are very accurate. For this reason, the Corp recom-
mends the use of the flow interval method in conjunction with a sampling
strategy to include high flow measurements.
23
-------
For streams with very limited data bases, the Corp developed the
Regional Phosphorus Load Model (PLM). PLM is basically a correlation between
(a): the flow times the difference between the measured phosphorus concen-
tration and the low-flow or base phosphorus concentration, divided by the
drainage basin area and (b): the flow divided by the drainage basin area
to the 0.85 power. The later quantity is a modified unit area contribution
of flow based on the idea that the most important periods of phosphorus flux
are storms and that during peak discharges only a fraction of the total area
near the stream contributes to the observed flow [18]. Data from the Maumee,
Portage, Sandusky, Huron, Chaguin, Vermilion, and Cattaraugus Rivers are
used in the correlation.
For streams with very little data, if the flow record and low flow
phosphorus concentration are known, fluxes may be calculated from the PLM
and used in the flow interval method to develop loadings.
Using the IJC method and OME data for the Detroit River; the flow inter-
val method and, where necessary, the PLM for other tributaries; and permit
information regarding direct discharges, the Corps has calculated total phos-
phorus loadings of various constituents to each basin of Dike Erie over the
period June 197^ to May 1975 [18].
The calculations to be discussed subsequently required monthly loadings
of additional parameters for the period 1970 to 1975. To develop this data
the Corps used a modification of the flow interval method and daily flow
information as well as the above information [19]. Loadings were calculated
for total phosphorus, ortho-phosphorus, organic nitrogen, ammonia nitrogen,
nitrate nitrogen, dissolved silica and chlorides, separated into point and
non-point sources. Plots are shown in figures 9-13. Note the peak which
consistently appears each spring for all the variables. The effect of this
peak is evident in the Western Basin concentration data for these variables
as shown in section (5). Spring basin concentrations especially for ammonia
nitrogen, total phosphorus and ortho phosphorus are significantly higher
than the rest of the year, reflecting this large winter-spring load and the
rapid response of the basin.
These loadings, developed by the Army Corp of Engineers, Buffalo Dis-
trict, are used in the subsequent calculation for both the 1970 calibration
year and the 1975 verification year. They are chosen among the available
loading estimates for two main reasons:
l) They provide monthly values, which is a reasonable; time scale
for the Western Basin.
2) Based on the information presented regarding the various methods
of mass loading estimates and the data sets existing for the
Detroit River, it appears that the methods and data used by the
Corp are the most consistent.
21*
-------
100.00
50.00
WESTERN BASIN
0.00
X
-------
20.00
10.00
o 0.00
x
(Q
O)
WESTERN BASIN
1970 ' 1971 I 1972 ' 1973 ' 1974 ' 1975
20.00
CO
ID
DC
I 10.00
Q_
CO
O
I
O 0.00
CENTRAL BASIN
1970 ! 1971 ' 1972 ' 1973 ' 1974 ' 1975
QC
O
20.00
10.00
0.00
EASTERN BASIN
1970 ' 1971
1 1975
Figure 10. Orthophosphorus Loading (10 kg/day) to Lake Erie
19TO-19T5.
26
-------
220.00
110.00
£ 0.00
^»
WESTERN BASIN
1970 ' 1971 ' 1972 ' 1973 ' 1974 ! 1975
(0
a
220.00 -
LJJ
o 110.00
cc
0.00
'
CENTRAL BASIN
1970 ' 1971 ! 1972 ' 1973 ' 1974 ! 1975
cc
O
220.00
110.00
0.00
EASTERN BASIN
1970 ' 1971 I 1972 ' 1973 ' 1974 ' 1975
Figure 11. Organic Nitrogen Loading (10 kg/day) to Lake Erie
1970-1975.
27
-------
60.00
30.00
2 o.oo
WESTERN BASIN
1970 ' 1971 ' 1972 ! 1973 ! 1974 ' 1975
CO
T3
O>
60.00
z
LU
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oc 30.00
o
* 0.00
o
oc
O
z
60.00
30.00
0.00
CENTRAL BASIN
1970 ' 1971 ' 1972 ' 1973 ' 1974 ' 1975
EASTERN BASIN
1970 ' 1971 ' 1972 ' 1973 ' 1974 ' 1975
Figure 12. Inorganic Nitrogen Loading (10 kg/day) to Lake Erie,
1970-1975-
28
-------
120.00
60.00
0.00
o
X
to
a
o>
o
_J
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Q
LLJ
WESTERN BASIN
1970 ' 1971 ' 1972 ' 1973 ' 1974 ' 1975
120.00
60.00
0.00
CENTRAL BASIN
1970 ' 1971 ' 1972 'l973 ' 1974 ' 1975
120.00
60.00
0.00
EASTERN BASIN
1970 ' 1971 ' 1972 ! 1973 ' 1974 ' 1975
Figure 13. Dissolved Silica Loading (10 kg/day) to Lake Erie,
1970-1975.
-------
ATMOSPHERIC LOADING
Estimates of the atmospheric loading to Lake Erie are given in the 1975
IJC report [l^]. The atmospheric phosphorus contribution, 560 MT/yr, is
insignificant in comparison to other loadings (^ 2%). However, the atmos-
pheric nitrogen load, 37,000 MT/yr, comprises about 15.5% of the total nitro-
gen load to the lake. The atmospheric loadings are assumed to be temporally
and spatially constant. The atmospheric phosphorus load is assumed to be in
the available form and the atmospheric nitrogen load is assumed to be ammonia
nitrogen. A summary of the loadings used in the calculation is given in
Tables A6-A9 of the appendix. They do not include the atmospheric loadings
which are inputted separately.
NITROGEN FIXATION
An additional source of available nitrogen is that fixed by the nitro-
gen-fixing algae themselves. Measurements of the rate of nitrogen fixation
have been made [20]. The maximum observed fixation rate is 0.057 mg N fixed/
mg TKN-day for a period of eleven days (day 233-2kh). For the TKN concentra-
tion calculated and observed to be present in the three basins it is possible
to compute the total mass of nitrogen fixed in the three basins at this maxi-
mal rate. These are shown in Table 6 below.
TABLE 6. NITROGEN FIXATION
Segment Mass of Nitrogen Fixed Percent of Lake
(MT/yr) Total N Load
Western Basin 2k 0.01
Central Basin
Epilimnion
Eastern Basin
Epilimnion
120
39
0.05
0.02
The quantities are negligible when compared with the other sources of
nitrogen to the lake.
30
-------
SECTION 5
DATA COMPILATION AND REDUCTION
HISTORICAL DATA
Declining fisheries, increasing algal production and anoxia have moti-
vated significant interest in the water quality of Lake Erie since the late
1920 's. As a result, a substantial historical data base exists.
The first intensive study of Lake Erie's water quality was conducted by
the Federal Water Pollution Control Administration (FWPCA) in 1963-1964. Of
the data collected prior to this, most are sparse both temporally and spa-
tially. In the FWPCA study, chemical, biological, and bacteriological
variables were measured at 158 stations located throughout the lake.
This study was followed in 1966 by the commencement of the Canada
Centre for Inland Waters (CCIW) yearly series of cruises. These cruises
included the entire lake and were carried out monthly from ice breakup to
the end of the year. A comprehensive set of chemical, biological, and bac-
teriological parameters are measured on each cruise. Table 7 lists the
sampling data collected for Lake Erie prior to 1970. Except for BOD data
compiled from 1967 CCIW cruises [21] , no data collected prior to 1970 are
used explicitly in this investigation. The 1967 BOD data are important
since they are the only available BOD data with extensive temporal and spa-
tial coverage.
CALIBRATION DATA SET
The criteria for selecting an appropriate calibration data set are
adequate temporal and spatial coverage with respect to the time and space
scales of the calculation, and data for all the state variables considered
in the calculation. For this investigation, data sets which provide esti-
mates of basin average epilimnion and hypolimnion concentrations of phyto-
plankton and zooplankton biomass, nutrients and dissolved oxygen at monthly
intervals are suitable. All major cruise work done on Lake Erie since 1970
has been concerned with variables associated with eutrophication and conse-
consequently comply with most of these requirements.
The majority of the lake data used in this project is obtained from two
sources:
1. STORET, U.S. Environmental Protection Agency (EPA)
2. Limnological Data Reports, Lake Erie, 1970, Canada Centre for
Inland Waters (CCIW)
31
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These data are supplemented with information reported in the literature.
The groups responsible for the collection of the data obtained through
STORET and the Limnological data reports are:
1. The Ohio State University Center for Lake Erie Area Research (CLEAR)
2. The State University College at Buffalo Great Lakes Laboratory (GLL)
3. Canada Centre for Inland Waters (CCIW)
Canada Centre for Inland Waters Cruise Data
CCIW has been involved in extensive data collection on Lake Erie since
1967. A set of cruises is carried out on the lake each year. These commence
following ice breakup and continue until December at approximately monthly
intervals. An extensive base of bacteriological, biological, chemical, and
physical parameters for the waters of Lake Erie are monitored on each cruise.
Spatial grids for these cruises are usually dense, 60 to 100 stations being
visited per cruise. At each station, samples are collected at the surface
and at three meter intervals to the bottom. At some of the deep stations in
the Eastern Basin, the sampling interval is five meters. A listing of the
data available from these cruises is shown in Table 8. The data for 1970
were obtained from the Limnological Data Reports [22]; 1975 data were acquired
through STORET.
Nutrient Control Program Cruise Data
The Nutrient Control Program is a three-year study sponsored by the U.S.
Environmental Protection Agency as part of the Large Lakes Program adminis-
tered by the EPA Grosse lie Laboratory. It is a comprehensive monitoring
program of the chemical and biological parameters of Lake Erie waters de-
signed to determine the effectiveness of nutrient control programs in abating
eutrophication. The groups responsible for the data collection and analysis
are CLEAR, the Ohio State University (Western and Central Basin) and GLL,
State University College at Buffalo (Eastern Basin). The program began in
1973. Each year a series of cruises are conducted, beginning after ice
breakup, and performed at monthly intervals during the spring and fall and
bi-monthly during the critical summer period. The parameters available from
these cruises are listed in table 8. All of the Nutrient Control Program
data used in this project were obtained through STORET.
Project Hypo Cruise Data
Project Hypo, an ambitious joint undertaking of the Canada Centre for
Inland Waters and the U.S. Environmental Protection Agency, was an exhaus-
tive study aimed at discovering the degree and causes of the depletion of
oxygen during the summer in the Central Basin of Lake Erie [23]. Sampling
work was performed from June through August 1970, although the bulk of the
data collected is the result of an intensive phase from July 27 to August 25.
Twenty-five stations in the Central Basin were sampled of which five were
termed "major stations". Sampling at these stations was intensive. During
the period July 27 to August 25, there were seven major cruises. Table 8
35
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includes a description of the data available from these cruises. At each
station, samples were taken one meter below the surface, one meter above the
thermocline and at one-meter intervals through the hypolimnion to one meter
above the bottom. In addition, at the five major stations, mid-thermocline
samples were taken where possible. The Project Hypo data were obtained
through STORET.
In addition to the cruise data, certain biological parameters have been
compiled from published information. The 1970 zooplankton carbon data are
from Watson [2k} 25]; 1970 gross primary production data are from Glooschenko
et al. [26], phytoplankton species composition data for 1970 are from Munawar
and Munawar [27].
All data are reduced to segment averages for each cruise. In addition,
the number of points, maximum value, minimum value and standard deviation
for each average are compiled.
These reduced data sets are:
1. 1970 Canada Centre for Inland Waters
2. Project Hypo, Canada Centre for Inland Waters and United States
Environmental Protection Agency, 1970
3. 1973 Lake Erie Nutrient Control Program, CLEAR; The Ohio State
University and GLL; State University College at Buffalo.
4. 197^ Lake Erie Nutrient Control Program, CLEAR and GLL
5. 1975 Lake Erie Nutrient Control Program, CLEAR and GLL
6. 1975 Canada Centre for Inland Waters
These data are compared in figures 1^-26 which present the segment averages
+_ standard deviations for 1970, 1973-197^ and 1975. Chlorophyll concentra-
tions are compared in fig. l4a for the Western Basin (WB) and Eastern Basin
(EB) epilimnion. The unimodal distribution of the WB sharply contrasts with
the bimodal distribution in the EB and Central Basin (CB) epilimnion, shown
in fig. 15. Also the similarity in the seasonal patterns for the three years
of data is apparent.
Total phosphorus for the WB, shown in fig. l6 exhibits a spring maxima
which then rapidly declines. As the mass loading data indicate, there is a
spring loading maxima as well. As shown subsequently, this is insufficient
to explain this observed peak and other sources of phosphorus, probably from
sediments resuspension, are involved. EB concentrations appear to be rela-
tively constant, as are CB epilimnion concentrations with the exception of
an early spring peak in 1975. CB hypolimnion concentrations, fig. 17, re-
flect increased phosphorus release during the anoxia in late July and
August. This is more clearly seen in the soluble reactive concentrations.
The WB and EB data for 1970, fig. 18, follow the expected pattern with
depletion during the periods of phytoplankton growth and the recovery due to
overturn and recycle. CB hypolimnion data, fig. 19, clearly show the effects
of anoxia commencing in late July in both 1970 and 1973-7^. The distinct
increase of soluble reactive phosphorus reflects the anaerobic release from
37
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the sediments. The absence of this phenomena in 1975 is due to the aerobic
conditions that prevailed throughout that year as shown subsequently.
Ammonia concentrations, fig. 20 and 21, are small and somewhat erratic
in the WB, EB and CB epilimnion, whereas the CB hypolimnion reflects the
anaerobic release as did the soluble reactive phosphorus. Nitrate concentra-
tions follow the pattern of seasonal uptake in the surface segments, as shown
in fig. 22 and 23, whereas an increasing pattern appears in the CB hypolim-
nion as nitrogen is recycled.
Reactive Silica concentrations reflect the uptake of silica by diatom
growth in the spring in all epilimnion segments as shown in fig. 2k and 25.
As diatoms are replaced by non-silica using algae the concentrations increase
due to aerobic regeneration. Anaerobic regeneration in the CB hypolimnion
is also clearly seen in 1970 with less regeneration occuring in 1975 during
aerobic conditions.
Dissolved oxygen concentrations for the CB are shown in fig. (26). The
epilimnion is nearly saturated but the average hypolimnion DO concentration
decreases to approximately 1.0 mg/£ in both 1970 and 1973-7^ just before
overturn. 1975 is unique, however, with an average concentration above k.O
mg/H. This accounts for the lack of anaerobic regeneration of nutrients in
1975, since as shown subsequently, if the average segment 5 concentration ex-
ceeds k mg/£, no anoxia is recorded at any of the sampling stations.
The overall impression of similarity between the different years indi-
cates that the total mass loadings of nutrients to the lake have not substan-
tially changed during this period. This is confirmed by the U.S. Army Corps
of Engineer estimates. However, there is a striking difference in the CB
hypolimnion dissolved oxygen between 1970, 1973-7^, and 1975. The cause for
this unusual behavior is discussed in the verification section.
The initial calibration of the kinetics is accomplished using the 1970
CCIW data and Project Hypo data. The data for 1973 and 197^ are also
included in the calibration because of the similarity of in-lake conditions
over the period 1970 to 197^. The 1975 data is used for the independent
verification of the model. The fact that the dissolved oxygen data for 1975
differs markedly from the 1970-197^ pattern makes this data set ideally
suited for verification. Both the data collected by CCIW and the data col-
lected by CLEAR and GLL are used in the calibration and verification calcu-
lations.
51
-------
SECTION 6
KINETICS
INTRODUCTION
The calculation employed in this investigation is made up of two, essen-
tially independent, components - the kinetics and the transport. This dis-
tinction is basic to mass balance calculations. The kinetics control the
rates of interconversions among the dependent variables and are independent
of location per se although they are functions of exogenous variables such as
temperature that vary with location. Thus the rate at which any reaction
proceeds is controlled only by the local concentrations of the variables that
appear in its reaction rate expression. The same expression is used for each
location be it the Western Basin or the Eastern Basin hypolimnion.
It is in this sense that the formulation is an attempt to utilize funda-
mental principles: biological and chemical reactions have no fundamental
relation to position, they are only affected by concentrations, light, and
temperature. Hence the kinetic constants which appear in these expressions
cannot vary from segment to segment.
The kinetics employed are designed to simulate the annual cycle of phy-
toplankton production, its relation to the supply of nutrients, and the
effect on dissolved oxygen. The calculation is based on formulating the
kinetics which govern the interactions of the biota and the forms of the
nutrients and applying them to the regions of Lake Erie within the context of
conservation of mass equations. The fifteen variables for which these calcu-
lations are performed are:
Phytoplankton
1. Diatom Chlorophyll-a
2. Non-Diatom Chlorophyll-a
Zooplankton
3. Herbivorous Zooplankton carbon
1*. Carnivorous Zooplankton carbon
Nitrogen
5. Detrital Organic Nitrogen
6. Ammonia Nitrogen
7. Nitrate Nitrogen
Phosphorus
8. Unavailable Phosphorus
9. Soluble Reactive Phosphorus
52
-------
Silica
10. Unavailable Silica
11. Soluble Reactive Silica
Carbon, Hydrogen, Oxygen
12. Detrital Organic Carbon
13. Dissolved Inorganic Carbon
Ik. Alkalinity
15. Dissolved Oxygen
An important criteria for the inclusion of variables in the calculation
is the existence of adequate field data for the variable, as well as its im-
portance in the processes being considered. For example it is clear that
the recycling mechanisms are dependent on the bacterial population and per-
haps the microzooplankton so that a direct calculation of the decomposer
population is desirable. However no comprehensive decomposer biomass data
are available and a direct calibration is not possible. Since the reaction
kinetic constants are not known a priori their values would be chosen based
on only indirect evidence, that is, on how well the calculation as a whole
reproduced the data for the other variables being considered. These addi-
tional constants and dynamical equations add significantly to the available
degrees of freedom in the calculation without an equivalent increase in the
data available for their estimation. Thus one's ability to uniquely specify
a consistent set of kinetic constants diminishes rapidly and the calculation
becomes more speculative as a consequence. Unfortunately no adequate statis-
tical theory of estimation is available to guide these choices so that the
exact point at which this division is to be made is uncertain at present and
is largely a matter of intuitive judgement.
The calculation presented below is based on a strict application of this
principle of parsimony, consistent with the objectives of the analysis. The
only variables for which no direct or indirect data are available are una-
vailable silica and the fractions of zooplankton biomass which are herbivor-
ous and carnivorous. Thus the zooplankton dynamics are necessarily somewhat
speculative and cannot be thought of as having a strong foundation in obser-
vational fact. The inclusion of unavailable silica is necessary in order
that the observed recycling can be calculated in a way consistent with mass
balance principles. In this instance that requirement is judged to super-
cede the requirement of parsimony.
This principle also provides the motivation for the inclusion of inor-
ganic carbon and alkalinity in the calculations. The concern is that the
reactions involving oxygen are correctly represented. Since total dissolved
carbon dioxide as well as oxygen is produced or consumed in these reactions,
if this variable is included it is possible to check the results against
observations of pH if alkalinity is also known. Alkalinity data are avail-
able and its production and destruction can be calculated for each reaction
based on the appropriate stoichiometric ratio, the only additional constant
required. Thus adding total inorganic carbon and alkalinity to the calcula-
tion actually decreases the degrees of freedom since more data are available
for comparison with no increase in constants that need to be estimated for
53
-------
the inorganic carbon component of the reactions. A computational complica-
tion is added since only C02(aq) and not total inorganic carbon is trans-
ferred across the air-water interface but methods for accomplishing this
calculation are available [29].
The interactions of the variables are shown in figures 27-28 which are
representations of the nutrient cycles. Consider phosphorus: the avail-
able phosphorus is utilized by both phytoplankton groups for growth. Phos-
phorus is returned from the biomass pools to the two phosphorus species
through several mechanisms. It is returned directly from the phytoplankton
through endogenous respiration and non-predatory mortality. Herbivorous
zooplankton obtain phosphorus through predation of the phytoplankton and
return it through excretion and death. Carnivorous zooplankton preying on
herbivorous zooplankton return phosphorus by the same mechanisms. The phos-
phorus returned from the biomass systems is partly in the available and
partly in the unavailable form. Unavailable phosphorus is converted to
available phosphorus at a temperature dependent rate.
The silica species interact in the same manner as the phosphorus species
with certain key differences: available silica is utilized only by the dia-
toms; all silica taken up by herbivorous zooplankton through predation is
excreted; and the silica returned from the biomass systems is assumed to be
entirely in the unavailable form.
The kinetics of the nitrogen species are fundamentally the same as the
other nutrients. Ammonia and nitrate are used by the two phytoplankton
systems for growth. The rate at which each are taken up is proportional to
its concentration relative to the total inorganic nitrogen available. The
nitrogen returning from the biomass systems follows pathways that are analo-
gous to the phosphorus.
Dissolved oxygen is coupled to the other state variables. The two
sources of oxygen considered are reaeration and the evolution by phytoplank-
ton during growth. The sinks of oxygen are oxidation of detrital carbon,
nitrification, and respiration by the phytoplankton and zooplankton. The
details of these reactions are presented below.
PHYTOPLANKTON GROWTH AND DEATH
The need to differentiate between diatom and non-diatom chlorophyll
arises from a silica depletion that occurs during the spring bloom as is
clear from the data presented previously. The formulation of the growth
kinetics for each class is identical to that employed in the Lake Huron-
Saginaw Bay analysis [30] with the exception that the silica cycle is explic-
it in the calculation and a nutrient limitation term for reactive silica is
included in the diatom growth rate equation. The Michaelis-Menton expression
is used which multiplies the other terms of the rate expression for diatoms.
The temperature and light dependences are identical to those employed pre-
viously and have been discussed at length [30].
The nutrient dependency employed relates the growth rate to the external
-------
DISSOLVED OXYGEN
1
DIATOM
CHLOROPHYLL
t
AMMONIA
NITROGEN
TOTAL
INORGANIC
CARBON
k
I
»
1
OTHERS
CHLOROPHYLL
t
NITRATE
NITROGEN
NON-LIVING
ORGANIC
CARBON
1
Reaeration
DISSOLVED
OXYGEN
I
Dee
Oxy
p sediment
gen demand
PHOSPHORUS SPECIES
CARNIVOROUS
ZOOPLANKTON
CARBON
DIATOM
CHLOROPHYLL
OTHERS
CHLOROPHYLL
*
UNAVAILABLE
PHOSPHORUS
*
AVAILABLE
PHOSPHORUS
Figure 27. Lake Erie State Variable Interactions. Representations
of dissolved oxygen (top) and phosphorus (bottom) nutri-
ent cycles.
55
-------
NITROGEN SPECIES
DIATOM
CHLOROPHYLL
CARNIVOROUS
ZOOPLANKTON
CARBON
HERBIVOROUS
ZOOPLANKTON
CARBON
OTHERS
CHLOROPHYLL
NON-LIVING
ORGANIC
NITROGEN
AMMONIA
NITROGEN
NITRATE
NITROGEN
SILICA SPECIES
UNAVAILABLE
SILICA
HERBIVOROUS
ZOOPLANKTON
CARBON
DIATOM
CHLOROPHYLL
AVAILABLE
SILICA
Figure 28. Lake Erie State Variable Interactions. Representations
of nitrogen (top) and silica (bottom) nutrient cycles.
-------
nutrient concentrations rather than the internal cellular concentration as
observed in laboratory chemostat experiments. Since the validity of this
approach is a point of some contention, a rather complete analysis of the
consequences of this approximation is presented below. The results show that
indeed the external nutrient concentration controls the growth rate, albeit
indirectly via the internal concentration. The dynamics of the cellular
uptake and internal concentration are important only at short time scales
which are not significant in seasonal calculations. The variation of cellu-
lar stoichiometry can be important, however, and the constant stoichiometry
used in the subsequent calculations must be regarded as an approximation.
Nutrient Dependence of Phytoplankton Growth
The relationships between the growth of phytoplankton and the concentra-
tion of the nutrients: carbon, nitrogen, phosphorus and, for diatoms, silica
have been extensively investigated [see 31, for a review]. A primary diffi-
culty in presenting a coherent quantitative description of the results is
that the nutrient uptake and growth kinetics at the short time scales char-
acteristic of laboratory batch experiments are quite involved and depend on
the previous history of the population. In order to circumvent this diffi-
culty to some degree, and to investigate relationships occurring at longer
time scales, investigations of growth kinetics in chemostats have been under-
taken. The population is subject to an influent of constant composition and
a constant dilution rate. At steady state the growth rate is reduced to the
dilution rate due to a depletion of the nutrient in shortest supply. By
operating the chemostat at varying dilution rates it is possible to observe
the state of the population at various growth rates and nutrient concentra-
tions.
For these experiments the most successful correlation of growth rate, y,
is to the cellular nutrient concentration or cell quota, q, in units of
nutrient mass /individual cell, and is expressed by the equation [32]:
V-* 1 - (1)
; «
where y' is the theoretical maximum growth rate at infinite cellular nutri-
m
ent content and q is the minimum cell quota at which growth ceases . These
two constants are directly estimated from a Droop plot of yq versus q and,
therefore, form a convenient starting point for the analysis. This equation
can also be cast in the form:
where the half saturation constant of the excess nutrient quota, q - q , is
q itself. Such behavior has been observed for phosphorus, nitrogen, silica,
iron, and vitamin B ? limited chemostats [31].
Nutrients are transported across the cell wall at a rate which depends
on both the internal and external nutrient concentrations [31,33] as well as
other physiologically important variables such as temperature and light
57
-------
intensity which, for this analysis, are assumed to "be constant and nonlim-
iting. If S is the external nutrient concentration, then it has been ob-
served, primarily using short term batch kinetic experiments, that the rate
of cellular uptake, v, in units of nutrient mass/cell/unit time, is related
to the external nutrient concentration, S, via a Michaelis-Menton equation:
v =
V S
m
K + S
m
(3)
where V is the maximum uptake rate at a particular cell quota and K is the
half saturation constant for uptake.
A simple relationship between growth rate and external nutrient concen-.
tration exists if the population is assumed to be independent of the internal
cellular concentration and if the population is at steady state since, for
this latter condition, the uptake rate of nutrients must balance the synthe-
sis rate of cells, i.e.
v = yq
from which it follows that [32]:
where:
and
y =
m
K + S
y'3
y = pm
m Ii?
K
K =
s
m
1+3
v
3 =
m
q y'
o m
(M
(5)
(6)
(T)
(8)
For this important special case the growth rate is related to the external
nutrient concentration as in the Monod theory of bacterial growth, with
the important modification that the half saturation constant for growth, K ,
is always less than the half saturation constant for cellular nutrient uptake
K , and y is always less than y'. In addition it has been assumed that the
parameters: y', q ,K , and V are constants. The validity of this assump-
tion is examined subsequently. What is important to realize is that for the
common case of cellular equilibrium for which eq. (k) applies, the growth
rate of the population is indeed related to the external nutrient concentra-
tion as shown by eq. (5), which is a direct consequence of the growth eq.
(l) and the uptake eq. (3).
The important dimensionless parameter of the theory is 3: the ratio of
V /q , the maximum specific uptake rate at q = q ; to y', the maximum growth
rate°at q -* °°. For large 3 the apparent half saturation constant for growth
is small relative to the half saturation constant for uptake: K - K /3«
s in
The reason is that, with 3 large, the rate of nutrient uptake is large rela-
-------
tive to the growth rate so that a large reduction in uptake rate can be
tolerated (i.e. S « K ) while still supplying adequate nutrient for growth.
Table 9 presents these experimentally determined parameters for various
phytoplankton species. With phosphorus as the limiting nutrient, 3 is quite
large, 3 ^ 50-200, indicating that K « K . For cases where both these con-
S HI
stants have been measured the estimate of K using eq. (7) compares quite
well with the observation with a single exception for which the measured
3 = 18 also seems small. Thus the half saturation constant for growth can be
expected to be in the range K - 0.01-0.05 yM (0.3-1.5 yg PO.-P/A) whereas
the cellular uptake half saturation constant is much larger: K = 0.4-1.9 yM
(12-60 yg PO^-P/£). m
With nitrogen as the limiting nutrient 3 is smaller, 3 - 5-25, although
still large enough for K < K . The half saturation constant for growth is
s m
estimated to be in the range K - 0.03-0.3 yM (0.42 - 4.2 ygN/£) and K has
s m
been observed in the range K - 0.3-3.0 (4.2 - 42.0 ygN/£).
For silica limitation 3 - 1-3 which indicates that K is approximately
S
one half of K . The ranges are K - 0.7-4.5 yM (20 - 130 ygSi/£) and K -
in s in
0.5-7-7 yM (14 - 220 ygSi/£). For species with both K and K measured, the
s m
application of eq. (7) again yields satisfactory agreement, as shown in
Table 9 indicating that the relationship between the cellular parameters and
the half saturation constant for growth are in reasonable accord with the
previous analysis.
A similar relationship, eq. (6), exists between the maximum growth rate
at infinite cell quota, y1, and the maximum growth rate at infinite external
substrate concentration y . An interesting point arises if these parameters
m
are estimated for an algal specie using different limiting nutrients. Two
examples are presented in Table 7» with y calculated from the appropriate
parameters in Table 9 and eq. (6). These experiments were conducted using
the same conditions and cell culture so the estimates are directly compar-
able. It is clear that the cellular maximum growth rates y' are quite dif-
ferent whereas the maximum growth rate corresponding to the jVIonod expression
y , are much closer.
m
There is a fundamental reason for this result. While it is possible to
increase the external nutrient concentration so that S » K and y - y , it
is not possible to arbitrarily set q » q so that y ~ y' since q is con-
trolled by the ratio of substrate uptake rate to growth rate. In fact it is
easy to see that at external concentration S, and steady state,
V K +S
m s
m m
which, for S » K , yields as the maximum cell quota at large external
nutrient concentration:
59
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61
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= qo(l+3)
(10)
Since 3 is quite different for phosphorus, nitrogen, and silica the maximum
cell quotas achievable, q^, are also different, corresponding to different
fractions of the y' in the cell quota equation, with the actually achievable
maximum growth rate being y . Thus it is the parameter of significance.
The dimensionless parameter 3, the ratio of maximum uptake to maximum
growth rate sets the ratio of the extremes of the cell quota, q^/q =1+3,
for the possible range of steady states. For phosphorus this ratio is cal-
culated to be ^ 100 whereas for nitrogen it is ^ 10 and for silica it is
^ 2. Thus calculated cell quota for phosphorus is quite variable, for nitro-
gen less so, and for silica even less. As shown below, however, the observed
TABLE 10. COMPARISON OF GROWTH RATES
y
m
(day'1)
y
m
(day'1)
Reference
Asterionella
formosa
Cyclotella
Meneghiniana
P
O.T
0.69
N Si
1.21
- 1.16
P N Si
0.70 - 0.85
0.65
0.77
Tilman & Kilham,
1976 [35]
Tilman & Kilham,
1976 [35]
variation in cell quota for phosphorus is actually much less than that pre-
dicted by eq. (10), because the uptake rate is a function not only of the
external nutrient concentration, S, but also of the internal cell quota q.
Nutrient Uptake
The principal assumption underlying the previous analysis is that the
are constant.
growth parameters: y' and q
and uptake parameters: K and V
^ mm
At constant light and temperature it appears that y' and q are reasonably
constant indicating that the Droop eq. (l) is a correct representation of
growth rate versus cell quota, although some recent data, for Skeletonema
costatus indicates that at high growth rates, y > 1.5 day" the correlation
changes slope abruptly (see Table 9).
The nutrient uptake rate varies not only with external concentration
but also as a function of cell quota. An analysis of this behavior by Rhee
[3^] indicates that the uptake rate of phosphorus for Scenedemus is of the
form:
62
-------
(11)
K +S i+K.
m i
where i is the cellular concentration of labile polyphosphates. If it is
assumed that i = q - q then eq. (ll) has the form:
V S K , ,
_m q (12)
K +S K + (q-q )
m q 0
where K =K.. Using Droop's eq. (l) for the growth rate, it is possible to
calculate, at cellular equilibrium, the magnitude of the possible varia-
tion in cell quota:
- - = f [ -1 + A + WC] (13)
%
where £ - K /q and q^ is the value of q as S -* °°. If K -» °° so that eq. (12)
reduces to the Michaelis-Menton expression, eq. (3), Aq^ = 8 which for the
large 3 characteristic of phosphorus uptake, implies an enormous variation.
For example, Scenedemus with 3 = 56 would be expected to exhibit a Aq^ =
56 with K -> °°. However, with K /q = 0.1+5 (using K = K. = 0.73 f-mol/cell
= h.8 using eq. (13), in comparison to an observed Aq^ =6.1
Thus the reduction of v with increasing q is quite significant in reducing
the variation in expected cellular composition of phosphorus from a fifty-
fold to a five-fold variation.
The variation of cellular stoichiometry based on eq. (13) is shown in
fig. 29a for a range of 3 characteristic of silica, nitrogen, and phosphorus
kinetic constants. Large variations in cellular composition occur for large
8 and large K /q whereas if the internal concentration tends to limit the
uptake rate then the range in stoichiometry is less. The effect is expected
to be most significant for phosphorus and least significant for silica.
Ah alternate expression for uptake as a function of external and internal
substrate concentration has been incorporated in a species-dependent analysis
of phytoplankton growth by Bierman [^2], A similar analysis of the variation
of internal stoichiometry yields Aq^ - k for phosphorus uptake of diatoms.
It is pertinent to note that the effect of the modification of the
uptake rate incorporated in eq. (12) has only a small effect on the shape
of the resulting growth versus external substrate concentration relation. In
fact, it is very well approximated by the Michaelis-Menton expression. To
see this, define an apparent half saturation constant for growth, K' , and
S
observe that the relationship between q and S can be obtained from the
equilibrium equation v = yq, yielding:
63
-------
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o
q
d
O
d
q
d
jo
q
d
oo
CN
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ID
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1 - <10
= i [ /I + 4 pg/C - 1 ]
d.
where p = S/K +S. The cell quota at which the growth rate is one half the
maximum, q ,, satisfies the equation:
which when substituted into eq. (1*0 yields S = K1 , the apparent half satura-
s
tion constant. The relevant formulae are given in the Appendix of this sec-
tion. The result is shown in fig. 29b. For the range of B characteristic
of silica limited growth and, to a less extent for nitrogen, the ratio is not
sensitive to the. magnitude of the internal cellular dependence of uptake. In
the region characteristic of phosphorus the effect is more pronounced. Note
that in all cases K1 is less than K .
s m
Since eq. (l^) relates q to S, and eq. (l) relates y to q, it is possi-
ble to calculate y as a function of S implied by these equations. A compar-
ison of these results to the Monod expression:
y s
is shown in fig. 30. The formulae are given in the Appendix to this section.
The variation of growth rate is well approximated by the Michaelis-Menton
expression regardless of the magnitude of |3. The value of K /q used in this
figure produces the largest calculated differences with other values showing
even less deviation. Thus the difference is of no practical importance, in-
dicating that the Monod expression with the proper half saturation constant
is a valid characterization of the growth rate of phytoplankton species at
cellular equilibrium.
The difficulty in using the classical Monod theory occurs in the uptake
expression, v = yq, since the cell quota can vary as the external concentra-
tions vary whereas in the conventional Monod theory the cell quota, or the
analogous nutrient to biomass stoichiometric ratios, are held constant. The
practical import of this phenomena, which is related to the choice of the
measure of biomass employed, is discussed below.
Validity of Cellular Equilibrium Approximation
The analysis of the previous section depends on the cellular equilibrium
approximation, that is v = yq. This is certainly true for steady state
chemostat experiments but its applicability to dynamic situations is open to
question. The dynamic equation for q(t) follows a mass balance of the cell
itself:
q = v - yq (IT)
65
-------
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o
where q = dq/dt. Consider a perturbation analysis about steady state values
* # *
q , S and let £ = q - q . Substituting these values and ignoring all
higher order terms in £ yields:
o *
e = - (y1 + - ~ - ) £ (18)
* m K + q* - q q
<1 o
* * *
where v is the uptake expression, eq. (12), evaluated at q and S . But
* *
from eq. (l),v = y' (q -q), and eq. (l8) becomes:
** - * (i * rw> ^
* *
where Aq = (q - q )/q and £ = K /q as before. Thus e has an exponen-
tially decaying solution:
£ = £ (o) e~t/Tq (20)
with time constant:
(21)
The relationship between y1 and the actual growth rate of the -population,
* m
y , involves two factors: the relationship between y1 and the actual maxi-
mum growth rate at infinite external substrate concentration, y ; and a
factor f, the reduction due to the finite external nutrient concentration:
*
y = f y . With these reductions the time constant becomes:
m
Ty (22)
*
where T = 1/y , the time constant of phytoplankton population growth. The
first two fractions in the above expression are both bounded between 1/2 and
1 for all possible values of Aq^ and £. Thus it follows that:
₯f Ty < \ < f Ty (23)
67
-------
Since growth at less than one-half the maximum rate is often the case;
f < 1/2, and:
The conclusion drawn from this analysis is that the cells equilibriate with a
time constant of approximately one-half to one-eighth the time constant for
population changes. It is, therefore, quite reasonable to assume that the
cells are at equilibrium even during population growth since the internal
concentration can equilibriate a good deal more rapidly than the rate at
which population can change.
Dynamic Equations at Cellular Equilibrium
In view of the preceding analysis consider the behavior of a population
which is responding to an abrupt increase in external nutrient concentration.
Initially as the concentration of limiting nutrient increases the cells are
no longer in equilibrium and begin to assimilate the nutrient thereby in-
creasing their cell quota and in turn their growth rate. The population then
starts to increase. The results of the previous analysis suggest, however,
that the cells reach cellular equilibrium more rapidly than the cells can
grow. In other words, the cell uptake occurs over a length of time during
which the population size is not changing appreciably. Thus it is possible
that the cells achieve a new cell quota which is in equilibrium, i.e. v = yq,
before the population size has reacted appreciably. Thus a picture of dyna-
mic equilibrium emerges where the cells are continuously in equilibrium with
the slowly changing external substrate concentration. As the population and
substrate concentrations change the cell quota responds a,nd changes also,
but always more rapidly, so that the cells continuously achieve a condition
of cellular equilibrium throughout the response to the external concentration
change. Since this is a considerably simplified picture of how phytoplankton
cells respond to substrate variations it is of interest to formulate the
equations which apply to this approximation, which can be called cellular
dynamic equilibrium.
The formulation of the growth kinetic equations at cellular euilibrium
follow from the general equations for dynamic cell quota by a simple trans-
formation. It is an example if a general technique that can be used to re-
move fast reactions from general mass balance equations [29]. Consider the
dynamic equations that apply to a chemostat :
N = y^(l - qQ/q) - N/tQ (25)
q (26)
S = - vN + (S. - S)/t (27)
i o
where N(t) is the number of cells /£; t is the detention time of the chemo-
stat, the reciprocal of the dilution rate; S. is the influent substrate
68
-------
concentration; and the dot implies a time derivative. The remaining variables
have been defined previously. In order to derive the equations which apply
at cellular equilibrium it is necessary to apply the equation v = yq. However
o
it is not true that q = 0 as would be expected from eq. (26), since this im-
plies a constant cell stoichiometry and as is clear from the description of
the expected behavior the cell stoichiometry is changing in response to the
changing conditions. Therefore, although v = yq id s good approximation,
o
q = o is not and it is necessary to find equations which are valid regardless
of the rate at which cell uptake occurs. The transformation that yields
equations which do not involve the dynamic cell uptake reaction requires that
a component variable be identified which is unaffected by that reaction. In
this case a suitable variable is:
ST = S + qN (28)
the sum of the substrate concentration in the chemostat and the equivalent
cell substrate concentration. The dynamic equation for S^ is obtained
directly from the time derivative of eq. (28);
o o o o
S = S + qN + qN
Using eqs. (25, 26, 27) and substituting, the result is:
(29)
Note that the total substrate concentration is a quantity unaffected by the
kinetics of growth and uptake. The growth equation is also unaffected by
the uptake reaction so it is applicable:
N = y^ (1 - qo/q)N - N/tQ ( 30 )
However in order to apply this equation the cell quota is required. It is
obtained by solving the cell equilibrium equation, v = yq, i.e.:
B- K 9 -.
m q o
and the component equation
S = qN = ST (32)
simultaneously for the unknowns, S(t) and q(t), given the total substrate
concentration S (t) and the cell numbers N(t). The solution involves four
simultaneous equations: two differential equations for N(t) and S (t) and
two algebraic equations for q(t) and S(t). The computational method is
discussed in the Appendix to this section.
Some typical results are shown in fig. 31 using the parameters character-
69
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istic of Scenedesmus with phosphate as the limiting nutrient [Ta"ble 9]. The
comparison is between the solution of the dynamic equations (25)-(27) and the
cellular equilibrium approximations, eqs. (29)-(32). The initial condition
is specified as the equilibrium which is reached for an influent of S. = 10
yg-PQ - P/i. At t = o the influent is abruptly changed to S. = 20 ygPO, -P/i
and the population reacts by increasing in size. The cell equilibrium solu-
tion reacts more quickly with an increase in q(t) relative to the dynamic
equation. As a consequence the phosphate concentration at cell equilibrium
decreases more rapidly than the dynamic solution. However the dynamic solu-
tion soon catches up and the error in q(t) for the cell equilibrium approxi-
mation decreases rapidly as predicted by the perturbation analysis. The
residual error in phosphorus concentration takes a somewhat longer time to
decay as can be seen.
The effect of 3, the ratio of maximum uptake rate to growth rate, can
be seen in fig. 32. As 3 decreases the error in q(t) tends to increase
slightly and the time for the error to decay increases. The largest dis-
crepancies occur for small 3 and large K /q for which T - f T and the cells
are not equilibriating very rapidly. In general, however the approximation
appears to be quite satisfactory with errors decreasing to less than 10% in
all variables rather rapidly.
Discussion and Conclusions
It is important to note that the cellular equilibrium approximation is
essentially a computational simplification. The rapid dynamics of the inter-
nal cellular nutrient concentrations are eliminated which in turn removes the
possibility of cells having different internal concentrations within a volume
segment, since, in this approximation, cells equilibriate immediately as they
are transported to other volume segments. However these kinetics still allow
the cell nutrient content to adjust to changes in external concentrations.
The question of the necessity of even this refinement in practical compu-
tation is still unresolved. If algal species are aggregated into functional
groups, e.g. diatoms, greens, etc., then the question of an appropriate bio-
mass measure arises since aggregated cell counts are no longer reasonable.
However in one case at least [3^] it has been shown that Droop kinetics do
not apply if the species biovolume is used to measure population size. Hence
the fundamental basis of these kinetics for biomass growth comes into question.
Further for aggregated populations the five kinetic constants that specify the
kinetics: y1 , K , V , q , K , are known only within ranges and their final
m' m' m' ^o' q' J e
values are dependent on a calibration to field data. By contrast the Monod
theory requires three constants: y , K' and a nutrient/biomass ratio. Since
m s
the data available for calibration is the same for both kinetic formulations
the two additional constants add additional degrees of freedom which may add
uncertainty to the final values obtained from the calibration. At present it
is uncertain whether the increased realism at the expense of an increase in
the number of kinetic constants that specify details of this more realistic
behavior is an advantageous tradeoff.
71
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It is likely that variable stoicbiometry is important in computing the
nutrient budget of a lake, particularly for phosphorous. Consider the nutri-
ent flux to the bottom sediment via algal settling. For that fraction of the
population which settles during periods of non-nutrient limitation the cell
quota is low in the epilimnion, as these cells settle into the hypolimnion if
they are exposed to higher external nutrient concentrations then their cell
quota will increase. Thus the nutrient flux to the sediment is larger than
what would be calculated if the minimum stoichiometry were employed through-
out the calculation.
For constant stoichiometry kinetics the minimum stoichiometry is, in
fact, often chosen since the yield of biomass at nutrient limitation is cal-
culated correctly. Hence it would appear that nutrient fluxes to the sediment
are underestimated for this choice of the nutrient/biomass ratio. If, on the
other hand a stoichiometric ratio is chosen which is more representative on
non-nutrient limited conditions then the computed maximum population will be
an underestimate. Hence it appears that variable stoichiometry can be im-
portant for refined nutrient budgeted calculations. The computational results
indicate that the cellular equilibrium approximation is adequate if the rate
of change of the substrate concentration is slow relative to phytoplankton
growth rate. This is typically the case in natural settings. However in
laboratory experiments, particularly in batch kinetic investigations, these
conditions are not met and the fully dynamic equations are necessary.
If this mechanism is judged to be important then the inclusion of vari-
able cell stoichiometry is necessary.
MULTIPLE NUTRIENT LIMITATION
The proper formulation of the growth and uptake kinetics for multiple
limiting nutrients is an area which is under investigation. The proposals
to date make use of the relationships obtained from single limiting nutrient
investigations and composite them in various ways. If y , yp , ..., y are
the growth rate expressions as functions of nutrient concentrations S S_ ,
..., S with maximum rates: y , y? , ..., y then the proposals are:
Multiplicative [k3,kk]:
y
1m 2m nm
Minimum [^5] :
y = Min(y1, y2, , yn) (3k)
Reciprocal [k6] :
ym _ 1 ,lm 2m ^nm^ ,, \
(35)
- n .»
73
-------
The experimental investigations have been concerned either with growth
kinetics or uptake kinetics. For the former, Droop [^7] proposed the multi-
plicative expression but found subsequently that experiments using vitamin
B, p and phosphorus corresponded to the minimum expression [^5], Bierman et
al: [33] in an anlysis of Rhee's data for nitrogen and phosphorus limitation
of Scenedesmus [hQ] showed that the minimum expression correctly reproduced
the experimental data. It is interesting to note that for the latter inves-
tigation the multiplicative expression is satisfactory if y is increased.
m
Thus in situations for which absolute growth rates are unknown the difference
between the two expressions is in the leading constant. For uptake kinetics,
the data of Ketchum [^9] was analyzed using the multiplicative expression
and found to conform.
The majority of evidence to date supports the minimum hypothesis. How-
ever the multiplicative formulation usually fits the variations in experi-
mental data as well, if not their absolute magnitude, as in the case of
Scenedesmus. Therefore from an operational point of view either form is a
reasonable representation of limitation, although the minimum is probably
preferable.
MEASURES OF BIOMASS
The kinetics discussed in the previous sections have been shown to be
applicable to a single species in well-controlled environments. For appli-
cation to natural settings it is necessary to aggregate the individual species
into manageable groupings and to characterize their kinetics as a whole. The
initial requirement is an appropriate measure of the aggregated population
size. For single species, the direct measure of population size is the number
of cells/unit volume. For naturally occurring populations even this measure
may be somewhat ambiguous: it is difficult to distinguish viable and non-
viable cells, and colonial species pose a problem since tie count usually does
not distinguish individual cells and the sizes of the colonies are quite
variable.
The sum of the numbers of each species, the total count, is a possibility
but since cell sizes vary substantially the nanoplankton would dominate such
an aggregation. To account for this, the total biovolume or wet weight of
phytoplankton, assuming unity density, can be calculated using characterises
volumes for each identified species. Unfortunately volumes can vary apprec-
iably as a function of nutrient status. And, in fact, the growth kinetic
equation (l) does not appear to apply, for Scenedemus at least, if cell quota
is calculated on a cell volume basis [3^]. Conversion to phytoplankton dry
weight and carbon involves further species dependent constants which are also
nutrient dependent and, therefore, are subject to variation and uncertainty.
Thus although the use of phytoplankton dry weight or carbon concentration is
an appealing solution to the issue of aggregation, it suffers from some prac-
tical difficulties.
An alternate to this problem is to measure a parameter which is charac-
teristic of all phytoplankton, namely, chlorophyll-a, and use this as the
-------
aggregated variable. The principle advantages are that the measurement is
direct; it integrates cell types, age, and it accounts for cell viability.
The principle disadvantages are that it is a community measurement vith no
differentiation of functional groups (e.g. diatoms), and that it is not nec-
essarily a good measurement of standing crop in carbon or dry weight units
since the chlorophyll to dry weight and carbon ratios are variable.
As can be seen from the above discussion, no simple aggregate measurement
is entirely satisfactory. Further, the use of detailed kinetics which are
satisfactory for single species are not necessarily appropriate with biomass
or chlorophyll as the population variable. Hence the analysis of naturally
occurring phytoplankton populations as aggregates of species involves a number
of simplifications and assumptions, with the result that the actual kinetics
employed, although guided by the results discussed in the previous sections,
are of necessity empirical.
From a pratical point of view the availability of extensive chlorophyll
data for the Great Lakes essentially dictates its use as the aggregate mea-
sure of the phytoplankton population. These data, together with somewhat lim-
ited species and biomass data, form the basis of the analysis to be discussed
subsequently.
STOICHIOMETRY AND UPTAKE KINETICS
If it is assumed that the growth kinetics discussed in the previous
sections are applicable to chlorophyll-a as the measure of aggregated popu-
lation, then a Monod expression for growth, with a suitable half saturation
constant, is appropriate. In order to specify the nutrient uptake accom-
panying this growth, it is necessary to specify the population stoichiometry
in units of mass of nutrient uptake/mass of population synthesized. For
chlorophyll as the population measure, the relevant ratios are the mass of
carbon, nitrogen, phosphorus, and silica per unit mass of chlorophyll-a. A
selection of these ratios has been presented [W], At first glance the vari-
ability of these ratios is large and the use of constant ratios in the analy-
sis is questionable. However, it is clear that the reason these ratios vary
is the varying cellular content of nutrients and chlorophyll which is, in turn,
a function of the external nutrient concentration and the past history of the
population. Large ratios correspond to excess nutrients and small ratios
correspond to that nutrient limiting the growth rate. Thus the choice of the
relevant ratio can be made with the situation of interest in mind. Since the
population to be discussed subsequently are primarily phosphorus and/or silica
limited, the stoichiometry chosen can reflect these facts. The operational
consequences of this choice is that the population stoichiometry under non-
limiting conditions will be underestimate but the maximum chlorophyll concen-
trations under limiting conditions should be correctly estimated. Hence the
trade-off is a probable lack of realism during a portion of the year versus
a correct estimate of population chlorophyll during the period of nutrient
limitation. Since this is usually the critical period, and most questions
to be answered are usually sensitive to the maximum population size, this
choice is a practical expedient.
75
-------
KINETIC FORMULATIONS
The kinetics employed for the calculations to be presented subsequently
are based on the considerations discussed in the previous sections as well as
the experience accumulated during the performance of similar calculations for
Lake Ontario [50], Lake Huron and Saginaw Bay [30] and elsewhere [51]. The
format of the presentation given below is designed to be complete and de-
tailed and yet provide an overview of the reactions.
Table 11 presents the formulation and kinetic constants used for the
phytoplankton growth reaction. The left hand side of the reaction stoichio-
metric equation specifies the quantity of the reactants which are consumed by
the reaction and the right hand side specifies the end products. The square
bracketed species represent concentrations, not necessarily in molar units,
but in any convenient set of mass units. The stoichiometric coefficients must,
of course, be consistent with these units. The specie with a unity stoichio-
metric coefficient corresponds to the units of the reaction rate. Thus for
phytoplankton growth the reaction rate has units yg Ch£-a/£/day consistent
with the stoichiometric equation. The growth kinetics chosen for this inves-
tigation incorporate the temperature and light intensity variations discussed
previously [30].' The nutrient dependency follows a Michaelis-Menton expres-
sion external substrate concentration with the proper half-saturation constant
which, as shown in Fig. 30, is an entirely adequate approximation to the re-
sults of the kinetics derived from Droop's growth equation, Rhee's uptake
equation, and the assumption of cellular equilibrium. Multiple nutrient lim-
itation is incorporated using the product formulation although the minimum
expression would yield comparable results. The stoichiometry of the popula-
tion is assumed to be constant, and, for the limiting nutrients, at values
close to their minimum requirement. As discussed above this appears to be a
practical solution to the problem of fluctuating cellular stoichiometry.
The stoichiometry of the reaction varies in two ways. Inorganic nitro-
gen uptake is either ammonia or nitrate depending on the ratio of ammonia to
total inorganic nitrogen. Nitrite nitrogen is ignored since its concentration
is so low. For ammonia and nitrate uptake alkalinity changes occur, consis-
tent with experimental observation [52]. Ammonia uptake causes a decrease in
alkalinity. For nitrate uptake the initial step is reduction to ammonia which
produces oxygen and alkalinity after which the reaction continues as before.
Gross primary production as measured in a shipboard incubator utilizes
the same reaction rates except that the light limitation term is not depth
averaged since the illumination is uniform for this measurement.
The reaction rate constants are comparable to those used for Lakes
Ontario and Huron although there are some differences due primarily to the
division of phytoplankton biomass into diatom and non-diatom fractions.
Phytoplankton endogenous respiration and non-predation mortality, Table
12, is essentially the reverse of the growth reaction. However only a por-
tion of the organic carbon is directly oxidized to COp. The remainder is
detrital carbon. The estimate used of the fraction of the algal carbon which
76
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is refractory is 30%. Similarly only a fraction of the cellular nitrogen and
phosphorus is released in the available form. This is estimated to be 50%.
It is assumed that all the silica released is in the unavailable form. The
sources for these estimates have been presented [30].
The rational and experimental data for the zooplankton formulations,
presented in Tables 13, lU, and 15 have been presented elsewhere [30]. Table
16 presents the formulations for nitrification. The rate is reduced as oxy-
gen concentration approaches the half saturation constant. Denitrification,
which is only effective at very low or zero oxygen concentrations is only
active in the sediment segments, as presented in Table 17.
The recycle reaction which transform the unavailable nutrient to their
available forms are listed in Table 18. Their rates are dependent on the
chlorophyll concentration as in the Lake Huron calculation [30].
Since dissolved oxygen is of major concern in Lake Erie, the rationale
for these reactions is detailed below.
DISSOLVED OXYGEN KINETICS
The reduction of dissolved oxygen in the Central Basin hypolimnion dur-
ing stratification is a consequence of the aerobic respiratory processes in
the water column and the anaerobic respirations in the sediments. Both these
processes are significant contributors and, therefore, it is necessary to
formulate their kinetics explicitly.
The methodology for the analysis of dissolved oxygen dynamics in natural
waters, particularly in streams, rivers, and estuaries is reasonably well
developed [53]- The long history of applications have focused primarily on
the use of biochemical oxygen demand (BOD) as the measure of the quantity of
oxygen demanding material and its rate of oxidation as the controlling kine-
tic reaction. This has proven to be appropriate for waters receiving a het-
erogeneous combination of organic wastes of municipal and industrial origin
since an aggregate measure of their potential effect is a great simplifi-
cation that reduces a complex problem to one of tractable dimensions.
The situation in Lake Erie is somewhat simpler in this respect since
the major source of oxygen demanding material is photosynthetic primary pro-
duction and the kinetics that control are the aerobic respiration of the
phytoplankton, and the bacterial oxidation of detrital organic carbon. The
complicating feature is the interplay of the aerobic reactions in the water
column and the anaerobic reactions in the sediment.
WATER COLUMN PRODUCTION AND RESPIRATION
A byproduct of photosynthetic carbon fixation is the production of
dissolved oxygen. The reaction stoichiometry is given in Table 11. The rate
of nutrient uptake and oxygen production are all proportional to the growth
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-------
rate of the population since its stoichiometry is fixed.
The reaction describing algal respiration is similar to the growth
reaction since it is the reverse process. However during algal respiration a
fraction of the organic carbon, f , is oxidized directly, while a portion
OX
is not, becoming detrital organic carbon. The reaction is given in Table 12.
The detrital organic carbon oxidation reaction proceeds separately and its
stoichiometry is given in Table 18. This is the reaction that is analogous
to the classical BOD oxidation reaction and it is formulated as first order
with respect to the organic carbon concentration. These three reactions,
together with the oxygen consumed during nitrification, are the water column
kinetic reactions being considered in this calculation.
It is important to realize that both the rates of oxygen production and
community respiration are comparable to direct field observation. Gross
primary production corresponds to the rate at which reaction 1 proceeds,
whereas the classical BOD measurement is, in this formulation, the rate of
oxygen consumption due to the respiration reaction, reaction 2, and the
organic carbon oxidation, reaction 8. Hence these measurements provide
direct calibration data with which the kinetic constants can be estimated.
The formulation for the sediment reactions involves a more detailed
explanation of sediment mass transport and kinetics and these are presented
in the next section.
87
-------
Appendix to Section 6: Mathematical Details
1. Apparent Half Saturation Constant
The equations which result from solving eqs. lU and 15 and which are
presented in fig. 29b are:
£"& (A1>
m
with:
a = £ a (1 + a)/B (A2)
a = V/(2 + v£) (A3)
V = | [ /I + 43/5 - 1 1 (AH)
Note that :
£v (A5)
2. Growth Rate - Substrate Variations
For the cellular equilibrium theory,
" "i (1 - TTT;' (A6)
and
with Aq and Aq^ given by eqs. lh and 13.
To obtain p = S/(K + S) in terms of the dimensionless substrate concen-
m
tration S/K' note that
S
S/K KS/K1
' m _ s
P ~ 1 + S/K ~ 1 + KS/K1
m s
The ratio of eq. (A6) is compared to
S/K'
m
88
-------
in Fig. 30.
3. Solution of the Cellular Equilibrium Equations
The dynamic equations for the cellular equilibrium approximation
ST = (Si - ST)/tQ (A9)
H = Um (1 - qo/q)N - N/tQ (A10)
are solved numerically using a simple Euler one step method. At each evalu-
ation of the derivatives, the value of q = q(N,S ) is obtained by solving
the nonlinear equations for y = Aq and S:
y (qQN) + S = ST - qQN (All)
m
using the Newton-Raphson method. For the solution at t = zero the initial
estimates are S = S and y = Aq^. For subsequent evaluations as the solution
of the previous time step is used as the initial estimate in the Newton-
Raphson method.
89
-------
SECTION 7
SEDIMENT-WATER INTERACTIONS: MASS TRANSPORT AND KINETICS
The interactions between sediments and lake waters can have profound
effects on the concentrations of oxygen and nutrients in a comparatively
shallow lake such as Lake Erie since the areal fluxes from the sediment are
substantial sources on a volumetric basis. Additionally the occurrence of
anoxia dramatically increases certain of these fluxes. The details of the
mechanisms responsible for this increase are as yet unclear but they are re-
lated to a set of complex redox reactions that change the state and concen-
trations of various nutrients and metals thereby releasing bound nutrients.
The anoxia itself is, in part, a result of sedimentary processes which re-
sult in the exertion of an oxygen demand at the sediment water interface.
The importance of these processes and their interrelation to the processes
in the overlying water requires that they be analyzed within a framework
that is consistent with that discussed in the previous sections. The analy-
sis is begun by considering the nitrogen components of the calculation since
there already exist reasonably well formulated methods for the description
of the ammonia distributions in sediments. The application of analogous
methods to oxygen and the mechanism of benthal oxygen demand generation re-
quires additional theory and is presented next. The result is a novel
approach to the analysis of benthal oxygen demand. The analysis of the
fluxes of phosphorus and silica is basically empirical and concludes the
discussion.
NITROGEN
The methods to be employed in this calculation are based on one-dimen-
sional formulations of mass transport and reaction kinetics in the inter-
stitial waters of the sediment. The simplest of these consider the concen-
tration distribution of substances which are either essentially unreactive
or, as in the case of radionuclides , have known decay ra.te [5^,55]- The
basis is the one-dimensional mass transport equation for the concentration
of a dissolved substance c(z,t):
3z
where D is the effective diffusion coefficient of the substance, w is the
sedimentation velocity induced by sedimentation, relative to a coordinate
system fixed with respect to the sediment surface, and S is the net source
or sink of the material. The equation is more complicated if porosity
variations and compaction are taken into account [56], but the principle
is the same.
90
-------
Consider the application of this framework to the nitrogen species in
a reducing sediment [57]. Particulate organic nitrogen is hydrolyzed to
ammonia by bacterial action within the sediment. If this reaction is assumed
to be first order with respect to the organic nitrogen concentration, the
equations are:
3[NH ] 82[NH ] 3[NH
£- - D - - + w - = K [Org-N] (38)
9z
where [ Org-N] is the concentration of particulate sedimentary organic nitro-
gen and [NH ] is the concentration of ammonia nitrogen in the interstitial
water. Note that the sedimentary organic nitrogen does not diffuse since it
is part of the sediment solid. The effects of stirring by benthic organism
are neglected in this simple analysis. At steady state the solutions are:
[ Org-N] (z) = [Org-N]Q e"Kz/W (39)
[Org-N] ,
[NH l(z) = [NH 1 + - (1 - e'KZ/W) (1*0)
where [ Org-N] and [NH ] are the concentrations at the sediment surface,
z=o. The result of the conversion is an exponential decrease in sedimentary
organic nitrogen and a consequent increase in ammonia. In a reducing sedi-
ment ammonia appears to be conservative so there are no other kinetic sources
or sinks.
The importance of this result is that ammonia and organic nitrogen mea-
2
surements provide a method of estimating the parameter groups K/w and KD/w .
If the sedimentation velocity is known from other information the decay rate
and effective diffusion coefficient can be estimated.
The outgoing flux of ammonia through the sediment water interface can be
calculated directly from eq.
= D
9z
= J
n (in;
z=o
Org-N 1+n
n
where n = KD/w and J is the flux of particulate organic nitrogen into
the sediment: J^ = w [Org-N] . As the dimensionless parameter I] in-
creases the fraction of this incoming flux of particulate nitrogen that is
released to the overlying water as an ammonia flux increases. This result
is quite interesting since it relates the fraction of the regenerated nitro-
91
-------
gen to three parameters: K, D, and v. As the reaction rate for the conver-
sion of particulate sedimentary nitrogen to ammonia increases, more ammonia
is generated which tends to increase its flux. A decreased sedimentation
velocity increases the residence time of aprticulate nitrogen in the sediment
which increases the time available for regeneration and therefore the frac-
tion of the incoming flux that is returned to the overlying water.
In the application to Lake Erie one layer of sediment is considered.
This simplification is introduced in order that the computations are of man-
ageable proportions, and are consistent with the rather crude vertical seg-
mentation of the water column. The choice of the depth of the sediment layer
is based on the depth of sediment mixing induced by the benthic organisms
(bioturbation) . Although this depth varies considerably it was judged at the
time that a depth of 5 cm approximated this depth in the Central Basin [58].
It is assumed that a similar depth applies to the other basins. More refined
computations would involve multiple sediment layers which include both the
organism induced mixing and the interstitial water diffusion. This was
judged to be beyond the scope of this initial computation of sediment behav-
ior.
For a one layer sediment, the mass balance equations are considerably
simplified. However the principle is the same. For one layer of sediment
with thickness H, the mass balance equations are:
3 _ 3] + K[Org-N] (Us)
H
where E' is the effective exchange rate. It is assumed in these equations
that the ammonia concentration in the overlying water is small relative to
the interstitial water concentration. The steady state solutions are:
[Org-H] = J°rg-" (MO
»(1 * S)
("5)
KH KH2
The outgoing flux of ammonia: J = E'[NH ]/H, becomes
(kg)
vH
92
-------
which is a function of the Peclet number: wH/E', and a dimensionless reaction
rate velocity ratio: w/KH, As shown subsequently in section 9, the appro-
priate values for Lake Erie are wH/E1 = O.OOM and w/KH = 2.9 so that ^ 25%
of the incoming detrital organic nitrogen flux is returned to the overlying
water as ammonia.
In addition to the ammonia produced by the hydrolysis of refractory or-
ganic nitrogen in the sediment, ammonia is generated by the anaerobic decom-
position of algae. In a study of this reaction [59] it has been shown that
the anaerobic rate of the decay of algae is substantial (0. 007-0. 022 day ).
However the end product is not exclusively ammonia. Rather a fraction of the
algal nitrogen, f , becomes refractory organic nitrogen. The equations for
this and the hydrolysis reaction are:
dc J we
tr - ir - TT - Kici
dc J we
dT = iT - -T - K2C2 + KlfrCl
dc ,
=-c-c+Kc + K(l-f)e (U9)
where c = [Algal-N], c = [Refractory Org-N], c = [NH ] , with the sub-
scripts also denoting the appropriate fluxes and reaction rates. The steady
state solutions for the sediment concentration of algal and refractory nitro
gen are straightforward:
c = - (50)
J2 , frJl
°
2 K H K H K
-+V ^+1^KH + K
The ammonia flux from both these decomposition reactions is:
J2 Jl fr
1 (52)
It has three parts: that due to the decay of the refractory organic nitro-
93
-------
gen; that due to the direct decay of the labile (l-f ) algal nitrogen; and
that due to the sequential decay of the remaining refractory algal nitrogen,
f . With the reaction rate appropriate for the algal decomposition, w/HK =
0.008, so that nearly all (99%} of "the ammonia produced by direct decomposi-
tion of the labile algal nitrogen, which is (l-f ) fraction of the total
algal nitrogen flux, is returned. However only 25% of the refractory algal
fraction is returned to the overlying water.
The analysis of the sediment nitrogen concentrations and the resulting
flux of ammonia are comparatively straightforward because of the simplicity
of the kinetics: hydrolysis and anaerobic algal decay produce a stable end
product, ammonia, which undergoes no further reactions in the anaerobic sed-
iment. This is not the case for the reduced carbon compounds and the other
nutrients which are released by the sedimentary decomposition reactions.
CARBON AND OXYGEN
The reactions which convert algal and refractory carbon to their end
products are more complex. The initial step in which the algal and refrac-
tory carbon are converted to reactive intermediates appears to be similar to
the refractory organic and algal nitrogen degradation and in the subsequent
calculations the rates for carbon and nitrogen decomposition are assumed to
be equal. However the reactive intermediates participate in further reac-
tions: for example volatile acids react to become methane, and the mechanisms
that control these reactions are somewhat uncertain. In addition few mea-
sueements of these intermediate species are available and a calculation which
incorporates their concentrations explicitly would of necessity be specula-
tive. Instead one is led to seek a simplified yet realistic formulation of
these reactions.
The method employed is based on separating the initial reactions that
convert sedimentary organic matter into reactive intermediates, and the re-
maining redox reactions that are occurring. The initial decomposition reac-
tions can be expressed as half reactions in terms of the components chosen
for the representation [60], For example, with CO H ,, e~ and HO as the
components for carbon, hydrogen, electrons, and water, the half reactions
are:
[Algal-C] > UI;L [C02] + u12 [H+] + u13 [e~] + u^ [H,,0] (53)
[Org-C] -> U21 [C02] + U22 [H+] + U23 [e~]
The half-reaction stoichiometric coefficients y.. are positive, zero, or
J ^-
negative as appropriate. If the detailed kinetics were being considered,
these half reactions would be combined with the half reactions of the inter-
mediates to obtain the stoichiometric equation for that reaction.
-------
In the reducing environment of sediments, it is expected that reduced
carbon compounds such as volatile acids and methane are the end products of
the reactions. Let [A.] i=l,...,N represent the concentrations of these N
IS S
intermediate and endproduct chemical species. Assume that there exists N
redox reactions involving these species with stoichiometric coefficients v
for the i species in the j reaction. Let the rate of reaction be R . .
J
These reactions can be represented as stoichiometric equations:
N R.
s J
Z v A 0 j=l,...,N (55)
i=l J
occurring at reaction rate R.. Consider the conservation of mass equations
J
for the interstitial water concentration of these species in a one-dimen-
sional vertical analysis. Let D. be the diffusion coefficient of species
A. and let w. be the advective velocity of A., as before. For the N re-
11 J i r
dox reactions involving the species A. , the rate at which A. is produced by
reaction j is V..R.. Let S. be the net source of A. due to the sedimentary
Ji J i i
decomposition reactions, which are not included in the redox reaction set.
For this situation the conservation of mass equations for the concentration
of each species, [A.], is
9[A.] 92[A ] 3[A ] Nr
- - D. - ~ + w. = S. + I V..R. i=l,. ..,N (56)
3t i 9z2 i 9z i .=1 ji j s
To solve these equations explicitly requires explicit formulations for
the sediment kinetics reaction rate expressions, R . Since these reactions
J
are quite complex a complete formulation would be difficult. Instead an
alternate method is proposed. The crux of the idea is to eliminate these
terms, Z. v.. R., which cause the difficulty and replace eqs. (56) with an
J J i J
alternate set of equations. The following fact [29] leads to a convenient
choice for the transformation: Let B., j=l,...,N be the names of N com-
j c c
ponents and let a. . be the quantity of component B. in species A., i.e. its
stoichiometry in terms of the components. The choice of the components of
the intermediates is arbitrary but the conventional ones are those used for
the half reactions of the decomposition reaction, eqs. (53) and (5M« With
the components established it can be shown [29] that, for a set of component
conserving reactions, the formula matrix with elements a. . , is orthogonal to
the reaction matrix with elements V ., i.e.: 1J
J -^-
a = ° *=!,.. .,N; J=l,..-*N (57)
95
-------
This fact suggests that the conservation equations be transformed by multi-
plying eqs. (56) by the transpose of the formula matrix yielding:
N 2 N N N
Es (* - D. + w. I-) a., [A.] = Is a..S. + Zr R, ESa.1v,. (58)
1=1 3t X 9z2 X 8Z lk X 1=1 lk X J=l J 1=1 lk Jl
But the orthogonality relation, eq. (57), implies that the terms involving
R. are zero. This is the fundamental simplification which makes the analysis
J
tractable. The transformed equations are no longer functions of the redox
reactions; they are influenced only by the species mass transport coeffici-
ents and the decomposition reaction sources, S.. It is clear from the devi-
ation of the transformed equations that the method is directly applicable to
more general equations which consider temporally and spatially variable para-
meters [56].
If D. and w. are not species dependent, the summations in the left-hand
side of the above equation become the component concentrations, [B ], by the
k
definition of a... Therefore the transformation yields conservation of mass
ij
equations for the N component concentrations:
9[B ] 32[B ] 3[B ] Ns
Z a S k=l,...,Nc (59)
i=l
The critical and surprising result is that these mass balance equations are
independent of the details of the redox reactions and, indeed, of the species
concentrations themselves. Rather they are only functions of the component
concentrations. It suffices, therefore, to compute only the component con-
centrations which can be treated in exactly the same way as any other vari-
able in the mass transport calculation, without regard to the redox reactions,
so long as the mass transport coefficients are independent of the species.
This later restriction is admittedly severe since certain species, e.g. meth-
ane, can form gas bubbles and would be transported quite differently. Thus
the results using this assumption must be regarded as preliminary.
The convenient choice of components for the calculation are those that
parallel the aqueous variables: [CO ] and [0_], for the carbon and oxygen com-
ponents of organic matter. Restricting the calculation to these components
eliminates the possibility of explicitly including the effects of other re-
duced species such as iron, manganese and sulfide which play a role in over-
all redox reactions and may be involved in the generation of "t;he sediment
oxygen demand. In addition the changes in alkalinity that occur, due to the
component H , are also neglected although marked changes in alkalinity do in
fact occur. These simplifications appear reasonable in light of the prelimi-
nary nature of the sediment calculation.
96
-------
The decomposition reactions vhich drive the component mass balance equa-
tions are the anaerobic decomposition of the algal carbon, eq. (53), and the
anaerobic breakdown of the sedimentary organic carbon, eq. (54), where the
stoichiometric equations can now be thought of as being in terms of the com-
ponents. The substitution for the electron component in terms of the oxygen
component is simply 1/4 0 = 1/2 H?0 - H + e~. Both reactions are sinks of
the component, [0 ], and rapidly drive its concentration negative, indicating
that the sediment is reduced rather than oxidized. The negative concentra-
tions achieved can be thought of as the oxygen equivalents of the reduced end
products produced by the chains of redox reactions occurring in the sediment.
Since the mass balance equations (59) are independent of the actual species
concentrations, it is not necessary to calculate the actual distribution
among the species. It should be pointed out that this result is a direct
consequence of the species-independent assumption, and a more realistic cal-
culation would require such a determination .
Using the one layer approximation the equations for algal and refractory
carbon parallel those for nitrogen. The equations for the components are a
one-layer version of the mass transport eq. (59):
d[CO ]
] ~ H [C°2] + yil V1'^^1^1'^ + y21K2[Org-C] (60)
F-
-££=- = - ~ [02] - | [02] + y12K1(l-fr)[Algal-C] + y22
H
where y , y are stoichiometric coefficients for [CO ] in eqs. (53) and
(5^); y1?» ypo are the oxygen stoichiometric coefficients which are one
fourth of the electron stoichiometric coefficients. Again the steady state
solution is straightforward and the component fluxes are
^ y21JOrg-C _ yilJAlgal-C (l _ f , fr ) (g
with exactly the same expression for the flux of the 0 component with y
and y replaced by y12 and y^.
The mass balance calculations in the sediment are in terms of the compo
nents. Strictly speaking the mass balance calculations in the water column
should also be transformed into component equations, with the species distri
bution obtained by a separate calculation involving an additional principle
such as the requirement that the species be at redox equilibrium. For the
carbon and oxygen system this is exactly what is done. Since the calculated
concentration of the oxygen component is positive in the overlying water it
97
-------
is assumed that the reduced carbon species that are transported across the
sediment water interface combine with the available oxygen and are oxidized
to CO and HO with a consequent reduction of 0 in the overlying water. No
remaining reduced species are assumed to be present so that the resulting 0
component is equal to the 0 species concentration if it is positive. If it
is negative the result is interpreted as the oxygen equivalent of the reduced
species produced by the redox reactions but no attempt is made to be more
specific. In this way the need for actually calculating the concentrations
of the reduced species is eliminated and the carbon and oxygen balance can be
calculated in a way that is exactly parallel to the nitrogen balance.
Within this framework the benthal oxygen demand exerted by the sediment
tie flux of the 0 component through the sed:
terms of the previous equations, the result is:
is the flux of the 0 component through the sediment water interface. In
J f
T _ I Org C Algal-C , r
J - a^ s ~ + ll-f +
2
(63)
where a is the oxygen to carbon ratio of algal and organic carbon. This
°2C
expression relates the benthal oxygen demand to the incoming fluxes of algal
and detrital carbon. The result is entirely parallel to the ammonia flux and
the underlying reason for this parallelism is the assumption that all elec-
tron exchanging reactions have species independent transport coefficients.
To the extent that this is not the case, e.g. gas bubbles of methane or sig-
nificant quantities of reduced precipitates forming within the sediment, this
equation is in error. It is, however, appealing in its simplicity and util-
ity. A detailed investigation of the applicability of this type of analysis
where the assumption of species independent transport is not made is cur-
rently underway. For the Lake Erie calculations presented subsequently in
this report the differential equations describing the approximation, eqs.
(60) and (6l) are numerically integrated. As shown subsequently the results
are quite close to the analytical steady state solutions.
The reaction stoichiometry and rate equations are presented in Table 19.
The rate constant, temperature coefficient, and fraction available for the
algal decomposition reaction are obtained from anaerobic decomposition exper-
iments [59]. The organic decomposition rates are obtained from the calibra-
tion as discussed in section 9-
The results of the dissolved oxygen and sediment calibrations indicate
that the sediment oxygen demand generated by the decomposition in the surface
layer is insufficient. This is not unexpected since the active depth in sed-
iments is usually considerably more than 5 cm. [58]- To remedy this defect
an additional flux of oxygen equivalents the deep sediment demand, is intro-
duced into the bottom of the sediment segments. It represents the flux of
reduced carbon from the deep sediment into the shallow sediment segments.
98
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100
-------
Its magnitude is estimated from the calibration as described in section 9-
PHOSPHORUS AND SILICA
The success of the above calculation depends directly on the degree to
which species independent transport is a realistic approximation. If the
majority of species being considered remain dissolved in the interstitial
water of the sediment then the above analysis is reasonable. For phosphorus
and silica it is well known [6l,62] that the formation of precipitates
affects the interstitial water concentrations and therefore, the species in-
dependent transport assumption is not valid. Further the dissolved concen-
trations are also affected by redox reactions which in turn affect the nutri-
ent fluxes that occur during aerobic and anaerobic conditions. Hence a com-
plete analysis of the phosphorus and silica fluxes requires a rather elabor-
ate computation of the solution-precipitate chemistry and its interaction
with the mass transport of the dissolved species. Such a calculation was
judged to be outside the scope of this project. Instead an empirical
approach is adopted. Observed interstitial water concentrations of phos-
phorus and silica are used rather than those which are calculated from the
integration of the mass balance equations. During aerobic conditions the
diffusive exchange of phosphorus and silica is assumed not to occur and E'
for these constituents is set to zero. During anaerobic conditions the dif-
fusive exchange is set to the same value used for ammonia, carbon, and oxy-
gen. The flux of phosphorus and silica is due to the exchange between the
overlying water and the observed interstitial concentrations of phosphorus
and silica. However since the area of anoxia is not equal to the entire area
of the sediment-water interface the flux is assumed to be proportional to the
ratio of the anoxic area to the total area of the interface. The computation
of the anoxic area is based on the segment dissolved oxygen concentration and
is discussed in section 9-
This approach, while not wholly satisfactory, is at least consistent
with the framework within which the fluxes of the other materials are being
generated. What is being assumed is that under aerobic conditions the fluxes
of phosphorus and silica are small relative to all other sources whereas
under anaerobic conditions the prevailing concentrations in the sediment
interstitial water exchange freely with the overlying water.
101
-------
SECTION 8
MASS TRANSPORT CALIBRATION
A realistic representation of the mass transport mechanisms is a basic
requirement for an analysis "based on conservation of mass. For a finite dif-
ference formulation the transport "between the homogeneous segments is requir-
ed. The transport mechanisms are of two types: the unidirectional net mo-
tion of the water between segments which is termed the advection; and the tur-
bulent mixing and circulations due to velocity and density gradients between
segments which is termed the dispersion.
Ideally the transport regime ought to be the result of a hydrodynamic
analysis. However the relevant calculations have not yet been made and, al-
though models of large lake circulation exist, a number of uncertainties in-
cluding their level of verification at seasonal time scales preclude a com-
pletely hydrodynamic analysis. The results of circulation measurements and
calculations are combined with an analysis of inert tracers and temperature
in order to estimate the exchange coefficients between segments. Their
values are chosen to be consistent with known details of the circulation
patterns and the observed concentrations of the tracers.
HYDRODYNAMIC CIRCULATION
A number of studies which include both measurements and calculations of
Lake Erie currents are available which set the framework for the specifica-
tion of transport patterns in Lake Erie [63,6U]. Unfortunately the time and
space scales of these results tend to be smaller than the seasonal and basin-
wide scales that are appropriate to this analysis. Thus these results must
be supplemented with calibrations based on the seasonal distribution of
basinwide averaged concentration of tracer constituents. Nevertheless the
hydrodynamic studies can be used to set the order of magnitude of the trans-
port coefficients.
The horizontal wind driven circulation of the lake dominates the inter-
basin exchanges and an analysis of these velocity fields provides the first
estimates of these exchange coefficients. Consider the calculated current
patterns [63] across the boundary of the Western and Central Basins shown in
fig. 33 for a wind direction of W6TS and a magnitude of 5-2 m/sec. based on
the prevailing direction and magnitude for Lake Erie winds [23]. Flow is
either from the Western Basin to the Central Basin or vice versa. For each
direction the component of the velocity normal to the boundary is estimated
at each depth for which current velocities are reported: O.U, 1.5, 3.0, ^.5,
6.0, 7.6 and 9-1 meters. The velocities are weighted according to the per-
centage of the total depth they represent and are averaged to produce the
average normal velocities which exchange flow across this boundary. The re-
sult is a bi-directional velocity of 3-05 cm/sec. The range over the depths
102
-------
Wind direction: W67S Wind magnititude: 5.2m/sec
WESTERN BASIN
Depth: 3.0 meters
/ S «. ^ / /
CENTRAL BASIN
WESTERN BASIN Depth: 9.1 meters
CENTRAL BASIN
Miles 0
10
0
1 ft/sec
Km 0 7.5 15
Distance
0 20 cm /sec
Current magnitude
Figure 33- Calculated current patterns across the "boundary of the Western
and Central Basins based on the prevailing direction and magni-
tude of Lake Erie winds (direction of W 67 S and magnitude of
5.2 m/sec) at a depth of 3 meters (top) and 9.1 meters (bottom).
103
-------
is l.U to U.7 cm/sec. For the purposes of comparison the unidirectional ad
vective component of velocity due to the Detroit River flow is 3-7 cm/sec.
Thus the circulating and advective velocities are of the same order of mag-
nitude.
By comparison the horizontal dispersion to be expected across the boun
dary is an order of magnitude less. Consider the empirical formula for
oceanic diffusion [65], expressed as an exchanging velocity:
V = 0.0103 £'
ex
for V = E/& in cm/sec and £ in cm. For a length scale on the order of the
ex
midpoints of the Western and Central basin, Si - 1.6 ' 10 cm and V =0.12
ex
cm/sec. Thus the principle mechanisms of mass transport across the Western-
Central Basin interface are due to the unidirectional advective velocity and
the bi-directional circulating velocities induced by "Che wind shear and modi-
fied by the coriolis force and lake geometry.
In order to proceed with a purely hydrodynamic analysis it would be
necessary to first verify that the circulation model employed indeed repro-
duces the observed circulation magnitude and direction. Then a simulation
of the seasonal circulation using observed wind patterns would be undertaken.
The exchanging flows could then be extracted from these results. Lacking
such a calculation, a more empirical approach has been adopted which employs
the observed distribution of tracers.
HEAT BALANCE
The temperature distribution in Lake Erie is the result of the inter-
action between the heat fluxes to the lake and the circulation. If the
former were known then the latter could be estimated by fitting the observed
temperature distribution to that calculated for a variety of horizontal ex-
changing velocities and vertical dispersion coefficients. The advantage of
such a precedure is that it is at the proper temporal and spatial scales
since a seasonal computation using the appropriate segmentation is compared
to the appropriately averaged concentrations. Hence the resulting exchang-
ing velocities and vertical dispersion coefficients are directly applicable
to other dissolved constituents.
Heat balance calculations can be attempted from first principles with
the exogenous sources specified from other measurements and the within lake
sinks parameterized as a function of air and water temperature, and relative
humidity [66]. Such a procedure is a reasonably large undertaking. Instead
a more direct estimation method has been developed [30]. The idea is to cal-
culate the net heat flux to the entire lake using the observed temperatures
directly. The observed heat change between cruises for each segment is com-
puted from the equation:
AH. = V. (T.. -T. .)pC (6U)
i i ik ij 'K p
-------
where
AH. = the change in heat content for segment i (calories)
1 -3
V. = the volume of segment i (cm )
.
1
o
where
T' = the volume average temperature of segment i for cruise k ( C)
IK o
T. . = the volume average temperature of segment i for cruise j ( C)
1J o
p = the density of water (gm/cm )
o
C = the specific heat of water (cal/gm- C)
The segment heat changes are summed to yield the total lake heat change:
n
AH = I AH. (65)
i=l
The net average daily flux between cruises is then:
AH
JT= A (J-t.) <66)
s k j
2
-------
COMPARISON OF CALCULATED HEAT FLUX FUNCTIONS
FOR 1970 AND 1975
LAKE ERIE CENTRAL BASIN
400 -
CALCULATED
--- ESTIMATED
1975
CALCULATED
-400 -
Figure
Calculated heat flux (cal/cm -day) to the Central Basin using
observed temperatures for each cruise. During the winter period,
the net heat flux shown is estimated.
106
-------
is known from observations. Hence it is possible to calculate the net heat
flux to each basin. In particular applying eq. (67) to the entire surface
area of the lake yields:
JT = r ^ ^ Asi = - K (T - E)
s i
where:
T = ~ Z T .A . (69)
A . SI 31
S 1
the areal surface average temperature. And substituting eq. (68) into eq.
(67) yields:
J.. = JT - K(Tsi - T) (70)
so that the surface temperatures can be used to calculate the net heat flux
into each basin provided that the heat transfer coefficient is known. It is,
in principle, possible to compute the heat transfer coefficient from meteoro-
logical data. However there are substantial uncertainties in such a proced-
ure. In particular the expression used for evaporation is open to question
and the lack of actual over lake meteorological observations make such a cal-
culation speculative. For these reasons it was judged that a constant value
was sufficient. For Lake Huron this coefficient is estimated to be 50-65
P
cal/cm /day [67]- A value of K/pC = 2 ft/day (0.67 m/day) corresponding to
2
6l cal/cm /day is chosen for the analysis. The results are listed in Table
20. These fluxes provide the forcing functions for the heat balance calcu-
lation.
The conservation equation for the segment average temperature T. is:
dT.
V. -- = Z E. . (T. -T. ) + Z Q. .T. + A .J. (71)
i dt . ki k i . Tti k si i
k k
where V. is the segment volume, E . is the exchanging flow and Q . is the
i ki Ki
advective flow between segment k and i. The integration of this equation- for
all segment yields the calculated segment temperatures. The method is app-
lied to both 1970 and 1975- The two years differed not only in net heat flux
but also in the depth at which the thermocline formed. Since the logical
place for the segment boundary between the epilimnion and hypolimnion is
within the thermocline, the volumes of the Central Basin segments are adjus-
ted to reflect this fact. The data used to establish these locations are
shown in fig. (35). The average temperatures in two meter sections of the
Central Basin are shown for the cruise dates as indicated. While 17 m. is
the appropriate depth of the epilimnion in 1970, a shallower thermocline
formed at 13 m. in 1975.
107
-------
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CALIBRATION AND VERIFICATION YEAR
TEMPERATURE PROFILES USED FOR
THERMOCLINE DEPTH SPECIFICATION
1970
PROJECT HYPO CRUISE 1
7/28-7/29
1975
CCIW CRUISE 4
6/24-6/29
0
0
10
TEMPERATURE, °C
20 0 10
20
E 10
i"
CL
gl5
20
25
Q
ASSUMED
THERMOCLINE
DEPTH
0
10
15
20
25
BT
ASSUMED
THERMOCLINE
- DEPTH Q
Figure 35.
Average temperatures (°C) in two meter sections of the Central
Basin for 1970, Project Hypo Cruise 1 (left) and 1975, CCIW
Cruise h (right). This data was used to establish location of
the thermocline for each year.
109
-------
The comparison between the calculated and observed temperatures for 1970
are shown in fig. (36). Included in the calculation are also the computed
and observed chloride distributions as shown in fig. (37). The mass input
rates of chlorides are based on the U. S. Army Corps of Eagineers work dis-
cussed previously. They are listed in Table A-9 of the appendix. The trans-
port coefficients corresponding to this calculation are shown in fig. (38).
Between the Western Basin and Central Basin epilimnion the circulating flow
is equivalent to a velocity of 2.72 cm/sec. A velocity of 3-^6 cm/sec is
used between the Central and Eastern Basin epilimnions. These velocities
are comparable to the current magnitudes based on hydrodynamic calculations
as indicated previously.
The vertical exchange is based on the unstratified diffusion coefficient
range reported by Csanady [68]. Unstratified exchange between the Central
2
Basin epilimnion and hypolimnion is set at 1 cm /sec. During stratification
2
the dispersion is reduced to 0.05 cm /sec. Between the main body of the hy-
polimnion and the lower hypolimnion layer (segments 5 & 6) the rate is kept
2
constant at 1 cm /sec. In the Eastern Basin the dispersion reaches a maximum
2 2
of 5 cm /sec in late fall-early winter and falls to 0.25 cm /sec during
stratification. These parameters are also shown in fig. (38) for 1970.
For the estimate of the 1975 transport, the horizontal exchanges are
assumed to be the same as for 1970. However, the vertical exchanges result-
ing from the calibration differ from 1970, probably as a result of the diff-
ering wind speeds. Between the Central Basin epilimnion and hypolimnion the
2 2
unstratified dispersion is estimated to be 2 cm /sec reducing to 0.1 cm /sec
during stratification. For the Eastern Basin dispersion is lower than for
2 2
1970, 1.5 cm /sec during unstratified periods and 0.1 cm "/sec during strati-
fication, during the entire year. In addition the stratification started
earlier in the year as evidenced by the significantly lower vertical disper-
sion coefficient during April and May. The results of the calculations are
shown in fig. (39) for the transport regime shown in fig, (Uo).
The transport regimes estimated using this methodology are applied to
the calculations of dissolved constituents in the next section. The only
modification is the addition of a sinking velocity for the particulates,
which cannot be estimated from the dissolved tracers examined in this section.
110
-------
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4,000 cfs
HORIZONTAL TRANSPORT
209,600 cfs
,600 cfs
000 cfs
100
10
1.0
0.1
100
SEGMENTS 2 AND 5
J'F'M'A'M'J' J'A'S'O"N '
(0
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J I F I M ' A ' M I J
S O N ' D
SEGMENTS 5 AND 6
Figure 38. 1970 Model Transport Regime. Horizontal Transport and Vertical
Exchanges (m2/day) resulting from the calibration.
113
-------
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HORIZONTAL TRANSPORT
220,600 cfs
100
10
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SEGMENTS 5 AND 6
|F'M'A|M|J|J'A| S'O'N'D
Figure Ud. 1975 Model Transport Regime. Horizontal transport are assumed
to be the same as for 1970 while vertical exchanges differ as
shown.
115
-------
SECTION 9
STRUCTURE AND COMPUTATIONAL DETAILS
The detailed discussions of the previous chapters present the back-
ground and frameworks adopted for the kinetics and transport mechanisms "being
considered in this calculation. In this chapter the overall structure is
presented together with the details of the implementation. A discussion of
the rationales which lead to the final choices is included in order to record
the motivations involved and the assumptions and compromises that were nec-
essary. The computational details are also discussed.
ONE GROUP AND TWO GROUP PHYTOPLANKTON KINETICS
The initial structure adopted for the Lake Erie kinetics followed up
kinetic framework used for Lake Ontario [50] and Lake Huron/Saginaw Bay [30],
For these lakes the phytoplankton are represented "by a single chlorophyll
concentration without regard to differing algal groups. However this struc-
ture was not satisfactory for Lake Erie, especially in the Central Basin.
The data display less evidence of the spring phytoplankton "bloom characteris-
tic of the other Great Lakes investigated. The Ontario-Huron kinetic struc-
ture consistently predicts a larger-than-observed spring chlorophyll bloom
if the growth kinetic coefficients are kept within reported ranges. Hence
this kinetic structure was judged to be inadequate and ;lt was necessary to
reevaluate the kinetics by examining the peculiar characteristics of Lake
Erie in an effort to determine the cause of the failure,,
The reactive silica distribution in the Central and Eastern Basins of
the lake reveals severe depletion in late spring. For 1970, the average
epilimnion reactive silica concentrations in the Central Basin during the
April and May CCIW cruises are 0.027 and 0.029 mgSiOp/& respectively. In
the Eastern Basin the values are O.OU and 0.021 respectively. Typical half
saturation constants for SiO range from 0.06 to 0.2U mg SiO /& (See Table
9). With Michaelis-Menton growth kinetics and the smallest reported half-
saturation constant, the result is approximately a 75% growth rate reduction
for silica dependent diatoms. Lake Erie phytoplankton biomass and species
composition in 1970 [27] indicates that the diatoms predominate in the early
spring comprising about 90% of the total crop by volume but by June have
declined to 10% of the crop. During this same period the total biomass de-
creases by greater than a factor of 2. These facts indicate that the lack
of a spring phytoplankton chlorophyll peak is due to a silica limitation of
diatoms. Therefore to properly account for the observed total chlorophyll
distribution it is necessary to separate the chlorophyll into that portion
associated with diatoms and that associated with other phytoplankton. This
is accomplished by including two chlorophyll concentrations in the calcula-
tion, one for the diatoms and one for the non-diatom algae. The relative
116
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concentrations of each chlorophyll type are calibrated by assigning chloro-
phyll to biomass ratios to the diatom and non-diatom chlorophyll and check-
ing the percent diatom biomass against observed data [27]. The chlorophyll-a
to biomass ratios, obtained from linear regressions of chlorophyll versus
diatom and non-diatom biomass using 1970 data are 1.2k and 2.32 yg Ch£-a/mg
dry wt. for diatoms and non-diatoms respectively. The necessity of using
different chlorophyll-a to biomass ratios for diatoms and other groups has
been demonstrated in Saginaw Bay as well [69].
The comparison of the one group and two group phytoplankton calculation,
the latter including unavailable and available silica as well is given in
fig. kl. For the Western Basin the use of two phytoplankton groups does not
materially improve the calibration. An examination of the reactive silica
data, fig. 2k, shows that this result is not unexpected. Silica limitation
does not play an important role in this basin. An earlier investigation of
the Western Basin confirmed this result [70]. In the Central Basin the use
of a two-group model has a dramatic effect on the results. Whereas the one-
group kinetics predicts a large spring bloom which does not appear in the
data, the use of the two-group kinetics eliminates this bloom and in fact
follows the spring chlorophyll data rather well. Silica limitation of the
diatoms prevents the bloom from occurring. However the fall bloom is simu-
lated equally well by both calculations. The two-group kinetics provides no
advantage here because silica does not have a significant effect in growth
limitation during this period.
DETERMINATION OF ANOXIA
The determination of the areal extent of anoxia in the Central Basin is
an important component of the calculation. The anoxic area controls the
fraction of the sediment surface from which nutrients are regenerated into
the water column. However the horizontal and vertical spatial scale of the
calculation is such that a direct computation of the anoxic area is not poss-
ible. A segment is either entirely oxic or anoxic. In Lake Erie anoxia
begins in a small area and increases until overturn. An empirical method
has been developed in order to simulate this effect.
Fig. k2 is a plot of the cruise average dissolved oxygen for segment 5
and segment 6 the Central Basin hypolimnion segments versus the minimum dis-
solved oxygen observed during that cruise. For segment 5 when the segment
average D.O. is approximately k mg/£ the minimum observed D.O. in that seg-
ment reaches zero. For segment 6 at a segment average of 2 mg/£ the segment
minimum reaches zero. The interpretation of these results is that as the
average D.O. of segment 5 reaches k mg/£ or as segment 6 reaches 2 mg/Jl there
will be an anoxic region somewhere within that segment volume.
In order to calculate the fraction of the segment which is anoxic, plots
of cruise average D.O. versus the fraction of stations anoxic for that cruise
for both segments 5 and 6 are constructed and shown in fig. k3. An anoxic
station occurs if the observed D.O. at one meter from the bottom is less than
0.50 mg/£. From these plots it is inferred that, to a reasonable approxima-
tion, the fraction of anoxic stations, which is assumed to be equivalent to
the fraction of the anoxic area, increases linearly with decreasing mean D.O.
117
-------
X
Q.
o
cc
o
51
22
O a-
I- 3
>- O
X CC
Q. o
? 6
eg 5
< h-
UJ O
u.
O
1
OC
<
Q.
O
o
8
o
o
(N
n
o
LO
O > ^1
O rS
-------
LU
X
o
Q
LU
_J
O
CO
CO
1970 DA TA
1973-74 DA TA
* 1975 DATA
1
0 2 4 6 8 10 12 14 16
MINIMUM DISSOLVED OXYGEN, mg/l
16
14
12
10
8
6
4
2,
n ,
- SEGMENTS
-
*A
"/
-A"
LVf
x»
"ill
J
~
7576>
*
£
DATA
1973-74 DATA
A 19 75 DAT A
I i
I I I
0 2 4 6 8 10 12 14 16
MINIMUM DISSOLVED OXYGEN, mg/l
Figure 42. Mean dissolved oxygen (yg/A) for the Central Basin hypolimnion
segments vs. the minimum dissolved oxygen (mg/£) observed for
each cruise.
119
-------
00
QC
is
Uj
5
CD
^J
00
>- Cj
3: ^
D O » 0)
o -p
CD w
O W
0} -p
CO 0
H 0)
CD 0)
bfl W
csJ
^ fl
-------
For segment 5 at a mean of k mg/£ anoxia is just beginning while at a seg-
ment average of 0 mg/£, 100% of the segment is anoxic. The regeneration of
phosphorus and silica is calculated based on these plots as discussed subse-
quently.
EXOGENOUS VARIABLES
The water temperature, incident solar radiation, and photoperiod are
specified externally. The extinction coefficient for each basin is computed
using an observed correlation between the non-algal associated extinction
coefficient and the wind velocity. The observed secchi disk depth, SD, is
converted to an extinction coefficient, K , using the relationship [71]:
Ke = If
The influence of the chlorophyll concentration is estimated using [72] :
K = K1 + 0.008 [Ch£-a] + 0.05^ [CM-a]2'3 (73)
e e
which yields, K', the non-algal related extinction coefficient. A compari-
son of K' to wind velocity [10] is shown in fig. (W). The effect of wind
suspension of particulate material is clearly seen. These correlations to-
gether with the wind data for each basin, also shown in this figure, are
used to calculate K'. The total extinction coefficient is then based on the
e
calculated chlorophyll concentrations, in addition to the wind related non-
algal effect. The reason for this somewhat elaborate procedure is that
under projected conditions of reduced chlorophyll concentration the extinc-
tion coefficient would be reduced and this effect is important in increasing
primary production and affecting the oxygen balance, as shown subsequently.
The effect of ice cover over the lake is included by reducing the incident
solar radiation by 70$ [73] during the periods of reported ice cover [7^],
from the first of the year to day 60 for the WB and CB and day 90 for the EB.
TIME SCALE
The intention of this calculation is to simulate the seasonal varia-
tions of the major water quality constituents involved in the eutrophication
process. Short term conditions such as storm events or diurnal variations
of phytoplankton or dissolved oxygen are not considered. Rather the sea-
sonal changes of plankton, nutrients and dissolved oxygen distribution are
characterized.
The differential equations which describe the kinetics and transport
are solved using finite difference techniques. The simple Euler-Cauchy
method, with a provision for avoiding negative concentrations, is used for
the integration. The time step used is 0.5 days. Results are filed every
15 days except for the anoxic period evaluation which is based on the 0.5
day time step. The fifteen state variables and ten segments result in 150
simultaneous differential equations which are solved at each time step.
Approximately 122 equivalent central processor unit (CPU) seconds of execu-
tion time on a CDC 6600 are used for each one year model run. This includes
121
-------
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CN
00
CO
a.
E
~£
CM
Uj
00
CN
O
.
a
E
CJ
O O
LLJ
00 >
Q
CO Z
5
1
CO
q
CN
'!N3IOIdd300 N0110NI1X3
I I
O
CO
Q
z:
o
CO
o
CN
a
o
CO
00
CO
sf:
00
CO
CN
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5
tt
CO
O
CO
-------
time for graphic display, overhead, and the chemical computation, as well as
transport and kinetics.
CARBON DIOXIDE AND ALKALINITY
The relevance of computing a carbon dioxide mass balance is discussed
in Section 6. As can be seen from the reaction kinetics stoichiometry car-
bon dioxide and oxygen play opposite roles during primary production and
aerobic oxidation. Therefore the measurements of pH and alkalinity provide
additional calibration information with virtually no additional degrees of
freedom included in the formulation. The only complication is that the
reactions involving the dissolved inorganic carbon species: CO (aq), HCO~
= + _
CO , and the hydrolysis products of water: H and OH must be considered.
Methods for including these reactions within the context of mass balance
computations have been developed [75] and generalized [29] so that any set
of fast reversible reactions can be accomodated. The basic idea has been
developed in Section 7: for species independent transport it suffices to
compute mass balances for the components of the species. In the case of
water column concentrations, the mixing is via large scale water motions and
the transport is clearly species independent. The most convenient choice of
components for the carbon dioxide-water species are total carbon dioxide,
C , and alkalinity, A, where:
CT = [C02(aq)] + [HCO~] + [C03=] (7*0
A = [HCO~] + 2[C03=] + [OH~] - [H+] (75)
These components are conserved during any fast reversible reaction, for
example:
C°2 + H2° + HC°3 + H+
It is clear that total carbon dioxide is conserved since any change in CO
is compensated for by a change in HCO . The same is true for alkalinity
_ + -1
since HCO and H have opposite signs in the alkalinity definition. There-
fore the mass balance equations for these components are independent of the
source-sink terms due to the fast reversible reactions. They include only
the sources and sinks of total carbon dioxide and alkalinity due to the bio-
logical production and consumption reactions as specified in Tables 11-1-9
It remains only to describe the method used to compute the species con-
centrations given the component concentrations as computed from mass balance
equations. This is a straightforward chemical equilibrium calculation for
which a number of methods are available [76], The algorithm chosen for this
application was developed by the RAND Corporation [77] and it has been found
to be quite reliable. The chemical thermodynamic constants required are
listed in Table A-l6 of the appendix. The temperature dependence is calcu-
lated assuming the enthalpy and entropy changes for each reaction, AH° and
AS0 are independent of temperature. It has been verified [50] that the
123
-------
equilibrium constants, K computed using the relationships:
AG° = AH° - T AS0 , .
r r r (77)
AG° = - RT £n K (78)
where T is in °K and R is the universal gas constant within 5% of experi-
mentally reported over the temperature range of 0 to 30°C,, The ionic
strength correction is ignored since, for Lake Erie surface waters it is
small.
There is a significant additional computational "burden associated with
the chemical equilibrium computation and in order to lessen its impact on
the overall computational requirement, the equilibrium calculation is done
only every ten days. The computed species concentrations are assumed to
be constant during the ten days and then they are updated. This has no
effect on the accuracy of the rest of the computation since no reaction
rates are functions of the species concentration. Only the air-water ex-
change of CO (aq) is affected by this procedure and it to a negligible
extent.
AIR-WATER INTERFACE EXCHANGE
For the segments with an air-water interface (segments 1, 2 and 3)
the dissolved gases: CO (aq) and 0 (aq) are both subject to exchanges with
their atmospheric counterparts: CO (g) and 0 (g). The exchanges can be
thought of as slow reversible reactions of the form:
C02(aq) £ C02(g) (79)
02(aq) £ 02(g) (80)
with first order reaction rates KT/H. where K is the air-water surface ex-
Jj i J_i
change coefficient and H. is the depth of segment i. The conventional for-
mulation for the source term in the mass balance equation for total CO in
segment i, C , is [78]:
dC,,
;C00(aq) , - C00(aq).)
'Ti KL
dt H.
where CO.(aq) . is the equilibrium (or saturated) concentration at the aque-
J Si
ous temperature in segment i corresponding to the atmospheric partial pres-
sure of CO (g). A similar equation applies for dissolved oxygen. The satu-
rated aqueous concentrations for C0? is computed from the chemical equilibrium
requirements for eqs. (79) and (80), that is that the change in Gibbs Free
energy, AG , for each reaction is zero:
-------
AG = Z , AG - Z . AG = 0 (82)
r products reactants
so that, assuming the activity coefficients are unity:
AG°c02(g) + RT £n [C02(g)] = AG^^ + RT £n [CO^aq)] (83)
and, therefore:
o o
2 si 2^' XP r R+ r i
where R is the universal gas constant and T. is the aqueous temperature in
degrees Kelvin. The appropriate thermodynamic constants are listed in table
A-1T in the Appendix.
The remaining parameter in eq. (8l) that needs to be specified is the
exchange coefficient, K , for CO and 0 . For lakes, the predominant mechan-
ism that influences the exchange coefficient is wind velocity and ice cover.
During the period of ice cover the exchange is set to zero (day 0 to 60 for
the WB and CB, and day 0 to 90 for the EB). Estimates have been made of the
relationship between K and wind velocity [79]- For the sake of simplicity a
constant exchange coefficient has been used for both CO and 0 : KT =2 m/day
d C. J-J
which is reasonable for the wind velocities over Lake Erie. As shown subse-
quently this simplification is of no consequence in the dissolved oxygen cal-
culation since the observations and calculations in the surface segments are
essentially saturated. For the carbon dioxide calculation including the
wind velocity effect may improve the results of the pH calibration in the
latter part of the year where, as shown subsequently, the computed pH is
below that which is observed. The effect on the hypolimnion computations
is neglibable since only small differences is total CO concentrations would
result from this modification.
SETTLING VELOCITIES
The settling of particulates from the water column to the sediment is
the principle mechanism by which nutrients are lost from the water column.
The inclusion of this advective term in the mass balance equations is
straightforward. Assume that segment 2 is below segment 1 and that for the
variable c the settling velocity is w. The mass balance equations for these
segments ignoring horizontal advection and reactions are:
dc
V - =E(c-c)-wAc (85)
Vl dt *12 l°2 V W A12 °1 W'
V dc?
V2 -f = E1? (cn - cj + w A c, (86)
dt It: 1 2 Ld 1
where E is the exchanging flow and A is the interfacial cross sectional
area. The term w A ? is an equivalent unidirectional flow which is due not
125
-------
to water motions "but rather to the settling of the participates.
The settling velocity used in this computation represents both the
settling through the thermocline and through the sediment water interface.
It is possible that these two velocities are different due to the effects
of resuspension which may be more pronounced at the interface. However it is
not possible to resolve this difference with the coarse vertical segmentation
of the water column used in this calculation.
A more detailed parameterization of settling into the sediment would
include not only a downward settling velocity but an upward resuspension
velocity as well. In this context the single settling velocity used in these
computations can be thought of as the settling velocity which represents the
net flux due to the difference between the downward settling flux and the
upward resuspension flux. Since the settling velocity is constant it cannot
account for increased resuspension flux during high wind periods. As shown
subsequently the computations underestimate the total phosphorus concentra-
tions during this period.
MASS BALANCE EQUATIONS
The formulation of the mass balance equations for ea.ch variable in each
segment is straightforward. For concentration C.. of variable £ in segment
i, the general mass balance equation is:
dc, .
Vi -dT = I Eki (G£k - C£i} + I \i C£k + SM Vi (8T)
k k
where V. is the segment volume, E is the exchanging flow between segments
i ki
k and i, Q, . is the advective or settling velocity flow from segment K to
ki
segment i, and S0. is the sum of the sources and sinks for variable £ in
x/l
segment i. The transport coefficients, which are determined from the tem-
perature balance computation described in Section 8, are listed in Tables
A-l and A-2 for 1970 and 1975 respectively. The settling velocities which
are determined from the calibration are listed in Table A-15.
The dynamic mass balance equations for both the water column and sedi-
ment segments are solved simultaneously to produce the concentrations. Thus
although the sediment-water interactions were analyzed on a steady state
basis in Section 75 and a remarkably simple expression, eq. 63, describes
the expected result, the dynamic mass balance equations are integrated for
the results that follow. These take into account the time variable nature
of the incoming fluxes of algal and detrital carbon and their subsequent
conversion to sediment oxygen demand.
The sources and sinks due to the kinetics are easily constructed from
Tables 11-19 corresponding to the reactions incorporated in the calculation.
For example consider variable 1, diatom chlorophyll a. It appears in the
stoichiometry of reactions I, II and III of the aqueous reactions and reac-
tion XII of the sediment reactions. Thus for the aqueous segments, the
126
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source term is:
Sli = Rli ' R2i ~ R3i
vhere the R's are the reaction rates in the tables evaluated with the diatom
coefficients and concentrations in segment i. The signs are determined by
the location of the variable in the stoichiometric equation: it is positive
if it occurs on the right hand side of the equation since it is being pro-
duced by the reaction; otherwise it is negative since it is being consummed
by the reaction. The stoichiometric coefficients must multiply the reaction
rates as they do in the stoichiometric equations. For diatom chlorophyll
they are unity but for other variables they are as indicated in the reaction
stoichiometric equations in Tables 11-19- It should be clear that following
the conventions it is possible to construct the complete source terms for
all variables in the computation.
127
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SECTION 10
CALIBRATION
The final calibration is the result of approximately 150 model runs
which were made in order to determine a consistent set of parameter values
that are reasonable and reproduce the observed data for all the variables
considered. The method employed is essentially trial and error. The start-
ing point is a set of parameter values which have been used in the previous
Lake Ontario [50] and Lake Huron/Saginaw Bay [30] computations. As described
in Section 9 it was necessary to consider diatom and non-diatom chlorophyll
separately. The coefficients for their kinetics were then adjusted to repro-
duce the observed behavior. Zooplankton kinetic constants were also simi-
larly adjusted. The remaining kinetic constants are essentially those used
previously. In particular the chlorophyll dependent recycle mechanism [30]
was found to apply to Lake Erie as well. The coefficient values used for the
final calibration are listed in Tables 11-19. The data used for the calibra-
tion are from the 1970 and the 1973-197^ survey years. The comparison fig-
ures for the 1970 and 1973-197^ data are presented subsequently The data
plotting symbols are means +_ standard deviations. The square symbol repre-
sents CCIW cruise data for 1970. The circle represents Project Hypo data
for 1970. The triangle represents 1973 data. The diamond represents 197^
data. The computed lines are the same for both years since the conditions
are similar for these years.
DISSOLVED OXYGEN
The dissolved oxygen comparisons for 1970 are shown in fig. ^5 and for
1973-7^ data in fig. ^6. The agreement is quite good with the computation
duplicating all the major features of the temporal oxygen distribution with-
in the various sections of the lake. As can be seen, the epilimnion D.O.
follows the saturation oxygen concentration very closely during the entire
year except for a small oxygen peak of supersaturation appearing in early
July. This is due to photosynthetic production resulting from the increase
in phytoplankton chlorophyll taking place at this time. In the Central Basin
hypolimnion the low oxygen values during the summer months are caused by a
combination of phytoplankton decay within the hypolimnion and the oxygen
demand exerted by the sediment. The increasing temperature which results in
increasing reaction rates, the sinking of the phytoplankton which have grown
in the epilimnion, and stratification which inhibits transfer of oxygen from
the epilimnion to the hypolimnion all contribute to the oxygen decline. The
decline continues until overturn in early fall. At this time there is a
rapid increase in the hypolimnion D.O. which continues to the end of the
year. The actual rate of vertical mixing during the latter stages of over-
turn appears to be more intense than the vertical mixing used in the calcu-
lation. These transport coefficients are estimated from the temperature
distribution as described in section 8. Since the temperature gradient is
128
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to
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K g
to E
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130
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very small during the latter part of the year the estimate of the vertical
mixing coefficient is uncertain, and, based on the results of the oxygen cal-
culations, it should be larger.
Fig. hi illustrates the relative importance of the individual oxygen
sinks. The bottom curve is the calibrated D.O. calculation for the Central
Basin hypolimnion. Proceeding up from this curve are the results of sequen-
tially removing the deep sediment oxygen demand, surface sediment oxygen
demand, organic carbon oxidation (actually organic carbon oxidation + nitri-
fication but the latter is insignificant) and finally phytoplankton respira-
tion. It is apparent that organic carbon oxidation represents the largest
oxygen sink in the hypolimnion, making up about k3% of the total depletion
at the time of minimum D.O. The two components of the sediment oxygen
demand (SOD) account for kO% of the total depletion at the time of minimum
D.O. Individually, the computed surface SOD accounts for 12% of the total
depletion and the deep SOD accounts for 28%. The remaining 1J% is due to
phytoplankton respiration.
A summary of the computed sources and sinks of dissolved oxygen in the
Central Basin hypolimnion during the summer stratified conditions is pre-
sented in Table 21.
TABLE 21. SOURCES AND SINKS OF DISSOLVED OXYGEN
Central Basin Hypolimnion - 1970
Reaction Average Volumetric Dissolved 0
Depletion Rate Change
Source (+) Day lQO-2hd
Sink (-) (mg 02/£-day) (mg 02/£)
Phytoplankton Respiration -0.020 - 1.20
Detrital Organic Carbon - 0.06U6 - 3.88
Oxidation
Sediment Oxygen Demand - 0.0597 - 3-58
Thermocline Transport 0.0305 1.83
Total 0 Change - 6.83
The aqueous reactions: phytoplankton respiration and detrital organic carbon
oxidation, consume a total of 5-08 mg 0/1 during the period of stratifica-
tion. The oxygen decrease attributed to sediment oxidation is 3-58 mg 01.
These sinks are balanced by the only significant source considered in the
computation: transport of dissolved oxygen through the thermocline, which
increases the concentration by 1.83 mg 0/1. The net result is a decrease
of 6.83 mg 0/1 during the sixty day period analyzed in this table.
131
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'N3DAXO QBAIOSSIQ
132
-------
An additional check on the validity of the oxygen calculation is avail-
able. As discussed earlier in this section, an empirical correlation has
been developed which relates the average hypolimnetic dissolved oxygen to
the anoxic area during a year. For the 1970 calibration year the calcu-
lated maximum anoxic area is 6819 km2. The anoxic area of the Central Basin
for 1970 is reported to be 6600 km2. Thus calculated anoxic area is essen-
tially the same as that determined from actual oxygen measurements.
PHYTOPLANKTON
The results of the calibration for phytoplankton chlorophyll are shown
in figs. 48 and h9- The calculated curve is the sum of the diatom chloro-
phyll and non-diatom chlorophyll. Calculated peak Western and Central Basin
chlorophyll is overestimated for the 1973-197^ calibration year but is in
reasonable agreement with 1970 observations, as are the calculations for the
other segments. The chlorophyll associated with each of these two groups
together with the estimated fraction of diatom biomass [27] are shown in
fig. 50 for 1970. Note that in all basins the diatoms begin to grow in early
spring but are declining by the beginning of summer. The growth and death
dynamics plots, fig. 51-53, reveal that this decline is due to a decrease in
growth rate caused by nutrient limitation which allows respiration and
grazing to exceed the growth rate. As shown subsequently the nutrient limi-
tation is caused by silica depletion.
Following the diatom decline there is a bloom of non-diatoms in all
three basins (segments 1,2 & 3). The dynamics plots show that this bloom is
delayed due to grazing pressure. The zooplankton population is high due to
the early diatom bloom. The result is that the phytoplankton are suppressed
until the herbivorous zooplankton decline due to predation by carnivorous
zooplankton. At this time there is a rapid increase in the non-diatom popu-
lation. In the Western Basin the population stabilizes at approximately
26 \ig Ch£-a/£. This is due to several factors. As would be expected, light
and nutrient limitation are significant in controlling the size of the bloom.
This can be seen in the nutrient limitation plots. Perhaps not as obvious
is the contribution of phytoplankton settling to population limitation in the
Western Bbasin. Examination of the dynamics plots for the Western Basin
shows that settling constitutes an important loss rate throughout the year
for both diatoms and non-diatoms. During the non-diatom summer bloom set-
tling is the largest non-diatom sink by approximately a factor of three.
The shallow depth of the basin is the reason for the importance of settling.
The dynamics plots of the deeper Central and Eastern Basin epilimnions show
that settling never reaches the importance it does in the Western Basin. In
these basins, nutrient limitation is the controlling factor for growth. The
decline of the phytoplankton in late fall is due to decreasing light and
temperature. The dynamics plots show that by this time there is very little
nutrient limitation.
Fig. 5^ illustrates the growth rate reduction factors due to silica,
nitrogen and phosphorus limitation effects. As discussed earlier, the spring
bloom in all three basins is terminated by the depletion of silica. These
plots show that for the Western, Central and Eastern Basins the growth rates
are reduced by a factor of 0.^5, 0.15 and 0.15 respectively due to silica
133
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limitation. The summer bloom in all three basins is phosphorus limited. As
the plots show the phosphorus factor is low while both the silica and nitro-
gen factors are high. For the Western, Central, and Eastern Basins, phos-
phorus depletion results in growth rate reduction factors of O.kO, 0.21 and
0.21 respectively while at the same time the nitrogen reduction factors are
0.83, 0.87 and 0.8T. From these plots it is clear that for all three basins,
the spring bloom is silica limited and the summer-fall bloom is phosphorus
limited. However, especially in the Western Basin, inorganic nitrogen is
almost exhausted by the bloom so that if phosphorus limitation did not ter-
minate the bloom, the nitrogen limitation would do so at essentially the
same peak chlorophyll concentration, assuming that nitrogen fixation did
not add significantly to the available fixed nitrogen sources.
These calculations of nutrient limitation effects also indicate that
nutrient limitation is much more severe in the Central and Eastern Basins
than in the Western Basin. In the Western Basin nutrient levels do not
reach the low values seen in the other basins. This is due to the contri-
bution of the other factors that limit growth in the Western Basin. As dis-
cussed previously, settling is important. In addition, the high turbidity
level of the Western Basin and the resultant high light attenuation factor
contribute to growth rate limitation.
NITROGEN
The distribution of the forms of nitrogen may be seen in fig. 55. The
total nitrogen concentrations within the Central and Eastern Basin epilimnia
are fairly constant throughout the year. In the Western Basin there is
variation due to loading effects. All three of these segments exhibit simi-
lar patterns throughout the year. The non-living organic nitrogen concen-
tration is stable until midsummer when there is an increase in concentra-
tion due to a combination of respiration of the phytoplankton and zooplank-
ton populations which have increased just prior to this time and the excre-
tion of organic nitrogen by predating zooplankton. There is a gradual
decrease of the non-living organic nitrogen toward the end of the year as it
is recycled to ammonia.
Ammonia concentrations are observed to be fairly constant in the epi-
limnia of the Central and Eastern Basins throughout the year. This is due
to two factors. First, the rate of nitrification is sufficiently rapid so
that equilibrium is reached quickly. Second, because nitrate concentrations
are much higher than ammonia it is used as the algal nitrogen source in pref-
erence to the ammonia. In the Western Basin as the nitrate decreases it
reaches the ammonia concentration level after which there is a depletion of
ammonia as the phytoplankton preference structure switches from nitrate up-
take to a more even nitrate-ammonia uptake. At this point inorganic nitro-
gen is approaching limiting values. Note that in all three basins during
the summer growth period, the nitrate concentration is lowered and the con-
centration of phytoplankton associated nitrogen increases. At this time the
total nitrogen remains approximately constant, but there is substantial vari-
ation within the individual nitrogen species: nitrate decreases, algal nitro-
gen increases and non-living organic nitrogen increases.
Ihl
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The increase in total nitrogen in the Central Basin hypolimnion is due
to the recycling in the water column and the sediment. The calibrations of
the individual nitrogen species are shown in figs. 56-61. Organic nitrogen
profiles, the sum of the computed phytoplankton and non-living organic
nitrogen, are in reasonable agreement with observations in 1970. The 1973-7^
data is more variable especially in the Eastern Basin. The ammonia profiles,
figs. 58 and 59 are in rough agreement with the observations. However the
computed phytoplankton uptake in the Western Basin is not observed in 1970
although it is suggested in the 1973-7^ data. The increase in recycled am-
monia in the Central Basin hypolimnion is reproduced but the observed rapid
mixing after overturn is not reproduced. This is consistent with the dis-
solved oxygen results and suggests a more rapid mixing at overturn.
The nitrate distributions in the surface segments are in reasonable
accord with observations for both years. The phytoplankton uptake is well
reproduced in each segment. However the hypolimnia are calculated to have
considerably more nitrate than is observed in 1970. However 1973-7^ data are
in reasonable accord with the calculation. The cause of this discrepancy is
uncertain at present.
PHOSPHORUS
The distribution of the phosphorus species, fig. 62, exhibit patterns
very similar to those of the nitrogen species. Total phosphorus is approxi-
mately constant throughout the year. During the summer growth period there
is an increase in phytoplankton associated phosphorus and an equivalent de-
crease in available phosphorus. Unavailable phosphorus is increased due to
respiration and excretion. In late fall and winter phytoplankton phosphorus
decreases gradually as it becomes unavailable phosphorus which is recycled
to available phosphorus that increases in the absence of uptake.
The results of the calibration for total phosphorus is shown in figs.
63 and 6k. A discrepancy is apparent in the 1970 data. In the Western Basin,
the data indicate high total phosphorus concentrations in the spring and fall.
In all of the other segments high total phosphorus concentrations appear con-
sistently in late fall. This trend in the data is not calculated to occur,
i.e. there is no mechanism incorporated in the model to account for this ob-
servation. A hypothesis for these high total phosphorus concentrations is
that they are due to wind resuspension of bottom material. Fig. kk shows
the monthly average wind velocity for the Western, Central and Eastern Basins
over the year. Note that in all three basins the wind velocity is highest in
the spring and fall. It is possible that these high winds stir the bottom
sediments bringing particulate phosphorus into the water column. The fact
that the total phosphorus distribution for the rest of the year is simulated
properly indicates that the high phosphorus concentrations in late fall do not
contribute to the following year's dynamics. If wind resuspension is the
cause then it is likely that the material will have settled again by the next
spring. Since the total phosphorus mass balance is essentially correct from
May to October, which is the critical time for phytoplankton growth and dis-
solved oxygen depletion, it appears that the wind induced resuspension is not
significant mechanism for supplying phosphorus to the phytoplankton. The
1973-7^ data for total phosphorus are more variable and the trends are more
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difficult to discern. This is especially true for the Eastern Basin where
the midsummer erratic data are suspect.
The calibration for ortho-phosphorus is shown in figs. 65 and 66. The
observed summer uptake is reproduced by the calculation in the three surface
segments but the spring concentrations are slightly higher than observed. A
substantial discrepancy occurs in the Central Basin hypolimnion. The spring
data indicate concentrations of about 1-2 ygPO,-P/£ while the calculation
yields concentrations in the range of 6 - 8 ugPO,-P/£. For this same time
period the Central Basin epilimnion calculations are close to the data. The
data indicate that the ortho-phosphorus during the spring is higher in the
epilimnion than in the hypolimnion. The calculation yields epilimnion and
hypolimnion concentrations of close to the same magnitude due to the large
exchange between the epilimnion and hypolimnion since stratification has not
started and the lake is fairly isothermal. Examination of the temperature
calibration plot, fig. 36, Section 8 shows that the fit to data is very good,
especially at this time. Apparently, a vertical ortho-phosphorus gradient
exists without an accompanying temperature gradient. The present structure
is unable to simulate this feature and therefore the hypolimnion calculation
of ortho-phosphorus is high. Fortunately this discrepancy appears to be less
significant for the phytoplankton dynamics in the epilimnion. During the
summer growth period the basin is stratified and the interaction between the
hypolimnion and the epilimnion is limited. At overturn, when the phosphorus
in the hypolimnion is transported into the epilimnion, the calculated concen-
tration agree with the observed concentration and the proper mass is trans-
ported.
However this discrepancy is disturbing since it indicates that a signi-
ficant mechanism is operating in the Central Basin hypolimnion that is not
incorporated in the calculations. The results suggest that either the re-
cycle in the hypolimnion is slower than that employed in the calculations,
or that there exists a selective removal mechanism for ortho-phosphorus.
Perhaps an adsorption mechanism is operating which binds the recycled ortho-
phosphorus to particulates. Since regeneration of reduced iron and manganese
occurs during anoxia, these metals would oxidize to form amorphous oxides and
hydroxides which have a high adsorption capacity. This might provide a suf-
ficient quantity of particles for this mechanism to be significant.
SILICA
The calibration results for reactive silica are shown in fig. 67. The
data for the Western Basin appears to confirm the early spring uptake by the
diatoms. The summer and fall data are quite variable and scattered. This
may reflect a highly variable loading. Since total silica data is lacking
there is no confirmation from a total mass balance calculation. The results
for the Central and Eastern Basin epilimnia are remarkably consistent and
reproduce the observed patterns: the spring uptake and the subsequent regen-
eration during the summer and fall. The Eastern Basin hypolimnion concen-
trations are larger than observed for an as yet uncertain reason. The Central
Basin hypolimnion results reproduce the regeneration of silica during anoxia
and the reduction during overturn as the silica rich bottom waters mix with
153
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the silica poor surface waters.
SECCHI DISK DEPTH
Fig. 68 gives the Western, Central and Eastern Basin calibration results
for secchi disc depth. In all three basins the calculation follows the data
very well. Note that the secchi depth is low in spring and fall and high in
summer. This indicates much higher turbidity in the spring and fall. As
discussed previously with regard to phosphorus resuspension, winds are highest
in spring and fall. The effect of the winds is incorporated into the secchi
disc depth calculation as discussed previously.
ZOOPLANKTON CARBON
The results of the calibration for zooplankton carbon are shown in fig.
69. The data used, reported by Watson [24,25], is part of the 1970 CCIW
data set. It represents hauls from 50 meters to the surface or from 2 meters
above the bottom to the surface at stations with a depth less than ho meters.
The calculation has been adjusted accordingly by weighting the surface and
bottom segment calculations. In all three basins the timing of the model is
consistent with the data. The zooplankton peak is earliest in the Western
Basin and latest in the Eastern Basin. This is due to the temperature dif-
ferences and resultant phytoplankton growth timing between the basins. The
zooplankton begin to grow after the diatom bloom. The growth of the non-
diatoms is initially inhibited due to the presence of the zooplankton. The
total zooplankton population is decreasing by midsummer in all three basins
even though the carnivorous zooplankton increase until August.
The dynamics of the zooplankton populations are shown in fig. 70. Res-
piration exceeds the growth rate until phytoplankton populations develop.
This occurs first in the Western Basin and later in the Eastern Basin. As
the herbivorous populations develop the carnivores commence growth and exert
grazing pressures which exceeds the growth rate and decreases the herbivorous
population. The assimilation and grazing rate reduction factors are smallest
in the Western Basin where chlorophyll concentrations are largest. This
reduces the absolute grazing and assimilation rates in the Western Basin
relative to the Eastern Basin. A similar effect is postulated for Saginaw
Bay and Lake Huron [30]. As pointed out in that study the zooplankton calcu-
lations are very sensitive to the parameter values used in the kinetics. Thus
the zooplankton results must be regarded as preliminary, particularly in view
of the sparcity of zooplankton data.
CARBON
Because of the importance of organic carbon oxidation.in the dissolved
oxygen balance in the Central Basin, both total inorganic carbon and non-
living organic carbon are included as state variables in the calculation.
Data for both these variables are not available. However, data for alkalin-
ity, pH and BOD is available. These variables are used, together with obser-
ved primary production, in order to calibrate the inorganic and organic car-
bon kinetics.
157
-------
Q_
LU
o
o
LU
8.00
4.00
0.00
8.00
4.00
0.00
8.00
4.00
0.00
WESTERN BASIN
_ Segment 1
j 'F'M'A'M'j 'J'A'S'O'N'D
CENTRAL BASIN
Segment 2
j 'F'M'A'M'J ' J'A'S'O^D
EASTERN BASIN
Segment 3
j 'F'M'A'M'j' J'A'S'O'N'D
Figure 68. Lake Erie model calibration for secchi depth (m) , 1970.
158
-------
0.80
0.40
5
0.00
WESTERN BASIN
_ Segment 1
CARNIVOROUS
_ ZOOPLANKTON
TOTAL
ZOOPLANKTON
^HERBIVOROUS
'ZOOPLANKTON
J IFTMTAIMI J ' J'A'S'O'N'D
O
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0.80
O
z
O 0.40
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CENTRAL BASIN
Segment 2
TOTAL ZOOPLANKTON HERBIVOROUS
ZOOPLANKTON
7Y
CARNIVOROUS
ZOOPLANKTON
TTI
J'F'M'A'M1J'J'A'S'O'N'D
EASTERN BASIN
Segment 3
CARNIVOROUS
ZOOPLANKTON
TOTAL ZOOPLANKTON
HERBIVOROUS
.ZOOPLANKTON
J'F'M'A
_A_^St~~/^ /
'M'j'j
A'S'O'N'D
Figure 69. Lake Erie model calibration for zooplankton carbon (mg/£), 1970,
159
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160
-------
The results for pH and alkalinity are shown in figs. 71 and 72. The
alkalinity calibration is not wholly satisfactory although the discrepancies
are emphasized due to the plotting scales employed. Uncertainties in loading
information contribute to the difficulties. However alkalinity is not of
direct concern; it is only useful for the computation of pH. The alkalinity
inaccuracies are not significant since they amount to only a few percent
error, and do not severly impact the pH calculation. The pH results are
quite good for the Western Basin and are reasonable for the Central and
Eastern Basin epilimnia. The Central Basin hypolimnion results are quite
dramatically reproduced.
In the epilimnion there is an increase in pH during the summer due to
the uptake of inorganic carbon by the phytoplankton. In the hypolimnion
there is a decrease in pH during the summer due to an increase in inorganic
carbon resulting from a combination of the respiratory oxidation of non-re-
fractory organic carbon, phytoplankton respiration, and the sediment release
of inorganic carbon. The rapid increase of pH in the hypolimnion in late
summer-early fall is due to overturn. Again the calculated mixing is not
quite rapid enough.
PRODUCTION AND RESPIRATION RATES
The calibration calculations are a method of inferring the kinetic con-
stants of the relevant rate expression from measurements of the state vari-
ables. This indirect procedure is necessary because direct measurements of
all the relevant rates of production, respiration, and recycle are not avail-
able and, in some cases, impractical. However, two conventional measurements
are measurements of rates: primary production and BOD. As pointed out in the
kinetics section gross primary production corresponds to the rate of algal
synthesis while net primary production subtracts the endogenous respiration
rate. Since both these reaction rates are being evaluated in the calculation
it is a simple matter to compare these to the observed rates. Fig. 73 pre-
1k
sents the results. The observations are shipboard C measurements. The four
fold decrease in primary production from the Western to the Eastern Basin is
mainly due to the chlorophyll differences. The calculated results are in
agreement with observations with the exception of the late fall measurements
which indicate more production is occurring than is being calculated.
The two major classes of sinks of dissolved oxygen in the hypolimnia of
lakes are the respiratory processes in the water column and the sediment.
For Lake Erie the former are dominated by algal respiration and organic
carbon oxidation, and a direct calibration of these rates is of major impor-
tance. Fortunately a series of BOD measurements were made in 1967 which
form the basis of this calibration. A BOD measurement can be regarded as a
direct experimental determination of the total water column respiration
exerted in five days in oxygen units. Since the relevant reaction rates are
being evaluated in the calculation a direct comparison is possible. The
algal respiration, and detrital organic carbon oxidation rates are adjusted
to 20 C, the incubation temperature of the BOD measurement, and are multiplied
by five days and compared to observation. The result is shown in fig. 7^'
Both epilimnia and hypolimnia observations are in accord with the calculation.
161
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WESTERN BASIN
GROSS
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GROSS
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Figure 73- Lak6 Erie model calibration for the rate of primary production
expressed as mg/m3/hr of carbon, 1970. Observations are ship-
board C11( measurements.
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This is significant in the Central Basin hypolimnion since the water column
components of the oxygen sinks are of major importance in the oxygen balance.
SEDIMENT CALIBRATION
The other major oxygen sink in the Central Basin is that generated by
the oxidation of the oxygen equivalents which result from the decomposition
of sedimentary organic carbon. The magnitude of the flux of oxygen demand
is determined by the incoming flux of detrital and algal carbon; the sedi-
mentation velocity and interstitial water dispersion coefficient; and the
rate constants for the algal and detrital organic carbon decomposition
reactions. For the steady state case the relationships between these para-
meters anci flux of oxygen equivalents is given by eq. (63). The institial
ammonia concentration and flux is similarly determined, eqs. (^5) and (U6),
by the incoming fluxes of algal and detrital nitrogen. Hence it is important
that these fluxes be reproduced by the computation.
The fluxes to the sediment are determined by the aqueous concentration
of total chlorophyll, organic nitrogen, and organic carbon and their settling
velocities (0.1 m/day and 0.2 m/day respectively). A comparison of computed
and estimated sedimentation fluxes are shown in Table 22.
TABLE 22. ANNUAL SEDIMENTATION FLUXES
VARIABLE COMPUTED ESTIMATED [80]
mg/m /day
Organic Carbon
WB 173. 279.0, 227.0, 53.^
CB 73.7 167.0, 87.8,
EB 37-8 ^57.0,
Nitrogen
WB kh.3 31 A, 23.7, 5-75
CB 30.0 11.6, 23.2,
EB 21.8 62.5
Phosphorus
WB 6.18 9.73, 9-59, 2.05
CB 3.03 3.15, 6.58,
EB 1.9U 19-5
The computed fluxes are simply the product of the yearly average concentra-
tions and the appropriate settling velocity; the estimates are from dated
cores [80], three in the Western Basin (WB), two in the Central Basin (CB) ,
and one in the Eastern Basin (EB). The computed WB and CB flux are slightly
lower for organic carbon and higher for nitrogen and phosphorus than the
means of the estimates from the cores. The EB estimate appears to be unreal-
166
-------
istically high for the three constituents.
The transport parameters of importance are the sedimentation velocity
and the dispersion coefficient. Table 23, shown below, lists the estimates.
TABLE 23. TRANSPORT PARAMETERS
This Work Estimates Ref.
Sedimentation Velocity 0.6l 0.61^0.^3 [8l]
(cm/yr)
Dispersion Coefficient 1.9 0.17 - 1-73 [82]
2
(cm /day)
The sedimentation velocity used is the mean of that determined from dated
cores [80,81]. The dispersion coefficient is in the high range of the esti-
mates from interstitial radionuclides profiles [82]. As can be seen from
eq. (63) the critical dimensionless parameter group is wH/E' which is much
smaller than one for even the smaller estimates of the dispersion coefficient.
This implies that, for the five cm layer at least, dispersion is not the rate
limiting step in the generation of the flux.
The reaction rate constants for the decomposition of algal and detrital
particulates determine the fluxes from these species. The former is chosen
using the results of anaerobic algal decay experiments. The rate used,
1 2
0.02 day at 20°C, corresponds to w/KH ^ 10 which is small relative to one
so that essentially all the labile sedimented algal carbon and nitrogen are
returned to the overlying water. Even an order of magnitude smaller rate
would cause a recycle of ^ 90% so it is reasonably certain that the labile
fraction, 70%, of the algal nutrients are returned.
The reaction rate constant for the detrital nitrogen and carbon are
assumed to be the same. The value is chosen so that the observed intersti-
tial concentration of ammonia nitrogen is reproduced by the calculation. A
reaction rate constant of 0.091 yr~ at 20°C reproduces the annual average
observed interstitial ammonia concentration (1.66 mg N/£) in the top 6 cm.
for 5 core samples at each of three stations in the CB taken from the late
spring through the fall [83]. For that rate the dimensionless number
w/KH = 1.3^ and h3% of the detrital nitrogen is returned to the overlying
water. However at the average sediment temperature (which is assumed to be
equal to the overlying hypolimnion temperatures) of ^ 8°C, the outgoing flux
is ^ 25% of the incoming flux.
If it is assumed that detrital carbon reacts at this rate as well, the
flux of oxygen equivalents for the CB is calculated by eq. (63) to be:
JQ = 125 mg 02/m /day at 20°C or k^.6 mg 02/m /day at 8°C, the temperature
167
-------
of the sediment. Observed sediment oxygen demands are appreciably higher.
Measured rates, corrected for "benthic algal photosynthetic activity yielded
o
estimates of ^00 to TOO mg/m /day during project Hypo [8U]. In addition the
CB dissolved oxygen calibration indicated that an additional sink was requir-
ed. If 550 mg 02/m2/day @ 20°C (or 218.H mg 02/m2/day §8°C) is introduced
into the sediment segment representing the flux of oxygen equivalents from
the deeper sediment layers, then the CB oxygen balance results are in agree-
ment vith observation. It is interesting to note that with this additional
source of 0_ equivalents to the surface sediment layer, the observed BOD and
COD of the interstitial waters bracket the calculated results as shown in
Table 2k.
TABLE 2h. COMPUTED AND OBSERVED SEDIMENT INTERSTITIAL
WATER CONCENTRATIONS - CENTRAL BASIN
VARIABLE COMPUTED Observed Ref.
Average +_ std. dev. Average +_ std. dev.
NH0 1.7U + O.U* 1.66 + 0.68 [83]
3 ~ ~
(mg
02 equivalents -U8.5+25.8 -2U. 5(0,-92)# [85]
(mg 0/fc) -T1.6(-36,-125)§
# Average (Range) of Interstitial Water BOD
§ Average (Range) of Interstitial Water COD
These measurements can be taken to approximate the total oxygen equivalents
of the reduced species in the interstitial waters. They therefore provide
another check for the calculation.
However, the total sediment oxygen demand used in the computation which
is the sum of the surface and deep sediment contribution amounts to 268
mg 0 /m /day (@8°C). This is still significantly less than the observed
2
range of ^00-700 mg 0_/m /day [8U]. The cause of this discrepancy is un-
certain at present.
168
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SECTION 11
VERIFICATION
As discussed in the data section of this report, 1975 is used for veri-
fication of the Lake Erie model because of the abnormal oxygen conditions
which occurred in that year. As is seen in the data comparison plots phyto-
plankton and nutrient concentrations in 1975 were approximately the same as
in the years 1970-197^. However oxygen conditions in '75 were radically
different. The Central Basin of the lake did not become anaerobic as it had
for the preceding years. This difference in behavior provides a good test
of the ability of the model to predict lake oxygen concentrations, a parameter
of prime concern in Lake Erie.
The verification calculations are made using 1975 conditions (loadings,
etc.) but with no changes in the kinetics. The major changes reflect the
altered physical conditions: the depth of the thermocline is altered and the
transport is recalibrated to reflect the unusual thermal condition of the
lake in 1975- Figure 35 shows the two meter vertical average temperature
profiles used to determine the thermocline depth for both the calibration
and verification years. For the calibration year the depth is 17 meters and
for the verification year the depth is 13 meters. It should be noted that
the 17 meter thermocline is the normal depth for the lake. The 13 meter
thermocline is due to a combination of unusually low wind speeds and low
net heat input to the lake in 1975-
The results of the verification for chlorophyll and the nutrients are
shown in figs. 75-81. The results are comparable to the calibration. West-
ern Basin chlorophyll concentrations are underestimated in the spring and
overestimated in the summer. An early Central Basin bloom is not reproduced
but the remainder of the year is reasonable although the hypolimnion concen-
tration is underestimated. Ammonia concentrations are slightly overestimated
but the nitrate concentrations are well reproduced in the epilimnion. With
the exception of the March-April conditions, which are attributed to wind
induced resuspension, the calculated total phosphorus concentrations are in
reasonable agreement with observation as are the orthophosphorus and silica
calculations.
The most striking feature of the verification is the dissolved oxygen
comparison, shown in fig. 82. The agreement in the Central Basin hypolim-
nion is remarkable especially when compared to the lower oxygen concentra-
tions in the 1970 and 1973-197^ calibrations. The higher oxygen concentra-
tions in 1975 are basically due to the larger volume of the hypolimnion in
that year. It is easy to see that the sediment oxygen demand, which is a
constant on an areal basis, would result in a smaller volumetric depletion
rate in 1975. However, as is apparent in Table 25, the sinks due to phyto-
plankton respiration and organic carbon oxidation are reduced as well. The
169
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TABLE 25- CALCULATED AVERAGED CENTRAL BASIN CONCENTRATIONS AND
OXYGEN CONSUMPTION RATES FOR 1970 AND 1975
Variable
Phytoplankton Chlorophyll
yg Ch£-a/&
(% change)
Organic Carbon
mg C/£
(% change)
Phytoplankton Respiration
mg 02/&/day
(% change)
Organic Carbon Oxidation
(% change)
Year
Epilimnion
Hypolimnion
19TO
1975
19TO
1975
1970
1975
1970
1975
8.32
7.25
(-12.9)
o.Hoo
0.365
(-8.8)
0.125
0.107
(-1U.10
0.108
0.098
(-9.3)
2.89
2.35
(-18.7)
0.310
0.212
(-31.6)
0.0200
0.0165
(-17.5)
0.0568
0.039^
(-30.6)
JJ
Average from day 180 to day 2^0
178
-------
estimated total phosphorus loading in 1975, 20,1400 MT/yr is 11.7$ lover than
that for 1970, 23,100 MT/yr. This is reflected in the 12.9$ reduction in
average Central Basin epilimnion chlorophyll during stratification and the
8.75$ reduction in organic carbon concentration. The reduction in the hypo-
limnion are more dramatic, however. This is due to two factors. The verti-
cal exchange in the Central Basin is less in 1975 than 1970, compare figs.
38 and 40, and the flux of chlorophyll and organic carbon due to the settling
velocity across the thermocline is diluted into a larger hypolimnion volume,
thus reducing the resulting concentrations. The combined effects produce a
reduction in hypolimnion concentrations of 19$ and 31.5$ for chlorophyll and
organic carbon concentrations and, since the rates of oxygen consumption are
proportional to these concentrations, a reduction in the utilization rates
as well. The interactions of these factors produces the 1975 dissolved oxy-
gen distribution and it is clear that an analysis of the mass balances and
changing sources and sinks requires a rather complex and complete calcula-
tion.
The successful oxygen verification indicates that the determination of
the contribution of the various components which interact to produce the
oxygen profile is reasonable. The difference between the verification and
the calibration oxygen profile is due to a combination of water column and
sediment sink reactions. The balance between these sinks is important
since the percentage reduction is not uniform for each sink. If the magni-
tudes were not correctly estimated by the calibration then the reductions
would not have resulted in the proper oxygen profile for the verification.
The contributions of each of the oxygen sinks to the Central Basin
hypolimnion 1975 oxygen profile are shown in fig. 83 and Table 26.
TABLE 26. PERCENTAGE CONTRIBUTION OF OXYGEN SINKS TO DISSOLVED OXYGEN
DEFICIT AT THE TIME OF MINIMUM D.O. IN THE CENTRAL BASIS
HYPOLIMNION
Oxygen Sink Percent Contribution
1970 1975
Deep Sediment Oxygen Demand
Surface Sediment Oxygen Demand
Organic Carbon Oxidation
Phytoplankton Respiration
28.7
11.5
42.5
17.3
22.8
8.2
46.9
22.1
At the time of minimum D.O. organic carbon oxidation accounts for almost 47$
of the depletion, sediment oxygen demand 31$, and phytoplankton respiration
22$. The comparable 1970 contributions are: 43$, 40$, and 17$ so that in
1970 the water column contributed 60$ and the sediment contributed 40$. In
1975 the water column contributed 69$ and the sediment contributed 31$.
179
-------
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180
-------
ANALYSIS OF CALIBRATION AND VERIFICATION ERROR
The comparisons of computed and observed concentrations for the calibra-
tion (1970, 1973-197*0 and verification (1975) years have "been made using
graphical presentations. These are easy to interpret and give a direct
visual impression of the quality of the comparisons. Hovever they lack a
quantitative dimension with which to assess statistically the extent of the
agreement achieved. Although a complete analysis has not been performed for
all variables, results are available for three principal variables in the
Central Basin: dissolved oxygen, chlorophyll ei, and total phosphorus. For
each year of calibration and verification the residuals, that is the differ-
ence between calculated and observed average concentration, in the epilimnion
and hypolimnion of the Central Basin have been analyzed. The results are
given in Table 27 The mean residual, and its standard error are tabulated,
as well as the standard deviations of the residuals. The median relative
error, where the relative error is [calculated - observed]/observed, is
also included. This latter variable has been analyzed in detail for a Lake
Ontario computation [l].
The means of the residuals for dissolved oxygen and chlorophyll are for
the most part not statistically different from zero (t-test at the 95$ level
of significance). Those that fail this test are indicated on the table.
This indicates that the model is not biased relative to the data. The total
phosphorus results, however, indicate a statistically significant bias.
The standard deviation of the residuals for dissolved oxygen are between
0.6 and 1.3 yg 0 /£, for chlorophyll a, between 2.2 and 3.^ ug/£, and for
c.
total phosphorus between 6.9 and 15 ug/&- The values for dissolved oxygen
will be used subsequently to establish a lower bound for the prediction error.
A comparison of the calibration and prediction error is instructive. In
nearly all cases the mean residual is larger for the verification (1975) than
for the 1970 calibration. The same is true for the residuals standard devia-
tion and the median relative error. This is not unexpected since the verifi-
cation calculation was not adjusted to fit the data. The average ratio of
verification ('75) to calibration ('70) residual standard deviation is 1.52
which suggests that the scatter is ^ 50% larger for the verification than the
calibration.
During the calibration procedure the major emphasis was placed on the
1970 rather than the 1973-7** data. This is apparent in the residual standard
deviation and median relative error results in Table 27. For dissolved oxy-
gen the 1975 verification residual standard deviations are actually smaller
than the 1973-7** result, for chlorophyll they are comparable and for total
phosphorus the 1975 residual standard deviation is larger than the 1973-7**
result. Again this is not unexpected. The transport calibration was made
using 1970 data. The 1973-7** data served as confirmatory rather than pri-
mary data for calibration. As a consequence it is not as closely reproduced
by the calculation.
181
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TABLE 27. CALIBRATION AND VERIFICATION RESIDUAL ANALYSIS
Residual Median Relative
Standard Standard Error
Mean Error Deviation (%}
Dissolved Oxygen
(mg/A)
Epilimnion
Hypolimnion
Chlorophyll a
Epilimnion
Hypolimnion
Total Phosphorus
Epilimnion
Hypolimnion
1970
73-7^
75
1970
73-74
75
1970
73-74
75
1970
73-74
75
1970
73-74
75
1970
73-74
75
-0.1308
-0.1649
-0.2439
0.5645*
0.6286
0.2943
0.0964^
-2.6924
1.8616
No
-0.9646^
3.2651
9-90**
18.42
1.56
14.02**
0.188
0.381
0.238
0.245
0.233
0.798
0.974
1.316
Observed
0.576
0.734
2.3
2.0
4.0
2.2
2.9
4.4
0.595
1.263
0.953
0.775
1.084
0.932
2.524
3.376
3.316
Data Available
2.154
2.934
7-21
6.88
13.46
8.92
10.16
14.5
3.6
7.
6.
12.2
15.
8.3
31.6
63
20.6
52
4o
M-l
l\ ~\
53.3
35-
21.2
40.5
failed t-test at 95% level of significance
failed t-test at 99$ level of significance
182
-------
SECTION 12
ESTIMATED EFFECTS OF PHOSPHORUS LOADING REDUCTIONS
In this section a series of calculations are presented which are an
attempt to estimate the consequences of phosphorus loading reductions to
Lake Erie. Nitrogen loading reductions are not considered because the pos-
sibility of increased nitrogen fixation by blue-green algae during periods
of excess phosphorus concentrations is not addressed in the calculation.
A number of uncertainties exist in the present model which limit the
accuracies of these estimates. Perhaps the most severe is in the dynamics
of the generation of the sediment oxygen demand. Only the surface layer is
considered, the deeper layers are included by externally specifying their
effect on sediment oxygen demand. Thus the calculation is not a complete
mass balance involving both sediments and water column. The vertical segmen-
tation of the Central Basin is not fine enough to resolve the vertical gradi-
ents in a comprehensive way which limits the resolution that can be achieved
in the calibration. The horizontal exchanges of the Central Basin hypolim-
nion are uncertain and may be significant. The water column oxygen consump-
tion data (BOD) are meager and not available for either the calibration or
verification year, nor are adequate sediment oxygen demand measurements
available. In light of these and other uncertianties, the estimates to be
presented in this section must be viewed with caution. Although they repre-
sent the most comprehensive attempt to date that directly addresses the dis-
solved oxygen mass balance together with the nutrients and phytoplankton
interrelationships, they are, as yet, only estimates whose accuracy is diffi-
cult to assess. Further efforts are required to refine the quantitative
formulations and to verify the kinetics from independent experiments. This
is an ongoing process and it is expected that as more data are collected and
analyzed within the general framework of these calculations the confidence in
their descriptive and predictive ability will increase.
In order to make at least some quantitative assessment of the error
associated with the predictions a crude analysis based on the errors of cali-
bration is included. It should be stressed that this analysis is very pre-
liminary and is intended more as a guideline than as a rigorous evaluation.
TIME TO STEADY STATE
Lake Erie appears to be in equilibrium with the current phosphorus load-
ing. This is not unexpected since the loadings have been essentially con-
stant for the last five years, see figs. 9-13, and the hydraulic residence
time which sets the upper bound on the time to reach steady state in the
water column is approximately 2.5 years so that in 5 years the lake would be
at 87% of equilibrium if only hydraulics control. Phosphorus sedimentation
decreases the time to approximately h years as shown in Table 28, a yearly
183
-------
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summary of a projection calculation to be discussed subsequently. Water col-
umn concentrations of phosphorus and chlorophyll have reached their equili-
briums by the 4th year. However dissolved oxygen concentrations and the
estimated area of anoxia are still changing in the 5th year. This is due to
the considerably longer time to steady state for the sediment segments. This
is discussed in more detail subsequently.
The loading to the lake is a combination, of point sources (industries
and municipalities) and diffuse sources (runoff). The point sources are
fairly constant while the diffuse sources are a function of tributary flow.
To determine the loadings for projections the average water year based on
USGS flow data is used to calculate diffuse loadings. These diffuse loadings
are combined with 1975 point source loadings to yield the current load. This
load is termed the base year load. It has been calculated, as have all the
loadings used in these calculations, by the U.S. Army Corp of Engineers,
Buffalo District. Tables A-10 to A-12 summarize these loadings.
The reductions of phosphorus loading follow guidelines provided by the
U.S. EPA. Initially, point sources are reduced to a 1 mg/£ effluent level
for all treatment plants. At this loading level two separate scenarios are
followed. In the first scenario point sources are reduced from 1 mg/£ to
0.5 mg/£ and then 0.1 mg/£. In the second scenario point sources are held at
1 mg/£ and reductions are made in diffuse source loadings exclusive of
Detroit River diffuse sources.
Depending on which of the two scenarios is followed there is a differ-
ence in the computed effect on lake conditions for the same total loading.
Removal of diffuse sources is computed to have less of an effect on the
important water quality parameters than does removal of point sources. This
is caused by the differences in the ratio of available or orthophosphorus to
total phosphorus for diffuse and point sources. Point sources have a higher
ortho to total phosphorus ratio than do diffuse sources. If point sources
are removed more orthophosphorus is removed per unit mass of total phosphorus
than if diffuse sources are removed. The result is less phosphorus entering
the lake which is immediately available for growth. Since a higher percentage
of the phosphorus load is now in the unavailable fraction, the percentage of
the phosphorus which is particulate is also increased. This results in a
greater phosphorus loss due to settling. Consequently, the concentration of
phosphorus in the water column is decreased. In other words, the removal of
point sources has a greater impact in improving water quality than diffuse
source removal because more orthophosphorus is removed and, in addition, a
greater fraction of the total phosphorus is in the unavailable form. Thus
the proportion of phosphorus that can be removed by sinking is also increased
resulting in lower lake phosphorus concentrations and lower phytoplankton
chlorophyll concentrations. These points are illustrated in fig. 84. This
figure shows the unavailable and orthophosphorus loads to the Western Basin
under two reduction schemes. The solid line is the phosphorus load with
point sources reduced to 0.5 mg/£. The dashed line is the phosphorus load
with the diffuse sources reduced by 25% after point sources are reduced to
1 mg/Jl. The total phosphorus load is approximately the same for both cases.
With the point source reduction the lake total P load is 12652 MT/yr and with
the diffuse source reduction it is 125^7 MT/yr. The plot of unavailable
185
-------
WESTERN BASIN ORTHOPHOSPHORUS LOADING
CO
CO
oc
o
I
70000
60000
50000
40000
30000
20000
10000
POINT SOURCES AT 0.5 MG/L
25% DIFFUSE REDUCTION
J'F'M'A'M'J'J'A'S' o ' N ' D
WESTERN BASIN UNAVAILABLE PHOSPHORUS LOADING
ro
70000
60000
50000
co
5: 40000
O
x
8> 30000
20000
10000
POINT SOURCES
AT 0.5 MG/L
25% DIFFUSE REDUCTION
J I F T M I A I M I J i J I A 1 S I 0 I M^ D
Figure 8U. Western Basin Loads (its/day) for orthophosphorus (top) and
unavailable phosphorus (bottom) under two reduction schemes
(a) with point sources reduced to 0.5 mg/& and (b) diffuse
sources reduced by 25% after point sources are reduced to 1
mg/£.
186
-------
phosphorus shows that this fraction of the load is higher with point source
reduction, and the orthophosphorus plot shows that this fraction of the load
is lower with point source reduction.
SIMULATION RESULTS
The results are a summary of the last year calculations for a five year
simulation for which the loadings are reduced to the indicated level at the
beginning of the first year. All loading reductions are uniform across the
lake. Except for a comparison to the Army Corps Plan IX, there are no basin
specific removals.
The annual whole lake phosphorus loading at a number of reductions are
shown below in Table 29.
TABLE 29
LOADING CONDITION
TOTAL PHOSPHORUS LOAD
MT/YR
#
Base Year 19182
1 mg/£ Point Sources 1^195
0.5 mg/£ Point Sources 12652
0.1 mg/£ Point Sources Ilkl9
s
1 mg/£ Point Sources + 10$ Diffuse Reduction 13535
20% " 128T6
kO% " 11558
" 60$ " 102^0
" No Diffuse Sources 7600
excluding atmospheric loading
§
excluding the diffuse source contribution to the Detroit River
The results for phytoplankton chlorophyll are shown in fig. 85. They
represent the epilimnion concentration averaged for June 15 to September 5-
For the reasons explained above there are separate responses for point and
diffuse source reductions. There is a greater effect in the Western Basin
than in the Central and Eastern Basins. This is due to the large percentage
of the load that enters in the Western Basin and its short detention time.
Conditions in the 'basin are dominated by the incoming load. Changes in that
load cause significant changes in the basin. An important conclusion is
that even with substantial load reductions the Western Basin average summer
chlorophyll will still be above what is generally considered the mesotrophic-
eutrophic level.
187
-------
25
20
O)
=L
I
CL
X
O
15
10
0
(JUNE 15-SEPTEMBER 5)
POINT SOURCE REDUCTIONS
DIFFUSE SOURCE REDUCTIONS
WESTERN BASIN
CENTRAL BASIN
I
EASTERN BASIN
I I I
I
6000 8000 10000 12000 14000 16000 18000 20000
TOTAL PHOSPHORUS LOAD, MT/yr
t » ft t »
NO DIFFUSE LOADS
DIFFUSE REDUCTIONS (%): 60
POINT REDUCTIONS (mg/l):
40
20
0.1 0.5
BASE YEAR
LOADINGS
1.0
Figure 85. Summer average epilimnion chlorophyll (yg/&) vs. lake total
phosphorus loading (mt/yr). There are separate responses for
point and diffuse source reductions and for each basin.
188
-------
The effect of phosphorus load reductions on the average epilimnion sum-
mer total phosphorus is shown in fig. 86. As is expected the pattern is
similar to the chlorophyll results. However, there is one major difference.
There is a larger difference in phytoplankton chlorophyll "between point and
diffuse reductions than in total phosphorus. If point sources are reduced so
that the total phosphorus load is reduced to 12000 MT/yr the resultant calcu-
lated chlorophyll and phosphorus concentrations in the Western Basin are
about 12 and 20 yg/£ respectively. To achieve these same levels with diffuse
source reductions a load of about 8,700 MT/yr is needed for chlorophyll and a
load of about 10,300 MT/yr is needed for phosphorus. In other words, an
extra 3,300 MT/yr of phosphorus must be removed under diffuse reduction to
achieve the 12 yg/£ chlorophyll level. An extra 1,700 MT/yr of phosphorus
must be removed under diffuse reduction to achieve the 20 yg/£ phosphorus
level. Both these levels are met with the same total load reduction in point
reduction.
Fig. 87 shows the effect of the phosphorus load reductions on light
penetration expressed as average summer secchi disk depth. Load reductions
will result in a significant increase in water clarity in the Central and
Eastern Basins. In the Western Basin there is only a small increase in
clarity as the load decreases. Also, as the load decreases the rate of
increase of Western Basin clarity declines, since most of the light attenua-
tion in the Western Basin is caused by high turbidity levels rather than
phytoplankton. Even with substantial decreases in phytoplankton light ex-
tinction rates remain high. The converse is true in the other basins.
The phosphorus discharge to Lake Ontario as a function of phosphorus
load is shown in fig. 88. This loading is calculated based on the Eastern
Basin epilimnion concentration and the existing flow to the Niagara River. A
factor accounting for the difference in concentration between the whole East-
ern Basin and the area near the outlet is included in the calculation. This
factor is the ratio of the average concentration at the four stations closest
to the Niagara and the Eastern Basin epilimnion average. A value of 1.18 has
been calculated using 1970 CCIW cruise data.
Perhaps the variable of most interest with regard to load reductions is
dissolved oxygen particularly the effect on anoxia in the Central Basin. The
methods used to simulate the Central Basin oxygen concentrations as well as
the area of anoxia has been discussed in Section 9- However, the structure
of the sediment calculation raises a question as to its applicability to oxy-
gen predictions under loading reductions. The sediment oxygen demand is com-
posed of two parts: a surface layer (5 cm) demand which is dynamically calcu-
lated, and a demand for the rest of the sediment which is externally speci-
fied. Under loading reductions the surface" segment responds to the input
changes and the oxygen demand dynamically adjusts. However the deep sediment d
demand, since it is externally specified, remains constant. This is likely
to be a good assumption during the short time period used for the simula-
tions (5 years). However, the deep sediment will ultimately also be affected
by the reduced load.
In order to estimate the change in water column oxygen conditions due
189
-------
40
30
C/3
§20
Q_
V)
O
0
(JUNE 15-SEPTEMBER 5)
i POINT SOURCE REDUCTIONS
- DIFFUSE SOURCE REDUCTIONS
CENTRAL BASIN
1
L
1
1
EASTERN BASIN
I L
NO DIFFUSE LOADS
DIFFUSE REDUCTIONS (%): 60
POINT REDUCTIONS (mg/l):
40
20
0.1 0.5
6000 8000 10000 12000 14000 16000 18000 20000
TOTAL PHOSPHORUS LOAD, MT/yr
BASE YEAR
LOADINGS
1.0
Figure
Summer average epilimnion total phosphorus (yg PO,-P/£) vs. lake
total phosphorus loading (mt/yr). mi-- -~'J
chlorophyll results.
The pattern is^similar to the
190
-------
8.0
I
t 6.0
UJ
Q
CO
Q
E
o
O
UJ
CO
4.0
2.0
0.0
(JUNE 15-SEPTEMBER 5)
POINT SOURCE
REDUCTIONS
DIFFUSE SOURCE
REDUCTIONS
EASTERN BASIN
CENTRAL BASIN
1
WESTERN BASIN
1
I
I
I
6000 8000 10000 12000 14000 16000 18000 20000
TOTAL PHOSPHORUS LOAD, MT/yr
NO DIFFUSE LOADS
DIFFUSE REDUCTIONS (%): 60
POINT REDUCTIONS (mg/l):
40
20
0.1 0.5
BASE YEAR
LOADINGS
1.0
Figure 87. Simmer average secchi disk depth (m) vs. lake total phosphorus
loading (mt/yr). Load reductions will result in a significant
increase in water clarity in the Central and Eastern Basins.
191
-------
\
O
P:
o
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tt
POINT SOURCE ,
1
|
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t
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JSE LOADS J
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ake Ontario vs. lake total phos-
oading is calculated based on the
tration and the existing flow to
o
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-------
to the eventual effect of the loading reductions on the deep sediment, a
coupling of the deep sediment oxygen demand to the surface sediment demand
has been incorporated into the calculations. The deep sediment demand is
reduced each year of the simulation by the same percentage as the oxygen
demand reduction that is calculated for the surface sediment. This adjust-
ment is meant only to simulate the magnitude of the reduction in deep sedi-
ment oxygen demand and not its progression in time. A fully dynamic calcula-
tion of this effect is beyond the scope of this initial effort. However it
is a key component in any calculation vhich attempts to estimate the time
required to reach an equilibrium with a changed loading. The projections
relating to dissolved oxygen have two sets of results: one to represent the
short term effect where only the surface sediment oxygen demand responds to
loading changes and the other to represent the long term or ultimate effect
when the entire sediment oxygen demand has responded to loading changes.
Figure 89 shows the effect of reducing phosphorus on the maximum anoxic
area to be expected in any year. Using the short term and ultimate effect to
bracket the estimate, the estimate is that anoxia will be eliminated when the
phosphorus load is reduced to between 10,500 and 12,600 MT/yr under point
source reduction and between 9,500 and 12,200 MT/yr under diffuse source
reduction. These loadings correspond to a point source effluent concentra-
tion range from about 0 to 0.5 mg/£ phosphorus and a reduction in diffuse
sources of between 30 and TO percent with a 1.0 mg/£ point source effluent.
Higher chlorophyll concentrations under diffuse reductions is the reason for
the greater anoxic area at the same total phosphorus loading. The extra
chlorophyll and detrital organic carbon settles into the hypolimnion where
it increases the oxygen demand through respiration and oxidation.
The oxygen consumption rate necessary to maintain oxic conditions may be
found in the phosphorus loading vs. oxygen depletion rate plot, fig. 90. The
consumption rate is the sum of all water column and sediment oxygen sinks in
the hypolimnion. It does not include the dissolved oxygen being transported
across the thermocline from the epilimnion. It is, therefore, larger than
the actual observed depletion rate. Using the loadings given above the con-
sumption at the point of zero anoxic area is found to be 0.089 mg/£/day.
The current depletion rate (base year) is 0.13 mg/£/day. Thus a reduction
of 0.0^1 mg/£/day, or 32%, is necessary to prevent anoxia.
The dissolved oxygen concentrations calculated in these simulations are
volume averages for each segment. For the two segments comprising the
Central Basin hypolimnion, it is straightforward to calculate the volume
average concentration for the entire hypolimnion at any point in time. Two
sets of calculations are presented: the volume average hypolimnion concentra-
tion just prior to overturn, fig. 91; and the temporal average during the
period of stratification of the volume average hypolimnion concentration,
fig. 92. The oxygen concentration just prior to overturn are used to estab-
lish the area of anoxia. This figure may be used in conjunction with the
minimum vs. volume average dissolved oxygen plot, fig. U2, to determine an
estimate of the absolute minimum dissolved oxygen to occur in the basin at
any loading rate.
193
-------
CM
8000
6000
LU
CC
<
o
X
o
4000
2000
0
POINT SOURCE REDUCTIONS
DIFFUSE SOURCE
REDUCTIONS
SHORT TERM
EFFECT
(5 YEARS)
I
ULTIMATE EFFECT
I 1
6000 8000 10000 12000 14000 16000 18000
TOTAL PHOSPHORUS LOADING, MT/yr
20000
t
NO DIFFUSE LOADS
DIFFUSE REDUCTIONS (%):
POINT REDUCTIONS (mg/l):
0
0
I
40
1 0.
20
5 1.
t
BASE YEAR
LOADINGS
Figure
Area of anoxia (km2) vs. lake total phosphorus loading
shoving both the short term and ultimate effects.
-------
0.16
* 0.12
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r (B
LU -Q
-i ^ 0.08
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Q £
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LU
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0.04
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SHORT TERM
EFFECT
(5 YEARS)
ULTIMATE EFFECT
POINT SOURCE
REDUCTIONS
DIFFUSE SOURCE
REDUCTIONS
1
1
1
1
1
1
6000 8000 10000 12000 14000 16000
TOTAL PHOSPHORUS LOAD, MT/yr
t
NO DIFFUSE LOADS
DIFFUSE REDUCTIONS (%): 60
POINT REDUCTIONS (mg/l):
40
20
18000 20000
BASE YEAR
LOADINGS
0.1 0.5
1.0
Figure 90. Central Basin hypolimnion oxygen consumption rate (mg/£-day) vs,
lake total phosphorus loading (mt/yr). The consumption rate is
the sum of all vater column and sediment oxygen sinks in the
hypolimnion.
195
-------
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ULTIMATE EFFECT
POINT SOURCE REDUCTIONS
- DIFFUSE SOURCE REDUCTIONS
SHORT TERM
EFFECT
(5 YEARS)
I
1
I
I
I
I
6000 8000 10000 12000 14000 16000
TOTAL PHOSPHORUS LOAD, MT/yr
NO DIFFUSE LOADS I
DIFFUSE REDUCTIONS (%): 60
POINT REDUCTIONS (mg/l):
0
40
1 0
20
5 1.
18000 20000
\
BASE YEAR
LOADINGS
Figure 92. Summer average Central Basin hypolimnion dissolved oxygen (mg/Jl)
vs. lake total phosphorus loading (mt/yr).
197
-------
Certain specific phosphorus reductions are of special, interest. These
include the reductions necessary to achieve zero- anoxic area, to achieve a
minimum dissolved oxygen of 1 mg/£, or to maintain phosphorus concentrations
at or below 20, 15 and 15 yg/& in the Western, Central and Eastern Basins
respectively. The effect of these reductions on conditions in the lake as
well as the effect of specific point source reductions is shown in Table
30. The ranges given in the table represent the span froir. the short term
effect to the ultimate effect. The results shown in this table may be
obtained directly from the loading reduction plots.
Simulations of the effects of reducing phosphorus loadings on the in-
lake phosphorus concentration has also been performed by the U.S. Army Corps
of Engineers, Buffalo District. The model used is a phosphorus mass balance
model considering each basin as a completely mixed unit with no dispersive
exchange between basins. Phosphorus exchange with the sediment is incorpora-
ted. A full description of the model as well as its results is presented in
r -, o 1
the Corps LEWMS Preliminary Feasibility Report, Volume I . Phosphorus
concentrations of 20, 15 and 15 yg P/A in the Western, Central and Eastern
Basins respectively are proposed as objectives in order to provide signifi-
cant rehabilitation of Lake Erie. A series of alternatives have been devel-
oped in order to meet these criteria and Plan IX succeeded in meeting the
objective. This plan calls for the reduction of treatment plant effluents
to 1 mgPO, -P/£ for all treatment plants with capacities equal to or greater
than one million gallons per day. In addition there is to be a 37 percent
diffuse source reduction in the Western Basin and a 2k percent diffuse source
reduction in the Eastern Basin.
This phosphorus reduction scheme has been simulated using the Lake Erie
model employed for the above calculations. A comparison with the Corps of
Engineers results is shown in Table 31. Except for the Eastern Basin the
results are fairly consistent. However, the results for dissolved oxygen
indicate that these phosphorus levels are not sufficient to provide the sig-
nificant rehabilitation the Corps is striving for. The short term analysis
indicates that the anoxic area will be about 1733 km2, reducing ultimately to
500 km2 of anoxic area. The calculated oxygen consumption rate ranges from
0.106 to 0.093 mg/£/day between the short term and ultimate conditions.
There will be substantial reduction in chlorophyll levels with the average
summer chlorophyll being reduced from about 37 percent in the Western Basin,
it 5 percent in the Central Basin and it 5 percent in the Eastern Basin. Secchi
depth is calculated to improve by 18, 29 and 27 percent in the Western, Cen-
tral and Eastern Basins respectively.
ON THE UNCERTAINTY OF THE PREDICTIONS
An important and, unfortunately, difficult problem is to evaluate the
level of uncertainty associated with the predictions. Broadly speaking there
are two classes of uncertainty which can be associated with the predictions.
These might be called structural uncertainty due to physical, chemical, and
biological processes which have been misrepresented or entirely ignored, and
parameter uncertainty due to miscalibration and data uncertainty. There
appears to be no general way of evaluating the level of structural uncertain-
ty since if one knew that processes were incorrectly specified one would
198
-------
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TABLE 31. COMPARISON OF RESULTS - THIS WORK AND THE U.S.
ARMY CORPS OF ENGINEERS PHOSPHORUS MODEL
Loading
Condition
*
Present
(Base Year)
Corps Plan
IX
Basin
Western
Central
Eastern
Western
Central
Eastern
Total Phosphorus Concentration, ug P<\ -P/&
This Work Corps of Engineers
36.
16.3
12.4
23.5
11.
8.4
37-
18.
22.
20.
12.
15-
Corps model uses the 1973 loadings = 19607 MT/yr.
This work uses the ~ba.se year = 19182 MT/yr.
presumably correct that specification. The same is true for important pro-
cesses that have been entirely ignored. An attempt has been made to evalu-
ate the range of prediction uncertainty due to the long term versus short term
behavior of the sediment under reduced phosphorus loadings but no general
method is available to cope with other sources of structural uncertainty.
The effect of parameter uncertainty can be evaluated using sensitivity
analysis. However it is not known a priori how uncertain the parameters are
and therefore what are reasonable ranges for the sensitivity analyses.
Consider another approach. If it is assumed that there are no data
errors then the calibration uncertainty is due entirely to parameter uncer-
tainty. Under projected conditions it is likely that projection uncertainty
is at least as large as calibration uncertainty since if a "projection" were
made using the calibration conditions the correct "projection" uncertainty
would be the known calibration uncertainty. Hence it seems reasonable to
assume that calibration uncertainty is the lower bound of prediction uncer-
tainty.
An application of this line of reasoning can be made to the uncertainty
of prediction associated with fig. 91» the minimum Central Basin hypolimnion
dissolved oxygen versus lake total phosphorus loading. This figure, together
with figs. 42 and 92 are combined to evaluate the area of anoxia to be expec-
ted for various total phosphorus loadings. As can be seen in fig. 91, the
relationship between minimum D.O., C
mm-
and phosphorus loading, W, is roughly
linear so that a Taylor series expansion about
W yields:
201
-------
9C .
C . (₯) = C . (W ) + (-5^) (W-W ) (89)
mm mm o 3W o o v * '
where C . (W) is the minimum D.O. projected to occur at loading rate W.
If C . (W) and W are thought of as random variables then the variance of
mm
these quantities are related by the equation:
f}P P
v{w}
For the calibration and verification conditions the statistical analysis of
the D.O. residuals gave a residual standard deviation of 0.5 - 1.0 mg/£
2 2
(variance of 0.25 - 1.0 mg /£ ). If it is assumed that this calibration
variance is a reasonable estimate of V{C . (W)} which is the projection var-
min
iance, then from fig. 91 3W/9C . = 2000 (MT/yr )/(mg/£) (at ₯ = 10,000 MT/yr)
and taking the square root of eq. (90) yields
.
mm mm
where 0" is the standard deviation of the projected loading and aC . is the
standard deviation of the projected minimum D.O.. For Oc . =0.5 mg/&*
mm
a = 1000 MT/yr and for o~C . =1.0 mg/Jl, 0 = 2000 MT/yr. Hence if the cali
bration standard deviation is taken as a measure of the projection uncertain
ty of minimum D.O., the corresponding projected loading to achieve a speci-
fied minimum D.O. is uncertain to between 1000 and 2000 MT/yr depending on
the choice of ac . . At W = 10,000 MT/yr this amounts to between a 10 and
20% uncertainty of predictions.
202
-------
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203
-------
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83. Weiller, R.R. The Composition of the Interstitial Water of Sediments
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209
-------
APPENDIX
LAKE ERIE MODEL INPUT INFORMATION
The purpose of this appendix is to summarize the input information neces-
sary to make the calculations discussed in this report. The kinetics are
described in the body of this report (Tables 11-19). Where this is not the
case, references are presented here.
MORPHOMETRY AND HYDRODYNAMIC REGIME
The morphometry and hydrodynamic regime for 1970 is presented in Table
A-l. The modified segmentation and transport regime for 1975 is presented in
Table A-2.
PHYSICAL EXOGENOUS VARIABLES
1. Water Temperature - The temperature distribution is constructed from
cruise averages for each segment using 1970 CCIW data. This data are the same
as that used for the calibration of the transport regime. Winter values have
been estimated due to the absence of cruise data for this period. These tem-
peratures are given in Table A-3.
2. Solar Radiation - Solar Radiation data for Lake Erie are available from
Mateer , included in the 1969 UC report, and Richards and Loewen . The
Mateer data are over the lake data while the Richards and Loewen data are
recorded at Cleveland and adjusted by a lake-to-land ratio,, The Richards and
Loewen data are used and these are listed in Table A-^ as well as the data
from Mateer.
3. Photoperiod - The photoperiod data used in the Lake Erie Model are taken
(3)
from the photoperiod information given by Thomann et al. ' and listed in
Table A-5. This data are calculated from sunrise, sunset Information reported
in the World Almanac.
U. Background Water Clarity - The non-chlorophyll related extinction coeffi-
cient is a function of wind velocity. This function is shown in fig. kk.
The wind velocity is also given in that figure.
5. Oxygen transfer Coefficient - The oxygen transfer coefficient used is
temporally and spatially constant. It is set at 2 meters/day which is within
the observed range for the lakes at the prevailing wind velocities.
BIOLOGICAL AND CHEMICAL EXOGENOUS VARIABLES
1. Nutrient loadings - Phosphorus, nitrogen and silica loadings monthly by
210
-------
basin have been provided by the Corps of Engineers, Buffalo District. These
inputs are discussed in section k. The tabulation for both the calibration
and verification years is given for the Western Basin in Table A-6, the Cen-
tral Basin in Table A-7 and for the Eastern Basin in Table A-8. These
nutrients are inputted as mass loadings. The loading information for the
chloride calculation is given in Table A-9. The phosphorus loading used for
the projections are given in Tables A-10, A-ll, and A-12. The loading input
of the other state variables as well as atmospheric nutrient loads are in
the form of boundary concentration conditions (mass/unit volume). Boundary
conditions, since they are always associated with a flow regime (volume/unit
time), can also be viewed as mass loadings into the water body. The boundary
conditions used are given in Table A-13.
2. Initial Conditions - the initial conditions are shown in Table A-1^. The
integration starts the first of January and these values reflect the measured
values for this part of the year.
211
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TABLE A-U
SOLAR RADIATION DATA
Month
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
Solar Radiation
@ Cleveland
Langleys/day
123
ITT
300
365
509
5U6
523
1*58
360
2U6
129
100
Lake to Land
Ratio*
0.8
0.8
0.8
0.89
1.09
1.19
1.15
1.28
1.33
1.37
1.08
0.9
Solar Radiation
on Lake*
Langleys/day
98
lk2
2l*0
325
555
650
602
586
1*T9
33T
iho
90
Solar Radiation
from Mateer
Langleys/day
110
190
290
390
1*50
550
550
1*70
3TO
21*0
130
90
Values used in the calculation
21T
-------
TABLE A-5
PHOTO PERIOD DATA
Julian Date
0
15
U5
75
105
135
165
195
225
255
285
315
31*5
365
Photo Period in days
0.375
0.385
0.1*35
0.50
0.57
0.626
0.658
0.61*6
0.599
0.532
0.1*66
0.1*05
0.370
0.375
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225
-------
TABLE A-13
BOUNDARY CONDITIONS
System
Boundary Condition
Western Basin Central Basin
Non-Diatom Chlorophyll
Herbivorous Zooplankton
Organic Nitrogen
#
Ammonia Nitrogen
Nitrate Nitrogen
Unavailable Phosphorus
*
Ortho Phosphorus
Carnivorous Zooplankton
Total Inorganic Carbon
Alkalinity
Segment 1
3.0
0.02
0
0.023
0
0
.0003^8
0.01
1.80
1-75
Segment 2
3.0
0.02
0
6.5^9
0
0
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0.01
1.76
1.9
Non-Living Organic Carbon 0.206 k.2h
Dissolved Oxygen
Diatom Chlorophyll
Unavailable Silica
Available Silica
8.0
5.0
0.2
0
8.0
5.0
2.0
0
Eastern Basin
Segment 3
3.0
0.02
0
3.^5
0
0
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0.01
1.82
1.9
5.06
8.0
5.0
0.2
0
Units
ygChl-a/£
mgC/£
mgN/£
mgP/£
mgC/Jl
meq/£
meq/£
mgC/£
mg02/£
ygChl-a/Jl
mgSiM
Atmospheric loads
226
-------
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-------
TABLE A-15
Settling Velocities
(m/day)
Diatom Chlorophyll 0.1
Non-Diatom Chlorophyll 0.1
Herbivorous Zooplankton 0.0
Carnivorous Zooplankton 0.0
Detrital Organic Nitrogen 0.2
Ammonia Nitrogen 0.0
Nitrate Nitrogen 0.0
Unavailable Phosphorus 0.2
Soluble Reactive Phosphorus 0.1
Unavailable Silica 0.2
Soluble Reactive Silica 0.0
Detrital Organic Carbon 0.2
Dissolved Inorganic Carbon 0.0
Alkalinity 0.0
Dissolved Oxygen 0.0
228
-------
TABLE A-16
CHEMICAL THERMODYNAMIC PARAMETERS FOR AQUEOUS PHASE COMPUTATION [ ]
§
Components
H20
+
H
CO (g)
c.
Species (Aqueous Phase)
#
H2C03
HCO~
co3=
H20
+
H
OH~
(kcal/mole) (cal/mole °K)
- 68.315 16.71
0.0 0.0
-9U.051 51.06
-167.22 W.8
-165.39 21.8
-l6l. 81* -13.6
-68.315 16.71
0 0
-5!*. 970 -2.57
#
H_CO_(aq) = H0CO_(aq) + CC
£ J d 3
>2(aq) * C02(aq)
+
to negative alkalinity.
229
-------
TABLE A-17
Equations for 0 and CO Aqueous Saturation
+ T(-O.Ul022 + T * (0.00791 - 0.00007777^ * T))
for 000 in mg/£ and T in °C
£-O
C000 = 3.16 10 exp(U.85VRT - 0.02297/R)
do
for CO in mole/liter
£-D
R = 0.0019872 kcal/deg mole
T = °Kelvin
230
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1 REPORT NO.
EPA-600/3-80-065
2.
3. RECIPIENT'S ACCESSION-NO.
4 TITLE AND SUBTITLE
Mathematical Models of Water Quality in Large Lakes;
Part 2: Lake Erie
5. REPORT DATE
JULY 198C ISSUING DATE.
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Dominic M. DiToro
John P. Connolly
8. PERFORMING ORGANIZATION REPORT NO,
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Manhattan College
Environmental Engineering and Science Division
Bronx, NY 10471
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
R803030
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Duluth, Minnesota 55804
13. TYPE OF REPORT AND PERIOD COVERED
Final
14. SPONSORING AGENCY CODE
EPA/600/03
15. SUPPLEMENTARY NOTES
Large Lakes Research Station, 9311 Groh Road, Grosse lie, Michigan 48138
16, ABSTRACT
This research was undertaken to develop and apply a mathematical model of the
water quality in large lakes, particularly Lake Huron and Saginaw Bay (Part 1) and
Lake Erie (Part 2).
A mathematical model was developed for analysis of the interactions between
nutrient discharges to Lake Erie, the response of phytoplankton to these discharges,
and the dissolved oxygen depletion that occurs as a consequence. Dissolved oxygen,
phytoplankton chlorophyll for diatoms and non-diatoms, zooplankton biomass, nutrient
concentrations in available and unavailable forms and inorganic carbon are considered
in the model. Extensive water quality data for Lake Erie was analyzed and statisti-
cally reduced. Comparison of data from 1970 and 1973-74 to model calculations served
for calibration of the model. A verification computation was also performed for 1975,
a year when no anoxia was observed.
Recent developments in phytoplankton growth and uptake kinetics are included in
this analysis. The methods of sedimentary geochemistry are expanded to include an
analysis of sediment oxygen demand within the framework of mass balances. Projected
effects of varying degrees of phosphorus removal on dissolved oxygen, anoxic area,
chlorophyll, transparency and phosphorus concentration are presented.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
Mathematical Models
Water Quality
Great Lakes
Lake Erie
Ecological Modeling
08/H
8. DISTRIBUTION STATEMENT
Release Unlimited
19. SECURITY CLASS (ThisReport)
21. NO. OF PAGES
2U9
20. SECURITY CLASS (Thispage)
22. PRICE
EPA Form 2220-1 (9-73)
231
HJS GOVERNMENT PRINTING OFFICE 1980-657-165/0069
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