Modeling of Thermal and Oxygen Conditions in Lake Kinneret (Israel)

Physical Oceanography, Vol. 12, No. 1, 2002

MODELING OF THERMAL AND OXYGEN CONDITIONS IN LAKE KINNERET (ISRAEL) V. A. Ivanov, S. P. Lyubartseva, É. N. Mikhailova, N. B. Shapiro, and B. S. Shteinman

UDC 551.465

Within the framework of a one-dimensional model taking into account the presence of an upper mixed layer, we compute the seasonal variation of temperature and the concentration of dissolved oxygen in the central part of Lake Kinneret. The temperature conditions of the lake are determined by heat exchange with the atmosphere, and the oxygen conditions depend on gas exchange with the atmosphere and oxygen consumption in sediments as well as on internal sources and sinks. The latter are connected with oxygen supply in the course of photosynthesis and its consumption for the oxidation of labile organic substance in the water thickness. In the period of winter convection from December to February, when the upper mixed layer reaches the bottom, complete aeration of water takes place. The presence of thermal stratification of the lake in the remaining time results in oxygen deficiency under the thermocline.

Increasing anthropogenic action on the environment, in particular, on lakes, makes the problem of their conservation and rational use the focus of attention of modern water-management policy. It should be emphasized that the strategy of environment-protective activity must be based on the results of ecological modeling. The rate of processes taking place in water ecosystems is governed, to a considerable extent, by the space-time scales of thermohydrodynamic phenomena. Hence, the development and improvement of thermohydrodynamic models represent a topical scientific problem. The solution of this problem makes it possible to pass to a higher level of understanding of the nature of lakes and of the development of methods for the diagnosis and prediction of their ecological state. From the ecological viewpoint, oxygen plays an important role in the processes taking place in lakes because it affects not only the chemical processes in the lake but also living organisms. Furthermore, oxygen is one of the fundamental factors of water quality. At present, there are numerous modern models describing oxygen distribution in lakes (Table 1) that are located in various latitudinal zones. Note that, up to now, a common theoretical model suitable for the description of the whole variety of experimental oxygen distributions in lakes has not been developed. Every quoted model is based on a certain, far from universal, hierarchy of processes characteristic of the dynamics and biogeochemistry of a specific water reservoir. This fact results in certain difficulties when one tries to compare different models. The model proposed in the present paper is close to the model [6] from the viewpoint of the given hierarchy of processes. Lake Kinneret is a warm monomictic lake, which, according to the classification [7], means that complete mixing in it occurs once a year, in winter, at a temperature higher than 4°C. The water reservoir is stratified from March to December. From December to February, convective mixing takes place [8, 9], and, during this period, the water temperature is minimum and constitutes 14 – 16°C. A specific feature of the stratification of the lake consists of the presence of a well-pronounced upper mixed layer (UML). In summer, its thickness is 10 – 20 m, and, in winter, practically the entire water thickness represents the UML. The oxygen conditions of the lake are determined by its thermal conditions. From January to March, oxygen distribution is practically conMarine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol; Israel Institute of Oceanology and Limnology (Kinneret Limnologic Laboratory), Israel. Translated from Morskoi Gidrofizicheskii Zhurnal, No. 1, pp. 47 – 58, January–February, 2002. Original article submitted March 30, 2000. 0928–5105/02/1201–0043 $27.00

© 2002 Plenum Publishing Corporation

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V. A. IVANOV, S. P. LYUBARTSEVA, É. N. MIKHAILOVA, N. B. S HAPIRO,

AND

B. S . S HTEINMAN

Table 1. Characteristics of Some Models of Oxygen Transfer in Freshwater Reservoirs Author

Dimensionality of the Model

Processes

Region

Bella [1]

One-dimensional

Vertical diffusion, photosynthesis and respiration of phytoplankton

Lake Sammamish USA (47°36′ N, 122°04′ W)

Umnov [2]

One-dimensional

Vertical diffusion and photosynthesis of phytoplankton

Rybinsk water storage (Russia) (59°10′ N, 37°07′ E)

Varga and Falls [3]

Box model

Flows, turbulent diffusion, and chemical reactions

Water storage Keystone (USA) (36°12′ N, 96°21′ W)

Newbold and Ligget [4]

One-dimensional two-layer

Photosynthesis, respiration of bacteria, phytoplankton, and zooplankton, sediment formation, and diffusion of organic substance

Cayuga Lake (USA) (42°37′ N, 76°38′ W)

Kurchenko and Kurchenko [5]

Three-dimensional analytical

Advection, turbulent diffusion, and oxidation of dead organic substance

Kremenchug water storage (Ukraine) (49°20′ N, 31°56′ E)

Stefan et al. [6]

One-dimensional

Vertical diffusion, photosynthesis and respiration of phytoplankton, oxidation of organic substance in the water thickness and sediments

Thrush Lake (USA) (47°41′ N, 92°28′ W)

stant along the depth, and its mean concentration is 8 mg / liter [8, 9]. In April, the epilimnion is supersaturated with oxygen, which is a result of the bloom of phytoplankton. Due to the presence of a clearly pronounced thermocline, the hypolimnion is depleted with oxygen from May to December. Note that the anaerobic condition of the hypolimnion is a sufficiently widespread phenomenon for stratified lakes [8]. In this sense, the critical factors are the duration of the period of stable stratification of the water thickness, the morphometry of the water reservoir, and the power of external sources of biogenic substances. A specific problem of our modeling consists of the investigation of the connection between the following processes: the process of formation of the seasonal thermocline and near-bottom hypoxia, on the one hand, and winter convection and complete aeration of the water reservoir, on the other hand. In addition to the practical aspect, this problem is interesting from the viewpoint of fundamental research on the nature of the anaerobic condition. For example, it turns out that the cause of the anaerobic condition of both the hypolimnion of Lake Kinneret and the deep waters of the Black Sea is common, despite the fact that the anaerobic condition has a seasonal character in the first case and is constant in the second. This feature can be explained by the failure to replenish the intermediate and deep waters with oxygen [9], which is a result of density stratification of the water reservoir (stratification has a haline nature in the Black Sea and a thermal nature in Lake Kinneret). Hence, it is reasonable to apply the experience accumulated in modeling of the hydrogen-sulfide zone of the Black Sea [10] to the investigation of the oxygen conditions of Lake Kinneret.

MODELING OF T HERMAL AND OXYGEN CONDITIONS IN L AKE KINNERET (ISRAEL)

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Within the framework of a one-dimensional diffusion model, we determine the seasonal variation of temperature and the concentration of dissolved oxygen. Our model is represented by the following equations for temperature T (°C) and oxygen concentration C (mg / liter ) as functions of time t and depth z: ∂T ∂  ∂T  , = Kz ∂t ∂z  ∂z 

∂C ∂  ∂C  + P – DBOD , = Kz ∂t ∂z  ∂z 

(1)

where Kz is the vertical turbulent diffusivity coefficient (we assume that it is identical for temperature and oxygen). In this formulation of the problem, the temperature distribution in the lake depends on diffusion processes only. At the same time, in the modeling of oxygen transfer, we also take into account internal sources and sinks describing the oxygen supply in the course of photosynthesis and its consumption for the oxidation of labile organic substance in the water thickness ( P and DBOD are the corresponding functions of power of sources and sinks). The power of sources is taken in the form P = ( 1 – r ) pmax α ( I ) γ1 ( T ) Chl,

(2)

where r is the index of efficiency of respiration of phytoplankton, pmax is the maximum rate of oxygen production due to photosynthesis, α ( I ) and γ1 ( T ) are functions modeling the dependence of the rate of photosynthesis on irradiance I and temperature, respectively, and C h l is the concentration of chlorophyll A ( mg / m3 ), which is considered a known function of time. We write the dependence of the rate of photosynthesis on irradiance I in the form α ( I ) = tanh [ a I ( z, t ) ],

(3)

where a is the index of photosynthetic efficiency and I ( z, t ) = IS exp [ – ( kW + kChl C h l )z ].

(4)

Here, IS is the irradiance (photosynthetically active radiation) on the water surface, kW is the coefficient of light extinction by pure water, and kChl is the specific coefficient of light extinction by chlorophyll. According to the Arrhenius law, we assume that the dependence of the rate of photosynthesis on temperature has the form γ1 ( T ) = θ1(T − 20) / T , *

(5)

where θ1 is the Arrhenius coefficient for photosynthesis and T* is a nondimensionalizing normalizing multiplier. For the power of sinks, we use the parametrization DBOD = bθ(2T − 20) / T BOD , *

(6)

46


AND

B. S . S HTEINMAN

where b is the constant of destruction of labile organic substance in the water thickness, θ2 is the Arrhenius coefficient for destruction of organic substance in the water thickness, and BOD is the equivalent of the concentration of labile organic substance in the water thickness in units of oxygen concentration (it is considered constant along the depth). We set boundary conditions in the following form: At the surface of the water reservoir, the condition of heat balance is satisfied. The total heat flow Γ0 is a sum of four components: falling short-wave radiation Γ1 , reflected long-wave radiation Γ2 , sensible heat flux Γ 3 , and latent heat flux Γ 4 . These quantities are calculated according to the formulas presented in [11] on the basis of experimental data on the seasonal variability of wind velocity, air temperature, cloud amount, albedo, and the pressure of saturated vapor in the atmosphere [12]. Further, we assume that the heat flow Γ0 is proportional to the difference between the effective air temperature 0 W 0 0 W T and the water temperature at the lake surface T : Γ = λ ( T – T ). The proportionality coefficient λ and 0 the effective air temperature T are determined according to the calculated components of heat balance and the surface temperature. The oxygen flux from the atmosphere is given by 0 sat W F = k ( C ( T ) – C ),

where the coefficient of exchange with the atmosphere k is taken as 1.64 k = lkU10 (mk + nk h).

Here, U10 is the wind velocity (in m / s) at a height of 10 m, h is the thickness of the UML (see below), and lk , mk , and nk are parameters. We approximate the relation between the oxygen concentration (mg / liter) corresponding to its partial pressure in water equilibrium with the atmosphere, and temperature (°C) by the polynomial C

sat

= 14.6244 – 0.40776 T + 0.0081136 T2 – 0.000078765 T3 ,

where CW is the oxygen concentration at the lake surface. At the bottom ( z = H ), we assume the absence of diffusion heat flow and specify the oxygen flow rate caused by its consumption in sediments

Kz

( T − 20 )/ T *

∂T ∂z

= 0, z= H

Kz

∂C ∂z

z= H

= – DSOD ,

(7)

C is the function of power of the source describing oxygen consumption for the where DSOD = f S ∆ S θ 3 oxidation of labile organic substance at the bottom, fS is the specific rate of oxygen consumption in sediments, θ3 is the Arrhenius constant for oxygen consumption in sediments, and ∆ S is the effective thickness of the active layer of sediments. The principal mechanism of heat and substance transfer in the ecosystem under consideration is vertical turbulent diffusion. To set an adequate and, at the same time, sufficiently simple parametrization of the vertical


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profile of turbulent diffusivity Kz, we a priori use the fact that, in the considered water reservoir, there exists a well pronounced UML of thickness h during the whole year. We represent the dependence of vertical diffusivity coefficient on depth in the form UML

 Kz Kz ( z ) =  H  Kz

for z < h,

(8)

for z ≥ h.

The thickness of the UML h = h ( t ) depends on time and is calculated from the one-dimensional balance equation of turbulent kinetic energy integrated along the vertical within the UML in the stationary approximation [13] g αh h [ αh Γ0 – w– ( TW – Th ) ] = 2mu*3 .

(9)

Here, TW is the water temperature at the lake surface, Th is the water temperature below the UML, αh = −

∂ρ ∂T

2 is the coefficient of thermal expansion, ρ is the water density, ρ = ρ0 [ 1 – ε1 ( T – 4 ) ] is the equation of state

( ε1 = 6.8 ⋅ 10– 6 ), g is the acceleration of gravity, – w = min ( 0, w ) =

w− w ≤ 0, 2

w = −

∂h ∂t

is the velocity through the lower boundary of the UML ( w < 0 corresponds to its deepening, i.e., entrainment, and, if w > 0, a decrease in the thickness of the UML, or subduction, takes place), u* = ( ρa c d) 1 / 2 U10 is the friction velocity, ρ a is the air density, and cd is the friction factor. Consider some useful corollaries of Eq. (9). Corollary 1. Cooling or the absence of heat flow through the lake surface ( Γ0 ≤ 0 ) is a necessary condi∂h > 0 ) because 2mu*3 > 0. tion of the entrainment mode, i.e., of the validity of the inequality w < 0 (or ∂t Corollary 2. If the water thickness is heated ( Γ0 > 0 ), two cases are possible: w ≥ 0 and w < 0. If w ≥ 0, then h =

and, for

2mu*3 , gαΓ 0

∂h ≤ 0, it is necessary that ∂t 3 ∂  u*  ≤ 0. ∂t  Γ 0 

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AND

B. S . S HTEINMAN

Note that, in the case u* = const, which can often be met in practice, the requirement ∂ ∂t is reduced to

3

 u*  ≤ 0  Γ0 

∂Γ 0 ≥ 0. ∂t

We solve the problem numerically. We divide the computational domain into 23 horizontal layers. The values of temperature and oxygen concentration are computed at the following depths: 0.2, 0.5, 0.9, 1.5, 2.5, 3.5, 5.0, 6.5, 8.5, 10.5, 13.0, 15.5, 18.0, 20.5, 23.0, 25.5, 28.0, 30.5, 33.0, 35.5, 38.0, 40.5, and 43.0 m. The boundaries between the boxes are located in the middle between these depths. We approximate the diffusion terms of Eq. (1) in an implicit way and the terms describing oxygen sources and sinks in an explicit way. We solve the obtained finite-difference equations by the sweep method and integrate with respect to time by using a two-layer scheme with a step of 15 min up to attaining a quasiperiodic state during five computational years. To find the vertical diffusivity coefficient, we preliminarily calculate the thickness of the UML h; for this purpose, we solve Eq. (9) together with the equation w = −

∂h , ∂t

which is approximated in an implicit way [11, 13]. Proceeding from the obtained thickness h, we refer it to the chosen vertical grid, namely, we take the box whose lower boundary is located closest to k as the last, k th box going into the UML. In this case, we can rewrite relation (8) in the form  K zUML Kz ( z ) =  H  K z

for z < z k , for z ≥ z k

and assume that the water temperature under the UML is Th = Tk + 1 . As the initial conditions, we took the values of temperature (15°C) and oxygen concentration (8 mg / liter) corresponding to the winter state when the UML reaches the bottom, and these values are independent of depth. Note that the seasonal variation of the heat flow Γ0 is computed on the basis of the monthly average data on wind velocity, air temperature, cloud amount, radiation heat flow, albedo, and the saturated vapor pressure in the atmosphere [12], interpolated linearly with respect to time. We evaluated the seasonal variation of the concentration of chlorophyll A, necessary for modeling, on the basis of the experimental data on primary production presented in [14]. The seasonal variation of chlorophyll concentration is characterized by its maximum value ( 200 mg / m3 ) in April (which corresponds to primary production 400 ± 200 mg Chl / m 2 ) and minimum value ( 26 mg / m3) in August and September ( 100 ± 70 mg Chl / m 2 ). We assume that the irradiance of the surface in the photosynthetically active part of IS spectrum is related to radiation heat flows Γ1 and Γ2 by the formula IS = 0.46 ( Γ1 + Γ2 ). The values of the semiempirical parameters of the developed model are presented in Table 2.


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Table 2. Semiempirical Parameters of the Model Parameter

Symbol

Value

Unit of measurement

pmax

13.2

mg O2 / mg Chl h

Index of efficiency of respiration of phytoplankton

r

0.12

—

Index of photosynthesis efficiency

a

0.06

m2 / W

Coefficient of light extinction by pure water

Kw

0.04

m –1

Specific coefficient of light extinction by chlorophyll

k Chl

0.0067

1 / (( mg Ch l / m 3 ) m )

Arrhenius coefficient for photosynthesis

θ1

1.036

—

Constant of destruction of labile organic substance in the water thickness

b

0.01

h –1

Normalizing multiplier

T*

1

°C

Arrhenius coefficient for destruction of organic substance in the water thickness

θ2

1.047

—

BOD

0.2

mg / liter

Constant of oxygen consumption in sediments

fS

20

h –1

Arrhenius constant for oxygen consumption in sediments

θ3

1.065

—

Effective thickness of the active layer of sediments

∆S

1

m

Vertical turbulent diffusivity coefficient in the UML

K zUML

0.5

cm2 / s

K zH

0.001

cm2 / s

Constant used for parametrization of k

lk

2.96

m / day


mk

0.04

—


nk

0.25

m –1

Parameter of the balance equation of turbulent kinetic energy

ah

1.0 for Γ 0 > 0 0.5 for Γ 0 < 0

—

Parameter of the balance equation of turbulent kinetic energy

m

1.25

—

Air density

ρa

1.25 ⋅ 10– 3

g / cm 3

Friction factor

cd

1.6 ⋅ 10– 3

—

Acceleration due to gravity

g

9.81

cm / s 2

Maximum water density

ρ0

1

g / cm 3

Maximum rate of oxygen production due to photosynthesis

Equivalent of the concentration of labile organic substance in the water thickness in units of oxygen concentration

Vertical turbulent diffusivity coefficient below the UML

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In Fig. 1, we show the obtained numerical results for the seasonal variation of temperature, heat flow through the surface, and the thickness of the UML. The winter period from December to February is characterized by homothermy with a temperature of 17°C, and the UML envelops the entire water thickness. In March, when heating begins ( Γ0 > 0 ), the seasonal thermocline is formed. During this period, the thickness of the UML is minimal and equals 6 – 8 m. From May to August, the thermocline is stabilized, becomes sharp, and deepens to 16 – 18 m. The horizon corresponding to the maximum value of the vertical temperature gradient is considered the thermocline depth. In June, the temperature difference in the layer between 15 and 23 m reaches 8 – 9°C, and the thickness of the UML changes from 9 to 13 m. The cause of the onset of thermal stratification consists of solar radiation (heating), and among the main factors weakening stratification, one should mention evaporation, emission of radiant energy by the lake surface, and convection. Note that the change in temperature is determined, on the one hand, by the factors mentioned above, which change the heat storage, and, on the other hand, by turbulent mixing, which results in heat redistribution along the vertical. Our computations show that the system absorbs heat approximately from the first decade of February to the third decade of August and gives it away during the remaining time. From the end of August, heat losses exceed its supply, and the process of cooling begins. With decrease in the temperature of the lake surface, convection currents arise, the epilimnion loses its stability, and the UML deepens gradually. Not only cooling of the epilimnion takes place but also the metalimnion is heated due to intense entrainment into the UML. As the lake cools down, the epilimnion temperature decreases so that it ceases to differ from the main water mass of the lake, and the period of complete autumn convection begins at this moment. In Fig. 2, we present the seasonal variation of oxygen concentration and oxygen fluxes in the system. From January to the second decade of February, the oxygen distribution is uniform, and its concentration is approximately 9 mg / liter. The depth of the euphotic zone zPH (the zone in which photosynthesis takes place) is determined from the condition I ( zPH ) / IS and constitutes 15 – 19 m in this period. A powerful atmospheric aeration of the water thickness is going on. The contribution of biological aeration is insignificant owing to the low concentration of chlorophyll A. Note that, during this period, oxygen absorption by sediments is maximal and equals 80 – 90 (mg / liter ) (m / day). From the middle of February to May, the bloom of phytoplankton takes place in the epilimnion, which results in significant biological aeration of this layer of the lake. During this period, the system gives oxygen to the atmosphere. The depth of the euphotic zone is 3 – 6 m. The thermocline, which has already been formed to this moment, hinders oxygen penetration into the hypolimnion, and, therefore, hypoxia arises here (deficiency of oxygen < 2 mg / liter ). Note that, in May, oxygen consumption by sediments is minimal. Hypoxia of the hypolimnion lasts up to complete destruction of the thermocline in December. The following small-scale specific feature of the investigated processes is not reflected in Fig. 2: From July to September, the depth of the euphotic zone is 19 – 21 m and exceeds the thickness of the UML, which is 10 – 16 m. This means that, in the hypolimnion, which is practically isolated from the epilimnion, photosynthesis takes place, and oxygen is released and “preserved.” Our computations demonstrate the presence of weak local maxima of oxygen concentration at these depths. A diffusion “ejection” of near-bottom waters with a concentration of 7 mg / liter into the water thickness with a concentration of 9 mg / liter in December attracts our attention. In such a way, the entrainment mode, characteristic of this period, manifests itself in the field of oxygen concentration. From March to August, the oxygen flux into the atmosphere and its insignificant absorption in sediments is compensated by oxygen supply as a result of photosynthesis. Later, in August and September, photosynthesis becomes weaker, and the flux into the atmosphere increases. Therefore, in October, the total oxygen flux changes its sign. Oxygen losses for the oxidation of labile organic substance in the water thickness are small and constitute – 0.1 (mg / liter ) (m / day ) on the average.


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Fig. 1. Seasonal variation of temperature (a), heat flow through the surface (b), and the thickness of the UML (c). An important advantage of the proposed model, first of all, is its simplicity, which was attained by means of the maximum use of a priori information on the water reservoir at the stage of statement of the problem. Note that, unlike models describing the dynamics of the ecosystem of the Black Sea [15, 16], we take into account the dependence of biogeochemical processes on temperature for the warm lake under consideration. Application of the concept of a UML in our model makes it possible to introduce a sufficiently simple parametrization of the profiles of vertical turbulent diffusivity and the coefficient of gas exchange with the atmosphere, which, nevertheless, admit a transparent physical interpretation. We did not take into account oxygen losses for

52


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B. S . S HTEINMAN

Fig. 2. Seasonal variation of oxygen concentration (a) and oxygen fluxes (b) in the system. the oxidation of reduced compounds in the hypolimnion. However, despite this fact, numerical results and experimental data [17, 18] agree well among themselves, which partly corroborates the known Mortimer paradox [19]: simple models possess better simulation properties than complex models. Furthermore, we studied the sensitivity of the model to variations of empirical parameters. We established that thermal stratification and hypoxia are determined, to a considerable extent, by the parameters of vertical turbulent exchange. The power of the zone of oxygen deficiency depends on atmospheric aeration and the rate of oxygen consumption in sediments and is weakly connected with photosynthesis intensity. This conclusion agrees with the classical concepts of Streeter and Phelps [20]. Thus, the developed mathematical model makes it possible not only to predict the seasonal variation of temperature and oxygen concentration but also to evaluate the role of different factors that govern these processes.


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In the future, it seems promising to take into account in our model the sink of oxygen due to its losses for the oxidation of reduced nitrogen compounds (ammonium ions, nitrogen of organic substances, and nitrites) in the hypolimnion. For this purpose, it is advisable to unite the proposed model with the model of the nitrogen cycle. This will enable one to describe correctly both the dynamics of primary production in the lake and the destruction of organic substance under conditions of oxygen deficiency. We also propose to answer the following questions: is eutrophication of the lake possible, what factors govern this phenomenon, and how does anaerobic condition of the hypolimnion affect the ecosystem of the lake? REFERENCES 1. D. A. Bella, “Dissolved oxygen variations in stratified lakes,” Proc. ASCE, J. Environ. Eng. Div., 96, 1129–1146 (1970). 2. V. A. Umnov, “Application of the method of mathematical modeling in the investigation of the role of photosynthetic aeration in a lake,” Ékologiya, No. 6, 5–12 (1971). 3. L. P. Varga and C. P. Falls, “Continuous system models of oxygen depletion in an eutrophic reservoir,” Environ. Sci. Technol., 6, 135–142 (1972). 4. J. D. Newbold and J. A. Ligget, “Oxygen depletion model for Cayuga Lake,” Proc. ASCE, J. Environ. Eng. Div., 100, 41–59 (1974). 5. T. S. Kurchenko and V. D. Kurchenko, “A model of oxygen dynamics in a water reservoir,” Gidrobiol. Zh., No. 14, 80–86 (1978). 6. H. G. Stefan, X. Fang, D. Wright, et al., “Simulation of dissolved oxygen profiles in a transparent, dimictic lake,” Limnol. Oceanogr., 40, No. 1, 105–118 (1995). 7. G. E. Hutchinson, A Treatise on Limnology. Geography, Physics, and Chemistry, London (1957). 8. G. K. Nurnberg, “Quantifying anoxia in lakes,” Limnol. Oceanogr., 40, No. 6, 1100–1111 (1995). 9. E. V. Yakushev, L. N. Neretin, and I. I. Volkov, “Mathematical modeling of the transformation of inorganic compounds of nitrogen in the c-layer of the Black Sea,” Okeanologiya, 32, No. 6, 1033–1041 (1992). 10. V. I. Belyaev, E. E. Sovga, and S. Lyubartseva, “Modeling the hydrogen sulphide zone of the Black Sea,” Ecol. Model., 96, 51–59 (1997). 11. Yu. N. Ryabtsev and N. B. Shapiro, “Modeling of the seasonal variability of the Black Sea,” Morsk. Gidrofiz. Zh., No. 1, 12–24 (1997). 12. C. Serruya (editor), Lake Kinneret, Junk (1978). 13. É. N. Mikhailova and N. B. Shapiro, “A quasiisopycnic layered model of large-scale oceanic circulation,” Morsk. Gidrofiz. Zh., No. 4, 3–12 (1992). 14. T. Berman, L. Stone, Y. Z. Yacobi, et al., “Primary production and phytoplankton in Lake Kinneret: a long-term record (1972–1993),” Limnol. Oceanogr., 40, No. 6, 1064–1076 (1995). 15. T. Oguz, H. Ducklow, P. Mallanotte-Rizzoli, et al., “Simulation of annual plankton productivity cycle in the Black Sea by a one-dimensional physical-biological model,” J. Geophys. Res., 101, No. C7, 16585–16599 (1996). 16. S. P. Lyubartseva, É. N. Mikhailova, and N. B. Shapiro, “An ecological three-dimensional numerical model of the Black Sea: the seasonal evolution of the ecosystem of the euphotic zone,” Morsk. Gidrofiz. Zh., No. 5, 55–80 (2000). 17. C. Serruya and H. Leveivier, “Lakes and reservoirs of Israel,” in: F. B. Taub (editor), Lakes and Reservoirs, Elsevier, Amsterdam (1984), pp. 357–383. 18. T. Berman and U. Pollingher, “Annual and seasonal variations of phytoplankton, chlorophyll, and photosynthesis in Lake Kinneret,” Limnol. Oceanogr., 19, No. 1, 31–54 (1974). 19. T. Platt, K. H. Mann, and R. E. Ulanowicz (editors), Mathematical Models in Biological Oceanography, UNESCO, Paris (1984). 20. H. W. Streeter and B. Phelps, “A study of the pollution and natural purification of the Ohio River,” Publ. Health. Bull., 146, 1–25 (1925).