Temperature computation for temperate lakes

Temperature computation for temperate lakes R. A. Waiters

USA Geological Survey, Menlo Park, Cali[brnia, USA

G. F. Carey

Texas Institute for Computational Mechanics, University of Texas at Austin, Austin, Texas, USA

D. F. Winter

Department o[Oceanography, University of Washington, Seattle, Washington, USA (Received 14 March 1977; revised 1 December 1977)

Introduction

The distribution and annual variation of temperature in lakes is of fundamental importance in its influence on biological primary productivity and nutrient cycling. Thus, an accurate determination of the thermal regime, particularly associated with thermocline formation, is necessary.

A qualitative understanding of the thermal characteristics of lakes has developed over a considerable period of time: from early observations, the amount and quality of temperature measurements have grown extensively. Over the past twenty years, attempts have been made to model the thermal behaviour of lakes; however, only within the last ten years has some success been achieved, using the two different theories typified by the work of Kraus and Turner I and Munk and AndersonL

To better understand the problem, a brief description of the thermal stratification cycle is necessary 3. Consider a deep, temperate lake which remains above 4°C throughout the year. During early winter the lake is subject to substantial vertical mixing and in late winter reaches its minimum temperature. During the spring, the surface heat flux and lake temperature are increasing and the lake displays a temperature profile that decreases approximately exponentially with depth while the thermal diffusivities remain large. At some time in the spring or early summer, a mixed layer forms and stratification is initiated. The diffusivity remains high at the surface, decreases to a minimum near the thermocline, and increases somewhat in the hypolimnion. The depth of the thermocline is controlled by an interaction between wind-induced stress at the surface and stable buoyancy gradients associated with surface heating. These relationships do not change appreciably until autumn,

when the surface cooling causes a deepening of the thermocline until the lake becomes isothermal again.

The theories used to describe the thermal cycle generally follow one of the two methods: the mixed- layer temperature is assumed to vary only as a function of time, and the depth of this layer is found from physical constraints such as the energy balance 1. Alternatively, the temperature is allowed to vary continuously and the interaction controlling the thermocline depth is considered in terms of a Richardson number 2"4. The first approach poorly describes the thermal cycle of lakes since the mixed layer is not in existence during the whole year, and even during the summer the mixed layer displays some thermal structure. Furthermore, there are significant changes in the hypolimnetic temperature which are not considered at all with this theory. However, this approach may be preferable during the fall convective period, since convection cannot be modelled by the second strategy. The other alternative is used in this study. As opposed to Sundaram and Rehm 4, the surface heat flux is expressed in terms of real data and the diffusivity in each of several regimes is formulated in a more consistent manner.

In the present investigation a finite element analysis of the one-dimensional thermocline problem is developed. A new form of the eddy diffusivity for temperate lakes is proposed. Here the eddy diffusivity is described in terms of either the gradient Richardson number or Brunt-Vais[il~t frequency in regimes corresponding to the epilimnion, clinolimnion, and bottom of the hypolimnion. A Galerkin predictor--corrector scheme is used to treat the non- linearities entering through the dependence of these diffusivities on temperature and temperature gradients.

Appl. Math. Modelling, 1978, Vol 2, March 41

Temperature computation for temperate lakes: R. A. Waiters et aL

This analysis has been programmed and used for numerical simulations of heat transfer in a lake. The results are compared with experimental data for Lake Washington. A series of comparative studies with finite difference solutions provide a practical evaluation of these two strategies. In general, the differences in the solutions are small, but an interesting feature is the significant effect on the diffusivity near the thermocline. A comparison is also made with the case where a constant diffusivity is used in the hypolimnion. This scheme poorly reproduces the temperature profile.

Mathemat ica l formulation

Let (x, y) be the horizontal coordinate distances and z be the depth measured positively downwards from the lake surface. Then the governing diffusion equation for lake temperature T(x, y, z, t) at time t may be determined from the conservation of heat energy. Including diffusion and advection of heat energy, and heat sources, we have the general expression for the mean temperature 5:

a t + + 7-y + 0--7-

+LI °TI azl azl + f (1)

\

Here u, v are the horizontal velocity components and w is the vertical velocity; J is a distributed heat source; and the eddy plus molecular (K=) diffusion coefficients are:

K = K , - u ' T , / a T , l a x

4 ~ K r = K = - v ' T ' l - - ,

l a y

K = K= - w ' T ' / - - / az

The prime is used to indicate the fluctuating portion of a quantity and the overbar denotes a time-averaged quantity; for example, with u = fi + u':

t+ (*/2) 'f = - u(x, y, z, q) dr/ T

t - ( r / 2 )

where z is the turbulent time scale of interest. Introduce the area average of a quantity F over the

lake as:

1 ( F ) = - - ~ f FdA

A(z)

where ( ) denotes an area averaged quantity and the

area at depth z is A(z). Applying this area average to equation (1) leads to:

+ (f> (2)

where we have used the divergence theorem and the condition that there be no mass or heat flux through the lake basin.

This equation is to be solved in the domain 0 ~ z ~< L, subject to appropriate boundary conditions and an initial condition. However, some simplifying assumptions may be introduced. During most of the year, horizontal homogeneity is observed in many deep, temperate lakes and except during the fall convective period, the vertical velocities are small over most of the lake. Thus, the advective term (1/A)(a/az)(A<g,T>) will be neglected, although its magnitude will be reexamined when the numerical results are considered. With the above assumptions, the governing equation may be written:

dr a__[arar I a t = A az ~ az I + j (3)

where T, K and J now denote mean quantities averaged over the area, ( ( ) and overbar will now be deleted, for notational convenience). In general, the diffusion coefficient K is a non-linear function of temperature and thermal gradients and, represents the turbulent mixing processes. These processes include an interaction between wind-generated turbulence and stable buoyancy gradients due to surface heating, and are usually expressed in terms of a gradient Richardson numberS:

= g ao / laul2 ar /la.12 Ri P a z / l a z I = -=g-~--/lx-loz/~ozl (4)

where g is gravitational acceleration, p is density, u is horizontal fluid speed, and:

= (1 /p ) (~p /aT) ~- ~o(T - 4)

is the thermal expansion coefficient. In another study 6"7, it is demonstrated that for deep

temperate lakes, the eddy diffusivity is best expressed as follows:

(1) In the epilimnion, the eddy diffusivity is assumed to be approximated by:

K -- K= + Ko(1 + trRi) -1 a = 0.048 (5)

where K= is the molecular diffusivity and Ko is the eddy diffusivity at the surface. This expression is analysed by Kent and Pritchard s and is used by Sundaram and Rehm 4 with a = 0.1. Alternative expressions, 2"9 were found to lead to unstable temperature profiles in the one-dimensional case with the boundary layer approximation. In fact, if the exponent in equation (5) is larger than 1, the temperature profiles will always exhibit strong discontinuities 6. The parameter Ko, was computed from an empirical relationship 2 of the form:

Ko(m2/sec) = 4.47 x 10-SU, (6)

where U is wind speed (m/sec).

42 Appl. Math. Modelling, 1978, Vol 2, March

Temperature computation for temperate lakes: R. A. Waiters et al.

(2) In the clinolimnion (the upper portion of the hypolimnion where the temperature varies exponentially with depth), we use the following expression:

K = KTc(N2c/N2) 1/2 (7)

where the subscript TC indicates that these quantities are evaluated at the thermocline (that is, where OT/~z attains a maximum), and N 2 is the Brunt-V~is~l~i frequency squared:

N2 _ g c~p _ c3T p Oz ag~z (8)

This expression has been discussed by PowelP ° and Jassby and Powell 11. Since the eddy diffusivity increases with depth in this regime, an upper limit must be placed on K. This limit is approximated as 10 Krc in accordance with existing data for Castle Lake 11.

(3) A bottom layer, when in existence, is then considered to have a constant diffusivity of 10 Krc.

The location of the interfaces between the different diffusivity regimes changes as a function of time.

The governing equation is seen to contain two dependent variables, T and u, and two independent variables, z and t. The problem statement can be completed in two ways: (1) the equations of motion can be solved along with the heat equation as a coupled system, or (2) effective values for u (or, actually, the velocity shear) can be estimated from boundary layer theory. The latter approach is adequate for the thermal analysis considered here.

Then, the velocity shear is assumed to "be given by the boundary-layer approximation for homogeneous water:

~U U. - (9)

~z kz

where u. = (Z/p) 1/2 or friction velocity (m/sec), z is the surface stress (Nt/m 2) and k is von Karman's constant. In practice, u . is calculated from the wind speed as u2. = CoU 2, where Co = 1.5 x 10 -a and U is the wind speed (m/sec) at a height of 10m. The expression for velocity shear is then used to eliminate u from Ri so that the governing equation is reduced to a time- dependent, one-dimensional equation for temperature. However, note that the value for a in the expression for the eddy diffusivity depends upon the method of computing u . . For the method adopted here, the optimal value was about 0.048. If another value of Co is used or wind speed is measured at a different height, tr would have to be adjusted accordingly.

The statement of the problem is completed by specifying the heat flux Q(T, t) at the surface, a zero flux at the bottom and the initial temperature distribution To(z).

Numer ica l procedure

The thermocline problem was solved using a modified Crank-Nicolson finite difference algorithm and a finite element Galerkin scheme, and the numerical results compared.

The preceding mathematical analysis determined the governing non-linear diffusion equation for heat transfer in a lake. This equation is then written in non- dimensional form as follows:

cdT 1~ IAH~OTI -~f = A-~z 1 cgz ] + f(z, t) (10)

where H is the non-dimensional heat diffusivity and is a function of the unknown temperature and temperature gradients; f(z, t) is a distributed heat source describing incident solar radiation; and C is a constant which arises when the variables are non- dimensionalized, C = z2/Koto where Zo is the depth scale, to is the time scale and Ko is the diffusivity at the surface. (Scales of Zo = 1 m and to = 1 day were used.)

The associated boundary conditions: -H(t~Z/t~z) = Q(T,, t) at the lake surface; (aT/az)(L, t) = 0, a zero heat flux at the bottom; and initial condition; T(z, O) = To(z), complete the description of the problem.

Finite d e m e n t analysis

Galerkin semi-discrete formulation A Galerkin finite element analysis 12 is applied to

approximate the solution of equation (10). Partition the lake depth z e [0, L] to n elements of variable length hi = zi - zi-1, i = 1 . . . . . n + 1. The Galerkin procedure requires that the weighted residual with respect to a set of test functions be zero. After appropriate integration by parts and introduction of the finite element approximation, this yields a system of ordinary differential equations.

Here, for consistency with the subsequent finite difference analysis we use linear approximation on an element:

T(z, t) = ~ Tk(t)pk(Z) (11) k

where pk(Z) is the piecewise linear patch function at node k. The general differential equation at arbitrary interior node i then becomes:

i+1 ~a dTk t)} [ ,k--~ + b,k(T)Tk( -- q = 0 (12)

k = i - - 1

where: g i + l

f c aik = C piPkdZ = ~ [hi2(hi + hi+l)hi+l]

z i - i

(13)

g i + I

g l - i

H(T) dz

and

z i+ L

cl = f f(z, t)pl dz (14) z i - I



The flux boundary conditions enter the extreme nodal equations (i = 0, n) directly through the integrated terms in the variational statement of the semi-discrete problem. The remaining contributions to these equations are defined in equation (13) with the integrations specified over the patches adjacent to the boundary. Details related to the integration of H(73 and f ( z , t) are summarized in the Appendix.

A predictor-corrector scheme is implemented to solve this system iteratively 1 a. Both predictor and iterative corrector lead to linear systems for solution, and the accuracy estimates are preserved with one or two iterations of the corrector. The predictor is obtained by evaluating the integral for b~k(T) using 7(z, t), the known solution at the beginning of the time step. Subsequent corrections are made with bik and c~ evaluated at the average of T k and the current estimate o f T k+l"

Let arJ+~ be the current estimate of T i+x Then, in matrix form with A = (aik), B = (bik), C = (ci) and T j - (T~), the scheme is defined by:

Predictor:

{A +-~B(Tk)}Tk, +1 = A t C ( T k)

+ {A - A-jB(Tk)t Tk (15)

Corrector:

/

Finite difference analysis Consider the mathematical statement of the

problem as specified by equations (5)-(10). In a previous investigation 7 several alternative finite difference formulations were examined. The following scheme was found to be the most effective of these for the particular class of physical problems that are of interest here. Only the main procedural steps are outlined here.

A uniform mesh of size Az is assumed. Consider an interior grid point i and time step j from t i to ti+ 1 = tj + At. The spatial derivatives on the right in equation (10) are centrally differenced across [zi-1, zi+ 1] and the time derivative on the left is approximated by a forward difference on [ti, tj+ 1J- Evaluating the spatial derivatives as an average of central differences at times t i and ti+~ (the mid-step value at tj + At~2 in a sense), we obtain an implicit finite difference scheme of the Crank-Nicolson form:

C (Ti+1 - Ti) m+1:2 _ "'i+1/2 ~,T/+ 1 T1+1) At 2(---~z)2 ~ ,+1 - + (T!+1 -- T,~)}

Hi+ 1/z ' -1/2{(Ti+1 -- --i-I,TJ+ I~

2(Az) 2

+ (T! - T,~_,)} + f~+,/2 (17)

Note that the mid-step values of H and f appear in the differencing and that the non-linearity again arises

through the dependence of diffusivity H on temperature T.

The flux boundary conditions are approximated by central differences at the lake surface and bottom by including additional grid points beyond the associated boundary. This may be then used in equation (17) in the standard way to complete the finite difference systemlS.

In this scheme the non-linearity is treated by direct iteration in each time step: the diffusivity H(T, T~) is evaluated at time t i (the beginning of the step); the difference system is then solved for Ti* 1 the current estimate of T i÷ 1, and used to determine W, ÷ 1/2: the solution and iterative improvement o._.¢ Ti+ 1,.,,rli+ 1/z is repeated to a specified convergence tolerance.

In the case studied the value of/3 in the distributed heat source f ( z , t) was large, resulting in small penetration distances for the insolation. Since the surface layers are generally well-mixed and the e-1 depth was of the order of the grid spacing, f was . included in the surface boundary condition.

Numerical simulations and comparison studies

Lake Washington was chosen to test the validity of the present model, since it has already been the subject of limnological study 16, and because field measurements are available for the stratification cycle. Lake Washington has a total surface area of approximately 86 km 2 and is located in the north-west United States. The lake is 35 km long and 1.5-6.5 km wide; it is deep wiih relatively steep sides (maximum depth, 65 m) and has few shoaling regions. The lake undergoes an annual stratification cycle characterized by a thermocline development at about 10m in late spring. During winter the lake does not freeze, but is isothermal with substantial vertical mixing by wind action and episodic, local convective overturn 17. The temperature of the epilimnion appears to be closely correlated with air temperature and insolation. Chemical and biological sampling at widely separated stations in the lake suggest a fairly high degree of lateral homogeneity.

The thermal cycle for Lake Washington is simulated using available Weather Service data to prescribe insolation and the heat flux boundary condition at the surface. Details of the latter condition may be seen from the literature s'6,la'19.

The different forms for the eddy diffusivity in the epilimnion display considerably different mathematical properties. On the one hand, some functions such as the Munk-Anderson and Mamayev form lead to unstable temperature profiles and layer formation, while others, e.g. the Kent and Pritchard form lead to smooth temperature profiles and tend to damp perturbations in temperature. To see this behaviour more clearly, consider the following functional form for the diffusivity:

K = Ko(1 + aRO-"

where Ri = bz2(dT/dz). The heat flux is then:

K OT K o a T [ - 2~T~ -n O = - ~ z - ~-~z[ 1 + = trbz ~z)

If the heat flux Q at a constant z is plotted against

44 Appl. Math. Modell ing, 1978, Vol 2, March

Temperature computation for temperate lakes: R, A. Waiters et al.

temperature gradient dT/~z, we find that Q asymptotically approaches a limiting value for n = 1, but has a maximum and then goes to zero for n > 1. The Mamayev form for the diffusivity behaves in a manner similar to that in the second case (see Figure 1).

In physical terms, these 2 cases can be considered in the following manner: for n ~< 1, Q increases monotonically for increasing gradient; if the surface heat flux increases, say, so that the gradient increases, more heat is fluxed into the deeper layers which has a tendency to counteract the increase in gradient: thus this system behaves as if it were damped due to the feedback exerted by the form for K.

For n > 1, Q increases until the gradient reaches a critical value when Q decreases to zero with increasing gradient. If the gradient increases but remains less than the critical value, the heat flux also increases, as in the first case. However, if the gradient is greater than the critical value, an increase in gradient leads to a decrease in flux which further increases the gradient. Thus this system proceeds to Q = 0 and large gradients which represents steps or layers in the temperature profile. Since these layers are of large scale (several metres), this behaviour in no way represents the macro-scale temperature being modelled here. As a result, a necessary conclusion is that the Munk and Anderson, and Mamayev forms for the eddy diffusivity are not adequate for a one-dimensional model using the boundary layer approximation. Numerical simulations using these forms confirmed this analysis 6.

For the Kent and Pritchard form of the diffusivity, n = 1 in the foregoing analysis and the heat flux asymptotically approaches a limit with increasing gradient which results in smooth temperature profiles. Thus, this form is preferred for modelling within the assumptions discussed previously.

A uniform mesh of 60 elements is used with fixed time steps of 10 steps per day. The corrector iteration ceases when the relative difference between successive temperature iterations is less than a prescribed tolerance, e. Here e = 5 x 10-3 was chosen. For the given time step of 0.1 day, convergence of the corrector was usually established within 1 iteration.

1.0

~075

.E

0.5

~ 0 25

0 3 Temperature gradient, BY/ Bz

Figure 1 Heat flux vs. temperature gradient. A, 2.598a(1 + G(dT/dz)) -~(dT/dz); B, a(1 + ~r(~T/~z)) - 1 (~T/~z); C, (ae) exp( - a(aT/dz) (dT/~gz))

23

v- I

7

L , , , i i

0 ' 90 180 270 360 Time, (doy)

Figure 2 Time series of temperatures at various depths for Lake Washington. ( ), finite element calculation; ( - - - ) , finite difference calculation; (/k), surface; (<~), 10m; (+) , 20m; (,), 40 m; (~) , 60 m or bottom

The results in Figure 2 show the computed and measured temperatures at several representative depths in the lake. The experimental values are drawn from data for 1963 provided by Edmondson 2°. In general, this method of examining the behaviour of the temperature as a function of time is much more critical than comparing isotherms on a depth time domain, since the former places emphasis on the temperatures near the thermocline where, of course, the important interactions that determine the thermal structure take place.

From Figure 2, the temperatures are seen to increase proportionately until day 90. During this period in the spring, the diffusivity is large and the temperature profile is approximately exponential with depth. At day 90, stratification is initiated and the epilimnion and hypolimnion become weakly coupled. After approximately day 240, the lake cools until it becomes isothermal again. As is seen in Figure 2, there is a particularly good fit with data with slightly large gradients between 20-40 m depth. The poorest agreement between the analysis and data is during the fall cooling period, from day 240 to approximately day 280. The data indicate that there is substantial penetration of warm water beneath the thermocline, either by convective penetration or by cooler water flowing under the mixed layer 17. In the model, the epilimnion deepens solely by erosion of the thermocline since the advective term in the governing equation has been neglected. A formulation including this term is feasible but beyond the scope of the present study 21.

To ensure a valid comparison, the same time step and discretization was used in the finite difference calculations. The finite difference and finite element solutions agree closely in the upper and lower layers. The only noticeable disparity occurs in the clinolimnion near the thermocline. (The finite difference solution at this depth is included in Figure 2.) In fact, this is the regime where the diffusivity varies most rapidly--from large values in the epilimnion to the minimum value at the thermocline. Since the diffusivity is highly non-linear in this region, integration over the element seems to provide a more accurate


Temperature computation for temperate lakes." R. A. Waiters et al.

representation of the system than using discrete midpoint values as in the finite difference scheme.

Results in Figure 3 show the variation of temperature with depth at several times during the annual thermal cycle. The corresponding finite difference solutions on days 90 and 180 are shown in Figure 4. The finite element scheme predicts a smaller heat flux into the hypolimnion, which is more in agreement with the temperature data. The difference in temperatures at 10m can be of the order of I°C. While the differences in temperature do not appear large, the differences in gradients cause significant variations in the diffusivity near the thermocline. Furthermore, the use of a constant value for the diffusivity in the hypolimnion leads to a poor representation of the temperature profile. In general, the diffusivity at the thermocline controls the average temperature in the hypolimnion while the vertical variation of the diffusivity controls the temperature profile. Using a constant diffusivity, the average computed temperature is seen to be approximately correct while the gradient is far too large due to insufficient mixing at depth.

After a number of numerical experiments, it was found that the value for tr which gave the most accurate thermocline depth was a ~ 0.048. It must be emphasized, however, that the parameter tr should not be considered as an isolated number since it is closely related to the choice of the value of the drag coefficient, CD. Consider:

tTC2(t~T/t~zJz2(T- 4) trRi =

(pa/pw)CoU 2

IC

20

E

-'::'30

4C

5C

6 0 5

8 13 7

7 9 II 13 15 17 19 21 Temperature, (oc}

Temperature profiles calculated using finite element ~um 3 method for conditions given in Figure 2. (1), 0 days; (2), 60 days; (3), 90 days; (4), 120 days; (5), 180 days; (6), 210 days; (7), 240 days; (8), 270 days; (9), 300 days

23

I ! 2

I0

2C

~'3C r~

r~

. . . - ' "

4 0

6O 5 7 9 II 13 15 17 19 21

Temperature, {°C)

Figure 4 Comparison of finite element ( - - ) and finite difference ( - - - ) calculations; (1) for 90 days; (2), for 180 days. ( . . . . ), corresponding calculation using a constant diffusivity in the hypolimnion

23

where C2 = --g~to k2, Pa, Pw are air and water densities, respectively and U is the wind speed. All quantities here are fixed by physical properties except tr and Co. Thus the value of tr for the best thermal results will depend upon the choice of CD such that:

tr/Co -'- 32

If Co = 2.6 × 10-3 is chosen {as with earlier researchers), then tr = 0.083.

Further, a set of sensitivity computations indicated that the model is not particularly sensitive to changes in tr or Ko. Changes in a of the order of 20~o resulted in a change of about 1 m in the thermocline depth, while 10~o changes in Ko had essentially no effect.

Typical depth profiles for the eddy diffusivity are shown in Figure 5. The different regimes can be clearly seen: the surface layer extending down to the diffusivity are found as follows: the bottom of the where the diffusivity increases; and the bottom layer with limited diffusivity values. As the changes in epilimnion depth show, the boundaries between the 3 different regimes move as a function of time. The location of the boundary and the values for the diffusivity are found as follows: the bottom of the epilimnion is taken to be at the thermocline, here defined as the point where the temperature gradient is a maximum. The value of the diffusivity is then matched across this boundary. The bottom of the clinolimnion is considered here to be that point where the diffusivity has increased by a factor of 10 or is equal to the surface diffusivity, whichever is less. While

4 6 Appl . Ma th . M o d e l l i n g , 1 9 7 8 , Vol 2, M a r c h

0

I0

20

E

i 30

~3

40

50

60 II o


50 IOO 150 2 0 0 Diffusivity, (m 2/doy)

Figure 5 Depth profiles of eddy diffusivity. (1), 180 days; (2), 150 days; (3), 60 days

limiting the value of the diffusivity in this manner is justified experimentally, the primary reason is to retain stability in the computational methods. The net effect is that the diffusivity over the entire water column as well as the thermocline depth are controlled only by surface processes once the geometry is specified.

When the diffusivity is large, such as during fall overturn, the temperature and diffusivity may occasionally exhibit small oscillations, particularly in the vicinity of the interfaces between differing diffusivity regimes where the functional form of H changes. The oscillations in temperature have typical amplitudes of approximately 0.01°C.

Numerical experiments with a variable number of quadrature points and with larger time steps were conducted to explore the computational efficiency of the finite element predictor-corrector. For reference, an 11 point quadrature was used initially. The number of points was then decreased to 5, and then 3. Errors became significant near the thermocline for the latter (AT ~ 0.7°C). These errors arise from the non-linear behaviour of the diffusivity and the fact that With few points a large value of H at one end of the element will dominate the quadrature. Thus, between the bottom of the mixed layer and the thermocline, H is over- estimated leading to higher calculated temperatures than the other quadrature methods.

The time step was increased to 0.2 day without any deterioration in the numerical solution. The thermal cycle calculation for 360 days with this time step and a five point element quadrature requires approximately 178 sec of CDC 6400 central processor time. In comparison, the finite difference method used 130 sec of central processor time for the 360 day run made with a timestep of 0.2 day and using one iteration of the corrector. The larger run times for the finite element method are almost entirely due to the use of a high-order multipoint quadrature to evaluate the intregral of H in the epilimnion. If this integral is evaluated analytically or a simpler quadrature used,

the computation times would be approximately the same for both the finite element and difference algorithms. As a further comparison, the CP times for the two methods were less than 10sec on the CDC 7600.

Conclusions

A mathematical and finite element analysis for computing the thermal cycle in deep, temperate lakes is described. The nature of the non-linearity arising through the heat diffusivity differs between three fundamental regimes corresponding to the epilimnion, clinolimnion and bottom layer of the hypolimnion. A Galerkin finite element approach is formulated and leads to an efficient predictor-corrector algorithm for treating the non-linearities. The different forms of finite element quadratures corresponding to the three regimes are determined as part of the formulation.

Numerical simulations give a good correlation with experimental data. A valid comparison with finite difference computations indicates that the strategies produce very similar results through most of the lake depth with noticeable variations evident in the clinolimnion, particularly near the thermocline. The finite element computations produce a smaller heat flux which more closely resembles observed experimental results. This may be attributed to the influence of integrating the thermal diffusivity across the regime interfaces and thermocline rather than using discrete values as in the difference approach. The formulation of the eddy diffusivities is shown to produce results more consistent with experimental observations than those corresponding to earlier diffusivity representations.

References

I Kraus, E. B. and Turner, J. S. Tel/us, 1967. 19, 98 2 Munk, W. H. and Anderson, E. R. J. Mar. Res., 1948, 7, (3),

276 3 Mortimer, C. H. Mitt. Int. Verein. Limnol., 1974, 20, 124 4 Sundaram, T. R. and Rehm, R. G. Tellus, 1973, 25, 157 5 Neumann, G. and Pierson, W. J. "Principles of physical

oceanography', Prentice-Hall, Inc., Englewood Cliffs, 1966 6 Walters, R. A. PhD Dissertation, Univ. Washington (1976) 7 Waiters, R. A. et al. Proc. Syrup. Appl. Comput. methods Eng.

(Ed L. C. Wellford, Jr.) Univ. S California, 1977 8 Kent, R. E. and Pritchard, D. W. J. Mar. Res., 1958, 18, 62 9 Mamayev, O. I. Akad. Nauk. SSSR Bull., Geophys Ser. 1958

10 Powell, T. Verh. Int. Verein. Limno/., 1975, 19, 104 11 Jassby, A. D. and Powell, T. Limnol. Oceanogr., 1975, 20, 4,

530 12 Martin, H. C. and Carey, G. F. "Introduction to finite element

analysis', McGraw-Hill, New York, 1973 13 Douglas, J. Jr. and Dupont, T. SIAM J. Numer. Anal. 1970, 7,

575 14 Ralston, A. bA first course in numerical analysis', McGraw-Hill,

New York, 1965 15 Ames, W. F. "Nonlinear partial differential equations in

engineering', Vol. 1, Academic Press, London and New York, 1965

16 Scheffer, V. B. and Robinson, R. J. Ecol. Monograph 1939, 9, 95.

17 Syck, J. M. MS thesis, Univ. Washington (1964) 18 Anderson, E. R. US Geol. Sur. ProJ~ Paper 269, 71-119, 1954 19 Laevastu, T. Soc. Sci. Fernica Commentations Phys. Math.,

1960, 25, 1, 136 20 Edmondson. W. T. Verb. Int. Verein. Limnol., 1972, 18, 284 21 Farmer, D. M. Quart. J. Roy. Meteorol. Soc., 1975, 101, (430),

869


Temperature computat ion for temperate lakes." R. A. Waiters et al.

Acknowledgements The authors thank Professor W. T. Edmondson for providing data for Lake Washington.

Appendix Evaluation of integrals:

The integral expressions of B and C in equations (15) and (16) are now examined. Terms in B(T) require calculation of three main integral types (see equation (5)):

(1) b

l e = f a

{H= + (1 + trRi)- 1} dz

in the epilimnion:

(2) Ic ~-

b

a (A1)

in the clinolimnion;

b

(3) lh = f Hb dz, a

constant Hb below the clinolimnion

In general a = 2i_ 1 and b = zi+ 1 for a patch entirely contained within a given zone. For convenience the interface between two zones is taken as a node, and can change position with time. In treating the patch centred on an interface node, a and b are the end nodes of individual elements.

In the epilimnion, consider the first integral above with Richardson number Ri = T T . 2 ( T - 4)(OT/Oz) where H=, tr and y are constants. Introduce the finite element approximation on the patch and simplify to obtain:

I e = H ~ ( b - a ) + f { l + o T z 2

x Tip i - 4 T i dz (A2) i - - i - 1

Since the values of {Tj{t)} are known in both the predictor and corrector algorithms of equations (15) and (16), then the integrand in equation (A2) is a function of z alone. A composite trapezoidal integration scheme was adopted in the computer analysis 14

The second integral in equation (A1) arises in the clinolimnion. Substituting the finite element approximation, with fl a constant and 0~ a prescribed function of time the integrand has the general form (ao + alz) -I/z and is evaluated analytically. The contribution for element e = [zi,Zi+x] is:

I~ e~ = 2~//- I/2(Ti+ I - T/)-3/2hi+ 1

X ((T/+ 1 -- 4) 1/2 -- (T/- 4) I/2)

Below the clinolimnion H is a constant Hb and the third integral is trivially Ih = Hb(b - a). A standard form for the insolation f(z, t) is f(z, t) = fo(t)e -p(''- where fo(t) is known from environmental data at the surface and fl(t) is prescribed from empirical data. Substituting f(z, t) into the integral for ci and evaluating the intregral analytically:

{e -#z~ - e-#-'°(l - flhl)}/flhl,

Ci(t ) /°(t) = ~

(A3)

i = 0

e pz,-, e-#'-'(h, + hi+ 1) e-# .... flh, flhih,+ l + flh,+------~l J"

l <~ i <~ n - 1

[e -# . . . . . e-#~"(flh. + 1)}/flh.,

i = n

(A4)

48 Appl. Math. Modelling, 1978, Vol 2, March

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