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WATER POLLUTION CONTROL RESEARCH SERIES 16130 DPU 02/71
RESEARCH ON THE PHYSICAL
ASPECTS OF THERMAL POLLUTION
ENVIRONMENTAL PROTECTION AGENCY WATER QUALITY OFFICE
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Series describes
the results and progress in the control and abatement
of pollution in our Nation's waters. They provide a
central source of information on the research , develop-
ment, and demonstration activities in the Water Quality
Office, Environmental Protection Agency, through inhouse
research and grants and contracts with Federal, State,
and local agencies, research institutions, and industrial
organizations.
Inquiries pertaining to Water Pollution Control Research
Reports should be directed to the Head, Project Reports
System, Office of Research and Development, Water Quality
Office, Environmental Protection Agency, Room 1108,
Washington, D. C. 20242.
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RESEARCH ON THE PHYSICAL
ASPECTS OF THERMAL POLLUTION
Cornell Aeronautical Laboratory,, Inc.
Buffalo, Nev York 1*1221
for the
WATER QUALITY OFFICE
EIWIRONMENTAL PROTECTION AGENCY
Project #16130 DPU
Contract #1^-12-526
February 1971
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $1.75
Stock Number 6501-0143
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EPA Review Notice
This report has been reviewed by the Water
Quality Office, EPA, and approved for publication
Approval does not signify that the contents
necessarily reflect the views and policies of
the Environmental Protection Agency, nor does
mention of trade names or commercial products
constitute endorsement or recommendation for
use.
11
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ABSTRACT
The mechanisms of formation and maintenance of the characteristic
thermal structure of deep, temperate lakes are investigated along with
the effects on the thermal structure of discharges of waste heat from
electric generating plants. It is shown that a thermocline is formed
by the nonlinear interaction between the wind-induced turbulence and
stable buoyancy gradients due to surface heating.
A theoretical description of the stratification cycle of temperate
lakes is given in which the interaction between wind-induced turbulence
and buoyancy gradients is included explicitly. The theoretical model
predicts all the observed features of stratification accurately. It is
shown that thermal discharges increase the temperature of the epilimnion
and also the temperature during spring homothermy. A lengthening of the
stratification period also occurs. In addition, the attendant transfer
of large quantities of water from one level to another has a significant
effect.
An exploratory experimental study is described on the nature of the
interfacial mixing between a flowing layer of warm water and an underlying
cooler pool of water. It is shown that the downward transfer of both
momentum and heat are severely inhibited at the interface by the stable
buoyancy gradients; momentum to a lesser degree.
This report was submitted in fulfillment of Contract Number 14-12-526
under the sponsorship of the Federal Water Quality Administration.
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TABLE OF CONTENTS
Section Page
ABSTRACT
LIST OF ILLUSTRATIONS viii
LIST OF TABLES xii
NOMENCLATURE xiii
FOREWORD xv
I. INTRODUCTION 1
II. A CRITICAL REVIEW OF THE STATE OF THE ART 3
II. 1 General Remarks 3
II. 2 Theories of the Thermocline 6
II. 3 The Seasonal Stratification Cycle 11
II. 4 Thermal Discharges at or Below the Level of
the Thermocline 12
II. 5 Interfacial Mixing 14
II. 6 Concluding Remarks 16
III. THERMOCLINE FORMATION 18
III. 1 General Remarks 18
III. 2 Basic Relations and Boundary Conditions 18
III. 3 Forms of the Eddy Diffusivity 20
III. 4 Turbulence in the Deeper Layers of a Lake 24
III. 5 Numerical Integration of the Basic Equations 27
III. 6 Concluding Remarks 30
IV. THE STRATIFICATION CYCLE 32
IV. 1 Effect of Changes in Surface Conditions on
Thermocline Behavior 32
IV. 2 Seasonal Variations in Environmental Conditions
Above the Lake 34
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Section £iS£.
V.
VI.
IV. 3
IV. 4
IV. 5
IV. 6
IV. 7
IV. 8
Qualitative Considerations on the Seasonal
Temperature Cycle
Free Convection
Numerical Results and Discussion
Cyclic Behavior of the Results
Comparison with Observations
Concluding Remarks
EFFECTS OF THERMAL DISCHARGES
V. 1
V. 2
V. 3
V. 4
General Remarks
Buoyant Plume
Effects of Discharge on Overall Thermal
Structure
V. 3. 1 Model When Discharge Remains
Below the Lake Surface
V. 3. 2 Model When Discharge Surfaces
V. 3. 3 Model with Pumping Only
Numerical Results and Discussion
V- 4. 1 Transient and Periodic Responses
V. 4. 2 Effects of Thermal Discharges
V. 4. 3 Effects of Pumping Alone
V- 4. 4 Comparison Between the Effects of
Thermal Discharge and the Effects
of Pumping
INTERFACIAL MIXING
VI. 1
VI. 2
VI. 3
VI. 4
Introduction
Theoretical Formulation for the Experimental
Determination of KM and KH
Design of the Flow System
Instrumentation
VI. 4. 1 Velocity Measurements
VI. 4. 2 Temperature Measurements
VI. 4. 3 Coordination of Temperature and
Velocity Measurements
36
39
41
47
50
52
53
53
54
C.G.
~J '
57
60
61
63
64
64
69
71
73
73
75
78
79
79
84
89
VI
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Section Page
VI. 4. 4 Motion Picture of the Experiment 90
VI. 5 Experimental Results 90
VI. 5. 1 Velocity and Temperature Records 90
VI. 5. 2 Interface Location 92
VI. 5. 3 Flow Starting Process 94
VI. 5. 4 End and Sidewall Effects 96
VI. 5. 5 Analysis of the Data RLO = 3. 0 97
VI. 5. 6 Effect of Richardson Number 102
VI. 6 Summary and Recommendations 105
VII CONCLUSIONS 108
APPENDICES
A. Perturbation Method of the Study of the Nonlinear
Behavior of the Basic Heat Transport Equation 111
B. Implicit Method of Considering the Interaction
Between Turbulence and Buoyancy Gradients 117
C. A Description of the Numerical Program 121
D. Verification of the Adequacy of the Basic Heat
Transport Equation When the Variability of the
Volumetric Coefficient of Expansion is Accounted
For 130
E. On the Value of the Semi-Empirical Parameter (3TJ 133
REFERENCES 134
FIGURES 142
vn
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LIST OF ILLUSTRATIONS
Figure Pa£e_
1 Temperature Structure of Cayuga Lake, 1952 142
2 Wind-Induced Currents in a Lake 142
3 Vertical Temperature Distributions for a
Constant Surface Temperature 143
4 Variation of the Depth of the Thermocline
for a Constant Surface Temperature 144
5 Distribution of Thermal Diffusivity for a
Constant Surface Temperature 144
6 Effect of Variations in Surface Conditions
on the Thermal Structure 145
7 Schematic Representation of the Annual
Temperature Cycle 146
8 The Stratification Cycle 147
9 Vertical Temperature Distributions 148
10 Seasonal Variation of Surface Heat Flux 149
11 Vertical Distributions of Heat Flux 150
12 Vertical Distributions of Thermal Diffusivities 151
13 Vertical Distributions of Temperature Gradients 152
14 Seasonal Variation of Thermocline Depth 153
15 Effect of Improper Initial Conditions 154
16 Cyclic Behavior of Temperature Variations 155
17 Cyclic Behavior of Temperature Distributions 156
18 Cyclic Behavior of the Eddy Diffusivities 157
19 Comparison of the Computed and Observed Stratification
Cycles of Cayuga Lake, New York 158
vin
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Figure Page
20 Comparison of Computed and Observed Temperature
Profiles for Cayuga Lake, New York 159
2la Schematic Representation of Thermal Plume 159 a
21b Comparison of Computed and Observed Thermocline
Depths for Cayuga Lake, New York
160
22 Effect of Thermal Discharges on Vertical Temperature
Distribution 161
23 Effect of Pumping on Temperature Cycle 162
24 Effect of Pumping on the Depth of the Thermocline 163
25 Effects of Thermal Discharge and Pumping Alone
on the Stratification Cycle 164
26 Effects of Thermal Discharge and Pumping Alone
on the Depth of Thermocline 165
27 Effects of Thermal Discharge and Pumping on
Thermal Diffusivity 166
28 Schematic Arrangement of the Flow System 167
29 Sketch for Derivation of the Refraction Error
on Photographs of Flow Traces 168
30 Refraction Error on Photograph of Flow Traces 168
31 Bridge Circuit to Measure Water Temperature 168
32 Enlarged Photograph of a Thermistor Probe Tip -
The Scale Divisions are 1 mm Apart 169
33 Design of Temperature Probes Not to Scale
(Only Two are Shown) 169
34 Photograph of Drive Mechanism for Temperature Probes 170
35 Circuit of Electronic Relay for Water Contact 170
36 Calibration of One of the Temperature Probes Between
25°C and 35 C Before Adjustment of the Calibration
Voltage 170
IX
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Figure Page
37a Velocity Profiles Ri0 = 3.0 (Upper Trace 2.67 sec,
Lower 30 sec) ^
37b Velocity Profiles R70 = 0.1 (All Traces 2.67 sec) 172
38a Vertical Temperature Profiles Ri0 =3.0 173
38b Vertical Temperature Profiles Ria =0.1 174
39a Temperature Extremes Vs Time Rit =3.0 175
39b Temperature Extremes Vs Time Ri0 - 0. 1 175
40a Position of Interface (max 9T/32) Ri0 =3.0 176
40b Position of Interface (max 2T/(?Z) R^ = 0. 1 176
41 Variation of Local Richardson Number, RL» ,
With Distance 177
42 Starting Process Ri0 =3.0 Warm Water Thymol Blue,
Cold Water, Sodium Hydroxide, (Dark Region Indicates
Mixing of Two Fluids) &0 = Depth of Warm Water at
Inlet 178
43 Peak Reverse Flow Velocity Ri0 =3.0 179
44 Velocity at Surface Ria = 3.0 179
45 Flow Velocity Vs Depth and Position R^ = 3.0, 30 min 180
46 Temperature Vs Depth and Position RiA = 3.0, 25 min 181
47 Temperature, Flow Velocity Vs Depth and Time,
Ri0 =3.0, Position 4 182
48 Solution of_Momentum Equation Ri - 3. 0, Position 4,
30 min, R^£ = 4. 1 ° 183
49 Temperature, Velocity Vs Depth and Time R~7^ = 0. 1,
Position No. 5 * 184
50 Temperature, Velocity Vs Depth and Position R' =01
5. 7 and 6. 7 min
185
51 Solution of Momentum Equation R^ = 0. 1, Position 5
5. 7 min, R^ = 2. 5 ' ' 186
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Figure Page
52 Vertical Velocity, Momentum and Thermal
Diffusivity Coefficients 187
KM KH
53 fj,/^ > .A / at Interface Vs Richardson
Number Vc 188
54 Prandtl Number at Interface Vs Richardson
Number 188
XI
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LIST OF TABLES
Table
Temperature Differences Produced by Thermal Discharges
with Different Discharge Temperatures but the Same
Heating Rate ' a
Temperature Differences Produced by Different
Pumping Rates 70 a
Flow Parameters '
XII
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NOMENCLATURE
/-) Surface area of the waterbody
flz Cross-sectional area of the waterbody at depth
C.J0 Constant-pressure specific heat
ot Depth
Acceleration due to gravity
a
f/ Depth of the interface
"H, Depth of the upper layer of water
fa vonKarman's constant^ 0.4
KH Eddy diffusivity for heat
K Eddy diffusivity for momentum
^( Heat exchange coefficient
/_ Monin-Obukhov length
P Potential energy of stratification of a lake
y. Heat flux
^ Volumetric flow rate through power plant
Re, Reynolds number
RL Richardson number
5 Dilution factor
S(Z) Heat source term
' Temperature
t Time
'£ Equilibrium temperature
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lL,V,i>J' Velocity components
U7* Friction velocity
H Vertical coordinate
Effective depth of discharge
Depth of the intake
Depth of lake
a
Depth of center of gravity of the lake
O^v Coefficient of volumetric expansion for water
9 Density
P~ Viscosity
^ Kinematic viscosity
^ Shear stress
]/C Molecular thermal diffusity
CT Normal stress
C^ , 0~z Empirical constants used in form of eddy diffusivity
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FOREWORD
This report describes the results of a study, on some problems on
the physical aspects of thermal pollution, that was carried out by Cornell
Aeronautical Laboratory under Contract No. 14-12-526 with the Department
of Interior, Federal Water Quality Administration. Technical monitoring
for the program was provided by Dr. Bruce A. Tichenor, National Thermal
Pollution Research Program, Pacific Northwest Water Laboratory, Con/all is,
Oregon.
The experimental part of the study, reported in Section VI was
conducted by Drs. G. Rudinger and G. E. Merritt, while the remaining
parts of the study were conducted by Drs. T. R. Sundaram and R. G. Rehm.
The authors would like to express their thanks to Mr. John Moselle for
his very capable programming of the numerical computations. The authors
also wish to express their appreciation for the valuable help given by
A. F. Gretch who built the flow system and assisted with the running of
the experiments, R. Hiemenz who designed and constructed the electronic
bridge circuit and relay for the temperature measurements and N. Kay who
prepared the thermistor probes.
The computer program developed during the project is not contained
in this report. Individuals interested in obtaining program documentation
should contact National Thermal Pollution Research Program, EPA-WQO,
Pacific Northwest Water Laboratory, Corvallis, Oregon 97330.
XV
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I. INTRODUCTION
One of the problems that has gained increasing importance in recent
years is the thermal pollution problem which is caused by the discharge of
waste heat from electric generating plants into various bodies of water and
the subsequent degradation of the quality of these waters. The degradation
of the body of water may occur either through the direct influence of the
increased temperature on aquatic life or through the lowering of the amount
of dissolved oxygen. In the last few years a number of investigations on the
effects of elevated temperature on various aquatic life have been carried
out, and a fairly extensive amount of literature on the subject exists.
These studies have also resulted in some criteria for judging water quality
requirements of aquatic life.
On the other hand, corresponding physical studies on how the heat
from a discharge is dispersed and distributed within a receiving body of
water are relatively few in number, and the state of the art has been sum-
7-9
marized recently by several authors. It should be emphasized that an
understanding of the changes in the thermal and current structure of the body
of water due to the effluent discharge is a necessary first step in assessing
the possible adverse effects of the discharge on the aquatic life in the
receiving waters. Only when these physical changes caused by the dis-
charge are known can the attendant ecological effects be predicted.
The present report describes the results of a study on two specific
problem areas connected with the physical aspects of thermal pollution.
The first problem area is concerned with the mechanisms of formation and
maintenance of the characteristic thermal structure of temperate lakes
and reservoirs, and the effects on this structure of thermal discharges at
or below the level of the thermocline. The second problem area is con-
cerned with the manner and rate of spreading of a warm-water wedge
overlying a body of colder water with specific reference to the behavior of
the interfacial mixing between the two. The latter problem is of impor-
tance in determining the extent of the mixing zone in river discharges and
the possibility of recirculation of the warm water through the condensers
1
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of the electric generating plants. The choice of these two specific problems
from the myriad of problems in the broad area of thermal pollution was
dictated, in part, by the priorities for research set forth in a recent
10
paper.
A critical review of the existing knowledge on the above two problem
areas is given in Section II of the present report. Specific emphasis is
placed not so much on giving detail descriptions of related work, but rather,
on pointing out those aspects of existing theories that are unsatisfactory.
A theory for the formation and maintenance of thermoclines in stratified
lakes and reservoirs is put forth in Section III. The theoretical model is
also used in Section IV to study the stratification cycle of temperate lakes
and reservoirs. The effects of thermal discharges, at or below the level
of the thermocline, on the stratification cycle are discussed in Section V.
The results of an experimental study to investigate the nature of the inter-
facial mixing between a flowing, warm layer of water and an underlying
cooler pool of water are given in Section VI. Finally some concluding
remarks and recommendations for further study are given in Section VII,
Most of the detailed mathematical discussions are given in the
appendices at the end of the report. In Appendices A and B detailed discussions
are given^espectively, of a perturbation-expansion technique and of an integral
technique for the analysis of the thermal structure of temperate bodies of
water. In Appendix C a discussion of the numerical procedure used in the
analyses is given, while Appendix D consists of a discussion of the adequacy
of the basic equations used. Finally, a discussion of a semi-empirical para-
meter used in the analyses is given in Appendix E.
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II. A CRITICAL REVIEW OF THE STATE OF THE ART
In the present section a brief review will be given of existing knowl-
edge on the specific problems mentioned above. General reviews of the
state of the art of thermal pollution research can be found in Refs. 7-9. It
should be emphasized that the brief review given below is not meant to be
an exhaustive one, but rather, it represents a selection of some of the more
important studies that are directly relevant to the present study.
II. 1 General Remarks
When considering the dispersion of thermal discharges from a
power plant into an aquatic environment it should be recognized that almost
all geophysical fluid-dynamic phenomena are dominated by turbulent trans-
port processes. A number of years ago, Jeffreys demonstrated that in
a channel much wider than deep, transition from laminar to turbulent flow
occurs at a Reynolds number of only 310, where the Reynolds number is
defined as
R
e =
Here £> is the density of the fluid, V~ is the velocity, d is the depth
and p, is the viscosity of the fluid. In other words, in a lake about ten
meters deep, transition to turbulent flow will occur at a velocity of about
-3 12
3x10 cms/sec. Thus as Hutchinson points out in his monumental
treatise, almost all limnological flows are turbulent, the only exceptions
being flows close to a smooth bottom or cases in which turbulence is
inhibited by a stable density stratification.
When a heated effluent is discharged into the top layers of a body of
water, part of the heat is lost to the atmosphere directly and part of the
heat is transferred to the lower, colder layers of water by turbulent diffu-
sion. The downward transfer of heat influences, and is in turn influenced
by, the prevalent circulation patterns in the body of water. Moreover, the
turbulence that causes the downward diffusion is the result of wind-
generated waves and is influenced strongly by various factors such as the
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prevalent thermal stratification in the body of water and the ambient metero
logical conditions. In some cases the turbulence may also be caused by
hydraulic factors such as those due to inflow. When the body of water of
concern is an estuary, the density stratifications resulting from varying
salinity also influence the turbulence (and hence the downward diffusion).
Thus, the effect of discharge of a heated effluent into the top layers of a
body of water is felt over the entire body of water through a series of com-
plicated, and coupled, processes.
Because of the complexity of the problem, most of the analyses that
have dealt with the physical aspects of thermal pollution are based on
13, 14
grossly simplifying assumptions. For example, in some of the analyses
complete vertical and lateral mixing of the effluent with the receiving waters
is assumed. The former of the above assumptions is invalid in a lake or
reservoir with a well-defined thermocline or in an estuary with significant
stratification. Estimates based on such assumptions on the effect of the
thermal discharge from the Marchwood power station into the Southampton
bay (which is an estuary with significant stratification) proved to be com-
pletely incorrect. It should be emphasized that approximations such as
that of complete vertical mixing cannot be uniformly valid in all cases.
Thus, the assumption of complete vertical mixing may be a realistic one if
the receiving water is a relatively shallow river. On the other hand, this
assumption is likely to lead to erroneous results if the receiving water is
an estuary, a reservoir or a deep lake with a well-developed thermocline.
Recently, several mathematical models have been developed
specifically for predicting thermal discharges into lakes and streams.
While some of these models have been able to predict fairly accurately the
thermal patterns in specific water bodies for which they were developed,
they are in general not applicable to other environments since these models
include empirical coefficients unique to the given environment for which
they were formulated. When developing methodologies and techniques for
predicting the effects of thermal discharges, it is desirable that they have
as general a validity as possible and not be confined to specific situations.
Therefore the use of empirical coefficients should be avoided whenever
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possible. However, because of the inherent turbulent nature of the trans-
port processes involved, a certain degree of empiricism would become
unavoidable. In this context, as pointed out by Brooks, it is necessary
to differentiate between empirical coefficients and physical coefficients.
Thus while empirical coefficients which are unique to given bodies of water
are to be avoided, the use of experimentally determined relations between
environmental conditions and the resulting turbulent transport properties
would become unavoidable. It should be emphasized that, if properly exe-
cuted, the latter step will in no way limit the general validity of the results.
It has been pointed out in the literature that generalized three-
dimensional models need to be developed. While the inclusion of spatial
variation in all three coordinate directions is certainly desirable when such
variations are important, it should be emphasized that this step has to be
in addition to, not instead of, the inclusion of the essential variable nature
of the eddy diffusivities. The primary difficulty in predicting the physical
effects of thermal discharges into a body of water is indeed that the turbu-
lent transport properties are complicated functions of the environmental
conditions above the body of water as well as the prevalent thermal and
current structures in the body of water. In our view, it is of utmost
importance to consider the complex interactions between the thermal dis-
charges and the turbulent transport properties if a satisfactory prediction
of the effects of thermal discharges is to be obtained. On the other hand,
as Sundaram et al point out, if the turbulent transport properties are
assumed to be constant (as has been done often in the literature), then there
is no basic conceptual difficulty in solving even the full three-dimensional
equations. However the limited usefulness of the results would hardly
justify the effort and expense that would be involved in such a procedure.
We interpret three-dimensional models as meaning models in which
transport in all three spatial directions are considered (when important),
and in which transport in a given direction is not neglected purely for mathe
matical convenience. However, in many practical problems diffusion in all
three directions do not become important simultaneously, and indeed over
significant regions of the flow diffusion in only one or two directions
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predominates. For example, when there is a thermal discharge to the top
of the epilimnion of a stratified body of water, the stable three-dimensional
plume can be treated as being essentially two dimensional. Conversely,
when considering the effects of the discharges on the entire body of water,
conditions are essentially quasi-homogeneous in the horizontal directions
and vertical diffusion predominates. Problems of this type can be
analyzed in terms of suitably chosen combinations of two- and quasi-one -
dimensional problems.
With the above general remarks in mind, we will now critically
review the literature relevant to the specific problems being considered in
the present study. Again, it should be emphasized that the review given
here is by no means exhaustive, the purpose here being mainly to discuss
those aspects of existing literature that are essential for placing the present
study in the proper perspective.
II. 2 Theories of the Thermocline
The term 'thermocline' was first proposed by Birge in 1897 (see
2 1
Fairbridge ) to describe the layer of intense temperature gradient that
separates the almost homogeneous upper layer from the colder bottom
waters of deep, stratified lakes. Because of its obvious importance in lim-
nology and oceanography, several studies on the physical mechanism of the
formation of the thermocline have been carried out since the time of Birge.
However, it is fair to state that a satisfactory quantitative theory of the
thermocline has not yet been found.
Before considering the salient aspects of various thermocline
theories, it is first relevant to review the actually observed characteristics
of stratification. During early spring, most temperate lakes exhibit a
nearly homothermal temperature distribution with a temperature of about
4°C (which is the temperature of maximum density for water) extending all
the way to the bottom. As the weather above the lake begins to warm, the
lake receives heat, mainly by solar radiation, at an increasingly rapid rate.
During the early part of the warming season the lake continues to remain
nearly homothermal, since the heat that is received at the surface layers
-------
by solar radiation is transported to the deeper layers by wind-induced
currents and turbulence. As the rate of heating of the lake continues to
increase, the rate at which heat is received at the surface layers soon
exceeds the rate of heat removal to the deeper layers, and the temperature
of surface layers begins to increase. During this early period the tempera-
ture decreases monotonically with increasing depth, with the bottom
temperature remaining close to that at the end of vernal circulation.
Figure 1 shows some typical plots of the vertical distributions of
temperature in Cayuga Lake, New York, during various parts of the strati-
fication cycle. The plots given in Fig. 1 have been constructed using the
monthly averages of the temperatures measured by Henson, Bradshaw and
22
Chandler. The initial isothermal distribution and the later monotonically
decreasing distribution can be seen in (a) and (b) of Fig. 1.
As the heating continues, a point of inflection develops in the tem-
perature profile and a well-mixed upper layer, with relatively intense
temperature gradients at its bottom boundary, is formed (as shown in (c)
of Fig. 1). The plane of the maximum temperature gradient is, of course,
the thermocline. During the remainder of the heating period, the thermo-
cline slowly descends into the deeper, colder layers of the lake. It should
be noted that once a thermocline forms, the deeper regions of the lake are
relatively uninfluenced by changes in surface conditions. In fact, the tem-
perature structure of the deeper layers below the thermocline changes very
little with time, so that the deeper layers often serve as records of the
processes, during the early parts of the heating period, which created the
prevalent thermal structure.
As the lake attains its maximum heat content and subsequently begins
to cool, the thermocline moves down rapidly into the deeper layers of the
lake as wind mixing is now augmented by convective mixing due to surface
cooling. The thermocline continues to move down rapidly as the well-
mixed upper layers cool further, as shown in (d) and (e) of Fig. 1, until the
whole lake again attains homothermy.
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In the brief description of the observed features of stratification
given above, two important points should be noted. The first is the trans-
formation of a smooth, monotonically decreasing temperature distribution
at the beginning of the warming season into a temperature distribution which
exhibits a well-defined thermocline. The second is the continuous downward
erosion of the thermocline into the deeper, colder layers of the lake. In
our view, a satisfactory theory for the stratification cycle in a temperate
lake must account for these important aspects. It is also our view that the
basic phenomena that are responsible for the formation and the maintenance
of the thermocline are the addition or loss of heat at the surface, turbulent
transport of heat from the surface to greater depths and the striking inter-
action of the turbulence with the temperature gradients. Thus, it is
essential to account for the variation of the turbulent transport properties
j*
with depth if a satisfactory theory of the thermocline is to be developed.
Some authors have used constant values for the eddy transport
properties when solving for the thermal structure of lakes. The practice
of assuming constant values for the eddy transport properties is especially
prevalent in oceanic-thermocline theories, where it has been customary to
interpret the thermocline, and the thermo-haline circulations related to it,
24
in terms of Ekman's famous theory (see Sverdup, et al) of wind-driven
currents on a rotating globe. Ekman's original analysis, as well as most
25 -28
of the related analyses that have been developed since, assumes a
constant value for the eddy transport coefficients, invariant with depth.
Therefore, these analyses predict a temperature distribution which
decreases more or less uniformly downward from the surface. In parti-
cular, they do not predict a uniformly mixed upper layer with a well-defined
lower boundary. Nevertheless, these results have been used to estimate
the depth of the thermocline below the surface by artificially postulating it
to be the depth at which the temperature has decayed to a value equal to
e times that at the surface.
The theory of the thermocline which is most satisfactory from a
qualitative point of view, but unsatisfactory from a quantitative point of
29
view, is due to Munk and Anderson. The key feature of Munk and
Some interesting qualitative features of "Motions in Thermoclines" have
been pointed out by Mortimer, Verhandlungen Internationalen VereinigUna
fur Theoretische und Angervandte Limnologie, Vol. 14, 1961, pp 79-8T;
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Anderson's theory is that it accounts for the decrease in eddy transport
coefficients with increasing depth due to the stabilizing influence of tem-
perature gradients. According to this theory, the sequence of events
leading to the formation and maintenance of the thermocline are as follows:
The heat from the topmost layers of a body of water are transferred to
greater depths by the turbulence that is produced at the surface due to wind
action. The turbulence that is produced at the surface gives rise to a
certain vertical distribution of eddy transport coefficients within the top
layers of the water body. However, these eddy transport coefficients are
not independent of depth and, in fact, get progressively small with increas-
ing depth due to the stabilizing influence of the temperature gradients.
The degree of inhibition of the turbulence by the stable temperature
30
gradients is determined by tne value of the Richardson number, which is
merely the ratio of the buoyancy and inertia forces.
29
Munk and Anderson accounted for the interaction between the
turbulence and the thermal stability by postulating the eddy transport coef-
ficients to be certain physically meaningful functions of the gradient
Richardson number. They introduced these functional relationships into the
differential equations governing the temperature and current distributions
and solved them simultaneously on a computer. They found that in general
the Richardson number increases with increasing depth, and that at a crit-
ical depth turbulence is severely inhibited by the stabilizing influence of the
buoyancy forces. Above the critical depth the water is kept well mixed and
homogeneous by the dominant turbulent transport processes, while the
layers below the critical depth are 'protected1 from surface heating. Below
the critical depth the temperature drops rapidly to a value appropriate to
that at the bottom of the water body. Thus the fundamental reason for the
existence of the thermocline is the inhibition of the turbulence by the stabil-
izing influence of buoyancy forces.
As Defant (in his monumental two-volume treatise on oceanography)
points out, Munk and Anderson's theory delves deeply into the fundamental
reasons for the formation of the thermocline and throws much light into the
-------
consequences of the nature of the interaction between turbulent transport
processes and thermal stability. However, as pointed out earlier, Munk
and Anderson's theory is unsatisfactory from a quantitative point of view
and predicts depths of the thermocline which are much smaller than those
actually observed. Moreover, since the theory is based on the assump-
tion of the existence of a steady state, certain inconsistencies arise at
large depths. The authors point out that "it appears that the distributions
of current and temperature cannot both be stationary at the same time. "
Since the time of Munk and Anderson's original paper, a consider-
able amount of work has been done on the nature of the interaction between
turbulence and thermal stability in connection with atmospheric -boundary-
layer flows (see Refs. 32 and 33, for example), and these studies have
enhanced our knowledge of the interaction considerably. One of the more
important developments in connection with atmospheric turbulence has been
the so-called "similarity theory" that was formulated by Monin and
34
Obukhov. These authors postulate that the influence of thermal stratifi-
cation on the turbulence at any depth 3? , can be characterized by the
parameter ( 2/L ) alone where L is a length scale given by
L =
In Eq. (II. 1) ~k. is the von Karman constant ( *= 0. 4) and C. is the
specific heat. The physical significance of the Monin-Obukhov length scale,
L , is that it represents the depth at which the rate of production of tur-
bulence by the Reynolds stress and that by the buoyancy flux are approxi-
mately the same. The significance of the Monin-Obukhov length in the
stratification cycle of lakes, as well as its application to predicting the
effects of thermal discharges have been discussed in detail by Sundaram
et al.
Thermocline theories based on mechanisms other than those described
above have been put forth by Ertel, Dake and Harleman36 and Li. 3? The
theories proposed by Ertel and by Dake and Harleman are based on the
10
-------
assumption that the primary mechanism responsible for the formation of
the thermocline is the differential absorption, at various depths, of the
o /
incoming solar radiation. It may be noted that Dake and Harleman
37
include the effects of turbulent diffusion in an indirect way. Li has put
forth a theory for the thermocline based on the so-called Burgers' equation,
and his theory does not consider buoyancy effects at all. Li's theory essen-
tially deals with the development of a turbulent layer in the upper regions
of a body of water due to the onset of strong winds. The speed of downward
propagation of the developing turbulent front is proportional to the wind
speed, and the front disappears when the wind subsides.
It was mentioned earlier that actual observations indicate that the
vertical temperature profile at the start of the stratification season in a
temperate lake is fairly smooth, and that a thermocline forms only after a
certain period beyond the time of maximum spring homothermy. However
the thermocline forms well before the start of the cooling season, and a
well-mixed upper layer already exists at the time the lake begins to lose
heat to the atmosphere. None of the existing theories predict the above-
noted features.
II. 3 Seasonal Stratification Cycle
In addition to the studies described above on thermocline formation,
there are also some theories for the seasonal stratification cycle of tem-
perate lakes. Some of these theories are based on the global (or integral)
forms of the conservation relations in which the vertical temperature pro-
files are described in terms of a given number of characteristic parameters.
The time evolution of each parameter is determined by using the conser-
vation relations themselves as well as suitable moments of these equations.
The most important theory in this category is the one put forth by
38
Kraus and Turner. Kraus and Turner have included the effects of con-
vection due to preferential heating below the surface or cooling at the surface
as well as the effects of wind stirring. Their theory predicts the formation,
immediately after maximum spring homothermy, of a thermolcine at great
depths. As the heating season proceeds, the thermocline moves upwards,
11
-------
and reaches its shallowest depth during summer solstice when the rate of
surface heating is a maximum. The thermocline then begins its descent,
and continues to do so till the time of fall homothermy.
Kraus and Turner's theory predicts many of the important observed
features of stratification discussed earlier. However, as pointed out earlier,
observations indicate that a thermocline does not form immediately follow-
ing spring homothermy. While the depth over which surface influences are
felt steadily decreases during the warming period, a thermocline or a
point of inflection does not form immediately following spring homothermy.
As discussed by Sundaram et al, the value of the Monin-Obukhov length,
which is a measure of the depth of penetration of surface influence,
decreases rapidly during the early parts of the warming season. However,
a thermocline does not form till around the time when the Monin-Obukhov
length reaches a minimum positive value.
39 40
Orlob ' has developed a method of describing the thermal struc-
ture of deep lakes and reservoirs in which information derived from actual
measurements is used in the analytical framework. He introduced the
values of the eddy diffusivities deduced from measured vertical tempera-
ture profiles into a simple unsteady, one-dimensional heat conduction
equation to solve for the seasonal changes in the thermal structure. While
the eddy diffusivity was allowed to vary with depth and the experimentally
determined values were fitted with empirical analytical expressions, the
essential nonlinear dependence of the eddy diffusivity on the thermal struc-
ture itself was not included. Orlob has compared the predictions of the
mathematical model with the measurements made in the specific body of
water for which the model was constructed, and has found good agreement
between the two.
II. 4 Thermal Discharges at or below the Level of the Thermocline
An important problem that arises in connection with thermal pollu-
tion is the effect of thermal discharges into a stratified body of water at or
below the thermocline level. Discharges below the thermocline have been
used extensively in connection with sewage disposal into marine environ-
12
-------
41 -44
merits. The advantages of this mode of discharge have been described
45
by Brooks. The primary advantage is, of course, that because of sewage
being heavier than the surface sea water, the sewage field can be held sub-
merged below the thermocline thereby preventing contamination of the shore
areas.
When considering the physical effects of thermal discharges into the
hypolimnion of a stratified body of water, two specific problems have to be
analyzed. The first is concerned with the behavior of the buoyant plume due
to the discharge and the second is concerned with the effect of the discharge
on the thermal structure of the entire body of water. While considerable
knowledge exists on the first problem, practically nothing is known about
the second.
When water that is withdrawn from the hypolimnion of a waterbody
is returned to the hypolimnion after cycling through a power plant's conden-
sors, it will tend to rise as a buoyant plume since it will be warmer than
41
the surrounding hypolimnic water. As Rawn, Bowerman and Brooks
point out in connection with sewage disposal, three different types of plumes
can be identified depending on the initial density and velocity of the effluent.
In the first type, the effluent rises to the surface and spreads as a surface
field (this was the most common type observed by Rawn et al). In the
second type, the jet penetrates the thermocline and rises to the surface
only to plunge down below the thermocline and spread as a thin submerged
field. This type of plume occurs when the density of the plume is higher
than that of the epilimnion and when the discharge velocity of the effluent is
such that the plume has enough residual kinetic energy at the thermocline
level to drive through it. In the third type, the plume rises to the thermo-
cline level and spreads as a thin field below the epilimnion.
Rawn et al state the approximate condition for the submergence of
the effluent field as
(ST - J ) 9(( + 9o
> e£ (ii. 2)
13
-------
where S ^ is the dilution factor at the point where the axis of the jet
crosses the thermocline and (pM ,
-------
The magnitude of interfacial friction at the boundary between a
warm layer and an underlying cooler body of water is strongly influenced
by the stable density stratification at the interface. Indeed the stable
density stratification tends to strongly inhibit the turbulent transfer of
momentum, as well as of heat and mass, between the two layers. This
phenomenon is not unlike that of the inhibition of the downward transfer of
heat from the epilimnion to the hypolimnion of a stratified lake.
Transport across stably stratified interfaces is of interest in con-
nection with a wide variety of environmental problems, the densimetric
currents in lakes and reservoirs and the katabatic winds of the atmosphere
being two important examples. Because of the above reason, experi-
mental studies of the nature of the mixing processes across a stably
stratified interface have been carried out by a number of authors. For
example, Keulegan has studied, in a laboratory flume, the mixing
processes at the interface of a fresh-water layer flowing over saline water.
Keulegan systematically varied the velocity of the upper layer and
found that at low velocities the interface was quite stable, with very little
mixing occurring across it. As the velocity was increased, internal
waves started to appear at the interface. At even higher velocities, the
crests of the internal waves began to break with the consequent slow mix-
ing of the upper layer into the lower layer.
Experiments analogous to the one described above have also been
conducted by Lofquist and Ellison and Turner. Lofquist considered
the case when salt water flows turbulently under a pool of fresh water,
while Ellison and Turner considered the case of a surface jet in a channel
in which a layer of fresh water flows over a pool of salt water. Lofquist's
experiments were primarily concerned with the internal waves at the
stable boundary, while Ellison and Turner studied the nature of the turbu-
lent entrainment process at the interface. These latter authors found that
the entrainment coefficient depended on a bulk Richardson number and
decreased with increasing values of this number. At some critical value
of the bulk Richardson number entrainment of the lower fluid into the
upper layer ceased altogether.
15
-------
Ellison and Turner ' have also studied the behavior of a layer
of dense salt solution on the floor of a sloping rectangular pipe in which
there is a turbulent flow. They measured both velocity and density pro-
files near the interface and correlated the turbulent transport coefficients
59
for momentum and salt in terms of a local Richardson number. They
found that the ratio of turbulent transport coefficient for salt to that for
momentum decreases with increasing values of the local Richardson
number. However, because of experimental errors, and the influence of
the shape of the pipe and the Reynolds number on the results, the results
are not generally applicable.
Thus while some knowledge exists on the nature of the transfer
processes at the interface between fresh and salt water, this knowledge is
not directly relevant to the problem of the interfacial friction between a
warm layer and underlying cooler water. Firstly, a complete analogy
between the transfer of salt and heat has to be assumed in order for exist-
ing results to be usable. Secondly, accurate quantitative information on
the interfacial friction is not available.
II. 6 Concluding Remarks
It can be seen from the above brief description that while some
previous studies of the problems that are being considered here have been
carried out, very few quantitative results exist. Thus while several
theories of the thermocline have been advanced, none of these theories
predict one of the most important observed features of stratification,
namely, the change of a monotonically varying temperature distribution
during the early parts of the warming season to one that displays a thermo-
cline during the later parts of the warming season. Again, none of the
existing theories have accounted for the important interactions between
the wind-induced turbulence and the buoyancy gradients due to surface
heating in an entirely satisfactory or consistent manner.
It should be emphasized that when considering the effects of thermal
discharges on the stratification cycle, it is of paramount importance to
account properly for the interaction between the turbulence and thermal
16
-------
structures. As pointed out by Sundaram et al, heated effluents not only
influence the thermal structure directly, but they also influence it
indirectly through their influence on the structure of the turbulence.
Again, as pointed out by the above authors, if the effects of the thermal
discharges on the mechanisms of mixing are altogether neglected (a. step
which cannot be justified), then the problem of predicting the effects of
thermal discharges on the lake becomes relatively simple. There are no
existing theories which account for the changes, due to the thermal dis-
charges, in the eddy diffusivities in a stratified lake.
A theory for the formation of thermoclines in deep lakes and reser
voirs is given in Section III. This theory includes a proper description
of the interaction between wind-induced turbulence and buoyancy gradients
due to surface heating. The analytical framework developed in Section III
is extended to study the stratification cycle in Section IV and the effects
of thermal discharges on the stratification cycle in Section V.
An experimental study of the interfacial mixing at a stable inter-
face is described in Section VI.
17
-------
III. THERMOCLINE FORMATION
III. 1 General Remarks
As pointed out in the last section, even though a number of theoret-
ical models have been proposed to explain the characteristic thermal
structure of temperate lakes, a satisfactory explanation for the formation
and maintenance of thermoclines is still lacking. In the present section,
it is shown that a satisfactory theory for the mechanism of formation and
maintenance of thermoclines in a temperate lake has to take into account
two essential aspects. Firstly, the mechanism of not only the formation but
also the maintenance of a thermocline is an unsteady process even when
conditions above the body of water under consideration are steady. There-
fore, a satisfactory theory for the thermocline must include this basic
aspect. Secondly, the formation of a thermocline is by the nonlinear inter-
action between the wind-generated turbulence and the stable buoyancy
gradients in the body of water under consideration. This nonlinearity, while
making the equations difficult to analyze, is an essential feature of the
interaction between the turbulence and prevailing temperature structure
and as such has to be retained.
III. 2 Basic Relations and Boundary Conditions
In a large, relatively deep lake, the transport of any property takes
place much faster in the horizontal directions than in vertical ones, so that
one can assume that horizontal homogeneity exists in planes parallel to the
surface. It should be emphasized that the above assumption can be valid
only in an approximate sense since, as pointed out by Wedderburn a
number of years ago, the existence of a current in the well-mixed upper
layer, or the epilimnion, must necessarily involve an upward tilting of the
isotherms toward the windward end of the lake. However, in the localized
theoretical model being proposed here, the small horizontal nonhomogeneity
due to the tilting of the isotherms is of no direct consequence since it does
not influence the mechanisms of formation or maintenance of the thermo-
cline. The implications of the assumption of horizontal homogeneity have
been discussed at length by Sundaram et al.
18
-------
Under the assumption of horizontal homogeneity, the equation
describing the vertical transport of heat is
where T is the temperature, t is the time, I is the distance meas-
ured downward from the surface and KH is the eddy diffusivity for the
vertical transport of heat. Molecular thermal diffusivity is not explicitly
accounted for in Eq. (III. 1) since it is in general smaller than the eddy dif-
fusivity, and if necessary, it can be incorporated into the definition of
KH . It should be noted that Eq. (III. 1) is nonlinear since, in general,
the diffusivity is a function of the thermal as well as current structure in the
lake.
Before discussing the dependence of the eddy diffusivity on the
thermal and current structures, it is appropriate to consider the initial and
boundary conditions that have to be used in conjunction with Eq. (III. 1). In
all of the calculations presented in the present section, the initial condition
will be taken as that corresponding to the end of spring homothermy; that
is, the initial condition will be taken as
T ( -i , o ) = To (in. 2)
where I0 is the temperature of the lake at maximum spring homothermy.
The boundary condition at the surface of the lake must describe the heat
exchange between the lake and the atmosphere, and this can be written in
. , 7, 16
the form,
- -'^K-- KIT' r'}
where Q is the heat flux (taken positive when downward), K is a heat
exchange coefficient, Ts is the surface temperature and T£ is a ficti-
tious surface temperature, called the equilibrium temperature, at which
there would be no net heat transfer to or from the lake surface. The
19
-------
equilibrium temperature and the heat-transfer coefficient are both functions
of the environmental conditions above the lake and can be expressed as
functions of the wind speed, air temperature and humidity and net incoming
(sky and solar) radiation. Methods of evaluating T6 and K are
described fully by Edinger and Geyer and by Sundaram et al.
It may be noted that Eq. (III. 3) is a statement of Newton's law of
cooling. It should be pointed out that implicit in Eqs. (III. 1) and (III. 3) is
the assumption that the bulk of the incoming solar radiation is absorbed
within a small layer near the surface. This assumption is in general valid
in most deep, turbid lakes. For example, as Ruttner points out, the
characteristic depth for absorption of the solar radiation in Seneca Lake,
New York is considerably smaller than the depth of the well-mixed upper
layer.
As mentioned earlier, theories of the thermocline based on the so-
called "internal-radiation-absorption model" have been given by Ertel
and by Dake and Harleman. These theories account for turbulent diffu-
sion only in an indirect and empirical way, and they are based on the assump-
tion that the primary mechanism responsible for the formation of a thermo-
cline is the differential absorption, at various depths, of the incoming
solar radiation. If necessary, the feature of the differential absorption of
the incoming radiation can be easily incorporated into Eqs. (III. 1) and
(III. 3). However, it is felt that the inclusion of this additional feature will
not add significantly to the nature of the conclusions derived in the present
paper.
III. 3 Forms of the Eddy Diffusivity
It was pointed out earlier that Eq. (III. 1) is nonlinear, since the
eddy diffusivity, KH , is a function of the thermal as well as the current
structure in the lake. One of the primary objectives of the present study
is to demonstrate that the nonlinearity of Eq. (III. 1) is an essential feature
of the interaction between wind-induced turbulence and buoyancy gradients
due to surface heating, and that, as such, it must be retained if a satisfac-
tory theory for the thermal structure of a stratified lake is to be developed.
20
-------
The interaction may be accounted for either explicity, or implicitly as in
two-layer models in which the upper and lower parts of the lake are
described by different, but constant, eddy diffusivities. It should be
emphasized that the interaction between the turbulence and thermal struc-
tures is crucial in determining the structure of each. It is this interaction
that makes the problem of predicting the effects of heated discharges on the
thermal structure of stratified lakes difficult, since the heated discharges
not only influence the thermal structure directly, but they also influence it
indirectly through their effect on the turbulence structure.
The major mechanism by which turbulence is generated in the upper
layers of a lake is the wind shear acting on the surface of the lake. Corre-
spondingly, the buoyancy gradients are produced in the lake by the heat
exchange, at the surface of the lake, between the environment and the lake.
When the mean buoyancy field in the upper layers of a lake is statically
stable, it tends to suppress the generation of wind-induced turbulence.
Conversely, when the mean buoyancy field is statically unstable, it adds to
the generation of wind-induced turbulence. The effects of the interaction
between the turbulence and the buoyancy field on the structures of each other
have been studied fairly extensively in connection with atmospheric and
oceanic turbulence. Recent developments on the interaction between the
turbulence and buoyancy fields in the lower atmosphere have been summar-
33
ized by Lumley and Panofsky, and those in the upper ocean have been
summarized by Phillips and Okubo.
In general, the eddy diffusivity under arbitrary thermal stratification
conditions can be written as the product of the eddy diffusivity under corre-
sponding neutral stratification conditions and a function of an appropriate
stability parameter characterizing the stratification. Thus, one can write
^H = KH(j f (stability parameter) (III. 4)
where KHo is the eddy diffusivity under identical environmental condi-
tions, but in the absence of stratification.
21
-------
One of the more commonly used forms of the stability parameter is
the gradient Richardson number, K t , which is defined as
(HL5)
where & v is the coefficient of volumetric expansion of water, ^
is the acceleration due to gravity and IX is the horizontal component of
the current velocity. The denominator of the right-hand side of Eq. (III. 5)
represents the rate of production of turbulence by Reynolds' stresses, while
the numerator represents the rate of production or suppression of turbu-
lence by the mean buoyancy field. The Richardson number is positive for a
stable stratification, negative for an unstable stratification, and its absolute
value increases with increasing stratification.
It was mentioned earlier that Eq. (III. 1) is, in general, coupled to
the equation describing the velocity field. It can be seen now that the coup-
ling occurs through Eqs. (III. 4) and (III. 5). The only existing theory of the
thermocline which accounts for the interaction between turbulence and
stratification, through a coupling between the velocity and temperature
29
fields, is the theory of Munk and Anderson.
As the wind blows over the surface of a lake, the turbulence in the
upper layers of the lake are generated both by mean shear and by breaking
of the waves. As Phillips points out, while the momentum flux from the
air to the waves is only a small fraction of the momentum flux transferred
to the current, the energy flux to the waves is usually comparable to or
greater than the energy flux to the current. Several Russian workers such
64
as Dobrolonskii and Kitaigorodoskii (see Ichiye ) have indeed characterized
the turbulence in the upper layers in terms of the predominant amplitudes,
wavelengths, and periods of the surface waves.
Thus it is reasonable to suppose that the mechanical generation of
turbulence in the upper layers of a lake can be characterized by the surface
22
-------
'{
conditions alone without an explicit consideration of the current structure.
In other words, the Richardson number characterizing the interaction
between the mechanically generated turbulence and the thermal structure
can be taken, instead of that defined in Eq. (III. 5), as
6)
where ur - ^Jt s / q> ' is the friction velocity, Ts is the surface shear
stress induced by the wind and £ is the density of water. Forms of the
Richardson number similar to that given above have been used by Pritchard
in his analysis of the dispersion of contaminants in tidal estuaries and have
also been used recently by Kato and Phillips.
o o L o 1 L
In the literature, ' ' a number of different forms have been
used for the function of f in Eq. (III. 4). Most of these relations are
obtained by making various empirical assumptions; for example, some of
the relations are extensions of Prandtl's mixing -length theory to include
the effects of stratification. Of the many existing relations, two have been
chosen for the purposes of the present study as being typical ones. These
relations are
KH = KHo 0 +
-------
The first of the above relations was deduced by Rossby and Mont-
68 69
gomery and the second relation was originally proposed by Holzman.
Kent and Pritchard have tested Eqs. (III. 7) and (III. 8) with their obser-
vations in a coastal plane estuary, and their results seem to indicate that
Eq. (III. 7) fits the experimental data better than Eq. (III. 8). However,
within the scope of the accuracy of the measurements and the other unknowns
involved, it is not possible to establish whether or not Eq. (III. 7) is a more
suitable form than Eq. (III. 8).
In the present study both Eqs. (III. 7) and (III. 8) have been used, in
conjunction with Eq. (III. 6), to express the eddy diffusivity Kw in Eq.
(III. 1) in terms of the local temperature gradient.
III. 4 Turbulence in the Deeper Layers of a Lake
Equation (III. 6) describes the eddy diffusivity due to the turbulence
generated by surface wind stress, and will not be valid for the deeper
layers of a lake, since the mechanisms by which turbulence is generated
in the deep layers are considerably different from those by which turbulence
is generated in the upper layers. It has already been pointed out that the
regions below the thermocline are little influenced by changes in surface
conditions. It was also mentioned earlier that, while an explicit coupling
between Eq. (III. 1) and the current structure was not retained in the present
study, an implicit accounting has to be included.
When a lake is unstratified, the wind-induced current structure in it
will be as shown in Fig. 2(a), with the entire lake being in circulation.
However, when the lake is stratified, the wind-induced drift is confined to the
upper layers with the maximum return current occurring near the thermo-
cline as shown in Fig. 2(b). This characteristic current structure was first
pointed out by Wedderburn and has since then been verified by a number
72 ]2 1 A
of others. ' ' It immediately follows from the current structure that
there is no mechanism by which the wind stress at the surface can directly
create turbulence in the hypolimnion. This point needs to be emphasized
since, in spite of all evidence to the contrary, some authors have assumed
high values for the eddy diffusivity in the hypolimnion. The only mechanisms
24
-------
16
by which turbulence can be created in the hypolimnion are indirect ones,
such as degradation of internal waves and currents produced by internal
73 74
seiches, and water withdrawal. None of the above indirect mechanisms
is explicitly accounted for either in the present study'''or in any existing
theory of the thermocline.
McEwen has proposed a very simple method of demonstrating that
the eddy diffusivities in the hypolimnion are small and nearly invariant with
depth; he notes that the temperature distributions in the hypolimnia of most
lakes can be well approximated by the relation
T T^ = (\ e (III. 9)
where T^ is a constant, and the parameters ft and d, are independent
of depth, but could be functions of time. If the assumption is made that the
eddy diffusivity is constant in the hypolimnion, so that Eq. (III. 1) will be
valid with a constant value V in place of KH , then Eq. (III. 9) leads
to
^ - £t £
Qe. (ill. 10)
2
at
If measurements, taken at various times, of the temperature pro-
files in the hypolimnion of a lake are available, and if log ( T - Tw ) and log
- are plotted against the depth, Z , then it is clear, that within the
a ~t
scope of the approximations made, two parallel straight lines must result.
Conversely, the straightness and parallelism of the two lines together
validate the assumptions that the eddy diffusivity is a constant and that Eq.
(III. 9) represents the temperature distribution in the hypolimnion. The
value of >* can be determined from the intercepts of the two curves on
the Z 0 axis.
McEwen used this method to evaluate the values of the eddy diffu-
sivities in the hypolimnion of Lake Mendota, and the method has been used
in various lakes by other authors. ' All these authors find that the eddy
It should be noted, however, that in the present study the convective mix-
ing due to water withdrawal from the hypolimnion by the power plant has
been included explicitly. See Section V.
25
-------
diffusivity is constant over an extended region of the hypolimnion (except
L^>
close to the bottom of the lake), and that the values of }J are quite small,
being only a few times larger than molecular diffusivity.
It is clear that Eqs. (III. 7) and (III. 8), which predict that the value
of the eddy diffusivity will approach KHo at large depth, cannot be valid
in the hypolimnion, because the value of KHo is typically two or three
orders of magnitude greater than the molecular diffusivity. Moreover the
dominant processes, such as seiches, by which turbulence is produced in
the hypolimnion are not accounted for in the present study.
Since an explicit coupling between Eq. (III. 1) and the current struc-
ture has not been included in the present study, it is necessary to implicitly
account for the observed change in the current structure with the onset of
stratification and the different mechanisms of generation of turbulence in
the epilimnion and hypolimnion. Of course, there are several implicit
procedures possible, and a relatively simple procedure of accounting for
the change in the current structure has been described by Sundaram and
1(**
Rehm. In the method used by these authors, Eq. (III. 1) was assumed to
be valid over the entire lake with the eddy diffusivity being described by
Eq. (III. 7) or (III. 8), so long as the minimum value of the eddy diffusivity
Wl
was greater than some specified value, ^ . When the minimum value of
^>
the eddy diffusivity becomes equal to or less than ^ , Eq. (III. 7) or
(III. 8) was assumed to be valid down to the point of minimum diffusivity and
V*
the eddy diffusivity was assumed to remain constant at the value l) below
this point.
A somewhat more realistic procedure than the one described above
can be formulated if it is recognized that the need for different methods of
descriptions of the diffusivities in the upper and lower layers of the lake
arises after the formation of the thermocline. Thus before the time of for-
mation of the thermocline, when the temperature decreases monotonically
with increasing depth, Eq. (III. 7) or (III. 8) can be used to describe the eddy
diffusivity over the entire depth of the lake. After the thermocline forms,
Eq. (III. 7) or (III. 8) can be an adequate representation for the eddy
26
-------
diffusivity only in the epilimnion since the deeper layers are "protected"
from direct wind effects by the stratification. Thus an appropriate proce-
dure for the representation of the eddy diffusivity after the formation of the
thermocline (that is, after the maximum value of the temperature gradient
occurs at some point below the surface) is to assume that Eq. (III. 7) or
(III. 8) is valid only in the region above the level at which the eddy diffu
sivity attains a minimum value. The value of the eddy diffusivity in the
*
hypolimnion is taken to be equal to the minimum value of the eddy diffu-
sivity predicted by Eq. (III. 7) or (III. 8). This procedure, which yields a
value of the hypolimnetic diffusivity which decreases continuously with the
progress of the stratification, is more in accordance with observations '
than the time-independent value assumed by Sundaram and Rehm. All the
calculations on stratification described in the present report are based on
the above implicit procedure for accounting for the change in the current
structure with the onset of stratification.
III. 5 Numerical Integration of the Basic Equations
As pointed out earlier, the analysis of the thermocline can be
divided for convenience into two phases, that of formation and that of the
continuous downward migration of the thermocline into the deeper layers of
the waterbody. Both of these phases are embodied in the solution of Eq.
(III. 1) with the proper boundary conditions and an appropriately selected
form of the thermal diffusivity. Quite general conditions relating the sur-
face temperature and the heat flux have been given in Eqs. (III. 3) and (III. 4).
However, to derive an understanding of the nonlinear and unsteady aspects
of the interaction under simplest conditions, solutions for steady boundary
conditions were obtained first. The formation process and the erosion of
the thermocline into the hypolimnion were obtained with a constant tempera-
ture T1 imposed at the surface or with a constant heat flux <^ n imposed
* It may be noted that the level of the plane of minimum eddy diffusivity is
slightly different from that of the thermocline. However, this slight dif-
ference is of no consequence in the approximate implicit procedure being
considered here.
27
-------
at the surface. With the understanding gained from these examples, inter-
pretation of solutions with the more general boundary conditions were
simplified.
It has been emphasized earlier that the nonlinearity of Eq. (III. 1),
which arises due to the coupling between the diffusivity and the temperature
structure itself, is an essential feature and cannot be neglected. To inves-
tigate the effects of nonlinearity, Eq. (III. 1) was solved by a perturbation
technique for the case when the effects of the nonlinearity are small. When
the Richardson number is small, so that conditions are near neutral sta-
bility, the eddy diffusivity can be expanded (around its value for neutral
stability) in increasing powers of RL . The details of the perturbation
procedure and the solution are described in Appendix A. The solution
clearly displays a distortion of the temperature profile due to nonlinear
effects. The reduction of the thermal diffusivity due to the increasing sta-
bility during the heating process tends to produce a thermocline, or an
inflection point, in the temperature profiles. The solution is valid only at
early times when the effects of the nonlinearity are small, and becomes
invalid at later times when nonlinear effects become dominant.
Sundaram and Rehm have investigated the effects of nonlinearity,
for a rather general form of the eddy diffusivity, by using dimensional
analysis. They found that both for the case of constant surface temperature
and for the case of constant surface heat flux, a characteristic length scale
and a characteristic time scale can be defined. The characteristic length
scale determines the depth at which a thermocline first forms, and the
characteristic time scale determines the time at which the thermocline
forms.
7^i
Sundaram and Rehm have also reported numerical solutions, in a
nondimensional form, of Eq. (III. 1) for the forms of the eddy diffusivity
given in Eqs. (III. 7) and (III. 8). The solutions display a nearly linear
behavior at small times, but with increasing departures from the classical
linear solutions at later times as the effects of nonlinearity become impor-
tant. The solutions display clearly the formation of the thermocline and its
28
-------
subsequent migration into the deeper layers of the lake. The character-
istics of the solutions have been discussed in detail by Sundaram and Rehm,
and they need not be repeated here. Here, some additional solutions will
be described which include some improved features which are not included in
the solutions reported by the above authors.
The solutions reported here differ from those reported by Sundaram
and Rehm in three specific features. First, as mentioned earlier, the
manner of treating the coupling between the current and thermal structures
is different in this report from the one used in Ref. 76. Second, in the
present solutions the variations in the volumetric coefficient of expansion
Oiv of water, which were not included in Ref. 76, are included. (The
changes in OLV during the early parts of the stratification process can
have a significant influence on the details of stratification. ) Third, the
boundary condition that there be no heat flux through the bottom boundary
(that is, the bottom mud) has been used in the present calculations.
Figures 3-5 show the results of a numerical integration for the form
of the thermal diffusivity given in Eq. (III. 7) when the temperature is main-
>\<
tained at a constant value. Note that unlike the results presented in Ref. 76,
which were in nondimensional units, the results in Figs. 3-5 are presented
in terms of dimensional quantities. The calculations are for the case in
which the waterbody, whose depth is taken to be two-hundred feet, is
o
assumed initially homothermal throughout at a temperature of about 3 C.
At a given instant of time the temperature of the surface is suddenly raised
o
to, and held constant at, about IOC. The time evolution of the vertical
distributions of the relevant quantities are then studied.
Figure 3 displays the temperature as a function of depth below the
surface for various times. If the eddy diffusivity, KH , in Eq. (III. 1)
were assumed to be a constant, then all the temperature profiles in Fig. 3
would be self-similar. In particular, the temperature gradient would always
decrease monotonically with increasing depth. The first and second profiles
shown in Fig. 3, which respectively correspond to conditions one day and
three days after the impulsive increase in surface temperature, do indeed
* The value of the semiempirical parameter (j^ , which is needed for
performing the calculations, is discussed in Appendix E.
29
-------
display temperature profiles which are nearly self-similar. It may be
noted that during these early times conditions in the upper layers of the
waterbody will actually be unstable, since the volumetric coefficient of
water is negative below 4°C. In other words, for these conditions heating
of the surface actually leads to an augmentation of the wind-induced turbu-
lence in the upper layers of the waterbody so that the eddy diffusivities in
these layers will be larger than those for neutral conditions. This phenom-
V A
enon, which was not considered by Sundaram and Rehm, is discussed in
detail in Section IV.
By the third time (corresponding to ten days) shown in Fig. 3, the
temperature throughout the waterbody is above 4 C and the stratification is
stable everywhere. The nonlinear behavior is clearly evident in this plot
which displays a point of inflection or the thermocline. The formation of
the thermocline can also be seen clearly in Fig. 4 which shows a plot of the
depth of the thermocline (that is, the depth at which the maximum tempera-
ture gradient occurs) against time. It can be seen from Fig. 5 that the eddy
diffusivity is drastically reduced near the thermocline level.
The last two plots in Fig. 3, as well as Figs. 4 and 5, show the
slow, continual progression of the thermocline to larger depths. Figures 3
and 4 clearly indicate the fact that the depth of the epilimnion increases
even when surface conditions are held steady. It can be seen from Fig. 5
that the position at which the minimum diffusivity occurs (that is, the posi-
tion at which the downward heat transfer is most inhibited), also propagates to
greater depth. Also, the minimum value of the diffusivity decreases with
increasing time.
Qualitatively similar results are also obtained for the case when the
heat flux at the surface is suddenly increased and then held constant.
III. 6 Concluding Remarks
Some results are provided on the mechanisms by which thermoclines
are formed and maintained in stratified lakes. It was pointed out that the
thermocline is formed by the nonlinear interaction between the wind-induced
turbulence and the buoyancy gradients created by surface heating. While
30
-------
the nonlinear aspect of the interaction makes the problem difficult, the
nonlinearity is an essential feature, and as such, has to be retained.
The influence on the eddy diffusivities of the interaction between
the turbulence structure and the buoyancy gradients was included by using
the techniques that have been successful in the study of atmospheric turbu-
lence. That is, the eddy diffusivities were assumed to be given by the
product of the eddy diffusivities under conditions of neutral stability and an
appropriate function of a stratification parameter such as the gradient
Richardson number. The nonlinear equations of the problem were then
solved using an electronic computer.
It was demonstrated that, while the vertical temperature profile at
the start of the stratification season in a temperate lake is fairly smooth, a
sharp interface (the thermocline) develops because of the nonlinear inter-
action between the turbulence and the temperature structure. It was also
shown that the continuous downward erosion of the thermocline into the
deeper layers of the lake is a necessary part of its sustenance. The above
conclusions were confirmed by considering the simple case of a semi-infinite
region, initially at a uniform temperature, which is subjected to a sudden
increase in surface temperature. In this case, the temperature distributions
were initially similar to the error-function distribution for the linear case,
but nonlinear effects soon came into play, and a thermocline was formed
some distance below the surface. The thermocline also propagated steadily
away from the surface.
The qualitative behavior of the solutions obtained here is in agree-
ment with the experimental observations of Turner and Kraus and Kato
and Phillips. The present solutions also confirm the features observed
y/
earlier by Sundaram and Rehm.
31
-------
IV. THE STRATIFICATION CYCLE
IV. 1 Effect of Changes in Surface Conditions on Thermocline Behavior
It was demonstrated in Section III that the mechanism responsible
for the formation and maintenance of thermoclines in deep, temperate
bodies of water is the nonlinear interaction between the wind-induced tur-
bulence and the stable buoyancy gradients created by the heating of the
surface layers by insulation. It was also shown that the thermocline
propagates steadily away from the surface even when the surface tempera-
ture and the wind stirring are kept constant. That is, the unsteady behavior
of the thermocline is inherent to the mechanism responsible for its for-
mation and maintenance. It is now relevant to consider the effects on
thermocline behavior of variations in the environmental conditions above
the waterbody.
As a first step in studying the effects of varying surface conditions,
a solution of Eqs. (III. 1), (III. 6) and (III. 7) was carried out for an imposed
sinusoidal variation in surface conditions. The surface temperature was
assumed to be well represented by the expression
Ts - TSuv + AT3 stn. cot (IV. 1)
where Ts^ and ATS are the average value and the amplitude of the
variations. The quantity w has units of reciprocal time and for an
annual cycle is equal to -Tjjy days
For the case of an imposed constant surface temperature, Sundaram
and Rehm have shown that a characteristic time for the formation of the
thermocline can be formed from the difference 7", T0 between the
imposed and initial temperatures, the eddy diffusivity KHo and the quan-
tit>r icr*2 For the present case, the characteristic time for thermo-
cline formation can be written as
r = ~^~^7 (iv. 2)
32
-------
r~~
Clearly the nondimensional quantity u; T , which represents the ratio of
the time for formation of the thermocline to the characteristic time for
variation of the imposed surface temperature, will have an important influ-
ence on the behavior of the thermocline. For example when this quantity
is vanishingly small, so that the time for thermocline formation is negligible
compared to the characteristic time for the changes in surface conditions,
then a thermocline will form immediately following spring homothermy.
As mentioned in Section II, the assumption that the above condition is valid
38
is implicit in the theory proposed by Kraus and Turner. According to
the theory proposed by these authors, a thermocline and a completely homo-
thermal epilimnion form immediately following spring homothermy.
However, as pointed out earlier, most temperate lakes do not exhibit a
thermocline, or a homothermal epilimnion until after several weeks
beyond spring homothermy. In other words, in most lakes, the approxima-
^*s
tion that UJ T 0 cannot lead to a proper representation of the thermal
structure during the early parts of the stratification cycle.
The results for the time variations of the vertical temperature dis-
tributions for the early part of the stratification cycle, obtained from a
solution of Eqs. (III. 1), (III. 6), (III. 7) and (IV. 1), are shown in Fig. 6. It
can be seen that as in the case for the steady surface temperature, the
temperature profiles are smooth at the initial times with a thermocline
forming only at later times. It can also be seen from Fig. 6 that the
thermocline propagates steadily downward into the deeper layers of the lake.
When considering the entire stratification cycle, it should be empha-
sized that the surface temperature is not specified a priori and that it has
to be calculated as a part of the solution from the given environmental con-
ditions above the lake. The primary purpose of the results presented in
Fig. 6 is to demonstrate that the qualitative features of the formation and
maintenance of a thermocline are the same for a gradually applied increase
in surface temperature as they are for a suddenly applied increase in sur-
face temperature or heat flux. Thus while changes in surface conditions
undoubtedly influence the formation and maintenance of thermoclines, these
33
-------
by themselves are not the primary reasons for the characteristic thermal
structures of temperate lakes.
IV. 2 Seasonal Variations in Environmental Conditions above the Lake
It has been pointed out above that the seasonal variation of the sur-
face temperature of a lake is not known, a priori, and that it has to be
determined from the known variations in the environmental conditions above
the lake. Thus when solving Eqs. (III. 1), (III. 6) and (III. 7) for the entire
stratification cycle, the necessary boundary conditions at the surface have
to be specified in terms of the known variations in the appropriate environ-
mental conditions. It is intuitively obvious that one of the boundary condi-
tions has to express the thermal energy exchange between the lake and the
environment above it. It has also been long recognized by limnologists that
the transfer of mechanical energy from the wind to the upper layers of the
lake plays a very important role in determining the thermal structure. '
Birge was the first to compute the effect of the work done by the wind in
distributing the heat that is received bythe surface layers of the lake into
the deeper layers. In the present section the boundary conditions express-
ing the thermal and mechanical energy transfers to the surface layers of a
lake are discussed.
The heat exchange across the surface of a body of water depends on
a number of complex factors. A detailed discussion of the methods of com-
puting the contributions to the heat exchange of a number of different
mechanisms has been given by Edinger and Geyer and Sundaram et al,
and it need not be repeated here. However, it is necessary to note that the
heat exchange at the surface can be represented surprisingly accurately by
the simple expression,
^s = K CT£ - Ts)
Both the equilibrium temperature, TE , and the heat-exchange coefficient,
K , are functions of the environmental conditions. Methods of deter-
mining T£ and K from the environmental conditions have been
34
-------
described fully by Edinger and Geyer and Sundaram et al. The latter
authors have also given various charts from which the values of IE
K can be readily read off.
The physical significance of the equilibrium temperature is that it
is the fictitious value of the surface temperature at which there will be no
heat exchange between the lake and the surrounding. The equilibrium tem-
perature is also the value to which the surface temperature of the lake would
tend to, if the lake adjusted instantaneously to changes in environmental
conditions. Thus the concept of equilibrium temperature is an extremely
important one, and its significance will become apparent from the discus-
sions to be presented in Sections IV. 3 and IV. 5.
The annual variation of the equilibrium temperature over most tem-
perate lakes can be represented by the simple sinusoidal relation,
6T£ SLn(iot + (p) (IV. 4)
where T£ is the average value of the equilibrium temperature over one
annual cycle, under neutral conditions as well as the
Richardson number (through the friction velocity \jJ ). Semi-empirical
relations between uJ and KH and the wind conditions above the lake
35
-------
are available and can be used to relate these quantities. For example,
Munk and Anderson29 have given a suitable semi-empirical relation between
these quantities.
In the present study, the friction velocity has been allowed to be a
cyclic function of time of the form,
ur* = B, + Bz si*. ( u>t + yr ) (IV. 5)
where B< , Ba and Iff are constants to be determined from the known
conditions above the lake, and i*J is the angular frequency as before. A
similar relation is assumed for the variation of KHo also.
IV. 3 Qualitative Considerations on the Seasonal Temperature Cycle
It has already been pointed out that the two important external con-
ditions that have to be specified for a determination of the thermal structure
of a lake are the degree of surface heating and the degree of wind mixing.
The effects of these two conditions on the lake are closely coupled, of
course, since the heat that can be received by the surface layers is
dependent on the depth to which wind action can mix this heat, and con-
versely, the depth of wind mixing is itself a function of the degree of surface
heating. The effect of wind mixing on the seasonal temperature cycle of a
lake can be interpreted in terms of the changes in the values of either the
Monin-Obukhov length or the over-all stability of the lake. Both of these
interpretations have been discussed in detail by Sundaram et al. In the
present section, some qualitative considerations on the effects of the varia-
tions in the equilibrium temperature on the seasonal temperature cycle will
be given.
As mentioned earlier, the variation, over an annual cycle, of the
equilibrium temperature over most lakes can be represented well by a
sinusoidal function of the form given in Eq. (IV. 4). A schematic represen-
tation of this variation over one annual cycle, starting from the time when
the heat content of the lake is a minimum, is shown in Fig. 7.v A schematic
representation of the features of the variations in the surface temperature
* Some of the features noted on this figure have also been noted independently
by Prof. K. B. Cady of Cornell University - private communication.
36
-------
and the deep-water temperature that are observed ' ' ' in temperate
lakes is also given in this figure. During winter, when the heat content of
the lake is a minimum, the lake will also be completely homothermal due
to the low over-all stability during this period. Since the heat flux at the
surface of the lake has to change sign during the time of minimum heat con-
tent, it is clear from Eq. (IV- 3) that the surface temperature, the deep-
water temperature and the equilibrium temperature will all coincide at this
time.
During the spring months, the equilibrium temperature increases
rapidly due to increasing insolation. While the surface temperature
also increases during this period, it increases much more slowly than the
equilibrium temperature, since the temperature of the surface layers
depends not only on the rate of heating of these layers but also on the rate
at which heat is removed, by turbulent mixing, from these layers to the
deeper layers of the lake. During the early period, the lake remains nearly
homothermal since, due to the low over-all stability, wind action is able to
mix the surface heat into great depths. However, as the heating continues
a thermocline forms, with the time of its formation usually coinciding with
the time when the heat flux through the surface is a maximum. As the
thermocline forms, the surface temperature begins to increase rapidly,
since now the upper layers are heated preferentially in relation to the deeper
layers (whose temperature increases only slightly).
After the formation of stratification, the surface temperature usually
increases at a more rapid rate than the equilibrium temperature so that the
heat flux into the lake begins to decrease. As the surface temperature con-
tinues to increase and the volume of the epilimnion increases, the lake
begins to approach its maximum heat content. However in many lakes, the
surface temperature begins to decrease before the maximum heat content is
*
reached because of a decrease in over-all stability of the lake and the
''" The over-all stability of the lake is not to be confused with the static sta-
bility of the stratification. This point is discussed in detail by Sundaram et al.
37
-------
12
attendant rapid descent of the thermocline. As pointed out by Hutchinson,
the above behavior is exhibited by the Lunzer Untersee. It has also been
found by Sundaram et al. to be characteristic of Cayuga Lake.
The lake attains its maximum heat content as the equilibrium and
surface temperatures once again coincide. Beyond this point the equilib-
rium temperature falls below the surface temperature, and the lake begins
to lose heat. During this period, the stratification in the upper layers of
the lake is statically unstable and the thermocline continues to descend
rapidly into the deeper layers of the lake. When, in late fall, the thermo-
cline descends to the bottom of the lake, the lake again attains homothermy
and cools uniformly while losing heat. The minimum heat content (and the
end of the cycle) is reached when the equilibrium temperature once more
equals the surface temperature. The cycle is then repeated.
It is clear from the above discussion that the concept of the equilib-
rium temperature is an extremely useful one for studying the characteristic
features of the stratification cycle of temperate lakes. The physical signi-
ficance of the equilibrium temperature is that the redistribution of heat
within a lake always tends to be in such a manner as to drive the surface
temperature towards the equilibrium temperature. The relationship between
the equilibrium temperature and the surface temperature is best illustrated
by considering the linear form of Eq. (HI. 1) in which the eddy diffusivity is
assumed to be constant and equal to KH(> . The solution to the linear
equation, subject to the boundary conditions given by Eqs. (IV. 3) and (IV. 4)
has been discussed in detail by Sundaram et al. They found that the rela-
tionship between the surface temperature and the equilibrium temperature
was governed by the nondimensional ratio of the rate of transfer of heat
from the environment to the lake and the rate of transfer of heat from the
surface layers to the deeper layers by turbulent diffusion. When this ratio
is a large quantity, the surface temperature tends to follow the equilibrium
temperature closely. In other words, for this case the conditions in the
surface layers of the lake adjust immediately to changes in surface condi-
tions. Conversely, when the above ratio is very small, the surface
38
-------
temperature changes very little and conditions in the lake do not respond
to changes in environmental conditions.
The qualitative considerations given above for the solution of the
linearized problem will remain valid for the nonlinear case also. The
primary purpose of the discussion given above is to emphasize that a suc-
cessful model for the stratification cycle of a temperate lake should be
capable of predicting the above characteristics.
IV. 4 Free Convection
The qualitative features of stratification that are displayed by most
temperate lakes have been described above. Before proceeding to verify
whether these features are predicted by the model being proposed here, one
other point needs to be considered. This point is concerned with the change
in character of the turbulence in the upper layers of the lake with the onset
of the cooling season. Thus, during the cooling period, the representation
given in Eq. (III. 7) for the eddy diffusivity is not an adequate one and has to
be modified to take into account the change in the character of the turbulence,
During late summer, as the lake begins to lose heat to the environ-
ment, the stratification in the upper layers of the lake becomes statically
unstable. In other words, in these layers the turbulence is augmented,
rather than suppressed, by the buoyancy gradients. When the rate of cool-
ing is relatively small, as in the early parts of the cooling season, the
degree of augmentation of the wind-induced mechanical turbulence by the
convective turbulence is still governed by the Richardson number. However
when the rate of cooling becomes large, as in the later parts of the cooling
period, the Richardson number no longer constitutes a meaningful param-
eter, since now the dominant transport mechanism in the upper layers is the
convective turbulence and wind-induced turbulence has very little effect on
the mixing processes. This does not mean that the quantity of mechanical
energy transferred to the water by wind shear is small compared with the
quantity of convective energy. Indeed, free convection is initiated when the
33
above two energies are approximately equal. As Lumley and Panofsky
point out, the structures of mechanical and convective turbulences are quite
39
-------
different, the latter being a far more efficient transporting agent. The
mechanical eddies are usually quite small while the eddies produced by con-
vection are relatively large, their size being of the order of the thickness
of the unstable layer. Thus the latter provide larger correlations between
the fluctuating quantities, and hence larger transport, than the former.
Formulae appropriate to free convection have been developed by
o o £ o
various authors, ' and these can be used to describe the transport
processes during the cooling period. However, it should be pointed out that
during the cooling parts of the stratification cycle both stable and unstable
conditions will exist simultaneously, though at different levels, in the lake.
Thus while the upper layers will be statically unstable, the lower layers,
including the thermocline region, will exhibit a stable stratification. There-
fore the appropriate procedure for representing the transport processes
during the cooling period would be to use a suitable free-convection formula
to represent the upper layers of the lake and to match this to Eq. (III. 7) at
the level separating the unstable and stable regions. Clearly, this proce-
dure will be quite complex, and it has not been adopted in the present study.
Instead a relatively simple procedure for representing free convection has
been chosen.
It can be seen from Eq. (III. 7) that, when the thermal stratification
in the upper layers of a lake is unstable, the maximum value of the eddy
diffusivity will occur at some depth below the surface. This depth repre-
sents the depth at which the rate of generation of turbulent energy by buoy-
ancy forces is a maximum. In other words, convective eddies of the size
of this critical depth provide the largest correlations between the fluctuating
quantities. In the present study, whenever the stratification in the upper
layers is unstable, the eddy diffusivity is assumed to be well represented
by Eq. (III. 7) below the critical depth, while above this depth the eddy dif-
fusivity is assumed to be constant everywhere and equal to the value at the
critical depth. That is, the region above the critical depth is assumed to
be kept well mixed by the convective turbulence.
While the above procedure for representing free convection is
admittedly crude, it has been found to give a fairly accurate representation
in practice. Other more sophisticated methods are, of course, possible.
40
-------
For example, the convection velocity due to surface cooling can be included
explicitly into Eq. (III. 1). However since, in the present study, the coup-
ling between the velocity and thermal structures is included only implicitly,
the effect of the free-convection velocity can also be included only implicitly.
Finally, it should be pointed out that in a temperate lake free con-
vection can occur not only during the cooling season, but also during the
early parts of the warming season when the temperature of the lake is below
4°C. The occurrence of free convection during the early parts of the
warming season is automatically accounted for in the present study by
including the effects of changes of temperature on the volumetric coefficient
of expansion, CLV , of water.
IV. 5 Numerical Results and DiscXission
When all the features described above are incorporated into the
analytical model, calculations for the entire stratification cycle can be car-
ried out. In the present program, calculations for complete annual cycles
were carried out for a limited number of values of the various input para-
meters. The required input parameters have already been described in
detail. These are:
i) The variation of the equilibrium temperature over the body of
water under consideration.
The quantities JE , 8 J£ and (£> in Eq. (IV. 4) have to be
specified. The quantities depend on the environmental conditions
above the waterbody, and methods of their determination have
been discussed in detail by Sundaram et al.
ii) The heat exchange coefficient, K .
The heat-exchange coefficient is also a function of the environ-
mental conditions above the water, and the method for its
determination has also been described by Sundaram et al.
41
-------
iii) The annual variation of the average wind speeds above the
waterbody.
The known wind conditions above the waterbody can be used to deter
mine the friction velocity, U/* , and the eddy diffusivity,
KH , for neutral conditions by utilizing an experimentally
HO 29
determined relation such as the one given by Munk and Andersen.
'p
The above three conditions are the only primary input conditions
required for the present analytical model. It should be noted that all three
of these input conditions are concerned with the environmental conditions
above the lake. Specifically, no knowledge of the temperature profiles
within the lake, or eddy diffusivities derived from them, are needed. Some
a priori knowledge of the thermal structure of the waterbody is a prerequi-
39 40
site in most of the existing models for the stratification cycle. '
The starting point for the calculations is from the time of the mini-
mum surface temperature of the lake, and at this time the lake is assumed
to be homothermal. However, it should be pointed out that the minimum
temperature of the lake and the time at which the minimum temperature
occurs are, a priori, not known. Nevertheless, because of the cyclic
nature of the imposed boundary conditions, the solution will ultimately tend to
a cyclic one (regardless of the initial conditions) provided that the computa-
tion is carried out over several cycles. On the other hand, if the initial
conditions are chosen appropriately, the solution can be expected to return
to the same conditions after just one cycle. Thus when information on the
temperature of the lake is available for some period during spring homo-
thermy, the calculations can be greatly simplified. This point will be
illustrated later with a specific example.
In the present study a number of calculations, corresponding to dif-
ferent input conditions were carried out. The details of the numerical
5C
In addition to these conditions, a knowledge of the variation of the cross
sectional area of the waterbody with depth is also needed. This point will
be discussed in detail later.
42
-------
computational procedure are described in Appendix C. The computer pro-
gram used had provisions for generating, automatically, plots of several
key variables of interest. For each case, seven such computer-generated
plots were produced. Three of these plots were concerned with the seasonal
variations of the temperatures at different levels, the surface heat flux and
the depth of the thermocline. The remaining four plots were for the vertical
distributions, at various times during the stratification cycle, of temperature,
heat flux, temperature gradient and eddy diffusivity.
Figures 8-14 show the above plots for a typical calculation. For this
calculation the following input conditions were used:
T£ = 11 + 1fo sin(--t + 0.531) /C
K = 180 BTU/ i± -A*} - °C
LJ* = 0.7 + 0.025 sin ( -^-t
31*5
*9 TV
M = 800+200 Sirt( t + Z.lel) , ft* /day
In addition to the above input conditions, the depth of the lake was
assumed to be 200 ft and the minimum temperature during spring homo-
thermy was assumed to be equal to 2. 9°C. All the conditions given above
correspond to conditions in Cayuga Lake, New York, which was chosen since
extensive data on the thermal structure of this lake are available.
The calculated seasonal variations in the thermal structure is shown
in Fig. 8 -which shows five temperatures, the equilibrium temperature
(which is an input condition for the calculations), the surface temperature
and the temperatures at depths of 50, 100 and 150 feet, as functions of time
for approximately 1-1/4 cycles. As mentioned earlier, the imposed equi-
librium temperature variation is sinusoidal with a mean of 1 1 ° C and an
43
-------
amplitude of 16° C. A solution for and a discussion of the response when
the problem is linearized (i. e. , when the thermal diffusivity KH is taken
as constant) has been given in Section VII 5. 3 of Reference 16. According
to this solution, for fixed values of the thermal diffusivity and of the heat
transfer rate at the surface, the temperature variation at any depth would
have a smaller amplitude and a phase shift with respect to the imposed
equilibrium temperature variation; the amplitude would decrease and the
phase shift increase with increasing depth. Figure 8 clearly displays the
nonlinear behavior resulting from the interaction of the turbulence with the
buoyancy field. Stratification in the lake can be seen to occur at around 60
days where the temperature plots at the greatest depths display severe
deviation from the sinusoidal behavior predicted from a linearized model.
As the thermocline descends, between 180 and 270 days, the temperature
plots at depths below the surface again follow a sinusoidal curve after the
thermocline has swept past that depth.
Some of the above features can be seen more clearly in Fig. 9 which
displays temperature as a function of depth for various times during the
annual cycle. Note that in all of the figures temperature is plotted in
degrees centigrade, depth in feet and time in days. The lake was assumed
to be homothermal at 2. 9°C initially, corresponding approximately to condi-
tions in March. The first plot, at 30 days, displays a nearly homothermal
profile at about 4°C resulting from the simultaneous heating at the surface
and the complete wind mixing of the lake. The next plot, at 90 days and
corresponding approximately to conditions which would be expected in June,
clearly displays the thermocline. It may be noted that the qualitative fea-
tures of the formation of the thermocline are analogous to those shown in
Fig. 6 for a sinusoidally varying surface temperature. The third plot (at
180 days) in Fig. 9 displays the thermal structure of the lake as it starts
into the cooling portion of the cycle. In this plot the thermocline is some-
what deeper than at 90 days and is much sharper due to the convective mixing
which now supplements the wind-induced mixing in the surface (nearly
homothermal) layer. The next three plots show the effects of progressive
cooling and the enhanced mixing due to cooling upon the temperature structure.
44
-------
At 240 days (corresponding approximately to November) the thermocline
has descended beyond 100 feet showing a completely mixed upper layer and
a characteristic temperature decay in the hypolimnion. At 300 days the
mixed layer extends to the bottom of the lake (200 feet) and a slight tempera-
ture inversion exists. At 360 days the lake is once again homothermal at
about 3°C. Two additional plots, at 420 and 500 days, display the heating
part of a new cycle with the stratification again forming.
Figure 10 is a plot of the heat flux measured in Btu/ft day at the
surface as a function of time. The variation in this quantity deviates
slightly from sinusoidal. Note that the heat flux changes sign (from the
heating portion of the cycle to the cooling portion of the cycle) around 180
days. Well before this time, stratification, with the formation of the
thermocline, has occurred. After this time, the wind-induced mixing in the
epilimnion is supplemented by convective mixing due to cooling, and the
depth of the thermocline rapidly increases. The corresponding vertical
distributions of the local heat fluxes are shown in Fig. 11. It can be seen
that at 30 days and at 90 days the maximum heat flux occurs at the surface
and decreases monotonically with increasing depth. However, beyond
summer solstice, the intermediate layers of the lake are heated relatively
more rapidly than the surface layers. Thus at 180 days, the surface heat
flux is zero while the downward heat flux at the layers close to the thermo-
cline is relatively large. At 240 days the upper parts of the lake are cooling
rapidly, but the region below about one-hundred feet continues to gain heat,
the maximum downward heat flux again occurring close to the level of the
thermocline. On the other hand, at 300 days, after the disappearance of
stratification, the entire lake is losing heat to the atmosphere.
The vertical variation of the eddy diffusivity for various times during
the stratification cycle are shown in Fig. 12. At 30 days, before the forma-
tion of the thermocline, the eddy diffusivity remains nearly invariant with
depth. However as the thermocline forms, there is a drastic reduction in
the values of the diffusivity in the lower layers as can be seen from the plots
for 90, 180 and 240 days. In the plot for 240 days, the initiation of free
convection (with the attendant increase in the value of the eddy diffusivity)
45
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can be seen in the upper layers. By 300 days, a large part of the lake is
dominated by free convection. At 360 days, the lake is again homothermal
and the eddy diffusivity is nearly invariant with depth. The last two plots
in Fig. 12, which are for the early parts of the second cycle, display the
same general features as the plots for the first cycle.
Figure 13 is a plot of the vertical variation of the temperature gra-
dients for the various times considered in Figs. 9, 11 and 12. It can be
seen that at all times, the upper layers of the lake are nearly homothermal.
On the other hand, relatively larger temperature gradients occur in the
hypolimnion. In other words, the hypolimnion is always considerably less
homothermal than the epilimnion.
Figure 14 is a plot of the depth of the thermocline versus time. The
thermocline is seen to form at about 60 days. The depth of the thermocline
is then seen to decrease to a minimum value of about 35 feet between 120
and 150 days. As noted above the depth of the thermocline increases
rapidly beyond 180 days when the cooling part of the cycle begins. The
thermocline reaches the bottom, and the lake attains homothermy, at about
270 days so that the length of the stratification period is about 210 days.
Clearly, the above results display all the observed characteristic
features of the stratification cycle of temperate lakes that were discussed
earlier. For example, in most temperate lakes the surface temperature
begins to decrease before the lake has reached its period of maximum heat
content. It can be seen clearly from Fig. 8 that the maximum surface
temperature of the lake is reached at about 150 days after the start of the
calculations, while the maximum heat content of the lake does not occur till
after thirty days later. The occurrence of this phenomenon in Cayuga Lake
and the reasons for it have been discussed by Sundaram et al. Again, in
accordance with observations, the present model predicts the formation of
the thermocline some time after the period of maximum spring homothermy
(with an intermediate period of monotonically varying temperature distri-
butions), but before the lake attains its maximum heat content and begins to
Q O
lose heat to the atmosphere. On the other hand, some theories predict
46
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the formation of the thermocline immediately following maximum spring
homothermy, and others predict the formation of a well-mixed upper
layer only after the start of the cooling season.
IV. 6 Cyclic Behavior of the Results
It has already been pointed out that the minimum temperature of the
lake during spring homothermy is, in general, not necessary for a solution
of the problem because of the cyclic nature of the solution. Thus, while an
unique starting point for the calculations does not exist, the solution will
always tend to a cyclic one, regardless of the starting conditions, provided
the solution is carried out over several cycles. In the example considered
in Section IV. 5, the calculation was started with the assumption that the
minimum temperature during spring homothermy was 2. 9°C since this is
known to be the case from actual observations in Cayuga Lake. Thus, for
this example, it can be seen from Fig. 8 that the solution exhibits a cyclic
behavior after just one cycle (with the temperature at the end of one cycle
returning to the initial temperature), because of the proper choice of the
initial condition. On the other hand, if the calculations had been started at
some arbitrary temperature, say 4°C, then the solution will not become
cyclic after the first cycle and the temperature at the end of one cycle will
be different from 4 C (and also 2. 9 C). However, the temperature at the
end of several cycles will automatically approach the correct temperature
of 2. 9°C.
The above cyclic behavior of the solution has important implications
for the practical applications of the theoretical model developed in the
present study. As pointed out earlier, the necessary input conditions for
the present analysis of the stratification cycle are the annual variations of
the equilibrium temperature, the heat exchange coefficient and the wind
speed above the lake. These quantities are the logical "external parameters'
for specifying the thermal structure of the lake since these determine the
exchange of thermal and mechanical energies between the lake and the
atmosphere. Indeed, concepts such as the Monin-Obukhov length scale
are based on "external parameter" considerations. However, in many
practical examples, a complete knowledge of the "external parameters" is
47
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not available. For example, the equilibrium temperature over a waterbody
depends on several complex factors such as the incoming (sky and solar)
radiation, the temperature and humidity of the air over the waterbody and
the wind speed. Complete and accurate information on all the above vari-
ables is often difficult to obtain.
The nonavailability of complete information on all of the external
parameters over a lake does not mean that information on the thermal struc-
ture of the lake cannot be generated using the present theory. In fact,
because of the requirement of cyclicity, the requirement of a complete
specification of the "external parameters" can be relaxed if other compatible
information on the thermal structure is available. For example, if some
information on surface-temperature variation(such as the minimum tempera-
ture at spring homothermy and the maximum summer temperature) is
available, then this information can be used instead of the equilibrium
temperature.
Consider, the instance, the example given in Section IV- 5. Suppose
that for the case considered, the minimum temperature at spring homothermy
and the maximum summer temperature are known from measurements, but
that no accurate information on the equilibrium temperature variation is
available. Information on the other input quantities is assumed to be known.
For such a case, all required information on the stratification cycle can be
obtained using the present theory with a trial-and-error procedure. Figure
15 shows a calculation for the stratification cycle for the same condition
given in Section IV. 5 except that now the equilibrium-temperature variation
is taken to be given by
= 74 + 13 sin. ( ± -1.03)*
34,5 '
';
Note that the phase difference, (j) , has always to be chosen in such a
manner that at t = 0, the equilibrium temperature equals the known
initial temperature at spring homothermy.
48
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The calculation is started at the known initial temperature at spring homo-
thermy of 2. 9 C, and the results for the first 420 days are shown in Fig. 15.
It can be seen from Fig. 15 that the solution does not display a cyclic
behavior over the first year, and that the conditions after one year are con-
siderably different from the starting conditions. Indeed, homothermy is not
achieved till after 390 days and the temperature at homothermy is about 6 C.
Thus, the assumed equilibrium temperature variation is clearly incompatible
with the lake attaining homothermy at 2. 9 C. On the other hand, it can be
seen from Fig. 15 that the maximum predicted summer surface temperature
for this case is not significantly different from that for the previous case.
In general it can be stated that, all other things being equal, the
temperature at spring homothermy depends strongly on the minimum value
of the equilibrium temperature and that the maximum surface temperature
depends strongly on the maximum value of the equilibrium temperature. In
the two examples considered above the maximum values of the equilibrium
temperatures are the same and hence the maximum values of the surface
temperature are not significantly different. On the other hand, the minimum
values of equilibrium temperatures for the two cases are significantly differ-
ent from each other, and hence the minimum temperatures at spring homo-
thermy are different also. Thus a more accurate representation of the
stratification cycle can be obtained by choosing an equilibrium-temperature
variation in which the maximum value of T£ is the same as before but in
which the minimum value of the equilibrium temperature is lower. A rough
rule of thumb for choosing a new value for the minimum equilibrium
temperature is that the difference between the new and the old values should
be somewhat larger than the difference between the minimum surface
temperature after one cycle and the initial temperature of spring homothermy
By successive repetition of the iterative procedure described above, the
correct choice for the equilibrium temperature, and the corresponding infor-
mation on the other relevant quantities (such as the vertical distributions of
temperature or the variation of the depth of the thermocline), can be arrived
at fairly rapidly.
49
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It should be emphasized that if the calculations for the above con-
ditions were carried out over several cycles, the solution will ultimately
tend to a cyclic behavior, but with the minimum and maximum temperatures
being quite different from the specified values. This is because for a given
set of 'external parameters' there is only one unique cyclic solution regard-
less of the initial starting conditions assumed. This is illustrated in Figs.
16-18 which show the results for nearly three years for the case when
T = rz + 11*
with all other conditions being the same as before. The calculations are
started assuming an initial temperature of 2. 9°C, as before, and it can be
seen from Fig. 16 that the solution tends to a cyclic state at a higher value
of the minimum homothermal temperature. The attainment of cyclicity is
clearly evident from Figs. 17 and 18 which show the vertical distributions
of temperature and eddy diffusivity for the same relative times during the
three years. While noticeable differences can be seen between the distri-
butions for the second and first years, the distributions for the third year
are indistinguishable from those for the second year.
IV. 7 Comparison with Observations
As mentioned earlier, in the present program numerical calculations
(using the analytical model) were performed only for a limited number of
combinations of the various input parameters. Many of the calculations
were designed to check the various aspects of the numerical program and
some calculations were performed to check that the model was predicting
the appropriate qualitative features. However, due to time limitations,
neither a comprehensive parametric study of the behavior of the solution
for various input conditions, nor calculations for conditions corresponding
to specific waterbodies were carried out. It is hoped that these calculations
can be performed in the near future.
For some of the calculations, the input parameters were chosen to
correspond roughly to those of Cayuga Lake, since extensive information
on the thermal structure of this lake is available. 16< Z2 However, it should
50
-------
be emphasized that no attempt was made to duplicate exactly the "external
parameters" for this lake. A detailed discussion of this specific example
has already been given in Section IV. 5 and the results are presented in
Figs. 8-14. Comparisons of the calculated and observed values for the
temperature cycle, the temperature profiles and the thermocline depth are
shown in Figs. 19-21. In all the figures, the measured values shown are
the averages of the values for the years 1950, 1951, 1952 and 1968.
It can be seen from the figures that the agreement between the
measured and computed values are very good both in a qualitative and in a
quantitative sense. However, the quantitative agreement has to be viewed
in the light of the fact that the "external parameters" used in the calcula-
tions represent the conditions over Cayuga Lake only in a rough manner.
It may well be that, in this specific example, the mismatches in different
"external parameters" produced opposing effects and that their net cancel
lation is responsible for some of the good agreement found in Figs. 19-21
between the measured and computed values. It should also be pointed out
that in a lake in which the area of cross section changes with depth, the
assumption of horizontal homogeneity necessarily implies a distortion of the
vertical scale. This fact was pointed out by Birge a number of years ago
in connection with the calculation of the heat budget of a stratified lake. '
Thus a direct comparison (that is, without taking vertical distortion into
effect) of the computed and measured values is not compatible with the
assumption of horizontal homogeneity.
The points mentioned above have to be resolved by carrying out sys-
tematic calculations using the present model before definitive conclusions
on the quantitative accuracy of the present scheme can be reached. Since
such calculations have not yet been performed, it can only be stated here
that the present scheme appears to give quite adequate quantitative accuracy.
However it should be reiterated that the present scheme gives excellent
qualitative agreement with all of the observed features of stratification in
temperate lakes. Indeed, many of these features have never before been
predicted analytically.
51
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IV. 8 Concluding Remarks
In the present section, the theoretical concepts developed in Section
III were used to generate an analytical model for the annual stratification
cycle of a temperate lake. The analytical model essentially viewed the
changes in the thermal structure of a waterbody in terms of certain
"external parameters" characteristic of the environmental conditions over
the waterbody. Specifically, the external parameters were used to charac-
terize the transfer of thermal and mechanical energies at the air-water
interface.
The results of the analytical model are in excellent agreement with
the observed qualitative features of the stratification cycles of temperate
lakes. In particular, the analytical model predicted accurately certain
characteristic features which were never before predicted analytically.
While because of the limited number of calculations that were carried out
an assessment of the predictive accuracy of the model was not possible,
comparisons of the calculations with observations in Cayuga Lake gave very
good agreement.
52
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V. EFFECTS OF THERMAL DISCHARGES
V. 1 General Remarks
In Section III it was shown that the mechanism responsible for the
formation of thermoclines in stratified lakes is the nonlinear interaction
between the wind-induced turbulence and the buoyancy gradients due to sur-
face heating. In Section IV, the above concept was extended to study the
entire stratification cycle. It is now relevant to enquire as to what the
perturbing effects on this basic thermal structure will be when thermal dis-
charges are introduced into the lake. Specifically, in this section we will
consider the effects of thermal discharges from power plants for the case
when the intake water is withdrawn from the hypolimnion, and the heated
effluents are injected back into the hypolimnion in such a manner as to trap
them below the thermocline during part of the stratified period.
When considering the effects of thermal discharges into the hypo-
limnion of a stratified waterbody, two specific problems have to be
analyzed. The first is concerned with the behavior of the buoyant plume due
to the discharges and the second is concerned with the effects of the dis-
charges on the entire waterbody. These two problems have been termed the
micro- and macro-scale problems by some authors. As mentioned in
Section II, the first problem (namely, the behavior of a buoyant jet in a
density stratified environment) has been studied extensively in other con-
nections. Therefore this problem is discussed only briefly in Section V. 2.
On the other hand, very little is known about the effects of thermal dis-
charges, at or below the level of the thermocline, on the entire waterbody.
The latter problem is discussed in detail in Sections V. 3 and V. 4.
It should be emphasized that, as pointed out by Sundaram et al,
the primary difficulty of predicting the effects of heated effluents on the
thermal structure of a stratified lake is that the discharges affect the
thermal structure of the lake not only directly, but also indirectly by their
influence on the turbulence structure in the lake. The effects of the thermal
53
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discharges on the structure of the turbulence, and the resulting changes in
the eddy diffusivities, are quite important since the manner in which the
added heat is dispersed into the lake is controlled by the values of the eddy
diffusivities. Thus one cannot assume, as has been done by some authors,
that the thermal discharges do not affect the mechanisms of epilimnial
mixing. This point is discussed in detail in Sections V. 3 and V. 4.
Finally, one additional point needs to be made here. When consid-
ering the effects of thermal discharges from a power plant with a hypo-
limnetic intake, it should be recognized that in addition to the effect
associated with the discharge of heat there is also an effect associated with
the transfer of large quantities of water from one level to another. This
latter effect arises due to a change in the potential energy of the stratifica-
tion and, as pointed out in Refs. 16 and 76, it can be viewed in terms of an
equivalent change in wind conditions above the lake. This effect is dis-
cussed in detail in Section V. 4.
V. 2 The Buoyant Plume
When thermal discharges are injected into the hypolimnion of a
stratified body of water, they will rise as a buoyant plume because of the
positive buoyancy with respect to the surrounding cooler medium (see Fig.
2 la). As the plume rises, it will entrain the cooler surrounding water into
it. The entrainment will increase the volume of the plume and also reduce
the temperature difference between the plume and the surroundings. In
addition, the temperature difference between the plume and the ambient
water also decreases due to the fact that the ambient temperature increases
with increasing distance from the point of discharge. Even after the plume
becomes neutrally buoyant, it will continue to rise because of its residual
kinetic energy and will come to rest only when this energy is fully dissi-
pated.
After this point the plume will spread rapidly in the horizontal direc-
tion as a relatively thin sheet. The initial horizontal spreading of the
thermal plume will create a well-mixed layer which is nearly homothermal
so that, due to the stable stratification at the (horizontal) boundaries of this
54
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layer, vertical mixing will be inhibited. The assumption of horizontal
homogeneity, which was used in Sections III and IV to study stratification,
will continue to remain valid when considering the effects of the discharges
on the entire waterbody. Indeed when considering the lake-wide effects of
the discharges, it is appropriate to assume that the added heat is injected
uniformly at the level at which the effluents spread.
79
Recently Baines and Turner have studied, both experimentally
and theoretically, the effects of continuous convection from a small source
of buoyancy enclosed in a bounded region. They point out that conservation
of mass requires that at any horizontal plane the upward volume flux
in the plume be balanced by an equal downward flux in the surrounding
region. Also, because of the stable stratification in the surrounding region,
the fluid that is entrained into the plume at any level (including the intake
level) can be assumed to come entirely from that level. In the present
work, these concepts have been adapted to study the lake-wide effects of
thermal discharges into the hypolimnion of a stratified lake. However, a
detailed consideration of the behavior of the buoyant plume itself has not
been included. Rather, the effluents have been assumed to be injected
directly into the level at which the lake temperature is equal to the discharge
temperature. It is planned to include the effects of dilution in future modi-
fications of the present model using existing knowledge ' on the behaviors
of buoyant plumes in density stratified environments.
V. 3 Effects of Discharge on Overall Thermal Structure
As mentioned earlier, there are two effects which must be included
when modeling the influence of a power plant effluent upon the overall
thermal structure of a body of water. Water withdrawn from lower, colder
layers of a lake, heated and discharged into the lake again, adds thermal
energy, and, by changing the vertical thermal structure of the lake, alters
the potential energy of the lake. Therefore a power plant both adds heat
and does work on the lake when using lake water for cooling. In the present
section, terms will be added to Eq. (III. 1) which will explicitly include the
far-field effects of heat added and of work done by the power plant.
55
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As discussed earlier, detailed consideration of the buoyant plume
created by the effluent will not be made. A proper accounting of interaction
of the plume and the surrounding lake water would entail a model in which
the current structure is explicitly treated, rather than implicitly treated as
in the present model. The effects of dilution of the effluent by entrainment
will not be considered either for the determination of depth at which the
effluent begins its horizontal spread or for the determination of the volume
of surrounding water entrained. Rather effluent water will be assumed to
be discharged directly at the depth at which the lake temperature equals the
effluent temperature, and this water will be considered to mix rapidly in
the horizontal direction. If the temperature of the discharged water exceeds
the temperature of the lake water at each depth, then the effluent will rise
to the surface and spread. Both the intake and the outfall will be assumed
not to affect the condition of horizontal homogeneity imposed by the basic
model without the effects of discharges.
It should be noted that the limitations on the model imposed by
neglecting dilution of the thermal plume will not change the qualitative
behavior determined. All of the effects discussed in this section for thermal
discharges into a stratified lake are qualitatively correct and are nearly
quantitatively correct provided that the discharge temperature used in the
calculations is chosen to be an effective discharge temperature. The major
effect of the reduction of plume temperature due to dilution of the thermal
effluents will be to change the level at which spreading of the discharge
occurs. Depending upon the outfall configuration and the lake conditions,
the quantity of lake water entrained into the discharge will generally be a
small fraction of the volume of the water used for power-plant cooling.
Therefore, the temperature at which the plume begins to spread will be
somewhat lower than the actual temperature of the discharge at the outfall.
The temperature at which the plume begins to spread, or the effective dis-
charge temperature, can be taken to be somewhat (a few degrees) smaller
than the actual discharge temperature to compensate for dilution within the
plume0 Throughout the subsequent considerations the temperature increase
AT^ due to the power plant will be taken to be the difference between the
water temperature at the intake level and an effective temperature at discharge.
56
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V. 3. 1 Model When Discharge Remains Below the Lake Surface
7 9
In the present study, following Baines and Turner, it has been
assumed that the effluents spread instantaneously into a thin sheet at their
level of neutral buoyancy, and the details of the actual horizontal spread
have been ignored. This assumption is justified in view of the relatively
long time scales involved in the consideration of the lake-wide effects of the
discharges. In the present model, the effect of the intake will be equivalent
to that of a uniformly distributed sink at the level of the intake, with the
total strength of the sink corresponding to the volumetric flow rate through
the power-plant condensers. There will be a corresponding uniformly dis-
tributed source of fluid at the effective level of injection of the discharge,
When considering the effects of thermal discharges from a power
plant with a hypolimnetic intake, in addition to the effect associated with the
discharge of heat there is also an effect associated with the transfer of large
quantities of -water from one level to another. This latter effect arises due
to a change in the potential energy of the stratification and, as pointed out in
Ref= 16, for a surface discharge it can be viewed in terms of an equivalent
change in wind conditions above the lake. In the present model the effect of
"pumping" water from a lower level to a higher one has been properly
accounted for.
Although there are different ways in which the effects of heat and
pumping work could be included, when the effluent is discharged below the
surface, these effects will be added to Eq. (III. 1) through a single term
s\ -r
(jj- ° , Here uJ-^ is the uniform, downward vertical velocity which is
induced (between the level of the intake and the effective level of the dis-
charge) by the power plant pumping; t^r^ can be assumed to be the volumetric
flow rate Q-^ through the power plant divided by the area R of the lake.
With the addition of this term Eq. (III. 1) becomes
j
If the level of the intake for water is fixed at Z^ and the effective level of
57
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discharge of the heated effluent is at H^ (see Fig. 21a), then Eq. (V. 1) is
appropriate for Z,. ^ H ^ 2d . Below the intake level, 2 > Z^ , and
above the discharge level 2 < Zd , the direct effects of the power-plant
pumping are absent, and the governing equation is Eq. (III. 1) again.
The meaning of the extra term in Eq. (V. 1) can be understood simply
by examining integrals of Eqs. (V. 1). Integrating these equations from the
surface of the lake, Z - 0 , to the bottom of the lake, Z = Z ^ , we obtain
Multiply by Q C^ ft and note the boundary conditions that have been imposed
at the bottom and at the surface of the lake.
The heat flux at the bottom ^ ^ 9 T - 0
of the lake is zero:
The heat flux at the surface
of the lake:
H -
- K(.Tf
Then Eq. (V. 2) can be re-written
W r*~
oC+TfldJE = PKCT, -Ts) -i- Q oC [T(Z4) - TfZj] (V. 3)
cU ,/ ^
^ O si O
Equation (V. 3) is simply and directly interpretable. It states that the time
rate of increase of thermal energy, integrated over the whole lake, is equal
to the rate of heat addition through the surface Bc^^ plus the rate of heat
addition by the power plant F\^ . Note that the rate of heat addition by the
power plant is simply the water volumetric flow rate 0.^ times the energy
per unit volume required to heat water from the intake temperature T ( i^ }
to the effective discharge temperature T ( Z^).
58
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Taking the first moment of Eq. (V- 1) and integrating again from Z= 0
to H^Z^ j we find, after some manipulation,
r*+
/ [T(Z) 7"
\ u
Equation (V. 4) can be interpreted in terms of the changes in the potential
energy of stratification.
The potential energy of stratification of a lake can be written in the
form
2>M,
C£ ?) (T-f) d Z (V- 5)
where Z- is the depth of the center of gravity of the lake and T is the
0
temperature the lake would attain at any given time if the wind energy were
able to mix it completely. Note that the potential energy of stratification is
always negative since wind energy is required to upset the stratification.
Thus the stronger the stratification, the more negative will be the potential
energy. The time rate of change of potential energy can be written as
' "*-
-------
of wind mixing. The third term in Eq. (V. 7) represents the change in
potential energy of stratification due to the thermal discharges. Note that
it depends only on the amount of heat added and the effective level of the
discharge. The fourth term in Eq. (V. 7) represents the effect of pumping
and it depends on the pumping velocity t*/. .
V. 3. 2 Model When Discharge Surfaces
When the effective temperature at which the effluent is discharged is
higher than the temperature at any depth within the lake, then the effluent
will surface and spread horizontally. The model discussed above is no
0 T
longer adequate since the term uJ, ., in Eq. (V- 1) requires the effluent
and the ambient temperature at the discharge level to be equal. Under these
conditions, then, heat in addition to that supplied to the lake by the term
2 T
UJ*, -y^- must be explicitly introduced. If AT_*, repres ents the effective temperature
change produced in the intake water by the power plant, then the total heat
added per unit time to the lake will be Q.^ 0 C^ A T^ t or the heat added Q
fi*
per unit area of the lake per unit time is Q = f o C. A "[*, - uT^., o C, AT*, .
When the effluent surfaces, the surface temperature Ts is such that
3T
I, - T(Z^) ^ AT o The term t*Sp ~ ^ in Eq. (V. 1) adds only w^ O C
r i d i. r r
[T5 - T (ij ] heat per unit area per unit time. The additional heat per
area per time, namely ^ Q Cp [ T (Z^) + A 7^ - TS ] , must be added
explicitly into Eq. (V. 1).
This heat is added by a source term in Eq. (V. 1) of the form
~) r°i f r!~~r/3\ AT ~T~1 ^ ^Vft\
oC. " ~P^~ e "-*
v ^ a. V-TT
where CL is a length scale for the distribution of the source term. The
factor arises so that the integral of the source term over depth will approxi-
mately yield the total additional heat flux to be added. Therefore, when Z^
reaches the surface, the explicit heat source term given by Eq. (V. 6) is
incorporated into Eq. (V. 1). (Note that .2^ is never allowed to become
less than Ou in the numerical program).
9 7 9 / 9T 9T
60
-------
if- if(i<»i>^> ^-*,(v
An alternate method by which the additional heat flux uJT^ o C^ [T(J^)
- /s ] can be added when the thermal discharge surfaces
is by changing the boundary condition at the surface. In the present calcula-
tions the heat flux at the surface is given by Eq. (IV. 3), and, when the
thermal plume surfaces, the additional heat due to the power plant is intro-
duced by means of the source term in Eq. (V.8). However, the additional
heat flux uy. ^ C^, [T(Z^) + AT^ - Ts ] could also be included in the cal-
culation by increasing the heat flux qs directly at the boundary. In future
computations, this alternate procedure will be tried, and the results will be
compared with those obtained by the present procedure.
V. 3. 3 Model with Pumping Only
In the preceding sections a model was introduced for calculating the
effects on the overall thermal structure of a stratified lake of thermal
effluents discharged at or below the level of the thermocline. In this model
the effects of dilution of the effluent due to entrainment during thfi rise of the
buoyant fluid have not been included. However, as noted before, the reduc-
tion in temperature of the effluent due to dilution by lake water entrained
into the plume can be implicitly accounted for in the model by selecting an
effective discharge temperature somewhat less than the actual discharge
temperature.
In this section a model opposite, in a sense, to that discussed in the
preceding sections is introduced to calculate the effects on the overall
thermal structure of a stratified lake of pumping cold water from the hypo-
limnion and discharging it and mixing it into the epilimnion. In this model
no heat is added to the water being pumped, and it is assumed that dilution
and mixing of the cold effluent by the surface layers of lake water are
complete. Dilution and mixing of the plume over a depth CL (the length scale
defined in the source term in Eq (V. 8)) can be accomplished practically by
supplying a sufficient number of ports in a multiple-port diffuser at the
outfall. This model is of interest when considering the effects on the thermal
61
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structure of the lake of the work done when water is pumped from the
hypolimnion to the surface.
In this model water is withdrawn at a depth Z^ , pumped to the
surface, discharged at temperature T(£^) and mixed with the water in the
epilimnion0 Such an example of pumping water from the hypolimnion into
the epilimnion with no heating of the water, can, be calculated as a special
case of the model Eqs. (V. 9) and heat source term (V. 8). When pumping
with no heating is simulated, the source term becomes negative, a sink.
The heat absorbed by this sink must balance the heat added by the term
gj
u^., ~T5~ . In an integral sense a balance will occur in Eqs. (V. 9) if
is set to zero in the source term in Eq. (V. 8).
62
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V-4 Numerical Results and Discussion
The numerical model for the effects of thermal discharges described
in Section V. 3 has been used to make several numerical integrations of
Eqs. (V. 1) and (V-9). The purpose of these calculations is to determine
the effects on the overall thermal structure produced by discharging a heated
effluent into the hypolimnion of a body of water. An additional objective of
these calculations is to examine the effects on the thermal structure of
pumping -water from the hypolimnion to the epilimnion without heating the
effluent. As discussed earlier, the thermal discharge of water into the
hypolimnion of a lake produces two alterations: heat is added to the lake,
and the potential energy of the thermal stratification is reduced by the
pumping work. The latter effect is isolated by the calculations which model
pumping alone, and for this reason calculations were performed with both
thermal discharges and pumping alone.
The calculations presented in this subsection by no means represent
a complete study of the effects of thermal discharges or of pumping upon
the thermal structure of a body of water. Rather the results show some of
the interesting features. A parametric study with this model of these effects
would be very informative and highly desirable.
In this subsection the calculations performed to determine the effects
of thermal discharges and of pumping alone are described. Comparison is
made between these calculations and those describing the basic annual varia-
tion in a stratified lake. The comparison shows many important features
which thermal discharge into the hypolimnion and pumping from the hypolim-
nion to the epilimnion can produce. Most of these features have never been
discussed in quantitative terms before.
First, a discussion is given of the transient and periodic responses
produced by a thermal discharge or by pumping alone. Then the general
effects of a thermal discharge and of pumping are presented. Finally, com-
parison between the base calculation, the response with thermal discharge
and the response with pumping alone is made.
63
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V. 4. 1 Transient and Periodic Responses
As discussed in Section IV, the variation in the temperature or the
thermal diffusivity within a lake will always be periodic because the surface
conditions are periodic. If the initial conditions are chosen properly, these
quantities will be periodic from the beginning of the calculation. In general,
if arbitrary initial conditions are chosen, the solution will require a transient
period before cyclic distributions of temperature are achieved.
To examine the effects of thermal discharges or of pumping alone,
one must start with a periodic base calculation. In Section IV a detailed
description was given of the procedure used to determine desired periodic
conditions. Figures 8 through 14 are plots of the solution for the base cal-
culation. With the parameters and the initial conditions chosen to be the
same as for the base calculation the terms representing a thermal discharge
can be introduced and the effects resulting from the discharge can be deter-
mined by numerical integration.
The base calculation represents the average annual distributions of
temperature prevailing in an undisturbed temperate lake similar to Cayuga
Lake. If a power plant were situated on this lake and if the waste heat dis-
charged by the power plant were taken to be constant with time, then the
response of the lake could be divided into a transient portion, during which
the lake adjusted to the new heat budget and to the additional work done by
pumping, and a periodic portion thereafter. The results presented in this
subsection concentrate on the transient response. Although the periodic
differences produced by a power plant are of great general interest, the
practical limitations introduced by the length of the calculation and by the
available time did not permit an adequate assessment of these differences.
V. 4. 2 Effects of Thermal Discharges
Since a power plant withdraws water, heats it and discharges it into
the lake again, the water temperature at any time and at any level can be
expected to be higher than the corresponding temperature without the dis-
charge. The magnitude of this temperature increase can be estimated from
the heating introduced by the power plant. It is convenient to consider the heat
64
-------
added per unit time by the power plant divided by the surface area of the
lake. This quantity, the heat flux added by the power plant, can be compared
with the heat flux at the surface of the lake in the absence of the thermal dis-
charge. In Fig. 10 the surface heat flux of the standard cycle has been
shown, and the maximum value of the heat flux is seen to be about 2000 Btu/
ft -day. The corresponding temperature variation at the surface of the lake
is about ZO C (see Figure 8). For the calculations discussed in this sub-
section the heat flux attributed to the power plant was taken to be 280 Btu/
ft -day. If this heat flux were added to that occurring naturally at the surface
and if the mixing processes were taken to be unchanged, a maximum tempera-
ture increase of about 2-3 C would be expected from the power plant.
Heat can be discharged at a specified rate in a variety of ways,
depending upon the flow rate of water used for cooling. As discussed before,
the heat added per unit surface area of the lake per unit time is Q = ~^~ ^C>, AT
For a fixed heat flux a the temperature increase AT. produced by the
waste heat is inversely proportional to the volumetric flow rate Q.^ of cooling
water. Therefore, specification of fie heat flux Q^ and of the volumetric
flow rate or of the pumping velocity uT. = Q /'f\ within the lake determines the
temperature increase. The temperature increase of the effluent water over
that of intake water determines the level at which the discharge spreads hori-
zontally within the lake and therefore the details of the change in the thermal
structure produced by the power plant. Two flow-rate and temperature-
increase combinations for the specified heat flux have been considered: in
one case the pumping velocity was taken to be 1/4 ft/day with a temperature
increase of 10 C at discharge and for the other case uJ"^ =1/6 ft/day and
AT^ = 15°C.
In each case the temperature increase resulting from the power plant
is small compared with the maximum natural variation of the temperature
at the lake surface, and the qualitative features of the temperature variation
with time at any depth will be the same as without the discharge. Therefore,
differences in temperature produced by thermal discharges are of interest.
A comparison between temperature profiles without and with thermal
discharge at three times during the year is shown in Fig. 22. The thermal
65
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discharge has a pumping velocity of o/^ = 1/4 ft/day and a discharge-tempera.
ture increase of AT = 10°C. The first set of plots, at 90 days, is chosen
to show the temperature change after formation of the thermocline during
the heating portion of the annual cycle. The curves show that the maximum
temperature increase occurs at the surface and that smaller increases
occur at greater depths. This behavior is to be expected for the following
reason. The intake depth for this calculation has been placed at about 125
feet. At this depth the temperature has varied from its initial value of 2. 9 C
to around 6°C during the 90 days. With a discharge temperature 10 C above
the temperature at the intake depth, the effluent is found to surface throughout
the first 90 days. Consequently, the temperature increase produced by the
discharge above the natural temperature will be expected to be greatest at
the surface. At some time after 90 days the effluent will no longer surface,
but will be trapped by buoyancy effects below the surface during part of the
stratification period.
The second set of plots in Fig. 22, at 180 days, occurs just as the
cooling portion of the annual cycle begins. The mixing above the thermocline
is very complete both without and with the thermal discharge. In this region,
above about 50 feet, the temperature increase produced by the thermal dis-
charge is smaller than in a good portion of the hypolimnion, even though the
discharge has again surfaced.
For this set of curves and for the curves representing 90 days, the
thermocline is found to be at a somewhat greater depth with the thermal dis-
charge than without. This result cannot be simply predicted a priori. The
two changes produced by a thermal discharge, namely the heat added and the
pumping work performed, have opposite effects on the thermocline. The
additional heat tends to increase the temperature, temperature gradients and
therefore the stability within the lake. On the other hand, the pumping work
tends to enhance mixing within the lake. The former effect would, by itself,
decrease the depth of the thermocline while the latter would increase its
depth. A detailed discussion of these two competing effects and of their
relationship to the Monin-Obukhov length has been given in Ref. 16. In
the present case the thermocline is driven somewhat deeper between 90 and
66
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180 days by the effects of the thermal discharge.
The last set of curves, for 300 days, occurs during the period of
greatest cooling within the annual cycle. The cooling causes efficient
convective mixing, which produces nearly uniform temperature distributions
both without and with the thermal discharge. The temperature increase
resulting from the thermal discharge is nearly 1. 5°C.
The qualitative effects produced by the thermal discharge with uJ^ =
1/6 ft/day and AT^ = 15 C are the same as those described above, with one
exception. A discharge-temperature increase of 15 C above the temperature
at the intake level forces the effluent to remain surfaced throughout the
annual cycle.
The quantitative effects are summarized in Table 1. In this
table the increases in temperature produced by the thermal discharge with
6oT =1/4 ft/day are listed along with the increases produced when o/v =
1/6 ft/day. The important feature common to both calculations is that the
additional heat loads introduced by the thermal discharges are the same.
The difference between the two results from, the change in pumping work
done. For the larger pumping velocity uJ"^ =1/4 ft/day, more mixing work
is introduced.
During the heating portion of the cycle the increased mixing produced
with u/L = 1/4 ft/day results in larger temperature increases in the deeper
waters and correspondingly smaller temperature increases in the shallower
waters. In each case a temperature increase over the naturally occurring
temperature can be expected as a result of the heat added by the thermal
discharge. However, for uJ~. =1/4 ft/day, heated waters near the surface
are more effectively mixed into the deeper waters by the larger pumping
velocity.
During the cooling portion of the cycle, when the lake is thoroughly
mixed, the temperature increase throughout is larger for the case when
OU^ =1/4 ft/day. Once again the increased mixing resulting from the larger
pumping rate explains this feature.
67
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Table 1
TEMPERATURE DIFFERENCE PRODUCED BY THERMAL DISCHARGES WITH
DIFFERENT DISCHARGE TEMPERATURES BUT THE SAME HEATING RATE
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
JAN
FEB
MAR
DAYS
0
30
60
90
120
150
180
210
240
270
300
330
365
ATS
Wp = 1/4
0.0
0.41
0.82
1.06
0.58
0.50
0.64
1.10
1.62
1.42
1.52
1.50
1.35
Wp = 1/6
0.0
0.41
0.84
1.22
0.84
0.67
0.84
1.17
1.51
1.25
1.38
1.39
1.28
ATBO
wp = 1/4
0.0
0.34
0.76
1.22
1.65
1.63
1.05
1.16
1.62
1.46
1.45
1.43
1.38
wp = 1/6
0.0
0.34
0.75
1.13
1.46
1.70
1.25
1.22
1.52
1 .30
1.31
1.33
1.31
AT100
wp = 1/4
0.0
0.28
0.46
0.54
1.07
1.75
2.30
2.68
1.67
1.47
1.39
1.37
1.39
wp = 1/6
0.0
0.28
0.45
0.46
0.80
1.31
1.77
2.05
1.54
1.30
1.26
1.27
1.31
AT, 5o
wp = 1/4
0.0
0.24
0.29
0.22
0.31
0.55
0.87
1.23
1.30
1.39
1.36
1.33
1.39
wp = 1/6
0.0
0.24
0.28
0.20
0.24
0.41
0.65
0.91
0.93
1.22
1.23
1.23
1.31
-------
The rate of heat added to the lake at any time is equal to the heat
flux K(TC - Tc) at the lake surface times the surface area R of the lake plus
C 3
the rate of heat added by the discharge. Under periodic conditions, i. e. ,
when the lake has adjusted to the increased thermal load provided by the
discharge, the integral over an annual cycle of the heat added at the surface
plus the heat added by the discharge must be zero. Thus
K
f (T£ - T5Ut = ' Q
vvhere Q. is the total heat added over a year by the thermal discharge and
where K is the constant heat transfer coefficient at the lake surface. For
both cases discussed above, the thermal discharge with u/^ = 1/4 ft/day
and the discharge with u/V, = 1/6 ft/day, the total heat added is 280 ftT ,
where R is the surface area of the lake and T is one year. Also, for
both pumping rates the equilibrium temperature variation is the same.
However, as discussed above, the larger pumping rate produces a lower
surface temperature during the heating portion of the annual cycle. There-
fore, since the integral of 4 (Te ~ Ts ) oi t is the same for both cases
and since Ts is lower during stratification for the higher pumping rate, this
temperature must be higher during the cooling portion of the cycle. During
cooling, however, the lake is thoroughly mixed so that the surface tempera-
ture is very nearly the temperature throughout.
This argument describes the lake quantitatively only when the
response has become periodic. However, it should still apply qualitatively
also when the response is transient. Therefore, the temperature increases
during the cooling portion of the annual cycle when uJ" =1/4 ft/day are larger,
as expected, than those increases when uj". =1/6 ft/day.
In this subsection a discussion of the effects of thermal discharges
has been given and a detailed comparison between the thermal structures
without and with thermal discharges has been made. In the next subsection
the effects of pumping alone are presented.
68
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V.4. 3 Effects of Pumping Alone
When water is withdrawn from the hypolimnion of a lake and discharged
and mixed at the surface, the thermal structure of the lake is changed. The
pumping work done to change the potential energy of the lake results in an
enhanced mixing within the lake. In this section the effects of pumping are
discussed; no heat is added to the lake.
In general, as discussed in Section V. 3. 1, heating at the surface of
a lake increases the absolute value of the potential energy of stratification,
or stability, of the lake whereas turbulent mixing tends to reduce it.
Mixing of the heated surface waters into the deeper waters by pumping
reduces the stability of the lake and produces temperature profiles for which
the thermocline is lower than would occur naturally. Because of the increased
volume of the epilimnion, its average temperature will be lower than that
which occurs naturally. Because of the effective mixing within the epilimnion
the temperature profiles will display a nearly uniform temperature throughout
the increased volume. As a result, the maximum surface temperature of the
lake is found to be lower than the corresponding natural maximum.
The pumping velocity for the plots shown in this section is (sf. = 2 ft/
day. The intake for water is situated at 125 ft, and the water is discharged
and mixed at the surface. This pumping velocity should be compared with a
typical velocity for descent of the thermocline of about 1 ft/day which occurs
naturally in Cayuga Lake during the cooling portion of the annual cycle. For
this pumping velocity the effects of the work done on the lake by pumping can
be vividly demonstrated.
In Fig. 23 the annual temperature variation at the surface and at
three levels, 50 feet, 100 feet and 150 feet, is shown when water is simply
pumped. With reference back to the standard annual cycle shown in Fig. 8,
one can see the features described above as consequences of pumping. For
example, Fig. 23 shows that the maximum surface temperature with pumping
included is approximately 18°C; Fig. 8 shows the natural maximum to be
about 22°C. The temperature variations at 50 feet and 100 feet follow within
a few degrees the temperature variation at the surface. This behavior is to
69
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be contrasted with that shown in Fig. 8, where the temperature at 50 feet
differs by several degrees from the surface temperature near its maximum.
The temperature difference at 100 feet is even more pronounced, being about
ten degrees for much of the stratification period.
The temperature plots at 50 feet and 100 feet are both for levels above
that of the intake. On the other hand, the plot of temperature at 150 feet is
for a depth below that of the intake, and this curve varies by as much as six
degrees from the others. Therefore, as might be expected in this case, the
mixing produced by the pumping between the levels of the intake and the
discharge is quite complete, whereas the mixing induced below the intake is
much smaller.
Another effect of pumping can be observed by a close comparison
between Figs. 8 and 23. After a cycle, the temperature at homothermy is
determined as the point at which the temperature curves for different depths
cross. With pumping effects included, the temperature at homothermy is
found to be somewhat less than a degree higher than that without pumping.
As discussed in the previous subsection, this increase in temperature at
homothermy results from the integrated change in surface heat flux induced
by pumping.
Figure 24 shows the effect of pumping on the depth of the thermocline;
this curve is to be compared with that shown in Fig. 14 for the seasonal
variation of the thermocline with no thermal discharge or pumping effects
introduced. The most dominant feature of Fig. 24 is the rapid descent of
the thermocline to the intake level at 125 feet. As discussed above, this
descent results from the rapid mixing between the intake and discharge
levels produced by the pumping.
In Table 2 the temperature differences produced by different pump-
ing rates, co^ = 2 ft/day and uJ^ =1/4 ft/day, are compared. For both
cases the intake level is at 125 feet and water is discharged and mixed at
the surface. A temperature increase over the natural variation is a number
greater than zero and a decrease is one less than zero. Both pumping rates
show the qualitative features described above. During the heating portion of
70
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Table 2
TEMPERATURE DIFFERENCES PRODUCED BY
DIFFERENT PUMPING RATES
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
JAN
FEB
MAR
DAYS
0
30
60
90
120
150
180
210
240
270
300
330
365
ATS
wp = 1/4
0.0
-0.005
-0.03
-0.32
-0.95
-0.98
0.58
0.0
0.57
0.57
0.1*7
0.34
0.19
Wp = 2
0.0
-0.04
-0.23
-1.79
-4.08
-3.79
-1.76
0.57
1.91
2.18
1.85
1.37
0.86
AT50
wp = 1/4
0.0
0.001
0.003
0.21
0.89
1.22
0.10
0.06
0.57
0.57
0.45
0.34
0.23
WP = 2
0.0
0.005
0.01
0.00
0.43
0.17
-1.18
0.63
1.87
2.21
1.76
1.33
0.99
AT,OO
wp = 1/4
0.0
0.003
0.01
0.19
0.66
1.31
1.90
2.24
0.64
0.60
0.44
0.34
0.26
wp = 2
0.0
0-02
0.16
1.57
4.87
7.20
7.26
4.73
1.97
2.27
1.71
1.29
0.99
AT, 5o
wp = 1/4
0.0
0.003
0.01
0.06
0.15
0.34
0.62
0.96
1.02
0.58
0.43
0.33
0.2b
wp = 2
0.0
0.02
0.09
0.44
1.37
3.05
4.64
5.39
4.73
2.28
..69
1.28
1.02
-------
the annual cycle, the temperatures in the upper levels of the lake in the
presence of pumping are reduced below the natural ones while the tempera-
tures in the deeper waters are increased. While the lake is cooling, on the
other hand, the temperatures at all levels are increased above the values
with no pumping. As expected this behavior is enhanced as the pumping
rate, as expressed by the pumping velocity ur^ , is increased.
V-4. 4 Comparison Between the Effects of Thermal Discharge and the
Effects of Pumping
The effects of thermal discharge and of pumping alone have been
discussed in the previous two subsections. It is of interest now to compare
these effects in a controlled fashion. In this subsection the results from the
base calculation, a calculation with only a pumping velocity of 1/4 ft/day and
a calculation with a thermal input of 280 Btu/ft -day and a pumping velocity
of 1/4 ft/day are compared. In both cases, with a discharge and with pumping
the intake level remains at 125 ft.
In Fig. 25 the temperature variations at the surface and at depths of
100 and 150 feet for all three cases are plotted. The seasonal variation of
the temperature, chosen as the standard case, is shown as a dashed line.
The temperature variations with the effects of a thermal discharge are shown
as a solid line and the variations with pumping effects only are shown as a
broken line. Throughout the cycle the thermal discharge is seen to increase
the temperature above the natural variation at the surface, at 100 feet and
at 150 feet. During the stratification period the effect of pumping at the
same rate, but with no thermal input is to decrease the surface temperature
below the natural variation. At depths of 100 and 150 feet the temperature
is increased by pumping only but by an amount smaller than that produced by
the thermal discharge. During the cooling portion of the cycle when mixing
is complete, the temperatures at the surface and at 100 and 150 feet are
always greatest when the effects of the thermal discharge are included.
When pumping alone is included, the temperatures at all three positions are
greater than the corresponding temperatures without pumping.
71
-------
In Fig. 26 the thermocline depth for each case is plotted as a func-
tion of time. The depth of the thermocline is defined for the program as
the position at which the magnitude of the temperature gradient is maximum.
With the effects of the thermal discharge included a thermocline, according
to this definition, immediately forms below the surface as a result of the
heated effluent. The depth at which this thermocline forms depends upon
the details of the heat-source term used to model the discharge; later,
however, when a true thermocline forms, the thermocline depth no longer
depends upon these details. The position of the thermocline discussed below
will refer to the depth determined during this latter period.
With the effects of the thermal discharge included, the thermocline
is found to occur at a greater depth than it does naturally for most of the
stratification period. This result indicates that, during this time, the
pumping work added to the lake by the discharge has a greater effect on the
depth of the thermocline than does the discharge-heat added. With pumping
alone, the thermocline occurs at still greater depths since work with no
additional heat is introduced in this case. A close examination of these plots
indicates that the stratification cycle is increased somewhat by the effects of
pumping alone and somewhat more by the effects of thermal discharges.
In Fig. 27 the effects are shown of the thermal discharge and of
pumping upon the thermal diffusivity. As discussed before, the thermal
diffusivity at any point is a local measure of the ability of the water to dis-
perse heat. It is determined by the local stability of the lake and by the
competing effect of turbulent mixing. Pumping alone increases the dispersive
capability of the lake. A thermal discharge introduces both of these competing
effects: the pumping work performed by the discharge increases the mixing,
while the additional heat introduced near the surface increases the stability
The first set of curves, at 90 days, display several differences
between the effects produced by pumping and those produced by a thermal
discharge. The thermal diffusivity calculated with pumping effects alone is
72
-------
larger than either the natural diffusivity profile or the diffusivity with thermal
discharge effects. Since mixing only is introduced by pumping, these results
are expected. On the other hand, with increasing depth, the thermal diffusivity
including the effects of thermal discharge is first somewhat larger and then
smaller than the natural thermal diffusivity. This result shows that increased
mixing is the dominant effect at smaller depths whereas increased stability
due to the additional discharge heat dominates at greater depths.
The second set of curves is shown for 240 days. At this time con-
vective turbulence due to cooling, as well as wind-induced turbulence, is
producing mixing. The thermal diffusivity with the effects of thermal dis-
charge and the diffusivity with the effects of pumping are very similar, and
considerably larger than the natural diffusivity. At this time a very poor
approximation to the diffusivity with thermal-discharge effects included is
provided by the natural diffusivity.
The last set of curves shows the thermal diffusivities at 300 days.
At this time the lake is mixed throughout. The thermal diffusivity with dis-
charge effects is largest now. Both this diffusivity and the one with pumping
effects included are somewhat larger than the natural diffusivity throughout
the lake.
In this section the effects on the thermal structure of thermal dis-
charges and of pumping water from deep levels to the surface have been
discussed. The effect of the plume induced by the thermal discharge has
been considered only in relation to its effects on overall thermal structure.
A model, which includes mechanical work and thermal energy produced by
a discharge, has been introduced to calculate the changes in thermal struc-
ture.
This theoretical model has been used specifically to study the effects
of thermal discharges at or below the level of the thermocline on the seasonal
stratification cycle of temperate lakes. The major advantage of a hypolimnetic
discharge is that the effluents can be trapped below the thermocline at least
during part of the stratification season, so that surface effects of the dis-
charges can be expected to be minimized. The calculations presented in
72a
-------
this section show that the increases in temperatures, due to the thermal
discharges, are indeed greater at deeper levels than at the surface. How-
ever; some increase in the average surface temperature of the lake is
unavoidable, since such an increase is essential (once the lake attains a
new thermal equilibrium) for the lake to dissipate the added power-plant
16
heat to the atmosphere.
Several other conclusions have been drawn from these calculations;
the most important ones follow. A thermal discharge increases the tem-
perature at any depth and at any time over the temperature occurring
naturally. However, the relative magnitudes of the increases at various
depths and times are dependent upon the specific mode of discharge. Pump-
ing alone will increase the temperature in the hypolimnion and decrease it
in the epilimnion during stratification. However, during the cooling portion
of the cycle when the lake is well mixed, the temperature will be increased
over the naturally occurring value. Pumping increases the depth of the
thermocline over most of the stratification cycle. However, it may increase
the stratification period. For the thermal discharge calculations performed,
the depth of the thermocline was increased for most of the stratification cycle
beyond the depth occurring naturally. The stratification period was found to
increase with a thermal discharge.
72b
-------
VI. STUDY OF INTERFACIAL MIXING
VI. 1 Introduction
This section deals with a study of the flow of a layer of warm water
over a body of colder water. The resulting interaction manifests itself by
heat and momentum transfer between the two layers. In laminar, homogeneous
flow, these processes are characterized by the thermal diffusivity ft and
kinematic viscosity 3} which are related through the properties of molecular
motion. The resultant similarity of these processes is generally known as
Reynolds analogy. Flows in natural bodies of water are almost always turbu-
lent, and the interaction of adjacent layers is then determined by eddies of
various sizes. Such flows may be analyzed if K and i) are replaced by
appropriate eddy diffusivities for heat, KH , and for momentum KMo ,
which may be several orders of magnitude larger than the corresponding
molecular quantities.
If the density of the upper layer is less than that of the lower body of
water, the stratification is stable, and buoyancy tends to inhibit the mixing
process. Consequently, a rather marked interface appears across which heat
and momentum transfer may be inhibited to different degrees. The Reynolds
analogy is therefore not applicable to stratified flow.
The stability of a stratified flow is customarily expressed by a
Richardson number R'(, which may be formulated in different ways. The
overall or bulk Richardson number is essentially the ratio of the buoyancy
to the inertia forces and is given by
= avAT uS the value of the (small) density difference
between the upper layer and the lower body of water, ~n is the depth andu,the
average velocity of the upper layer, and Q the gravitational acceleration.
73
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If the density difference is expressed by the corresponding temperature
difference AT , the second form is obtained where CLV is the coefficient of
volumetric expansion.
In a stratified flow, the eddy diffusivities KHo and ^MO of a flow
of uniform temperature have to be modified to KH and KM which can be
16,68
related to the Richardson number in the general form
KH = KHo f (R.) (VL 2)
KM ^ K»o q(K) (VI. 3)
a
where Rx, is defined by Eq. (III. 5) and the functions f (Rt) and a (Ri) have
the property that f (o) = <3(o) = '.
The purpose of the present study is to determine the values of KH
and KM at the interface for different bulk Richardson numbers. At first, an
analytical approach was considered, based on linearization of the Navier-Stokes
equations, but a direct experimental approach seemed to be much more
promising at the present time.
The experimental approach is related to that used by Ellison and
Turner who determined entrainment parameters rather than eddy diffusiv-
ities. They studied the spreading of a thin layer of low-density fluid flowing
over a weir onto a channel of stagnant fluid of higher density. The duration
of their experiments was limited to a few seconds, and velocity distributions
were inferred from motion pictures of rising small plastic particles released
from the bottom of the tank. A somewhat different approach to measure
entrainment was used by Kato and Phillips who used an annular tank at
the surface of which a constant stress was applied by a slightly immersed
ring rotated by an external drive. The shear stress was measured by the
torque applied, and fluid motion was determined from motion pictures of the
flow visualized by dye injection. Ellison and Turner found that entrainment
74
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is substantially suppressed if the Richardson number exceeds approximately
*
0.8. Since Ellison and Turner produced stratification by the use of salt
solutions of various concentrations, no temperature measurements were
involved in their experiments.
In the present study, stratification is produced by temperature
gradients under conditions which allow observations over periods ranging
from a few minutes to about one hour. The techniques and results obtained
so far are discussed in the following sections.
VI. Z Theoretical Foundation for the Experimental Determination of KM
and K H
The quantities KM and KH cannot be measured directly but must
be computed from observable data on the basis of definitions and basic rela-
tionships .
Consider a horizontal channel of constant width in which a layer of
warm water flows over initially cool water in the lower part of the channel.
The only vertical velocities W in this system are produced by entrainment
of the cool water by the warm water (see Refs. 57 and 66) and by effects of
the end -walls of the channel.
The determination of KM involves solving the continuity and momen-
tum equations for two-dimensional flow.
Continuity:
+ = 0
Horizontal (%) Momentum:
* _L il = ±12. >t± +ut± + u,<>± (vi. 5)
9 32
-------
Vertical ( 2 ) Momentum:
KH is defined by
T
3%
- it.
(VI. 6)
^
2
(VI. 7)
and
by
C~ _ u u. _ - a
" M ~ "" ~
(VI. 8)
where % is the horizontal coordinate, positive in the direction of the warm
water flow, Z is the vertical coordinate, measured positive upward from the
tank bottom, u. is the % -component of velocity and uJ the ~i -component,
U.' and u) ' are the turbulent fluctuations of the velocity, ~b is pressure,
£> is density, t is time, T is the shear stress, CT is the normal stress,
and Q is the acceleration due to gravity.
a
For the experimental system described in Section VI. 3 all terms in
the vertical momentum equation (VI. 6) are small, except for the pressure
gradient and the acceleration due to gravity because of the near horizontal
homogeneity of the flow in the present experiment. Hence, (VI. 6) becomes
9-f> _ (VI. 9)
"31 " '
-------
The first step in the calculation of KH is the determination of the
vertical velocity to . From equation (VI. 4), and the experimental velocity
f\
gradient , uJ can be calculated as a function of 2. with the boundary
0 %
conditions that UJ = 0 at Z 0 , the tank bottom. From measured velocity
profiles, the substantial derivative
n)
Dt at 9%
f\ -to
can then be calculated. The pressure gradient . v is determined from the
Q fa
experimental data and equation (VI. 9) as discussed in Section VI. 5. Integra
tion of equation (VI. 10) then yields the shear stress T and use of equation
(VI. 7) gives KM .
The determination of the coefficient of thermal diffusivity KH
involves solution of the energy equation
4 1 st 0 T" 3 T O T M7" T 1 9 \
104. ff I d I 0 I (\L.i£)
* -j- UL + tO
P -to 9Z 0 ~t 0 % 3^
where <^. is the heat transfer rate and T the temperature. The various
gradients and parameters can be calculated from the experimental data as a
function of depth. The energy equation is then integrated from the tank
bottom upward to give the heat transfer ^ at each depth and KH is deter-
mined from the definition:
(VI.
The preceding analysis indicates that KH and KM can be evaluated
if T and u. are measured as functions of t , # and 2 . Practically, this
requires suitable probes which allow the temperature and velocity to be
measured as a function of time over1, the entire depth at several stations along
the flow.
77
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VI. 3 Design of the Flow System
An experimental setup is needed which can produce a substantially
two-dimensional flow of a layer of warm water over a pool of cool water.
This requirement can be met by a relatively long tank of rectangular cross
section. The warm layer should be sufficiently deep to allow observations
within the layer and its velocity high enough to make the flow turbulent. Flow
in shallow water becomes turbulent if the Reynolds number based on the depth
of the water exceeds about 300 . The Reynolds number of the layer of warrr
water is given by
Ro = Al± = -A, (VI. 14)
where Q is the volume flow rate and v the width of the tank. Thus, the
depth of the layer can be varied without change of Reynolds number if the
flow rate is kept constant.
An existing tank (210 cm long, 10 cm wide and 10 cm deep) with a
transparent lucite front wall was adapted for the experiments, and Fig. 28
is a schematic layout of the flow system. Two partitions form the cool-water
reservoir which is 170 cm long and 7 cm deep. The warm water enters into
the right end section where a baffle insures uniform flow over the partition.
After flowing over the left partition into the other end section, the water
returns into a warm-water reservior. The flow rate can be controlled by
valve No. 1 and measured by a rotameter. It was initially planned to control
the depth of the layer by means of a gate at the left partition. When the gate
turned out to be too difficult to adjust, the partition, as shown in the figure,
was raised simply by attaching blocks of fixed height on top of it.
The warm-water reservoir (5 gallons) was kept well stirred by an
electric stirrer and the temperature controlled by a thermostatically regulated
electric immersion heater (450 watt). The water was circulated by a small
78
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pump. An auxiliary return from the right end section to the reservoir was
provided through valve No. 2. The purpose of this by-pass is to bring the
water and pipes to the desired temperature without disturbing the cool water
in the main tank. When all conditions are established, valve No. 2 is closed,
and flow over the partition begins. The maximum flow rate that can be
measured with the present rotj
mum Reynolds number of 700.
measured with the present rotameter is 60 cm /s corresponding to a maxi
The cool water was kept at approximately room temperature. In
the course of an experiment, the cool water becomes partly mixed with warm
water, and it would require several hours to let it cool down for another
experiment. To reduce this waiting time, a supply of cool water was kept
in a second tank where it was maintained at the desired temperature by means
of a thermostatically controlled immersion heater, a cooling coil through
which cold tap water flows and a stirrer. This reservoir and a second
circulating pump were separated from the main tank by valves No. 3 and 4.
Overflow pipes and an auxiliary container (not shown in the figure) prevented
accidental spilling of the fluid.
The temperature of the warm water was measured by a thermometer
upstream of the inlet partition and that in the main tank by a thermometer near
the outlet.
VI. 4 Instrumentation
VI. 4.1 Velocity Measurements
Local flow velocities can be measured by means of tracers which
can be observed photographically. Because of the density stratification, it
is important that such tracers are neutrally buoyant regardless of their
location in the fluid. A technique which seems particularly suitable for this
purpose is based on the color change of an indicator solution if the hydrogen
ion concentration (Jo H ) is changed at a test point.
79
-------
67a
The technique used is an adaptation of a system described by Baker
Consider an indicator solution which exhibits a marked color change at a
-jo hi value that is characteristic for the indicator. If two electrodes connected
to a battery are inserted into the fluid, the hydrogen ion concentration is
reduced at the surface of the cathode corresponding to a local increase of
Jfr H . Under properly selected conditions, the fluid adjacent to the surface
of the cathode then changes color and subsequently moves with the rest of the
fluid. Since nothing is added to the fluid, this tracer is always neutrally
buoyant. Gradual diffusion increases the -p W value within the tracer fluid
which, eventually, returns to its original color. A solution of the following
composition was found to be satisfactory:
Distilled Water 1000 cm
3
Thymol Blue (1% solution) 10 cm
Hydrochloric Acid (0. 1 normal) approx. 1 cm
Before the acid is added, the thymol blue solution is deep blue. Acid is then
added drop by drop under continuous stirring until the solution suddenly
changes to a bright orange at a -io H value of about 8.
To produce as sharp a tracer line as possible, fine tungsten wire
(about 25jUm diameter) is used as cathode. One end is cemented into a
short piece of ceramic tubing (about 3 mm diameter and 5 mm long) which
is pressed into a hole in the bottom of the tank. The other end of the wire is
soldered to a thin rod held in an adjustable clamp above the surface of the
water. In this manner, the wire can be stretched vertically from the bottom
to the top of the water without disturbing the flow. The anode is formed by
a stainless-steel wire running along an inside edge of the tank.
If a 90-volt battery is briefly connected to the two electrodes, a
column of dark tracer fluid is formed on the surface of the tungsten wire
(cathode) and subsequently moves with the fluid. To produce a well-defined
tracer, the current pulse must be sufficiently short (about 1/3 second).
80
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Furthermore, long pulses lead to considerable formation of gas bubbles by
electrolysis; these are almost completely absent with short pulses. The
motion of the tracer is recorded photographically as discussed below.
The probe wires had an unexpectedly short life which varied between
a few weeks and a few days. They apparently corroded under the combined
influence of the electric current and the slight alkalinity of the solution. The
exact reason for their failure is not known. In the future, heavier wires or
wires of a different material, such as platinum, will be used. Current
pulses are produced by one cam of a multiple-cam switch (Industrial Timer
Corp. ). The cams are adjustable so that a microswitch is closed once
during each revolution for a selected time. The speed of the cam can be
adjusted in steps by a choice of gears between the drive motor and the cam
shaft. By means of an auxiliary switch in the cam-drive circuit, it is possible
to let the cam rotate continuously or for one revolution only. For continuous
rotation, the selected interval between pulses is 2.67 seconds. This time is
convenient to record the velocity in the warm upper layer of water, which is
of the order of 1 cm/s. The velocity in the lower layers is about ten times
smaller and would therefore require either many current pulses or a longer
separation between pulses. Since many tracers on one photograph make the
records difficult to evaluate, the following method was adopted. At the time
tf , one pulse was produced by a single revolution of the cam. About 30
seconds later, two more pulses were produced by two revolutions of the cam
at the times tz and t3 = t2 + 2.67 where the interval t3 - t, was measured
with a stopwatch. The camera shutter was operated by a solenoid which has
two switches in series in its power supply. One of these is another cam
switch which was set so that it closes always at the instant when a flow
visualization pulse is produced by the first cam. The other switch was
closed manually after the time t2 and before t3 . In this manner, the
third pulse delineated the probe wire which otherwise would not be visible
in the photographs. Thus, the low velocities in the lower layers of the water
are determined by the displacement of the first tracer during the time t3 t1
and the high velocities near the surface by the displacement of the second
-------
tracer during the time t3 ~ t2 .
Seven probes were installed along the centerline of the tank. Probe
No. 1 was located about 5 cm downstream from the inlet partition and probe
No. 7 about 25 cm upstream from the outlet partition. These probes served
only to indicate the flow near the ends of the tank and were not photographed.
Probes No. Z to 6 were mounted 20 cm apart in the center of the tank.
An exploratory attempt was made to observe the vertical entrainment
velocity by producing interrupted tracers. This was achieved by slipping a
short piece of a finely drawn glass capillary (about 0.1 mm outside diameter)
over one of the tungsten wires and annealing it on the wire with a fine flame.
Although this probe indicated a vertical velocity, this configuration was not
satisfactory for quantitative measurements.
To take photographs of the flow patterns, the tank was illuminated
by two 500-watt photoflood lights. The camera was placed at the level of the
center'of the water at a distance of about 140 cm from the front of the tank
and opposite the center probe. A Wollensak Raptar f/4. 5 lens of 162 mm
focal length was used. Photographs were taken on 4 in. by 5 in. sheets of
Plux-X film (ASA 125) at f/5. 6 and 1/50 s. The relevant portion of the
photograph was enlarged for evaluation, and several such records are
shown in Section VI. 5.
Taking photographs of an object under water when the camera is in
air leads to a displacement of the object on the photograph because of the
refraction of light at the air-water interface. Care must therefore be
exercised in the evaluation of the photographs to avoid errors. Analysis
of this problem is based on Fig. 29. The lucite front wall of the tank is
sufficiently thin (6 mm) that its effect on the path of the light rays need not
be considered.
82
-------
r
Let the camera lens be located at C at the distance L+ -~- from
the center of the tank 0 . If the line OC forms a right angle with the
centerline of the tank, the point 0 is not displaced on the photograph, but a
point P located at the distance X from 0 appears on the photograph as P
at an apparent distance X . The angle of incidence of the ray PC is &
on the side of the air and /3 on the side of the liquid. Let J. be the distance
of the point of incidence on the interface from the line OC . The law of
refraction then yields
Sin a (VI. 15)
= N
Sin /3
where N is the refractive index of the liquid. Elementary trigonometric
relationships show that
X = L *an OC + (VL
a
X' = (L + ~] ia,n. OC
These two equations may be combined to give the relative refraction error as
AX = X- X _ 1 /t _ -ttw.fi . (VI. 17)
X' " X' " I + 2 '
For any value of Od , the corresponding value of p then follows from equation
(VI. 15), the distance X' from the second of equations (VI. 16), and the
refraction error from equation (VI. 17). Some results are shown in Fig. 30
for water ( M = 4/3) and the present experimental conditions ( £ =10 cm).
For the magnitudes of the errors involved here, it is immaterial whether
the refraction error is referred to X or to X .
83
-------
Figure 30 shows that a tracer located at 50 cm from the center of
the tank appears displaced by about 0. 5 cm (1%) if the photograph is taken
from a distance of about 140 cm. It is therefore not permissible to assume
that a photograph represents merely a size reduction in the ratio of the
image distance to the object distance from the camera lens. Fortunately,
the maximum distances from the center of the tank that are of interest here
do not exceed about 50 cm, and within this range, the relative refraction
error is practically independent of X if the camera lens is located 100 cm or
more from the tank. The known distance between probe wires therefore may
serve as a scale with which tracer motions can be measured on the photographs
and which is already corrected for the refraction error. For this reason, it
is important to have the probe wires outlined on the photograph, and this is
achieved by synchronizing the camera shutter with the tracer pulses in the
manner described in the foregoing.
If photographs of the five central probes are taken at various times
during an experiment, both the velocity and its derivatives d^/3t and
du./d% can be obtained as functions of the depth below the water surface in
accordance with the requirements indicated by the equations in Section VI. 2.
VI. 4.2 Temperature Measurements
The requirement that the temperature be measured at several
stations over the entire depth of the flow can be satisfied by small probes
which traverse the depth of the water at selected locations. The time
dependence of the temperature distribution can then be obtained by repeating
the traverses at regular intervals. A convenient method is to mount a small
thermistor bead at the tip of a thin tube and record the resistance changes
that result from temperature variations.
The temperature sensitive element of the probes is a bead thermistor
(VECO No. 51A32) which has a diameter of about 0. 3 mm.
84
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The bridge circuit constructed to measure the resistance of the
thermistor while it traverses the flow is shown in Fig. 31 . Two arms of
the bridge are the probe resistance R and a load resistance RL ; the other
two arms are formed by R., and Re with the additional potentiometer /?3
to provide a zero adjustment. A conventional power supply, regulated by
two Zener diodes, provides 6. 2 volts to the bridge through the potentiometer
circuit formed by R4 , Rg and R^ . The setting of Rs controls the actual
bridge voltage and thus allows the sensitivity to be adjusted to a desired
value. The purpose of the condenser across the recorder input is to
eliminate high-frequency noise pickup by the probe circuit.
The variations of the thermistor resistance with temperature are
highly nonlinear as indicated in the following table
T R
°C Ohms
0 330,000
25 100,000
50 35, 000
A satisfactory linear response can be obtained by proper choice of the load
resistance RL For three selected temperatures T0 and T0 ± AT ,
which cover the range of interest, let the corresponding values of the
thermistor resistance be R0 , R+ and R_ . The fraction of the bridge
voltage which is delivered to the recorder from the thermistor side of the
bridge is given by RL /( R + RL ). Therefore, if RL is chosen so that the
condition
is satisfied, the output voltages, which correspond to the temperatures
T0 and T0 ± AT , are linearly related to these temperatures. If the entire
range is sufficiently small, intermediate points are also close to the
linear calibration. The exact value of RL is not critical because a
85
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small change merely places three other points on a straight line. As will be
seen below, a value R|_ = 100,000 ohms is satisfactory.
The short platinum-iridium leads to the thermistor are welded to
thin copper wires which are then threaded through a ceramic tube of about
2 mm diameter. The bead is attached to the tip by Conap epoxy and is also
given a thin coating of the same material to insulate it electrically when the
probe is immersed in water. An enlarged photograph of the probe tip and a
millimeter scale are shown in Fig. 32. It was found that water gradually
penetrated through the walls of the ceramic tube. To prevent the resultant
falsification of the thermistor resistance, the entire probe was coated with
epoxy. For the last probe made, the ceramic tube was replaced by a glass
tube of same size. This probe is more delicate than the others but does not
require coating. Although the probes are electrically insulated from the
water, the circuit resistance is sufficiently high to pick up noise signals
which are eliminated by grounding of the water.
The design of the probes is shown in Fig. 33 . The ceramic tube
referred to in the foregoing is about 12 cm long to reach from the bottom of
the flow to a little distance above the surface of the water. The top end is
cemented into a glass tube of about 4-mm diameter and 27-cm length. This
tube can slide through a 10-cm long stainless steel bearing tube which is
mounted vertically at the desired probe location several centimeters above
the water surface. The wires from the thermistor are threaded to the top
of the glass tube and are then connected to the described bridge circuit. By
pulling at a string attached to the top of the glass tube, the probe can be
raised or lowered. A metal rod is mounted horizontally above each probe,
and the strings from all the probes can be brought close together by guiding
them over these rods. A groove cut into the rods prevents side slipping.
The rods are slightly staggered, as indicated in Fig. 33 , to prevent
entangling of the strings. It is important that the length of the strings, which
is different for each probe, remains constant and not be affected by humidity.
A fine stainless-steel wire (about 0.1 mm) is therefore used except for a
86
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short piece of soft nylon string for the part that slides over the guide rods;
the steel wire turned out to be too stiff to follow the required bend without
jerking. A small weight is attached to the top of each probe to keep the
string under tension and insure a smooth motion.
A photograph of the drive mechanism is shown in Fig. 34 . The
reversible, variable-speed motor (Gerald K. Heller Co. , Model 2T60-6)
drives a screw which is long enough for the required travel of the probes.
The moving element is guided by two rigid brass rods. Two adjustable stops
activate a reversing switch, so that the probes traverse the water tank up
and down (at a rate of . 3 cm/s) as long as the motor is kept running. One
stop is adjusted to bring the probes to within about 2 mm from the bottom of
the tank, and the other insures that they are out of the water at the highest
point of their travel. All probe tips must touch the water surface at the same
time, the necessary fine adjustment of the wire length is provided by attaching
each wire to the drive mechanism by means of a screw which can be threaded
in or out as needed. After the adjustment is made, the position is held by a
locknut.
Six temperature probes are installed side by side with the velocity
probes Nos. 1 to 6 (see preceding section). Each is connected to its own
bridge circuit, and the outputs are recorded on a six-channel pen recorder
(Brush-Clevite Mark 260). This instrument is also provided with four
event markers which operate with an internal supply of -32 volts with respect
to the chassis; closing of a contact produces a small deflection of an auxiliary
pen. One of these markers is used in conjunction with the temperature
records to indicate the instant when the probe tips touch the water and to
provide reference marks that represent a known travel of the probes. Two
more event markers are used to coordinate the temperature and velocity
measurements (see Section VI. 4.3).
The instant of probe contact with the water is established by a
special probe which is similar to the temperature probes except that the
87
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stem with the thermistor tip is replaced by a steel needle. This probe is
located at the position of the velocity probe No. 7. When the needle tip
touches the water surface, the resistance between the needle and the grounded
water is about 5000 ohms which is too high to operate the event marker. An
electronic relay was therefore constructed based on the circuit shown in
Fig. 35 where the values of the various components are indicated. The main
element is the silicon controlled rectifier (SCR) 2N2323 which has a high
resistance in the untriggered state. The current through the needle probe
while it is immersed in the water is amplified by a simple transistor
amplifier (2N697) and fed to the gate electrode of the SCR, the resistance
of which then drops to a low value. As soon as the probe touches the surface
of the water, the voltage on the condenser drops then suddenly from ground
potential to approximately -32 volts. This pulse is transmitted to the recorder
and produces a brief deflection of the event marker. As soon as the condenser
is charged, the residual current through the 22, 000 ohm resistance is smaller
than the holding current of the SCR which then reverts to its untriggered state.
If the gate current is still on the SCR triggers again momentarily, but the
resultant oscillations are too small to operate the event marker. These
oscillations stop when the probe tip leaves the water, and the initial condi
tions are restored. In this manner, the event marker indicates only the
instance when the probe touches the water but not when it is withdrawn.
The moving element of the probe drive also operates a microswitch
by two adjustable knobs visible in Fig. 34 . This switch is connected to the
same event marker and thus indicates the instant when the knobs pass the
switch. They are adjusted to produce marks on the record which represent
a probe travel of 5 cm.
The probes were calibrated by immersing them in water of known
temperature, and the results for one of them are shown in Fig. 36 for the
range from 25 °C to 35°C needed for the experiments. The calibration lines
for the other probes are substantially similar. Each recording channel is
50 divisions wide (4 cm), and the slope of the calibration line varied between
88
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5.3 and 5.5 div/°C for a recorder sensitivity setting of 2 mV/div. After
this calibration was obtained, the calibration voltage for each probe (see
Fig. 31 ) was adjusted with the help of a voltmeter to make all calibrations
equal to 5.0 div/°C. The probe output is therefore 10 mV/°C, and a tem-
perature range of 10 °C corresponds to the full width of each channel.
Another probe characteristic of importance is the rate at which the
thermistors can follow changes in temperature. This information was obtained
by dipping the probes quickly into warm water and recording their response at
the highest writing speed of the recorder (125 mm/s). The response of all
probes is approximately exponential with a relaxation time (the time to reach
1-1/e of the total rise) that lies between 56 and 68 milliseconds. A discontinu-
ous temperature would therefore be correctly indicated after about one-fifth
of one second.
In view of the foregoing results, temperature traverses were made
at about 0. 3 cm/s with a writing speed of 125 mm/min. Under these condi-
tions, errors caused by the lag of the thermistor response are negligible.
The probe drive exhibited some hysteresis in traversing the water
in the upward and downward direction. Since the instant of contact with the
water surface is recorded during the downward motion, all temperature
records are evaluated only for this part of the traverse. Typical temperature
recordings are shown in Section VI. 5.
VI. 4. 3 Time Coordination of Temperature and Velocity Measurements
After starting the flow of the warm water, the experiment continued
for many (up to 90} minutes. During this time several photographs and
temperature traverses had to be taken. It was therefore important to
record when each record was obtained. For this purpose, an electric clock
was mounted over the water tank so that it appears on each photograph. The
clock was started when the warm-water flow begins.
89
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In addition, the paper of the temperature recorder was kept running
during the entire experiment, and the instant of taking a photograph was
marked by the second of the four event markers. To minimize the amount
of recording paper to be handled, the writing speed was reduced to 25 mm/
min between temperature traverses. A record of time was kept by means
of a third event marker operated by a timer which closed the contact of this
event marker briefly once every minute.
In the described manner, the time at which each temperature and
velocity record was obtained can be easily identified.
VI. 4.4 Motion Picture of the Experiment
To demonstrate the instrumentation and to provide a vivid impression
of the processes, a 16-mm color motion picture was taken. The movie camera
was set at about the same place as the camera used for the flow visualization
photographs. The movie was taken on Ektachrome film (ASA 125) at 24 frames
per second at f/8.
The temperatures and velocities are about the same as for the experi-
ment discussed in Section VI. 5.5. Several sequences were filmed, each
lasting about 30 seconds with intervals of about 10 minutes between sequences.
The gradual progress of the mixing process is clearly demonstrated. In
addition, a temperature traverse was photographed to demonstrate the
operation of the recorder.
VI. 5 Experimental Results
VI. 5.1 Velocity and Temperature Records
A selected portion of the data obtained during the experiments is
presented in Figs. 37 and 38 for two values of the initial Richardson
number Rio =3.0 and 0. 1. The Richardson number Ri is defined in
90
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equation (VI. 1) as
Rl
(VI. 1)
The initial value Rio is determined using the temperature difference AT ,
the hot water depth "& and the flow velocity U. at the inlet of the cold water
tank. The local Richardson number at any point, Ri£ , can be considerably
different from Ri0f but R~LC, will be retained as a label to differentiate
between the two sets of data. The values of the various parameters in the
two experiments are given in Table 3.
TABLE 3
FLOW PARAMETERS
Initial Richardson number
Temperature difference
Hot water height at inlet
Hot water flow rate
Flow velocity at inlet
Tank width
Reynolds number
3. 0
8
2. 8
37
1. 3
10. 1
440
0.1
2.7
1. 7
59
3.4
10. 1
700
- -
°C
cm
cc Is ec
cm/ sec
cm
_ _
Figures 37a and b consist of photographs taken at different times of
the flow between station 2 and 6. Station 2 is 45 cm from the inlet and the
others are spaced at 20-cm intervals. Time is measured from the moment
the warm water begins to flow over the inlet weir. The neutrally buoyant
pulses of dark thymol blue vividly trace the motion of the water. The time
between pulses was 2. 67 sec in the upper layer of warm water and 30 sec
for the low-velocity reverse flow in the cool water. The velocity in the upper
layer is slightly greater than 1 cm/sec while the reverse flow in the lower
layer is about .05 cm/sec.
91
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The refractive index of the warm water is different from that of the
cold water, and hence the warm layer is visible in Fig. 37a as a slightly
darker region. For the Ri0 = 3.0 tests, the layer of warm water penetrates
very slowly into the cold water and at 90 minutes has reached halfway to the
tank bottom. For comparison, the R J<> = 0. 1 photographs in Fig. 37b
reveal that the warm water has penetrated more than halfway to the bottom
by 5. 7 minutes.
The temperature records given in Fig. 38 show the growth of the
warm water region at the expense of the cool water at selected times and
positions in the tank. The location of the surface and the 5-cm calibration
marks discussed in Section VI. 4.2 are shown in the figure. Station 1 is 5 cm
from the inlet weir. The rapid decay of the temperature gradient for the
R<^0 =0.1 tests compared with the Ri6 =3.0 ones is striking. At 16.2
minutes, for R i0 =0.1 (Fig. 38b ), the warm water extends uniformly to
the bottom at station 1, whereas for Rl. = 3. 0 at 15 minutes (Fig. 38a )
a sharp temperature gradient roughly 1 cm in depth distinctly separates the
4-cm deep layer of warm water flowing over the cool water.
The temperature extremes between the hot and cold water are
plotted in Fig. 39 for the two experiments as a function of time. In both
cases, the hot water temperature remained fairly constant. The temperature
of the cold water at the tank bottom started to increase significantly at about
35 minutes for Ri0 - 3. 0 at position 4 and 10 minutes for Ri = 0. 1 at
position 5. At later times, the increase in temperature of the water at the
bottom indicates a significant heat transfer to the floor of the tank, and
data reduction at these locations should therefore be restricted to earlier
times.
VI. 5.2 Interface Location
The depth at which the maximum temperature gradient occurs is
defined as the interface in the present report and corresponds with the
boundary between the light and dark regions observed in Fig. 37a
92
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The location of the interface is plotted in Fig. 40a for Ri0 = 3.0 and
Fig. 40b for Ru0 =0. 1. At the top of each figure, a schematic drawing
of the cool-water tank gives the true slope of the interface relative to the
flow. In the main portion of the figures, the vertical dimensions are
exaggerated by a factor of ten compared with the horizontal dimension. The
exit weir was made 1. 9 cm higher than the inlet for the RLO = 3.0 tests and
0. 7 cm for Ri0 = 0. 1 to obtain the warm water heights at the inlet of 2. 8 and
1.7 cm respectively.
As shown in Fig. 40 , the depth of the interface between the warm
and cool water increases with time and distance downstream from the inlet
weir up to about position 2, The rate of increase in interface depth for the
R10 = 0. 1 experiment was about ten times that for the R i 0 =3.0 tests. By
position 2, the interface depth, although still increasing with time, begins to
decrease with increasing distance from the inlet. This is caused by the fact
that the exit weir is considerably higher than the inlet. Consequently, the
flow streamlines near the interface will be inclined upwards and the warm
water at the top will accelerate to satisfy the equation of continuity.
A local Richardson number is defined as
_ a -A ,
where the subscript J. denotes local values. Using values for -&^ from
Fig. 40 , RLi was calculated as a function of distance from the inlet
3.4 minutes after the start of flow. The results given in Fig. 41 show
that for the Ri0 =3.0 experiment, the local Richardson number R^ does
not change much with distance because mixing is strongly inhibited by
buoyancy. However, for RLO =0.1, mixing is so rapid that Ri ^ at
position 2 is greater than 3. 0.
The implication of this result is that measurements at very low
Richardson numbers require more closely spaced instrumentation much
nearer to the inlet than in the present experimental setup. Measuring
stations every 2. 5 cm in the first 15 cm downstream of the inlet weir would
93
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make it possible to study the details of the mixing process for local
Richardson numbers varying from 0. 1 or lower at the inlet to 1. 0 at 15 cm.
The existing apparatus is best suited for the study of flows with Richardson
numbers greater than about one.
VI. 5.3 Flow Starting Process
The manner in which the warm water flow is initiated, coupled with
the confining walls of the water tank, has a large effect on the velocity and
temperature profiles at later times. Consideration must be given to these
perturbations before interpretation and analysis of the data is possible. The
starting process for Ri0 =3.0 can be visualized through use of the photo-
graphs in Fig. 42 . For this experiment, the cold water was replaced by a
clear, slightly alkaline solution containing no thymol blue. The orange-
colored warm water, containing thymol blue, turns dark blue, as shown in
Fig. 42 , when it mixes with cold water.
The experiments were started by allowing the warm water to overflow
the inlet weir (see Fig, 4Qa ) at the volume flow rate Q. which was determined
by the 1. 3 cm/sec velocity required at an inlet warm water height of 2. 8 cm to
give a Richardson number of 3. 0. Because the height of the warm water was
zero at time zero, the initial flow velocity was considerably higher. For
example, in Fig. 42 , by 1/4 minute the dark region indidating the extent of
the warm water has travelled slightly past position 3, or a distance of about
70 cm from the inlet. This corresponds to an average flow velocity of nearly
5 cm/sec. For the observed warm water height of 0.45 cm at 1/4 minute in
Fig. 42 , the average Richardson number during the first 15 seconds of the
experiment from equation (VI. l)was approximately 0.04. This very low value
explains why the mixing region in Fig, 42 extends all the way to the tank
bottom upstream of position 3 at 1/4 and 3/4 minutes.
As the level of the warm water rises, the forward velocity of the flow
decreases. Between 1/4 and 3/4 minutes, the level in Fig. 42 increased from
0.45 to 1.15 cm, and the flow velocity dropped from 5 cm/sec to 1 cm/sec.
94
-------
The resultant increase in average Richardson number from 0. 04 to about
2.2 inhibits mixing between the warm and cold water, and the dark region
between positions 3 and 5 does not penetrate as deeply into the tank, as
shown in Fig. 42 . During this time, the cold water is displaced to the
downstream end of the tank and its level rises as the warm water is added
because the exit weir, Fig. 42a, is 1. 9 cm higher than the inlet weir. At
about 11/2 minutes, the water overflows the exit weir, and some cold water
is lost before the warm water reaches the exit at approximately 2 minutes.
The initially large mixing near the inlet during starting coupled with
the extra height of the exit weir and the rapid damping of the mixing as the
water level rises produces a triangular wedge of cold water in the tank as
shown in Fig. 42 . The vertical base of the triangle is along the exit weir
and the apex at 3/4 minute is located at about position 2. This situation cannot
persist because of buoyancy. The heavy cold water at the exit pushes up-
stream along the bottom displacing the hot water upward. This can be
observed in Fig. 42 at 1 1/4 and 21/2 minutes as a propagation upstream
of the light region with a corresponding decrease in depth of the dark region.
In fact at 3 1/2 minutes, the movement of the cold water at the bottom from
the exit to the inlet appears to have overshot the equilibrium level. There are
indications that warm water has mixed with cold at quite large depths at
station 6. This can also be observed in the velocity photographs, Fig. 37a
at 3 minutes where the dark region at station 6 is quite a bit deeper than at
station 3, demonstrating the reversal of warm water at the exit weir. By
about 61/2 minutes, Fig. 42 , indicates that equilibrium between the
warm and cold water has been reached, and the interface between the light
and dark regions occurs at the depth at which the temperature in Fig. 38a
first begins to rise above the cold-water value.
Large reverse flow velocities result from the upstream displacement
of the cold water at the tank bottom as observed in the velocity photographs of
Fig. 37a at 3 minutes. The peak reverse flow velocity at station 4 is plotted
95
-------
in Fig. 43 as a function of time after the start of flow. Transient velocities
of 0.4 cm/sec at 2 1/2 minutes die out to about .05 cm/sec by 5 minutes.
This large circulation in the tank during starting is caused by the high exit
weir and the low Richardson number during the first 15 sec of the experiment.
After the data had been obtained using the present experimental
apparatus, trial tests were performed with the exit and inlet weirs at the same
height and a reduced flow rate during the first few minutes. Reduction in the
starting transient was observed and future tests will use this technique
possibly with additional modification such as a flat plate mounted on top of the
inlet weir. For low Richardson number tests, the significance of the starting
effect will be considerably reduced since the Richardson number at which data
are required can be made the same as that attained during starting. In addi-
tion, measurements probably can be made before the influence of the exit
weir is felt.
VI. 5.4 End and Sidewall Effects
From the velocity photographs in Fig. 37a , it is clear that the
warm water flow accelerates with distance from the inlet. The surface
velocity in Fig. 44 was determined from records such as Fig. 37a
and by timing surface floats over measured distances. An increase from
0.6 cm/sec at position 1 to 1. 3 at position 6 was observed. However, below
the surface the variation in velocity with distance from the inlet was not as
large.
The flow accelerations can also be seen in Fig. 45 where the
velocity-depth profiles are plotted for Ri0 = 3.0 at positions 3, 4, and 5 at
30 minutes. The velocity increases with distance from the inlet in the upper
layer and decreases slightly with distance in the lower portion of the shear
layer. For an assumed constant-width channel, the equation of continuity from
the water surface to the tank bottom is clearly not satisfied from positions
3 to 4 and 4 to 5 in Fig. 45 . On this basis, the conclusion was reached
96
-------
that the sidewalla, which are only 10 cm apart, influenced the velocity
profiles in the upper layer. Essentially, the velocity profile in a horizontal
plane across the tank above the interface develops in a manner similar to
that at the entrance of a pipe causing the apparent acceleration in flow at
the top.
There will be an acceleration due to the exit weir being higher than
the inlet. From the interface trajectories plotted in Fig. 40a , most of
this acceleration occurs after position 6. The slight decrease in velocity
with distance from the inlet at a fixed depth in the shear layer between
positions 3 and 5 in Fig. 45 is a result of the upward motion of the inter-
face induced by the exit weir. This decrease in mass flow would be balanced,
to satisfy continuity, by a slight increase in velocity in the surface layer quite
a bit smaller than that observed in Fig. 45 .
Because the velocity profiles in the surface layer are influenced by
the sidewalls, the analysis of the data must rely on the velocity profiles below
the interface where the assumption of two dimensional flow is valid since the
velocities are low.
VI. 5.5 Analysis of the Data Ri0 = 3.0
Based on the velocity profiles Fig. 37a , the temperature profiles
Fig. 38a, 39a and the investigation of the starting process, the data
obtained at position 4 and 30 minutes for Ri0 =3.0 appears to be the most
suitable for the determination of the coefficients of thermal and momentum
diffusivity. From Fig. 40a , the interface at position 4 and 30 minutes is
located at a depth of 4. 5 cm from the surface. The interface defined as the
depth at maximum temperature gradient, corresponds with the boundary
between the light and dark regions in the velocity photographs, Fig. 37a .
Velocity-depth profiles as a function of probe position are presented
in Fig. 45 . As discussed previously, the profiles below the interface must
97
-------
be used in the data analysis because the high velocities above are influenced
by the sidewalls. The velocity at the interface at position 4 is about 0.3 cm/
sec so that the assumption of two-dimensional flow below the interface is
reasonable.
Temperature-depth profiles plotted from the data such as Fig. 38a
are given in Fig. 46 for different positions. The maximum temperature
gradient which locates the interface at 4. 5 cm depth occurs at a temperature
of about 27. 6°C, only 2. 5°C below the hot water value compared with an
overall difference of 8°C. Below the interface, a gradual fall in temperature
occurs to the cold-water value near the bottom. This spreading out of the
temperature drop over a considerable depth is attributed to the initial mixing
that occurs on starting.
Between the water surface and a depth of 5 cm, in Fig. 46 , the
temperature of the water decreases slightly with distance from the tank
inlet. However, below 5 cm, the temperature of the water increases with
distance from the inlet. This occurrence, the significance of which will be
made clear later, is caused by the starting process and by the reversal
of warm-water flow at the high exit weir. Heat is transported to lower depths
near the exit and then carried forward towards the inlet by the reverse flow.
The temperature-depth and velocity-depth profiles at position 4 are
given in Fig. 47 at different times after the start of flow. The interface
descends at the rate of about 1. 5 cm per hour as the warm water erodes the
body of cool water. Buoyancy inhibits the penetration of heat, and the
temperature above the interface at a given depth increases slowly. Below
the interface, the more rapid increase of temperature at times greater than
15 minutes in caused by the transport of heat by the reverse flow. The velocity
profiles in the shear layer and reverse flow region show little change with
time in Fig. 47.
98
-------
The basic objective of the experiment is to use the data given in
Figs. 45, 46 and 47 to determine the coefficients of momentum
diffusivity KM and thermal diffusivity KH . As outlined in Section VI. 2,
the first step in the calculation of KM is the determination of the vertical
velocity oo . At the interface where calculations were terminated, (^
reached a value of 1. 2 x 10 cm/sec.
As shown in Fig. 48, the substantial derivative Du./ Dt
(Eq. VI. 11), calculated from the velocity profiles is never greater than
-3 2
0.2 x 10 cm/sec . For comparison, the minimum shear stress gradient
which occurs for completely laminar flow can be calculated from
9 2 YYVuM.
the data, known values of viscosity ji> for water and by using the equation
(VI. 19)
In Fig. 48 , . -, , whether positive or negative, is always considerably
J-*' ~
i A ~b
greater than Du./Dt . If the pressure gradient term -=- - were negligible
in the equation for the x-momentum, equation VI. 10 then
9 9Z Dt
However, the minimum possible value for $T / 2^ is always about one order
of magnitude larger than Pu./ Dt . Hence, the x-momentum equation cannot
be balanced without including the pressure gradient term.
Integration of the z-momentum equation (VI. 9) with the surface
pressure at Z = 10. 1 equal to atmospheric -^ gives
,10.1 (VI. 21)
99
-------
The density of the water varies with depth as a result of the variation in
temperature. If the temperature at each depth were independent of the
horizontal location %, the variation in pressure with horizontal position
9-J&/9* would be zero. However, referring to Fig. 46, the temperature
is observed to vary with % . From position 4 to 5, the temperature above
the interface drops typically about 0. 1°C in 20 cm. The significance of
this small temperature gradient can be estimated by noting that from
equation (VI. 21)
' -
where ~ = -c*o,(X = 2.4x 10 / °C is the coefficient of volumetric expan
o I *" 2
sion at the temperature of the water, and Q = 980 cm/sec . For a constant
° ~
fl T
~- = -
t
= -5 x 10 °C/cm and integrating from the surface to the
92 20^..
interface at a depth of 4. 5 cm gives -5- % ^ = 5. 3 x 10 J cm/sec'1'. The
D
maximum value of the substantial derivative below the interface is
_ o o Iy "t
0. 2 x 10 cm/sec which is insignificant compared with the pressure
gradient term.
97
Use of equation (VI. 22) with values for calculated at each
. O-Jrt
depth determines p- -^-S- as a function of depth as shown in Fig. 48 by
the curve labelled
-fr i A mavimnm valnp of -C
- 6. 85 x 10 cm/sec
' V V A* / -5
is reached at a depth of about 5 cm followed by a drop to about 2. 6 x 10~
cm/sec at the bottom. This decrease at depths greater than 5 cm results
from the fact that at 5 cm, as shown in Fig. 46, the temperature
gradient - changes sign from negative to positive. Physically,
0 A-
above 5 cm depth, the water downstream is heavier than that upstream
because it is colder and hence the pressure increases with distance.
However, below 5 cm the water downstream is lighter than that upstream
because it is warmer, but the heavier water above 5 cm reladve to that
upstream must be supported so that the pressure still increases with
distance but at a rate that decreases with increasing depth.
100
-------
Up to this point, the water surface has been assumed to be
3 H
exactly horizontal. It would be anticipated that the surface slope -5
0 As
would be zero or negative in the flow direction. The pressure gradient
~r~ 5 due to the slope would also be zero or negative and equal to
II H
oq f independent of depth. In Fig. 48, the pressure gradient
v o /!/
calculated from the temperature gradient is positive at
I «
all depths. However, above 7. 6 cm
is positive and between
7, 6 cm and the tank bottom, it is negative. To satisfy the momentum
equation (VI. 10), J? must also be negative in this region since
\ 0 J&
the contribution from Dw-/_Dt as shown in Fig. 48 is very small, and
hence - . = . . At 1, 6 cm depth
(7 «-
~3
= 3. 5 x 10
cm/sec and decreases with increasing depth. Therefore, as a first
Q -^i ^ 7
approximation, if -E is taken equal to 3.5 x 10 cm/sec , the
3-fc _ 3f
v
/*--
-
pressure gradient - = g v + - will change sign as required
£ *--
0
to balance the shear stress gradient at depths greater than 7.6 cm. Note
3 H -3
that this corresponds with a surface slope y = -3. 5 x 10 /g - -3. 5 x
/ (7
10
-4
surface amounts to only 6 x 10 cm.
/ (7
cm/cm. For the 170-cm length of the water tank, the drop in the
A more rigorous determination of the slope pressure gradient
can be deduced by referring to the velocity-depth profile given
S
in Fig. 48. At depths of 6. 5 and 9. 2 cm, the velocity goes through a
local maximum, and hence the shear stress is zero at these points. The
only forces acting on a control volume bounded by these two depths are
those due to pressure since the convective terms are zero. The
requirement that the pressure forces cancel between depths of 6. 5 and
9.2 cm means that the shaded areas in Fig. 48 be equal which
s
= -3.5 x 10 cm/sec
establishes the slope pressure gradient -^
c %
in agreement with the first estimate. Since the shear stress at the
surface is zero, an additional check is provided by the condition that
the areas labelled A in Fig. 48 also must be equal. This condition
is also satisfied.
101
-------
In future experiments, the surface slope might be determined
directly by noting on a high-speed recorder the instant that the temperature
probes contact the water at several positions with and without flow. The
no-flow case would locate the horizontal axis and the flow case would give
the deviation from horizontal at each station from which the surface slope
1 M
S-H- would be calculated.
2*
With all of the forces acting on the water below the interface now
determined, the momentum equation VI. 10 was solved for the unknown
shear stress T . Integration was performed in steps from the zero
shear stress point at 6. 5 cm depth to the interface at 4. 5 cm. The
coefficient of momentum diffusivity KM was then calculated using
-2 2
equation (VI. 7). KM increased from about 1. 3 x 10 cm /sec in the
low-speed reverse flow to about 2.6 x 10 at the interface compared with
the dynamic viscosity for water of about 0. 85 x 10
The coefficient of thermal diffusivity KH was determined
through the use of equations (VI. 12) and (VI. 13) as described in Section
VI. 2. The required gradients were calculated from the experimental
data as a function of depth at position 4 for Ri0 = 3,0. The thermal
_ Q
diffusivity KH increased from the molecular value of 1. 5 x 10
2 -3
cm /sec at the tank bottom to about 2.1 x 10 in the low-speed reverse
flow followed by a subsequent increase to 2. 6 x 10 at the interface. Note
that the turbulent Prandtl number Pr = KM/KH in the low-speed reverse
flow was unchanged from the laminar value of 6 since both KM and KH
are about 40% greater than their respective laminar values. The
Prandtl number at the interface was about 10 indicating that the transfer
of heat is inhibited more by buoyancy than the transfer of momentum.
For these tests, the local Richardson number was approximately 4.1
compared with an initial value at the inlet of 3.0.
VI. 5. 6 Effect of Richardson Number
With a view to determining values of KM and KH at a Richardson
number lower than the 4.1 of the previous test, the data obtained for RTo= 0.1
102
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were examined. Based on Fig. 37b, and 38b, the optimum conditions
for data reduction appear to be at position 5 at 5. 7 minutes after the start
of flow. As discussed in Section VI. 5. 2 and shown in Fig. 41, the
Richardson number increases rapidly from a value of 0.1 at the inlet
to 3.7 at the first measuring station, position 2. However, because the
interface rises with distance from the inlet (Fig. 40) as a result of
the starting process and the exit configuration of the tank, the local
Richardson number R it decreases to about 2. 5 at position 5. This
is sufficiently lower than the value 4.1 at which KM and KH have been
determined to anticipate that there would be a change in KM and KH that
would at least indicate the trend.
For the R^0- OA tests, temperature-depth and velocity-depth
profiles are given at different times in Fig. 49 and at different
positions in Fig. 50. Comparing these profiles with the corresponding
ones at Ri0 3.0 in Figs. 45, 46 and 47 shows the much more
rapid penetration at low Richardson number of the warm water flow into
the pool of cold water reaching very quickly into close proximity of the
tank bottom.
The profiles for Ri0- 0.1 differ from those for Ri0 3.0 in that there
is no longer a slow flow at the bottom of the tank in the direction of the
surface flow. In addition, as shown in Fig. 50, at all depths the
temperature upstream of a given point is greater than or equal to that
downstream and never falls below the temperature downstream as in the
Ri0 = 3.0 tests, Fig. 46. This means that, as given in Fig. 51, the
pressure gradient ^ resulting from the temperature gradient never
. 9 -f
decreases with increasing depth. Instead a plateau in -£; is
O A- T
reached at a depth of 8 cm corresponding to the fact that 2r/?£ in
Fig. 50 goes to zero at about 8 cm.
The pressure gradient from the slope of the surface -7^
~
was
s
calculated as in Section VI. 5. 5 except that for Ri0=-0.1 there are only two points
of zero shear stress - at the surface and at a depth of 7.7 cm. For this
103
-------
experiment as shown in Fig. 51, ~£\ - - 2 x io"3 cm./s«2 compared with
_ 0 JL- IS
-3. 5 x 10 for the RJ0 = 3.0 experiment. The substantial derivative
term "^ is quite large since the velocity profiles change rapidly with
time t and the two positions coordinates % and Z . The pressure gradient
0-4-» ^3-fc 0 Hb
below the interface ^~-=-^ + -5^7 never goes negative corresponding to the
d% 4* T if-1 s -
fact that the gradient in shear stress y=- is never negative because there
is no forward flow at the bottom of the tank.
The vertical velocity u> and the coefficients of momentum
diffusivity Kn and thermal diffusivity K M were calculated as for the
Rfc'a 3 3.0 tests in Section VI. 5. 5. The results for both Richardson
numbers are plotted in Fig. 52. Because of the necessity for a
considerable amount of interpolation between rapidly varying parameters, the
results for RLO - 0.1 (.Rij_ 2.5) are not considered to be as accurate as
those for Ri0 - 3.0 ( R^ - -4 .1 ) ,. For Rit=-2.5, the vertical velocity
UJ reaches a value of about 7 x 10 cm/sec at the interface compared with
-3
1.2 x 10 for Rii = 4.7 . This should not be confused with an entrainment
velocity since the flow is strongly influenced by the configuration of the
exit weir.
The momentum diffusivity, the uncertainty of which is shown in
Fig. 52 exhibits an interesting feature for Ri^Z.5 . KM decreases from
-2 2 -2
a value around 6. 8 x 10 cm /sec in the reverse flow to about 5. 0 x 10~
at the interface. This is attributed to the fact that the velocity in the
reverse flow is the same magnitude as that at the interface for R'LJL 2,5 .
Consequently, the turbulent fluctuations in the flow below as well as
above the interface are inhibited by buoyancy introduced by the stabilizing
temperature gradient. The thermal diffusivity KH for Rit=Z.5 shows
a similar although not as pronounced behavior with a value of about
_3 2
6. 2 x 10 cm /sec at the interface. The results for Ri£ = 4.7 because of
the much slower reverse flow and the stronger stability at the higher
104
-------
Richardson number show a more conventional behavior of K and KH equal
to near molecular values at the bottom rising after a slight dip to
? 7 o o
Z. 6 x 10 cm /sec and Z. 6 x 10 cm /sec respectively at the interface.
The ratios at the interface of KM and Kw to their corresponding
molecular values are plotted in Fig. 53 against local Richardson number
Rit with maximum observed values of 5. 6 for , and 4.1 for
*
Both ratios show the anticipated decrease with increasing Richardson
number toward unity at high values of RiL as buoyancy completely
inhibits turbulent mixing. Over the limited Richardson number range of
the present experiments, thermal diffusivity is inhibited by buoyancy
slightly more than momentum diffusivity. As a consequence, the turbulent
K
Prandtl number " of 10 at Ri 4.1 and 8.Z at Rit = 2.5 are somewhat greater
* H
than the laminar value of 6 for water as shown in Fig. 54.
VI. 6 Summary and Recommendations
Experimental techniques have been developed and successfully
applied to the measurement of temperature and velocity profiles in the
controlled flow of a layer of warm water over a pool of cool water. The
measurements confirm the dominating influence of Richardson number for
the inhibition of turbulent mixing by buoyancy. The rate at which the
depth of the interface between the warm and cold water increases was
about twenty times faster at an inlet Richardson number of 0.1 than at
3. 0.
The present experiments were influenced, to varying degrees,
by the starting process, the configuration of the exit weir , the location
of the instrumentation, and the tank sidewalls and bottom, For these
reasons, the existing apparatus was best suited for measurements at
Richardson numbers greater than one. Under these conditions, the
coefficients of momentum ( KH ) and thermal ( KH ) diffusivity at the
interface and below were determined from the temperature and velocity
profiles. At the interface, KM increased from about 3 to 5. 6 times the
laminar value and KH from 1.8 to 4.1 times as the Richardson number
105
-------
decreased from 4.1 to 2. 5. The turbulent Prandtl number varied from
10 to 8.2 compared with the laminar value of 6 for water.
In the low-speed reverse flow near the tank bottom for the highest
Richardson number, the turbulent Prandtl number was equal to the laminar one
with both KM and KH about 40% greater than their molecular values. At
the tank bottom, Kri decreased to the molecular value.
Analysis of the experimental results revealed the strong influence
on the low speed flow at high Richardson numbers of horizontal pressure
gradients generated by temperature gradients and a slight slope of the
water surface. The various reversals of flow direction unexplained in
earlier work because of insufficient data could be explained in the
present experiment in terms of tank geometry and the interaction between
the variable temperature-pressure gradient and the constant slope-pressure
gradient.
Future work at low Richardson numbers (less than one) require
modifications to the starting process and the exit weir, a wider, deeper
and possibly longer tank and perhaps direct measurement of the surface
slope and vertical velocity. The measuring probes will have to be
located much nearer to the inlet and be more closely spaced. In addition,
measurements will have to be taken at considerably shorter running times
in order to determine the gradients in velocity and temperature required to
calculate the coefficients of momentum and thermal diffusivity. The use
of back lighting with both sidewalls transparent instead of front lighting
would greatly improve the quality of the velocity photographs.
Most of the work has been devoted to the development of suitable
experimental techniques, but the few data points that have already been
obtained demonstrate that the approach merits further experiments. In
addition to obtaining data for lower Richardson numbers, it should also
be established whether the Richardson number uniquely determines the
106
-------
effect of buoyancy on the mixing process or whether other parameters enter.
In such experiments, the same Richardson number should be produced by
different combinations of velocity, depth of the warm layer and temperature
differences.
107
-------
VII. CONCLUSIONS
The present study has been concerned with theoretical and experi-
mental analyses of some problems on the physical aspects of thermal
pollution. Specifically a theoretical study of the mechanisms of formation
and maintenance of thermoclines in stratified lakes was carried out. The
theoretical concepts developed in this study were then used to investigate
the stratification cycle of temperate lakes and the effects of thermal dis-
charges at or below the thermocline on the stratification cycle. An experi-
mental study of the turbulent transport of heat and momentum across a
stably stratified interface between a flowing layer of warm water and an
underlying pool of cooler water was also carried out.
It was demonstrated that thermoclines are formed in temperate lakes
by the nonlinear interaction between wind-induced turbulence and buoyancy
gradients due to surface heating. While the nonlinear aspect of the inter-
action makes the problem difficult, the nonlinearity is an essential feature,
and as such, has to be retained.
The influence on the eddy diffusivities of the interaction between the
turbulence structure and the buoyancy gradients was included by using the
techniques that have been successful in the study of atmospheric turbulence.
That is, the eddy diffusivities were assumed to be given by the product of
the eddy diffusivities under conditions of neutral stability and an appropriate
function of a stratification parameter such as the gradient Richardson
number. The nonlinear equations of the problem were then solved using an
electronic computer.
It was demonstrated that, while the vertical temperature profile at
the start of the stratification season in a temperate lake is fairly smooth, a
sharp interface (the thermocline) develops because of the nonlinear inter-
action between the turbulence and the temperature structure. It was also
shown that the continuous downward erosion of the thermocline into the
deeper layers of the lake is a necessary part of its sustenance. The above
conclusions were confirmed by considering the simple cases of a half-space,
108
-------
initially at a uniform temperature, which is subjected to a sudden increase
in surface temperature. It was shown that the temperature distributions
were initially similar to the error-function distribution for the linear case.
But nonlinear effects soon came into play, and a thermocline was formed
some distance below the surface. The thermocline also propagated steadily
away from the surface. In other words, the problem is inherently unsteady;
the continuous downward movement of the thermocline is inherent to the
mechanisms responsible for its formation, and will occur even when the
surface conditions are held steady. It may be noted that previous attempts
to treat the thermocline as a steady-state phenomenon have led to serious
29
inconsistencies in conditions at great depths.
The theoretical concepts developed were then applied to study the
entire stratification cycle. It was shown that the seasonal stratification cycle
of a temperate lake can be viewed as the response of the lake to certain
imposed "external parameters" which specify the exchange of mechanical and
thermal energies between the lake and the environment. Moreover, the
stratification cycle can be described in terms of these "external parameters"
alone without the need for specification of additional information on the
thermal structure. In particular, no knowledge of arbitrarily defined "eddy
diffusivities " peculiar to a given body of water is required.
The results of the theoretical model are in very good agreement
with the observed qualitative features of the stratification cycle. For the
one case in which quantitative comparison were made, good agreement with
observations was obtained.
The effects on the stratification cycle of thermal discharges at or
below the level of the thermocline were also assessed by using the analytical
model. It was found that the effect of the thermal discharge was to increase
the length of the stratification period as well as the temperature of epilimnion.
It was shown that in addition to the effect due to addition of heat, there is
also an important effect on the thermal cycle due to the transfer, by the power
plant, of large quantities of water from one level to another.
109
-------
An exploratory experimental study was also carried out of the
interfacial friction between a flowing layer of warm water and an underlying
pool of cooler water. It was found that the transfer of heat across the
stable interface was inhibited more strongly than the transfer of
momentum.
While the studies described here have yielded considerable amount
of information directly relevant to the problem of thermal pollution, much
needs to be done. A detailed parametric study of the effects of various
environmental conditions on the thermal structure of a lake should be carried
out by using the analytical model developed here. The model can also be
used to assess the relative effects of various modes of discharge. The
model proposed here can also be improved further by an explicit accounting
of the coupling between the current and thermal structures and by a more
accurate treatment of free convection.
In the experimental study, further systematic studies of the behavior
of the transport processes at the interface for various values of the Richardson
number have to be carried out.
110
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APPENDIX A
PERTURBATION METHOD FOR THE STUDY OF THE NONLINEAR
BEHAVIOR OF THE BASIC HEAT TRANSPORT EQUATION
It has been emphasized in Section III that while Eq. (III. 1) is non-
linear and as such difficult to solve, the nonlinearity is an essential part
of the mechanism responsible for the formation of the thermocline and that
therefore it cannot be neglected. Numerical integration of this equation
was selected as the technique by which maximum information could be
obtained about the solutions under general conditions, and these solutions
have been reported in Sections III, IV and V. However, to gain an under-
standing of the basic nonlinear behavior of Eq. (III. 1) and to check the
validity of the numerical solutions under limiting conditions, we analytically
calculated solutions to Eq. (III. 1) when these solutions could be expected
to deviate only little from the solution to the linear equation. Thus a per-
turbation technique was developed in which the difference between the
solution to the linear equation and that for the full nonlinear equation was
assumed to be small. In this appendix, a linearized equation for this per-
turbation is derived, and an analytical expression for the perturbation is
obtained.
In the basic equation
?T d ( ., VT }
77 ' n (K"^] (A-l)
it is assumed that the eddy diffusivity can be expressed in the form
KH = KHo + JU.K,;0 (A. 2)
where KH is the eddy diffusivity under neutral conditions (correspond-
ing to the linear problem), \j~ is a small perturbation parameter and
(i ) -i-
K H is a function of the dependent variable /
111
-------
and the general solution is
ll)fz,t) = (T,-TJ 1
(A. 6)
where jE/i/ (%) = e cL
The initial and boundary conditions for Eq. (A. 5b) are
T ~ 0 for t - 0 and H - 0
(A. 7)
JC - 0 f°r Z = 0 and t ^ 0
The functional form for K H must now be selected in order to
proceed with the solution. As discussed in Section III, a very reasonable
form for KH is
KH = ^' ' l~~ ^2 ' 'NH° (A. 8a)
This functional relation for the eddy diffusivity is similar to the form sug-
gested by Holzman. It can also be regarded as the relation obtained
when any functional form suggested for the eddy diffusivity is linearized
for small magnitudes of the nonlinear term.
For generality take
K^" - Z* JI (A. 8)
where n. is an arbitrary integer. Solutions for 7U> C H , t ) will be
given for YI = 0, 1 and 2 , and the meaning of the complete solution for
T C £ t t ) will be discussed briefly in each case.
112
-------
The nonlinear behavior of Eq. (A. 1) enters through Eq. (A. 2). The
solution for Eqs. (A. 1) and (A. 2) is sought in the region i ^ 0 for
t - 0 with the initial and boundary conditions that
T = T0 at t 0 for Z ^ 0
(A. 3)
7 = T, at Z _ Q for £ - 0
The solution for T is expanded in powers of the small parameter
fc , as
T - T0 + jxT'"* /T(E)+ -- (A. 4)
Equations (A. 2) and (A. 4) are substituted into (A. 1). All of the terms in
Eq. (A. 1) can be grouped as coefficients of powers of p. .If each coef-
ficient is separately equated to zero, the following sequence of equations
is obtained:
g , 3J (A.5b)
" ( j
etc.
The initial and boundary conditions for Eq. (A. 5) can be derived
from those given in Eq. (A. 3). For Eq. (A. 5) the initial and boundary con-
ditions are
TC" = 0 for t - 0 and £ - Q ,
rn) = T - T for Z. 0 and t ± 0
' ' * ' O *
113
-------
n = 0:
When Yl = 0 , the solution for T (£ , t ) can be written, after
some algebra, as
or
_
T
(2 -
r° ,
1 (
&
2KH t
(A. 9)
where I is obviously defined.
Yl = 1 :
T
T (Z, t) =
2 KM.
2 -
2 -
X
*
or
(A. 10)
T >
where 1 is defined by these relations
114
-------
n = 2;
T-CZ;
T CH.t) =
rr
X
- 3
(1 -
(2-
%'
(2-Z)
or
-(a)
(A. 11)
Note from Eq. (A. 6) that Tf"C£,t) can be written as
Therefore, the solutions for T
written
For n = 0
, through second order in p. , can be
For n
For n - 2
T = T0f /UL(T(-T0)f
/KH/1t'
115
-------
The important feature to note from these solutions is the time
dependence of the second-order terms. Both the zero and the first-order
term can be regarded as functions of zf/VK^t' , the linear diffu-
sion parameter. The step increase in temperature imposed at the surface
2 = 0 at t - 0 will diffuse into the medium in a self-similar fashion
as time increases. For n = 0 , the second-order term is proportional
to //\/KMot times a function of the similarity parameter Z/vKHoi .
As time increases, therefore, the nonlinear effects, as embodied in the
second-order term, will decrease relative to the zero and first-order terms
For Yi ~ \ , the second-order term is simply a function of the similarity
parameter and will neither grow nor decay with increasing time. For
n = 2. the second-order term is a product of VKHe * and a func-
tion of the similarity parameter. Nonlinear effects can be expected to
increase as time increases, and a perturbation solution will break down
when the nonlinear term in (A. 8) becomes comparable to unity.
Direct numerical calculations of Eq. (A. 1) with initial and boundary
conditions (A. 3) have verified the predictions made from the perturbation
solution. In addition, temperature profiles calculated from the perturbation
solution above have been used to check the details of the numerical calcu-
lations for this special case.
116
-------
APPENDIX B
IMPLICIT METHOD OF CONSIDERING THE INTERACTION
BETWEEN TURBULENCE AND BUOYANCY GRADIENTS
It was mentioned in Section III, that many of the characteristic
features of stratification can be predicted by accounting for the interaction
between turbulence and buoyancy gradients either explicitly or implicitly.
The explicit procedure, described in Section III, involves accounting for
the variability of the eddy diffusivity due to the interaction. In one form of
the implicit procedure, which has also been discussed, the lake is divided
into two or more layers described by different, but constant, values of the
eddy diffusivity. In another form of the implicit procedure used by Moore
33 38
et al and by Kraus and Turner, the vertical distribution of temperature
in the lake is characterized by a given number of parameters; and the time
evolution of these profile parameters is determined by using moments, over
the entire depth of the lake, of Eq. (III. 1).
Integration of Eq. (III. 1) and its first moment yield
(B. 1)
(B. 2)
where £ ^ is the maximum depth of the lake. The right-hand side of
38
Eq. (B. 2) can be expressed somewhat differently by using the turbulent-
energy balance, so that
117
-------
where the first term on the right-hand side represents the mechanical
energy input from the wind, and the second term represents the dissipation
of energy, over the entire depth of the lake by viscous forces.
Equation (B. 1) represents the conservation of heat, while Eq. (B. 3)
represents the rate of decrease of the potential energy of stratification due
to the net work done by the wind. The significance of various forms of
Eq. (B. 3) has long been appreciated by limnologists, who have computed
the wind work required to distribute a given heat income at the surface.
33
Moore et al have used Eq. (B. 1), along with the assumption that
the depth of the epilimnion remains fixed, to solve for the seasonal varia-
tions in surface temperature. They have also used Eq. (B. 3) to derive a
on
simple stratification criterion. Kraus and Turner have used a two-
parameter description of the vertical temperature profile, along with Eqs.
(B. 1) and (B. 3), to solve for the seasonal variations of the temperature and
of the depth of the upper, mixed layer.
It should be emphasized that implicit methods, because of their
integral nature, cannot be used to predict the formation of a thermocline in
the sense that was discussed in Section III. They can be used only to pre-
dict certain features of the maintenance of a thermocline. An assumption
that is implicit in all integral methods is that the unsteady aspects of the
problem arise primarily out of the unsteadiness in the boundary conditions
at the surface. However, it was seen in Section II that unsteadiness is
inherent to the mixing processes in the lake and that, when they are accounted
for properly, the conditions in a lake will be unsteady even when conditions
above it are steady.
Moreover, integral methods assume that the time scales for the
mixing processes in a lake are very small compared to the time scale for
the unsteadiness in the boundary conditions; that is, it is assumed that con-
ditions in the lake adjust instantaneously to changes in surface boundary
38
conditions. For example, in the method proposed by Kraus and Turner,
a steady heat flux applied to the surface will lead to the instantaneous forma-
tion of a mixed, upper layer whose depth is proportional to the Monin-
Obukhov length scale.
118
-------
In the present study, it is assumed that the vertical temperature
profile can be represented by a mixed upper layer of temperature, Ts ,
and depth, "& , with the temperature profile below the mixed layer being
given by Eq. (III. 11). That is
7 CZ , t) - Ts for 0 < Z ± -fu
-0.1
-------
If appropriate initial and boundary conditions are specified, then
Eqs. (B. 5) to (B. 7) can be used to solve for the time evolution of the quan-
tities Ts , "A. and a* . However, our objective here is to merely
illustrate the behavior of the thermocline during the early phases of the
stratification cycle.
Equations (B. 5) to (B. 7) can be used to derive the relation
J,
a. dt
(B. 8)
/--(-)
«- X -> 2 /
where the dissipation .D has been neglected and where L is the Monin-
Obukhov length defined in Eq. (III. 8). Thus it can be seen that the move-
ment of the thermocline is governed by the eddy diffusivity of the underlying
layer as well as by the value of the Monin-Obukhov length. The first term
on the right-hand side of Eq. (B. 8) is always positive, but the second term
is negative since in general, ~n. > L . Thus it is seen that the movement
of the fully developed thermocline can be either upward or downward depend-
ing on the relative magnitude of the two terms on the right-hand side of
Eq. (B. 8).
120
-------
APPENDIX C
A DESCRIPTION OF THE NUMERICAL PROGRAM
The numerical program used for the calculations described in this
report was modified extensively during this work. As a result the program
has not been tested under all conditions which might be desired. However,
the program has been found to perform simply and adequately under all
conditions attempted.
Knowledge of the performance of the program has been gained by work-
ing with it, and this knowledge cannot be conveyed in a report. However, few
warnings need to be given about operating the program. The program rep-
resents a satisfactory, working package for numerically integrating the
basic heat transport equation.
In this appendix a description of the details of the program will be given.
First the equations are presented. Then a discussion is given of all input
quantities for the program. Next the algorithm is described for computing
the temporal change in the temperature and thermal diffusivity profiles.
Finally, the output plots and typical run times are discussed.
1 . Equations
The program has been written to calculate, as a function of depth,
thermal diffusivity and temperature profiles over a complete annual cycle.
The equation integrated by the program is the model equation discussed in
Sections III and V of the report
|f - £- <*» TT> - "> W + ** 1C. la)
for 2 1 Z £ ^ Z^
37 3 , ,/ 3 T ,
for ^ * * > ^ or
121
-------
Note that 2^ is the specified intake depth, Z^ is the discharge depth which
is found as described in Section V, ur^ is the specified pumping velocity
and 2)
where CT is a dimensionless constant and the Richardson Ri number has
the form
-+ N-1
P s _/_i±_) a 2 J_T_
us** V 3Z (C.3)
The limitations of this form for the thermal diffusivity and the justification
for the alterations to this form discussed below have been given in the body
of this report. Note that this form of the thermal diffusivity can become
infinitely large when the Richardson number becomes negative. During the
cooling portion of the annual cycle, the Richardson number will become
negative, and the form given by Eq. (C.2) for the thermal diffusivity is no
longer valid (see Section IV). Under these conditions the thermal diffusivity
is limited by a maximum value denoted as Fm^.^ . Similarly, from the
form of KH given by Eq. (C.2), the thermal diffusivity can become infini-
tesimally small as the Richardson number increases. During the heating
portion of annual cycle the Richardson number can become very large, and
the form (C.2) is no longer valid (see Section IV). Under these conditions
the thermal diffusivity is limited by a minimum value ^~ m.in,
Equations (C. 1) apply over the complete lake 2 = 0 to Z = Z^,
where Zm is the depth of the lake. Integration of these equations requires
specification of initial conditions and boundary conditions. Initially the lake
is always taken to be homothermal at temperature ~T0 . At the bottom of the
lake the heat flux is taken to be zero, and this condition implies that
= 0
(C.4)
122
-------
At the surface of the lake one of three boundary conditions is applied:
either (i) the surface temperature is specified
Ts = 7 (if 0) = fi + B sivitSlt +) (C.5a)
where n is a. mean and 9 is an amplitude for the surface temperature
variation, JC = 3^5 (day ) ^s the frequency of the variation and 4> is a
phase angle, (ii) the heat flux Q at the surface is specified
= 0 ** Zr* (C.5b)
where n and B are now the mean and amplitude of the heat flux variation,
(iii) the equilibrium temperature T£ is specified
fl t B (C.5c)
2. Input Quantities
Since there does not appear to be a more efficient or understandable
method for presenting details of a numerical program, reference to names
of variables used in the Fortran program will be made. In particular to
describe the input quantities for the program, we will list the Fortran names
and give a brief description of the function of each of these quantities. Many
input quantities are dimensional, and the dimensions of each will be given in
square brackets after its description. If no dimensions appear, the input
quantity is dimensionless . All input variables beginning with letters I , K ,
N or M are integer variables and all others are floating point, as is
usual for Fortran programs.
Input variable
NRUN Run number
N Exponent in the definition of the Richardson number given
in Eq. (C.3)
K Number of spatial or depth intervals
NDAYS Time span of calculation [ days ]
123
-------
NBC
IP LOT
MINUTS
IPUMP
NDAYPL
SIGMA
FMAX
FMIN
A
PHI
CON1, CON2
Boundary condition indicator:
NBC = 1, Surface temperature Ts is specified to be
fl + 9 sin (lit + 0) , see Eq. (C.5a)
NBC = 2, Heat flux <£ at surface is specified to be
fl + S SLXI (J2t+ (j>) and the surface temperature
is derived from Eq. (C.5b)
NBC = 3, The equilibrium temperature T£ is specified
to be ft + B stxu (J2* + 0} and the surface
temperature is derived from Eq. (C.5c)
Number of days at which spatial plots are desired. Spatial
plots are always made on the final day.
Time in minutes allotted for the run
An indicator which is used in calculations involving thermal
discharges. If IPUMP = 0, the thermal discharge is added
in a normal fashion. If IPUMP ^ 0, then the model includes
the effect of pumping water from lower levels to the surface
with no thermal input to pumped water.
An array containing IPLOT elements that specify the days
on which spatial plots are desired.
The constant CT which appears in the Rossby-Montgomery
form for the thermal diffusivity, see Eq. (C.2)
Maximum allowed value for the thermal diffusivity
Minimum allowed value for the thermal diffusivity
Mean value of the surface temperature if NBC = 1 [°CJ ,
of the heat flux if NBC = 2 [fitu/ft -day] , of the
equilibrium temperature if NBC - 3 [°C]
Amplitude of the surface temperature if NBC = 1 [°c] ,
of the heat flux if NBC = 2 [fitu/ft -day] , of the
equilibrium temperature if NBC = 3 [°Cj .
Phase angle in surface boundary condition
Constants GI and GZ used in the definition of KHo in the
expression for the thermal diffusivity given in Eq. (C. 2)
K,
* (C, *C,Z)
124
-------
Bl, B2, PSI
DELTAZ
XK
TEMPO
CTSTEP
WPP
TEMPD
ZINTAK
ALEN
QPP
CON 1 has dimensions [ft] and CON2 is dimensionless
Constants B, , B and Vr is the definition of uJ
Bl and B2 have dimensions of [ft/sec] and Iff is
dimensionless .
Spatial mesh size AZ [ft] . The lake depth is the integer
K times AH , Z^ = KA Z .
The heat transfer coefficient K in Eq. (C.5c)
[Btu/ft2-day-°C]
The initial uniform temperature T0 [°c]
A nondimensional constant C t used in the determination
of the time step size in the numerical integration. For
stability 0 * CTSTEP * 0.5.
Power plant pumping velocity cJl used in Eq. (C. la)
[ft/day]
Temperature difference AT.^ produced by the power plant
Depth Hd of intake for power plant [ft]
A length denoted a defining the thickness of the gaussian
heat source C^ , see Section V [ft]
Heat Q per unit area per time added by the power plant
2
to the lake [fitu/ft -day] . Provided the discharged water
remains below the surface,
In addition to the input quantities described above certain constants are
defined within the program. A list of these follows:
Si = 5 [ days j - the frequency of variation of the boundary
condition at the surface,
O^ - 32.2 [ft/sec j - acceleration of gravity,
oC =112.32 [fitu/°C - ft J - heat capacity per cubic ft of water,
125
-------
A = 0.0 - Constants which determine the quadratic
1 c
A = 1.538 x 10 expression that approximates the coefficient
7
A = -2.037 x 10 for volumetric expansion for water
OL = A + A (T - 4°) + A (T - 4°)2
3. Algorithm
Numerical integration of Eqs. (C. 1) requires that these equations be
replaced by a set of difference equations. These equations are parabolic and
mathematically represent a diffusion process. Integration of the linear form
of such parabolic equations has been studied extensively, and many finite
difference equations have been used to obtain solutions for such equations
(for example see Refs. 81 and 82). Integration of nonlinear forms of the
diffusion equation by finite-difference methods has been widely used, but is
much more poorly understood.
a. Difference Equation
In this program a straightforward explicit finite-difference scheme has
been used. The total depth of the lake is divided into K equal increments
each of length A 2 . The time step is chosen at each time on the basis of
stability considerations. The following notation is used:
, _p th
^^k - the time increment chosen at the ~H. integration step,
''
(C.6)
c*^
The difference equation which replaces Eq. (C.la) can be written
V = Tr*AH' **> =
where
(KBATr;, K
126
-------
A similar equation with the last term deleted arises from Eq. (C.lb). The
meaning of the terms like KH[(^. + -y- ) AZ , t^J is somewhat ambiguous
because of the nonlinear form of Kri . If it is assumed for the moment
that the thermal diffusivity KH is specified at each depth by Eqs. (C.2) and
(C .3) alone, then
(C.8,
since KHf> = KH<( (£ ) and &-v = oLv(~T) However, the simple
forms (C.2) and (C.3) are not always valid for the thermal diffusivity and
therefore are not used at all spatial mesh points and at all times. The
algorithm used in the computation to determine the diffusivity is discussed
below .
b. The Thermal Diffusivity
Justification for the procedure for calculating the thermal diffusivity
has been given in Section III and Section IV. Therefore only brief mention
of the reasons for the procedures will be made here. At any time the form
given by Eqs. (C.2) and (C.3) for the thermal diffusivity is calculated at
^s.
each spatial mesh point. Denote this function by KH . It is on the basis
s*.
of this function KH that the value of the thermal diffusivity KH at each
point is chosen.
As discussed in Section III, the formation of the thermocline implies
that wind-induced turbulence is suppressed at levels below the thermocline.
Therefore, during the heating portion of the annual cycle once the thermo-
cline has formed, the values of KH below the thermocline do not repre-
sent the thermal diffusivity KH .A cutoff procedure is used to eliminate
9T
this problem. Thermocline formation is defined by the condition that ,
reaches a minimum value which is not at the lake surface 2. =0. After
,A
thermocline formation the minimum value with respect to Z of KH is
determined. This value is denoted by KH and the position at which
127
-------
the minimum occurs is denoted by Z ^Uvx. The thermal diffusivity K
A 'A'
is chosen to be KH for all £^ £ min. and ^H ^w. for a11
As discussed before the thermal diffusivity cannot decrease below a value
determined by the molecular thermal diffusivity, and often this absolute
minimum value will be somewhat larger than the molecular thermal diffusivity
because of ambient turbulence within the hypolimnion. A value rmiiyt is
specified as input to each numerical calculation. If KH becomes smaller
than FmiKt at some depth, then for all ~£ greater than this depth,
IX _
^
During the cooling portion of the annual cycle, again KH becomes
inappropriate in the upper layers of the lake. Because of cooling at the
surface of the lake, convective mixing becomes important within the
epilimnion. This mixing can be accounted for within the model by an in-
creased value of the thermal diffusivity. Again a cutoff procedure is used
,*.
to eliminate this problem. If KH reaches a local maximum with respect
to z? during cooling, then the value of this maximum KH ^a*t and the
position ? yxu>.^ are determined. For Z i fr^-*. > the thermal diffusivity
KH is chosen to be KH ^^ , and it is chosen to be K H for
Z Z. r>la_^ However, the thermal diffusivity is never allowed to exceed
the value /"">vua.x specified for each calculation.
c. Time Step Determination
At any time t^, the temperature profile is known, and from this profile
the thermal diffusivity as a function of depth is determined in the manner
described above. Then the maximum value < K.H)Tyvft_ifc of the thermal diffusivity
is ascertained. From this value, the constant C± specified for the calcula-
tion and the spatial mesh size Aj? , the time step size A-t is computed
* i
(C.9)
3. Output and Run Times
A very important feature of the numerical program is its ability to
generate graphs directly from the computer- This feature has been found to
128
-------
be very useful since much time and labor are saved when the plots are made
by the computer. However, and even more important, these graphs provide
immediate visualization and efficient understanding of the solutions.
Two types of graphs have been programmed, spatial and temporal
graphs. Spatial graphs are those for which a dependent variable is plotted
as a function of depth at a specified time. Temporal graphs are those for
which a dependent variable is plotted as a function of time. Four spatial
graphs and three temporal ones are generated for each numerical computa-
tion. On each spatial graph up to eleven plots can be made of the dependent
variable; each plot is made for a specified day so that plots can be made at
eleven different days. The temperature, the thermal diffusivity, the tempera-
ture gradient and the heat flux are the dependent variables which are graphed.
Temporal graphs are made of temperature (surface temperature, temperature
at three depths and the equilibrium temperature), depth of the thermocline and
surface heat flux.
The calculations presented within this report were performed on an
IBM 360/65 computer- The time required for each computation depended
upon the number of days requested for the calculation. Typically a computa-
tion reported in Section IV or Section V ran for 400 days, or slightly over a
complete yearly cycle. Such a run required 50,000-60,000 time steps and
about 10-15 minutes of computer time. Calculations reported in Section III
ran for 30-40 days requiring less than 4000 time steps and less than one
minute of computer time. These times include the computer time required
for performing the plots.
129
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APPENDIX D
VERIFICATION OF THE ADEQUACY OF THE BASIC HEAT TRANSPORT
EQUATION WHEN THE VARIABILITY OF THE VOLUMETRIC
COEFFICIENT OF EXPANSION IS ACCOUNTED FOR
Throughout this report the equation
II . J_(K11) - -±
(D.I)
and its solution, for various boundary conditions, have been discussed at
length.
There was some concern, however, that when a consistent account of
the variation in the volumetric coefficient of expansion ( dv ) was made,
Eq. (D. 1) would not be an adequate representation for the temperature field.
This uncertainty about the adequacy of Eq. (D . 1) was primarily due to
the fact that Phillips has made extensive use of the buoyancy equation
38 9N 9
tuJ't')
3t 92 9Z (D.2)
which is consistent with Eq. (D. 1) only when (X^ is a constant.
In Eq. (D.2), 6 is the mean buoyancy and ~lr is the fluctuation in
buoyancy, with
£ = B + -£' = -a is a reference density. Note that under the Boussinesq approxi-
mation
' 9'
* = -? ^ = a-?7'
(D.4)
and
= C6 o
3t v ? 3t
so that Eq. (D.2) becomes
, 37- 9 .
130
-------
Thus, it is clear that Eq. (D.5) can be compatible with Eq. (D.I)
only if (X^ is a constant. It should be pointed out that Eq. (D.2) was
obtained by Phillips from the mass-conservation (continuity) equation while
Eq. (D.. 1) follows from the energy equation. However, since under the
Boussinesq approximation both Eqs. (D. 1) and (D.2) describe the temperature
field, it is necessary to establish the reasons for the incompatibility between
the two relations. After a detailed investigation it has been established that
Eq. (D.2) is not an accurate representation of the mass-conservation relation
when OL. is variable.
The continuity equation gives
(n\ ) = 0
3t (D.6)
where V is the velocity vector. If ^ = ^ -» o , uJ = vJ + uJ etc.,
and if horizontal homogeneity is assumed to exist, then Eq. (D.6) gives
li. ' - 0
CP
gt 92 92 V (D.7)
In the literature it has been customary to assume that, that under conditions
of horizontal homogeneity, the last term in Eq. (D.7) is identically equal to
zero. It should be emphasized that this assumption is not justified when
&v is a variable. Note that the same heat flux passing through two
different horizontal planes a small distance apart will cause a relative motion
between the two planes because of the different rates of expansion at the two
locations. The last term in Eq. (D.7) indeed represents the mass flux into a
fixed control volume (between two horizontal planes) due to the above differ-
a GQC
ential expansion. This term can also be written in the form - _ r - £. ,
eS az
so that Eq. (D.7) becomes
131
-------
Thus the correct form of the buoyancy equation is
a
*
(D.9)
It can be easily verified that Eq. (D.9) reduces to Eq. (D. 1) when
Eqs. (D.4) are used to relate buoyancy to temperature. Thus under the
approximation of horizontal homogeneity, Eq. (D.I) represents the correct
description of the temperature field and this equation has been used through-
out.
132
-------
APPENDIX E
ON THE VALUE OF THE SEMIEMPIRICAL PARAMETER <5~,
In Section III, the form of the eddy diffusivity given by Equation
(III. 7) was used to demonstrate that the thermocline is formed by the non-
linear interaction between wind-induced turbulence and buoyancy gradients
due to surface heating. In Section IV, the same form of the eddy diffusivity
was used to study the various features of the stratification cycle. Before
the solutions given in these sections can be interpreted in a quantitative
fashion, it is necessary to choose a numerical value for the parameter , Equations (E-l) and (III-7) will be equivalent if the
value of 0^ were 5 and if the form of the Richardson number given in Equation
(III-5) was used. It was pointed out in Section II that one of the more impor-
tant developments since the time of Munk and Anderson's theory is the
34
formulation of the so-called similarity theory of Monin and Obukhov.
Monin and Obukhov have proposed a relation for the eddy diffusivity of the
form.
KH - KH.(T + /3-f) (E-2)
133
-------
where L is defined by Equation (II-l) and /S is a semi- empirical
parameter with a value of about 0. 6.
If Equations (II-l) and (E-2) are combined with the relation
9T
one obtains, after some manipulation,
where R^ is given by Equation (III-6). Since the eddy diffusivity under
neutral flow conditions is given by the relation
KMo ^ -k u3*i (E-5)
Equations (E-4) and (III-7) would be identical if Q", = K /& or, since the value
of the von Karman constant K. is approximately equal to 0.4, o~J & 0. 1.
It nnay be pointed out that, in the literature, there is no unanimity of opinion
as to the value of /S . The theoretical analyses of Swinbank yield a value of
0. 5 for /3 , while other authors have obtained values as high as 7.0 based
' 33
on atmospheric data.
Based on extensive measurements in the atmosphere, Elliot
suggests a relation for the eddy diffusivity of the form,
KH = KHoOt7^)-V2 (E-6)
with R_i given by Equation (III- 5). The difficulty in comparing Equations (E-l)
and (E-6) with (E-4) is indeed in the different forms of the Richardson number
used in these relations. This difficulty would not have arisen if in the present
analysis the coupling between the current and thermal structures had been
included explicitly, rather than implicitly. If it is assumed that the velocity
gradient is given by the relation
133a
-------
then the value of
-------
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141
-------
40
TEMPERATURE (°FI
80 40 80
40
80
100
200
1 THERMOCLINE
(a) APRIL
(b) MAY
(c) JULY
100
200 L
(d) AUG.
(e) SEPT.
(f) DEC.
Figure 1 TEMPERATURE STRUCTURE OF CAYUGA LAKE, 1952
(AFTER HENSON, ET. AL. REF. 22)
WIND
VELOCITY PROFILE TEMPERATURE PROFILE
(al UNSTRATIFIED CASE
THERMOCLINE
EPILIMNION
HYPOLIMNION
VELOCITY PROFILE TEMPERATURE PROFILE
(bl STRATIFIED CASE
Figure 2 WIND-INDUCED CURRENTS IN A LAKE
142
-------
25
50
75 100 125
DEPTH OF THE LAKE, ft
150
175
Figure 3 VERTICAL TEMPERATURE DISTRIBUTIONS FOR A CONSTANT SURFACE TEMPERATURE
200
-------
80
^ 60
_i
o
o
20
10
TIME, days
20
30
Figure 4 VARIATION OF THE DEPTH OF THE THERMOCLINE FOR A
CONSTANT SURFACE TEMPERATURE
800
600
>-
h;
>
«t
z:
ac.
200
days
days
j
50 100 150
DEPTH OF THE LAKE, ft
200
Figure 5 DISTRIBUTION OF THERMAL DIFFUSIVITY FOR A
CONSTANT SURFACE TEMPERATURE
144
-------
9.0-
8.0-
7.0-
6.0-
1.0-
3.0--
"40
80 1 20
DEPTH OF LAKE, ft
(DAYS)
10.
23.
36.
65.
82.
97.
200
Figure 6 VERTICAL TEMPERATURE DISTRIBUTION FOR A SINUSOIDAL SURFACE TEMPERATURE
-------
ce.
LLJ
Q_
TIME OF MAXIMUM HEAT CONTENT
EQUILIBRIUM TEMPERATURE, TE
SURFACE TEMPERATURE, Tg
BEGINNING OF
STRATIFICATION
STRATIFICATION PERIOD
DEEP WATER TEMPERATURE, TH
END OF STRATIFICATION
IME OF MINIMUM
HEAT CONTENT
TIME
Figure 7 SCHEMATIC REPRESENTATION OF THE ANNUAL TEMPERATURE CYCLE
-------
~>
CD
LU
on
CL
err
1 _
0
0.00
"50.00
100.00
150.00
270
T1ME(DRYS
450
Figure 8 THE STRATIFICATION CYCLE
-------
00
O
L.LI
or
CE
LT1-
I_LJ
LU
0
TIME I DRY'S
JlXOU
90.01
180.00
240.00
300.00
360.00
420.01
500 .-01
DEPTH, ZX1Q-
Figure 9 VERTICAL TEMPERATURE DISTRIBUTIONS
-------
vO
m
i
O
x
C_J
a:
u_
§ i
en
LU
a:
2 _
cr
UJ
0
1 _
-2
0
90
180
"1 1 T
270
TTMElDRYSl
GO
450
Figure 10 SEASONAL VARIATION OF SURFACE HEAT FLUX
-------
TIMriBi-li'3)
en
i
a
X
ZJ
LL
I
a:
90.01
1-80.00
240.00
300.00
360.00
420.01
500.01
0
1
DEPTH, ZX1Q
Figure 11 VERTICAL DISTRIBUTIONS OF HEAT FLUX
-------
CO
I
o
X
CO
ID
Li_
LJ_
or
01 2:
~ or
UJ
0
1 _
0
1
DEPTH, ZXLO~2
Figure 12 VERTICAL DISTRIBUTIONS OF THE THERMAL DIFFUSIVITIES
30.00
90.01
180.00
240.00
300.00
360.00
420.01
500.01
-------
o n
*
i
LU
i <
- r
a:
or
o
LJJ
i
a:
Q_
LJJ
i
t
~\
T
f ^~.._
j=
' \
"^
v
\
\
\
\
\
1
.
-^
\
\
\
\
\
\
\
\
\
\
\
\
\
\,
/
/
/
~* ' " _v- _A
^_ n . -m e _. - dfc=
j. " "\ - ~1'r-~l- - "" ^
\ .~~-*~ ^~ ^^~~ ' ' _^-'~~"
*^ \ ^^^ '^~ -s**^'
''^^ \ ^"^ *** ^f*~
s^ >--"' ^ ^'
^ .'' \ .i-r ,.--^
/' / \ X"'
- * \ ,--' _^-"
,.- " \--^"
X -x
/ / iTj'
/ / ,X TIM
° 30.
^
,/' / * 90.
/ PX/ ° 180
/
// - 240
// * 3UU
^ ^ 360
/ * 420
n/
4 -k SOO
^5==:^= ^i:^"**
^r^^^,--'- '"
-'^ ^"
^"'
E (DHVS!
no
01
. 00
,00
. 00
.00
.01
.01
0
DEPTH,, ZX10'
Figure 13 VERTICAL DISTRIBUTION OF TEMPERATURE GRADIENTS
-------
tn
CM
O
LLJ
I - 1
R
d:
cr
CD
_
ocr
tr
LU
o_
i>_
LLI
I ___
1 J
0
0
90
80 270
TJMElDHYSl
Figure 11 SEASONAL VARIATION OF THE THERMOCLINE DEPTH
450
-------
(Jl
0.00
50-.-00
100.00
150.00
0
180 270
TIMF (DRY'S)
Figure 15 EFFECT OF IMPROPER INITIAL CONDITIONS
360
-------
LT1
\
CD
X
LU
or
a:
or
LU
a_
CD
1 _
0 _
U.UU
50.00
100.00
150.00
_!__ _.-r_r .^^^
->4fl I?) 30 7?0 810 900
1 IMFIDRY'3
Figure 16 CYCLIC BEHAVIOR OF THE TEMPERATURE VARIATIONS
-------
l/l
I
o
X
LU
or
or
or
LU
Q_
LU
0
DRYS
D
A-
T
9U.01
180.01
240.00
545.111
820.00
910.00
970.00
1026.50
DEPTH. 7XICV
Figure 17 CYCLIC BEHAVIOR OF THE TEMPERATURE DISTRIBUTIONS
-------
ni
I
CD
X
LJ_.
Lt_
__ I
g:
of
LU
1 _
0
9U.01
180
240
545
820
910
\
01
00
01
00
00
970.00
1026.50
-------
30
oo
COMPUTED
MEASURED (REFS. 16 AND 22)
^ (. L + + +
30 60 90 120 150 180 210 240 270 300 330 360
Figure 19 COMPARISON OF THE COMPUTED AND OBSERVED STRATIFICATION CYCLES OF CAYUGA LAKE, NEW YORK
-------
TEMPERATURE, °C
0 5 10 15 20 0 5 10 15 20 0 5 10 '5 20 0 5 10 15 20
100
200
0
100
MARCH
200 - '- J~
APRIL
100
200
MAY
0
100
200
JUNE
0 5 10 15 20
01
= 100
200
0 5 10 15 20 0 5 10 15 20
Oi - ' Oi
0 5 10 15 20
100
200
JULY
AUGUST
SEPTEMBER
OCTOBER
0 5101520 0 5 1015 20
0 5 10 15 20 0 5 10 15 20
100
200
, ,
t
,
1
/
1- 1
u
100
9f>n
_.
L
u
100
onn
\j
1 f\C\
700
NOVEMBER
DECEMBER JANUARY
COMPUTER MEASURED (FROM REFS. 16 AND 22)
FEBRUARY
Figure 20 COMPARISON OF THE COMPUTED AND OBSERVED TEMPERATURE PROFILES FOR CAYUGA LAKE, NEW YORK
-------
un
sD
Z:
LAKE SURFACE
TEMPERATURE
i
EPI LIMN ION
,THERMOCLINE
Jt.
THERMAL PLUME
\
Wp
DISCHARGE
I EFFECTIVE
1 LEVEL
OF
DISCHARGE
: o -------
INTAKE
LAKE BOTTOM
Figure 21a SCHEMATIC REPRESENTATION OF
THE THERMAL PLUME
-------
200 i
o
o
100
0.
COMPUTED
MEASURED:
O REF. 16 (WEEKLY AVERAGES)
A REF. 22 (MONTHLY AVERAGES)
!i
90
180 270
TIME (days)
360
Figure 21b COMPARISON OF MEASURED AND OBSERVED THERMOCLINE DEPTHS FOR CAYUGA LAKE, NEW YORK
-------
30 -r
WITHOUT THERMAL DISCHARGE
WITH THERMAL DISCHARGE
100
Depth (Feet)
Figure 22 EFFECT OF THERMAL DISCHARGE ON VERTICAL TEMPERATURE DISTRIBUTION
-------
ro
i
O
X
LU
cr
or
or
UU
Q_
CD
1 J
0
0
z
0.00
50=00
100.00
150.00
EQUIL.
270
90 180
TIME(DRYS)
Figure 23 EFFECT OF PUMPING ON TEMPERATURE CYCLE
J60
-------
0-J
I
o
LU
CC
01'
o
LU
LE:
LLJ
LU
ro.i
0
-B-
-B-
-B-
-B-
-B--
-B-
-B-
0
90
180
270
TTMFITIfiYS
360
-------
o-
SEASONAL VARIATION
WITH THERMAL DISCHARGE
WITH PUMPING ONLY
180
270
360
TIME, days
Figure 25 EFFECTS OF THERMAL DISCHARGE AND OF PUMPING ALONE ON STRATIFICATION CYCLE
-------
WITH THERMAL D SCHARGE
360
Figure 26 EFFECTS OF THERMAL DISCHARGE AND OF PUMPING ALONE ON DEPTH OF THERMOCLINE
165
-------
2000-1-
a
T3
WITHOUT THERMAL DISCHARGE
WITH THERMAL DISCHARGE
WITH PUMPING ALONE
100
DEPTH, ft
Figure 27 EFFECT OF THERMAL DISCHARGE AND PUMPING ON THERMAL DIFFUSIVITY
200
-------
PARTITIONS
COOLING COIL
STIRRER
X VALVES
COOL WATER
WARM WATER
FLOW
METER
Figure 28 SCHEMATIC ARRANGEMENT OF FLOW SYSTEM
-------
TANK
CAMERA LENS
Figure 29 SKETCH FOR DERIVATION OF
THE REFRACTION ERROR ON
PHOTOGRAPHS OF FLOW
TRACERS (PLAN VIEW)
0 10 20 30 TO 50
DISTANCE FROM CENTER OF TANK, X IN cm
Figure 30 REFRACTION ERROR ON PHOTOGRAPHS
OF FLOW TRACERS
+6.2v
R5 (CALIBRATION)
PROBE
Figure 31 BRIDGE CIRCUIT TO MEASURE WATER TEMPERATURE
R = RESISTANCE OF THERMISTOR PROBE, RL = R] = Rg = 0.1 MS2,
R2 = 50 kfi, RY = 5.1 kfi, R5 = R6 = 1 kfi, C = 0.33AiF.
168
-------
GUIDE ROD
STRING
Mil!!
TO PROBE DRIVE
ELECTRIC CONNECTION
TO THERMISTOR
BEARING TUBE
.GLASS TUBE
CERAMIC OR
'GLASS TUBE
THERMISTOR
Figure 32 ENLARGED PHOTOGRAPH OF A THERMISTOR
PROBE TIP - THE SCALE DIVISIONS ARE
1 mm APART
Figure 33 DESIGN OF TEMPERATURE PROBES
NOT TO SCALE (ONLY TWO ARE SHOWN)
-------
Figure 3H PHOTOGRAPH OF DRIVE MECHANISM FOR TEMPERATURE PROBES
WATER
NEEDLE
ELECTRODE
-32V
Figure 35 CIRCUIT OF ELECTRONIC
RELAY FOR WATER CONTACT
27 29 31 33
TEMPERATURE, °C
35
Figure 36 CALIBRATION OF ONE OF
THE TEMPERATURE PROBES
BETWEEN 25°C AND 35°C
BEFORE ADJUSTMENT OF
THE CALIBRATION VOLTAGE
170
-------
TIME FROM
START - min
lO
30
60
90
Figure 37a VELOCITY PROFILES R, = 3.0 (UPPER TRACE 2.67 sec, LOWER 30 sec)
-------
TIME FROM
START - min
5-7
9.7
12.7
15.9
20.2
31.9
Figure 37b VELOCITY PROFILES Rj - 0.1 (ALL TRACES 2.67 sec)
-------
TEMPERATURE
SURFACE
21°C
31°C
( i __,
5 cm
DEPTH
00
TIME
FROM
START
min
15
35
SURFACE
21°C
31°C
t 1 6 N-
POSITION POSITION
Figure 38a VERTICAL TEMPERATURE PROFILES Rj = 3.0
TIME
FROM
START
min
65
95
-------
TEMPERATURE
SURFACE
DEPTH
21°C
26
SURFACE 21°C
26°C
1
K
/
1
/
1
(
f
s
/
f
/
^
s
\
/
X
1
/
J
1
\
5
rr
(
X
X
r
J*
H
^
^
^
x
^
^
'
x
/
j
<
)
x
'<
3
POSITION
t
5 cm
/
/
f
~-
/
>
d
-=
\i
-
i
^
c
/*
_-
^
r
^
»
«i
L
?
I
»
^
5
TIME
FROM
START
min
3.6
6.9
lO.f
f
V
1
t
V
^
/
X1
x
X
X1
^
V
J
n o
5
NJ
"v
x
7 '
^
i
r
T
'5 cm
3
POSITION
HH
71
I
1
^
k
1
TIME
FROM
START
min
16. f
35. H
Figure 38b VERTICAL TEMPERATURE PROFILES R: =0.1
'o
-------
30
10
30 40 50 60
TIME AFTER START OF FLOW - min
70
6.5°C
80
Figure 39a TEMPERATURE EXTREMES VS TIME R: = 3.0
10 20 30
TIME AFTER START OF FLOW - min
M.O
Figure 39b TEMPERATURE EXTREMES VS TIME R-. = 0.1
o
175
-------
VERTICAL SCALE = HORIZONTAL
EXIT
1 INLET
LOCATION ALONG TANK
Figure 40a POSITION OF INTERFACE (max dT/dZ) Rj = 3.0
VERTICAL SCALE = HORIZONTAL
\ INLET
LOCATION ALONG TANK
Figure Mb POSITION OF INTERFACE (maxdT/dZ) R= = o. 1
176
-------
o
V3
O
CJ
O
10
20 30 HO
DISTANCE FROM INLET - cm
50
Figure 11 VARIATION OF LOCAL RICHARDSON NUMBER, R , WITH DISTANCE
177
-------
TIME FROM
START - min
I/I
h0= O.it5 cm
oo
h = 1.15 cm
l-l/i*
hQ= 1.70 cm
2-1/2
hQ= 2.8 cm
3-1/2
hQ = 2.8 cm
6-1/2
h0 = 2.8 cm
Figure 12 STARTING PROCESS Rj = 3.0 WARM WATER-THYMOL BLUE, COLD WATER-SODIUM HYDROXIDE,
(DARK REGION INDICATES MIXING OF TWO FLUIDS) h = DEPTH OF WARM WATER AT INLET
-------
0.5
u
I
3 0-3
o
0.2
0.
LU
CL.
....
....
10 20 30 40 50 60
TIME AFTER START OF FLOW - min
70
Figure 43 PEAK REVERSE FLOW VELOCITY R; = 3.0
80
o
0)
1.6
1.2
o 0.8
UJ
o
Q±
=>
CO
SURFACE FLOAT
.O.THYMOL BLUE TRACE
20 cm
EXIT
543
LOCATION ALONG TANK
'
INLET
Figure 44 VELOCITY AT SURFACE R, = 3.0
179
-------
WATER SURFACE
POSITION 3
oo
o
J TANK BOTTOM
///////////// r ////// 1 //
0.4
// f //////////// > ! it i / >n> i } n } n n f } n n n n
0.6 0.8 1.0 1.2 ] ,
FLOW VELOCITY - cm/sec
Figure 15 FLOW VELOCITY VS DEPTH AND POSITION Rj = 3.0, 30 min
-------
WATER SURFACE
oo
E
O
o
-------
oo
ISJ
20
22
WATER SURFACE
HO min 30
0.6
FLOW VELOCITY - cm/sec
1.2
TEMPERATURE - °C
Figure 47 TEMPERATURE, FLOW VELOCITY VS DEPTH AND TIME, R= = 3.0, POSITION
' rt
-------
WATER SURFACE
e
o
o
<
CO
X
o
~uJ
oo ai
o.
LU
Q
r -0.8-
- cm/sec
EQUAL AREAS
ZERO SHEAR
/////////////////////////////////////////,
10123 15
ap ^ ar , ^
3x ' az MIN ' Dt
' ~ x 103 cm/sec2
7 -0.04 0 0.01
FLOW VELOCITY - cm/sec
Figure 48 SOLUTION OF MOMENTUM EQUATION R: = 3.0, POSITION 4, 30 min, R
-------
WATER SURFACE
oo
o
I
en
o
22
23
LLiiiiirJiiiiiii hi 11 r-i 11 n 111 k 111 /:////'/////
25 -0.2
0.2
0.6 0.8 1.0
1.2
TEMPERATURE - °C
FLOW VELOCITY - cm/sec
Figure ₯9 TEMPERATURE, VELOCITY VS DEPTH AND TIME Rj = 0.1, POSITION NO. 5
-------
WATER SURFACE
00
u
CO
o
22
23
2t 25 -0.2
TEMPERATURE - °C
0.2
FLOW VELOCITY - cm/sec
Figure 50 TEMPERATURE, VELOCITY VS DEPTH AND POSITION R, =0.1, 5.7 AND 6.7 min
-------
WATER SURFACE
oo
E
O
UJ
O
CO
:£
o
mm
FLOW VELOCITY cm/sec
Figure 51 SOLUTION OF MOMENTUM EQUATION RJQ =0.1, POSITION 5, 5.7 min, Rj? =2.5
-------
u - FLOW VELOCITY cm/sec
0 0.25 0.50 0.75 1.00 1.25
WATER SURFACE
O
I
3C
0.
UJ
O
21 23 25 27 29
T - TEMPERATURE - °C
0 0.5 1.0 0 0.5
w - VERTICAL VELOCITY
cm/sec x 10
1.5
KM - MOMENTUM DIFFUSIVITY
nr/
cnrsec x 10
2.0
FU
2
2.5 0
0.5 1.0 1.5 2.0 2.5 3.0
KH THERMAL DIFFUSIVITY
2 / 3
cm /sec x 10
00
u - FLOW VELOCITY cm/sec
0 0.1 0.8 1.2 1.6
WATER SURFACE
E
o
8
RJ = 0.1 Rj = 2.5
POSITION 5,5.7 min
k/pc
! I
NTERFACE
21 22 23 24 25
T - TEMPERATURE - °C
0
w - VERTICAL VELOCITY KM - MOMENTUM DIFFUSIVITY KH - THERMAL DIFFUSIVITY
/ , «o i . o " *), o
cm/sec x 103
cm
2/sec x 102
cm /sec x 103
Figure 52 VERTICAL VELOCITY, MOMENTUM AND THERMAL DIFFUSIVITY COEFFICIENTS
-------
o
0.
Figure
2 3 M_
LOCAL RICHARDSON NUMBER R;
k/pc
AT INTERFACE VS RICHARDSON NUMBER
12
o
<
<
as
o_
4
j
\.
>
A
. .
R;
'z
LARGE
0123^56
LOCAL RICHARDSON NUMBER R;
1 0
Figure 51 PRANDTL NUMBER AT INTERFACE VS RICHARDSON NUMBER
188
-------
1
5
Accession Number
r* Subject Field &. Group
02H, 05B
I i- - -
1 SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Organization
Cornell Aeronautical Laboratory
Title
RESEARCH ON THE PHYSICAL ASPECTS OF THERMAL POLLUTION
10
Authors)
Sundaram
Rehm, R.
Rudinger
Merritt,
, T. R.
G.
, G.
G. E.
16
21
Project Designation
FWQA Contract
#14-12-526
Note
22
Citation
Water Pollution Control Research Series, Report Number 16130DPU02/71, Water Quality
Office, Environmental Protection Agency
23
Descriptors (Starred First)
*Stratification, *Path of pollutants, *Thermal pollution, Lakes, Reservoirs, Water
temperature, Mathematical models, Thermocline, Hydrodynamics, Diffusion, Laboratory
tests, Thermal power plants
25
Identifiers (Starred First)
*Temperature prediction, Warm water wedge, Interfacial mixing.
27
Abstract
The mechanisms of formation and maintenance of the characteristic thermal structure of
deep, temperate lakes are investigated along with the effects on the thermal structure of
discharges of waste heat from electric generating plants. It is shown that a thermocline
is formed by the nonlinear interaction between the wind-induced turbulence and stable
buoyance gradients due to surface heating.
A theoretical description of the stratification cycle of temperate lakes is given in
which the interaction between wind-induced turbulence and buoyancy gradients is included
explicitly. The theoretical model predicts all the observed features of stratification
accurately. It is shown that thermal discharges increase the temperature of the epilimnion
and also the temperature during spring homothermy. A lengthening of the stratification
period also occurs. In addition, the attendant transfer of large quantities of water from
one level to another has a significant effect.
An exploratory experimental study is described on the nature of the interfacial mixing
between a flowing layer of warm water and an underlying cooler pool of water. It is shown
that the downward transfer of both momentum and heat are severely inhibited at the inter-
face by the stable buoyancy gradients; momentum to a lesser degree.
This report was submitted in fulfillment of Contract Number 14-12-526 under the
sponsorship of the Federal Water Quality Administration.
Abstractor
T. R. Sundaram
Institution
Cornell Aeronautical lahn^t-n
WR:'02 (REV. JULY 1969) SEND TO: W A"TE"R ~R~Es6 U R C ES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR
WASHINGTON, D. C. 20240
* SPO: 1969-359-339
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