DETERMINATION OF VERTICAL TURBULENT DIFFUSIVITIES OF HEAT
IN A NORTH FLORIDA LAKE
By
Jerry A. Steinberg
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1975
S UNIVERSITY OF FLORIDA
3 1262 08666 218 5
To my parents
Louis Steinberg and Bette King Steinberg
who have given to me,
directly and indirectly,
in ways I know
and in ways I will never comprehend,
much of the ability necessary to accomplish
what is represented here.
ACKNOWLEDGMENTS
This work was made possible in part by the Environmental
Protection Agency which provided stipends and other financial
aid and by the National Science Foundation which furnished funds
for much of the research apparatus. I gratefully acknowledge
the assistance of these two agencies.
I also appreciate the cooperation of the University of Florida
School of Forest Resources and Conservation, its director, Dr. John
Gray, and Professors J. W. Miller, Jr. and Don Post, in permitting
the use of Lake Mize in the Austin Cary Memorial Forest as the
field research site.
Many people helped accomplish the field investigation,
unfortunately all cannot be mentioned here. However, special recog
nition should be given to Mr. Howard McGraw and Mr. Bob Batey who
provided advice, materials, the use of shop equipment and tools,
and much personal assistance; Mr. Lloyd Chesney who worked on the
design of the electronic portion of the data acquisition system;
Dr. Robert Sholtes who was instrumental in my acquiring the paper
tape punch at no cost; and Mr. Alan Peltz, who provided scuba diving
services needed to install the research tower in the lake. These
people made it possible to conduct the field experiments on a meager,
and otherwise insufficient, budget.
Dr. Robert Dean provided advice toward the development of various
analytical techniques at a time when progress was virtually nil
and this impetus was most helpful.
I would also like to acknowledge the continuous support and
assistance provided by Dr. Wayne Huber, Chairman of my supervisory
committee. His encouragement and friendship have been as valuable
as his efforts in his role as my mentor.
The remainder of my supervisory committee, Dr. Edwin Pyatt,
Dr. Omar Shemdin, and Dr. John Cornell, have also been most helpful,
and I want to express my thanks to them.
The production of this volume has required exceptional effort
from a number of people who deserve special recognition. Ms. Mary
Polinski and Ms. Donna Hagen have typewritten both intermediate and
final manuscripts; Mr. Rocke Hill and Mr. Jim Cauthorn provided
drafting services; and fellow students, Miguel Medina and Henry
Malec, gave invaluable general assistance.
;Finally, the influence and effect of my wife, Emily Gill
Steinberg, must be acknowledged, for as significant as all other
aid I have received has been, hers is the only assistance I could
not have done without. Her love, her patience, her support, her
encouragement, and her companionship have both directly and indirectly
been crucial to the fruition of this research effort.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS. . . . . . . . . i
ABSTRACT ... . . . . . . . iii
CHAPTER
I INTRODUCTION. . . . . . . ...... 1
II TURBULENT TRANSPORT CONCEPTS. . . . . . 6
Introduction. . . ... .......... . .. 6
Transport in Receiving Waters . . . . . 6
AdvectiveDiffusion Equation . . . . . 8
Introduction of Diffusion Coefficients. ... .13
Heat Transport in Lakes . . . ..... 17
Motion in Lakes . . . ..........18
Evaluation of Diffusivities . . . ... .20
Results of Previous Studies . . . . ... 23
The Problem of Convective Mixing. . . . ... 25
Averaging . . . . . . . . * 27
Goals of this Study . . . . . . .. 30
III METHODOLOGY OF DIFFUSIVITY EVALUATION . . .. .32
Introduction.. .. ...... .. . . . 32
FluxGradient Method. ... . . . . . 33
Evaluation of Heat Flux and Gradient Terms. . 35
Evaluation of Solar Flux Term ;.. . . ... 37
McEwen's Method .......... . .... 38
Analytical Representation of Temperature
Profiles . . . . . . . . . 41
Cayuga Lake . . . . . ... . . . 44
Correlation of Fluctuations Method. . . .. .56
IV SELECTION OF A TURBULENCE MEASURING DEVICE. . 58
Introduction. ... .. .. . . . . . 58
Equipment Requirements. . . . . ...... 59
HotWire, HotFilm Anemometers . . . ... 60
LaserDoppler Velocimeters . . . . ... 62
Rotating Element Velocity Meters. . . . ... 63
Acoustic or Ultrasonic Devices. . . . ... 65
Lagrangian Methods. . . . . . . . 68
Electromagnetic Flowmeters. . . . . ... 70
Other Devices . . . . . . * * 71
v
TABLE OF CONTENTS (continued)
Page
CHAPTER
Price Comparisons ... . . . . . . .
Requirements of the Lake Mize Study. . . .
Selection of the Electromagnetic Flow Meter. .
V DESIGN OF THE DATA ACQUISITION SYSTEM . . .
Introduction . . . . . . . . .
Data Requirements for FluxGradient Method .
Data Requirements for Correlation of
Fluctuations Method . . . . . . .
Other Data Requirements . . . . . .
Aggregate Requirements . . . . . . .
Data Recording Requirements . . . . .
Additional Data Acquisition Needs . . . .
Commercial Equipment . . .
Design of Data Acquisition Sysl
Velocity Measurement . . .
Temperature Measurement . .
Solar Radiation . . . .
Cup Anemometers . . . .
Wind Vane . . . . .
Relative Humidity Sensor . .
Velocimeter Calibration. .
Wind Anemometers and Direction
Solar Pyranometer . . .
Relative Humidity . . .
Thermistors . . . .
Punch Paper Tape Recorder. .
tem. . . .
Indicator
VI ORGANIZATION OF THE FIELD STUDY AND DATA
REDUCTION TECHNIQUES. . . . .
Introduction.. . . . . . . . .
Selection of Lake Mize as the Research Site.
The Instrument Installation . . . .
Data System Operating Procedure . . .
Data Reduction . . ... . . . .
. . 105
TABLE OF CONTENTS (continued)
Page
CHAPTER
VII RESULTS OF ANALYSIS. ... . . . .. 121
Introduction ..... . . . .. . . .121
FluxGradient Analysis ........... .122
Inadequacy of the FluxGradient as Typically
Applied. ... . . . . . . .. 122
Detailed Examination of Convective Mixing. . 125
Manual Calculations of Diurnal Values of
Diffusivities. . . . . . . . 132
Correlation of Fluctuations. . . . ... 137
Comparison of FluxGradient and Correlation
of Fluctuations Results. . . . .. . 156
Variations of Diffusivities with.Time and
Depth . . . ... . . . . . 168
Comparison with Other Lakes . . . ... 181
VIII SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR.
FUTURE RESEARCH. . ... . . . . 184
Summary . . . . . ... ... . 184
Conclusions. . . . . . . . . 185
Suggestions for Future Research. . . .. .187
APPENDIX A FORMAT OF STORED DATA.. ... .189
APPENDIX B DATA FOR DIFFUSIVITY CALCULATIONS. 194
LITERATURE CITED . . . . . ... .202
BIOGRAPHICAL SKETCH. .......... . 206
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment'of the
Requirements for the Degree of Doctor of Philosophy
DETERMINATION OF VERTICAL TURBULENT DIFFUSIVITIES OF HEAT
IN A NORTH FLORIDA LAKE
By
Jerry A. Steinberg
August 13, 1975
Chairman: Wayne C. Huber
Major Department: Environmental Engineering Sciences
Knowing or predicting the distribution of water quality parameters
in water bodies is essential to efficient management of wateras.a
valuable natural resource. Theoretical concepts which describe the
transport of substances throughout the water have been developed, and
the effects of turbulence have been accounted for by the semiempirical
approach of relating turbulent diffusive flux to the average gradient of
concentration through coefficients of diffusivity. Because of the
significance of vertical temperature distributions in lakes and reser
voirs, this research has focused on evaluating vertical diffusivities
of heat in a deep Florida lake.
A complex data acquisition system was designed, built, and installed
at Lake Mize, Florida, to gather field data to be used to evaluate
diffusivities. The investigationoinvolved measuring both the actual
turbulent motion responsible for the transport, and the change in heat
content as a result of the motion. Emphasis was also given to making
viii
observations rapidly enough to enable the study of lake motion due
to convective cooling.
The results indicate that motion in the lake was at most times
too low to be detected by the velocity measuring equipment used.
Other results point out the need for closely spaced temperature
measurements near the surface in order to adequately define the con
vective cooling process. Detailed analysis of convective cooling
revealed that it may be described by the semiempirical concepts pro
vided that suitably short averaging intervals are used. Finally,
diffusivities found in Lake Mize are much smaller in the upper layers
than those observed in other, larger lakes, but in the lower strata
values are more nearly equivalent, and in most cases, two orders of
magnitude ,greater than values for molecular motion.
CHAPTER I
INTRODUCTION
The problem of determining the distribution of a substance in
a water body is often encountered as an aspect of managing the water
as a resource. The everincreasing need for water for industrial,
recreational, and potable uses, among others, requires that the
management of it be given more attention. For example, these needs
may dictate that available water be reused several times and often
essentially shared by successive users as it progresses through the
hydrologic cycle. Thus, one important facet of water resources
management is determining the fate of substances introduced into
receiving waters as discharges from some user. Another important
aspect is to understand and predict the behavior of naturally occurring
physical phenomena in fresh water and marine environments. Both
mancaused and intrinsic variations in physical, chemical, and
biological water quality are important because of the effects they
may have on other uses.
In order to obtain better knowledge of variations in water quality,
it is necessary to better understand the hydrodynamics of receiving
water bodies. This is because both artificial and naturally occurring
substances are transported by the movement of the water. By defining
.thismotion, a description of the levels of substances existing at
various times and places can be obtained. Knowing or predicting
these levels is valuable for purposes of planning uses of waterways,
administering regulations aimed at maintaining minimum standards of
water quality, and otherwise managing the resource in a way which
provides optimum usefulness.
Most water motion existing in nature is turbulent, that is, much
of the movement is characterized by randomness which is usually
described in terms of temporal and spatial averages. To describe
this turbulent flow,equations of motion, continuity, and conservation
of mass, among others, may be used to express the existing hydrodynamic
phenomena. The equations are used to develop a model for making
predictions of concentrations, and included in the model are usually
coefficients which relate to the turbulent nature of the water motion.
The lack of information regarding these coefficients is usually a
serious impediment to the application of the hydrodynamic concepts
to engineering and management problems. In their most common form,
the coefficients are called diffusion coefficients or merely
diffusivities and may be functions of direction and location and the
type of transported quantity (i.e., heat, mass, or momentum). The
focus of this study is to present a better understanding of diffusivities,
especially diffusivities of heat in the vertical direction.
In particular,transport of heat in lakes and reservoirs is a
subject which has received much emphasis for a variety of reasons.
The intrinsic relationship between temperature and density makes the
vertical distribution of temperature important because it reveals the
structure of existing thermalstratification. In reservoirs the
3
vertical distribution and transport of heat affect the withdrawal
operations of the impoundment due to considerations of thermal water
quality downstream. Another example of the importance of evaluating
heat transport is the need to solve problems of recirculation of
heated discharges. Because of the widespread use of lake and reservoir
waters for cooling in many industrial processes, especially power
production, this matter has received much attention.
The particular problem of transport in relatively deep,
quiescent water bodies like most lakes and many reservoirs is one
which has received much attention and is continuing to receive
attention. This is primarily due to the occurrence of stratified
conditions which develop in lakes and reservoirs annually. In many
cases the stratification exists with such stability that vertical
transport throughout the entire depth is greatly restricted, and the
effect on water quality is significant. Because mixing between upper
layers (epilimnion) and lower layers (hypolimnion) is restricted,
undesirable accumulations of chemical and biological substances can
take place in the hypolimnion, and effluents discharged into the
stratified waters will move to depths of equivalent density and tend
to concentrate there. An increased knowledge of the magnitudes and
variability with time and depth of the diffusion coefficients will
provide improved application of the turbulent transport concepts.
This work is a report of a study of vertical turbulent transport
of heat in a small, deep sinkhole lake in North Central Florida.
Specifically, mathematical methods are presented which express the
transport but require the evaluation of diffusion coefficients.
The usefulness and propriety of the concept of diffusion coefficients
are considered. In addition, special attention is directed to the
phenomenon of convective mixing due to surface cooling in an effort
to learn whether or not the notion of diffusivities is compatible
with this type of turbulent water motion. Because the usefulness of
the transport equations is often limited by the lack of knowledge
about the diffusivities, the evaluation of these by two conceptually
different methods is described and demonstrated. One method entails
the assessment of the effects of vertical turbulence by monitoring
the changes in heat content of the lake, and the other method seeks
to measure the turbulent motion in the lake by directly monitoring
vertical water velocity. The study requires a large amount of field
data collection; and themeasurement of velocity requires the use of
a relatively new type of velocimeter because of the very low levels
.of motion encountered. A complex and highly automated data acquisition
system which is capable of fulfilling the rigorous data requirements
is developed.and discussed. A key feature of the system is its
ability to record data that can be read directly into a digital computer
for.subsequent analysis.
The results of the analysis of the data collected at the lake
intermittently over a period of eight months are given. Use of
diffusivities during periods of convective mixing is shown to be
hazardous because of the nebulous definition of the temperature
gradient under these conditions. Included are computations showing
the magnitudes and variability of the diffusivities over both short
and long time intervals and at various depths; however, the extreme
5
quiescence of the experimental lake makes extrapolation to other
conditions difficult. Suggestions for future research and guidelines
for necessary data acquisition are provided.
CHAPTER II
TURBULENT TRANSPORT CONCEPTS
Introduction
The general direction of this study is presented in this chapter.
Included is a discussion of the various types of studies in which turbulent
transport has been modeled using solutions to fundamental equations. After
a development of the general equation and accompanying theory for defining
transport due to turbulent diffusion, the presentation is narrowed to con
sideration of the transport of a specific quantity, heat, in lakes.
Summaries of causes of lake motion and schemes for evaluating turbulent
diffusivities are given along with a review of earlier efforts to evaluate
the diffusivities.. Specific problems encountered in these previous efforts
are recounted. The latter portions of the chapter present arguments for
accepting the usefulness of semiempirical descriptions of turbulent trans
port. The final section discusses the efforts of this study to support
those arguments.
Transport in Receiving Waters
Knowing the distribution of substances or heat in a water body can be
of great importance. The concentrations of various water quality param
eters throughout a region of the water body may be due to a nearby
municipal or industrial discharge or due to normal physical conditions
which cause spatial variations. Regardless, it is often necessary from a
management point of view to be able to know or predict the levels which
exist or might exist at various locations and times. Historically, much
attention has been given to making such determinations, and in most
instances use has been made of hydrodynamic relationships which describe
the transport of substances in the water in terms of the dynamic and
physical characteristics of the liquid and the substances being transported.
Glover (1964) described the dispersion of sediment in a flowing stream
using solutions to the convective diffusion equations for turbulent flow.
The same type of mathematical approach has been used to predict distri
butions of nutrients in lakes and reservoirs (Chen, 1970), and Munk (1966)
studied the vertical variations of temperature and salinity in the oceans.
Studies of oxygen content in natural waters have been concerned mostly
with reaeration processes in rivers and streams, and vertical variations
in oxygen levels have been disregarded or found to be negligible in all
but a few studies. The work of Eloubaidy and Plate (1972) is one excep
tion. Numerous investigations have been directed toward finding specific
point concentrations of various pollutants which have been introduced into
receiving waterways. Attention has been given to estuaries (Pyatt, 1964),
lakes (Liggett and Lee, 1971), rivers (Cleary and Adrian, 1973), and
reservoirs (Morris and Thackston, 1969).
In addition to studies of substance concentration, a myriad of
attention has been given to spatial and temporal distributions of heat in
natural and impounded waters. The study by Dake and Harleman (1966)
focused on the temperature distributions in lakes and the work of Huber
(Huber and Harleman, 1968) involved predicting temperatures in reservoirs.
In most of these efforts the essential approach was to use advective
diffusive principles of transport which were modified and in most cases,
simplified to accommodate the particular situation. Some of the.works
cited above adopt and use mathematical expressions which include prior
assumptions regarding the behavior of turbulent diffusion terms without
calling attention to the significance of those assumptions. Others of
the works take care to stress the acceptance of certain assumptions and
nonetheless are forced to accept the inadequacies they offer. While for
a given analysis any of the components of the transport equations may
present evaluation problems, the diffusive terms are invariably a source
of uncertainty and difficulty.
Further discussion of the use of the turbulent advectivediffusion
equation should be preceded by a development and presentation of the
equation itself. The transport of mass, momentum, or heat in a water body
can be described mathematically by considering the fluxes of any of these
quantities into and out of an elemental volume of water. The ensuing
derivation may be found in any of several published works on transport
processes and is offered here for the sake of completeness.
AdvectiveDiffusion Equation
Consider the elemental volume shown in Figure 21. Through each face
of the volume passes a quantity of a substance of concentration, s. If
the effects of molecular diffusion are assumed negligible relative to the
effects of turbulence, the net flux of material into the volume in the
vertical direction, for instance, is
z (psw)axay
I I (psv)axaz
(psu)ayaz IIy
y [su + ( su)a x]acz
 [ U
psv+ (psv)ay] xaz [psw+(psw)a z] AAy
FIGURE 21. FLUXES OF SUBSTANCE OF CONCENTRATIONS THROUGH THE FACES
OF AN ELEMENTAL VOLUME
Net flux (z) = (psw)AxAy + [psw (psw)Az]AxAy
(21)
+ rs xAyAz
where p = density of water (mass volume)
s = concentration (mass substance mass solution1)
w = instantaneous vertical water velocity
(distance time1)
x,y,z = orthogonal distances
r = net rate of production of s inside the volume
(mass substance volume1 timel)
After simplifying, equation 21 becomes
Net flux (z) = (psw)AV + r AV (22)
where AV = element volume = AxAyAz
Given that the velocities in the x and y directions are u and v, respec
tively, the fluxes in these directions can be similarly expressed so that
Total flux = [ (psw) (psu) aypsv)]AV + rs V (23)
Now, the total flux must also equal the time rate of change of mass of
substance in the volume, therefore
Total flux (psAV) (24)
at
where t = time
For incompressible fluids changes in density and volume are negligible so
that when equations 23 and 24 are equated, the result is
a(s) a a a r
a) = (sw) (su) (sv) + (25)
Dt Bz ax By p
The instantaneous values can be expressed in terms of timeaveraged means
plus an instantaneous fluctuations from the mean, as
S s + ', w = w+w', u = + u', v v + v' (26)
where the overbar denotes mean values and the prime denotes fluctuation
values. Average values over a time interval T are defined by
t+T
Af/ tAdt (27)
t
The total transport over the time interval is found by averaging equation
25 over the interval, which results in the timeaverage of each term in
that expression. Taking the vertical flux term for example and recalling
the meaning of the overbar
(s) = ( + s')(w + w') = (sw + s'w' + s'w + sw') (28)
Z as as
Because the coordinate is not a function of time, the averaging can be
performed on the quantities before differentiation. Also, equation 27
implies that the averaging may be done on each term separately, and
furthermore, that s'w = sw' = 0 and sw = sw. Therefore, equation 28
becomes
(sw) = (sw + s'w') = (sw) + (s'W') (29)
Notice that the term involving the product of two fluctuations does not
vanish. This is because a positive vertical velocity fluctuation must
transport a small amount of substance downward. Depending on the local
gradient of the substance, a small increase or decrease of substance
concentration will occur, and hence, a positive or negative fluctuation
of s will be detected. A reverse, upward velocity fluctuation would cause
a reverse substance fluctuation provided the average gradient were the
same. Therefore, the product of the two fluctuations will always have
the same algebraic sign regardless of the direction of water motion and
unless there is no concentration gradient will always be nonzero. The
flux represented by the term s~' is known as the turbulent diffusive
flux. It is so described because it accounts for the transport due to
the irregular variations from mean values.
Applying a similar averaging process to other terms in equation 25
using equation 26 yields
( a ) _~ )  (< S
at az az ax ax
(210)
j(s) (s'v')
Substituting the quantities of equation 26 into the continuity equation
and time averaging produces the continuity equation for turbulent flow as
a(u) a(v)~a(w) (211)
ax + y + S = o (211)
3x ay 3as
By substituting this into equation 210, the result is
S= as) a(s) a(s) a
 W  V  ( t
at z a a y az
(212)
r
_ aTf)+
which is the advectivediffusion equation for turbulent flow. It states
that the time rate of change of average concentration over some interval
at a point is due to a flux of material advected by average velocities
and diffused by fluctuations of velocity in each of these directions plus
any production of the substance which occurs during the interval.
Introduction of Diffusion Coefficients
As noted above the terms containing the time average of the product
of fluctuation quantities accounts for transport due to turbulent
diffusion and are called the turbulent fluxes. The usefulness of equation
212 is limited because the fluctuations are difficult to evaluate; however,
conventionally the flux terms are replaced by the relationship
s'w' = Ks (s/az)
s'u' = K x(s/ax) (213)
s'v' = K ay(s/3y)
where the K values are known as coefficients of turbulent diffusion or
turbulent diffusivities. Generally, Ksz 4 Ksx # Ksy, and they will
only be equal in the special case of isotropic turbulence. Analogous
diffusivities exist for the cases of heat, mass, or momentum transport,
and they are identified accordingly. By relating the turbulent flux to
mean concentration (or velocity or temperature) gradient,the solution of
equation 212 is facilitated; yet the theoretical basis for such action is
questionable. This should be restated more specifically: to substitute
equation 213 into equation 212 does not inherently pose any theoretical
questions because the unknown fluctuation product is merely replaced by a
different unknown K, but at the sqme time no progress is made toward
improving the usefulness of the equation. If on the other hand, the
substitution is made and some specific assumption is made regarding the
variation of diffusivity with coordinate distance (e.g., K = constant),
then the theoretical structure becomes much less rigorous.
The need for more useful forms of equation 212 have stimulated and
perpetuated the existence of,the less rigorous use of equation 213.
Originally, Boussinesq developed the coefficient of eddy viscosity to
express turbulent momentum transport as an analogy to the Fickian equation
for molecular transport (p. 25, Hinze, 1959); however, the theoretical
basis of the molecular process does not exist for the turbulent process
because the nature of the turbulence is a property of the flow field and
not the transported quantity. A semiempirical basis for the form of
equation 213 was presented by Prandtl as his mixing length theory
(p. 277, Hinze, 1959). A review of his salient.points will be presented
at this time.
The onedimensional transport in the x direction of a substance of
concentration s taking place in the presence of a gradient of average
'concentration s is shown in Figure 22. The transport through a y z
plane at x = 0 does not result from the effects of the gradient but from
the effects of turbulence. During a time interval, t, any elemental
volume of fluid may pass through the plane of area BAyz but the average
distance a volume will travel is hi before the initial character of the
volume is lost. The average rate of transport in the positive x direction
is given by
Fs A A t f psdxdzdy s dxdzd (214)
Fs El y z l xj yz o d J 2]
PLANE OF AREA 6YAZ
ELEMENTAL
VOLUME
\ ELEMENTAL
VOLUME
I
I I I
XI 0 XI
x 0 x
FIGURE 22. TURBULENT TRANSPORT OF SUBSTANCE THROUGH A XY PLANE
Expanding the concentration in a Taylor series about x = 0 gives
s(x) = s(0) + x + 1/2x2 a + ... (215)
ax O x2 I
x0
Prandtl assumed the distance x to be small enough to cause the higher
order terms to be negligible, so after dropping all nonlinear terms, and
integrating equation 214
Fs = ax O(s) (216)
By substituting 2
K = (217)
t
into equation 216, the resultant direct proportionality between flux
of substance through a plane atx = 0 to the gradient at x = 0 is realized,
and the similarity to equation 213 is apparent.
Other semiempirical formulations have been offered. Two of the most
notable of these are by Taylor and von Karman and are discussed by Monin
and Yaglom (1971) who note that attempts to find improved forms of these
relationships were in progress as recently as the time of their writing.
They also indicate the suspect nature of semiempirical attempts at
simplifying transport equations and offer the possibility that radical
new approaches to the question may provide more nearly correct solutions.
On the other hand, the statistical solutions to transport problems are
usually very unwieldly, and the concept of diffusion coefficients continues
to be considered. This.is supported by Okubo (1962) who notes the useful
ness of the semiempirical approach in relation to the more theoretically
supported turbulence doctrines.
The theoretical concepts of turbulent transport summarized by
equation 212 and the semiempirical concepts of turbulent diffusion
given by equation 213 have been presented and discussed. These equations
form the basis for the development of methods for predicting the spatial
and temporal distribution of concentrations of substance in water bodies.
Having established this basis, it is now possible to direct attention
toward specific areas of application. In particular, the transport of
heat by turbulent diffusion in the vertical direction in lakes will be
considered.
Heat Transport in Lakes
The focus of this study is on the transport of heat in natural lakes.
The thermal structure of these water bodies may be studied using the
turbulent transport concepts presented in the preceding discussions. As
indicated then, the conveyance of heat is analogous to the conveyance of
substance. To adapt equation 212 for use in the study of thermal advec
tion and diffusion it is only necessary to substitute the scalar quantity,
heat concentration, for the scalar, substance concentration. Also,
analogous expressions to those of equation 213 may be written. The
resultant equations will contain the turbulent diffusivities of heat in
each of the three directions. It should be noted that since temperature
is the physical indicator of heat content, it is the quantity most often
used in equations. The substitution is a simple one because the relation
ship between heat and temperature is given by
H = ce
(218)
where H heat concentration (energy mass solution )
1
c = specific heat of water (energy mass solution 
temperature degree1)
8 = temperature
The value of c varies only slightly over ambient temperatures and is con
sidered constant. Its value is one calorie per gram mass per degree
centigrade. Care must be taken to express the production term in the proper
units depending on the transport quantity used.
The use of the equation to describe heat transport in lakes is pre
sented after a discussion of the factors affecting motion in lakes is
given.
Motion in Lakes
Water movement in natural lakes is nearly always turbulent
(Hutchinson, 1957). The turbulent and advective transport in lakes may
be caused by several factors. In larger lakes the dominant factor is
induced movement from wind. Energy is transferred from the air to water
by the shear at the interface. Surface currents caused by wind action
transport water to the leeward shore where it piles up, and soon a return
current begins to flow due to the tilted water surface. The extent of
the motion varies according to individual situations. In the case of
stratified lakes there may exist additional induced currents at interfaces
between layers of different densities. The shear profile existing at the
surface will increase the levels of turbulence in the water and thereby
contribute to increased vertical transport. Also, the formation of sur
face waves adds to the turbulent character of the near surface zone. The
degree of water movement caused is directly related to the magnitude of
wind velocity at the water surface and the length of fetch; therefore,
the surface area and surrounding topography are significant factors as
regards wind effects.
Another factor affecting lake motion is convective mixing. Surface
heat exchange at various times may cause the water to cool and become more
dense than water below. The gravitational instability will soon result
in the sinking of the cooler water to a level of equal density. The down
ward displacement necessarily causes an upward movement of other water
parcels, and if the surface cooling continues, pronounced mixing currents
will form. In most lower latitudes where significant solar heating occurs
during daytime, the convective process will occur nightly to some extent
all year. In other climates this phenomenon may happen on a diurnal
basis during only brief periods, and other lakes may only experience
significant convection currents on a seasonal basis. The converse is not
nearly so common,yet the situation does exist in which bottom waters are
warmed by sediments, especially in shallow regions, causing convectional
mixing.
Seiches are another type of lake motion, however, because seiches
are caused by the relaxation of a windtilted water surface or atmospheric
pressure variations, they are phenomena existing in large lakes only.
Internal seiches or waves are similarly stimulated and likewise are
encountered only on large water bodies.
Most lakes exhibit some density change with depth during most or all
of the year. The result is a dampening of vertical motion relative to
horizontal motion. The effects of stratification on the vertical structure
of lakes and other water bodies are usually considered in terms of the
Richardson number
R=  (219)
where U is average horizontal velocity and the other terms are as previously
used. The number represents the ratio of the rate of increase in potential
energy due to buoyancy to the rate of turbulence generation by energy trans
fer from the mean flow. It is a measure of the stability of the lakes.
Higher values of Ri indicate that vertical mixing is being inhibited because
the energy necessary to overcome stratification is not available from the
mean flow. Use of the Richardson number for evaluating turbulent motion in
lakes is difficult because of problems of defining vertical change in
average velocity, and in smaller lakes which are characterized by very low
velocities, only crude evaluations can be made which are not very useful.
Although the Richardson number has been used in some cases of atmos
pheric turbulence to predict the behavior of vertical diffusivities, the
most workable approach in lakes is to consider the concepts of turbulent
diffusion presented in equations 212 and 213. From these expressions,
methods may be developed which will yield values for the diffusivities
without the need to know velocity profiles.
Evaluation of Diffusivities
Various methods may be used to define the behavior and magnitudes of
the diffusivities of heat to be used in the transport equations. One such
approach is to assume the diffusion coefficients are constant which permits
an exact solution to the transport expressions. Measured data can then
be used to find the values of the K's that produce the optimum agreement
with data. The insufficiencies of this method are apparent, since obser
vations clearly indicate that the levels of vertical turbulent mixing vary
with depth. Dutton and Bryson (1962) used this method to describe the
vertical temperature structure of Lake Mendota. When their constant value
of diffusivity failed to produce satisfactory results, they divided the
lake into two layers roughly corresponding to the epilimnion and hypolimnion
and calculated separate diffusion coefficients.
A second and somewhat similar method involves assuming an alternative
manner of variation of the diffusivity with depth (e.g., exponentially
decreasing). This also permits a solution of the transport equation,
either exact or numerical, to be used in conjunction with measured data in
an effort to determine magnitudes of coefficients by fitting predicted
values to measured ones. McEwen's method (p. 468, Hutchinson) is of this
type and has been used extensively to evaluate hypolimnic diffusivities.
A more detailed discussion of this approach will be given in Chapter III.
Another method of diffusivity evaluation entails finding the quotient
of the heat flux and the heat gradient at a particular depth. This may be
accomplished in different ways. For instance, the expression for vertical
heat flux analogous to equation 213 is
(w')d = Kd (e/z)d (220)
where 6 = water temperature
Kd = diffusivity of heat
d = depth
z = vertical distance
r
Therefore, if the product of the fluctuations can be determined and the
gradient of temperature also determined, then the diffusivity of heat
can be found. This method may be labeled a first principles type of
evaluation because the flux is considered in terms of the turbulent water
motion actually effecting the heat transport.
Another option is to use the one dimensional form of the advective
diffusion equation for heat transport into which equation 220 has been
substituted. Care must be taken to include proper production terms. By
integrating the equation over the vertical distance between depth, d, and
the bottom, an expression results which may be used to evaluate diffusivity
at d over a specific time interval using vertical temperature data taken
over the interval. The principle is to assess the turbulent phenomena in
the water by ascertaining the amount of transport of heat they caused.
This method is often referred to as the fluxgradient method and will be
discussed further in Chapter III.
Finally, it may be noted that diffusivities may be evaluated by
monitoring the variations in content of any of a number of other substances
in the water body and then applying an analysis procedure similar to the
one just discussed. The diffusivities of substance and of heat may then
be related by the fact that they are approximately equal (Hinze, 1959).
Accordingly, the diffusivities for substance transport may be obtained
from studies of thermal transport. That the two diffusivities are nearly
equal can be seen by comparing the turbulent Schmidt and turbulent Prandtl
numbers. The turbulent Schmidt number is the ratio of the diffusivity of
momentum (eddy viscosity) to the diffusivity of mass substance, and the
turbulent Prandtl number is the ratio of eddy viscosity to the diffusivity
of heat. Empirically, the two numbers have been evaluated, and both have
been found to have values of about 0.7. Therefore, the diffusivities of
mass and heat are known to be approximately equal.
Results of Previous Studies
The evaluation of diffusivities has been the goal of many research
efforts because as already discussed the values are necessary to the
application of the advectivediffusion equation for predictions of trans
port. In most of these studies,heat or some substance, either introduced
into the water or naturally occurring, was used as a traceable quantity.
The volume of these various investigations prevents a comprehensive review
of them here. Most deal with horizontal transport, and many tracer studies
regardless of title are in fact efforts in dispersion measurement. As far
as finding reported values of vertical diffusivities as computed from con
siderations of the conservation of either mass or heat, the number of
reports is meager. Bowden (1964) presents results of deep sea studies which
include values of from 1 to 2000 square centimeters per second (9 to 17000
square meters per day) and show much variability with depth. Ryan and.
Harleman (1973) compile results of similar studies in lakes and reservoirs.
They list several cases with values ranging from 0.3 to 13 square feet
per day (0.03 to 1.17 square meters per day) which do not apply to epilimnic
regions. Morris and Thackston (1969) used dye as a tracer in reservoir
studies and found that the vertical diffusion coefficients varied with time
and depth. They obtained values of from 0.5 to 6.3 square meters per day.
In a study of the effects of thermal discharges on receiving waters
Sundaram et al. (1969) used both the conservation of energy and McEwen's
method and found values in the lower lake strata of 18 square feet per day
(1.6 square meters per day). Morris and Thackston (1969) summarize the
results of several studies performed by Orlob and Selna (e.g., Orlob and
Selna, 1970) who used a modification of the conservation of heat approach.
The values reported are grouped according to depth and vary from 0.81 to
9.2 square centimeters per second (7 to 80 square meters per day) at the
surface, 0.02 to 0.17 square centimeters per second (0.2 to 1.4 square
meters per day) at the thermocline, and 1.2 to 1.7 square centimeters per
day (10 to 15 square meters per.day) at selected hypolimnic locations.
The values reported vary greatly. Part of the reason for this variation is.
that the values apply to different lake depths. Epilimnion values are
much higher than values in lower strata. Also, it is interesting to note
that despite the smaller hypolimnic values reported, all of the diffusivities
are at least two orders of magnitude greater than the molecular diffusivity of
heat in water, 0.012 square meters per day.
Regarding investigations of vertical turbulent transport by measuring
velocity fluctuations, the comment by Wiseman (1969, p. 8) is apropos:
In the existent literature one finds not only a lack of
turbulent spectra, but especially a dearth of information
concerning vertical motions in natural bodies of water.
Bowden and Fairbairn (1956) used a custommade electromagnetic flow meter,
Wiseman (1969) used a custommade dopplershift meter, and Grant et al.
(1962) used a hotfilm anemometer to measure shear stress in estuaries under
conditions of brisk currents; however, they reported only momentum fluxes
and no diffusivities.
The Problem of Convective Mixing
Earlier a discussion of various phenomena causing turbulent motion in
lakes included mixing due to convective cooling currents. In addition to
wind these currents account for much of the motion in the epilimnion, and
nearly always, this nearsurface movement is far greater in magnitude than
the movement occurring below the thermocline in stratified waters. The
lack of knowledge regarding the behavior and vertical variations of
turbulent transport mechanisms has caused much uncertainty in studies of
predicting lake temperatures. Orlob and Selna (1970) developed a methodology
for matching observed and predicted temperatures in reservoirs by using
values of effective vertical diffusivities of heat substituted into a predic
tive model based on conservation of heat principles. The usefulness of
their model has been questioned because the diffusivities were originally
calculated from the observed data using an inverse solution of the model.
This type of complication prompted Dake and Harleman (1966) and later Huber
and Harleman (1968) to describe the thermal transport in lakes and reser
voirs alternatively. They have reasoned that the stability conditions
existing in lakes and reservoirs due to density gradients are strong enough
to inhibit virtually all turbulent transport except for the regions influenced
by convective mixing. Their methodology entailed specifying a very small,
constant value of diffusivity throughout the lower strata. In some cases,
the value of molecular diffusivity of heat (0.012 square meters per day)
was used. The upper strata were characterized as completely mixed and
isothermal. Their procedure for defining the single surficial temperature
and the depth to which it extended used thermal energy budgets to predict
a temperature profile. Whenever an unstable nearsurface profile was
predicted, a unique average temperature was calculated which indicated
an amount of energy equivalent to the predicted condition over a
particular depth. The depth was specifically determined by the depth
existing in the stable portion of the predicted profile at which the
unique mixed temperature occurred.
The results obtained by comparing predicted temperatures to observed
values were good in both studies, and the method was deemed satisfactory.
However, Huber's results mainly showed that vertical advection was
dominant in the reservoir, and this condition is not applicable to most
lakes. In the lake study by Dake, much of the success of the model
application depends on the assessment of the vertical distribution of
absorbed solar radiation. This is a highly variable phenomenon as Dake
points out and may affect significantly the predictive performance of the
model when applied to various lakes, especially ones with high or variable
turbidities.
These authors as well as others (Sundaram et al., 1969 and Powell
and Jassby, 1974) have all questioned the ability of the concept of
turbulent diffusivities to adequately describe the process of convective
mixing. Although not stated explicitly in any of these investigations,
much of the problem with describing convective cooling using turbulent
transport equations is as follows. The normal annual thermal cycle of
lakes is reviewed by Hutchinson (1957), among others. When climatic condi
tions cause more heat to leave the water than enters, the lake cools. This
process is usually a diurnal one wherein most of the cooling occurs at night
and most of the heating is due to daytime solar radiation. If during the
cooling portion of the cycle, the thermal profile of the lake is examined
over a period of days or weeks, the loss of heat is apparent. Yet, in many
cases the shapes of the profiles observed will be very similar, and
temperature structure will be seemingly stable. This is very likely to
happen if, for instance, noontime temperature measurements are made days
or weeks apart. Such behavior cannot be readily explained by conservation
of heat methods because the loss of heat observed is incompatible with the
stable gradients (i.e., a loss of heat indicating surface cooling and
profile slope indicating a downward heat transport). Two possibilities
exist. Either the semiempirical approach to turbulent transport problems
is unsatisfactory or the method of application is in error. The process
of convective mixing seems to have a specific length scale associated
with it. The Dake and Huber models both compute a characteristic depth
over which the mixing extends. Furthermore, the nature of the process
suggests the formation of eddytype currents. It, therefore, seems
appropriate to describe convective mixing in the semiempirical manner. On
the other hand, there may be some basis for assuming that the usual methods
of applying conservation of heat approaches to conditions of mixing are
in error. This stems from the fact that the mixing is a diurnal process
and cannot be considered properly without giving attention to this fact.
Specifically, the aspect of the length of the time interval used to make
calculations must be considered because there may be marked influences
of this interval on the values of diffusivities calculated. This line
of thought will be developed in the following section.
Averaging
Equation 27 indicated the averaging method used to determine mean
quantities; nevertheless, the subject of proper averaging requires some
additional commentary. Although equation 27 was presented without
discussion, certain facts are implied by its form. The equation states
that the average value of a quantity, say concentration, in a turbulent
flow is found by time averaging over some interval T. This is true only
as a result of assuming ergodic behavior of the quantity. Ergodicity
implies that the mean value of any individual sample taken over a finite
time interval will be.approximately the value obtained by taking an ensemble
average (i.e., the average of different samples taken at the same time).
Such behavior is in certain cases strictly provable, and in other cases,
safely hypothesized (i.e., the ergodic hypothesis) (Monin and Yaglom, 1971).
In addition to this question of theoretical propriety is the practical
matter of what averaging intervals to use. Generally the guidelines
for selecting the proper interval entail using a time spah which is some
what longer than the largest frequencies of turbulent motion being
studied and short enough so that the mean does not vary during it. Con
ceptually these requirements present no problems; yet in practice they
may. In studies of large scale, naturally occurring motions, it may be
much harder to establish which are the largest scales of motion existing
than it is to do so in a laboratory study. Large eddies with frequencies
lower than 2 r/T will be excluded from analysis using T as the averaging
time (Okubo, 1962). Furthermore, as Bowden (1964) notes, the method of
getting mean values depends on the scale of movement occurring and also
the particular aspect of that movement being considered. Conversely,
selecting and applying a certain averaging interval to a set of measure
ments may have an effect on the nature and quality of results obtained,
which is the premise presented at the conclusion of the preceding section.
The emphasis of this discussion so far has related to averaging
procedures for records of fluctuating turbulent quantities versus time;
however, certain parallels may be drawn between computation of diffusivities
by the first principles approach and by integration of the conservation of
heat (fluxgradient) methods. In particular, if the averaging interval
used on a time record of a quantity is long relative to a.specific period
of .fluctuation of the quantity, then the effects of the shorter scale
variations are diminished. If the magnitude of the shorter scale
fluctuations is less than that of the longer scale variation, the
effects may be diminished to the point of not being detected at all. Satis
factory description of the transport by calculating and using diffusivities
depends greatly on using an averaging interval pertinent to the turbulent
motion. As regards diurnally variable convective mixing, the use of a
oneday averaging time to determine diffusivities by equation 213 will
yield the same inappropriate results as would be found using the flux
gradient method calculated over a twentyfourhour period. For example,
should the turbulent motion in a lake over the period of one day be
caused exclusively by convective mixing,then the velocities occurring in
the lake would fluctuate (about a zero mean) during the period of cooling.
During the remainder of the day no velocity would exist. .Now the
temperature in the lake at a depth affected by the convective currents would
also fluctuate during the mixing period and coincidently cool. Considering
the daily period, the cooling during the mixing and heating during the
day (regardless of periods of no change should they exist) would produce
a daily mean temperature. If the daily mean of velocity (zero) were used,
even the convective velocities,regardless of size,would be calculated as
instantaneous fluctuations. On the contrary, if the daily mean.of
temperature were used, the entire period of cooling would be repre
sented by only negative fluctuations. The results of the subsequent
crosscorrelating (i;e., w'e') over the averaging period or even a
portion of the day, would not reflect the shorter scale convective
process suitably. Analogously, using a daily time step to calculate
diffusivities in the lake by the fluxgradient approach would not reflect
the shorter scale convective process suitably. Therefore, using a
daily time step to calculate diffusivities in the lake by the flux
gradient approach would yield poor results due to the insensitivity of
the method to the shorter scale motions.
Goals of This Study
The preceding discussion of averaging requirements tends to support
the suggestion that the inability of the semiempirical theory to satis
factorily describe convective mixing is due to improper use of the method.
It is a goal of this research to determine whether or not the transport
in a lake can be adequately described by the semiempirical theory of
turbulent transport. Emphasis will be given to the convective mixing
phenomenon. Specifically, the vertical transport of heat will be monitored,
and from it the vertical diffusivities of heat will be determined using the
fluxgradient method. At the same time, the fundamental soundness of the
semiempirical theory will also be tested using the more difficult but more
direct firstprinciples approach of measuring the turbulent water motion in
the lake. As indicated by equation 220 timeaveraging the product of
fluctuations of vertical water velocity and temperature at a given depth will
produce the heat flux term needed to find K. The gradient term at the
same depth can be obtained from the fluxgradient study.
Additionally, the computation of the diffusivities by two different
methods could yield some worthwhile insight into the relative merits or
deficiencies of each. Furthermore the study will seek to examine the
behavior of the diffusivities as a function of various vertical positions
in the lake. Not only will diurnal variations of the diffusion coeffi
cients be evaluated as the convective mixing process is studied, but also
longer term variations which may occur will be documented and analyzed.
A final objective involves the effort to establish to what extent the
vertical diffusivity information obtained can be related to the environ
mental conditions which exist coincidently. Such relationships might prove
most beneficial to the application of transport theory to practical problems,
especially problems of predicting pollutant transport. The ascertainment
of the diffusivities by separate methods will provide the many values of
diffusivities necessary to accomplish this goal.
CHAPTER III
METHODOLOGY OF DIFFUSIVITY EVALUATION
Introduction
The overall goals of this research were stated in Chapter II.
Specific use is to be made of the fluxgradient method and the correlation
of fluctuations method to evaluate diffusivities of heat in a lake. To
perform the planned analysis,values of diffusivities are needed at various
depths, several times per day, and over a span of several months. This
indicates that a large volume of data must be gathered andsubsequently
analyzed.
This chapter presents the conceptual basis for the fluxgradient
method which is followed by discussions pertaining to the use of the
concepts. Included in the description is a scheme for approximating the
measured temperature profiles with analytical expressions so that the
requisite calculations for the fluxgradient analysis need not be performed
manually. The feasibility of diffusivity computation by this approach is
determined by whether or not such a scheme is workable. Therefore, an
account of a trial of the methodology on Cayuga Lake data is given. The
result of the test is that the procedures developed work well, and that
when applied to data gathered for this study, they should enable many
accurate computations of diffusion coefficients using a digital computer.
Following the discussion of the fluxgradient method is a presentation
of the correlation of fluctuations method for diffusivity evaluation. The
measurement of water velocitiesis essential to this approach and the
entire following chapter is devoted to the subject of making such
measurements.
FluxGradient Method
The basis for the fluxgradient method is the onedimensional form
of the advectivediffusion equation for the turbulent transport of heat.
This expression may be obtained by first modifying equation .212 to des
cribe the transport of heat using equation 218 to express heat concen
tration as temperature, then substituting equation 220 so that the
turbulent fluxes are represented as the product of diffusion coefficients
and gradients, and finally by amending the threedimensional expression
to a onedimensional form. This is done by assuming the lake is
horizontally homogeneous which is valid ifthe isotherms in the lake are
horizontal. This condition is often not met in large lakes which may be
affected by seiches; however, for small lakes studies (e.g., Smith and
Bella, 1973) this has been found to be a reasonable assumption. Also the
production term must be included to account for the absorption of inci
dent solar radiation which varies vertically. The result of these
manipulations is
ad a I6 1 Rd
K ) (31)
St z d az pc z (3
where 8 = temperature
t = time
z_ vertical distance
Kd = vertical turbulent diffusivity of heat at depth d
distance2 time1)
p = density of water (mass volume)
1
c = specific heat of water (energy mass 
degree .temperature1)
Rd = flux of absorbed solar radiation at d
(energy distance2 time1)
This equation describes in differential form the vertical transport
of heat in a water column in which no horizontal transport takes place.
Furthermore, the only sources or sinks for energy are boundary fluxes of
heat and solar radiation, and turbulent transport is assumed proportional
to the gradient of the average.heat content.
An explicit expression for the diffusion coefficient is obtained'by
integrating equation 31 from depth zl to the bottom'depth z = h,
h Dd h h DR
dz = =dz (32)
z1 zI z1
which gives
h 1, K (d e h Rd h h
S dz = K II (33)
at1 daz Pc
1 zz1 El
Assuming that there is no heat transport or solar flux through the bottom
gives,
h ae
h d = K e (34)
az zlz pc
z1 1
Solving equation 34 for K yields
z1
2 Zl h 8d 1 I5)
zI 1 (
Because the limits of the integral, z1 and h, are not functions of time
equation 25 may also be written
R h
pc Bt d (36)
K = zl
1
The first term in the numerator of the righthand sides of equations 35
and 36 represents the energy content due to absorbed solar radiation.
The other numerator term represents the rate of change, or flux, of the
total heat stored in the water column below zl, and the denominator is the
gradient of the mean concentration of heat at z1, hence, the nomenclature:
fluxgradient method.
Evaluation of Heat Flux and Gradient Terms
The evaluation of the heat flux term in equation 36 may be attempted
by either analytical, numerical, or graphical methods, or combinations of
these. If analytical expressions are known for temperature as a function
of depth at two times, then
h z
z1
may be solved analytically or numerically. Alternatively, plotting 6
versus z and determining the area under the curve between z1 and h can
be done. Analytical or numerical integration is usually easy when
digital computers are used,while plotting points, drawing curves, and
measuring areas are usually time consuming. However, arriving at
satisfactory analytical expressions for 6 = f(z) may not be feasible,
and the graphical approach may be required. Dividing the integral quantity
by the time increment, At, which is determined by the times the two
temperature profiles were measured, yields the flux term.
Another alternative when 6 = f(z) is not known is to evaluate
h
j (a3d/at)dz
z1
as is indicated by equation 35. Using temperature versus depth data at
two times, it is possible to approximate 3ad/3t at various depths by
computing Aed/At at depths between zI and h. The flux is computed by
evaluating the integral by analytical, numerical, or graphical means.
Both of these methods will be described in greater detail below as they
are applied to data from Cayuga Lake.
Determining the gradient value a /az is more difficult. Whereas
knowing the amount of stored heat at the end points of a given time span
is sufficient for quantifying heat flux, the gradient of the average
temperature over the same time span may or may not be accurately found
by considering only the end points. If the variation of the gradient
with time is not linear, significant errors may result unless additional
temperatures measured during the original time span are used to improve
the accuracy of the averaging process. Computation of the gradient should
be made using data taken at periods over which the changes in temperatures
are known to be linear, and the likelihood of error should be recognized
when a longer time span is used.
Evaluation of the gradient term entails computing the time average
of the temperatures at various depths, thereby determining an average
profile. If an analytical expression can be found for the average profile,
differentiation gives the gradient at any zj. Otherwise, plotting the
profile and graphically determining the slope at z = zI or adopting some
numerical technique such as straight line interpolation between data points
may be used.
Evaluation of Solar Flux Term
Of the solar radiation striking the water surface, the longer wave
portion is absorbed at the surface while the shorter wave portion is
absorbed within the water body in a manner which decreases exponentially
with depth. The degree and extent of subsurface penetration depends on
wavelength and the physical and chemical quality of the water. Dake and
Harleman (1966) described this twofold behavior with the expression
R = R (1 8)e1z (37)
z 0
where R = solar flux at depth z (energy area time1
z
R = solar flux at surface (energy area time)
o
B = fraction of R absorbed at the surface
dimensionlesss)
n = extinction coefficient (depth )
They note that values of 8 and n vary greatly depending on the water quality
and suggest that the coefficients be determined for a given lake using
solar absorption data applicable to specific conditions.
The average solar flux over the time span At must be used, and as noted
earlier regarding the suitability of the use of endpoint values to compute
averages, the variation of solar flux should be linear during At; otherwise
interim calculations should be made.
McEwen's Method
McEwen (1929) offered an approach for calculating diffusion coeffi
cients in the hypolianion of lakes which has received much attention.
Hutchinson (1957) presented a description of this scheme which is often
called McEwen's Method, and Powell and Jassby (1974) have restated its
development while including solar radiation terms which are usually
neglected. In many stratified lakes the thermal profile exhibits an expo
nential form in the region of the lower metalimnion and upper hypolimnion.
The metalimnion is the zone of lake depth wherein the temperature variation
with depth is greatest; it separates the epilimnion and hypolimnion. This
region of exponential temperature decrease is known as the clinolimnion,
and McEwen's Method pertains to diffusivities in it.
Let the depths z1 and z2 indicate the extent of the clinolimnion.
The exponential form of the temperature profile for z1 < z < z2 may be
expressed as
(38)
8d =.1 + 82e
where 8d = temperature at depth d (degrees)
11 B2 = constants (degrees)
= constant (depth )
The onedimensional conservation of energy equation (equation 31) derived
earlier as the basis for the fluxgradient method also forms the basis for
this approach. Assuming the diffusion coefficient to be constant in the
clinolimnion, this equation becomes
aed 27 1 aRd
 d p (39)
where each term is as previously defined except that since K does not
vary with depth, the subscript denoting its value at a specific depth is
superfluous and is not used. The assumption that K is constant is made
so that some value for it may ultimately be found. The validity of the
assumption is checked by a parallel line criterion that is discussed below.
2 2
Differentiating equation 38 to get 3 d /3z and substituting into
equation 39 and then rearranging yields
ad 1 BRd 2
at pc az K(8O22e ) (310)
Taking the natural logarithms of equation 38 and 310, respectively,
gives
ln(ed 81) = nB2 z (311)
and
ad 1 aRd 2
In(  ) =ln(K8 ) =z (312)
at pc az 2
McEwen concluded that if plots of In(6d 81) versus z and 
In[3ed/at (l/pc)8Rd/az] versus z produced straight, parallel lines for
a given set of temperature data, then the assumption of constant diffu
sivity was justified, and the value of diffusivity could be obtained by
finding the slopes and intercepts of the two plots. Powell and Jassby
demonstrated mathematically that this hypothesis was incorrect. They
showed that permitting diffusivity to vary with depth produced a more
complex expression but one which was still resolvable to a form that
gave straight and parallel plots. The result of their analysis was that
the diffusivity could have the form
Kz r1 + r2e= (313)
where rl,r2 = constants (distance time)
The crux of this argument is that assuming K constant defines r2 as zero,
which is not generally justifiable.
Powell and Jassby point out that in some lakes using r2 equal to zero
is appropriate. In other lakes this assumption cannot be made and
applying McEwen's Method leads to erroneous conclusions because the value
of r2 remains to be evaluated. They also use some values of diffusivities
found by the fluxgradient method to lead to values of r2 and show that
about fifty percent of the magnitude of K is due to the r2exp(=z) term.
Powell and Jassby do not offer a new method of finding diffusivities, but
they do show conclusively that in many cases using constant hypolimnic
diffusivities is inappropriate.
Analytical Representation of Temperature Profiles
As already discussed one alternative for evaluating the heat flux
and heat gradient terms in equation 36 is to integrate and differentiate,
respectively, an analytical expression which defines lake temperature as
a function of depth. Analytical analysis utilizing digital computers
seems preferable to other methods requiring manual operations, especially
when there are many diffusivities being calculated. The crux of this
approach is to be able to represent the measured temperature profile by
an analytical expression because once such an expression is found, inte
gration and differentiation may be performed handily. There are at least
two techniques which might be used to approximate measured temperatures
analytically: a polynomial fit and a least squares fit. Both are types
of statistical regression.
Initially, a polynomial fit of the temperature data was tried. This
involved the multiple regression of successive powers of depth, z, on to
temperature, T, thusly,
n
T' = aii1 (314)
where the prime denotes a predicted value and the a's are regression
coefficients. Digital computer programs prepared as part of the Scientific
Package by the International Business Machines Corporation (Anon., 1968)
were used as the basis for the computations. The results of the effort
were unsatisfactory because the fits obtained did not approximate the
measured values well at all depths. Also, the predictedprofiles in some
cases assumed erroneous shapes in regions between data points.
Upon the suggestion of Professor Robert Dean (1974), a new approach
involving the nonlinear least squares fit of a series of terms was tried.
Computer programs included in the UCLA (University of California at
Los Angeles) Biomedical Computer Programs Library (Dixon, 1973) served as
the basis of the analysis,although some custom modifications were made.
A function of the form
n
T' = bicos[(i=l)nz/h] (315)
where h is the depth of the .lake bottom, was subjected to least squares
analysis to determine the values of bi giving the best fit to the measured
temperatures. Problems arose because this function yielded zero slopes
(i.e., BT'/az = 0) at both water surface and bottom, so a modification was
made which extended the primary period slightly to permit nonzero slopes
at the water surface. This was accomplished by substituting z h'/h'
for z/h where h' = h + c and c = constant (meters). The results seemed
improved, and further modification was made by making c one of the
regression constants. The effect of this was not helpful because the
regression procedure would not converge to satisfactory values of c
consistently.
It was then decided to revert the cosine expression to its initial
form and amend the entire function to be fit.by adding an exponential term
which was weighted such that it only had an effect near the surface. The
shape of the temperature profiles suggested an exponential term; further
more, during periods of surface cooling, the profile shape suggested a
skewed function of the general form
T' = zXez (316)
where X = constant
A more general expression for the upper layer
b2 b z
T' = bz e (317)
was combined with the Fouriertype terms and used as the function to be
fit by the least squares scheme. The coefficient b3 could not be deter
mined consistently by the scheme and was removed as a regression
coefficient; however, it was kept as a constant. Its primary effect was
to regulate the extent of depth over which the entire term is influential,
and tests made using both integer and noninteger values for it indicated
that best overall performance of the curve fit was realized when the
value six (6) was used. Therefore, the ultimate form of the nonlinear
function to be fit by the least squares scheme became
b n
T' b1z e + = b cos[(i3)wz/h] (318)
1 i=i3
Other slight modifications were tried but none noticably improved the
predictive performance of equation 318.
Throughout the process of selecting the optimum function to be used,
the quality or goodness of fit produced by any one function was evaluated
by both analytical and observational criteria. Estimates of error,
residuals, and correlation levels were provided as part of the calculation
of the regression coefficients; however, these parameters only indicated
the relative ability of the particular coefficients to predict temperatures
at the data points used in the analysis. Consequently,each set of
coefficients was used to calculate temperatures at short intervals
throughout the water column. Therefore, erroneous oscillations and
other forms of erratic behavior could be detected. The criteria for
satisfactory temperature profile prediction were that the measured
temperatures should be approximated to within 0.1 degree centigrade,
that no extrinsic behavior exist in the predicted profile, and that
the performance of the regression scheme be repeatable and not erratic.
Temperature versus depth data for three days, one each from October,
November., and December, 1973, were used to test the various prediction
functions. Both daytime and nighttime values were used to insure a
variety of near surface profile shapes. The ultimate choice (as given
by equation 318) yielded results which most closely met the criteria
described above. For the October, 1973, profiles the predicted tempera
tures differed from those observed by as much as 0.2 degrees centigrade
at some of the points. The procedure was deemed a qualified success.
Cayuga Lake
It was desirable to better evaluate the merits and suitability of
using a nonlinear least squares fit of equation 318 to analytically
define measured temperature profiles. One possibility was to apply the
regression procedure and subsequent flux and gradient computations to
data from which calculations of diffusivities by the flux gradient
method had already been made by manual means. The recent article by
Powell and Jassby (1974) appeared to provide such an opportunity. In
cluded in the article is an analysis of data taken at Cayuga Lake,
New York which was originally examined by Sundaram et al. (1969).
Data for periods in 1950 and 1968 were considered.
Although the original data for 1968 were not tabulated by Sundaram
et al., they were presented in graphical form (Figure 49a, p. 277). The
1950 data which were tabulated in another publication were averaged
and perhaps otherwise manipulated by Sundaram before being used by
Sundaram et al. Therefore, only the graphs of lake temperatures for the
weeks of August 1420 and August 2127, 1968, were used to tabulate
temperature profiles to which least squares approximations could be
made. It should be noted that the resolution of these plots is poor
due to small size and reproducing effects, and that temperatures read
from them may disagree with the data used to construct the plots
originally. Nevertheless, the value obtained should indicate the
ultimate utility, or lack of same, of the procedures devised for this
study.
Before a comparison of results can be made, it is necessary to
consider in greater detail the alternative methods for calculating
fluxgradient quantities. As discussed earlier in this chapter, the flux
term may be considered in two different ways as given by equations
35 and 36, respectively. If the form indicated in equation 36
is used, temperature data taken at times t1 and t2 are used to
approximate 38/8t by figuring AS/At (At = t2 tl) at various depths.
It is then necessary to integrate this newly created set of data below
the depth being considered, z1. The form of the flux indicated by
equation 36 requires that the temperature data taken at t1 be integrated
below z, and the data for t2 be treated likewise. The flux is obtained
by dividing the difference between the two integrals by At. Either
method should yield a proper result; however, one or the other may be
preferable when questions of implementation are considered.
Evaluation of the gradient (at zl) may be made by either one of
two similar methods,assuming the gradient changes linearly during At.
If this is the case, then the average of the slopes of the two temperature
profiles taken at ti and t2 should yield the desired quantity. Alter
nately, the two sets of temperature data may be averagedto form a new
set of data the slope of which may be used to provide values of the
gradient.
Sundaram et al. calculated diffusivities of heat by both the flux
gradient method and McEwen's Method. The emphasis of his work is on the
latter which may explain why the data are presented in a manner which
makes fluxgradient calculations difficult. It should in fact be
noted that their computations of diffusion coefficients by the flux
gradient method are in error. The text (p. 117) refers the reader to
Figure 54 (p. 284) which shows plots of AG/At, f(AO/At)dz, and K
diffusivityy). The values of K shown (and also discussed in the text)
appear to have been calculated from the quotient of the other two sets of
data in the figure, i.e., I(AO/At)/(AB/At), when in fact K should be given
as the quotient of the quantity /(A8/At)dz and 39e/z (not 38/at).
Fortunately, Powell and Jassby independently compute diffusivities using
Sundaram's data, and their values appear to be more precise..
As part of the presentation of the results of the useof McEwen's
method, plots are given (Figure 56a, p. 286) by Sundaram et al. which are
easy to discern and which provide data suitable for fluxgradient
calculations. Powell and Jassby tabulate in their Table 3 (p. 196)
these data,which include 6 f8 (cf., equations 38 and 311) and
A6/At at 1.52 meter (5 foot) intervals to a depth of 27.4 meters. These
data are presented in Table 31. Also in Table 31 are computed values
for other quantities required to calculate fluxgradient diffusivities.
Powell and Jassby do not elaborate on their methods of obtaining these
other quantities; however, after examining their computed values of
diffusivities, it seems that the gradients are obtained for a specific
depth by taking differences between values on either side of the point
in question. Therefore, gradients calculated in this manner have been
included under the heading A6/Az of Table 31. For example, the gradient
Ae/Az, at z = 7.62 meters is given by (13.3 12.2) degrees centigrade
divided by (6.10 9.14) meters = 0.36 degrees centigrade per meter.
While this method may be employed easily, it may not be accurate; there
fore, a representation of the temperature profile as given by (6 8 )
was plotted and slopes evaluated by visually constructing tangents to the
profile curve at the specified points. This information is listed in
Table 31 in the column headed (/z)slope so that comparisons may
be made.
Again,because Powell and Jassby do not describe their methods for
evaluating the flux term, it was assumed they used graphical techniques.
At any rate the data for 36/3t were plotted versus z, and the integrals
were evaluated by measuring areas on the plot. The results of this
procedure are given in Table 31 also. Some question remains about the
Powell and Jassby technique because they do not show calculations of
diffusivities (see Table 33) at those depths where values of AS/At are
TABLE 31. FLUXES AND GRADIENTS IN CAYUGA LAKE 8/14/68 8/27/68
DETERMINED GRAPHICALLY
(1) (2) (3) (4) (5) (6)
za 8 A6/At* A6/Az lh(AO/At)dz (Oe/z)slope
m C *C/day .C/m 'Cm/day *C/m
0.366  6.96
0.293 0.10 6.51
0.27 5.96
0.352 0.17 5.55
 + 0.10 4.98
0.382 0.36 4.36
0.85 3.73
0.539 0.75 2.96
0.92 2.22
0.311 1.36 1.65
0.259 1.02 1.19
0.173 0.75 0.85
0.138 0.55 0.62
0.075 0.42 0.32
0.065 0.26 0.20
0.045 0.20 0.19
0.025 0.19 0.02
0.0  0.0
 0.67
 0.05
 0.21
 0.48
+ 0.25
 0.34
 0.31
 0.62
 0.58
 0.93
 0.89
 0.76
 0.65
 0.37
 0.22
 0.17
 0.21
 0.10
*From Sundaram et al. (1974, p. 196).
**Values at z = 30.5 meters added by extrapolation.
0.0
1.52
3.05
4.57
6.10
7.62
9.14
10.70
12.20
13.70
15.20
16.80
18.30
21.30
22.90
24.40
27.40
30.5**
14.1
13.5
13.8
12.7
13.3
13.0
12.2
10.4
9.9
7.6
5.7
4.5
3.4
2.0
1.5
1.2
0.6
0.0
TABLE 32. FLUXES AND GRADIENTS IN CAYUGA LAKE 8/14/68 8/27/68
.FROM LEAST SQUARES FITS
(1) (2) (3) (4) (5) (6) (7)
At tl (8/148/20) At t2 (8/218/27)
z 6* ae/az fedz 8* a8/az /edz
m oC C/m Cm OC "C/m Cm
23.1
0.30
22.4 0.46
0.42
21.2 0.38
0.56
19.4 0.96
1.30
15.6 1.23
0.75
13.4 0.25
0.18
12.5 0.56
10.0 0.88
0.46
8.4 0.11
8.1 0.36
7.2 0.0
446.92
411.66
377.04
342.73
310.13
278.46
248.14
219.32
194.17
171.79
151.15
130.46
111.02
76.10
62.02
49.05
23.11
0.0
23.7 108.00
0.32
22.2 0.21
+ 0.04
22.2 + 0.10
0.21
21.5 0.66
0.94
18.7 1.00
1.03
15.6 1.04
1.03
12.5 0.74
11.2 0.34
0.56
9.4 0.74
8.1 0.20
7.1 2.0
*From graphs given by Sundaram et al. (1964, Figure 49a, p. 277).
0.0
1.52
3.05
4.57
6.10
7.62
9.15
10.70
12.20
13.70
15.20
16.80
18.30
21.30
22.90
24.40
27.40
30.50
477.55
442.82
408.83
374.57
341.02
307.32
274.14
241.72
212.20
155.01
160.22
137.19
117.25
81.08
64.74
49.69
23.11
0.0
TABLE 33. DIFFUSIVITIES IN CAYUGA LAKE 8/14/68 8/27/68
(1) (2) (3) (4) (5)
z /(de/At)dz K f/(ae/t)dz AflAt
Ae/Az (ae/az) slae/z
m m/day m/day m/day /day
a m2/day m2/day m2/day a2/day
0.0
1.52 (
3.05
4.57
6.10 
7.62 1
9.14
10.70
12.20
13.70
15.20
16.80
18.30
21.30
22.90
24.40
27.40
30.50
*After Powell
5.0
2.0
33.0
i0.0
L2.0
4.4
3.9
2.4
1.2
1.2
1.1
1.1
0.76
0.76
0.95
1.1
 10.4
68.0 130.0
 28.0
32.0 12.0
 20.0
14.0 13.0
 4.6
4.3 4.7
 3.8
1.4 1.8
1.5 1.3
1.6 1.1
1.6 0.95
1.6 0.87
1.9 0.90
2.3 1.1
 0.09
and Jassby (1974, p.
14.0
13.0
24.0
31.0
11.0
4.5
2.9
2.3
2.1
1.9
1.6
1.4
1.2
0.76
0.21
0.0
196).
missing (see Table 31). Since a plot of the data available should permit
evaluation of the integral quantity at any desired point (provided a
smooth curve may be drawn),and gradient values, A6/Az, are easily
determined at every depth, it is hard to understand why some diffusion
coefficient values are missing.
Table 33 listsvalues of diffusivities of heat at 1.52 meter
(5 foot) intervals. Column 3 shows the results of Powell and Jassby's
analysis. Column 2 shows the quotient of the pertinent quantities from
Table 31,as indicated by the column heading. The close agreement of
these two columns suggeststhat, in fact, Powell and Jassby performed
their analysis identically. Column 4 also is calculated directly from
entries in Table 31. A comparison of these values against the first
two sets reveals the effects of more precisely evaluating gradients as the
slopes of a plotted profile. Also, of interest is the negative value of
the diffusion coefficient at 6.10 meters which was among the values
omitted by Powell and Jassby. Such a value should not exist, and dis
cussion of such occurrences will be presented in a later section.
The final column of Table 33 lists diffusivities as computed by
the least squares fit of the temperature data taken from plots in
Sundaram et al. These data and intermediate quantities are shown in
Table 32. By comparing items in both Table 31 and Table 32 it can be
seen that the information is not completely compatible. Values of 6 at
tI and 6 at t2 when averaged do not differ from e L by. some constant
amount (i.e., Bi) and values of 6 at t1 minus e at t2 divided by At equals
7 days do not correspond to Ae/At. These differences may be caused by
either the inability to accurately read the plots from which the data
were taken or an error in the reporting or calculation of e $land
AB/At. Despite the discrepancies,the data were nonetheless subjected
to analysis by the least squares fit method. Each set of temperature
data was approximated by the function given in equation 318. The
analytical expression containing the resultant regression coefficient
was then used to evaluate the integral, fedz, at various depths by a
numerical procedure. The mathematical derivative was computed
analytically. Integral and derivative quantities computed at t1 and
t2 are shown in Table 32. The difference between the two integrals,
Af, was divided by At = 7 days to obtain the average gradient,
The quotient is presented in Table 33 in column 5.
The values tabulated generally agree with those calculated manually
by the alternative approach. The value for the.surface isnot shown.
The entries in Table 32 could be used to arrive at a diffusivity;
however, the quantities in Table 32 are meaningless because the function
as given in equation 318 is not usable at z = 0. The negative values
encountered at 6.10 meters are not predicted by this method. Even
though the slopes att2 in that region (see Table 32) are positive,
which would yield negative diffusivities, the corresponding slopes at t1
are sufficiently negative to cause the average slope to be negative,
thereby predicting diffusivities that seem satisfactory. As' mentioned
earlier the question of negative diffusivities will be considered below.
The overall agreement between the diffusivities calculated manually
and calculated by analytically representing temperature profiles lends
credibility to the latter approach. Furthermore, the closeness of fit
of the calculated temperatures to curves drawn by eye to the observed
data indicates the satisfactory performance of the nonlinear least
squares approach. The observed and calculated points are shown in
Figure 31 for both the week of August 14 and the week of August 20,
1968. The diffusivities shown in column 5 of Table 33 were determined
using the analytical expressions and using the digital computer for all
calculations,including finding the analytical expressions. These
results are also plotted in Figure 32. The plot indicates a marked
variation of diffusivity over the depth of the lake and although values
below ten meters are much lower than the maximum, there are definite
variations throughout the lower depths.
The preceding discussion indicates the manner in which the flux
gradient method can yield values of diffusivities of heat at various
depths in a lake using two observed temperature profiles. Different
interpretations of the flux quantities and gradient quantities can
produce different values of diffusivities as was shown. Also, pointed
out was the effort required to make determinations of diffusivities by
graphical methods. The use of a computeroriented method requiring
significantly less manual effort was demonstrated, and the results com
pared to those obtained graphically. The generally satisfactory behavior
of the method and apparent suitability of its results suggest the
overall usefulness it can have in applications employing large amounts
of temperature profile data to calculate many diffusivities.
TEMPERATURE DEGREES CENTIGRADE
e7
LEGEND
0 OBSERVED 8/14/68 8/20/68
x LEAST SQUARES FIT 8/14 8/20
r OBSERVED 8/21/688/27/68
A LEAST SQUARES FIT 8/218/27
S LEAST SQUARES FIT AT DEPTHS OF
OBSERVATION COINCIDENT WITH OBSERVATIONS
I I I I IN CAYU I
FIGURE 31. OBSERVED AND CALCULATED TEMPERATURES IN CAYUGA LAKE
A. I
U)
w
w
I0
I15
x
0.
w
C
20
I
en. ..
.
VERTICAL DIFFUSIVITY METER2 DAY'
0 4 8 12 16 20 24 28 32
0 I I I I I I
5
I0
10
I
tiU
I 1
I
FIGURE 32. DIFFUSIVITIES IN CAYUGA LAKE
Correlation of Fluctuations Method
In Chapter II the correlation of fluctuations of velocity and
temperature was shown to be equal to the turbulent heat flux. The
timeaveraged term in the fundamental convectivediffusion equation,
equation 212, was related to the diffusivity by equation 220. By
rearranging equation 220, the expression for defining vertical
diffusivity ofheat by the firstprinciples approach results as
Kd  (319)
d ^d
where the terms are as defined in equation 220. Over a given time
interval, T, the individual records of w and 9 are averaged using
t+T t+T
w = t wdt, = J Odt (320)
t t
Then the records are used again to determine w' and 6' by
w' = w w, e' = (321)
from which the product w'6' is calculated. The flux is the timeaverage
of the product over the same integral. Thusly,
t+T
w'' = +T w'9'dt (322)
t
The gradient term is determined by the slope of the average temperature
profile at depth d during the same time interval. By computing the
quotient of the two terms the average diffusivity over the time interval
at the depth of the measurements is found.
57
The most important aspect of this firstprinciples approach is
the determination of the fluctuation terms, especially the vertical
water velocity fluctuations. The successful application of the method
to the problem of evaluating diffusion coefficients depends on the
ability to detect the turbulent water motion. In the following
chapter a detailed survey of turbulence measuring equipment is given
in an effort to determine the manner in which the correlation of
fluctuations method may be used in this study.
CHAPTER IV
SELECTION OF A TURBULENCE MEASURING DEVICE
Introduction
As discussed earlier, the evaluation of turbulent diffusion
coefficients describing the transport of mass or heat may be
accomplished by either a firstprinciples approach or the flux
gradient method. The former requires the measurement of turbulence
caused fluctuations of both a directional water velocity component
and the quantity being studied. The latter requires the measurement
of spatial and temporal concentrations of the quantity being studied.
Depending on the method used the evaluation of turbulent transport.
of heat or mass may or may not require the detection of water velocity;
however, sensing water velocity is an important aspect of most
studies of turbulence, or more specifically, turbulent intensity.
So, although relatively few studies have been made of correlations
between measured fluctuations of velocity components and transportable
quantities, many evaluations have been made of velocimeters from
the standpoint of suitability for use in .studies of turbulent inten
sity.
The criteria applicable to the selection of a velocimeter used
in a firstprinciples approach to transport measurement are similar
to those required in a study of intensity. Many of these criteria,
in a general sense, are pertinent to the selection of any device .to
be used in a scientific investigation.
This chapter presents an overview of the factors which must
be considered when selecting turbulence measuring equipment, a
survey of the various types of velocimeters available with comments
about the potential usefulness of each in turbulence measurement
and lastly, a discussion of the specific requirements for velocity
sensing in the study and the ways that the device ultimately chosen
satisfies those requirements.
Equipment Requirements
The criteria which must be satisfied when selecting a device
for measuring turbulent water velocity fluctuations are similar
to those criteria pertinent to the selection of most devices used
in scientific investigations. Consideration must be given to
instrument performance in the areas of range, sensitivity, stability,
ruggedness, power requirements, noise generation, calibration
requirements, and frequency response, among others. Turbulent
velocity fluctuations may be of a small magnitude when compared
to the average velocity, and velocimeters used in such studies
must have sufficient range to measure the mean and be sensitive
enough to detect the fluctuations. The frequency of the fluctuations
may vary from near zero hertz to several thousand hertz. Frequency
response limits indicate the maximum frequency at which the device
can accurately detect the existing velocity. Stability is a parameter
which may be better considered in three categories: lack of drift,
retention of calibration, and the ability tn be unaffected by
changes in the water other than velocities. Drift, short term
deviation, must not occur to a significant degree during the length
of time of continuous measurements. The same may be said for a
permanent alteration of the device calibration, and if water quality
changes during measuremen; the device must not respond to the change.
The nature of the turbulence will dictate the length of time stability
must persist.
Hotwire, Hotfilm Anemometers
Hotwire anemometers have been used for many years as turbulence
measuring tools. A short wire of platinum or similar metal is
heated above ambient temperature by an electrical current, and subse
quently cooled by fluid flowing across it. The amount of cooling
can be related to the velocity. The hotwire probe is more useful
in gas flows than in liquid flows because the greater electrical
conductivities usually experienced in liquids interfere with normal
operation. This problem stimulated the development of hotfilm
probes for use in liquids. Hotfilms are essentially hotwires
coated with an electrically insulating substance such as quartz or,
more recently, Teflon. The application of the film is done in such
a manner as to preserve as much of the thermal response of the probe
as possible. Also, increased structural strength resulting from the
film coating makes them preferable for use in water where more impact
force is exerted on the probe.
Hotwire and hotfilm probes are used in two basic electrical
models: constant current and constant temperature. The constant
current method maintains a uniform current through the probe, and
changes in velocity cause changes in wire temperature which cause
changes in wire resistance. Therefore, the voltage drop through
the probe is an indication of the velocity across the probe. An
electronic feedback network helps constant temperature anemometers
to maintain the probe at a uniform temperature. Hence, the amount
of current through the wire is indicative of level of velocity
passing it. Problems of electronic stability within the feedback
circuits inhibited the early usefulness of constant temperature
anemometers; however, now sufficient electronic capability exists,
and the constant temperature method is prevalent among commercial
anemometer manufacturers.
Because turbulence measurements in water are nearly always
made using hotfilm probes, the remainder of this discussion will
focus on anemometry using them. Hotfilm probes are available in
various geometric shapes and configurations. The shape of the probe
may make it more or less suitable for a particular application.
Furthermore, probes and associated electronics are now available
which sense water velocity in the three component directions. Thermal
response and electronic response of modern anemometer systems are
such that very high frequency turbulence can be accurately sensed.
Also, electronic instruments are available which perform correlation
and averaging operations as the velocity measurements are being made.
Although, hotfilms have been used successfully in many studies
of turbulent flow, they do have some drawbacks. The physical size
of the probes is very small; however, the somewhat larger size of
the probe supporting apparatus restricts the use of them in some
instances. The use of hotfilms in natural water causes probe
deterioration due to accumulations of dirt, biological growth, gas
bubbles, and scale. The result of this degradation is, among other
things, loss of calibration and thermal response (see Morrow and
Kline, 1971). Another problem arises from the heating of the prohe.
The quality of output signal is proportional to the amount over
ambient the probe is heated, and in some cases this heating may 
in fact inject significant amounts of heat into the water being
studied.
LaserDoppler Velocimeters
LaserDoppler anemometry is one of the newest velocity measuring
techniques and has only recently been applied to studies of turbulence.
A beam of light is focused on a small volume of fluid in which
particles reflect and scatter the beam. If the particles are moving,
then they reflect the light with an apparent difference in wavelength,
the Doppler effect. An indication of the'particle velocity is
obtained by determining the frequency shift. This is done by hetero
dyning (i.e., producing a new frequency by adding or subtracting,
the shifted and the unshifted signal). In some cases,two signals
shifted in a controlled manner are heterodyned. Lasers provide
63
illumination sources with very narrow bandwidths, which permit
measurement of even relatively low velocities. If the positions
of the laser and the scattered beam sensor are fixed, the: .
velocity detected represents a specific direction of particle motion.
Arranging three mutually perpendicular sensors to detect the.scattered
light enables a determinationof the three velocity components of
the particle (Fridman, Huffaker, and Kinnard, 1968).
The most significant advantage oflaserDoppler anemometryis
realized in studies of fluid flow through transparent ducts or. 
pipes. In these cases thevelocity'sensing equipment need notbe
inserted into the flow,"and no alteration or other disturbance of
the flow should occur. This is especially useful for investigating
fluid motion very near conduit walls. Investigations of flow in
opaque conduits or natural waters would require that much of the
laserDoppler equipment be enclosed .in a submersible package which
would make satisfactory measurements unlikely.
Many problems encountered in the formative stages of laser
Doppler velocity measurement have been resolved or are being resolved.
Developments in electronic capability, beam splitting techniques,
and heterodyning procedures are providing better equipment perfor
mance than in the past. Indications are that this method of anemometry
will permit much progress in laboratory studies of turbulence.
Rotating Element Velocity Meters
Velocity meters which use rotating elements such as propellers,
turbines, and cup assemblies are very common. Propellers and turbines
are similar to each other; however, propellers usually have fewer
blades and propeller blades usually have more curvature. Turbines
are used to measure high velocities in pipes, and lower pipe velocities
and open channel (freesurface) currents are measured by propeller
type meters. Cuptype meters are used for free surface flow. Propeller
and cuptype velocimeters sense water motion in a plane parallel to
the propeller axis or pendicular to the cup axis of rotation. However,
the response of each to the direction ofvelocity within the plane
being considered is different. The propeller is affected only by
flow components parallel to its axis while the cup is affected byall
velocity components in the plane of its rotation. For these reasons
the cup type meter is often preferred for use in studies of total
planar velocity, and propeller type meters are required when it is
desirable to distinguish various directional components of velocity.
There are several techniques used to indicate the rate of revo
lutions of the propeller or cup element. The most common is pulse
generation by means of electrical contact closure. The pulses are
counted as clicks heard through ear phones or as digital signals
which trigger electronic counters. Other meters operate by sensing
changes in electrical resistance or direct current voltages generated.
at different rotational speeds. The design.of these meters has been
refined to the point that they are highly reliable and rugged, and
they are used extensively in both field and laboratory studies. They
have been used mostly in studies of flow or current measurement and
similar applications requiring detection of average velocities.
Attention has been given to the effects of turbulent water velocities
on propeller meters, and besides improving the reliability of average
flow data obtained using them, the better understanding of propeller
response to turbulence has permitted them to be used in studies of
turbulent flow structure (Plate and Bennett, 1969).
Propellers do not foul or degrade or otherwise lose calibration
when used in natural waters. Furthermore, they resist damage due
to impact from foreign matter. Using two meters positioned at right
angles enables resolution of separate velocity components from
simultaneous measurements. Advanced designs using lightweight materials
and low friction pulse generation techniques allow quick, accurate
response to even turbulent velocity fluctuations. However, the limit
of this response due to inertia effects of even improved designs
precludes the use of propellers in investigations of very low velocities.
The lowest threshold velocity of meters currently being used is in
the range of onehalf centimeter per second. Slower water motions
are not detected. The physical size of the propeller also limits the
minimum eddy size which it can detect.
Acoustic or Ultrasonic Devices
Methods have been developed and improved which detect fluid
motion using acoustic signals whereby the behavior of ultrasonic
waves emitted into the flow is monitored and processed into velocity
information. Acoustic devices function according to either one of
two basic principles: time of travel or Doppler shift. Measuring
66
water velocity by measuring the Doppler shift of highfrequency
signals reflected from particles in the flow is the same principle
of operation used by laser velocimeters. The chief difference is
the wavelength of the emitting source. Time of travel devices
sense the incremental propagation of the emitted signal caused by
the fluid motion. In other words, the ultimate velocity of the
acoustic signal is the vector sum of its generated velocity plus
the spatially integrated velocity of the fluid in the region
through which the signal passes. Indication of water velocity is
obtained by discerning that portion of the received signal altered
by the water motion.
Various techniques may be employed to detect the alteration of
emitted waves. The simplest in concept is to produce a pulse or
wave form and transmit it toward a receiver unit which senses it.
By accurately measuring the time of travel of the pulse and sub
tracting it from the expected time of travel in still water, calcu
lated from a knowledge of the speed of sound for ambient water
conditions and senderreceiver separation distance, the water velocity
along the senderreceiver path may be determined. However, the
necessity of knowing the ambient speed of sound inhibits the usefulness
of the technique, and it may be averted by adopting a dual path
procedure. If the times of travel of two signals traveling the
same path but in opposite directions are measured, then the difference
between the two should indicate twice the water velocity only, due
to the canceling out of the speed of sound component common to each.
A modification of this elementary concept is the singaround circuit
which operates by emitting an initial signal that is received by
a secondary unit and upon reception triggers the emission of a
return signal which,when detected by the primary unit,triggers yet
another signal. The process continues for a selected time or number
of rounds. It uses the dual path concept and is essentially a
summation of individual measurements. This design yields improved
resolution or sensitivity while indicating an average velocity over
the repetition period. Another modification of the time of travel
method is the differential time circuit which also uses a dual path
instrument arrangement. Two signals are originated at each end of
the path coincidentally. The reception of one signal at the other
end triggers the start of a timing circuit which is stopped when
triggered by the reception of the slower signal. Knowing the
difference in time and which signal arrived first permits computation
of water speed and direction along the instrument path.
Acoustic methods are desirable because they may be nonintrusive;
hence they do not disturb the water at the point of measurement. They
do not foul or degrade rapidly when used in "dirty" waters, and they
are rugged. They have given very satisfactory results when used
as flow quantity meters because they intrinsically average flow
between sensors. Stream flows have been measured using timeoftravel
acoustic systems installed on opposite banks. Doppler shift systems
may be the more satisfactory in studies of turbulence which examine
local flows. It is possible to construct a Doppler shift device
using properly oriented sensors which will detect water velocity in
three directional components. This meter used in a study of estuarine
turbulence was calibrated over a wide velocity range of from about
onehalf centimeter per second to thirty meters per second (Wiseman,
1969). Problems with acoustic velocimeters are caused by electronic
noise, sensing of spurious waves, inadequate alignment of emitter
and receiver units, and interference from foreign matter in the
water.
Lagrangian Methods
An alternate method to measuring water velocity past a fixed
sensor is to mark water parcels and follow the movement of the
markers. Such studies of the Lagrangian nature of turbulence are
carried out by a wide range of methods among which are flow visual
ization techniques, tracer tests, and drogue studies. Flow visual
ization methods are often used in laboratory investigations of
turbulence. They involve making a succession of photographs, which
record the positions of fluid markers at the time of exposure.
Markers must be detectable by the photographic equipment and possess
physical properties which permit them to behave as suitable indicators
of fluid behavior. Types of markers include small spheres, hydrogen
bubbles, and small volumes of dyes, among others. Merritt and
Rudinger (1973) studied flume turbulence by using a solution of
water and pH sensitive dye. A pulse of a current through a thin
metallic wire positioned in the flow created a surplus of hydrogen
ions at the wire surface which changed the pH and consequently the
color of a thin line of water particles. The fate of these particles
was then recorded photographically and subsequently analyzed.
The travel of water parcels may also be monitored by tracer
tests. Tracers have been used in both laboratory and field studies
of turbulence. They may occasionally be detected visually, but more
often, the nature of the tracer is such that it may be sensed in
invisible concentrations by equipment designed for such a purpose.
Fluorescent dyes, radioactive substances, and salt solutions,
among others,have been used extensively in investigations.
Field studies of larger.scale motions have been performed
using drogues or similar devices which are transported by water
currents. Various schemes have been employed to reduce unwanted
interference, for example' those of wind. The primary problem
in such studies is that drogues which can be located and monitored
successfully are often affected by wind or buoyancy or inertia and
do not accurately indicate water motion.
Studies of turbulence by Lagrangian means require constant
monitoring throughout the course of the experiment. Observations
of the entire flow field must be made often enough to detect the
effects of the smallest eddies being studied and over sufficient
total times to satisfy statistical requirements. Laboratory studies
of small scale eddies may last only several seconds while field
studies of larger scale motions may take days or weeks. Improvements
in photographic techniques, electronic drogue tracking capabilities,
and tracer detection equipment have made the performance of Lagrangian
studies easier; however, they are still difficult and laborious to
carry out because they are not adaptable to high levels of automation.
Electromagnetic Flowmeters
The movement of charged particles through a magnetic field
induces currents. This was discovered by Farraday in 1831 with
ordinary solid electrical conductors. The same principle can be
used to measure fluid flows, and since all particles have at least
atomic level electrical charges, they are affected by magnetic
fields regardless of whether or not the fluid is an electrically
conductive medium. Electromagnetic flow meters have been used
primarily to measure pipe flow (Grossman, et al., 1958), and blood
flow in humans and animals, but instruments for measuring local
velocity have also been developed (Bowden and Fairbairn, 1956).
There are two methods of electromagnetic flow meter operation.
The first senses the induced voltage generated by the flow of charged
particles through a magnetic field. The second senses an induced
magnetic field generated by a conducting fluid flowing through an
established magnetic field. The second method is often used to
measure flow in electrically conductive media because it need not
be in electrical contact with the media. The induced voltage method
requires electrical contact between the sensing probes and the flow.
Cushing (1958) showed by theoretical arguments and experimental
results that the sensitivity of induction flow meters is not affected
by variations in water conductivity as long as the conductivity is
above a threshold value of about 105 mho's per meter. The vector
nature of the induced voltage field enables the determination of
velocity direction, and instruments exist which can measure at least
two velocity components simultaneously.
In theory, velocities down to zero can be detected; however,
electronic noise in the signal processing portion of electromagnetic
meters prevents detection of motion below some threshold level.
Also, some error in measurement is caused by a "transformer effect"
voltage which is due to the use of alternating current to create the
magnetic field. Electronic circuitry is capable of diminishing the
resultant variations in zero baseline due to this "transformer
effect", but some drift still occurs. Boundarylayer effects at
the probewater surface cause minor errors in the response of the
instrument; however, the probe is affected by water motion over a
distance equal to two or three times the probe radius and the boun
darylayer is small in thickness relative to the overall distance
of influence. The device has no moving parts, is rugged, and does
not lose calibration quickly because of fouling.
Other Devices
Several other types of devices have been developed for measuring
turbulent flow. Some are altogether different from those already
discussed, and some are similar to or use similar principles to those
mentioned previously. Several kinds of pressure transducers have
been devised to gage instantaneous changes in water velocity. The
dynamic pressure at the probe is often.sensed by electronic means;
for instance, Ippen and Raichlen (1957) used a diaphram which moved
slightly due to pressure change. The diaphram also served as a
variable capacitance in an electrical circuit, and the response
of the circuit indicated flow velocity. Force transducers have
also been used to measure water velocities. The force of the fluid
impinging on a small sensor surface is converted to an electrical
signal which is calibrated to velocity (Earle et al., 1970; Siddon,
1971). Pressure and force transducers cannot measure very low water
velocities, and the use of them is limited to velocities above 1
centimeter per second. 
Electrokinetic transducers have also been used in studies of
turbulence (Binder, 1967). This type of sensor is unusual because
it is not energized. Fluctuations in water velocity cause a thin,
small wire probe to emit detectable electrical impulses. However, the
wire does not respond to laminar flow and can only be calibrated
dynamically, which makes it useful only in studies of the relative
intensities of turbulence. Other studies have been performed using
thermal methods which impart small quantities of heat to the flow
and then attempt to sense the heat nearby. Similarly a device known
as the Deep Water Isotropic Current Analyzer (DWICA) has been
developed for use in investigations of reservoir transport and
turbulence. It omits a small quantity of radioactive material near
the center of a ring of sensors. Flow direction and magnitude are
determined by which sensor detects the radioactivity and the time
lag between emission and detection.
Price Comparisons
No information on prices of the various types of devices has
been given, although this is an important consideration when
selecting suitable instrumentation. Some of the reasons for not
stating prices follow. Many devices reported in the literature
are custommade, and often cost information is not given. Even
when costs are noted,they may only reflect a fraction of the
total resources necessary to produce the equipment. Furthermore,
many research systems are integration of commercially available
devices and custommade ones. Also, quoting prices for commercially
available equipment is of itself difficult because optional or
auxiliary equipment is usually available. The desirability and,
more importantly, the necessity of any or all of these supplemental
devices varies greatly according to the specific application.
Requirements of the Lake Mize Study
As discussed in Chapter VI, Lake Mize,near the University of
Florida campus,was selected as the site for the field study of
vertical turbulent transport. The nature of the lake together with
the overall objectives of the study imposed a set of rather rigorous
constraints on the selection of a velocity measuring device. The
primary requirementwas for the instrument to be able to measure
water velocities in two or three directional components at a point
in the lake. It had to be suited to field use, although battery
operation was not necessary because of the 110 vac supply at the lake.
The scheme of the field study was to take measurements over a long
period of time, so stability of calibration was essential. The
time, scales of turbulence were suspected to be such that stability
over at least a period of one day was necessary while the maximum
frequency response needed would be 1 hertz. Furthermore, an
acceptable flow meter had to be unaffected by changes in chemical
and physical water quality such as ion concentration, temperature,
and concentrations of dissolved substances,since significant variations
occurred with depth in the lake and with time; Probably the most
stringent criterimwas that of sensitivity. Although definite values
of water velocities were not known, it was known that the rate of
movement would be very low. A target value of 0.3 cm/sec (0.01
ft/sec) was established. This value represented a compromise between
the desire to sense large scale water motion and the awareness that
ultralow velocity measurement was not feasible at the time. For the
approximate lake depth of 25 meters,a 0.3 cm/sec velocity was
indicative of vertical time scales on the order of 1 day. Another
requirement involved instrument output. Because the entire field
study would be automated,the device had to indicate water velocity
in analog form. Manual operation was unacceptable. It was felt
that the design and fabrication of an adequate velocimeter was beyond
the capabilities of the personnel and physical resources available on
hand. Funds for the purchase of a suitable instrument were limited
to well under ten thousand dollars,which was the total amount allotted
for all equipment expenses of this study.
Selection of the Electromagnetic Flow Meter
Of all the types of devices available for use in turbulence
measurement only one, the electromagnetic flow meter, seemed capable
of satisfying the criteria established above. LaserDoppler meters
were unacceptable for field work. No Lagrangian technique could
be used for a continuous, highly automated study. Pressure and
force sensors and propellers could not respond to low enough water
velocities. Hotfilm probes would foul too quickly. The acoustic
Doppler meters seemed to exhibit promising characteristics, but
they were not available commercially, and similarly dualpath
acoustic devices were only supplied in massive units for measuring
streamflow. The electromagnetic meters were previously available
only for pipe flow measurements; however, it was learned that a
meter for..local velocity measurements in natural waters was newly
available. The meter measured two perpendicular flow components and
seemed to meet all the other criteria established. The lower
limit of its sensitivity was 0.3 cm/sec; however, drift and noise
limits were on the same order. Despite these possible shortcomings,
it was decided that the electromagnetic meter was acceptable and
possessed superior overall capabilities to any other device being
considered. Its cost of less than five thousand dollars was also
acceptable.
CHAPTER V
DESIGN OF THE DATA ACQUISITION SYSTEM
Introduction
Because the ultimate objectives of this study involved the
collection of large amounts of data over a period of at least one
year, there existed a need for acquiring this information in an
automated fashion. This chapter describes the projected requirements
for a data acquisition system capable of satisfying these needs.
The design of such a system was made by incorporating the data needs
for each phase of analysis planned. The result was a systemcapable
of performing a combination of many types of in situ measurements
and recording them in a compatible fashion on a single punched paper
tape recorder. Brief descriptions of the principal components
of the system are given in this chapter, and a discussion of the
physical installation of these components at the field research
site is presented in the following chapter. Also included in this
chapter is a description of the calibration procedures used on each
of the sensors in the data acquisition system.
Data Requirements for FluxGradient Method
Evaluation of heat flux and gradient quantities is done by using
temperature profile data taken at two separate times. Temperature at
various lake depths must be known, and the times at which the
temperatures occur must also be known.
The number of locations within the water column at which
temperatures are recorded and the frequency at which the obser
vations are made which will provide sufficient information for
fluxgradient analysis depend: on the individual situation. For
instance, the lake temperature may be nearly constant over several
meters of depth in certain regions (e.g., the hypolimnion) and
demonstrate large spatial gradients at other levels.(e.g., the
epilimnion).
Furthermore, the zones of highest gradient may change with
time. Similarly, changes in temperature at a point may occur
regularly over a long time span or may become irregular relatively
quickly. The adequacy of using a given vertical spacing or time
interval is indicated by whether or not actual temperatures occurring
between measurements may be inferred from the data collected.
Since the field experiment also involved measuring water
velocities, the use of a single temperature sensor mounted on a
.moving support was not feasible. Therefore, a survey of water
column temperature required multiple sensors at fixed positions.
Consideration of temperature profiles taken by Nordlie (1972)
at Lake Mize indicated vertical spacing of less than one meter near
the surface and about one or more meters in the lower region would
be advisable. Diurnal changes in temperature were foreseen so the
sampling frequency was specified as at least six times per day.
The required precision of temperature measurements was established
as 0.1 degree centigrade. While resolution to minute fractions of
a degree would have been desirable, preliminary study indicated a
practical limit of resolution of commercially available temperature
sensors of about 0.1 degree centigrade.
Data Requirements for Correlation of Fluctuations Method
Evaluation of heat flux by the correlation of fluctuations
method entails the measurement of the vertical component of water
velocity (i.e., magnitude and direction) and coincident water
temperature at the place of velocity detection. The gradient
evaluation requires knowledge of the change in temperature with
depth at the depth of the velocity measurement. Although'discussed
in detail in the preceding chapter an additional comment regarding
velocity measurement is in order here. Of particular import is the
ability to sense water velocity in only one direction and at low
levels. A magnitude of water velocity was not known a prior
however, the small size and sheltered nature of Lake Mize suggested
that very low levels of water speeds might be'expected. By estimating
distance scales of motion in the lake to be from less than one to
about ten meters and time scales of from several seconds to several
minutes, a crude estimation of a minimum velocity of 0.003 meters
per second (0.01 ft/sec) was made.
The requirement for temperature measurement was established as
0.1 degree centigrade due to prior knowledge of equipment limitations.
Since evaluation of the gradient was to be accomplished as part of
the fluxgradient method analysis, it was felt that sufficient
gradient data would exist from those measurements, and no new data
would need be collected.
As noted above, the time scales of the turbulent motion were
estimated to be as low as several seconds. Therefore, a maximum
sampling frequency of one second was specified for the velocity
temperature correlation data.
It was also desired to evaluate diffusivities by the correlation
of fluctuations method at various depths throughout the water column.
To do this the velocity probe and its companion temperature sensor
had to be capable of being positioned at any depth. Of course, once
at a specified depth the apparatus had to be rigid so that even
slight water movement might be detected.
Other Data Requirements
Other data needs were those related to environmentalmeteorological
conditions. Wind speed measurements at three different heights were
planned, and wind direction was also specified. Incident shortwave
solar radiation, relative humidity, and air temperature complete
the list of required parameters. Since these parameters all change 
daily and most exhibit marked diurnal fluctuation, a sampling frequency
of several times per day was deemed desirable. After the study
began,it became apparent that instantaneous measurements of these
variables were subject to great error. A discrete observation of wind
speed, for instance, could not accurately be applied to a lengthy time
interval as.the average value over the interval. Therefore,
during the field study the requirements for wind speed and solar
radiation were changed such that summed or integrated values .of each
were needed. Instead of wind speed in meters per second, wind in
total meters past the sensor during the interval since being last
recorded was specified. Similarly, total langleys of solar radiation
were called for. The variations in air temperature, relative
humidity, and wind direction were such that readings taken halfhourly
(as will be discussed later) sufficed.
Wind velocity up to about five meters per second was judged
suitable. The capability to measure air temperature from zero to
forty degrees centigrade and relative humidity from twenty to one
hundred percent was specified. 1.2 to 1.5 langleys per minute was
estimated as the maximum levels of solar radiation it would be
necessary to measure.
Aggregate Requirments
Before considering other factors associated with the plan of
the data acquisition system, a recapitulation is in order. Two
groupings of data needs existed due primarily to different requirements
in frequency of collection. One was velocitytemperature correlation
data sampled as often as once per second. As the study developed,
this group became known as Group A. The other, a larger group of
data signals, monitored the environmental conditions at a rate as
slow as several times per day. Thislatter group, eventually labeled
Group B, included water temperatures at a series of depths, air
temperature, solar radiation, wind speed at a series of heights
above the water surface, wind direction, and relative humidity.
Finally, although not stated earlier, a precise knowledge of the
time of sampling of each parameter had to be available.
Data Recording Requirements
The amount of data to be collected was forecast to be very large
because not only was the overall length of the field study projected
at one year but also the number of separate data sources was great
and the frequency of sampling high. The need to employ special
procedures to accommodate this volume of information was evident.
Certainly manual data collection was not feasible. A high level
of automation was required. Due to the number of inputs (parameters),
some type of multiplexing or switching was indicated, because a
different recording device for each signal would necessitate the
use of well over ten units. The satisfactory recorder must then be
able to accept data from several sources either in parallel or
serially and also be able to operate accurately at the rate of at
least one measurement per second. Finally, the form in whichthe
data were recorded should permit easy access for subsequent analysis;
*in other words,converting the data to a form which could be read
into digital computers should be a process requiring as little effort
as possible.
Additional Data Acquisition Needs
The major requirements of the data gathering equipment have
already been stipulated or alluded to; however, some factors should
be discussed further.
The speed of operation of the system had to be at least one
data item per second to accommodate the velocitytemperature correlation
information. Yet, since the actual time scales of motion in the lake
were not known and could perhaps be very much slower, say, on the
order of minutes, the ability to select from a range of sample rates
was desired. This sample rate flexibility could permit optimum
system operation by enabling the selection of the slowest sample
rate providing adequate data and thereby reducing the volume of
data collected.
The data collecting equipment had to be suitable for use in
the field. Furthermore, the ability to function on, in, or very
near the water was requisite of most of the system components.
There were few requirements on the various sensors to be used
other than that they be capable of measuring the levels of the
respective parameter of each which were specified earlier. It was
necessary, however, that each of the types of sensors be compatible
with the overall data gathering effort.
The budget for the procurement of field equipment was ten
thousand dollars. This figure was less than requested and considered
to be a significant constraint on the ultimate comprehensiveness of
the data acquisition system.
Commercial Equipment
As a general rule,equipment needs can most efficiently be met
by purchasing commercially available equipment whenever possible.
The necessity to use custommade devices arises whenever there is
no satisfactory equipment on the market or when there is not enough
money to afford equipment which is otherwise available. A survey
of manufactured data acquisition equipment was made. This included
a study of sensors capable of measuring each of the physical parameters
required and also the available alternatives for gathering and
recording these measurements. In summary, as could well be expected,
no single device capable of performing all of the stipulated tasks
was found.
As described in detail in the proceeding chapter, the search
for a suitable velocity meter culminated in the purchase of an
instrument which detected velocities in two perpendicular directions
and the output of which was an analog (voltage) signal at a level
of one volt out per foot per second of water speed. Other instruments
were found which could measure each of the other parameters; however,
several drawbacks were encountered. Among' these was the fact that
the total cost of all of these exceeded available funds. Additionally,
the aggregate of instruments would occupy much space if they all were
operated at the same location and, besides, there was much duplication
of duty because most were designed to function independently.
The data assembling and recording phases of the acquisition
system could also be performed by commercially available equipment.
Disadvantages encountered included very high costs, incompatibilities
with certain types of sensors, and unworkable requirements of power
and operation. No system was found which offered satisfactory
performance yet was affordable. It should be noted that the high
costs usually were associated with instruments which provided
greater sampling speeds or capacities than needed for this project.
As could well be expected,considering the range of performance
requirements associated with this field study,very little of the
necessary equipment could'be purchased already assembled. Most of":
the sensors were available commercially, but even they had to be
acquired in a form which would permit the incorporation of them into
a mostly customfabricated instrument package.
Design of Data Acquisition System
The preceding actionss have discussed in detail requirements
of the data acquisition system. What follows is a description of
the system as it was ultimately fabricated within the constraints
of available resources (especially money).
To stay within the budget of ten thousand dollars it was
necessary to amend the goals of the project by limiting its scope
or modifying technique,or find alternate sources of supply of
equipment,or make better use of material already on hand. The
result was an optimization procedure which combined all of these
alternatives to various extents. While every step in the decision
making process cannot be explained, suffice it to say that many
characteristics of the completed system were dictated by the
availability of instruments at low or no costs, even when preferences
would have been otherwise. However, in most cases,the requirements
outlined earlier were met or exceeded.
The system consisted of several stages. One stage was the
sensors which sensed the level of a physical parameter (e.g.,
temperature, wind speed) and reflected that level as an electrical
signal (e.g., resistance, current). Another stage of the system
received the sensor outputs and conditioned them, individually, to
yield compatible voltage signals. The voltages from the various
conditioners were multiplexed or switched one at a time into the
recording device. The multiplexing and recording stages were supported
and controlled by a stage which contained all the digital logic
circuitry. This digital stage was also the source of the system
timing.
As a rule the sensors were manufactured devices and the signals
from each were conditioned by custommade circuits. An exception
was the water velocity meter which sensed and conditioned its own
signal so that the output was a voltage directly related to water
speed and direction. The only other system component not custom
made was the paper tape punch which included an analogtodigital
converter.
The requirements for a suitable recording device outlined earlier
indicated the use of telemetry, magnetic tape recording, or punched
paper tape recording. Other alternatives such as photographic methods
and manual recording seemed infeasible, and even stripchart recording
was thought to be acceptable only as a last resort. The degree
of complexity and the high cost of installing and operating a
telemetry system eliminated that option. Magnetic tape recording
satisfied all of the stipulations; however, its cost was high
and could only be justified for data collection at much higher
speeds (thousands of items per second). Also, magnetic tape was
thought to be susceptible to extraneous electronic noise, especially
when used at slow rates. The capabilities of punched paper tape
seemed matched to the needs of this study. Data could be recorded
sequentially at rates up to tens of items per second, and the stored
information could be read directly into digital computers for sub
sequent use. No cumbersome data handling was required.
It was determined that the local computing facility had the
capability of reading the punched paper tape, so the decision to
procure such a device was made providing costs were within budgetary
limitations. Fortuitously it was learned that an apparently suitable
piece of equipment might be obtained gratis from a nearby govern
mental agency which no longer used it. Ultimately this tape punch
was acquired and after testing proved most satisfactory; therefore,
the remainder of the data acquisition system was planned such that
optimum use could be made of it.
The recorder was manufactured by Towson Laboratories (Model
DR100P35) and contained an analogtodigital converter as part
of the unit. Zero to ten volt signals of either polarity were stored
in digital form by punches in a oneinch wide paper tape. The device
punched whenever a pulse command from the digital control stage was
received.
The nerve center of the system was the digital control stage.
This stage used integrated circuits of transistortransistor logic
(TTL) type to generate pulses which controlled all the system
functions. Precise timing was achieved by using a sixty cycle per
second signal from the alternating current power line as a source
for a digital timing chain. The chain was composed of several
serial stages each dividing the frequency into it to produce a lower
frequency. Any one of several time chain outputs could be selected
as the time control for the velocitytemperature correlation data,
Group A. By actuation from shore,any of a range of sample frequencies
from one sample per second to one per three minutes, could be selected.
A one pulse per minute signal was used to control and activate
the Group B sensors. Ultimately, there were twenty thermistor
probes which sensed water and air temperatures, a pyranometer for
measuring solar radiation, three cup anemometers detecting wind
speed, a wind direction vane, a polymertype sensor for relative
humidity, a voltage signal indicating depth of the Group A probes,
another voltage signal indicating the rate of Group A sampling, and
two signals indicating power supply levels in the system. These
thirty parameters were sampled at the rate of one:per minute making
a complete scan every halfhour.
The timing chain controlled the switching and recording of Group
A and Group B so that they were sampled in an integrated, coincident
manner. The design of the chain was such that the Group B signal
88
could be interspaced between Group A recordings even at the fastest
Group A sampling rate.
The switching in the system was done by solid state FET (Field
effect transistor) switches. The analog signals were amplified
using integratedcircuit operational amplifiers. The amplifiers
were of a common style (741) but specially selected for low levels
of signal drift due to ambient temperature changes.
Figure 51 gives a conceptual diagram of the system in its
ultimate form. The various analog and digital control phases are
as indicated in the preceding discussion. No further description
of the electronic control circuitry will be presented, but the
sensors used to detect parameter values will be discussed in greater
detail. Brief descriptions and identifications of each of the sensor
types used and the circuits used to condition the sensor outputs
are given. Following that is a summary of calibration procedures.
Velocity Measurement
The velocity meter purchased was made by the EngineeringPhysics
Company, Rockville, Maryland. The unit selected was Model EMCM3BX
(Serial number 620). The meter has been discussed earlier and is
not described in detail here. It produces a voltage output of one
volt per foot per second.water velocity in either of two perpendicular
directions. The algebraic sign of the output voltage indicated the
orientation of the movement in either direction.
Island bridge) "...... o..ga
.9 r .
26_^  ''>* p
ir
Thermistor A/D
S String Temperature g Final Convertor
S. Multiplexer Bridge 6opp Multiplexer
19 
20
r^______^..
,o _ KEY
Wind DATA SIGNAL
2 .Speed
Counter   AMPLIFICATION
S, Group B
fEnvironmental ._
Sensors
Wind Direction Multiplexer DATA COLLECTION
Solar Radi.on nearer AND
Others (inl. rel. hum.) ACQUISITION SYSTEM
FIGURE 51. CONCEPTUAL DIAGRAM OF THE DATA ACQUISITION SYSTEM USED AT LAKE MIZE
Temperature Measurement
The choice for temperature detection was between using thermistors
or thermocouples. Thermistors were picked because the voltage
outputs of thermocouples was lower than that which could be tolerated
due to electronic noise associated with boosting.the signal. Originally
thermistors manufactured by the Yellow Springs Instrument Company
(Part No. 44018) were selected because they offered linear change
in resistance with change in temperature. Each thermistor was
connected to a multiconductor cable, and the joints sealed with potting
resin. One by one the joints leaked, and new probe making techniques
were attempted. Before a satisfactory technique was developed nearly
all of the thermistors deteriorated to the point of uselessness. New
thermistors were bought from Fenwal Electronics, Inc. (Part No.
GA51J1). They were conventional nonlinear devices but were ordered
because of lower cost. The revised probes were constructed by
connecting the thermistor to twoconductor, vinylcovered wires
(thereby using one wire per probe) and covering the thermistor and
joint with a short (23inch) piece of copper tubing which had been
crimped and soldered at one end. The region of juncture between the
tubing and wire covering was sealed by wrapping the area with over
lapping turns of plastic electrical tape and then repeating with
another wrapping of tape. These probes performed satisfactorily even
though with time some failed for various reasons.
The probes were initially spaced at intervals of two feet
(0.610 meters) from lake surface to bottom, but when the first cable
