• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 List of symbols
 Abstract
 Introduction
 Theory
 Experimental equipment and...
 Results
 Discussion of results
 Conclusion and recommendations
 Appendix A: Diffusivity of NaCl...
 Appendix B: Negative buoyancy generation...
 Bibliography






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 74/033
Title: Halocline erosion due to wind induced stress
CITATION PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00076159/00001
 Material Information
Title: Halocline erosion due to wind induced stress
Series Title: UFLCOEL
Physical Description: ix, 73 leaves. : illus. ; 28 cm.
Language: English
Creator: Humiston, Kenneth Kirby, 1946-
University of Florida -- Coastal and Oceanographic Engineering Laboratory
Publication Date: 1974
 Subjects
Subject: Fluid dynamics   ( lcsh )
Turbulence   ( lcsh )
Wind-pressure   ( lcsh )
Ocean-atmosphere interaction   ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (M. Eng.)--University of Florida.
Bibliography: Bibliography: leaves 71-72.
Statement of Responsibility: by Kenneth K. Humiston.
General Note: Typescript.
General Note: Vita.
Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
 Record Information
Bibliographic ID: UF00076159
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida
Resource Identifier: oclc - 14119084

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Title Page
    Acknowledgement
        Acknowledgement
    Table of Contents
        Table of Contents
    List of Figures
        List of Figures 1
        List of Figures 2
        List of Figures 3
    List of symbols
        Unnumbered ( 8 )
        Unnumbered ( 9 )
    Abstract
        Abstract
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    Theory
        Page 6
        Page 7
        Page 8
    Experimental equipment and procedures
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
    Results
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
    Discussion of results
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
    Conclusion and recommendations
        Page 63
        Page 64
        Page 65
    Appendix A: Diffusivity of NaCl in water
        Page 66
        Page 67
    Appendix B: Negative buoyancy generation at the surface due to evaportation
        Page 68
        Page 69
        Page 70
    Bibliography
        Page 71
        Page 72
Full Text
r()
CIO















HALOCLINE EROSION DUE TO WIND INDUCED STRESS


By

KENNETH K. HUMISTON


















A THESIS PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF ENGINEERING







UNIVERSITY OF FLORIDA

1974


Coastal Engineering Archives
University of Florida















ACKNOWLEDGEMENTS


I wish to thank Dr. Omar H. Shemdin, Chairman of my supervisory

committee, for his encouragement and guidance in the preparation of

this thesis, and for his part, as Director of the Coastal and Oceano-

graphic Engineering Laboratory, in making the excellent internal wave

facility there available for this study. I also extend my thanks to

the other members of my supervisory committee, Dr. Robert G. Dean, Dr.

D. Max Sheppard, and Dr. Yu-Hwa Wang for their instruction and inspira-

tion during the period of my graduate learning and thesis preparation.

Special thanks are due Dr. Wang and Dr. Ronald J. Lai for their assist-

ance with the instrumentation during the experimental investigation.

Finally a word of thanks to Fena Jones and Evelyn Hill for typing,

Denise Frank for drafting, and the personnel at the Laboratory for their

cooperation and assistance in obtaining and setting up the experimental

apparatus.















TABLE OF CONTENTS


ACKNOWLEDGEMENTS ii

LIST OF FIGURES iv

LIST OF SYMBOLS vii

ABSTRACT ix

I. INTRODUCTION 1

II. THEORY 6

III. EXPERIMENTAL EQUIPMENT AND PROCEDURES 9

IV. RESULTS 18

V. DISCUSSION OF RESULTS 56

VI. CONCLUSIONS AND RECOMMENDATIONS 64

APPENDIX A DIFFUSIVITY OF N C1 IN WATER 66
a
APPENDIX B NEGATIVE BUOYANCY GENERATION AT
THE SURFACE DUE TO EVAPORATION 68

BIBLIOGRAPHY 71

BIOGRAPHICAL SKETCH 73
















iii


Coastal Engineering Archives
University of Florida















LIST OF FIGURES


Schematic drawing of the facility showing the overall
dimensions and the location of the various components.

Wind velocity measurement instrumentation, pitot tube,
pace transducer, signal conditioner and voltmeter.

Hot film anemometry instrumentation; anemometer power
supply and linearizer, integrator, voltmeter, tape
recorder, and rms meter.

Hot film calibration by specific gravity, 0 21 cm/sec.

Hot film calibration by specific gravity, 0 1.5 cm/sec.

Hot film calibration by specific gravity, 0 10 cm/sec.


Figure 7.


Figure 8.


Figure 9.


Figure 10.


Figure 11.


Figure 12.


Figure 13.


Density profile
wind velocity 2

Density profile
wind velocity 2

Density profile
wind velocity 2

Density profile
wind velocity 2

Density profile
wind velocity 2

Density profile
wind velocity 2

Density profile
wind velocity 2


at Station 6 for wind duration 3 hours,


mps.

at Station
mps.

at Station
mps.

at Station
mps.

at Station
mps.

at Station
mps.

at Station
mps.


6 for wind duration 4 hours,


12 for wind duration 3 hours,


12 for wind duration 4 hours,


12 for wind duration 5 hours,


18 for wind duration 3 hours,


18 for wind duration 4 hours,


Figure 14. Density profile at Station 6 for wind duration 3 hours,
wind velocity 2.5 mps.


Figure 15.


Density profile at Station
wind velocity 2.5 mps.


6 for wind duration 4 hours,


Figure 1.


Figure 2.


Figure 3.


Figure

Figure

Figure








Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


Figure


16.


17.


18.


19.


20.


21.


22.


23.


24.


25.


26.


27.


28.


29.


30.


31.


at Station
mps.

at Station
mps.

at Station
mps.

at Station
mps.

at Station
mps.

at Station
mps.

at Station
mps.

at Station
mps.

at Station


wind velocity 3 mps.


Figure 32.


Lines at constant density for
wind velocity of 2 mps. (See


wind of 4 hour duration,
Figures 8, 10, 13)


V

Coastal Engineering Archive
University of Florida


Density profile at Station
wind velocity 2.5 hps.

Density profile at Station
wind velocity 2.5 mps.

Density profile at Station
wind velocity 2.5 mps.

Density profile at Station
wind velocity 2.5 mps.

Density profile at Station
wind velocity 2.5 mps.

Density profile at Station
wind velocity 2.5 mps.

Density profile at Station
wind velocity 2.5 mps.


Density profile
wind velocity 3

Density profile
wind velocity 3

Density profile
wind velocity 3

Density profile
wind velocity 3

Density profile
wind velocity 3

Density profile
wind velocity 3

Density profile
wind velocity 3

Density profile
wind velocity 3

Density profile


6 for wind duration 5 hours,


12 for wind duration 3 hours,


12 for wind duration 4 hours,


12 for wind duration 5 hours,


18 for wind duration 3 hours,


18 for wind duration 4 hours,


18 for wind duration 5 hours,


6 for wind duration 3 hours,


6 for wind duration 4 hours,


6 for wind duration 5 hours,


12 for wind duration 3 hours,


12 for wind duration 4 hours,


12 for wind duration 5 hours,


18 for wind duration 3 hours,


18 for wind duration 4 hours,


18 for wind duration 5 hours,


28


29


30


31


32


33


34


35


36


37


38


39


40


41


42


43










Figure 33.


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure


34.

35.

36.

37.

38.

39.

40.

41.

42.


Figure 43.


Lines of constant density for wind of 3 hour duration,
wind velocity of 2.5 mps. (See Figures 15, 18, 21)

Density profiles for turbulence measurement data.

Mean velocity profiles for wind velocity 2.5 mps.

RMS velocities for wind velocity 2.5 mps.

Wind velocity vs. log z, (height z) above SWL.

One dimensional scalar energy spectra at a depth of 2 cm.

One dimensional scalar energy spectra at a depth of 10 cm.

One dimensional scalar energy spectra at a depth of 15 cm.

One dimensional scalar energy spectra at a depth of 33 cm.

Velocity distributions induced by wind over a closed basin
with a stable stratification.

Experimentally observed entrainment velocity vs. predicted
entrainment velocity.















LIST OF SYMBOLS


A A constant

D Depth

E(k) Scalar energy

K Curvature

N Brunt-Vaisala Frequency

R. Richardson number

U Velocity

a A constant

b Fluctuating component of buoyancy force

g Gravity

i Subscript, i = 1, 2, 3

k Wave number

1 Length

L Mixing length

m Unit vector upwards

n Mixing frequency

p Pressure

r Separation vector

t Time

U Mean velocity

u' Perturbation velocity

ue Entrainment velocity








u* Shear velocity

u Horizontal velocity components

w Vertical velocity fluctuations

ij. Velocity spectrum tensor

a Subscript, a = 1, 2

P Angle between direction of velocity U and wave number k

E Viscous dissipation

p Density

Pa Density of air

p Density of water

po Reference density

T Shear stress

Viscosity

e Eddy viscosity


viii








Astract of Thesis Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Engineering



Halocline Erosion Due to Wind Induced Stress

By

Kenneth K. Humiston

August, 1974


Chairman: Omar H. Shemdin
Major Department: Coastal Engineering

A laboratory investigation of the effects of wind on the depth of

formation of a halocline is presented. A wave tank measuring 24.39 m

long by 1.83 m high by 0.61 m wide was filled to a depth of 0.75 m with

water so that a continuous density stratification was obtained, the vary-

ing density being achieved by varying salt concentrations. This strati-

fied system was subjected to a wind induced stress at the surface and

the changes in the velocity and density profiles and horizontal turbulent

fluctuations were measured. It was found that on a macroscopic scale a

discontinuous density jump initially forms below the surface a short time

after the wind starts blowing, thereafter the interface is depressed by

entrainment of the more dense lower layer in the turbulent upper layer.

The rate of entrainment is dependent upon an overall Richardson number.

Comparison of the horizontal turbulent energy spectra for stratified and

unstratified situations shows a greater decreasing level of energy with

depth in the presence of a stable stratification.





Chairman















I. INTRODUCTION


A shear stress applied to a fluid boundary will induce motion

and, if the Reynolds number is high enough, turbulence. In a fluid

with a stable stratification the turbulent mixing results in the for-

mation of a layer nearly homogeneous with respect to density. This

layer extends from the surface of applied stress down to a level at

which a discontinuous jump in density marks the border between the

turbulent region and the underlying region where the original density

distribution may remain unchanged. The extent of the homogeneous layer

is dependent upon the magnitude and duration of the applied stress,

physical parameters of the fluid, and characteristics of the flow. The

formation of the discontinuous density jump inhibits further extension

of the homogeneous layer by acting as a barrier to turbulent mixing.

This phenomenon is observed in many areas of significance and

economic importance. Heavy atmospheric layers which form and are re-

sistant to mixing may enhance air pollution problems. Methane can

accumulate in high areas of a coal mine shaft where the extent of

mixing with ventilating air has an important bearing on safety

(Ellison and Turner, 1959). The degree of mixing of thermal discharges

from water cooled power plants is important to the ecology of the sur-

rounding area.

This thesis examines the formation of a discontinuous density jump

in a stably stratified body of water when a wind induced shear stress

1

Coastal Engineering Archives
University of Florida









is applied to the surface. ~n the experiments conducted here the for-

mation of a halocline is analagous to the formation of a thermocline

in nature. Varying salinity was used to achieve density stratification

rather than temperature since the latter is more difficult to control

in the laboratory situation due to the high rate of thermal diffusion

as compared to molecular diffusion. (Turner, 1968). Knowledge of the

process involved will be useful in the study of internal waves as well

as in situations such as quiescent basins and thermal discharges where

a stable stratification may be altered by climactic conditions and form

a barrier to vertical mixing.

The concept of entrainment is important in the process being studied.

The turbulent region grows as the underlying non-turbulent region becomes

entrained in it. The rate of entrainment can be described by an entrain-

ment coefficient. Morton, Taylor, and Turner (1956) were the first to

use such an entrainment coefficient in their examination of entrainment

in buoyant plumes rising through a stably stratified ambient body of

water. They found the entrainment constant to be proportional to the

axial vertical velocity of the plume. Ellison and Turner (1959) per-

formed a series of experiments on stratified flows induced by a heavy

salt solution flowing down the sloping bottom beneath a layer of fresh

water, and the spread of a surface jet of lighter fluid over a heavier

ambient fluid. They measured a characteristic velocity at the inter-

face by timing the movement of entrained sheets of fluid and found that

the ratio of the entrainment velocity to the characteristic mean flow

velocity is a function of a stability parameter having the form of a

Richardson Number, g(p p )h
Ri po Vz

where V is the characteristic velocity, p is the density of the fluid,





3


g is the acceleration of gravity and h is the thickness of the turbulent

layer and the subscript o refers to the reference fluid. Turner (1968)

did some experiments on turbulent entrainment with no mean shear to de-

termine the effects of molecular diffusivity. He found that it is not

always possible to neglect molecular effects when the stability is high

and the Reynolds number is not very large. By comparison of experiments

using salinity to experiments using temperature to obtain stratification,

he showed however, that below a critical value of the Richardson Number

the entrainment is independent of diffusivity. This value is Ri = 1,

where R. is defined

i R=

where n is the frequency of the mechanical mixing generating the turbu-

lence Ap is the difference between the densities of the turbulent and

non-turbulent regions and t is the mixing length. The results of these

experiments also show an inverse proportionality between his Richardson

number and entrainment velocity ue, namely ue~RiR. ue is the rate of

depression of the interface.

Kato and Phillips(1969) did an experiment in which they applied a

constant shear stress to the surface of an annular tank containing water

with a linear stratification. A motor in the center of the tank rotated

an annular screen at the water surface, constant shear was obtained by

mounting the motor on a bearing then restraining it from turning with a

spring, and adjusting the controls so that the tension in the spring was

constant. They observed that the mean velocity varied most rapidly near

the surface below the screen and immediately above the interface which

formed shortly after the constant shear was applied. The central region

remained relatively uniform. When the interface reached the lower quarter









of the tank the density jump had become so large that entrainment

virtually ceased, yet the fluid below the interface was set in motion

by the purely viscous stress across it, the rate of diffusion of momen-

tum by molecular viscosity being greater than that of salt by molecular

diffusion.

The Reynolds number UD/v, where U is the speed of the screen, D

the depth of the layer and v the kinematic viscosity, during these ex-

periments ranged from 5 x 103 to 2 x 104, large enough, so that the

large scale structure of the turbulence would be independent of the

molecular viscosity. The properties of the turbulence are characterized

by the friction velocity u. = T-/p and D. Their entrainment constant E

defined as the ratio of the entrainment velocity ue, to the friction ve-

locity u., is thus a function of; the fractional change in buoyancy

across the interface g-- where po is the density of fresh water, the

friction velocity.itself; and the .depth D of the interface, character-

izing the scale of the turbulence. Grouping these three variables in

a dimensionless group gives an overall Richardson number

R. = g 6p D
1o Po u.
Furthermore, since the upper layer is nearly homogeneous, a salt balance

computation gives

ap 2 az o
where ( i) is the initial density gradient, so that
az 0
R g( ) D2
10o 2po u*

Thus the entrainment coefficient is a function of the Richardson Number,

E e = f(Rio)
E* 0u






5


Employing the same functional relationship as Turner, ueaRi-1, Kato

and Philips proposed the empirical relationship


E = ue = 2.5 PO u*2
u g 6p D '

where the numerical constant is stated to be uncertain to within 30

per cent.
















II. THEORY


The mechanisms involved in halocline (or thermocline) erosion can

be described in terms of the turbulent energy equation for horizontally

homogeneous turbulence (see Phillips,1969)


uat at {w'(p/po + 1 U2) =__- (1)

where u. is the fluctuating component of the velocity, w' is the vertical

fluctuating velocity, p is pressure, U is the mean velocity, b' is the

fluctuating component of the buoyancy force and E represents viscous dis-

sipation. The subscript i = 1, 2, 3 and subscript a = 1, 2 specifying

velocity components. The first term on the right is the rate of transfer

of energy from the mean flow to turbulence by the work of the Reynolds

stresses. The second term on the left represents the convective redis-

tribution of turbulent energy in physical space by the turbulence itself.

In the presence of a halocline some of the energy imparted to turbulence

would be redistributed in the region of the interface where turbulent

eddies entrain fluid from the more dense lower layer and mix it with the

turbulent upper layer. This increases the density of the upper layer

and therefore increases the potential energy of the mean buoyancy field.

The transfer of kinetic to potential energy is represented by the covari-

ance between the fluctuations in vertical velocity and buoyancy in Equa-

tion (1). The continued entrainment of fluid from the lower layer results

in the erosion of the interface at a rate referred to earlier as the











entrainment velocity ue. Referring again to the results of Kato and

Phillips(1969), this entrainment velocity can be shown to be a func-

tion of an overall Richardson number characterizing the stability of

the system.

Turbulence is frequently described in terms of spectral quantities.

Phillips(1969, chapter 6) shows how the scalar energy spectrum, E(K),


E(K) = f ii(t ) d S(K) (2)
E(K) 1

can be obtained from the velocity covariance tensor of the velocity

field at a given instant, where ..ij() is the wave number spectrum

tensoror the Fourier transform of the covariance with respect to the

separation vector between the two points under consideration. Philips

then postulates that the energy density of turbulence is reduced in the

presence of a stable stratification. This is a consequence of a reduc-

tion in the covariance between vertical velocity fluctuations which re-

duces the apparent eddy viscosity defined as


Pe(z) = A pi cos2B P33(t, o=-K'*(Z), Z) dk (3)
4- 4
where A is numerical constant and is the angle between K and U. This

results in a reduction in the Reynolds stress gradient

dTr d2U
dz e dz (4)

and the stress itself if = 0 at some point in the flow, so for a

given velocity gradient the energy flux T- is also decreased. Since

the energy flux is of the order u'3/Z (Tennekes and Lumley, 1972) the

energy density of the turbulence is also reduced, reducing the eddy

viscosity still further.







8


For this postulate to be correct the turbulent energy spectrum in Eq.

(2) should show a lower level in stratified flow when compared to the

spectrum in an unstratified flow, all other conditions being equivalent.

For isotropic turbulence the longitudinal contribution to the total

kinetic energy of the turbulence is just one third of that value, so

that the above postulate could be verified with one dimensional velocity

measurements. A major aim of the experimental study described here is

to verify the postulate proposed above.















III. EXPERIMENTAL EQUIPMENT AND PROCEDURES


The experiments were conducted in the internal waves tank, 24 meters

long by .6 meters wide and 1.22 meters high, with a wind section above the

water .95 meters wide and .6 meters high (Figure 1). A complete description

of the tank was given by Sheppard, Shemdin, and Wang (1973). The tank was ini-

tially filled to a level of 37.5 cm with fresh water, then filled slowly

from the bottom with .02% salt water to bring the level to 75 cm. From

this an approximately linear profile was obtained by dragging a board

the length of the tank, at the interface between the salt and fresh water.

The board was positioned perpendicular to the length of the tank and the

interface, so that as it moved along the interface the turbulent eddies

which formed behind the board caused mixing between the two layers. This

process was continued until the desired degree of mixing was obtained.

The density profiles were measured initially with a single electrode

conductivity probe (Gibson and Schwarz, 1963) but due to the low frequency

drift, calibration was impossible, so the probes were modified by inser-

tion of a length of tygon tubing into each probe so that samples of water

could be siphoned from the wave tank at any level. The density could then

be measured with a hydrometer to an accuracy of four decimal places, al-

though the procedure is rather tedious.

Once the desired density profile was set, wind was produced by pull-

ing air through the tank with an axial flow variable pitch fan, manufactured

by Joy, and an integral part of the wave tank unit. Three fans settings

were used over the course of the experimentation, corresponding to wind

velocities of 2, 2.5, and 3 meters per second. Wind velocity profiles were

























SOUTH
END


NORTH.
END

AIR FILTE
AIR IN'




HYC
r ----


Fig. 1. Schematic drawing of the facility showing the overall dimensions and the location of the
various components.










measured above the water surface with a pitot static tube, manufactured

by United Sensor and Control Corp., a Pace Model P90D differential pres-

sure transducer, a Disa Type 52B30 True Integrator signal conditioner and

a Darcy Model 440 digital multimeter (Figure 2).

For each wind velocity, observations were made for wind durations

of 3, 4, and 5 hours. In each case density profiles were measured before

alteration by the wind, for the dynamic situation with the wind blowing,

and after the wind was turned off allowing sufficient time for internal

oscillations to die out. The tank was refilled after each run in an at-

tempt to recreate the initial density structure as accurately as possible.

Velocity profiles were measured with hydrogen bubble wire and tracer

dye, but these methods proved inadequate due to the fluctuating salinity.

The hydrogen bubble wire failed because bubbles too large formed in re-

gions of high salinity while regions of low salinity saw no bubbles at

all. The tracer dye method was inadequate due to buoyancy effects.

Because of the failure of the velocity profile measurements, it was

decided to run more experiments to obtain velocity profiles with a hot

film anemometer system. A Thermo-systems, Inc. hot film sensor, model

1233 NACL was mounted on a vertical traverse mechanism anchored firmly at

the top and bottom to reduce vibrations. The sensor was moved vertically

with a pulley-cable setup. The anemometry system consisted of a Thermo-

systems model 1051 power supply and indicator, and a model 1055 linearizer.

The output was channeled simultaneously through the Disa integrator and

Darcy DVM to obtain the mean voltage directly, and through a Thermo-systems

correlator, model 1015C. The correlator amplified the signal by a factor

of 10 to reduce any subsequent interference by noise when recorded on a

Midwestern Instruments tape recorder. The recorded signal was analysed























I i3 ii' II'C'-U&ri'f i iie a .

+ .r ,r r h

r7



^ ~---


.- -. .. .. Is

O LL
* ^ tf -. y
i^ -- -"-
QO' '^ ^ .


1?

.nar '
-*ci~s;F; *TSi~; i Z .


Wind velocity measurement instrumentation, pitot tube,
pace transducer, signal conditioner and voltmeter.


Figure 2.


Cli~-t-~T C-l. P-P-t--l. CC -.--~/--~ ~
I









.." "'F"W low In


Hot-film anemometry instrumentation; anemometer power
supply and linearizer, integrator, voltmeter, tape
recorder, and rms meter.


I


Figure 3.


E .61 toa










on a Thermo-systems 1060 rms meter, and the Weston-Boonshaft and Fuchs

series 711 spectrum analyser. Figure 3 shows the anemometry instrumen-

tation.

Calibration of the velocity sensor was carried out in a plexiglass

tow tank. Calibrations were done in water of different salinities in

order to determine the dependence of response on salinity. Figure 4 shows

the calibrations at various salinities. Bubbles sometimes formed on the

tip of the sensor, probably due to disolved gases coming out of solution.

It was found that the tendency for bubbles to form at the sensor tip in-

creased with exposure time to the salt water, so immediately prior to

the data taking run the overheat ratio was reduced and a calibration curve

was obtained in water of specific gravity 1.0070, which is close to that

expected in the top layer during a run. The calibration is displayed in

Figures 5 and 6.





























































































it) LO 0 LO
Cn d

SIOA 3


*
C
10
L



4.-
-


O

o


0






E
-o


E

*r



ICC
u -


i '.;^













16




























4,





'4-
u
U
a-
0
,0 C


tt
E
O C
O
*r-









U,,
S *- CL












Un

LA-


SIOA '3


4*










17













o
N













>.1

u
'4J


S-
u

4)

0



u Ea
EE








4-) 1
~0
0 ,-
o







--




=5





LL


(D in


S41oA '3















IV. RESULTS


The results presented are from two phases of experimentation.

In the first phase density profiles (density, p vs. depth, z) were measured

to determine the depth of formation and rate of lowering of the density

jump, or "interface" between the homogeneous upper layer and the stably

stratified bottom layer, and are necessary for the determination of the

Richardson number. In the second phase velocity profiles (mean velocity,

U vs. depth, z) were measured for both the stratified and unstratified

situations to show the differences in flow distribution. At the same time

turbulent fluctuations u' were recorded to determine the horizontal tur-

bulent energy distribution (u12 vs. depth, z).

Nine separate data taking runs were performed in the first phase,

three runs of different wind duration for each of three different wind

velocities. Density profiles were measured simultaneously at three loca-

tions along the tank, 8 meters, 15 meters, and 22.5 meters from the upwind

end of the tank, designated stations 6, 12, and 18,respectively. The

profiles show the position of the interface and comparison of two profiles

under wind of the same velocity but different durations g.i.ve the entrain-

ment velocity. Figures 7 through 31 display the density profiles;each

Figure shows the profile of the starting conditions, a "steady state"

profile at a designated time after the wind was started, and a final pro-

file taken after the wind was stopped and internal oscillations had ceased.

Comparison of the corresponding profiles at the three positions along the

tank gives the horizontal density structure of the internal setup, illus-



























































1.005 1.010 1.015 1.020
Specific Gravity


Figure 7.


Density profile at Station 6 for wind
duration 3 hours, wind velocity 2 mps.


44 v.*


0'
1.000


1.025











Legend:
Initial
A Final
Steady State


1.010 1.015
Specific Gravity


i 020


1.025


Density profile at Station 6 for wind
duration 4 hours, wind velocity 2 mps.


E
0 40
>


1.005


Figure 8.





























-4O
E
S40


01 I I L I A
1.000 1.005 1.010 1.015 1.020
Specific Gravity


Figure 9.


Density profile at Station 12 for wind
duration 3 hours, wind velocity 2 mps.


1.025











Legend:
Initial
A Final
Steady


1.015 1.020 1.025


Figure 10.


Specific Gravity
Density profile at Station 12 for
duration 4 hours, wind velocity 2


State


E
3 40


1.005


1.010


wind
mps.











Legend:
Initial
A Final
Steady


State


E
S40-




30-




20-




10-




0
1.000 1.005 1.010 1.015 1.020
Specific Gravity
Figure 11. Density profile at Station 12 for wind
duration 5 hours, wind velocity 2 mps.


1.025






















































1.000 1.005 1.010 1.015 1.020 1.025
Specific Gravity


Figure 12.


Density Profile at Station 18 for wind
duration 3 hours, wind velocity 2 mps.










Legend:
Initial
A Final
Steady State


1.010


1.015


1.020


1.025


Specific Gravity
Figure 13. Density profile at Station 18 for wind
duration 4 hours, wind velocity 2 mps.


60




50



E
S40




30




20




10


1.005










Legend:
Initial
A Final
Steady State


S40 -




30-




20-




10-



01
1.000 1.005 1.010 1.015 1.020 1.025
Specific Gravity
Figure 14. Density profile at Station 6 for wind
duration 3 hours, wind velocity 2.5 mps.










Legend :
Initial
Final
Steady State


E
S 40
>>


1.005


1.010


1.015


1.020


Figure 15.


Specific Gravity
Density profile at Station 6 for wind
duration 4 hours, wind velocity 2.5 mps.


1.025


t
































































Figure 16.


1.010 1.015 1.020 I.C
Specific Gravity
Density profile at Station 6 for wind
duration 5 hours, wind velocity 2.5 mps.


r-IP




















































OL- I I I
1.000 1.005 1.010 1.015
Specific Gravity


Figure 17.


Density profile at Station 12 for wind
duration 3 hours, wind velocity 2.5 mps.


1.020


1.025
























































Figure 18.


I I I n
1.010 1.015 1.020 1.0
Specific Gravity
Density profile at Station 12 for wind
duration 4 hours, wind velocity 2.5 mps.


E
401










Legend:
Initial
A Final
a Steady


1.015


Specific Gravity


Figure 19.


Density profile at Station 12 for wind
duration 5 hours, wind velocity 2.5 mps.


State


1.005


1.010


1.020


1.025












SY1k _


60F-


50


40-


[L'I


1.000


1.005


Legend:
Initial
A Final
Steady


1.010


1.015


i".


1.020


Figure 20.


Specific Gravity
Density profile at Station 18 for wind
duration 3 hours, wind velocity 2.5 mps.


State


1.025









Legend:
Initial
A Final
q Steady State


0 1 I 1 1
1.000 1.005 1.010 1.015 1.020
Specific Gravity


1.025


Figure 21.


Density profile at Station 18 for wind
duration 4 hours, wind velocity. 2.5 mps..









Legend :
Initial
A Final
a Steady State


1.015


1.020


Figure 22.


Specific Gravity

Density profile at Station 18 for wind
duration 5 hours, wind velocity 2.5 mps.


. 40


)0 1.005


1.010


1.025



















































1.005 1.010 1.015 1.020
Specific Gravity
Figure 23. Density profile at Station 6 for wind
duration 3 hours, wind velocity 3 mps.


4
40




























































Figure 24.


1.010 1.015 1.020 1.(
Specific Grovity
Density profile at Station 6 for wind
duration 4 hours, wind velocity 3 mps.


36


- 40
:1%









- -- -
\.


\


V


Legend:
Initial
A Final
a Steady State


1.005


Figure 25.


1.010


1.015


1.020


1.025


Specific Gravity
Density profile at Station 6 for wind
duration 5 hours, wind velocity 3 mps.


37









Legend:
Initial
A Final
Steady State


1.010 1.015
Specific Gravity


1.020


Figure 26.


Density profile at Station 12 for wind
duration 3 hours, wind velocity 3 mps.


70





60


1.025


t























































Figure 27.


1.010 1.015 1.020 1.(
Specific Grovity
Density profile at Station 12 for wind
duration 4 hours, wind velocity 3 mps.









Legend:
Initial
A Final
a Steady


01 I
1.000 1.005


Figure 28.


I I I _-
1.010 1.015 1.020 1.025
Specific Gravity

Density profile at Station 12 for wind
duration 5 hours, wind velocity 3 mps.


N~


State








Legend:
Initial
A Final
a Steady State
(no data)


\%N


1.005


Figure 29.


1.010


1.015


1.020


Specific Grovity
Density profile at Station 18 for wind
duration 3 hours, wind velocity 3 mps.


50


E
340



30


1.025


i
























































Figure 30.


1.010 1.015 1.020 I.
Specific Gravity
Density profile at Station 18 for wind
duration 4 hours, wind velocity 3 mps.









--


Legend:
Initial
A Final
Steady


State


S40-




30- \










10-




0
O --
1.000 1.005 1.010 1.015 1.020 1.025
Specific Gravity
Figure 31. Density profile at Station 18 for wind
duration 5 hours, wind velocity 3 mps.








treated for two situations in Figures 32 and 33 as plots of lines of con-

stant density in a two dimensional tank where points with the same den-

sity are connected by straight lines for easier identification, although

the actual lines of constant density are nonlinear.

The initial density profile for the second phase of experimentation

was determined in the same way and is shown in Figure 34, along with a

stepwise approximation to the interface for the profiles after the wind

had been blowing for 1 and 5 hours. The density of the top layer and

the position of the interface were measured; below the interface the

density profile was assumed unchanged, which is a reasonable assumption

based on the data from the initial experiments.

Figure 35 is the mean velocity profile for the unstratified situ-

ation, as well as the stratified case for two different times during

the run. This illustrates the change in the flow which takes place as

the interface is eroded. Figure 36 is the rms velocity profile, {(u'2)2

vs. z} for the same situations as Figure 35, but does not show much more

than a general trend of decreasing magnitude just below the surface.

A logarithmic plot of height above the water surface vs. mean wind

velocity is shown in Figure 36. This was used to determine the shear

velocity in air u, directly, and the shear velocity in water from
a
(Shemdin, 1972)

U*2 P 2
w w w
W 'W W

The turbulent velocity fluctuations u' were analyzed on the spectrum

analyzer. The output is plotted in Figures 38 through 41 as the turbulent

energy spectra in volt2-sec vs. frequency. Figures 38, 39, 40 and 41 are

the spectra at depths of 2 cm, 10 cm, 15 cm, and 33 cm, respectively. Each

figure shows the difference in the horizontal component of the turbulent
























0)
I
I n
I 0)s-
I --- 3



o
I I G)




0 t- o




C*

I-- 0
4* r--



*n cc

c -




cm
I r -




















0 0 0 0 0
O- -

jI-



J^l ____I I I-\---






























ISE
.0


clr







Ic o
CD)

4--
I 1 ->



W v.

4-- I-1

.0 ) 0
D >


43 C >



o 40-
O c 0.-
I 4- /.- c "
0 c" 0
+II- 0 *_- 0



OO J -

L Ln
*r-







-J,








0 0 0 0 0 0 0 0
(wo) -
1I
1 --
1c
1
N~=









I--%-- -- -


I i\r
Ihr 4 hr \


1.005


1.010
Specific


.-Initial





\


1.015
Gravity


Figure 34.


Density profiles for the experimental
run during which turbulence measure-
ments were taken.


04-.4


sr i


SWL


60k


50s


E
S40
N


Profile


30 1-


20 F-


10o-


AL


1.000


1.020


1.025


I I I I~L


- ii






























20- 4


30



2- Legend
Fresh Water
Stratified I Hour of V
A Stratified 5 Hours of


-I O i 2 3
Mean Velocity U cm/sec

Figure 35. Mean velocity profiles for
wind velocity 2.5 mps.




U- w


80
SWL

70- "


60



50


E
-40
N


30



20
Legend:
Fresh Water
10- Stratified I Hour of Wind
A Stratified 2 Hours of Wind


0 I 2 3
RMS Velocity cm/sec


Figure 36. RMS velocities for wind
velocity 2.5 mps.

















































0.1 I I
120 140 160


Figure 37.


180 200 220 240 260
U, cm/sec


Wind velocity vs. log z,
(height z) above SWL.


'JU




4. -,


0.1 1.0 O0
Frequency, Hz


Figure 38.


One dimensional scalar
energy spectra at a
depth of 2 cm.


Volts 2


100































Volts2-sec


0.I 1.0 10 100
Frequency, Hz
Figure 39. One dimensional scalar
energy spectra at a
depth of 10 cm.


















.01






.001


Volts2 -sec


.0001






.00001


1.0 10
Frequency Hz

Figure 40. One dimensional scalar
energy spectra at a
depth of 15 cm.


100






























2
Volts -sec


0.1 1.0 10 100
Frequency Hz

Figure 41. One dimensional scalar
energy spectra at a
depth of 33 cm.






55


energy between the stratified aid unstratified situations at the same

depth. The position of the interface during the recording of the tur-

bulence was just below 33 cm, a depth which develops after 1 hour of

wind at 2.5 mps. The bandwidth for the analysis was .1 Hz for the range

.1 Hz to 1 Hz, 1 Hz for the range 1 to 10, and 10 Hz for the range 10 Hz

to 100 Hz. For 80% confidence the results in each of these three ranges

were within 70%, 20%, and 10% of the true values, respectively (Schiesser).














V. DISCUSSION OF RESULTS


In the initial phase of experimentation the density profiles (Figures

7-31) lend themselves to examination by continuity. If the original pro-

file is integrated to give the mass of a column of water of unit surface area,

the result should be the same for the profile under wind. This was found

to be untrue; in general the integral of the density profile at the down-

wind end of the tank is less than that of the initial profile, while at

the upwind end it is greater. Figures 32 and 33 illustrate the density

structure with an internal setup due to the wind stress. Determination

of rate of entrainment is thus complicated by the internal setup and re-

turn flow.

The shear-velocity above-the interface at the downwind end of Lhe

tank might normally be expected to be higher than at the upwind end due

to the boundary conditions of a tank of finite length. However, the

greater potential energy of the buoyancy field at the upwind-end of the

tank presents an unstable situation which results in a counter current

driven by buoyancy forces. Figure 42 illustrates the varied velocity

profiles which occur along the length of the tank. This figure was drawn

from observations of the movements of particles suspended in the water.

It is not a steady state situation because of the constantly changing

buoyancy forces, but it is typical. Unfortunately not enough velocity

measurements were made to verify this, but making the assumption that the

velocity gradients at the interface are of the same order of magnitude

along the length of the tank results in a Richardson number at the down-







































I



-a,-




-~o

4v)
4-J





0
3:rr








-C,
>) '



()4-)




O0
u L





=3 V)


u)a)
a 0

4-


0 %0
r- w --
a) > 4-
::: 0 --

al)c




S-



LL.


cj




%.. -S


58

wind end of the tank about 500 times greater than at the upwind end. In

addition the relative shallowness of the interface and smaller density

jump across it at the upwind end would increase the entrainment coeffi-

cient of Kato and Phillips. From this it is hypothesized that the majority

of mixing takes place very near the upwind end of the tank. The evidence

is by no means conclusive, but the Richardson numbers calculated from the

velocity data taken in the second phase of experimentation support this

hypothesis.

The overall Richardson number 16 meters from the upwind end of the

tank was 200 with the interface at a depth of 33 cm, 4 hours later it was

267 with the interface at 36 cm. The shear velocity term in the overall

Richardson number was determined from Figure 37 and

u*w = (p U2a/pw)

The interface receeded 3 cm in the 4 hours between the recording of velo-

city profiles giving an experimental u^ of 2.1 x 10 cm/sec. The rela-
S-2
tionship found by Kato and Phillips predicts ue = 1.1 x 102 cm/sec.

Some differences between the results of Kato and Phillipsare to be

expected due to the differences in the scale of the experiments but the

same parameters are used in defining the Richardson number so this great

difference indicates that some other mechanism may be responsible. There

are several possibilities which will be discussed but first a word about

the quality of the data is in order.

In the first phase of the experiments an attempt was made.to reproduce the

starting density profile as consistently as possible so that comparisons

could be made between the different runs. For a given wind velocity the

position of the interface at the end of three, four, and five hours should

give an estimate of the average rate of entrainment over two periods of








-ne hour each. In general this is correct but the quantitative aspects

:F the entrainment rate is poor due to slight variations in the slope of

:ne linear approximation to the density profile, and perhaps even more

significant is the variation in curvature. For example, considering the

density profiles at station 18 for a wind velocity of 3 mps (Figures 29,

30, and 31), a greater entrainment velocity is indicated from 4 to 5

tours than from 3 to 4 hours. Due to the increasing density gradient

and depth of the interface with time, this should not be so according to

the expression for the entrainment constant. A close look at the initial

profiles reveals that the 5 hour case had a gradual curvature concave

upwards while the other two were less curved and exhibited concavity both

up and down at different levels. Mixing of a finite element of a fluid

with a profile concave upward will result in positive buoyancy for that

element (Long, 1970), consequently, it will rise into the turbulent region

and be completely mixed with the upper layer and produce a lowering of the

interface. The same element in a fluid with the concavity of the profile

downwards would have a negative buoyancy and sink back towards the inter-

face rather than being immediately mixed with the turbulent layer, thereby

slowing the entrainment. This in itself is an interesting observation,

but the presence of this anomaly makes it difficult to determine an en-

trainment constant from the data.

In addition to the relative magnitudes of the entrainment in the

two experiments, there is a non-uniformity in the entrainment velocity

along the length of the tank. An expression for the entrainment might

look like 2
PU*w
E(R., x) = f(x) g6pD (5)
1 Ip


onere Po is the mixed layer density and








f(x) = 1-x (6)
1 1
where 1 is the length of the tank and x is the horizontal distance from

the upwind end of the tank. It does not matter that f(x) vanished at x=l

since at the boundary u, also becomes zero. The resulting form of the

entrainment constant is 2
E(R. x) = C l-x U*w (7)
1 1 g6pD

where the proportionality constant c must be determined empirically from
S pU*3
a plot of the experimentally observed entrainment velocity vs. ( 1) ~ g)

A plot of the data from the first phase of experimentation, after elimina-

ting some of the obviously bad data points and averaging the entrainment

velocities, results in a least squares fit to a straight line with a slope

of .108 as shown in Figure 43. Equation (7) becomes
2
E(Ri, x) = .108 ('-) P u* (8)
1 g6pD

Application of this to the second phase of experimentation results in a

prediction of ue = .000198 with the interface at a depth of 35 cm, as

compared with a value of .00021 experimentally, averaged over the whole

4 hour run. The agreement is not so good over the earlier and latter

parts of the run where the experimental values are respectively higher

and lower than the predictions. The initial density profile shows some

curvature which would effect entrainment as mentioned earlier. Since neg-

ative curvature increases entrainment and positive curvature retards it,

the entrainment constant might be modified to include a curvature term K,

and Equation (8) may assume the form

E(Ri, x, K) = .108 ( x) P* (-Ka) (9)
I g6pD















.001


.0009


.0008


.0007


.0006


.0005


.0004


.0003


.0002


.0001


0
(


.002 .003 .004 .005 .006 .007 .008 .009

PU3
Ue gTp-( ) cm/sec


Figure 43. Experimentally observed entrainment
velocity vs. predicted entrainment
velocity.


I I

0-





S



0














I I I I


3 .001


.010


a i N | i |




4%-"'


where a is a constant. Since a very small curvature has a pronounced

effect on entrainment, a will be small. Approximating the curvature as

circular over a small region and obtaining a radius of curvature R =

from the initial profile it is estimated, a = .03.

The range of eddy sizes supporting the Reynolds stresses is the

same as that containing the bulk of the kinetic energy of the turbu-

lence. From the spectra it is obvious that the stable stratification

reduces the transfer of energy to the large scale turbulence, as pre-

dicted by Phillips hypothesis by reducing the Reynolds stress. The

small scale turbulence, i.e. the equilibrium and dissipation range is

at a much higher frequency than the N = / for the upper layer,
p 3Z
which would be in the neighborhood of .1 sec Near the interface,

N = 1 and a corresponding reduction in turbulence in this range is

observed just above the interface.

It would be useful to compare these turbulent energy spectra Lu

the corresponding situation with no return flow. With return flow for

the same surface shear there would be a larger horizontal velocity gra-

dient in the vertical direction which would increase viscous dissipation.

There must also be a level at which -u = 0, which would inhibit the
az
energy transfer by Reynolds stresses. Considering these two factors

we would expect a lower level of turbulent energy near the interface when

a return flow is present, and therefore a lower rate of entrainment.

Besides these two reasons for expecting slower entrainment with

return flow, there are also surface and internal setups and the shear

stresses required to maintain those setups. The viscous forces would

contribute to the energy dissipation term. The internal setup represents

a transfer of kinetic energy to potential energy within the system, but















VI. CONCLUSIONS AND RECOMMENDATIONS


The functional relationship between the overall Richardson number

and the entrainment is verified but additional factors must be consi-

dered when the applied shear stress is a function of the distance along

the surface of applied shear, and a return flow is present. The mech-

anism by which return flow modifies the entrainment is due to the shear

flow at the interface and the mean shear flow in the turbulent layer

with its associated effect on the Reynolds stresses. Due to the den-

sity structure of the internal setup, buoyancy forces play an addi-

tional role not seen in the situation with no return flow, and consi-

derations of curvature in the initial density profile indicate that

buoyancy effects are very important when the initial profile is not

linear.

The first recommendation is that in future research in this area,

less time should be spent measuring initial and final density profiles,

and more effort go into studying the dynamic situation. More accurate

measurement of entrainment rates is needed to substantiate the empirical

results presented here. The same is true for the entrainment constant

involving the curvature term, although for what was originally intended

as the object of this study it would have been better if a better method

for establishing and reproducing linear density profiles had been avail-

able, leaving the curvature effects to be studied separately.








Perfection of the conductivity probe and the associated electronics

would facilitate the collection of density data. The method used here

for measuring density profiles was so time consuming that the profile

would change during the time required for measurement, thus introducing

an additional source of error.

The effects of stratification on turbulent energy transfer are

verified in that the one dimensional scalar energy spectra presented

here show a greater decrease in turbulent energy over depth for the

stratified case. The development of the theory behind this depends

on the correlation of the vertical velocity fluctuations at two sep-

arate points so that a quantitative discussion would need such measure-

ments. Determination of the scalar energy spectrum (Equation (2)) also

requires turbulence measurements from two points with separation r, so

it is recommended that additional studies in this area should include

measurement of at least two fluctuating components at two separate

points. These measurements would eliminate the need for the isotropic

assumption, and along with the information which could be gathered

with two conductivity probes it would be possible to determine the net

local buoyancy flux.









is unlike the potential energy increase due to entrainment in that it is

recoverable, and manifests itself in the form of an internal seiche when

the surface shear is removed.














APPENDIX A

DIFFUSIVITY OF NaCI IN WATER


The assumptions were made that molecular diffusion and salinity

increase from evaporation were insignificant compared to turbulent

mixing and overall salinity. The mass transfer due to molecular dif-

fusion is given by

D
NA= AB () (X XA2)
A ZX BM Al A2
BM

NA = Mass flux of salt

DAB = Molecular diffusivity

Z = Thickness of diffusion layer

XB XB2
XBM In(XB2/XBl)

XAl XA2, XB1, XB2 = Mole fractions of salt and water in


p = Density of salt

M = Molar weight of


the salt and fresh water layers,

respectively

water

salt water.


An exaggerated estimate of the diffusion can be given by assuming

XA2 = 0, therefore XB2 = 1 and DAB = 1.3 x 10-5 cm2/sec (Treyball, 1968)

which results in









1.3 X 10-5 cm2/sec (.06 g mole (009) 5.3 g
A (.1 cm)(3.1) cm G 'g mole)

= 1.113 x 10-6 g/cm2/sec

As a comparison, a 1 cm erosion of the interface across which c =

.03 g/cm would result in the transfer of .03 g of salt to the upper

layer, the ratio of this to the transport due to diffusion over one

hour is approximately 10. In the experiment the salinity jump across

the interface is less than for the above calculated diffusion, it is

therefore safe to say that molecular diffusion is at the very most one

order of magnitude smaller than the turbulent diffusion, and may be

neglected.















APPENDIX B

NEGATIVE BUOYANCY GENERATION AT THE

SURFACE DUE TO EVAPORATION


Mass transfer per unit area between a flat gas-liquid interface

is given by (Treyball, 1968)

1 PAi/t
NAMA = MAFG(In 1 PAG/t) (Bl)

where

NA = Molar flux

MA = Molecular wt = 18 Ib/lb mole

FG = Mass transfer coefficient

PAi = Vapor pressure of the liquid (A) at the
interface = 17.535 mm Hg

PAG = Partial pressure of the liquid in the gas

pt = Total pressure = 760 mm Hg.

The mass transfer coefficient is obtained by the heat-mass transfer

analogy with the empirical relationship (Treyball, 1968)

h = 0.072 G'0.6

where h is the heat transfer coefficient between an air water interface

and G' is a superficial air mass velocity. The heat transfer dimension-

less group defined by









H- Cph p 2/3
jH C Up r


(B2)


c = Heat capacity of gas

U = Velocity of gas

p = Density of gas

P = Dimensionless Pradtl number = Cpviscosity
r thermal conductivity

can be related by the heat-mass transfer analogy to the mass transfer

dimensionless group
kGPBM M 2/3
D pU 2/3

where

KG = Mass transfer coefficient per concentration

difference

PBM = Log mean partial pressure difference
M = Molecular weight of the gas

Sc = The dimensionless Schmidt number

Kinematic viscosity
Diffusivity of water vapor in air

Equating (B2) and (B3)


h
C Up
p


K P M
p 2/3 =G BM S 2/3
r pU C


The mass transfer coefficient FG is determined,


F h p 2/3 PU S 2/3
G = KG PBM G r M c

For air at 5 fps the value of FG is .0114 lb mile/hr ft2. Substituting this

into (Bl) and making the extremely exaggerated assumption that the air

is initially at zero humidity, the evaporation after one hour would be






70


.0023 grams H20/cm2. This woulc leave behind

.0023 gqH?0 .006 qNaCl) = 1.38 x 10-5 gaC
cm hr g H20 1.38 x

when the top layer is at a concentration of .006 g NaCl/gH20 as in the

experiment. The ratio of the amount of salt left behind to the concen-

tration in solution is 2 X 10-4; evaporation can therefore be neglected.













BIBLIOGRAPHY


Ellison and Turner, 1959, "Turbulent Entrainment in Stratified Fluids,"

Journal of Fluid Mechanics, Vol. 6, pp. 423-448.


Gibson, C. H. and Schwarz, W. H., 1963, "Detection of Conductivity

Fluctuations in a Turbulent Flow Field," Journal of Fluid Mechanics,

Vol. 16, pp. 357-364.


Kato and Phillips, 1969, "On the Penetration of a Turbulent Layer into

a Stratified Fluid," Journal of Fluid Mechanics, Vol. 37, pp. 643-655.


Long, R. R., 1970, "A Theory of Turbulence in Stratified Fluids,"

Journal of Fluid Mechanics, Vol. 42,.pp. 349-365.

Morton, B. R., Taylor, Sir Goeffrey, and Turner, J. S., 1956, Proceedings

of the Royal Society of London, Series A, Vol. 234, Jan. 24 Mar. 6.


Phillips, 1969, The Dynamics of the Upper Ocean, Cambridge University

Press.


Schiesser, W. E., Statistical Uncertainty of Power Spectral Estimates.


Shemdin, 0. H., 1972, "Wind-Generated Current and Phase Speed of Wind

Waves," Journal of Physical Oceanography, Vol. 2, No. 4.

Sheppard, D. M., Shemdin, 0. H., and Wang, Y. H., 1973, "A Multipurpose

Internal Wave Facility," Technical Report No. 19, Coastal and Oceano-

graphic Engineering Laboratory, University of Florida, June.








Tennekes, H. and Lumley, J. L., 1972, A First Course in Turbulence,

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