20086-6001-RO-00

DIFFUSION OF A PASSIVE SCALAR

Quarterly Technical Report

by

D. R. S. Ko and i. E. Alber

1972 February

NATIONAL TECHNICALINFORMATION SERVICE

Sponsored byAdvanced Research Projects Agency

ARPA Order No. 1910

Contract No. N00014-72-C-0074

"A 21 •• -7

TRWBSYSTEMS GRO!UP OF TRW INC - ONE SPACE PARK. REDONDO &EACH. CatPCOAN:A 902-8 - 213, $3543Z?V.I

20086-6001-RO-00

DIFFUSION OF A PASSIVE SCALAR

Quarterly Technical ReDort

0972 February

ARPA Order No. 1910 Contract No. N00014-72-C-0074Program Code No. 438 Scientific Officer -

Contractor - TRW, Incorporated Director, Fluid Dynamics ProgramsDates of Contract - 8/15/71 - 4/15/72 Mathematcal and informationContract Amount - S49,952 Sciences DivisionPrincipal Investigators - Office of Naval Research

Dr. Denny R. S. Ko Department of the NavyDr. Irwin E. Alber Arlington, Virginia 22217

Telephone No.: (213) 536-4422

This research was supported by the AdvancedResearch Projects Agency o, the Departmentof Defense and w~s nonitored by ONR underContract No. :400014-72-C-%474.

The views and conclusions contained in this document are thoseof the author and should not be interpreted as necessarilyrepresenting the official Dolicies, either expressed or implied,of the Advanced Researcn Projects Agency or the U. S. Government.

TRW

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20086-6001 -RO-O-

Prepared forAdvanced Research Projects AgencyUnder Contract N00014-72-C-0074

Prepared 7 ..

D. R. . Ka/

Prepared 4 , ... j-.•Lj_' < -

I. E. Alber

Approved - '

L. A. Hromas, ManagerFluid Mechanics Laboratory

TRW

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3 Se Po0T TITLE.

Diffusion of a Passive Scalar

4 of0ICRIPTIVE 91OE-s (T77" or a rf -00 cWa! h. do#"o;

Quarterly/ Technical Report - August 1971 - November 19715 AUTkOIUS) (14141 5%0-& .0a. sInjaJ)

Ko, Denny R.Alber, Irwin E.

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Adviced Resarh P cts Agency

13 AS*-.UACT

An analytical model for the prediction of mixing and decay if apassive scalar within the turbulent wake of a submerged body operatingbeneath the ocean surface. The scaling parameters are identified fromthe governing equations. Moreover, a review of the existing theoreti-cal and experimental studies of diffusion in an ocean environment isgiven. The apnlication to the oresent problem of interest is brieflydiscussed.

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i IWakefomentuml essT;rbulent entrairnmentStr3ti ficationParsive scalarOcca". turbulence t

Diffusion

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iv

20086-6001-RO-00

SUMMARY

The mai;, objective of this study is to develop an analytical model

for the prediction of the mixing and decay of a passive scalar within

the turbulent wake of a submarine operating beneath the ocean surface.

Even though the interest is on the passive scalar, the essence of the

problem is the .'etermination of the dynamics of a submarine wake. Once

the wake flow field is determined, the decay of the concentration of a

passive scalar can then be easily obtained.

Depending on the distance behind the submarine, two somewhat

different technical problems are encountered in the analytical modeling

of the wake. For the rtyion relatively close to the submarine, (wake age

of the order of an hour), the flow field is inflbenced by the effects directly

caused 'y the body. The oroper analytical model must take into consideration

the fol'awing characteristics of a submarine wake: the turbulence

generated by the body and the propeller; the momentunrless wake behind a

submarine moving at a constant speed, and the stratified nature of the fluid

media. For the region far behind the submarine, the sub-generated

perturbations decay below the natural background turbulence levels.

Further diffusion of the passive scalar in this region of the wake is

expected to be dominated by the ocean background turbulence and the

character of the diffusion plume.

An analytical model for the near wake region, based on the concepts

of local similarity and turbulent entrainment of the external fluid, is

formulated in Section 2. A turbulent model is introduced which pays

particular attention to the effect of stratification. Using basically

an integral conservation approach, the problem is reduced to solving a

set of five ordinary differential equations with given initial conditions.

By properly normalizing the governing equations, two primary sca'ing

parameters for the submarine wake are obtained. The most important

scaling parameter is the turbulent Froude number Fr = u' /NL, where u2- 1/ mo

denotes the initial turbulent intensity, N ( the Brunt-asailadz

frequency and L is the initial wake width. To a lesser degree, the initidl

turbulent intensity level Umo/U 0 provides ar.olher governing parameter.

V

S~ 20086-6001-R0-00

Theoretical or empirical estimates of the value of these parameters, for

any particular experiment or field condition is not available at the

present time. However, a parametric study -Aill soon be performed and

comparisons with existing laboratory experiments will be attempted. These

results will be discussed in the final report. Meanwhile, effort, both

analytically and experimentally, should be directed toward a logical and

consistent determination of the value of these governing parameters.

The effects of the background turbulent ocean dynamics has been

completely ignored is, the formulation presented in Sectiun 2, which is

valid fcr the relatively near wake region. As a first step toward

increasing the understanding of this important effect, an in-depth review

is presented in 5ection 3 of existing theoretical and experimental

studies of diffusion in an ocean environment.

In analytically modeling the diffusion of a passive scalar, thediffusion coefficients K. must be known for both vertical and horizontal

1transport. The review in Section 3, concludes that horizontal diffusion

in the ocean dominates over vertical diffusion (Kh > 104 Kv) because of

the effect of the stratified media in retarding vertical turbulence

fluctuations. Thus diffusion from a moving continuous source proceeds

as a thin fan in the opposite direction from the source trajectory.

The horizontal diffusion coefficient is estimated with the aid of the

theories of Taylor, Batchelor and Richardson, showing that Kh is proportional

to the 4/3 power of the scale of diffusion t. Examination of a rnumberof dye diffusion experiments in the ocean indicates that Kh is approximatelyrepresented by the Richardson 4/3 law, (K " z.1s) and that the lateral

dimension a of the dye plume increases rapidly with time a %, t.7 (a t

for the Richardson law). This rate of grawth indicates, for example, that

the size of a dye patch would be on the order of several kilometers after

a period of diffusion of one day.

In Section 3, the diffusion coefficients and rate of spread are

used to estimate the concentration decay behind a continuous source.

For a point source, it is shown that the concentration , falls off

inversely as the diffusion time to the 1.17 power (or 3/2 power for the

Richardson law). Far field solutions are also presented for diffusion

vi

20086-6001 -RO-00

from a finite bounded concentration field. Such initial conditions can

be su.pplied by the near field analysis of Section 2 where sub-generated

turbulence is important. Further numerical evaluaticns of the concentration

distributions for typical ocean and sub conditions will be presented in

the final report of this study.

[iI(

IJ

vii

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CONTENTS

Page

1. INTRODUCTION ...................................... 12. SUBMARINE WAKE MODEL ....... ............... 3

2.1 Non-dimensional Governing Equations ........ 8

3. PRELIMINARY CG'NSIDERATIONS OF TURBULENT DIFFUSIONOF A PASSIVE SCALA•. IN THE OCEAN ..... .......... 103.1 Diffusion Coefficient-s . .. .. .. .. . .. 103.2 Two Dimensional Horizontal Diffusion ..... ... 13

3.3 Three Dimensional Diffusion with Decay . ... 163.4 Vertical Diffusion . .. .. .. .. .. ... 17

R E F E R E N C E S . . . . . . . . . . . . . . . . . . . . . ... 2 6

viii

r -- 20086-6001-RO-00

ILLUSTRATIONS

Page

1. Coordinates and Density Field . . . . . . .. . . . 28

2. Variance, ac, of Dye Distribution against DiffusionTime (Data from 1961 Onwards) (after Okubo,1968, Ref. 7) .......... .................... 29

3. Apparent Diffusivity, Ka, against Scale of Diffusion,S= 3akc (Data from 1961 Onwards) (after Okubo,1968, ef. 7) ........ .................... 30

4a. Concentration Profiles c Behind a Point SourceMoving at a Velocity U in an Ocean Environment,Eq. (3-21) ............ ... ... ... ... ... ... 31

4b. Concentration Profiles c Behind a Point SourceMoving at a Velocity U in an Ocean Environment,Eq. (3-21) ...... ...................... ... 32

4c. Centerline Concentration Behind a Point SourceMoving at a Velocity U in an Ocean Environment,Eq. (3-21) (y = 0) ....... .................. 33

5a. Thermal Diffusivity from Stratified FlowExperiments in a Water Channel (after Merrittand Rudinger, Ref. 13) Comparison with TheoreticalModel, Eq. (3-31) (after Sundaram and Rehm,Ref. 12) ........ ....................... 34

5b. Thermal Diffusivity (Neutral Stability) as aFunction of Flow Velocity (after Merritt andRudinger, Ref. 13) ....... .................. 34

ix

----- ~---~---- -------- ~/

20086-6001 -RO-00

1. INTRODUCTION

The main objective of this study is to develop an analytical model

for the prediction of the mixing and decay of a passive scalar within the

turbulent wake of a submarine operating beneath the ocean surface. Even

though the interest is on the passive scalar, the essence of the problem

is the determination of the dynamics of a submarine wake. The decay of the

concentration of a passive scalar could then be easily obtained from the

conservation equation for a known flow field.

Numerous theoretical and experimental studies on the wakes behind

unpropelled bodies in a uniform medium have been reported. The knowledge

acquired through these studies is not readily applicable to the present

case of interest because of the following two characteristics of a submarine

wake:

l) For a submarine moving at a constant velocity, there is nonet axial momentum being deposited in the wake. The phenomenaassociated with these momentumless wakes are, in many respects,different from a wake having a momentum deficit.

2) The ocean environment is generally stably stratified. Becauseof the turbulent mixing in the wake, fluid particles withinthe wake are out of equilibrium with the stratified epvironment.This results in an increase of the potential energy of thewake. Gravitational restoring forces continuously convert partof this potential energy into kinetic energy and the wake tendsto collapse vertically and spread out horizontally.

These characteristics of a submarine wake have not been subjected to

any rigorous theoretical treatment to date. In a recent report, Ko1) has

presented a simplified phenomenological model with the main emphasis on

identifying the important physical mechanisms which affect the wake

dynamics. Even though a fairly good agreement with some existing laboratory

data have been obtained, this simplified model must be improved before

it can be used as a reliable prediction tool. This simplified model along

with several improvements is described in Section 2 of this report.

Anothler related problem having to do witit the effect of the ocean

background turbulence on the wake diffusion has never been addressed

previously. Based on the previous calculations (see Ko(1)•, the turbulence

20086-6001 -RO-0I

generated by the submarine and its propulsion system persists only for a

relatively short distance because of the stabilizing effect of the

stratified environment. Therefore, we might expect the far wake diffusion

to be dominated by the background turbulence. However, our present level

of understanding in this regard is rather limited. As a first step, we

will present in Section 3 a partial survey of the existing understanding

of the oceanic turbulent diffusion. The application to the submarine wakewill also be indicated.

2.

20086-6001 -RO-O0I

2. SUBMARINE WAKE MODEL

The basic approach follows closely the analysis of Reference 1 with

a few improvements. Some of the assumptions in the previous analysis will

be relaxed in order to achieve a more reliable first-order predictive tool,

dt least for the case of a negligible background turbulent diffusion.

Therefore, most of the details in the derivation of the governing equations

will be omitted here and reference may be made to the previous analysis.

For a submarine moving with e uniform speed Uo at a depth H below

the ocean surface, a Cartesian sub-fixed coordinate system is chosen as

sh(crn in Figure 1. The undisturbed medium outside the wake is assumed to

be horizontally stratified with the density given by

Ro = (l-cxz) (2-1)

with p denoting the mean density at the submarine depth and C being the

given constant gradient. The density field inside the wake is taken to be

"i= •( - 8z) (2-2)

with s, the internal density gradient, reflecting the degree of mixing

inside the wake. The effects of density discontinuity that exists at the

wake boundary for this assumed density distribution will be considered in

a later study. One of the main improvements of the present modeling will

be the inclusion of a suitable mixing equation to account for the variation

of 8.

The concepts of local similarity and turbulent eintrainment of the

external fluid will be us'.d Es before. Assuming that there is no direct

effect of turbulence on the return of a displaced fluid pa'ticle inside the

wake to its equilibrium position, the ,1 .lowing governing equation is

obtained from the momentum equations

b2df1 b2 1(l+a -U) - (I f1 (2-3)

a a a2

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where f, is the local velocity scaling factor such that

v yfl and w = -zfl (2-4)

The values of a and b represent the major and minor axis of the elliptical

wake boundary given by

2 z

+ 1 (2-5)a' b2

The governing equations for a and b are given by

daUo!L-= af1 + Ey

(2-6)

dbUo rx af, + Ez

with E and Ez denoting the turbulent entrainment velocities in the y and z

directions at the wake boundary y = a and z = b. If the entrainment

velocity is assumed to be proportional to the distance from the center of

the wake, then we have

b ~Uo- b EY - (2-7)a Y 2AdX

where A = rab denotes the area of the wake. The entrainment velocity is

then related to the local turbulent intensity and scale using the experimental

results of Naudascher, which gives

Uo dA K:lum'= (2-8)

A dX a

with K1 , 2.8.

The formulation presented thus f3r is identical to that given in

Reference 1. The intent of what follows here is to more accurately account

for the effect of density stratification on the rate-of-change of the

turbulent energy and its effect on the wake density gradient s.

4

20086-6001 -R1-O0

The total integrated turbulent energy in the wake is given by

Et = U U;2 A (2-9)

where u; is the averaged turbulent velocity over the wake. An equation for

the rate-of-change of the turbulent energy within the wake in a stratified

medium can be written as

dEU- = P -D- S (2-10)od

where P = rate of production

D = rate of dissipation

S = stratification effect on turbilent energy exchange

The modeling of the production and the dissipation terms follow3 the same

line of reasoning as Reference 1 in which it is shown

2 2nP = K2 ipa- (2-11)

and

%D = Cip un 3 a (2-12)

The constants are determined from the experiment of Naudascher to be K2 = 140

and C = 8. The modeling for the stratification effect on the turbulent

energy balance marks the main difference of the present analysis from the

previous one. The previous analysis was based on an overall potential and

kinetic energy conservation concept which led to a representation of the term

ff wgdA

which appears in the turbulent energy equation. However, the fact that

the effect of stratification on turbulence vanishes when s =- t leaves

some doubt as to its validity. Furthermore, because of the need to model

the (-Pw') term for the mixing equation, there is little reason not to moJei

the effect of strati,"ication on the turbulent kinetic energy directly.

5

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Since this modeling was not considered in the previous report, we

will go into some detail in its formulation here.

The modaling is basea oai the concepts of similarity and eddy

diffusivity. Let's assume

- = (-Pwr)rax Fc(Y*. z*) (2-13)

where y* y/a, z* = z/b, and Fc is a distribution function which vanishes

at the wake boundary and has a maximum at the center of the wake. One

simple form of F can bB taken as

F=1 y*2 z* 2 (2-14)

Using an eddy diffusivity formulation, we let

max 3z

-'ub~p (2-15)

The value of K' can be estimated from the diffusivity for an incompressible

turbulent wake (e.g., Townsend) to be about 0.2. With the above assumptions,

evaluation of the term S is

S g fAf 2~dA =4 g K u~pab f, b dzf1 -oý (1 - y*2 -Z*2) dy

= a ab 2 (2-16)

Therefore, the turbulent energy equation is then

d(u; 2ab) 5

Uo dx K2a ;- Cu 3a - s Bab 2gum (2-17)

6

20086-6001 -RO-00

"The proper mixing model for determining the local density gradient

within the turbulent wake is obtained from the averaged density

conservation equation, which gives

Pi +- ' - -i T-+ v + =w] (2-18)0ax 2zax ay 3Z j

Assuming T- =u is small and noting that

Pi =(l - BZ), v = yfý, W = -zfj

an integrp'tion of the LHS of equation (2-18) over the upper half of the

wake gives

bi ] dab

dab l (2-19)

where e* = ab and S* = Ob. The integration of the RHS of equation (2-18)

results in only one non-vanishing term

T =f (z 0) dy (2-20)

which represents the transport of mass across the interface z 0. With

equations (2-13) - (2-15), this transport term ca, be readily evaluated

to be

T Kum p Bab (2-21)

3 m

Then, by equating (2-19) and (2-21), we obtain the equation for mixing

EZ o : 2K'Umo (2-22)(•zt~ ~ -B)°db - *- U°

ab dx: b 0d

7

20086-6001 -RO-OO

2.1 NON-DIMENSIONAL GOVERNING EQUATIONS

By using the following characteristic quantities

Length ... L (characteristic dimension of the flow)

Velocity ... Uo or Uref U=o (initial turbulent intensity)

Time ... N-1 (N = a.. Vaisala-Brunt frequency)

and let

a*= a/L, b* = b/L

urn*=u,/Uref, E z/Uref, Ey* =Ey/Uref

fi* = f1/N. X*= XN/U 0

and define two non-dimensional numbers

UrefFr - ... tu-bulent Froude number

UN = Uref/Uo

Then, the governing equations can be sumuarized as

daf_* + FrE*dx* r~l y

db* __~ldx * f + FrE*

dx* ziK1 b*

with Ez* -• ur*-

K1Ey* = -ur*y 2 m

+*2 dfj* B* 2 a*.

u *5 *3 um*3d ur*2 r UN2 *C K,

I- 1K'Fr2 T Um*b*]

8

20086-6001 -RO-00

d,* un* [• b*d*= Fr [3 - K1 a* - (2K' + K1 b*

The fact that the turbulent energy equation allows a non-vanishing

derivative of Um* when u = 0 requires a constraint to the numerical

program such that uV is set to be identically zero once it reaches

zero. Another somewhat artificial constraint is placed on the magnitude

of 8 such that it will not exceed a.

Now assuming the wake to be initially circular and taking the

characteristic length L to be equal to the initial wake radius, the

proper set of the initial conditions is

ao= bo* 101

f1o*= 0

and a specified o

The equations point out that the proper scaling parameters for

the submarine wake are the turbulent Froude number Fr . u and ther NIinitial turbulent intensity level UN = U~o/Uo. In addition, the initial

degree of mixing, represented by 8o/a appearing in the form of an initial

condition, constitutes an additional parameter in the problem. The

value of these parameters for any particular experiment or field

condition is not apparent at the present time. Even though these parameters

are not completely independent of one another, the actual determination

of their magnitudes will have to rely quite heavily on empirical results.

While searching for the proper way of choosing these parameters,

parametric studies will be performed an%- the results will be presented

in the final report.

9

20086-6001-RO-00

3. PRELIMINARY CONSIDERATIONS OF TURBULENT DIFFUSIONOF A PASSIVE SCALAR IN THE OCEAN

I

To describe the diffusion of a passive scalar (e.g., a radioactive

trace contaminant) in the ocean, one has to quantita.ively understand

the kinematics of ocean turbulence. If one seeks to find the meanconcentration of a contami'aant, -i(x,y,z), diffusing from a fixed source

in an ocean flow field, the usual approach is to seek solutions of the

time averaged species conservation equation

Dt axj 13

where cZuj represents the species flux in the j direction and •i represents

the time averaged rate of pr:.duction (or loss) of species i by chemical

reaction or radioactive decay. 1t represents a material derivative.

The derivation of Eq. (3-1) assumes that the nature of the flow

field is described by a stationary random function of time. That is,when the turbulence properties are averaged over a suitable time increment

T, a stationary value for the turbulence properties is obtained. In a

meandering ocean flow field, various averaged turbulence properties mayresult depending on the magnitude of the time scale T. For example, large

scale horizontal ocean diffusion properties only a(.hieve a stationary valuewhen averages are taken over periods of several hours to a day.

For the analysis tu be presented in this report, stationary values

"of the turbulence properties will be assumed in order to provide a basicestimate of the diffusional characteristics of oceanic turbulence.

3.1 DIFFUSION COEFFICIENTS

To solve the species conservation equation, Eq. (3-1), expressions

for the species flux ZT, must be adopted. Using the Boussinesq approximation,

turbulent diffusion coefficients can be defined by the equation

Kj 1_ 3 F( i1 3xj)

10

20086-6001 -RO-00

These coefficients do not necessarily assume a constant value and may be

functions of the distance x from a source (or time t = x/U). Specific

forms K = K(x) have been suggested by Sutton(2) for various stability

conditions corresponding to the atmospheric diffusion of smoke plumes.

The change of the diffusion constant with time is best calculated by

Taylor'; 3) diffusion theory of continuous movements in a homogeneous

turbulent field.

Taylor defined the function R(-r) as the coefficient of correlation

between the turbulent velocity ut of a particle of fluid at time t and

the velocity ut+ of the same particle at a time T later, such that,

R(r) = utu+ý/u2 (3-3)

where u2 represents the mean square velocity fluctuation intensity at time

t. if E is the displacement of a particle of fluid at time t from its

original position at time t = 0, Taylor proved that

t t'

-02 = 2 f-tf t R(r)drdt' (3-4)

0 0

For small values of t, such that R(T) 1.0

= o2 t i t2 (3-5)

For times t >> t*, (where t* denotes the Lagrangian time scale =f R(T)dT)

the correlation R(r) is effectively zero. Hence if diffusion proceeds for

times much greater than t*, the integral in Eq. (3-4) will reach a

limiting value I. In that case, the mean square displacement is proportional

to the time of diffusion, i.e.,

02 = 2 Iv2 Itor (3-6)

. where A= f R(T)dT is defined as the Lagrangian integral length.L 0

If the particles had been dispersed by a diffusion process obeying thediffusion equation

11

20086-6001-RO-00

SK 2(3-7)

the solution of Eq. (3-7) for the concentration c would be

c - const exp (-y 2/4Kyt) (3-8a)

orce 2_xp [(3-8b)

y ly

Equations (3-8a) and (3-8b) yield a mean square displacement oy of fluid

particles in the y dimension; given by

ay2 2 Kyt (3-9)

Based on this relation, one can define a diffusion coefficient fromEq. (3-9) in terms of the mean square displacement, i.e.,

1 d~y 2 (-02 d (3-10)Ky dt

The diffusion coefficient •. given by Eq. (3-10) can thus be used directlyy

in a simple diffusion equation of the form given by Eq. (3-7).

Batchelor(4) extended Taylor's ideas with the aid of dimensional

arguments. He obtained expressions for the diffusion coefficients K(or da 2/dt) applicable to the horizontal diffusion of a passive scalar(or traze contaminant) in a homogeneous ocean. The characteristic velocity

scale is related to the rate of turbulent energy dissipation per unit

"mass c, and to the standard deviation of the initial source field ao by

the expression

f ., (cc)E"/ (3-11)

Batchelor finds three regimes of relative diffusion, the two given byTaylor's relations [Eq. 1,3-5) and (2-6)] and one important intermediate

regime given by the following rates of growth12

20086-6001-RO-00

1 da2 Clt(Co 2/3K = C • =) (initial) t <- t* (3-12a)

K =1 d22 = C2c(t - (intermediate) t ' t* (3-12b)

with tj = C3 002/3C/3

K =12 -2 = C, /u-2 AL (asynptotic) t >> t* (3-12c)2 dtL

For the intermediate times as given by Eq. (3-12!)

K a Ct2 (3-13)

and

a2 " Ct3 (3-14)

Thus the diffusivity grows as the 4/3 power of the particle field (or

plume) size, and the plume grows as the 3/2 power of the time. The 4/3 law(5)

was first derived by Richardson~s/ in 1926. Richardson's law of relativediffusion indicates that diffusion is an accelerating process, the rate

of growth increasing with the size of the field, at least while the supply

of eddies of the requisite size lasts.

3.2 TWO DIMENSIONAL HORIZONTAL DIFFUSION

Richardson's law has been found to describa fairly well the

horizontal diffusion in the ocean. Experimentally, horizontal diffusion

has been studied extensively using the fluorescent dye technique. An

overall discussion of such experiments is presented in the review articleof Bwden(6)of Bowden~ In some cases the dye has been released continuously from

a point source into a current and the dispersion of the plume observed.

In other cases the dye has bheen released as instantaneously as possible

and the dispersion 3f the resulting patch observed as a function of

horizontal coordinates and time. In both cases the initial dispersion

is usually three dimensional but after a relatively short time the vertical

dispersion becomes severely restricted by the presence of a thermocline,

and the subsequent spreading is effectively two dimensional. In that

case, the diffusion equation for a point source in a uniform turbulent

flow of velocity U, assumes the form

13

20086-6001 -RO-00

J - 'ci- (3-15)ax y y

For the case where an isolated source emits a mass per unit time of species i

at a constant rate Qi* per unit distance in the vertical direction,

Eq. (3-15) has the solution

=Qi*/py 21

ci I- exp 2(ay. J (3-16)

where

1 d2(3-17)2 i-~- Ky,

Okubo(7) in 1968 has made a review of 20 sets of data on instantaneous

releases of dye in the ocean over the last ten years. They include

experiments in the North Sea and in the Cape Kennedy area, and off of

Southern California. Okubo has tabulated the variance a2c, and theapparent diffusivity K a as a function of time t for each of the 20 releases.

These results are shown in Figures 2 and 3 where G2c is shown plotted

against the diffusion time and K is plotted against the scale of diffusion,aS= 3rc These diagrams cover time scales ranging from 1 hr. to 1 month

and length scales from lOm to 100km. For the increase of variance with

time, Okubo four:d that a best fit of the data could be obtained if

a 2 = .0108t 2 . 3 (3-18)rc

This is a somewhat smaller rate of growth than given by the t 3 taw based on

the Richardson theory, Eq. (3-14). The relation between the apparent

diffusivity and scale was found by Okubo to be

K 0 = .it1.1 (3-1ga)Ka

compared with the theoretical t law. However, the differences with

the Richard.,on theory are not large.

14

"20086-6001 -RO-00

If one assumes that the horizontal diffusivity can be written in

the form

nAK atn

and that (3-19b)

t=ba

then by Eq. (3-10) one can readily show that

ay [(2 n-a)]1(2- (3-20)

where

a =bna

The concentration distribution of a passive scalar downstream of a continuous

point source, moving at a velocity U, in a quiescent ocean, will be given

by Eqs. (3-16) and (3-20), i.e.,

Qi*/pUexp Y2/2 3-21)c. = */U ex,, [2/-n1i /2- [(2 - n)-ax/U]i/(2-n) L (2 'ax U 2 2-n 1

For the constants n = 1.15 the concentration drops off rapidly with

distance from the source x as

S.-(3-22)S1.17x

and the size of the field increases as

y •1"17 (3-23)

If the Richardson's 4/3 law is assumed to be valid, n = 4/3, hence

- 15cii .xp/.'oy • x (3-24)

Calculations of concentration versus distance behind the source are

presented in Fig. 4 for both the n = 1.15 and n = 4/3 laws.

15

20086-6001-RO-O0

gIIn tne following section more general solutions are presented of the

diffusion equation including the effects of vertical diffusion, andradioactive decay. In addition solutions will be presented which allowfor arbitrary initial conditions based on the near field submarine wakesolutions of Xo) in a stratified medium. Diffusion calculations fortypical submarine wakes will be included in the final report of thisstudy.

3.3 THREE DIMENSIONAL DIFFUSION WItH DECAY

The semi-empirical steady state diffusion equation is written belowfor a three dimensional Cartesian coordinate system x, y, z with stream

velocity U in the x direction

ac. c.3c. C.ac' ac1.)2-(K + -(K (3-25)ax BYy y3y 3z za3z T

Here the z axis is directed vertically downward, K and Kz are the turbulentdiffusivities defined by Eq. (3-2) in the y and z directions.* To take

into account a possible exponential decrease of Fii with characteristic

decay time Tc (e.g., caused by radioactive decay), the last term on the

right of Eq. (3-25) has been chosen to represent the production term, wi,

in Eq. (3-1).

A special solution of the diffusion equation (3-25) is possible for

an isolated fixed source of strength Qi (mass per unit time) located at

(0, 0, h) [a distance h below the water surface as shown below]

0

xxy(source)

z

ac.*A term of the form • (K •L) is neglected in Eq. (3-25) in comparison to

acx xaxU - This assumption is quite good for problems involving continuousax*sources.

16

20086-6001 -RO-00

For the case of a uniform stream velocity U, and for values of Kvy

and K which are constants or functions of x, the solution of Eq. (3-25)is found to be(8)

c(x, y, Z) 2 a exp - c exp y

""expTe 2(z- h)2]r + e [- 2]1 (3-262 exp 2- a z o.• )]I-6

2KvX 1/2 2KIx 1/2

The dispersion coefficients are ay =(--) and az 1 -0-) (3-27)

This solution corresponds to the follhwing initial and boundary conditions

imposed on Eq. (3-25).

x : • Q 6(y) S(h - z,'1 10:

ac.z =0: 3- 0 (non-catalytic surface)z W: - 0

If the source is far below the surface h >> az, then Eq. (3-26) takes the

simple Gaussian form

:2azU a ~exp x° U-- exp - (c-) -2 ( 2]) (3-28)yz UC 2 y

h >> cz

where z = z - h.

3.4 VERTICAL DIFFUSION

In order to make quantitative use of Eq. (3-26), in the calculation

of three dimensional concentration distributions, the diffusion coefficients

Ky, and Kz must be given or the dependence of ay and az on the distance x

must be specified. In the previous section it was shown that the horizontal

17

20086-6001-RO-00

diffusion coefficient K in the ocean could be characterized by the

yRichardson 4/3 law or a closely related empirical expression of the formKy= at . Typical ccean measurements (see Fig. 3) yield values of K in

ythe range

K O(lOt lo0) Iy sec

Corresponding values of the vertical diffusion coefficient Kz are usuallymuch lower and are reported to lie in the rangetg)

K = 0(1 to 103) CM2

The principal mechanism by which turbulence is generated in the upperlayers of the ocean is by the action of the wind shear acting on thesurface layer. Front nbservational data, Sverdrup(iO) proposed the

following empirical relationsliip between Nzo [the surface vei'ticalturbulent eddy stress coefficient = - uwr/(97u/az)J and the wind speedw(m/sec);

pNo 4.3w2 for w > 6 m/sec (3-29)

For most studies, Reynolds' analogy between momentum and diffusive

transport is assumed, i.e.,

Kz0 Nz0

In the case of a stably stratified ocean (the density increasing with

depth) the mean buoyancy field acts to suppress the generation of turbulence.The criteria used to determine the fluid conditions for which turbulenceis completely suppressed is based on the Richardson number

R. =.. /z1 p (9u;az) 2 (3-30)

where p is the density of the fluid, u is the current velocity and g theacceleration of gravity in the vertical z direction. The Richardson

18

20086-6001 -RO-O0

number represents the ratio of the rate of suppression of turbulence by

the buoyancy field to the rate of production by the action of Reynolds

stresses. If Ri is greater than some critical Richardson number, Ricrit,

(Ri of order .15) then the theory of Ellison(11) indicates that the(Rcrit tdiffusivity Kz should vanish.

Sundaram and Rehm( in their studies of lake thermoclines have

recently adopted the following empirical expression to determine the

effect of stability on the vertical diffusivity of heat

S= (I + alRi)- 1 (3-31)

where ol is a constant found to be approximately .6. In Sundaram and

Rehm's analysis, the velocity gradient au/az is assumed to be determined

by the expression for a constant stress turbulent layer

au!az =U*/gcZ (3-32)

where K is Karman's constant = .4 and u* = s is the friction velocitys

with Ts being the surface shear stress. Equations (3-31) and (3-32) were

shown to adequately represent both the field measurements in Cayuga Lake(13)

and the laboratory measu.'ments in a stratified channel flow (see Fig. 4).

An overall empirical expression for tht vertical diffusivity coefficient

Kz can be obtained by combining Eqs. (3-29) - (3-32), i.e.,

K43w 2 (3-33)

pu*2

with w in m/sec. For typical values of the stratified density gradienta- (a l0-7g cm-4) and shear velocity u* (w 33 cm/sec), encountered in

pZthe ocean, it is found from Eq. (3-33) that the vertical diffusivity Kzis reduced by a factor of one-half by a depth z ; 100 meters (near the

top of the ocean thermocline). At a depth of 300 meters, the magnitude

of Kz is only one tenth of its value near the surface, thus demonstrating7 -

that a density gradient as small as 10- gcm- can virtually eliminate

vertical transport in the thermuocline.

19

]

20086-6001-R0-00

.(14)Ozmidov reports a striking observation of this effect in hisexperiments of the diffusion of patches of dye (rhodamine C) in the Black

Sea during the fall of 1964:

"In its initial phase, the diffusion of the dye froman instantaneous point source is symmetric in all

directions. But after the patch reiches dimensionsof several meters it begins to "flatten up" sharply,and whereas tne increase of the patch along thevertical has practically stopped, the patch continuesto increa!;e rapidly along the horizontal directions.After several hours had elapsed, the patch reachedhorizontal dimensions of the order of Ion, whereasits vertical extent did not usually exceed 15 to 20 m(in this case the layer of discontinuity was locatedat a depth of u6fm)."

Ozmidov(14) hits proposed a simple theory for turbulent diffusionin the ocean, which seeks to quantitatively model the anisotropic effectsnoted in his ocean dye diffusion experiment. His arcjuments are as follows:

'f the energy supply of tne ocean is sufficiently large, an i::ereval

of scales fw Io to Icr will exist in which the turbulence is isotropic.The lower bound to Is equivalent to the smallest dissipation length scale

of the inertial subrange, i.e.,

1/4

to= (Wi/)1/ (3-34)

where v is the kinematic viscosity of sea water and £ is the rate ofturbulent energy dissipation. The upper bound for which the turbulencewill be isotropic is characterized by the eddy size tcr for which the

stratification density barrier is just overcome (tcr is related to theRichardson number based on c), i.e.,

3/'.2 -/3 11

icr [ /:z1j (3-35)

In the interval Io < t < Icr the diffusion of impurities from a instantaneous

point source is identical for all directions and a cloud of diffused matttwill have a spherical shape. Eddies with dimensions greater than icr can

no longer take part in the vertical diffusion transfer, thus limiting the

20

-L-

20086-6001 -RO-00

effective vertical diffusion coefficient Kz. The limiting value of Kz

can be estimated from the 4/3 power law for isotropic turbulence

1/3 4/3 CIpC

Kzmax le t cr - g(p/3z) (3-36)

where C1 is a constant approximately equal to .1.

Ozmidov argues that in the range of scales from z to Icr in which

the turbulence is isotropic the values of the vertical diffusion coefficient

Kz and of the horizontal coe ficia.t K• are equal (and are given by the

4/3 law). However, for pheni...,a with scales greater than zcr' K• and

Kz become markedly different as illustrated in the sketch below.

rzKKmax

S/K

-Get

zmaxlaminar 00

0o tcr

For scales ý > Ecr' K reaches its maximum value Kzmax (given by Eq. (3-36))

determined by ouoyancy. The horizontal coefficient, %, jumps to a new two

dimensional 4/3 law curve (with constant C2 ) and continues to increase

with I up to some maximum scale L, which may be of the order of several

kilometers. Sample calculations for typical values of ap/az and C in the

ocean, yield limiting values of Kzmax -in the range of 10 to 100 cm2/sec and(14.)

values of tcr in the range of 1 to 10 meters ). These numbers confirm

the relatively small extent of vertical diffusion in a stratified ocean.

One essential feature pointed out by this wodel is that while far

field diffusion effects may well be given by the 4/3 law [e.g., Eq. (3-24)]

for horizontal diffusion and the corrected Sverdrup law [Eq. (3-33)] for

vertical diffusion, these models are not adequate in the near field where

diffusion changes from isotropic to anisotropic behavior.

21

20086-6001 -RO-00

To estimate the effect of the near field behavior on the far field

solution, it is possible to seek alternate solutions of the two dimensional

diffusion equation [Eq. (3-15)], assuming that the initial field concentration

and size are prescribed as distributed initial conditions (rather than

point source initial conditions). Appropriate nea.- field solutions for a

collapsing submarine wake have recently been set forth by Ko(1) and

modifications presented in Section 2. These solutions would provide the

needed initial wake dimensions and initial concentration levels.

The reformulated two dimensional (vertical transport neglected)

diffusion equation and boundary conditions can be written as

U Ly- K 2 (3-37a)ax y2 TC

with boundary conditions

"0 0c. c o at x =0 < V (3-37b)(virtual origin) 2 <y (-3-

c• 0 as x ÷®, c ÷ 0 as y -+ ± for all x

where t 0 represents the initial width of the difiJsion layer at the joining

pzint with the near field solution.

The solution of Eq. (3-37) will follow the previous analysis of

Brooks(15) for diffusion of sewage effluents in an ocean current and the

thermal discharge analysis of Sur.daram et. al.(16).

Equation (3-37) can be converted to a simple diffusion equation

without a decay term by the transformation

Sc exp [- x/(UTc)] (3-38)

Then Eq. (3-37) reduces to the form

U K (X) c (3-39)ax y ay 2

22

20086-6001 -RO-0I

It will be assumed as in Eq. (3-19b) that the horizontal diffusivity is

related to the characteristic length t(x) by the relation

K = at n(x)

and (3-40)

I(x) ba(x)

where a is the standard deviation of the concentration distribution

defined by

)j

.x)2 y2Zdy (3-Ay)Ndy Co~o

If one requires that I = 0 at x = 0, where c is taken to be uniform and

equals to co in the domain -1o/2 < y < 1o/2, then b 2," 3.4.

The solution of Eq. (3-39) coul', be written down immediately if K

were a constant, independent of x. However, for K being a function of x,a simple transformation of the x coordinate can be used such that

Ko dx' = K (x)dxy

2[b 2dy] n/2 (3-42)cc 2-Zud:y_ dx

Then Eq. (3-39) reduces to the well known diffusion equation with constant

coefficient Ko = aton), i.e.,

U a-= Ko ! (3-43)

For the initial conditions given by Eq. (4-37b), the closed form solution

of Eq. (3-43) given by Crank(17), is

_ 1erf erf - (3-44)

Z10 23 4 f x 4K!0 0

23

20086-6001 -RO-00

The integral in Eq. (3-42) which is used to relate the physical

x and transformed x' variables, cat. now be evaluated by employing the

error function solution given in Eq. (2-44), i.e.,

2 _ b2 b-2 2K 0 K x i

0- o f ycdy = + 2-o2 U (3-45)0 c00 0

Hence the transformed axial dimension x' is found in terms of x by

integrating Eqs. (3-42) and (3-45),

x, 3 , + (1 2/(2-n) 1 (-

2b2IL ' O LUJ 3-6

For large values of x, far from the source, and for n = 4/3 (Richardson law)

x' ' x3 X/Lo >>

'lnder tr,t ;3ne circumi:stances, the cenceptrati'n s""tio'., Er. (3-..

takes the limiting form,

C (x') x/o >>

Hence, the point source behavior, given by Eqs. (3-21) and (3-24)

• (x)-3/2

is recovered from the solution, Eq. (3-44), for diffusion from a finite

near wake source.

In summary, the complete solution for diffusion from a finite

source of dimension to and concentration Eo" with decay Tc is written

in the following non-dimensional form

Co exp -X/LD lerf -4T err ----f4•• (3-47)

24

20086-6001 -RO-00 0

-;IS • • _ __K o = a t o -

where x = y = y/1t, L• = Ut c/L o, and K0 = o 0

In the final report of this investigatio.i, a parametric study wijl be

performed to determine the influence of the various prominent parametersin Eq. (3-47) on the magnitude and variation of c..•taminant concentration

behind a typical submarine in an ocean environment. I

25

I

I)

I

2: 2 5

20086-6001 -RO-O0

REFERENCES

1. Ko, D. R. S., "Collapse of a Turbulent Wake in a Stratified Medium,"TRW Report 18202-6001-RO-00, Vol. II, 1971.

2. Sutton, 0. G., "A Theory of Eddy Diffusion in the Atmosphere,"Proceedings of the Royal Society, Series A, Vol. 135, No. A826,February 1932, pp. 143-165.

3. Taylor, G. I., "Diffusion by Continuous Movements," Proceedings ofthe London Mathematical Society, Vol. 20, (1921).

"4. Batchelor, G. K., "Diffusion in a Field of Homogeneous Turbulence.II The Relative Motion of Particles," Proceedings of the CambridgePhilosophical Society 48, 1952.

5. Richardson, L. F., "Atmospheric Diffusion Shown on a Distance NeighborGraph," Proceedings of the Royal Society London A110, (1926), pp. 709-727.

6. Bowden, K. F., "Turbulence II," Oceanography and Marine BiologyAnnual Review, Vol. 8, edited by H. Barnes, (1970), pp. 11-32.

7. Okubo, A., Technical Report Chesapeake Bay Institute, 38, Ref. 68-6,(1968).

8. Hoffert, M. j., "Atmospheric Transport, Dispersionn and Chemic.lReactions, A Review," AIAA Paper 72-82, January 1972. See AlsoFrankrel, F. N., Advances in Applied Mechanics, Academic Press,(1953), pp. 61-107.

9. Munk, W. H., and Anderson, E. R., "Notes on the Theory of the T'nermocline,"Journal of Marine Research 1, 276, (1948).

10. Sverdrup, H., Johnson, M., and Flening, R., "The Oceans, Their Physics.Chemistry and General Biology," Prentice Hall, N. Y. (1942).

11. Ellison, T. H., "Turbulent Transport of Heat and Momentum from anInfinite Rough Plane," Journal of Fluid Mechanics, Vol. 2, (1957), p. 456.

12. Sundaram, T. R. and Rehrm, R. G., "The Effects of Thermal Discharges onthe Stratification Cycle of Lakes," AIAA paper 71-16, January 1971.

13. Merritt, G. E., and Rudinger, G., "Thermal and Momentum DiffusivityMeasurements in a Turbulent Stratified Flow," AIM paper 72-S),January 1972.

14. Ozmidov, R. V., "On the Turbulent Exchange in a Stably StratifiedOcean," Izvestia, Atmospheric and Oceanic Physics Series, Vol. 1,No. 8 (1965).

15. Brooks, N. H., "Diffusion of Sewage Effluents in an Ocean-Current,"Proceedings of First International Conference on Waste Disposal inMarine Envirotment, (ed. E. Pearson) Perganmn Press.

26

3-

REFERENCES (CONTINUED)

16. Sundaram, T. R., Easterbrook, C. C., Piech, K. R., and Rudiager, G.,"An Investigation of Physical Effects of Thermal Discharge intoCayuga Lake," Cornell Aero Report CAL No. VT-2616-O-Z, November 1969.

17. Crank, J. bhe Mathematics of Diffusion, Clarendon Press (1956), p. 13.

27

20086-6001 -RO-00

x

IQI

Fig. 1. Coordinates and Density Field

28

20086-6001 -RO-00

!I2I

* RHENO I/

0 1962 X NORTH SEA01962n

3 1961 I 3 0

A. CAPE*4 KENNEDY

0e Aii

1oI2 0 NEW YORK BIGHI 9 IOkm

a0Gb 0 oc OFF

( 21 -CALIFORNIA

o BANANA RIVER

M MANOKIN RIVER o

100- Ikm

0

-/9A

I Joe-- loom"HOUR 0 DAY W EK mONTH

Fig./, .3 1I, . . , .. 0

Fig. 2. Variance, o~c, of Dye Distribution against DiffusionTime (Data from 1961 Onwards) (after Okubo, 1968, Ref. 7)

29

20086-6001 -RO-00

I " * I I " " I " " I "

0 RHENOA 1964 IL 1962 m NORTH0196211 SEA

io o1961 I L1o? - 0.1e2 OFF -U

:30 CAPE j6054 KENNE.DY

*e6

IO6 - 0 NEW YORK SIGHT

*Go

E "

0*ed (CAUFORIIAo~f

0 BANANA RIVER

QOOf

I 0

I0po

1010o14 1o5 706to

1 (cm)

Fig. 3. Apparent Diffusivity, Ka, against Scale of Diffusion, z 30re(Data from 196 Onwards) (after Okubo, 1968, Ref. 7)

30

20086-6001-RO-00

cI

tI

, Diffusion Coefficient K -&,n

"n =411102

a =o

"1 . Qt/PU 1.0 ft

X 10" 1

%%

'.4

%I

C \

10 sTANDARD DEVIATION

10"•"1 o-

"4.

x/U 10 10 10u.C2• 103 -

16xlO-* 1 ",

o-- ..

-800 -400 0 400 8(0

y (ft)Concentration profile in 10 M2 103 10) l10 ylinear coordinates

x/U (sec)

Fig. 4a. Concentration Profiles c Behind a Point Source ;loving at a Velocity

U in an Ocean Environment, Eq. (3-21)

31

20086-6001-RO-00

rI

C

Diffusion Coefficient K - atn

n -1.1510"2 ,•\-8

a- .164sec\%

- 1 1 Qti/oU 1.0 ft

[ x/U103i " I

F%

[ 10 1J \\

2

10"10'

x/U1O 10 103O~

10 1 Y(ft)

S10"WSTANDARD DEVIATION•

110-04

3x10• [0 I

-400 -200 200 400 ...Y (ft) x/U = 105 10 102 i 104 Y

Concentration profile in linear (sec)

coordinates

Fig. 4b. Concentration Profiles c -ihind a Point Source Mov;ng at a VelocityU in an Ocean Environment, Eq. (3-21)

32

20086-6001 -RO-00

100

R n =4/3

n I 1.1

C11

"' 101.1

o

I--2

O Diffusion Coefficient

0-3- K = atn

Source Strength

QI/pU = 1.0 ft

10-

102 03 hr 10 (sec) 1day

DIFFUSION TIME

Fig. 4c. Centerline Concentration Behind a Point Source Moving at aVelocity U in an Ocean Environment, Eq. (3-21) (y = 0)

33

20086-6001-RO-00

0.10 .•-... ;.4."0..110

O .05 : • • :- • "---I ." .____."___ _ ,,_ ." ! ."" . '- i -o.c. z-8 "

-0.02 . - . -

oCo1 . - .. 1 = :! • -. •••

..,., I -oacv Z _--_

0.-2

,- 0.012

0.; 1. -0 2 -- 9 .Z2') o 0

2 z

K. '!. ..... . .

21e .i, !

0.002-- : 2 .-." .

- 1° 0.11 CM. . ."0.001 0 EXPERIMENT o~i=!

0.; 1.0 aT10 100

Fig. 5a. Thermal Diffusivity from Stratified Flow Experiments in a WaterChannel (after Merritt and Rudinger, Ref. 13) Comparison withTheoretical Model, Eq. (3-31) (after Sundaram and Rehmn, Ref. 12)

I0~ // OCEAN

CD 10~

_CAYUGA LAKE

S10-i

-102. PRESENT

S~TESTS:- 10-3

10 100 1.000

FLOW VELOCITY (u). cm/sec

Fig. 5b. Thermal Diffusivity (Neutral Stability) as a Function ofFlow Velocity (after Merritt and Rudinger, Ref. 13)

34