by D. R. S. Ko and i. E. Alber - DTIC20086-6001-RO-00 DIFFUSION OF A PASSIVE SCALAR Quarterly...
Transcript of by D. R. S. Ko and i. E. Alber - DTIC20086-6001-RO-00 DIFFUSION OF A PASSIVE SCALAR Quarterly...
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
-! MI *!Car 1vxiic rlob=
ila ,lkl
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
mf mSI
Unclassifiedeciuatity Classification
DOCUMENT CONTROL DATA - R&Daew,..e, C5.a..nsatacsest~i.. taste body as# .bfsattta ad Waestadss~ ~ef be arvr~r -64, As.,l .s.. Cie .. I.',E
042C.nATING ACTIIVUVY Ce~•-l.e ) Jaeo *CWOayT .LCuS iTv C A.ASSIVICATOft
TRW Systems Group Unclassifiedfng Spacl Park NEIKeaon0o Beach, Calif. 90278
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.
6 REPORT OATE 170 TOTAL. NO OW V*AG $S ' 1?& Oar gap's
March 1972 43 17ase COsaT.Q.C uso GRnA.ty sto!i O- ft'awOa*5l [M00014-72-C-0074I
Fa PRnOJECT ssARP ,,oru. ,o 20086-6001-R0-00ARPA Orie.r 1910
-0 AVA ILAI•SITVY'LINITATION X0'ICSs 4
it JuPPI•&EirTAfWrrOl 13 ISPONSIORNG O LITAPY ACTIVITY
SDi rector
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.
DD 1473 lnrlal4-ified
secgu.ty Classificaivon
ii;
----------------
F Unclassifiedbtcu:ttv Classitlcation
Is LINK A LINK a LINK C
gW0_ 3030 -. AC!I?*LIE i 3, SOLE O7
i IWakefomentuml essT;rbulent entrairnmentStr3ti ficationParsive scalarOcca". turbulence t
Diffusion
ii
INSTRUCTIONS
1. OR.,I.%ATI%CG ACTIVITY Enter tht nae and aiddress imposed bII securit, :lassisiciation. using standard statementsof the, om-.a'tor. s~b-onttriv'or. granter. (l~pstrnet of •e- such asj iOnse at t. s cr other qrgAnuflticfl 'comb rathor) Isting (1) "QUallfied requesters ritay obtain copies of this
rhe repir- report from Dot"*Pl PORT SECURITY CLASSIFICATCON Enter the oter- (2) "Foreign announcement and dissemination oi this
all secru.:-, dlasiifit ation of the repOrt. Indicate whether by"-Restricted Data** is sncludet. Marking is to be It accord- rePort Dy is t authorired7'aice with appr,'tratr sec.ritt ula"ti',ts, t3) "t S. Governmenrt agencies Ma) obtest. copies of
22• CR01'P A~o.uInrri -.k-.cralirt is specifie'd in DoD l* this report directly from DDC. Other qualified )DCrec-i7e 52.a. 10 and Ar-.ed For-es Indastriatl ManuaL Eer users shll request throught.e iro,,4p ru-'er Asso, at n-. a-,icatb:e. shot tha: .ipts•'r:ln'.aksnts hate been used fit Gr,,, I and Group 4 as sethot- W4, -' S -"-ilitar) agencies ma) obtain copies of thisseea report directly from DDC. Other qualified users
3 REPORT TITLE Enter the co-'plete rspon title in all shal! request throughcr 1 trtl letters. Titles in all cases slould be wtclassifs :• e.If a m.7ean••ngful title rannot he selected without ,-lasstfica-tsoi. sht•w title cla-ssficat-'- in all 4 apitals in parent."esis , All disttibutsin of this report is cott-olled. Quel-i•-tediate!y followinng the title- ified DDC users shall request through
4. DESCRIPTIVE NOTES U appropriate. enter the type of .'"report. e.g.. t'er... progress. summary, annual, or final. If the report has been furnushed to the Office of TechnicalGive the inc:usiue dates when a specific :eport:ng period is Services. Deps-;=tero of Commerce. for sale to the puhlic. sods-
osered. cate this tact and erter the price. if known.
S. AUTHOR S? Etner the r.ameis of author(s) as shown or. 1I. SUPPLEMENTARY %.TES Use for additional esplana-at in the repý'r. EntIr 'lest n•am. firs name, middle initzia to", n-te,.'. riitta•,. ,ho. rank nJ branch of service. The name ofthe printipa1 . -thor .% an absolute minimum tequiremnt. 12. SDONEilRING MILITARY ACTIVITY. Enter the name of:he depar••renta protect office or laboratory &ponsoring !par6. REPORT rAT7. Enter thr date of the reporrt a day. on r !0,, the research and develop.,ent. Include addr"ess.month. year. or month. year. ft ••ore than one date appearson the report. Use date of p;hbl' attin. I I ABSTRACT Enter ar abstract giving a brief and farc tus
--- n ,-f the .-x-jment indicative of the report. even thouvgh7a 1XITAL ?.UMIII-.R OF PAGES TI total page cuwr it -v•. - ap-.ar elehere in rie body of the, hti cal reshould l.!. n•er•al paK.rAtt.on prt,-eu. *. I.... ertr -Ite -ilt If asddies -nt: -•,a - is required a . ont-:rts.i-.r sheet hallnu.'ber of pages conta.ning ;ntrti. . artas hed
7b NUMBER OF REFERENCES F'-er the total nu%-ber of It is h:ghls desirab:e that the abtra t of , tsstified reportsreferent-s cited in the report. ; be unc:sai•zfied Each paragraph of the abstract she': end w.th
slir COTRACT OR GRANT NU• BER If appropriate, enter an indication of the izl:tary secur.ry classiflicatico of :h,! inthe asppocable number of the contract or grant under whsi:h formaston mr the paragraph, represented as fTs) "s, 'c,at (u)the rcpont was wtittti. The-e is no hlcita:ion on the length of the abstract Mteow-8b. tk. &t ad. PROJECT NUMBER Enter the appropriate ever. the suggested length Is from 1S0 t1 225 wordsmilitary department identiftcattsar. Such as prOject mr.-*ssbee.sutroject number. system numbers. task number. ec14 KEY WORDS 4ey words are technscslsy meaningful tems
ecr short phrases that characterize a report and may be used as9a. ORIGINATOR*t REPORT NUISERiS) Enter the offs- rindex cnmes for cataloging the report Key words must becia: tipet number by which the document will be identified selected so that no security classification is required IdeaU-and controlled by the originating er.tivity. This •numbr must ,rfsea. such as eqdipment model designation, trade nrame, mllitarybe urique to this report. pro:ecl code name, geographic location. mea be used as key
b OTHERt RETORT NUMBERISI ;f the cponl hat bee words but will be fol:owed by an indicatzoi of -halcal Can-asslgt•id any other report numbers 'either by the origdniaor test T l1 ssignmrer of links. ru:e. and we . its is optional.or by the sponsor). sdoo enter this nurmbers).
10. AVAILAB.ITY'IAMITATfON .OTICES, Enter any Its-itatisns on bather dissemsnstson of the repot,. other thain thoseI
UnclassifiedSecurity Classificatiso
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
20086-6001 -RO-00
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
3
20086-6001 -RO-00
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
20086-6001 -RO-O0
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