anisotropy of turbulence in stably stratiﬁed mixing layers

PHYSICS OF FLUIDS VOLUME 12, NUMBER 6 JUNE 2000

Anisotropy of turbulence in stably stratified mixing layersWilliam D. Smytha) and James N. MoumCollege of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon 97331-5503

~Received 12 February 1999; accepted 18 February 2000!

Direct numerical simulations of turbulence resulting from Kelvin–Helmholtz instability in stablystratified shear flow are used to study sources of anisotropy in various spectral ranges. The set ofsimulations includes various values of the initial Richardson and Reynolds numbers, as well asPrandtl numbers ranging from 1 to 7. We demonstrate that small-scale anisotropy is determinedalmost entirely by the spectral separation between the small scales and the larger scales on whichbackground shear and stratification act, as quantified by the buoyancy Reynolds number.Extrapolation of our results suggests that the dissipation range becomes isotropic at buoyancyReynolds numbers of order 105, although we cannot rule out the possibility that small-scaleanisotropy persists at arbitrarily high Reynolds numbers, as some investigators have suggested.Correlation-coefficient spectra reveal the existence of anisotropic flux reversals in the dissipationsubrange whose magnitude decreases with increasing Reynolds number. The scalar concentrationfield tends to be more anisotropic than the velocity field. Estimates of the dissipation rates of kineticenergy and scalar variance based on the assumption of isotropy are shown to be accurate forbuoyancy Reynolds numbers greater thanO(102). Such estimates are therefore reliable for use inthe interpretation of most geophysical turbulence data, but may give misleading results whenapplied to smaller-scale flows. ©2000 American Institute of Physics.@S1070-6631~00!02106-1#

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I. INTRODUCTION

This paper is one of a sequence reporting results frdirect numerical simulations of turbulence arising from tdynamic instability of stably stratified shear layers.1,2 Thework is motivated by the need to properly interpret measuments of turbulent events in the Earth’s oceans; howeverresults have important implications for a wide range of tbulent flows. Our focus in the present paper is on issuelocal isotropy, the tendency of the smallest scales of moto be isotropic despite the anisotropy of the large scales

In the view of turbulence that originated witKolmogorov,3 it is thought that the cascade of energy frolarge to small scales of motion is accompanied by a losinformation about the geometry of the large scales. Inlimit of high Reynolds number, the smallest eddies havstructure that is independent of the large-scale flow, and ithat sense universal. In other words, the small-scale strucdepends only on the equations of motion, and not on inior boundary conditions. In the absence of body forces,Navier–Stokes equations are isotropic, i.e., preferred ortations enter only through initial and boundary conditionTherefore, a universal state dependent only on the equathemselves should exhibit the property of isotropy.

Turbulence occurring in nature is usually influencedsome degree by density stratification and background shThe addition of buoyancy forces introduces a preferencethe vertical direction~the direction of the local gravitationa

a!Author to whom all correspondence should be addressed; [email protected]

1341070-6631/2000/12(6)/1343/20/$17.00

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field! into the equations. However, it is believed that buoancy acts preferentially upon motions larger than the Ozdov scale4,5 ~to be defined in the following!, and that scalesmuch smaller than this are unaffected. We therefore expthat, at high Reynolds number, motions on sufficiently smspatial scales will be isotropic despite the action of buoyaforces. In sheared turbulence, the background shear indanisotropy in the large eddies. As in the case of buoyanhowever, the effect is expected to become negligible at sficiently small scales.

Isotropy of the small scales is not proven. While maexperiments have shown that one or more turbulence sttics become consistent with local isotropy at large Reynonumbers~e.g., the wind tunnel experiments of Saddougand Veeravalli6!, the hypothesis asserts thateverystatistic isconsistent with local isotropy. To the contrary, Durbin aSpeziale7 have argued from theory that the small scales cnot be isotropic at any Reynolds number in the presencesteady background strain. Both laboratory experimentsgeophysical measurements~in which the Reynolds number ihigh! have revealed distinct departures from isotropy.8,9 Sta-bility analyses of many flows10–12 have shown that steadylarge-scale structures may inject energy directly intosmallest scales of motion regardless of the Reynolds numYeung and colleagues13,14 have conducted numerical explorations of the mechanisms through which large-scale aniropy may be transferred to the smallest scales, even atReynolds number. So, although the likelihood that local isropy holds at sufficiently large Reynolds number is wideaccepted, we cannot yet discount the possibility that soforms of small-scale anisotropy persist at all Reynolds nuil:

3 © 2000 American Institute of Physics

license or copyright; see http://pof.aip.org/pof/copyright.jsp

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1344 Phys. Fluids, Vol. 12, No. 6, June 2000 W. D. Smyth and J. N. Moum

bers of practical interest, or even to the limit of infinite Renolds number.

The importance of isotropy in the small scales is nlimited to our conceptual understanding of turbulence. Tmolecular dissipation of kinetic energy, as well as thatscalar fluctuations, takes place primarily at the smalscales of motion, in the so-called dissipation range ofwave number spectrum. The rates at which these crucialcesses occur are very difficult to measure, as they requiresimultaneous measurement of numerous spatial derivatThe problem becomes greatly simplified, however, if oassumes that motions in the dissipation range are isotroIn that case, the number of simultaneous measuremneeded to infer dissipation rates is greatly reduced. But eif the hypothesis of local isotropy is valid, the smallest scaof motion are expected to be isotropic only as the Reynonumber increases to infinity. In naturally occurring flows, wneed to account for the consequences of finite Reynnumber. For a given large-scale flow geometry at a givReynolds number, how anisotropic are the smallest scai.e., what are the uncertainties associated with the assution of isotropy in the dissipation range? Is local isotroreally a function of Reynolds number only? What are tphysical mechanisms through which information aboutgeometry of the large scales is transmitted down the sptrum?

The present work was motivated by the need to propeinterpret observations of temperature and velocity micstructure in the ocean. Such measurements almost alwhave the form of one-dimensional profiles. Dissipation raare inferred from this limited information using isotropapproximations,15 even though the Reynolds number is finand both background shear and density stratification are tcally present. Using measurements of flow over a sill, Ggettet al.16 have shown that certain properties of the inertsubrange are consistent with isotropy for values of the buancy Reynolds number,Rb ~to be defined in the following! inexcess of 200. Itsweireet al.17 tested isotropic approximations for dissipation range statistics using data from numcal simulations of turbulence with uniform background shand stratification, and found large errors forRb,102. Here,we extend the analyses of Itsweireet al. to more realisticflow geometries and consider the physics of local anisotrin greater detail.

Our primary tool in these analyses is direct numerisimulations~DNS! of breaking Kelvin–Helmholtz~KH! bil-lows. KH billows are commonly observed in geophysicfluid systems, and are an important source of geophysturbulence.18 They provide a useful model for turbulenevents in the ocean interior, and for sheared, stratified tulence in general. KH billows occur when a verticalsheared, horizontal flow coexists with stable density stracation such that the gradient Richardson number~a measureof the strength of the stratification relative to the shear! issmaller than a critical value.19,20 Billows take the form ofline vortices aligned horizontally and perpendicular to tmean flow, and result in an increase in the gravitationaltential energy of the fluid. As the billows reach large amptude, transition to turbulence is initiated via a combination


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convective and shear-driven secondary instabilities.21,22Evenafter the onset of turbulence, the large eddies reflectstructure of the original billows.23 The result is a flow whichis wave-like on the largest scales but highly chaoticsmaller scales. The disturbance then decays, leaving aof weak gravity waves propagating on a stable, parallel shflow.

Our ultimate goal is to assess the validity of varioisotropic formulas over a range of Reynolds numbers.preparation for this, we first examine the relationship btween the different scales of motion during each phase offlow evolution, paying particular attention to the degreewhich the anisotropic structure of the large scales is reflecin the small scales. Our dataset consists of eight simulatin which both the initial flow configuration and the Prandnumber of the fluid are varied. The Reynolds number varin time as turbulence grows and subsides, and is also gerned by the initial conditions that are specific to each simlation. The resulting dataset spans a wide range of flowgimes, with buoyancy Reynolds numbers as large as 2Our analyses enable us to draw general conclusions abouphysics of anisotropy in sheared, stratified turbulence,well as assessing the validity of popular isotropic appromations.

At all Reynolds numbers, signatures of anisotropy adetected throughout the spectrum. When turbulence is mvigorous, however, small-scale anisotropy is detectable oby the most sensitive tests. A significant finding is that, fowide variety of turbulent flow states, quantities associawith anisotropy in the dissipation scales can be predicquite precisely based only on knowledge of a suitablyfined Reynolds number. The isotropic approximations vin accuracy, but are generally valid for buoyancy Reynonumbers in excess ofO(102), which encompasses the largmajority of ocean turbulence measurements to date~e.g.,Moum24!.

Section II of this paper contains a brief review of thnumerical model and a description of the diagnostic methused in the present analyses. In Sec. III, we give an overvof the life cycle of turbulence originating from KH instabiity. In Sec. IV, we describe the geometry of the large-sc~i.e., energy-containing! flow structures. Section V is devoted to measures of anisotropy in the dissipation rangerevealed by the statistical characteristics of velocity graents. We examine the physical mechanisms responsibleanisotropy in these small-scale motions, the manner in whanisotropy vanishes as the Reynolds number increases,the impact of anisotropy on estimates of the dissipation rfor turbulent kinetic energy. We also show that our moturbulent states exhibit a distinct inertial subrange. In SVI, we carry out a similar analysis of anisotropy in spatgradients of density. Results are summarized in Sec. VII

II. METHODOLOGY

A. DNS methods

The numerical methods employed to generatepresent dataset are described in detail in the compa


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1345Phys. Fluids, Vol. 12, No. 6, June 2000 Anisotropy of turbulence in stably stratified mixing layers

paper;2 here, we provide a brief summary. Our mathematimodel employs the Boussinesq equations for velocity, dsity, and pressure in a nonrotating physical space measby the Cartesian coordinatesx, y, andz:

]u

]t5uÃ~“Ãu!2“P1gu k1n¹2u; ~1!

P5p

r01

1

2u"u. ~2!

Here, p is the pressure andr0 is a constant characteristidensity. The thermodynamic variableu represents the fractional specific volume deviation, or minus the fractional desity deviation, i.e.,u52(r2r0)/r0 . In a fluid where den-sity is controlled only by temperature,u is proportional to thetemperature deviation~with proportionality constant equal tthe thermal expansion coefficient!. The gravitational accel-erationg has the value 9.8 m/s2, and k is the vertical unitvector. Viscous effects are represented by the usual Lapian operator, with kinematic viscosityn51.031026 m2/s.

The augmented pressure fieldP is specified implicitlyby the incompressibility condition

“"u50, ~3!

and the scalaru evolves in accordance with

]u

]t52u"“u1k¹2u, ~4!

in which k represents the molecular diffusivity ofu.We assume periodicity in the horizontal dimensions:

f ~x1Lx ,y,z!5 f ~x,y1Ly ,z!5 f ~x,y,z!, ~5!

in which f is any solution field and the periodicity intervaLx andLy are constants. At the upper and lower boundar(z56 1

2Lz), we impose an impermeability condition on thvertical velocity:

wuz56~1/2!Lz50, ~6!

and zero-flux conditions on the horizontal velocity compnentsu andv and onu:

]u

]zUz56~1/2!Lz

5]v]zU

z56~1/2!Lz

5]u

]zUz56~1/2!Lz

50. ~7!

These imply a condition onP at the upper and lower boundaries:

F]P

]z2guG

z56~1/2!Lz

50. ~8!

The model is initialized with a parallel flow in whichshear and stratification are concentrated in the shear layhorizontal layer surrounding the planez50:

u~z!5u0

2tanh

2z

h0, u~z!5

u0

2tanh

2z

h0. ~9!

The constantsh0 , u0 , andu0 represent the initial thicknesof the shear layer and the changes in velocity and denacross it. These constants can be combined with the fl


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parametersn andk and the geophysical parameterg to formthree dimensionless groups whose values determine thebility of the flow at t50:

Re0[u0h0

n, Ri0[

gu0h0

u02 , Pr[

n

k. ~10!

The initial macroscale Reynolds number, Re0, expresses therelative importance of viscous effects. In the present simutions, Re0 is of order a few thousand, large enough that tinitial instability is nearly inviscid. The bulk Richardsonumber, Ri0 , quantifies the relative importance of shear astratification. If Ri0,0.25, the initial mean flow possesseunstable normal modes.25 The Prandtl number, Pr, is the ratiof the diffusivities of momentum and density. In orderobtain a fully turbulent flow efficiently, we add to the initiamean profiles a perturbation field designed to excitemost-unstable primary and secondary instabilities. Detailsthe initial perturbation are given in the companion pape2

Care is taken to ensure that the initial perturbation is weenough to obey linear physics. In that case, the precise fof the perturbation has little effect on the statistical quantitof interest here.

Because the horizontal boundary conditions are periodiscretization of the horizontal differential operators is acomplished efficiently in Fourier space using fast Fourtransforms. In the vertical direction, we employ seconorder, centered finite differences, in order to retain flexibilin the choice of upper and lower boundary conditions. Tfields are stepped forward in time using a second-orAdams–Bashforth method. Implicit time stepping of the vcous operators has turned out to be unnecessary, as thestep is limited by the advective terms. For accuracy andmerical stability, we limit the time step in accordance wit

Dt,0.12mini 51,3

S Dxi

Ui1 D , ~11!

in which Ui1 is the maximum speed in thexi coordinate

direction. At each time step, the augmented pressure fieobtained as the solution of a Helmholtz equation whforces the flow to be nondivergent at the next timestep.

A well-resolved DNS model must have grid spacinggreater than a few~3–6! times the Kolmogorov length scalLK5(n3/e)1/4 in which e is the volume-averaged kinetic energy dissipation rate~to be defined in the following!.26 Forflows with Pr.1, the Kolmogorov scale is replaced by thBatchelor scale:LB5LK /Pr1/2. In order to take advantage othis information, one must be able to estimate the maximvalue ofe in advance. We do this by employing the scaline15cu0

3/h0 . Through trial and error, we have establishthe value 6.531024 for the proportionality constantc. Thegrid spacing is then set to 2.5LB

2 , whereLB2 is the estimated

minimum Batchelor scale (n3/e1)1/4Pr21/2. The present ver-sion of the model uses isotropic grid resolution, i.e.,Dx5Dy5Dz. ~The largest of the simulations described heemployed array sizes of 5123643256, and required 800


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TABLE I. Parameter values describing a sequence of eight simulations of breaking Kelvin–Helmholtz biNx , Ny , andNz are the array dimensions,Lx , Ly , andLz are the domain dimensions, andDx (5Dy5Dz) isthe grid interval. Initial conditions are characterized by the length scaleh0 , the velocity scaleu0 , and thetemperature scaleu0 . Lsc5u0

2/4gu0 is an estimated lower bound on the final layer depth based on the asstion that the bulk Richardson number asymptotes to 1/4~see I for further details!. Pr is the Prandtl number; Ri0

and Re0 are the initial bulk Richardson and Reynolds numbers, respectively. Ref5Re0/2Ri0 is a lower bound onthe final value of the Reynolds number based on the assumption that the bulk Richardson number asyto 1/4. Simulations are labeled as RxxPy, where ‘‘xx’’ represents Ref /1000 and ‘‘y’’ represents Pr.

Run R06P1 R16P1 R04P1 R10P1 R03P1 R04P4 R06P4 R04

Nx 256 512 256 512 256 512 512 512Ny 64 64 64 64 64 64 64 64Nz 128 256 128 256 128 256 256 256Lx (m) 3.29 5.24 3.29 5.24 3.29 3.29 3.29 2.7Ly (m) 0.82 0.65 0.82 0.66 0.82 0.41 0.41 0.3Lz (m) 1.63 2.62 1.63 2.60 1.63 1.63 1.63 1.3Dx(1022 m) 1.29 1.03 1.29 1.02 1.29 0.64 0.64 0.5h0 (m) 0.24 0.38 0.24 0.38 0.24 0.24 0.24 0.2u0(1023 m/s) 8.34 13.27 8.34 13.27 8.34 8.36 8.34 8.3u0(1026 K) 2.41 3.83 3.61 5.75 4.81 3.70 2.41 2.00Lsc ~m! 0.74 1.17 0.49 0.78 0.37 0.48 0.74 0.8Pr 1 1 1 1 1 4 4 7Ri0 0.08 0.08 0.12 0.12 0.16 0.12 0.08 0.0Re0 1965 4978 1967 4978 1967 1967 1967 1354Ref 6140 15 557 4098 10 371 3074 4119 6147 4232Symbol s ^ , n L v s d

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Mbytes of core memory and 100–200 h of CPU time on32-node Connection Machine CM5.! Spectra confirming theaccuracy of the numerics, along with comparisons with laratory experiments, are provided elsewhere.1,2

The dataset employed here consists of eight simulatiodesigned to reveal the effects of changing Ri0 , Re0, and Pr.In order to adequately sample the regime of KH instabiwhile maintaining accuracy, Ri0 was set to 0.08, 0.12, an0.16. Prandtl numbers of 1, 4, and 7 were employed. Tchoice of Re0 was strongly constrained by the need to matain adequate resolution of the small scales; values ranfrom 1354 to 4978. As the evolution progressed, bothbulk Richardson and Reynolds numbers increased in protion to the increasing depth of the shear layer.2 For all runs,the time scaleh0 /u0 was set to 28.28 s. Parameter valuesall eight simulations are summarized in Table I.

B. Diagnostic methods

For the present analyses, we require statistical quantwhich give the clearest possible picture of the structure ofturbulence, the structure of the large-scale environmenwhich the turbulence evolves, and the relationship betwthe two. We begin with a definition of the averaging volumOur simulated turbulence is of infinite horizontal extent~viaperiodic boundary conditions!, but is confined in the verticato a finite layer. Outside that layer are regions of laminflow that extend to the upper and lower boundaries ofcomputational domain. The locations of the upper and lowboundaries of the turbulent layer,zU and zL , are to somedegree arbitrary. Our objective is to define these locationa maximally straightforward manner consistent with the neto exclude the outer, laminar layers. After some experimtation, we have definedzU(x,y,t) and zL(x,y,t) to be thesurfaces upon whichu takes the valuescu0/2 and2cu0/2,

l 2010 to 128.193.70.43. Redistribution subject to AIP

a

-

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r

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.

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respectively, wherec5tanh(1). The value ofc is largeenough to ensure thatzU andzL remain single-valued functions of horizontal position. The volumezL,z,zU is re-ferred to asVs . The efficacy ofVs in isolating the turbulentregion of the computational domain is demonstrated be~cf. Fig. 3 and the accompanying discussion!. In this paper,volume averages are taken overVs , with a few exceptions tobe specified explicitly.

We also require values for the shear and stratificatcharacteristic of the large-scale flow at any given time. Tbackground scalar gradient,u ,z , is defined as the verticaderivative of the horizontally averaged scalar concentratiu(z,t), averaged vertically over the layer in which2cu0/2, u,cu0/2. ~Subscripts preceded by commas indicate ptial differentiation.! This layer is related toVs , but is definedin terms of horizontally averaged quantities, so thatboundaries are independent ofx and y. Background stratifi-cation is then characterized by the buoyancy frequenN(t)5Agu ,z. In similar fashion, we define the backgrounshear,S, to be the average ofu,z over the layer in which thehorizontally averaged velocityu(z,t) satisfies2cu0/2,u,cu0/2. Note that bothS and uz are positive for the flowsconsidered here.

To characterize relationships amongS, N, and the turbu-lence statistics at any given time, we choose from sevscalar quantities that are used commonly in the interpretaof turbulence data. These are the buoyancy Reynolds nber, the shear Reynolds number, the shear number, andCox number. The buoyancy Reynolds number is given b

Rb5^e&nN2 . ~12!

Here,^e& represents the average overVs of the kinetic energydissipation rate,e52nsi j si j , where


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si j 512~ui , j8 1uj ,i8 ! ~13!

is the strain tensor. Primes denote fluctuations about the hzontally averaged fields.

A second measure of spectral range is the shear Rnolds number:

Rs5ê&nS2 . ~14!

The shear Reynolds number is equal to (LC/LK)4/3, whereLK is the Kolmogorov length scale, (n3/ê&)1/4, and LC

5(ê&/S3)1/2 is referred to here as the Corrsin scale,27 anestimate of the smallest scale at which eddies are strodeformed by the mean shear. In similar fashion,Rb equals(LO/LK)4/3, whereLO5(ê&/N3)1/2 is the Ozmidov scale, ameasure of the smallest scale that experiences significanformation by the mean density gradient.

Rs is the inverse square of the quantity referred toSaddoughi and Veeravalli6 asSc* , the latter being the largescale shear scaled by the strain rate of Kolmogorov eddAê&/n. A useful measure of the strength of the large-scshear relative to the strain of the energy-containing scalethe shear number:

S* 5Sq2

ê&, ~15!

in which q25ûi8ui8& is twice the specific kinetic energy othe velocity fluctuations.

Finally, we include the Cox number,

Cox5û“u8u2&

~ u ,z!2 . ~16!

Rs may be interpreted as the ratio of kinetic energy dissition due to turbulence to that accomplished by the mflow. The Cox number may be interpreted in analogous faion; it is the ratio ofpotential energy dissipation by turbulence to that accomplished by the mean flow.

A convenient way to characterize anisotropy is throuthe use of symmetric, traceless tensors whose elementsish in isotropic flow. Anisotropy in the energy-containinscales is quantified using the velocity anisotropy tensor:

bi j 5ûi8uj8&

ûk8uk8&2

d i j

3~17!

in which d is the Kronecker delta and summation overpeated indices is implied. With some exceptions~to be dis-cussed in due course!, volume averages are taken overVs

and are indicated by angle brackets. Again, primes indicfluctuations about the mean~i.e., horizontally averaged! ve-locity profile. ~The mean flow is removed before calculatinbi j because, otherwise, it dominates the tensor so complethat the properties of the velocity fluctuations cannot be dcerned. In other anisotropy tensors, to be described infollowing, removal of the mean profile is unnecessary.! Thestructure of the velocity anisotropy tensorbi j is convenientlyrepresented using the two invariants

IIb5bi j bji , III b5bi j bjkbki . ~18!


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Becauseb is traceless, IIb and IIIb are proportional to thesecond and third principal invariants ofb, respectively.28

Here, we follow previous terminology~e.g., Lumley andNewman29! by referring to IIb and IIIb as the second andthird invariants, respectively, ofb. IIb is positive definite, andrepresents the degree of anisotropy in the velocity field.b

can take either sign; its value contains information aboutnature of the anisotropy. When IIIb.0 the flow tends to bedominated by one Cartesian component that is much lathan the other two, whereas when IIIb,0, two componentsdominate. We refer to these two classes of anisotropy‘‘prolate’’ and ‘‘oblate,’’ respectively, in analogy with theusage of those terms to classify ellipsoids.~Other authorshave used the terms ‘‘rod-like’’ and ‘‘disk-like’’ for thesame concepts.30! For a given value of IIIb , IIb is boundedfrom above by the two-component limit IIb5 2

912IIIb , andfrom below by the axisymmetric limit IIb561/3uIII bu2/3 ~cf.Fig. 6!. Lumley and Newman29 provide further details on thephysical interpretations of IIb and IIIb . George andHussein31 provide a detailed discussion of the axisymmetcase.

Anisotropy in the vorticity-carrying motions is represented by the vorticity anisotropy tensor,

v i j 5^v iv j&

^vkvk&2

d i j

3, ~19!

and its invariants IIv5v i j v j i and IIIv5v i j v jkvki . Here, vrepresents the total vorticity,v5“Ãu. The interpretation ofthe invariants is similar to that given in Sec. II A for thinvariants of b, with one important exception: in twodimensional flow, only a single component of the vorticitynonzero. That flow configuration therefore correspondsthe single-component limit, (IIv ,III v)5(2/3,2/9).

Anisotropy of the scalar gradient field is representedsimilar fashion by the tensor

Gi j 5û ,iu , j&

û ,ku ,k&2

d i j

3, ~20!

and its invariants IIG5Gi j Gji and IIIG5Gi j GjkGki .An anisotropy tensor pertaining to the dissipation rate

defined in a slightly different manner:

di j 5ûi ,luj ,l&

ûk,luk,l&2

d i j

3. ~21!

Again, commas before subscripted indices indicate pardifferentiation. The invariants ofd are denoted IId and IIId .

An alternative method of quantifying anisotropy, ofteused in the interpretation of laboratory data,31,32 is via theratios

K152û,x8

2&

^v ,x82&

; K252û,x8

2&

^w,x82&

;

~22!

K352û,x8

2&

û,y82&

; K452û,x8

2&

û,z82&

.

In isotropic flow, each of these ratios is equal to unity.


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FIG. 1. ~Color! Life cycle of a breaking Kelvin–Helmholtz wave train from simulation R06P1~see Table I for parameter values!, visualized by isosurfaces

of the enstrophy,Z512vkvk . Each frame shows three isosurfaces, with correspondingZ values given in the legend. Blue, green, and yellow meshes indi

low, intermediate, and high values, respectively. Simulation R06P1 is chosen for this demonstration because the moderate Reynolds number permraphicalrepresentation of the smallest scales of motion.

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The most sensitive tests for anisotropy employ the odimensional~streamwise! cospectraEi j (k1), in which k1 isthe streamwise wave number.Ei j is normalized such that

E0

`

Ei j ~k1!dk15uiuj , ~23!

where the overbar indicates a streamwise average. Theresponding correlation-coefficient spectra are given by

Ci j 5Ei j

AEii Ej j

. ~24!

~In the foregoing equation only, repeated indices aresummed.! To compensate for the distinction between lontudinal and transverse one-dimensional spectra,E11 can bereplaced by its isotropic transverse equivalent:33

E118 51

2 S E112k1

dE11

dk1D . ~25!

In a region of the spectrum where motions are isotropE118 5E225E33 andE125E235E135C125C235C1350.


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III. OVERVIEW OF TURBULENCE EVOLUTION

In this section, we give an overview of turbulence evlution in representative simulated turbulent events. Thelife cycle includes the initial growth and pairing of the twodimensional primary billows, breaking and the transitionturbulence, and an extended period of slow decay durwhich the flow relaxes to a stable, parallel configuration.

We begin with a direct visualization of enstrophy isourfaces in physical space~Fig. 1!. At t51131 s@Fig. 1~a!#,two KH billows have grown to large amplitude and are in tprocess of merging. Byt53111 s@Fig. 1~b!#, the transitionprocess is complete. While the range of scales is necesslimited by computer size, the small scales are in manyspects characteristic of fully developed turbulence~also seeFig. 3!. The spatial signatures of the three-dimensional sondary instabilities that drive the transition remain visiblestreaks in the braid regions at the left and right of the dmain, but the vorticity field within the billow is highly dis-ordered. Figure 1~c! typifies the early part of the decaphase. Turbulence is well-developed, with horseshoe-


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vortices visible near the top of the turbulent layer. By the eof the simulation@Fig. 1~d!#, turbulence has decayed almocompletely, leaving behind a dynamically stable, parashear flow perturbed by laminar gravity waves.

Next, we examine the volume-averaged perturbationnetic energies associated with the three Cartesian comnents of the velocity@Fig. 2~a!#, along with the buoyancyReynolds number, the shear Reynolds number, and thenumber@Fig. 2~b!#. Note that the perturbation velocities reresent fluctuations about the horizontally averaged flow pfile. These fluctuations include wave-like motions as wellturbulence. To aid interpretation, we divide the life cycle inthree stages. Fromt50 to t51200 s, the flow consists of thoriginal parallel shear flow, plus a growing, two-dimensionKH wave train. From t51200 to t51800 s, three-dimensional secondary instability grows until its enerreaches a level somewhat below that of the streamwisevertical flow components.~The growth of the secondary instability actually begins beforet51200 s. The boundaries othis interval are defined based on geometrical propertiethe velocity gradients to be discussed in Sec. V.! The growthof the three-dimensional secondary instability coincides wa dramatic increase in turbulence intensity, as quantifiedRs , Rb , and Cox. This is the period we identify with thonset of turbulence. After the billows reach maximum a

FIG. 2. Overview of flow evolution in a typical simulation~R10P1!. ~a!Components of the volume-averaged disturbance kinetic energy. Solid

Kx512û82& ~where the prime denotes the departure from the horizo

average!, dashed line:Ky512^v82&, dotted line:Kz5

12^w82&. ~b! Measures

of turbulence intensity. Solid line: buoyancy Reynolds numberRb

5ê&/nN2, dashed line: shear Reynolds numberRs5ê&/nS2, dotted line:Cox numberu“uu2/(u ,z)

2. Labels between~a! and ~b! indicate stages inflow evolution. The lower horizontal axis shows the time in seconds.upper horizontal axis gives the time in buoyancy periods, i.e.,tN

5*0t N(t8)dt8/2p.


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nd

of

hy

-

plitude, the KH waves exhibit a quasiperiodic exchangeenergy between streamwise and vertical motions which ainvolves the potential energy reservoir~not shown!. This os-cillation corresponds to the nutation of the elliptical vortcore.34

Turbulence develops largely during the phases ofoscillation in which potential energy is being convertedkinetic energy, a process very reminiscent of the breakinga surface wave. Early in the turbulent phase, vertical kineenergy decreases rapidly until it reaches a level very simto that of the spanwise component~neart53600 s!. Beyondthis point, the oscillations cease and the disturbance demonotonically. The simulation was continued until the dturbance energy was smaller than the energy of the nfield with which the model was initialized. Energy convesions that occur during the KH life cycle will be discusseddetail in a separate paper.

The relationship of our simulated flows to the classicpicture of fully developed turbulence is evident in the evoing probability distribution of the local turbulent kinetic energy dissipation rate,e ~Fig. 3!. At t50, the initial shearlayer plus perturbation appears as a smooth distribution oe.The maximum neare51029 W/kg represents the paralleshear layer. Byt5565 s, the distribution has taken on a bmodal form complicated by numerous small peaks. Rainitial spreading of the shear layer has decreased the cosponding local maximum from 1029 to 5310210W/kg. The

e:

l

e

FIG. 3. Probability density functions describing the spatial distribution oeat selected times during run R10P1. The area under each curve is ucurves computed at successive times are displaced vertically by 0.5.appearance of a nearly lognormal peak att52262 s shows that transition haoccurred. Att53111 s, the additional thin curve isolates values occurringthe averaging volumeVs .


al

-

,


FIG. 4. Horizontally averaged properties of a typicturbulent flow ~R10P1, t52828 s!. Horizontal dashedlines delineate2cu0, u,cu0 , the layer over whichthe background temperature gradient is computed.~a!Solid line: horizontal velocity; dashed line: temperature. ~b! Diagonal componentsb11 ~solid line!, b22

~dashed line!, andb33 of the velocity anisotropy tensoraveraged in the horizontal.~c! Off-diagonal compo-nentsb12 ~dashed line!, b13 ~solid line!, andb23 of thevelocity anisotropy tensor.~d! Invariants of the depth-dependent velocity anisotropy tensor.

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broad distribution of smallere values has been replaced byjagged peak neare510211W/kg as the effects of weak flowdistortions from the initial perturbation diffuse through thdomain. The jaggedness evident during the rollup phas~t5565 s, t51131 s! reflects the coalescence of regionsquasiuniforme in the cores and braids of the merging Kvortices. Between t51131 s and t52262 s, three-dimensional secondary instabilities trigger the transitionturbulence.

By t52262 s, the distribution is dominated by a singsmooth peak centered neare5331029 W/kg. The shape ofthis peak approximates the lognormal form, predictedfully developed turbulence by Kolmogorov.35 ~The history ofthe lognormal distribution and its modern variants isviewed by Frisch.3! The jagged distribution of lowere valuesrepresents the laminar flow surrounding the turbulent layFor the caset53111 s, we also include the distribution funtion pertaining only to the averaging volume,Vs , to demon-strate thatVs effectively isolates the most-turbulent regionthe flow. The skewness of log10(e) within Vs is 20.25, in-dicating that the departure from lognormality is small.slight, negative skewness was also observed by Wanget al.36

in simulations of stationary, homogeneous, isotropic turlence with Reynolds number larger than we achieve hWang et al.’s parameterQ, which is unity for a perfectlylognormal distribution, is equal to 0.997 withinVs . As thesimulation is continued, turbulence decays, and the pshifts gradually to lower values. In the long-time limit, thdistribution loses its lognormal character as turbulence isplaced by laminar gravity waves ande is dominated by themean shear, which is by this time nearly uniform in spac

We have seen that the probability distribution functifor e exhibits a dramatic change between the early andphases of the simulation. Early on, when the flow is domnated by KH billows and associated transitional flow strutures, the distribution is jagged. Later, the distribution exhits a smooth, nearly lognormal peak indicative of trandom, multiplicative character of a turbulent cascad37

We conclude that, although the Reynolds number of tsimulation is decidedly finite, the dynamics of the dissipatrange exhibit important characteristics in common with fudeveloped turbulence. In Sec. V, we will conduct detaicomparisons of dissipation range statistics from our simutions with those expected in fully developed turbulence,


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order to assess the applicability of the latter idealizationnaturally occurring flows. As preparation for this, we devoSec. IV to a discussion of the manifestly anisotropic struture of the large scales.

IV. ENERGY-CONTAINING SCALES

In this section, we describe the geometry of the largeddies in terms of the velocity anisotropy tensorb and itsinvariants~17! and~18!. We begin by examiningb as a func-tion of the vertical coordinate for a sample flow field. Wdisplayb computed as a depth-dependent tensor, i.e., byterpreting the angle brackets in~17! as representing horizontal averages~Fig. 4!.

For reference, Fig. 4~a! shows the mean temperature avelocity profiles. The velocity fields tend to be highly anistropic near the boundaries. This anisotropy is prolate in fo@III b.0, as shown in Fig. 4~d!# due to the dominance of thstreamwise component of the velocity fluctuations. Nearcenter of the domain, the degree of anisotropy is considably reduced. The vertical velocity becomes similar in manitude to the streamwise velocity, while the spanwise coponent remains smaller than the other two@Fig. 4~b!#. Thestrong negative correlation between the streamwise andtical velocity fluctuations@Fig. 4~c!# indicates that these fluctuations are configured so as to draw energy from the baground shear. The second invariant ofb remains significantlynonzero even at the center of the domain. To put theserved degree of anisotropy in perspective, we note thalaboratory studies of the return to isotropy in initially anistropic turbulence,29,38 a typicalstarting value of IIb is 0.05.That value is typical of the closest approach to isotropyour simulations. The third invariant oscillates between potive and negative values, reflecting fluctuations in the relatmagnitudes of the streamwise and vertical velocity comnents@Fig. 4~b!#.

We next examine the time-dependent behavior ofb overan entire simulation~Fig. 5!. ~Hereafter,b is defined usingaverages overVs .! Near t50, b11 is the largest of the diagonal components,b33 is somewhat smaller, andb22 is closeto 21

3. This indicates that the flow is two dimensional. Thstreamwise velocity dominates, while the spanwise velocis close to zero. During this time,b13 is negative, indicatingthe presence of a significant Reynolds stress working aga


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the mean shear to transfer energy into the growing disbance. During the 3d growth and transition phases,b11 andb33 oscillate, as do the kinetic energy components to whthey correspond@cf. Fig. 2~a!#. During this time, spanwisekinetic energy~expressed here throughb22! grows in re-sponse to secondary instability,22 although it does not represent a significant fraction of the disturbance kinetic ene(b22.21/3) until the 3d growth phase is complete and ttransition to turbulence is underway. During 3d growth atransition,b13 also oscillates, revealing a quasiperiodic echange of energy between the KH vortex and the mean flIn the initial growth phase,b13 reaches values near20.30.During the decay phase, as was seen in Fig. 2, the spanand vertical velocities are very similar in mean magnituTransition leaves the value ofb13 near20.10, and that valuesubsequently decays to zero as the turbulence ceases toenergy from the mean flow. Values ofb13 found here are

FIG. 5. Evolution of the elements and invariants of the disturbance veloanisotropy tensor,b, for run R10P1. Volume averages are taken overVs . ~a!Diagonal elements: solid line5b11 , dashed line5b22 , dotted line5b33 . ~b!Off-diagonal elements: solid line5b13 , dashed line5b12 , dotted line5b23 .The horizontal dashed line indicates the valueb13520.13 that is typical ofwall layer turbulence~Ref. 39!. ~c! Square root of the second invariant.~d!Cube root of the third invariant. In~c! and ~d!, the shaded region indicatethe range of possible values~see the text and the caption of Fig. 6 fodetails!. The lower horizontal axis shows the time in seconds. The uphorizontal axis gives the time in buoyancy periods, i.e.,tN

5*0t N(t8)dt8/2p.


r-

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d-

.

ise.

raw

comparable with the value20.13 that is typical in near-walturbulence.39

A striking aspect of this anisotropy analysis is the degto which the invariants, IIb and IIIb , adhere to the limitingvalues associated with two-dimensional and axisymmeflow @Figs. 5~c! and 5~d!#. During the 2d and 3d growthphases, IIb and IIIb have values characteristic of twodimensional flow, with IIb near its upper limit and IIIb nearits lower limit. Early in the transition phase, the invariandepart from their two-dimensional limits, moving to a staof oblate axisymmetry in which the spanwise velocity tento be smaller than the other two velocity components. Tvelocity field makes its closest approach to isotropy~as mea-sured by IIb!, at the end of the transition phase. By this poithe invariants have settled into a configuration characteriof prolate axisymmetry~with streamwise velocity dominant!,where they remain throughout the decay phase.

Figure 6 shows values of IIb and IIIb for all eight simu-lations. In general, the progression in time is counterclowise on the IIIb– IIb plane. The energy-containing scales rmain distinctly anisotropic in all cases. Also, the tendenctoward oblate axisymmetry in the transition phase and plate axisymmetry in the decay phase are present in all silations.

We have examined the form of the asymmetry of tlarge scales and how that asymmetry controls the fluxenergy from the mean shear to the turbulence. In the remder of the paper, we assess the degree to which this anropy is reflected in the structure of the smaller scalesmotion.

V. VELOCITY GRADIENTS

We turn now to an examination of the structure of eddin the dissipation range. Anisotropy in these small-scale m

ty

r

FIG. 6. Second and third invariants, computed at discrete times duringsimulations. Symbols indicate the different simulations as listed in TablThe shaded region is inaccessible. The lower limit of IIb for a given IIIbrepresents axisymmetric flow, in which the mean squares of two Cartevelocity components are equal. When IIIb,0, the third component issmaller than the other two~oblate axisymmetry!; when IIIb.0, the thirdcomponent is greater than the other two~prolate axisymmetry!. The upperlimit of II b for a given IIIb is two-dimensional flow, in which one component vanishes. Both IIb and IIIb achieve their maxima in the limit of onedimensional flow.


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tions is revealed through the vorticity and dissipation anisropy tensors and through streamwise cospectracorrelation-coefficient spectra of the velocity field. We thassess the accuracy of some popular approximations to^e&based on assumptions regarding isotropy in the dissiparange.

A. Tensor measures of anisotropy

Figure 7 shows the evolution of the vorticity anisotrotensor~19! during a typical simulation. Throughout the simlation, the vorticity field exhibits significant anisotropy. Athough the mean-square vorticity components in this reence frame approach a common value betweent52400 andt54200 s, the off-diagonal elementv13 is positive duringthis interval. As a result, the second invariant remains snificantly nonzero. IIv is, however, closer to zero during thmost-turbulent interval of the simulation than is IIb . ~Thiscomparison is not entirely fair, becausev contains the meanprofile whileb does not. If mean profiles were retained in t

FIG. 7. Evolution of the elements and invariants of the vorticity anisotrotensor,v, for run R10P1.~a! Diagonal elements: solid line5v11 , dashedline5v22 , dotted line5v33 . ~b! Off-diagonal elements: solid line5v13 ,dashed line5v12 , dotted line5v23 . ~c! Square root of the second invarian~d! Cube root of the third invariant. In~c! and ~d!, the shaded region indi-cates the range of possible values~see the caption of Fig. 6 for furthedetails!.


t-d

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r-

-

calculation ofb, the difference between IIv and IIb would begreater.! The positive value ofv13 suggests a preference fovortices aligned midway between the streamwise and vertdirections, i.e., in the direction of the principal axis of extesional strain for the mean shear.

We now look more closely at the form of the anisotropTime-averaged values of the elements in the vorticity anisropy tensor taken shortly after transition (t52400– 3600 s)are

v5F 0.023 0.001 0.104

0.001 0.001 0.001

0.104 0.001 20.025G .

~Figure 7 shows thatv is statistically stationary during thisaveraging interval.! With its dominant components residinin the ~1,3! and ~3,1! positions, this tensor is nearly propotional to the mean strain tensor:

S5F 0 0 12uz

0 0 0

12uz 0 0

G .

The eigenstructures of the two tensors are thus vsimilar. There are three eigenvalues which sum to zero:positive, one near zero, and one negative. The first eigentor ~i.e., that corresponding to the positive eigenvalue! lies inthe x–z plane at a 45° angle above thex axis. The thirdeigenvector, corresponding to the negative eigenvaluealso in thex–z plane, directed 45° below thex axis. Themiddle eigenvector is parallel to they axis. In the coordinatesystem defined by these eigenvectors, the vorticity fieldpears as three orthogonal components: the principal comnents. The strongest of the three is aligned with the extsional strain, and suggests the tilted vortices thatubiquitous in sheared turbulence.40–43 The weakest component is aligned with the compressive strain. The middle coponent represents the spanwise vorticity that dominatesflow in its early and late stages.

A crucial property of the principal component vorticitieis that the mean square of the middle component is neequal to the arithmetic average of the other two compone~and thus IIIv is near zero!. It is not clear why this should beso. Because of this property, this manifestly anisotropic flexhibits an important property in common with isotropflow: In the original reference frame~oriented with the meanflow!, the mean squares of the three components of theticity field are nearly equal@Fig. 7~a!#. In other words,knowledge of one Cartesian component of the mean sqvorticity allows one to deduce the other two components jas if the vorticity field were isotropic.

Typical evolution of the invariants ofv for the transitionand decay phases of all eight simulations is shown in F8~a! and 8~c!. To facilitate comparison between runs widifferent inherent time scales, we now measure time in buancy periods, i.e.,tN5*0

t N(t8)dt8/2p. In most cases, thepattern is similar to that seen in Fig. 7. IIv and IIIv remainclose to their one-component values~2/3 and 2/9, respectively! throughout the initial period of two-dimensiona

y


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are

theron-


FIG. 8. Invariants of the vorticity anisotropy tensocomputed at discrete times for all runs. In~a! and ~c!,the abcissa is time in buoyancy periods and symbolsas given in Table I. In~b! and ~d!, the abcissa is thebuoyancy Reynolds number. Symbol sizes indicatePrandtl number, with Pr.1 represented by the smallesymbols. The time is indicated by shading, as shownthe bar above~a! which is scaled according to the abcissa of~a!, ~c!.

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growth. The invariants then decrease rapidly during the trsition to turbulence, which occurs after 2–3 buoyancy peods. ~The time to transition may be a strong functioninitial conditions.! The subsequent phase of minimal anisropy, during which IIv

1/2 is in the range 0.1–0.2, lasts foanother 2–3 buoyancy periods. The invariants then increback to their one-component values as the flow gradurelaxes to the stable, two-dimensional end state. The oexception to this pattern occurs in the simulation R03shown with diamonds in Figs. 8~a! and 8~c!. This was theleast turbulent of our simulations, in the sense that Ri0 wastoo large ~0.16! and Re0 too small ~;2000! for vigorousturbulence to develop.

The dependence of local isotropy upon turbulence intsity is illustrated more vividly in Figs. 8~b! and 8~d!, whichshow the vorticity invariants as functions of the buoyanReynolds number. Time is indicated by symbol shaditime progression is clockwise, from the lightest shade todarkest. After the transition to turbulence is complete andRb

has attained its maximum value~the lower right-hand corneof the frame!, the evolution of the invariants is tied verclosely to the evolution ofRb . In other words, the invariant‘‘forget’’ the effects of initial conditions~and the Prandtlnumber of the fluid! and subsequently dependonly on Rb .This result strongly supports the notion that isotropy ofdissipation range is controlled primarily by the spectral seration between the smallest and largest scales of motion

The results shown in Fig. 8~b! permit us to guess at whamight occur at Reynolds numbers higher than those attain the present simulations. IIv cannot approach zero~withinthe scatter of the data! for Rb much smaller thanO(105),unless the form of the dependence changes qualitativelyRb.O(102). On the other hand, it is easy to imagine thatvcould asymptote to some small but nonzero value asRb

→`.


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We now return to a consideration of the nature of tanisotropy exhibited by the vorticity field after the transitioto turbulence. In the slightly idealized picture describedconnection with Fig. 7, the invariant IIv is proportional tov13, while IIIv is zero. In fact, the middle eigenvalue ofv isnot quite zero, but rather fluctuates about that value. Tcauses IIIv to fluctuate about zero in a similar manner. Figu8~d! reveals a common tendency for IIIv to become negativenear the end of the turbulent regime, just before the flstarts to relax towards the two-dimensional configuratiThe sign change is associated with values ofRb close to 10.

This intriguing regularity in IIIv can be understood in aqualitative sense via closer examination of Fig. 7~a!. Shortlyafter transition, whenRb is large, flows tend to develop postive III v as the middle principal component of vorticit~which is generally close to the spanwise direction! becomessmaller than its isotropic value. During this phase, the vticity field is slightly prolate, being dominated by its streamwise ~x! component. At later times, streamwise and vertivorticity decay, so that spanwise vorticity increases in retive magnitude as the flow relaxes toward the twdimensional end state. The latter state is also prolate, sthe vorticity is dominated by its spanwise component. Btween these two phases, when the turbulence is just bening to decay, there is inevitably an intermediate statewhich streamwise and spanwise vorticity are similar in manitude, while vertical vorticity is considerably smaller. Thintermediate state is oblate, and is therefore characterizeIII v,0.

The invariants IId and IIId behave similarly to theircounterparts IIv and IIIv in many respects~Fig. 9!. Both in-variants are large, indicating the dominance of a singlegenvector of the anisotropy tensor, at early and late timThe smallest values of IId are attained in the interval 3,tN,4, and are somewhat smaller than the smallest va


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are

theron


FIG. 9. Invariants of the dissipation anisotropy tensocomputed at discrete times for all runs. In~a! and ~c!,the abcissa is time in buoyancy periods and symbolsas given in Table I. In~b! and ~d!, the abcissa is thebuoyancy Reynolds number. Symbol sizes indicatePrandtl number, with Pr.1 represented by the smallesymbols. The time is indicated by shading, as shownthe bar above~a!.

ot

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ldalda

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withlestup-n a

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is

of II v . The third invariant, IIId , displays very little tendencyto take negative values during the interval of minimal anisropy. Like the vorticity invariants, IId and IIId are accuratelypredicted by the buoyancy Reynolds number once the tsition to turbulence is complete@Figs. 9~b! and 9~d!#.

We turn now to an examination of the dependence ofdissipation anisotropy invariant IId on parameters expressinthe relative strength of the large-scale shear. Corrsin27 andothers have suggested that the dissipation subrange shoueffectively insulated from anisotropy due to the large-scshear if Rs is sufficiently large. In contrast, Durbin anSpeziale7 have argued that the dissipation scales must remanisotropic regardless ofRs , except possibly in the limit thathe shear number,S* , goes to zero. In the present results,Rs

evolves in a manner similar toRb , growing to a maximumthen decaying@cf. Fig. 2~b!#, while S* decreases from starting values of order 102 to asymptotic values not far from thvalue 6 found previously in both laboratory and numeriexperiments.7 Rs proves to be an excellent predictor of disipation anisotropy~as quantified by IId! in the present ex-


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in

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periments@Fig. 10~a!#, whereas IId shows very little correla-tion with S* @Fig. 10~b!#.

The present results are not necessarily inconsistentthe predictions of Durbin and Speziale. Even the smalvalues of IId achieved here indicate nonzero anisotropy. Spose that the present simulations could be performed ocomputer of arbitrary size, so that the limitRs→` could beapproached. It seems likely thatS* would remain of orderunity or larger in such simulations@Fig. 10~b!#. ~In fact, in astate of production-dissipation balance,S* must remainlarger than unity.7! The results of Durbin and Speziale therfore predict that the small scales would remain anisotropicRs→`. Although Fig. 10~a! suggests otherwise, it does nrule out the possibility that IId would asymptote to somesmall, but nonzero value. Another possibility suggestedthe present results is that IId asymptotes to zero in the higReynolds number limit, but IIv remains nonzero@Fig. 8~b!#.This would indicate anisotropy in the vorticity field thatnot reflected in the dissipation tensor.

ot-

eron

FIG. 10. The second invariant of the dissipation anisropy tensor vs shear Reynolds number~a! and shearnumber ~b!, for all runs. Symbol sizes indicate thPrandtl number, with Pr.1 represented by the smallesymbols. The time is indicated by shading, as shownthe bar above Fig. 9~a!.


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B. Spectral measures of anisotropy

We now examine anisotropy over the entire rangelength scales as revealed by streamwise Fourier spectrthe velocity field~23! and ~24!. Figure 11~a! shows stream-wise spectra of kinetic energy production and~approximate!dissipation for a strongly turbulent flow. This example wtaken from simulation R16P1 att53111 s, shortly after transition. It is clear that production and dissipation are cotrolled by motions in different spectral ranges. Productoccurs mainly in the spectral rangek<kC, where eddies arestrongly deformed by the mean shear.~Wave numbers arescaled by the Kolmogorov wavenumber,kK51/LK .! Thedissipation range begins at about 0.1kK , as is typical in tur-bulent flow.39 The Ozmidov wave number, below which edies are strongly damped by stratification, is located nearsmall-scale end of the production range. Regions of negaproduction indicate that the large scales are still substantwavelike.

In Fig. 11~b!, we display the component energy specE118 , E22, andE33, scaled byk1

5/3 so as to test for the presence of an inertial subrange. At low wave numbers, the thcurves differ significantly; energy associated with spanwmotions ~dashed curve! is by far the smallest. The threcurves converge neark50.05kK and remain nearly horizontal until k'0.20kK , a range of about one half decade. Withthis range, the spectral level is consistent with a value 1.5the Kolmogorov constant. The ratios of the transverse sptra to the longitudinal spectrum@Fig. 11~c!# compare wellwith the theoretical value of 4/3 in the inertial range~cf. Fig.10 of Gargettet al.16!. At higher wave numbers, the component spectra roll off@Fig. 11~b!# while remaining nearlyequal. In summary, Fig. 11 indicates a highly anisotroproduction range at large scales, a nearly isotropic disstion range at small scales, and an intervening inertial raextending for about one half decade.

The flow shown in Fig. 11 represents one of the mstrongly turbulent states achieved in these simulations, wRb5100. We turn next to an examination of a more typiccase~Fig. 12!, taken from the same simulation~R16P1! longafter transition (t56222 s). By this time,Rb has decreasedto 25. This flow exhibits distinct production and dissipatiranges@Fig. 12~a!#, though they are less well separated thin the previous case. The component spectra are nearly ein the dissipation range, which begins neark50.1kK as be-fore. Near this wave number, the spectral slope is clos25/3, the amplitude is consistent withCK51.5, and thetransverse/longitudinal ratios@Fig. 12~c!# are not far differentfrom 4/3. However, this inertial subrange-like behavior donot extend over any significant region of the spectrum,rather appears only tangentially at the beginning of thesipation range. Thus, the lowerRb case shown in Fig. 12exhibits a mildly anisotropic production range and a neaisotropic dissipation range, separated by the merest hint oinertial range.

A more stringent criterion for isotropy thanE118 5E22

5E33 is the vanishing of the Reynolds stress spectraE12,E23, andE13. In our simulations, the first two of these vaish in the dissipation range as expected.E13, however, does


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not vanish but rather changes sign; over most of the disstion range, we find a small positive Reynolds stress. Tsign change can be seen from the production spectrumFig. 12~a!, which is proportional toE13. The sign of theReynolds stress in the dissipation range is shown mclearly by the correlation-coefficient spectrumC13 @Fig.12~c!#.

This correlation between streamwise and vertical velity fluctuations in the dissipation range suggests thatstraining effects of the large-scale shear are felt even atsmallest scales of motion. Similar sign reversals have bnoted previously in results from both numericsimulations44,45 and laboratory experiments.6,44,46–48 Theseexperiments have covered a range ofRs extending from lessthan 100 to nearly 3000. Comparison between these varstudies is complicated by differences in the form of the sptra used~E13, k1E13, andC13, in order of increasing sensitivity at small scales!, but it seems that the effect is lespronounced at higherRs . For example, the DNS experi

FIG. 11. Streamwise spectra of velocity fields for run R16P1 at

53111 s, averaged over7

16Lz,z,9

16Lz . ~This layer represents the centraturbulent region in which the dissipation rate is nearly uniform.! Wave num-bers are scaled by the Kolmogorov wave number,kK51/LK . ~a! Productionand dissipation spectra in area-preserving form. Closed curve: produspectrum2k13SE13 ; dashed curve: dissipation spectrumk1

332.5n(E1181E221E33). Both spectra are scaled by the mean dissipation rate. Notethis form of the dissipation spectrum is quantitatively valid only in tisotropic limit. Symbols at the top of the frame indicate the Ozmidov sc~O! and the Corrsin scale~C!. ~b! Component kinetic energy spectra. Solcurve:E118 , dashed curve:E22 , dotted curve:E33 . Each is scaled by 1/2 torepresent kinetic energy and by^e&2/3k1

25/3 to test for the presence of aninertial subrange.E11 is transformed to its equivalent transverse spectruE118 , so that coincidence of the three curves is consistent with isotropy.horizontal line shows the value 0.33, which is equivalent to KolmogoconstantCK50.33355/1251.5. ~c! Ratios of the two transverse componespectra to the longitudinal spectrum. Dashed curve:E22 /E11 ; dotted curve:E33 /E11 .


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ith

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thegin-

on-w

ed by

ctra

ing-

dis-t of

alesthe

dr.est

alousnceap-

adeuesrticalfectcon-ingfect

theen-ith

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ntl.tru


ments of Holtet al.45 employedRs of order 100 and less. Inthose results, the sign reversal is visible in theE13 spectrum,even though the amplitude of that spectrum is charactercally small at high wave numbers. The sign reversal is mvisible in the results of Antoniaet al.44 ~derived from DNSand small-scale laboratory experiments! and Shafi andAntonia48 ~from laboratory experiments!, who used similarlysmall Rs but displayed the compensated spectrumk1E13.Much higher values ofRs have been reached in large-scawind tunnel experiments. In these cases, the sign reversaonly be discerned in the highly sensitiveC13 spectrum, and isoften obscured by underresolution of the dissipation ranThe earliest such results were presented by Mestayer,46 whoachievedRs;2800 in a wind tunnel. The sign reversal in hresults is slight, but visible in theC13 spectrum@his Fig.13~a!#. Similarly, wind tunnel results of Saddoughi anVeeravalli,6 with Rs;1000– 2500, show a hint of a revers

FIG. 12. Streamwise spectra of velocity fields for run R16P1 at

56222 s, averaged over7

16Lz,z,9

16Lz . Wave numbers are scaled by thKolmogorov wave number,kK51/LK . ~a! Production and dissipation spectra in area-preserving form. Closed curve: production spectrum2k1

3SE13 ; dashed curve: dissipation spectrumk1332.5n(E118 1E221E33).

Both spectra are scaled by the mean dissipation rate. Note that this forthe dissipation spectrum is quantitatively valid only in the isotropic limSymbols at the top of the frame indicate the Ozmidov scale~O! and theCorrsin scale~C!. ~b! Component kinetic energy spectra. Solid curve:E118 ;dashed curve:E22 ; dotted curve:E33 . Each is scaled by 1/2 to represekinetic energy and by e&2/3k1

25/3 to test for the presence of an inertiasubrange. Coincidence of the three curves is consistent with isotropy~c!Ratios of the two transverse component spectra to the longitudinal specDashed curve:E22 /E11 ; dotted curve:E33 /E11 . ~d! Correlation-coefficientspectrumC13 .


ti-e

an

e.

in C13 @their Figs. 20~b! and 20~c!#. Antonia and Raupach47

observed a sign reversal inC13 in a wind tunnel withRs

ranging from 300 to 1500. In each of these cases, thereversal occurred atk1 /kK;0.1, the beginning of the dissipation subrange. Saddoughi and Veeravalli6 also investigatedcases with much higherRs @their Figs. 20~a! and 20~c!# inwhich there is no evidence of sign reversal, but this resuinconclusive since the dissipation range was not wellsolved. Thus, the evidence to date indicates that shearedbulence withRs less than a few thousand exhibits positicorrelation between streamwise and cross-stream velocomponents in the dissipation range.

The results shown in Fig. 12 are entirely consistent wthose quoted earlier. Our values ofRs are toward the low endof the range of previous investigations, and the sign chais visible even in the relatively insensitivek1E13 form of thespectrum@Fig. 12~a!#. In the C13 spectrum@Fig. 12~c!#, thepositive correlation at small scales is more obvious. As inprevious investigations, the sign reversal occurs at the bening of the dissipation range.

We now test the generality of the above-suggested cclusions by examining results from several turbulent flosolutions ~Fig. 13!. The zero crossing, at which point thcospectrum becomes positive, appears to be determinethe Kolmogorov scale. Specifically, it occurs atk1 /kK

;0.1– 0.2, the beginning of the dissipation range. Speplotted againstk1 /kC, where kC5(S3/^e&)1/251/LC showno such consistency in the location of the zero cross@compare Figs. 13~a! and 13~b!#. The fact that the wave number of the zero crossing scales withkK and not withkC sug-gests that the anomalous correlation is a property of thesipation subrange, and its spectral extent is independenstraining by large-scale motions. In contrast, theamplitudeof the anomalous correlation is clearly a function ofRs

~shown in Fig. 13 by symbol size!. When Rs is large, i.e.,there is a wide spectral separation between the large scand the dissipation range, the correlation is small. AtlargestRs attained here,C13 peaked near 0.15.~In the muchlarger Rs attained in the wind tunnel experiments citeabove,C13 tended to peak at values near 0.05, or smalle!As the turbulence decays, however, the effect of the largeddies on the dissipation range increases, and the anomcorrelation also increases. In the late stages of turbuledecay, the peak in the correlation-coefficient spectrumproaches unity.

This agreement between results from experiments musing a wide range of numerical and laboratory techniqsuggests that the correlation between streamwise and vevelocities at large wave numbers is neither a numerical efnor a measurement error. The correlation appears to betrolled by a combination of viscous and large-scale straineffects. The degree of correlation is determined by the efof large-scale straining on the dissipation scales, whereasspectral range over which the correlation occurs is indepdent of the large-scale straining, and coincides closely wthe dissipation range.

Toward a physical interpretation of the anomalous crelation, we note that the deformation of spanwise vorticby the large-scale shear is expected to lead to expansio

of

m.


nnca

llstania

tha

echat

i

metheofby

e

lve

wiseely

tl

per-itynrrorsonpo-theled.thent

n-iveda

he

ultslasIn

tedare

ot-

ision.

try

ns


the direction of the principal axis of extensional strain~andcompression in the perpendicular direction!. The resultingdeformed eddies will exhibit a positive correlation betweeuandw. At large scales, this effect competes with the tendefor eddies with the opposite tilt to be amplified by sheinstability. Due to the finite bandwidth of that instability25

the mean shear cannot drive unstable vortices on the smascales.~Shear associated with larger vortices leads to inbility on small scales, but those motions lack the highly uform orientation of the mean flow, and are therefore lessto induce coherent anisotropy.! Another competing influenceis the vortices that arise due to vortex stretching alongextensional strain of the mean flow. Such vortices are chacterized by negative correlation betweenu and w. In con-clusion, it is not difficult to see why motions with positivC13 exist, but it is also easy to identify mechanisms whifavor the opposite correlation and which could dominsome areas of the spectrum.

C. Estimates of the kinetic energy dissipation rate

The kinetic energy dissipation rate is often estimatedterms of just one of its terms, i.e.,

ê&'Ci j ûi , j82& ~26!

FIG. 13. Correlation-coefficient coefficient spectraC13(k1) vs streamwisewave number scaled using~a! the Kolmogorov wave numberkK

5(ê&/n3)1/4, and~b! the Corrsin wave numberkC5(ê&/S3)1/2. Data per-

tain to six typical realizations of turbulence, averaged over7

16Lz,z

,9

16Lz . Symbol size is proportional tokK /kC5Rs3/4 .


yr

est-

-pt

er-

e

n

~no summation oni, j!, in which the constantCi j is equal to15 if i 5 j , 15/2 otherwise. These approximations becoexact in isotropic flow, and are expected to be accurate ifdissipation subrange is isotropic. Figure 14 shows eachthese approximations, applied to our data and normalizedthe true value of e&. All nine of these approximations araccurate to within 10%–20% at the highest values ofRb

attained here~near 100!. At lower Rb , however, such ap-proximations may err by an order of magnitude or more.

The poorest approximations are those that invostreamwise derivatives@Figs. 14~a!, 14~d!, and 14~g!#. Thebackground shear tends to elongate eddies in the streamdirection, so that the corresponding derivatives are relativsmall. The best approximations are based on]u8/]y and]v8/]z @Figs. 14~b! and 14~f!#. The only approximation thaconsistently overestimatese& is that based on the verticashear of the streamwise velocity@Fig. 14~c!#. Although thebackground shear has been removed, this correspondingturbation shear is significantly larger than the other velocgradients. Errors ine& due to anisotropy in the dissipatiorange are particularly dangerous because they lead to ein Rb ~since the latter is itself proportional to the dissipatirate!. In other words, the quantity we use to assess thetential for anisotropy is only estimated accurately whenturbulence is isotropic! The data analyst may thus be misas to the degree of anisotropy to be expected in the flow

Comparisons between the relative contributions ofvarious terms to e& are in excellent qualitative agreemewith the results of Itsweireet al.17 ~cf. their Table I, casesRi50.075 and Ri50.21!. The latter were obtained from DNSof turbulence in a uniformly sheared and stratified enviroment. The present results also agree well with those derby Gargettet al.16 from spectra of turbulence occurring infjord. On the basis of those data, Gargettet al. concludedthat ê& can be reliably estimated from a single term of tform ~26! when Rb.200. While that value ofRb is not at-tained in the present simulations, extrapolation of the resshown in Fig. 14 suggests that all of the isotropic formuwill be accurate to within about 10% above that value.practice, many of the isotropic approximations evaluahere are accurate enough that errors due to anisotropylikely to be smaller than other sources of uncertainty forRb

as small asO(10).We next examine the ratiosK1 , K2 , K3 , andK4 ~22!,

which have proved useful in the characterization of anisropy in laboratory data.31 In isotropic flow, all of these ratiosshould be unity. An alternative idealization for shear flowthe the state of axisymmetry about the streamwise directFlows exhibiting that symmetry haveK15K2 andK35K4 .The results displayed in Fig. 15 suggest that axisymmemay be a useful approximation for these flows at lowRb .

In Fig. 16, we test the accuracy of four approximatioto ê&, each of which is exact in axisymmetric flow:31

ê1&5n@ 53û,x8

2&12û,y82&12^w,x8

2&1 83^w,y8

2&#, ~27!

ê2&5n@2û,x82&12û,z8

2&12^w,x82&18^w,z8

2&#, ~28!

ê3&5n@ 53û,x8

2&12û,z82&12^v ,x8

2&1 83^v ,z8

2&#, ~29!


-

tl

dve

eron


FIG. 14. Contributions to the kineticenergy dissipation rate,^e&, from eachof the mean squared velocity derivatives, averaged overVs and expressedas a fraction of the true value of^e&.Each ratio is unity in isotropic flow.Symbol sizes indicate the Prandnumber, with Pr.1 represented by thesmaller symbols. The time is indicateby shading, as shown on the bar aboFig. 9~a!. For clarity, data from t,1800 s ~i.e., preturbulent cases! areexcluded.

FIG. 15. Isotropy ratios as defined in~22!. Each ratio isunity in isotropic flow. Symbol sizes indicate thPrandtl number, with Pr.1 represented by the smallesymbols. The time is indicated by shading, as shownthe bar above Fig. 9~a!.

Downloaded 21 Jul 2010 to 128.193.70.43. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

a-

-in-

g,


FIG. 16. Approximations to the kinetic energy dissiption rate, ê& as defined in~27!–~30!, expressed as afraction of ê&. Each ratio is unity when flow is axisymmetric about the streamwise direction. Symbol sizesdicate the Prandtl number, with Pr.1 represented bythe smaller symbols. The time is indicated by shadinas shown on the bar above Fig. 9~a!.

th

eb-dnt

daron

lt

ax

nntiene

. AgtuaaI

ientsnalheretureisenalthell inee-ot-lts

at

sor,

ê4&5n@2û,x82&12û,y8

2&12^v ,x82&18^v ,y8

2&#. ~30!

All formulas are comparable in accuracy to the best ofisotropic [email protected]., Figs. 14~b!, 14~e!, 14~f!, and14~i!#.

These results compare well with similar analyses pformed on data from a laboratory experiment on duct flowAntoniaet al.30 ~cf. their Figs. 9 and 12!. In that case, departures from isotropy occurred not as the turbulence decayetime ~as in the present case! but rather as the measuremelocation approached the wall.K1 and K2 were greater thanunity while K3 and K4 were less than unity. George anHussein31 list numerous other experiments in which similresults have been obtained. As in the present case, Antet al.30 found thate1 underestimatese& ande2 overestimatesê& as the flow departs from isotropy. Antoniaet al. alsocomputed the isotropic approximation based on]u8/]x andfound it to be a considerably poorer approximation toê&than the axisymmetric approximations. Our own resuagree with this finding@Fig. 14~a!#, but show in addition thatother formulas based on isotropy do just as well as thesymmetric approximations.

VI. SCALAR GRADIENTS

We complete our analyses with an investigation of aisotropy in the scalar gradient field. In the most-turbuleregimes, the scalar gradient is isotropic except for a negacorrelation between the streamwise and vertical compon~Fig. 17!. This indicates a preference for gradients alignwith the principal compressional strain of the mean shearin the case of the vorticity field, the orientation and strenof the anisotropic structures is such that the root mean sqcomponents of the temperature gradient vector remain neequal in a reference frame aligned with the mean flow.other words, the mean squared gradient in the spanwise


e

r-y

in

ia

s

i-

-tvets

ds

hre

rlyndi-

rection is close to the average of the mean squared gradin the orthogonal directions defined by the compressioand extensional eigenvectors of the mean strain tensor. Tis, however, an important difference between the temperagradient and vorticity fields in this respect: The spanwvorticity component is associated with the two-dimensioflow that dominates in weakly turbulent regimes, whereasspanwise component of the temperature gradient is smathose regimes, and is instead associated with fully thrdimensional flow. Note also that the phase in which anisropy is smallest is very brief in comparison with resufound for the velocity gradients.

The variance of the scalar gradient is relatively smallthe center of the turbulent layer@Fig. 18~a!#. This reflects the

FIG. 17. Evolution of the elements of the scalar gradient anisotropy tenG, for run R10P1.~a! Diagonal elements: solid line5G11 , dashed line5G22 , dotted line5G33 . ~b! Off-diagonal elements: solid line5G13 ,dashed line5G12 , dotted line5G23 .


y

ice

la

rtei

nttur-n-rdi-

oniso-

ari-ofhen

lyve

anipa-

s to

rueum-

a-hatrre-

pyall,theults,red

nt

p

.on


fact that scalar mixing at the center of a stratified shear latends to proceed more rapidly than momentum mixing.2,49

As has been found in previous investigations,50–52 the skew-ness of the streamwise temperature gradient is near21,while that of the vertical temperature gradient is near11@Fig. 18~b!#. This is another expression of the anisotropstructures imposed by the mean strain acting on the mscalar gradient.

The minimum value of the second invariant of the scagradient anisotropy tensor, IIG , is considerably larger thanthose of IIv and IId @cf. Figs. 19~a!, 8~a!, 9~a!#, and the inter-val over which that minimum is attained is relatively shoas was seen in the single case shown in Fig. 17. This r

FIG. 18. Profiles of statistical moments of the temperature gradient comnents, taken over horizontal planes, for run R10P1,t54242 s.~a! Standarddeviations, normalized by the initial scalar fluctuation and length scales~b!Skewnesses. In both frames, solid, dashed, and dotted curves correspu ,x , u ,y , andu ,z , respectively.


er

an

r

,n-

forces the previous finding8,53 that the scalar field is intrinsi-cally less isotropic than the velocity field. The third invariaof G takes both positive and negative values during thebulent phase@Fig. 19~d!#. The dependence of the scalar aisotropy invariants uponRb displays considerably greatescatter than the corresponding results for the velocity graents @cf. Figs. 19~b!, 19~e!, 8~b!, 8~d!, 9~b!, and 9~d!#. Notsurprisingly, this scatter reveals a significant dependencePrandtl number, with high-Pr cases tending to be moretropic than low-Pr cases for a givenRb . This is becausehigh-Pr flows feature enhanced small-scale temperature vance, which is relatively immune to the aligning effectsgravity. This Prandtl number dependence is removed wone plots the invariants against the Cox number@Figs. 19~c!and 19~f!#, but the overall degree of scatter is only slightreduced~and preturbulent cases now lie far from the cursuggested by the turbulent cases!.

We now test isotropic approximations to the mesquared scalar gradient, to which the scalar variance disstion rate is proportional viax52ku“u8u2. In isotropic flow,the three Cartesian components ofu“u8u2 are equal, so that

3û ,x82&53û ,y8

2&53û ,z82&5u“u8u2. ~31!

Therefore, measurement of a single component sufficedetermine the mean squared gradient~and hencex!.

In Fig. 20, we display averages overVs of the threeisotropic approximations, expressed as ratios of the tvalue of the mean squared gradient. At the highest Cox nbers attained in these simulations, the isotropic approximtions are accurate to within a few tens of percent. Note tthese isotropic estimates are unaffected by the strong colation between]u8/]x and ]u8/]z shown in Fig. 17. Inweaker turbulence, the effects of departures from isotroare clearly evident. Horizontal gradients are relatively smwhile vertical gradients are relatively large, even thoughmean vertical gradient is subtracted out. From these reswe conclude that isotropic estimates of the mean squascalar gradient~and thus ofx! based on a single compone

o-

d to

-.-

h-,

FIG. 19. Invariants of the scalar gradient anisotropy tensor for all runsAbcissae are the time in buoyancy periods ~a!, ~d!, the buoyancy Reynoldsnumber~b!, ~e!, and the Cox number~c!, ~f!. In ~b!, ~c!, ~e!, and~f!, symbolsizes indicate the Prandtl number, witPr.1 represented by the smaller symbols. The time is indicated by shadingas shown on the bar above~a!.


ch

r--s

-,


FIG. 20. Contributions to the meansquared temperature gradient fromeach of its Cartesian components. Earatio is unity in isotropic flow.~a!, ~b!,~c! The streamwise, spanwise, and vetical components, respectively. The abcissa is the Cox number. Symbol sizeindicate the Prandtl number, withPr.1 represented by the smaller symbols. The time is indicated by shadingas shown on the bar above Fig. 19~a!.

ox.mb-teea

eessu

iowsn

ur

ttonhv

wp

ityhop

t-idatithe

s-ey

pea

la-ntski-eanisrre-

with

ateof

m-s,

thatay

e

thenot

ceto

tudeccu-

to

in-ap-ex-the

rethe59.ineal

ed,

ed

are accurate to within a few tens of percent when C.O(102), but may be highly misleading at smaller CoEstimates based on spanwise gradients seem to be therobust in this respect. Laboratory experimentsThoroddsen and Van Atta54 on inhomogeneous grid turbulence reveal similar results; streamwise scalar gradientsto be considerably smaller than spanwise and cross-strgradients.

VII. SUMMARY

Turbulence in a stratified shear layer at moderate Rnolds number exhibits significant anisotropy on all scaldue to the straining action of the mean shear and thepression of vertical motions by buoyancy forces. The flowstrongly anisotropic during the preturbulent phase of the flevolution. Anisotropy decreases dramatically with the trantion to turbulence, then increases again as the turbuledecays and the flow relaxes to a stable, parallel configtion.

Large-scale anisotropy of the velocity field is such asgenerate Reynolds stresses which extract energy frombackground shear flow. During the turbulent phase, the vticity field exhibits anisotropy in the form of a correlatiobetween streamwise and vertical vorticity components. Tanisotropy leaves the three Cartesian components of theticity ~in a reference frame aligned with the mean flo!nearly equal in magnitude, so that an estimate of enstrobased on knowledge of a single component and the~incor-rect! assumption of isotropy would give a valid result.

After the onset of turbulence, anisotropy of the velocgradients is controlled entirely by the Reynolds number. Tdependence is illustrated by the tight collapse of anisotrparameters when plotted againstRb andRs @Figs. 8~b!, 8~d!,9~b!, 9~d!, and 10~a!#. This result is one of the central oucomes of the present study, and strongly supports thethat small-scale anisotropy depends on the spectral seption between the largest and smallest scales. Extrapolafrom presently accessible Reynolds numbers suggeststhe vorticity field may become isotropic at sufficiently largvalues@Rb;O(105)#. However, we cannot rule out the posibility that some anisotropy persists even in the high Rnolds number limit.

Scaled energy spectra~Figs. 11 and 12! indicate anisot-ropy in the production range, but are consistent with isotroin the dissipation range. At the highest Reynolds numbachieved here, the production and dissipation ranges


x

osty

ndm

y-,p-s

i-cea-

oher-

isor-

hy

isy

eara-onat

-

yrsre

separated by an inertial range of about one half decade~Fig.11!. Correlation-coefficient spectra reveal a positive corretion between streamwise and vertical velocity componewithin the dissipation range, indicating a net transfer ofnetic energy from fluctuations on these scales to the mflow. The spectral range in which this correlation is foundindependent of the mean shear, but the strength of the colation is controlled by the mean shear and decreasesincreasing shear Reynolds number.

Estimates of the turbulent kinetic energy dissipation rbased on a single velocity gradient and the assumptionisotropy are generally accurate for buoyancy Reynolds nubers greater thanO(102). Estimates based on certain termsuch as]w/]z, are valid down to much lower values ofRb .In terms of oceanic observations, these results suggestvertical profilers equipped to measure vertical velocity mprovide particularly good estimates ofê&, while horizontalprofiling in the direction of the mean current is likely to givserious underestimates ofê& unlessRb.O(102). Estimatesof ê& based on the assumption of axisymmetry aboutstreamwise direction are more accurate than some, butall, of the isotropic estimates tested.

Anisotropy of the scalar gradient indicates a preferenfor isoscalar surfaces tilted in the direction perpendicularthe mean compressional strain. Estimates of the magniof the scalar gradient based on a single component are arate for Cox numbers greater thanO(102). SinceRb5100and Cox5100 are near the lower end of values consideredbe significant in geophysical flows,24 the effects consideredhere do not in general present a serious problem for theterpretation of field observations. However, the use ofproximate dissipation rates in the analysis of laboratoryperiments should be done with careful attention topotential for error due to small-scale anisotropy.

ACKNOWLEDGMENTS

Valuable critiques of early versions of this paper weprovided by L. Mahrt. This research was supported byNational Science Foundation under Grant No. OCE95213Computations were performed on the Connection MachCM5 facility at Oregon State University’s EnvironmentComputing Center.

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anisotropy of turbulence in stably stratiﬁed mixing layers

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