Numerical study of large-scale vorticity generation in shear-flow turbulence

2.2 MB·English

by  Petri J. Käpylä

Numerical study oflarge-scalevorticitygeneration inshear-flow turbulence
Petri J. Ka¨pyla¨
Observatory, Ta¨htitorninma¨ki (PO Box 14), FI-00014 University of Helsinki, Finland∗
Dhrubaditya Mitra
Astronomy unit, School of Mathematical Sciences, Queen Mary,
University of London, Mile End Road, London E1 4NS, UK†
Axel Brandenburg
NORDITA, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden‡
(Dated:)
Simulationsofstochastically forcedshear-flowturbulenceinashearing-periodic domainareusedtostudy
9
thespontaneousgenerationoflarge-scaleflowpatternsinthedirectionperpendiculartotheplaneoftheshear.
0
Basedonananalysisoftheresultinglarge-scalevelocitycorrelationsitisarguedthatthemechanismbehind
0
thisphenomenoncouldbethemean-vorticitydynamoeffectpioneeredbyElperin,Kleeorin,andRogachevskii
2
in2003(Phys.Rev. E68,016311). Thiseffectisbasedontheanisotropyoftheeddyviscositytensor. Oneof

n itscomponentsmaybeabletoreplenishcross-streammeanflowsbyactinguponthestreamwisecomponentof
a themeanflow. Shear,inturn,closestheloopbyactinguponthecross-streammeanflowtoproducestronger
J streamwisemeanflows. Thediagonalcomponent oftheeddyviscosityisfoundtobeoftheorderoftherms

2 turbulentvelocitydividedbythewavenumberoftheenergy-carryingeddies.

] PACSnumbers:PACSNumbers:47.27.tb,47.27.ek,95.30.Lz
h
p
-
o I. INTRODUCTION sor was found to be incompatible with that required for the
r shear–currentdynamo[12]. Ontheotherhand,theanalogous
t
s hydrodynamic effect has not yet been explored in sufficient
Theimperfectanalogybetweentheinductionequationand
a detail. This effect may explain the generation of large-scale
[ the vorticity equation has always raised questions regarding
vorticityinhomogeneousshear-flowturbulenceandwasfirst
theextentofthisanalogy. Whileitiswell-knownthattheav-
2 studied analytically in a seminalpaper by Elperin, Kleeorin,
eragedinductionequationforthemeanmagneticfieldadmits
v andRogachevskii[14]. Severalrecentstudiesdiscussnumer-
3 self-excitedsolutionsfora turbulentflowwith helicity,anal- icalevidenceforthespontaneousformationofmeanvorticity
3 ogoussolutionsto the averagedvorticity equation only exist [11, 12, 13]. In those papers the main objective is to study
8 in the compressible case [1, 2]. An exception is the case of
thegenerationoflarge-scalemagneticfieldsbyshear-flowtur-
0 flowsthataredrivenbyanon-Galileaninvariantforcingfunc-
bulence,whilethesimultaneousgenerationofmeanvorticity
.
0 tion, which can give rise to the so-called anisotropic kinetic was merely an additional(but interesting) complication. On
1 alphaeffect[3,4,5,6]. Thiseffectproducesmeanflowsthat
theotherhand,inviewofthedisappointingexperiencewhen
8 are helical and of Beltrami type. Another example of mean
tryingtoverifytheoperationoftheshear–currentdynamous-
0 flowgenerationistheΛeffect[7,8],wherebylarge-scalenon-
ing the test-field method, one should be careful in view of
:
v uniformflowscanbe producedin rotatinganisotropicturbu- earliernegativeresults[15]concerningboththeshear–current
i lence.
X effectandthemean-vorticitydynamoeffect. Theaimofthis
Inthe last fewyearsanotherexamplehasemerged,where paperisthereforetodiscussturbulentshearflowsimulations
r
a theanalogybetweenvorticityandinductionequationsismore withoutmagneticfieldsinordertodemonstratetheexistence
striking. This example applies to the case of shear-flow tur- of the mean-vorticity dynamo and to analyze its connection
bulence. Infact,ithasbeenarguedthatlarge-scalemagnetic withtheeddyviscositytensorinmoredetail.
fieldgenerationispossibleviatheshear–currenteffectthatre-
Followingearlierwork[11,12,13],periodicboundarycon-
sultsfromnon-vanishingoff-diagonalcomponentsofthetur-
ditionsareusedinthe streamwisedirectionandinthe direc-
bulentmagneticdiffusivitytensor[9,10]. Thiseffectpredicts
tion perpendicular to the plane of the shear, while shearing-
large-scalefieldgenerationinhomogeneousshear-flowturbu-
periodicboundaryconditionsareusedinthecross-streamdi-
lencewithnon-helicaldriving,whichhasindeedbeenseenin
rection. Thismeansthatmassandmeanmomentumarecon-
severalsimulations[11, 12, 13]. However,thereisthe prob-
served.Furthermore,ifalarge-scaleflowemerges,itwillalso
lem that, according to the test-field method, the sign of the
be periodic correspondingto a simple sine wave. The mean
relevantcomponentoftheturbulentmagneticdiffusivityten-
vorticityisthereforealsoalong-wavelengthsinewave. How-
ever, although the original analysis was based on mean vor-
ticity, we discuss in the followingmainly the mean velocity,
becausethecorrespondingequationsaresimplerandmorein-
∗Electronicaddress:petri.kapyla@helsinki.fi
tuitive.
†Electronicaddress:[email protected]
‡Electronicaddress:[email protected] For a proper analysis of the hydrodynamicmean-vorticity
2
dynamoeffectonewouldneedtoproceedanalogouslytothe scale,2π/kf,isreferredtoastheenergy-carryingscaleofthe
hydromagneticcasewhereitwaspossibletodetermineallrel- turbulence. Moreover,thetime dependenceoff isdesigned
evant components of the turbulent magnetic diffusivity ten- tomimicδ-correlation,whichisasimpleandcommonlyused
sor using the test-field method. One would then need to de- formofrandomdriving[17].
termineallrelevantcomponentsoftheeddyviscositytensor. Therearetwoimportantdimensionlesscontrolparameters,
However, in the absence of a properlydeveloped“test-flow” theReynoldsnumberReandtheshearparameterSh,
methodforhydrodynamics,wemustresorttomoreprimitive
measuresforestimatingcomponentsoftheeddyviscosityten- Re=urms/(νkf), Sh=S/(urmskf), (4)
sor. Using decaycalculationsofa large-scalevelocitystruc-
ture,itwasfoundthateddyviscosity,νt, andturbulentmag- that quantify the intensity of turbulence and shear, respec-
neticdiffusivity,ηt,areapproximatelyequal,i.e.νt ≈ηt,and tively.WenotethatthevaluesofReandShcannotbechosen
around(0.8...0.9)×urms/kf [16]. Here,kf isthewavenum- aprioriduetothestrongeffectthatthevorticitydynamohas
ber correspondingto the scale of the energy-carryingeddies on the value of urms in the saturated state. Thus we always
and urms is the rmsvelocity of the turbulence. On the other refertovaluesofurms,Re,andShthatapplytothesituation
hand,amoreaccuratedeterminationofηt ledrecentlytothe where the vorticity dynamois absent, i.e. early stagesof the
ηt = ηt0 ≡ urms/(3kf), where ηt0 is just a referencevalue. runoranon-shearingsimulation. Theratioofthesizeofthe
In thispaperwe use an analogouslydefinedreferencevalue, domain,L,tothesizeoftheenergy-carryingscaleisalsoan
νt0 ≡ urms/(3kf), but note that there is no strong case for importantcontrolparameterthatwe callthe scale separation
assumingthatνt willbeclosetoνt0. ratio,writtenhereaskfL/2π = kf/k1, wherek1 = 2π/Lis
thesmallestwavenumberthatfitsintothedomain.
We employthe PENCIL CODE [18] with sixth-orderfinite
II. THEMODEL differencesin space and a third ordertime stepping scheme.
Weusetriply-periodicboundaryconditions,exceptthatthex
Inthepresentworkweconsiderweaklycompressiblesub- directionisshearing–periodic,i.e.
sonicturbulenceinthepresenceofalinearshearflow,
U(−1L ,y,z,t)=U(1L ,y+L St,z,t). (5)
2 x 2 x x
S
U =(0,Sx,0), (1)
Thisconditionisroutinelyusedinnumericalstudiesofshear
so x is the cross-stream direction, y is the streamwise direc- flowsinCartesiangeometry[19,20].
tion, and z is the direction perpendicularto the plane of the
shearflow. Sincetheeffectoftemperaturechangesisnotim-
portantinthiscontext,weconsideranisothermalequationof III. RESULTS
state. Inthefollowingweworkwiththedeparturesfromthis
S
meanflow,sothetotalvelocityisU +U,andthegoverning Theinitialvelocityiszero,butthevolumeforcingdrivesa
equationsforU arethen[12] randomflowthatsoondevelopsaturbulentcascadewherethe
spectral energy follows an approximate k−5/3 inertial range
DDUt =−SUxyˆ−c2s∇lnρ+f +Fvisc, (2) wbeatvweeneunmtbheerkfodr=cinhgS2w/aνv2ein1u/4m.ber kf and some dissipation
InFig.1weshowimagesofthestreamwisecomponentof
U at the periphery of the computationaldomain from a run
Dlnρ
=−∇·U, (3) with Re ≈ 100 and Sh ≈ −0.2 (hereafterRun A). At early
Dt
times the velocity pattern is dominated by structures whose
whereD/Dt=∂/∂t+(US+U)·∇istheadvectivederiva- scaleiscomparablewiththeforcingscale,whichisaboutone
fifthofthedomainsize. However,atlatertimesthereisaten-
tive with respect to the total velocity, cs is the isothermal
dencytoproducelarge-scaleflowpatternswithalongwave-
soundspeed, hereconsideredasconstant,ρ isthe massden-
sity, f is a randomforcing function, Fvisc = ρ−1∇·2ρνS lengthvariationinthez direction. Thisflowpatterntendsto
isthe viscousforce,andS = 1(U +U )− 1δ ∇·U be unstableand keepsdisappearingand reappearing. Thisis
ij 2 i,j j,i 3 ij
seenalsoforotherrunswithsmallerReynoldsnumber.
isthetracelessrateofstraintensorandcommasdenotepartial
Giventhesystematicvariationinthezdirection,itisuseful
derivatives.
to consider averages over the x and y directions, denoted in
The forcing function is δ-correlated in time and consists
thefollowingbyoverbars. So,U =U(z,t)dependsonlyon
of random plane waves with wavevectors k in the interval
z andt. Figure2showsU andU asfunctionsoftimeand
4.5 ≤ k/k1 ≤ 5.5 [17]. During each time step, f is a sin- x y
gletransverse(solenoidal)planewaveproportionaltok×e, z. InFig.3weplotthez dependenceofUx andUy atatime
wherethewavevectork istakenrandomlyfromasetofpre- near the maximum vorticity. Note that the amplitude of Uy
definedvectorswithcomponentsthatareintegermultiplesof is about4 times as big as that of U , and thatthe two fields
x
2π/Landwhosemoduliareinacertainintervalaroundanav- areessentiallyinphase.ThefactthatU andU areinphase
x y
eragevalue,h|k|i,whichwedenotebykf,andeisanarbitrary isanimmediateconsequenceofthefactthatS < 0,andthat
random unit vector not aligned with k. The corresponding thereisaminussigninfrontofS inEq.(2).
3
FIG. 1: (Color online) Representation of Uy on the periphery of the computational domain for Run A at six different times showing the
occasionalgenerationoflargescaleflowpatternswithasystematicvariationinthezdirection. Dark(blue)shadesrefertonegativevaluesof
Uywhilelight(yellow)shadesrefertopositivevalues.Notethatattimetcsk1 =900(correspondingtoturmskf ≈480)theorientationofthe
flowpatterninthezdirectionisreversedcomparedtothepreviouseventattcsk1 =600(correspondingtoturmskf ≈300).
FIG.2: (Coloronline)Ux(a)andUy (b)asfunctionsoftimeandz
forRunA.
FIG.3: FourtimesUx (solidline)andUy (dashed)fromRunAat
t=500(csk1)−1.
In Run A with 5123 meshpoints there is one particularly
pronouncedeventduringthetime interval200 < turmskf <
′
400, where Uy reachesan extremumat turmskf ≈ 280, fol- withrespecttozaredenotedbyaprime.
′
lowed by an extremum of Ux a bit later at turmskf ≈ 300; The occasional extrema in the components of U and its
seeFig.4fortheirrootmeansquarevalues. Here,derivatives derivatives are accompanied by strong enhancements in the
4
′ ′
FIG.4: RootmeansquarevaluesofU (solidline)andU (dashed
x y
line)forRunA.Notethemaximaatturmskf ≈280andturmskf ≈
260,respectively.
rmsvalueofthetotalvelocity,Urms,whichincludesthemean
flowaswell. Thisfacthasbeenofsomesignificancein pre-
vious studies of hydromagnetic dynamo action from turbu-
lentshearflows[11,13,21],because,dependingonthevalue
of the sound speed, this can lead to numerical difficulties if
theMachnumberexceedsunityduringthesestrongenhance-
mentsofUrms. Thesedifficultiesarehereavoidedbychoos-
ing a smaller shear parameter Sh, regulatedby the inputpa-
rameterS.
TheeffectofincreasingS isdemonstratedinFig.5,which
showsthermsvaluesofthelarge-scalevelocitiesforfourruns
whereSisvariedwhiletheotherparametersarekeptconstant.
The amount of shear is here quantified by the value of Sh,
whichisbasedontheurmsvaluefromarunwithoutshearand
thuseffectivelyquantifiesthestrengthoftherandomforcing.
TheserunsaredenotedbythelettersBtoE,withthestrength
ofthe shearincreasingfromSh = −0.08inRun B toSh =
−0.33inRunE.ThebottompanelofFig.5showsthatUrms
increasesalmost in proportionto the shear for −Sh > 0.25.
TheflowinthelargeShrunsisalsohighlyfluctuatingduring
periodsofvigorousvorticitygeneration,seeFig.6foraspace-
timediagramofthelarge-scalevelocitiesfromRunE.Evenin FIG.5: Root meansquare values of Ux (a) and Uy (b), and Urms
(c) for Runs B toE with different shear as indicated in the legend
thelowestshearrun(RunBwithSh≈−0.08),whichisvery
similartothenon-shearingcaseduringmostofitsevolution,a inpanel (a). TheReynoldsnumber basedontheurms fromanon-
shearingrunis≈24.
weaklarge-scalepatternisdiscernibleattimes,seetimesafter
turmskf >600inFig.7.
plane,denotedherebyanoverbar,wehave
∂U ′′
=−SU yˆ+F +νU , (6)
x
IV. INTERPRETATION ∂t
where F = −u·∇u is a term that results from the nonli-
In orderto shedsome lightonthe mechanismresponsible nearity of the Navier-Stokes equations, and primes denote a
forthegenerationoflarge-scalevorticity,weconsidermean- z derivative. Note that we have assumed solenoidality, i.e.
field equations [14, 22]. Adopting averages over the (x,y) ∇·U =U3,3 =0,soU3 =const=0byasuitablechoiceof
5
to the x and y directions and u3 is the z component of the
velocityfluctuation.IntegratingEq.(7)overz,wehave
′
R +ν U ≡const. (9)
i ij j
Given that R and U can be obtained from the simulations,
we can then find all four components of ν by considering
ij
momentequationsoftheform
′
hR U i+ν M =0, (10)
i k ij jk
where we have introduced the correlation matrix M =
jk
′ ′
hU U i. We have also assumed that ν is independent of
j k ij
z,andthat,owingtoperiodicboundaryconditions,themean
flow and its z derivatives have zero volume average, i.e.
′
hU i=0foranyi.Thecomponentsofν canthenbewritten
FIG.6: (Coloronline)Ux(a)andUy (b)asfunctionsoftimeandz as i ij
forRunEwithSh≈−0.33.
′
νi1 =−M−1 hRiU1′i , (11)
(cid:18)νi2 (cid:19) hRiU2i!
wherei=1or2.
′
ItturnsoutthatthecomponentsofthecorrelationshR1Uji
aresmallcomparedwiththoseofM . Thismakestheeval-
jk
uation of the components of ν1j using Eq. (11) ill-behaved
(seeFig. 8). Thisproceduredoes, however,yieldreasonable
resultsfor the ν2i components: ν21 is highlyfluctuating, but
with an averageof the orderof roughlyhalf of thereference
valueνt0 ≡ 31urmskf−1, whereasν22 ispositiveandbetween
one and two times νt0 in the quiescent phases of the simu-
lation and peakingat roughly5νt0 when the vorticity peaks.
Thedefinitionofνt0isanalogoustoacorrespondingreference
valueforthemagneticdiffusivity[23],butitisnotclearthat
νtshouldbeexactlyequaltoνt0inanylimit.Instead,accord-
ing to the first order smoothing approximation, νt = 0.4νt0
FfoIrGR.u7n: (BCowliotrhoSnhlin≈e)−U0x.0(8a.)andUy (b)asfunctionsoftimeandz [16],andhencethemagneticPrandtlnumberwasexpectedto
be0.4.
′
If both componentsof hR1Uki for k = 1 and 2 were ex-
theinitialcondition.Thus,onlythexandycomponentsofU actlyzero,wecouldcalculateν12/ν11intermsoftheratios
arenon-vanishing.Therefore,U·∇U =0. Furthermore,the
pressuregradienttermdoesnotenterinEq.(6),becauseany ν12 =−M1k (12)
horizontally averaged gradient term can only have a z com- ν11 M2k
ponent. Usingmeanfieldtheory[14],F canbeexpressedin
fork =1and2. Yetanotherpossibilityistotakethegeomet-
termsof derivativesof the meanflow. In the presentcase of
ricmeanofthetwoexpressions,so
one-dimensionalmeanfieldsthisrelationshipreducesto
Fi =νijU′j′, (7) νν1112 ≈− MM1211 MM1222 1/2 ≡− MM2121 1/2, (13)
(cid:18) (cid:19) (cid:18) (cid:19)
where ν is the eddy viscosity tensor. We also assume in-
compresisjibilityof the small-scale velocity field, ∇·u = 0, where we have used the fact that M21 = M12. The results
shown in Fig. 9 indicate that the two ratios in Eq. (12) give
whichisagoodapproximationforsmallMachnumbers,and
consistentlynegativevalues,althoughtheirmoduliarediffer-
recallthathorizontalaveragesdependonlyonz,i.e.thej =3
coordinate.Thereforewehave ent. Assuming that ν11 is positive, which is reasonable, this
resultsuggeststhatanegativeν12ispresentinthesystemwith
′
Fi =−∇3uiu3 =−Ri. (8) a modulusthat is between 0.2 and 0.4 times the ν11 compo-
nent.
Here we have denoted the two relevant components of the Finally,acompletelydifferentapproachforobtainingesti-
Reynolds stress tensor by Ri ≡ uiu3, where i = 1,2 refer mates between the componentsof νij is to use the resulting
6
FIG.8: Components of theeddy viscositytensor asobtained from
Eq.(11)normalizedbyνt0 = 13urmskf−1 forRunA.Notethatthe
averagevalueofν21isnegative(seethedashedlineinthethirdpanel
forturmskf >100).
mean-fieldequations,Eq.(6),andapplythemtoahypotheti-
calsteadystate. Theseequationsarelinear,whichisaconse-
quenceofassumingthecomponentsofν tobeconstant. In
ij
that case we can Fourier transformand obtain the two equa-
tions
(ν+ν11)k2Uˆ1+ν12k2Uˆ2 =0, (14)
(S+ν21k2)Uˆ1+(ν+ν22)k2Uˆ2 =0, (15)
where Uˆ1 and Uˆ2 are the Fourier amplitudes of the x and y FIG.9: ScatterplotsofM12/M22 (a),M11/M21 (b),M11/M22 (c)
componentsof themeanflow. Sincethese equationsarelin-
forRunA.
ear,theycannotdescribenonlinearsaturationofamean-field
vorticitydynamoinstability. However,itisplausiblethatthe
assumptionofconstancyofthecomponentsνij breaksdown impliesthatD =1. NotethatEqs.(14)and(15)yield
when the resulting mean vorticity has become large enough.
The resulting modifications of νij may then explain satura- ν12 Uˆ1 ν+ν22
tion. =− = . (18)
Equations(14)and(15)showthatanecessaryconditionfor ν+ν11 Uˆ2 S/k2+ν21
the mean-vorticity dynamo to be excited is that the product
Thisallowsustocalculateν +ν22 intermsofurms/kf,pro-
ν12S is positive. This is indeed the case; in our case both
videdν21isnegligibleorknown:
ν12 andS are negative. A sufficientconditionfor the mean-
vorticitydynamotobeexcitedisthattheparameter
ν+ν22 Uˆ1 kf 2 ν21kf
2 2 2 =− Sh + . (19)
D ≡ ν12(S/k +ν21)+ǫ /νT ≥1, (16) urms/kf Uˆ2 " (cid:18)k1(cid:19) urms #
whereνT =ν+(cid:2)νtwith (cid:3) The amplitudes Uˆ1 and Uˆ2 for Run A are shown in Fig. 10.
νt = 12(ν11+ν22), ǫ= 21(ν11−ν22). (17) Puttinginnumbers,Uˆ1/Uˆ2 =0.23,kf/k1 =5,weobtain
The parameter D plays the role of a mean-vorticitydynamo ν+ν22 ν21kf
=1.15−0.23 , (20)
number.TheassumptionofasteadystateinEqs.(14)and(15) urms/kf urms
7
FIG. 11: Turbulent viscosity for Run A, as obtained from the
ReynoldsstresscomponentRxy,dividedbytheestimateνt0.
VI. CONCLUSIONS
The present work has demonstrated quite clearly that in
non-helical shear-flow turbulence a large-scale flow pattern
emergesspontaneously. Inthepresentcase,whereintheab-
FIG.10:(a):Uˆx(solidline),4Uˆx(dotted)andUˆy (dashed)asfunc- sence of shear the turbulencesaturates at a Mach numberof
tions of time for Run A. (b): scatter plot of Uˆx versus Uˆy for the order 0.01, the large-scale flow becomes exceedingly strong
samerun.Thedashedlineshowsalinearfittothedata. andsaturatesataMachnumberof0.1–0.2. Thisbehavioris
seenbothatsmallandatthelargestReynoldsnumbersconsid-
eredhere(Re = 100,basedontheinverseforcingwavenum-
so theuncertaintyinν21 entersonlyweakly. Note,however, ber).
thatν22 ismorethanthreetimeslargerthantheoriginalesti- Theflowpatterncanbeparticularlywellpronouncedatcer-
mateofνt0. tain times and shows a long wavelength variation in the di-
rectionperpendiculartotheplaneoftheshearflow (herethe
z direction). For negativeshear, the x and y componentsof
the shear flow are in phase in a way that is compatible with
V. EDDYVISCOSITYFROMTHEIMPOSEDSHEAR an interpretation in terms of a large-scale vorticity dynamo,
asexploredfirstbyElperin,Kleeorin,andRogachevskii[14].
Thismeansthatthelarge-scaleflowisdrivenbyananisotropic
We have so far only looked at the components of the
eddyviscositytensor. Particularlyimportantisitsxycompo-
Reynolds stress tensor that enter the horizontally averaged
nent, ν , which describes the production of a cross-stream
equations. However, there is at least one other component xy
large-scale flow component, U (z,t), from a z-variation of
thatdoesnotenterEq.(6),butthatcanalsobeusedtodeter- x
thestreamwiselarge-scaleflow,U (z,t). Themean-vorticity
minetheeddyviscosity(see,e.g.[24]).Thiscomponentisnot y
“dynamocycle”iscompletedbyasuitableactionoftheshear
drivenbythederivativesofU,butbytheimposedshearflow,
itself, which producesa streamwise large-scale flow compo-
S
∇ U , itself. Indeed, one expects that this imposed shear
x y nent,U ,fromthecross-streamcomponent,U ,bytheterm
leadstoanxystress y x
−U ·∇US.
Themean-vorticitydynamocyclecanonlyworkifthesign
S S
uxuy =−νt ∇xUy +∇yUx =−νtS. (21) of ν is the same as that of the shear, ∇ US. The present
xy x y
(cid:16) (cid:17) investigationssuggestthatthisisindeedthecase. However,it
Thisisindeedthecase;seeFig.11. Itturnsoutthattheνtde- isdesirabletoverifythesignofνxy usingatest-flowmethod
terminedinthiswayisrathersimilartothevalueofν22 esti- analogously to the test-field method used in magnetohydro-
matedfromEq.(20).Again,thereisnogoodreasonthatthese dynamics. Some care in using the correlation method is in
valuesarethesame,becausetheeddyviscosityobtainedfrom order,becausethereareexamplesinmagnetohydrodynamics
Eq. (21) belongsto a differentcomponentof the full rank-4 where the correlationmethod give incorrectvaluesfor some
eddyviscositytensorandisnotpartoftherank-2tensorcon- componentsof the magnetic diffusiontensor, although other
sideredabove. componentswerecorrect[25]. Forexample,whenwe apply
8
a method analogousto that in Eq. (18) to the magnetic field one-dimensional,the eddyviscosity reducesto a rank-2ten-
ofa simulationofshearflow turbulence(see, e.g., Figs. 7or sor.Finally,forastrophysicalapplicationsitshouldbepointed
8 of Ref. [12]), the components of the magnetic field scat- out that the gas in many shear flows is ionized and electri-
ter almost isotropically about the origin. This is compatible callyconducting,givingrise toefficientdynamoaction. The
with an interpretation in terms of an incoherent alpha–shear resulting mean Lorentz force from the small-scale magnetic
effect [12, 26]. On the other hand, there is still a weak cor- fieldmodifiestheeddyviscosityinawaythatsuppressesthe
relation with a negative slope. This would suggest that the mean-vorticitydynamo.Detailsofthisneedtobeinvestigated
shear–currentdynamomightalsobeatwork,eventhoughthe further. Another effect that can suppress the mean-vorticity
test-fieldmethodindicatesthatthisshouldnotbethecase. dynamoisrotation[13]. Thiscanbeunderstoodfromthedis-
Clearly,therealityofthelarge-scaleflowfoundinsimula- persionrelation in that the addition of rotationleads, among
tionsismorecomplicatedthanwhatissuggestedbythesim- other terms, to a −4Ω2 term inside the squared brackets of
plemean-vorticitydynamoproblem.Firstly,incontrasttothe Eq.(16)thatalwayssuppressesthemean-vorticitydynamo.
magneticdynamonokinematicstagecanbedistinguished,i.e.
thelarge-scalepatternsarevisibleonlyaftertheyarealready
of dynamical importance. Secondly, the mean flow can re-
Acknowledgments
verse sign in randomintervals which is notanticipated from
thelinearmean-vorticitydynamomodelwithanisotropiceddy
viscosity, whereself-excitedsolutionswouldalwaysbenon- We thank an anonymous referee for offering suggestions
oscillatory. Another question that needs to be addressed in regarding the break-down of the linearity of equations (14)
future work is the saturation levelof the large-scaleflow, its and(15). Thecomputationswereperformedonthefacilities
relation to the saturation level of the small-scale flow, and a hostedby the Center of Scientific Computingin Espoo, Fin-
possibledependenceontheMachnumber. land,whoareadministeredbytheFinnishministryofeduca-
For more realistic applications it will be important to get tion. This work was supported by the Academy of Finland
information about the full eddy viscosity, which is a rank-4 grantNo.121431(PJK),theLeverhulmeTrust(DM),andthe
tensor [8]. In the presentwork, where the averagesare only SwedishResearchCouncil(AB).
[1] Moiseev,S.S.,Sagdeev,R.Z.,TurA.V.,Khomenko,G.A.,& 016311(2003).
Yanovskii,V.V.,Sov.Phys.JETP58,1149(1983). [15] Ru¨diger, G., & Kitchatinov, L. L., Astron. Nachr. 327, 298
[2] Kitchatinov,L.L.,Ru¨diger,G.,Khomenko,G.,Astron.Astro- (2006).
phys.287,320(1994). [16] T.A.Yousef,A.Brandenburg,G.Ru¨diger,Astron.Astrophys.
[3] Frisch,U.,She,Z.S.,&Sulem,P.L.,Physica28D,382(1987). 411,321(2003).
[4] Sulem,P.L.,She,Z.S.,Scholl,H.,&Frisch,U.,J.FluidMech. [17] A.Brandenburg,Astrophys.J.550,824(2001).
205,341(1989). [18] http://www.nordita.org/software/pencil-code
[5] Galanti,B.&Sulem,P.-L.,Phys.FluidsA3,1778(1991). [19] J.Wisdom,S.Tremaine,Astronom.J.95,925(1988).
[6] Brandenburg,A.,&vonRekowski,B.,Astron.Astrophys.379, [20] J. F.Hawley, C.F.Gammie, S.A.Balbus, Astrophys. J.440,
1153(2001). 742(1995).
[7] G.Ru¨diger,Geophys.Astrophys.FluidDynam.16,239(1980). [21] P. J. Ka¨pyla¨, A. Brandenburg, submitted to ApJ,
[8] G. Ru¨diger, Differential rotation and stellar convection: Sun arXiv:0810.2298(2008)
andsolar-typestars.Gordon&Breach,NewYork(1989). [22] T.Elperin,I.Golubev,N.Kleeorin,andI.Rogachevskii,Phys.
[9] I.Rogachevskii,N.Kleeorin,Phys.Rev.E68,036301(2003). Rev.E76,066310(2007).
[10] I.Rogachevskii,N.Kleeorin,Phys.Rev.E70,046310(2004). [23] S.Sur,A.Brandenburg,K.Subramanian,Mon.Not.R.Astron.
[11] T.A.Yousef,T.Heinemann,A.A.Schekochihin, N.Kleeorin, Soc.385,L15(2008).
I.Rogachevskii,A.B.Iskakov,S.C.Cowley,J.C.McWilliams, [24] P.J. Ka¨pyla¨ & A. Brandenburg,, Astron. Nachr. 328, 1006
Phys.Rev.Lett.100,184501(2008). (2007).
[12] A.Brandenburg,K.-H.Ra¨dler,M.Rheinhardt,P.J.Ka¨pyla¨,As- [25] A.Brandenburg, D.Sokoloff, Geophys. Astrophys. FluidDy-
trophys.J.676,740(2008). nam.96,319(2002).
[13] T. A. Yousef, T. Heinemann, F. Rincon, A.A. Schekochihin, [26] E.T.Vishniac,A.Brandenburg,Astrophys.J.475,263(1997).
N. Kleeorin, I. Rogachevskii, S.C. Cowley, J.C. McWilliams,
Astron.Nachr.329,737(2008).
[14] T.Elperin,N.Kleeorin,andI.Rogachevskii, Phys.Rev.E68, Id:paper.tex,v1.742008-12-3108:57:22brandenbExp