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Astronomy&Astrophysicsmanuscriptno.6864text (cid:13)c ESO2008 February5,2008 Effect of rotation on the tachoclinic transport NicolasLeprovostandEun-jinKim 7 0 DepartmentofAppliedMathematics,UniversityofSheffield,SheffieldS37RH,UK 0 2 Received/Accepted n a ABSTRACT J 9 Aims.Westudytheeffectofrotationonshearedturbulenceduetodifferentialrotationinthesolartachocline. Methods.By solving quasi-linear equations for the fluctuating fields, we derive turbulence amplitude and turbulent transport coefficients 2 (turbulent viscosity and diffusivity), taking into account the effects of shear and rotation on turbulence. We focus on the regions of the v tachoclineneartheequatorandthepoleswhererotationandshearareperpendicularandparallel,respectively. 4 Results.For parameter values typical of the tachocline, we show that the shear reduces both turbulence amplitude and transport, more 8 strongly in the radial direction (parallel to the shear) than in the horizontal one, resulting in anisotropic turbulence. Rotation further 0 2 reduces turbulence amplitude and transport at the equator whereas it does not have much effect near the pole. The interaction between 1 shear and rotation is shown to give rise to a novel non-diffusive flux of angular momentum (known as the Λ-effect), possibly offering a 6 mechanismfortheoccurrenceofastrongshearregioninthesolarinterior.Furtherimplicationsforthetransportinthetachoclinearediscussed. 0 / h p Keywords.Turbulence–Sun:interior–Sun:rotation - o r t 1. Introduction The purpose of this Letter is to investigate the effect of s a (global) rotation on the tachocline transport. In our previous Solarrotation,bothglobalanddifferential,playsacrucialrole : works,wehavestudiedtheturbulenttransportinthetachocline v inthedynamicalprocessestakingplaceintheSun.Forexam- by taking into account the crucial effect of shearing, the so- i X ple,mean-fielddynamotheory(Moffatt1978;Krause&Ra¨dler called shear stabilisation, due to a strong radial differential 1980) uses these two ingredients to explain the presence of r rotation (Kim 2005; Leprovost&Kim 2006). We have also a thesolarmagneticfield.Firstly,thedifferentialrotationinthe incorporated the interaction of this sheared turbulence with tachocline is responsible for the in-situ generationof toroidal different types of waves that can be excited in the Sun due fieldviatheΩeffect.Secondly,convectionundertheinfluence to magnetic fields (Leprovost&Kim 2007) or stratification of (global) rotation leads to helical motion in the convection (Kim&Leprovost2006).InthisLetter,weelucidatetheeffect zonewhichpermitsthecreationofpoloidalfieldsviatheαef- of globalrotation on sheared turbulenceby studyinga (local) fect. Cartesianmodelvalidneartheequatorandthepoles.Wecon- To understand the differential rotation in the convective sideraturbulencedrivenbyanexternalforcingsuchasplumes zone,manyauthorsinvestigatedthe structureof turbulencein fromtheconvectionzoneandperformaquasi-linearanalysisto rotatingbodies.Themain featureof thistypeofturbulenceis derivethedependenceofturbulenceamplitudeandtransporton the appearance of non-diffusive term in the transport of an- rotation and shear. Compared to two-dimensional turbulence gular momentum which prevents a solid body rotation from studied in Leprovost&Kim (2007), global rotation supports being a solution of the Reynoldsequation (Lebedinsky1941; the propagation of inertial waves in three dimensions (3D), Kippenhahn1963).StartingfromNavier-Stokesequation,itis whichinteractwiththeshearflow,playinganimportantrolein possible to show that these fluxes arise when there is a cause theoverallturbulenttransport.Inparticular,wereportanovel ofanisotropyinthesystem,eitherduetoananisotropicback- resultthatthemomentumtransportcanbenotonlyduetoeddy groundturbulence (see Ru¨diger 1989, and referencestherein) viscosity but also to non-diffusive Λ-effect in the tachocline. or due to inhomogeneity such as an underlying stratification Non trivial Λ-effect here results from an anisotropy induced (Kichatinov 1987). To explain the observed internal differen- byshearflowontheturbulenceevenwhenthedrivingforceis tialrotationandthedepletionoflightelementsonthesurface isotropic,incontrasttothecasewithoutshearflowwherethis oftheSun,itisalsoofprimeimportancetounderstandthein- effectexistsonlyforanisotropicturbulence(Kichatinov1987). fluence of rotation on turbulent transport coefficients such as theturbulentheatconductivityandthe turbulentdiffusivityof particles. 2 NicolasLeprovostandEun-jinKim:Effectofrotationonthetachoclinictransport z Reynolds stress. One can formally Taylor expand R with re- specttothegradientofthelarge-scaleflow: Ω˜ Ri =ΛiU0−νT∂xU0δi1+...=ΛiU0+νTAδi1+... . (3) whereweintroducedtwocoefficientsΛ andν .Theeffectof i T theturbulentviscosityν issimplytochangetheviscosityfrom T θ themolecularvalueνtotheeffectivevalueν+νT.Incompari- son,thefirsttermΛU inEq.(3)isproportionaltotheveloc- i 0 ity ratherthan its gradient.Thismeansthatit doesnotvanish y for a constant velocity field and can thus lead to the creation x ofgradientin thevelocityfield.Thistermisequivalenttothe α-effect in dynamotheory (Parker 1955; Steenbeck&Krause 1966),andhasbeenknownastheΛ-effect(Krause&Ru¨diger A = −dUy 1974) or anisotropic kinetic alpha (AKA)-effect (Frischetal. dx 1987). Similarly,the transportofspeciesinthe largescales is governedbyanadvectiondiffusionequationwherethemolec- Fig.1. The configuration in our model: A, Ω˜ and θ are the ular diffusivity is supplemented by a turbulentdiffusivity, de- shearingrate,therotationrateandtheco-latitude,respectively. finedashvni=−Dij∂ N . i T j 0 Tocalculatetheturbulenceamplitudeandtransportcoeffi- 2. Model cients(Reynoldsstressandturbulentdiffusivity),weprescribe theforcinginEq.(1)tobeincompressible,isotropicandshort To elucidate the effect of rotation on sheared turbulence, correlated in time (modelled by a δ-function) with a power we model the tachocline by an incompressible fluid in local spectrumF.Specifically,weassume: Cartesian coordinates (x,y,z) in a rotating frame (see Fig. 1) with average rotation rate Ω˜ making an angle θ with a mean hf˜i(k1,t1)f˜j(k2,t2)i = τf (2π)3δ(k1+k2)δ(t1−t2)× (4) shear flow in the azimuthal direction:U = −xAˆj. We study F(k)(δ −kk /k2). 0 ij i j the effect of this large-scale shear on the transport properties Theangularbracketsstandforan averageoverrealisationsof of turbulence by assuming the velocity as a sum of a radial theforcing,andτ isthe(short)correlationtimeoftheforcing. shear (i.e. in the x-direction) and fluctuations: u = U +v = f 0 We solve Eqs. (1) and (2) in the case of unit Prandtl number U (x)ˆj+v = −xAˆj+v.Weresorttothequasi-linearapprox- 0 (ν = D)forsimplicity,andusetheresultsoftheseandEq.(4) imation(Townsend1976;Moffatt1978),alsocalledfirst-order to computethe turbulenttransport.InthisLetter,wefocuson smoothing(FOSA)orsecond-ordercorrelationapproximation thecaserelevanttotheSunanddiscusstheimplicationsofour (SOCA),wheretheproductoffluctuationsisneglectedtoob- findings for the dynamics of the tachocline. Note that in the tainthefollowingequationsfortheevolutionofthefluctuating tachocline,|A| ∼ 3×10−6s−1 byusingtheupperlimitonthe velocityfield: tachoclinethickness(5%ofthesolarradius)whiletheaverage ∂v+U ·∇v+v·∇U = −∇p+ν∇2v+f−Ω×v, (1) rotation rate of the interior is Ω˜ ∼ 2.6×10−6s−1. Therefore, t 0 0 the ratio of rotation to shear is very close to unity. However, ∇·v = 0, ifthetachoclineisfoundtobethinner,thevalueoftheshear- where p and f are respectivelythe small-scale componentsof ingrateAbecomeslarger,with|Ω/A|< 1.Furthermore,with thepressureandforcing,andΩ≡2Ω˜. molecularviscosityofν ∼ 102cm2s−1,ξ = νk2/|A|isasmall y To study the influence of rotation and shear on the parti- parameterforabroadrangeofreasonablelengthscalesinthe cle andheattransport,we havetosupplementEq.(1) withan azimuthaldirection,L > 104cm.Thus,thepresentresultsare y advection-diffusionequation for these quantities. We here fo- validforslowrotation(Ω≪|A|)andstrongshear(ξ ≪1). cusonthetransportofparticlessinceasimilarresultalsoholds fortheheattransport.Thedensityofparticlesisalsowrittenas 3. Neartheequator(θ = π/2) asumofalarge-scalecomponentN andfluctuationsn.Using 0 thequasi-linearapproximation,thefluctuatingconcentrationof Neartheequator,therotationandthedirectionoftheshearare particlesisgovernedbythefollowingequation: orthogonal.ExpandingthevelocityinpowersofΩ¯ =Ω/A,we computetheturbulenceamplitudeandtransportcoefficientsup ∂tn+U0·∇n+v·∇N0 = D∇2n, (2) tosecondorderin|Ω¯|≪1. whereDisthemoleculardiffusivityofparticle.Notethat,inthe case of heat equation, D should be replaced by the molecular 3.1.Turbulenceintensity heatconductivityχ. In the strong shear limit (ξ ≪ 1), using similar algebra as in Asthelarge-scalevelocityisinthey(azimuthal)direction, Kim (2005),we obtainthe turbulentintensityin theradialdi- wearemostlyinterestedinthemomentumtransportinthatdi- rectionasfollows: rection. The equation for the (large-scale) azimuthal velocity U0 isthengivenbyNavier-Stokesequationwiththecontribu- hv2i= τf d3k F(k) L (k)+Ω¯ kz2L (k) . (5) tion fromfluctuationsgivenby ∇·R, whereR = hvvyi is the x A Z (2π)3  0 ky2 1  NicolasLeprovostandEun-jinKim:Effectofrotationonthetachoclinictransport 3 Here,k2 =k2+k2isthe(squared)amplitudeofthewavenum- contrasted to the case withoutshear flow where non-diffusive H y z berinthehorizontalplane,andL andL aretwopositivedefi- fluxes emerge only for anisotropic forcing (Ru¨diger 1980; 0 1 niteintegralsdependingonlyonthewavenumberk.Notethat Kichatinov 1986). In our case, the anisotropy in the velocity thedetailsof L and L arenotimportantforourdiscussions. field is not artificially introduced in the system but is created 0 1 Therefore,theturbulenceintensityhv2iinEq.(5)increasesfor bytheshearandcalculatedself-consistently. x Ω¯ > 0whereasitdecreasesforΩ¯ < 0.Thisisbecauseaweak rotation destabilises sheared turbulence for Ω¯ > 0 whereas it 3.3.Transportofparticles stabilisesforΩ¯ <0(Bradshaw1969;Salhi&Cambon1997). Fortheothercomponentsoftheturbulenceamplitude,we Inthestrongshearlimit(ξ ≪1),thecomputationofhnvigives: obtain: τ d3k k2 −lnξ Hhve2zrie,∼ΓτAisf tZheΓ((G2d1π3a/k)m33)mF+a(Ωk¯f)uLkknzy222c(Γkt(i)o4 n/233aξ)n(!−d1/l3Ln×ξ)is.a positive defin(i6te) wDDhTxzTxzer∼∼e LAAτff22anZZd L((22dππ3ka))33reFFp((kko))sLLit65i(v(kke)) d21e3fiξ+n!i23Ω¯tek1yz2f+un2c3Ωt¯iokknyz22s.−,El3nqsξ.(9,(-11(900))) 2 5 6 function of k. Here again, the turbulence amplitude can in- showthattheeffectofrotationonthetransportofparticlesde- creaseordecreasedependingonthesignofΩ¯.Furthermore,the pends on the sign of Ω¯: for Ω¯ > 0, the transport is increased correctionduetorotationhasalogarithmicdependenceonthe whereasitisreducedwhenΩ¯ <0.Thisisagainbecauseaweak shear,contrarytothecaseoftheradialamplitude.Notethatthe rotation destabilises sheared turbulence for Ω¯ > 0 whereas it anisotropyinthefirstcorrectionislesspronouncedthaninthe stabilises for Ω¯ < 0. Note that a similar behaviour was also leadingorderterm.Asaresult,theturbulenceduetoshearing found in turbulence intensity, given in Eqs. (5) and (6). The becomeslessanisotropic.Thisillustratesthetendencyofrota- correctiontermduetorotationinDxxandDzzinEq.(9-10)de- T T tiontoleadtoalmostisotropicturbulence.Anequationsimilar pendsweaklyontheshearbyalogarithmicfactor|lnξ|,which to (6) is also found for hv2i with a slightly different function cannotbetoolargeevenforξ ≪1,andisofthesameorderfor y L . However,performingthe angularintegration,we find that transportindifferentdirections.Thus,thescalingoftheturbu- 2 hv2i is larger than hv2i, in agreement with numerical simula- lentdiffusivityisroughlythe sameasthatinthecase without y z tions(Leeetal. 1990). Thiseffectis presentforΩ¯ = 0andis rotation:theradialtransport(∝ A−2)ismorereducedthanthe thuscausedbytheshearonly. horizontal one (∝ A−4/3). This result should be contrasted to therapidrotationlimitwherethetransportintheradialdirec- tionwaslarger(butonlybyafactor2)thantheoneinthehor- 3.2.Transportofangularmomentum izontal direction (Ru¨diger 1989). These results thus highlight In the strong shear limit (ξ ≪ 1), the transport of angular thecrucialroleofshearintransport,inparticular,byintroduc- momentum can be found as a sum of two terms hv v i = inganisotropy. x y ν A+Λ Ω,thefirstandsecondbeingevenandoddwiththero- T x tationrate,respectively.Thefirstterm,theturbulentviscosity, 4. Nearthepoles(θ = 0) takesthefollowingform: Nearthepoles,thedirectionsofrotationandshearcanbetaken τ d3k 1 f ν ∼ F(k) − +L (k) , (7) tobeparallel.Contrarytothecaseattheequator,wefindthat T A2 Z (2π)3 " 2 3 # theleadingordercorrection(proportionaltoΩ¯)vanishesinthe whereL isapositivedefinitefunctionofk.ForΩ¯ = 0were- case of an isotropic forcing (due to the fact that these terms 3 covertheresultofKim(2005)showingthattheturbulentvis- areoddinoneofthecomponentsofthewavenumber)forall cosity is reduced proportionallyto A−2 for strong shear. The the previously calculated quantities. Consequently, the turbu- turbulentviscositycanbeeitherpositiveornegativedepending lent amplitude, viscosity and diffusivity are givenby Eqs. (5- ontherelativemagnitudeofthetwotermsinsidethe integral. 6), (7) and (9-10) with Ω¯ = 0, respectively. Similarly, the Inthe2Dlimit(whereL = 0),wecaneasilyseethatthetur- Reynolds stress involving the radial component (hv v i) van- 3 x y bulentviscosityisnegative.Onthe contrary,inthecase ofan ishes. However the componentof the Reynolds stress involv- isotropicforcingin3D,theturbulentviscosityispositive. ingthelatitudinalvelocity(hv v i)doesnotvanishandisodd y z ThecorrectionduetotherotationisproportionaltoΩand inΩ.Thus,theΛ-effectappearshereinhv v i(recallthatatthe y z isthusoddintherotation.Thisistheso-calledΛ-effect,anon- equator,theΛ-effectwaspresentonlyinhv v i),andtakesthe x y diffusivecontributiontoReynoldsstress,whichcanbeshown followingform: tobe: τf d3k Λ ∼−τf d3k F(k) 3 2/3[L (k)−L (k)] . (11) Λx ∼ A2 Z (2π)3 F(k)L4(k)(−lnξ). (8) z A2 Z (2π)3 2ξ! 7 8 It is important to emphasise that this non trivial Λ-effect re- Eq. (11) shows that Λ is of indefinite sign, as L and L are z 7 8 sults from an anisotropy induced by shear flow on the turbu- twopositivedefinitefunctionsbutappearwithdifferentsigns. lenceevenwhenthedrivingforceisisotropic.Thisshouldbe However,asforEq.(8),inthecaseofanisotropicforcing,the 4 NicolasLeprovostandEun-jinKim:Effectofrotationonthetachoclinictransport function L dominatesover L , leadingto a negativeΛ . This As both ν and Λ are positive, A must be negative. 7 8 z T Λ-effect arises even in the case of an isotropic forcing as the Interestingly, this is exactly what the observation indicates (a shearfavoursfluctuationsinthey-directioncomparedtothatin rotationratedecreasingtowardstheinteriorattheequator).We the z-direction(see §3.1), leadingto anisotropicturbulencein cannot use such a simple equation to predict the sign of the theplaneperpendiculartotherotation.Eq.(11)alsoshowsthat shearatthepoleastheΛ-effectnowappearsinthehv v icom- y z Λ islarger(scalingasA−4/3)thanΛ intheequatorialcase. ponentof the Reynolds stress and is found to scale as A−4/3, z x whichislargerthanΛ (∝A−2)neartheequator. x 5. Implicationsforthetachocline Furthermore,wefindthatthetransportofchemicalspecies is reducedby shear and moreseverely in the the radialdirec- Table 1 summarises the scaling of our results with the shear tionthanthehorizontalone(byafactorA−2andA−4/3respec- AandrotationΩinthelimitofweakrotation(Ω ≪ |A|)and tively), with a further, but weak reduction due to rotation at strongshear(ξ =νk2/|A|≪1). y theequatoronly.Asaresult,theradialmixing(inthe xdirec- tion)ofchemicalsisslightlymoreefficientatthepolethanat theequator.Theseresultsthussuggestthatturbulentmixingof Equator Pole lightelementsis weak(dueto shearing)in the tachoclineand hv2i A−1 1+cΩ¯ A−1 alsothatthedepletionoflightelementsonthesolartachocline x hv2i∼hv2i A−2/3 1h+cΩ¯|lniξ| A−2/3 depends on latitude. However, this latitudinal dependency is y z h i unlikelytobeobservedonthesurfaceduetotherapidmixing ν A−2 A−2 T intheconvectionzone(Spruit1977).Sinceasimilarresultap- Λ A−2|lnξ| 0 pliestoheattransport,afasterheattransportatthepolecould x contributetomakinghotterpolesthanequatoratthesolarsur- Λ 0 −A−4/3 z face. Finally, we note that all our results here are valid in the Dxx A−2 1+cΩ¯|lnξ| A−2 absenceofstratificationandmagneticfields(Kim&Leprovost T Dyy∼Dzz A−4/3h1+cΩ¯|lnξi| A−4/3 2006)andwillextendourstudytoincludetheseinfuturepub- T T h i lications.OfparticularinterestwouldbethesignofνT andΛ, Table 1. Scaling of turbulenceamplitude,turbulentviscosity andthustheresultingsignofA. (ν ), Λ-effect and turbulent diffusivity with the shear A and T rotationΩ¯ =Ω/A.cisapositiveconstantoforderunity. Acknowledgements. ThisworkwassupportedbyU.K.PPARCGrant No.PP/B501512/1. The first two rows show that, both near the equator and References poles,theturbulenceamplitudeinthe radialdirectionismore Bradshaw,P.1969,J.FluidMech.,36,177 reducedbytheshearthaninthehorizontalone,byafactorof Frisch,U.,She,Z.S.,&Sulem,P.L.1987,PhysicaD,28,382 A−1 and A−2/3, respectively. These results thus imply an ef- Kichatinov, L. L. 1986, Geophys. Astrophys. Fluid Dyn., 35, fectivelystrongerturbulenceinthehorizontal(y-z)planethan 93 theoneintheradial(x)direction.Furthermore,attheequator, Kichatinov, L. L. 1987, Geophys. Astrophys. Fluid Dyn., 38, Ω¯ <0astheshearincreasesfromtheinteriortowardsthecon- 273 vectivezone.Thus,theturbulenceamplitudeisfurtherreduced Kim,E.2005,A&A,441,763 by the rotation in all directions, but the turbulence in the ra- Kim,E.&Leprovost,N.2006,submittedtoA&A dialdirectionismoresuppressedthantheoneinthehorizontal Kippenhahn,R.1963,ApJ,137,664 directionbyafactorlnξ. Krause,F.&Ra¨dler,K.-H.1980,MeanfieldMHDanddynamo AnovelresultisalsofoundfortheReynoldsstress,which theory(Pergamonpress) involvestwo contributions:the turbulentviscosity and the Λ- Krause,F.&Ru¨diger,G.1974,Astron.Nachr.,295,93 effect. The latter, a source of non-diffusive flux, is present Lebedinsky,A.I.1941,AZh,18,10 evenwithisotropicforcingduetotheshear-inducedanisotropy Lee,J.M.,Kim,J.,&Moin,P.1990,J.FluidMech.,216,561 (Kim 2005). Note that for an isotropic forcing, the turbulent Leprovost,N.&Kim,E.2006,A&A,456,617 viscosity is found to be positive and therefore cannot act on Leprovost,N.&Kim,E.2007,ApJ,654,1166 its own as a source of differential rotation in the tachocline. Moffatt,H.K.1978,Magneticfieldgenerationinfluids(CUP) Theradialdifferentialrotationcan,however,becreatedbythe Parker,E.N.1955,ApJ,122,293 Λ-effect:since it does notdependonly on the gradientof an- Ru¨diger,G.1980,Geophys.Astrophys.FluidDyn.,16,239 gular velocity, it does not vanish for a uniform rotation and Ru¨diger, G. 1989, Differential rotation and stellar convection thuspreventstheuniformrotationfrombeingasolutionofthe (GordonandBreach) large-scale momentum equation. To determine the sign of A Salhi,A.&Cambon,C.1997,J.FluidMech.,347,171 asaresultoftheΛ-effect,weseekasolutionofthelarge-scale Spruit,H.C.1977,A&A,55,151 turbulentequationsbydemandingtheReynoldsstresstovanish Steenbeck,M.&Krause,F.1966,Z.Naturforsch.,TeilA,21, (e.g. Kichatinov1987). In the equatorialcase, we then obtain 1285 thefollowingequationfortheshear: Townsend, A. A. 1976, The structure of turbulent shear flow, ν (A)A+Λ(A)Ω=0. (12) 2ndedn.(CUP) T