Nonlinear Turbulent Magnetic Diffusion and Mean-Fi

arXiv:astro-ph/0206261v116Jun2002PHYSICALREVIEWE,v.64,056307(2001)Nonlinearturbulentmagneticdiffusionandmean-fielddynamoIgorRogachevskii∗andNathanKleeorin†DepartmentofMechanicalEngineering,TheBen-GurionUniversityoftheNegev,POB653,Beer-Sheva84105,Israel(Dated:February5,2008)Thenonlinearcoefficientsdefiningthemeanelectromotiveforce(i.e.,thenonlinearturbulentmagneticdiffusion,thenonlineareffectivevelocity,thenonlinearκ-tensor,etc.)arecalculatedforananisotropicturbulence.Aparticularcaseofananisotropicbackgroundturbulence(i.e.,theturbulencewithzeromeanmagneticfield)withonepreferentialdirectionisconsidered.Itisshownthatthetoroidalandpoloidalmagneticfieldshavedifferentnonlinearturbulentmagneticdiffusioncoefficients.Itisdemonstratedthatevenforahomogeneousturbulencethereisanonlineareffectivevelocitywhichexhibitsdiamagneticorparamagneticpropertiesdependingonanisotropyofturbulenceandlevelofmagneticfluctuationsinthebackgroundturbulence.Thediamagneticvelocityresultsinthefieldispushedoutfromtheregionswithstrongermeanmagneticfield,whiletheparamagneticvelocitycausesthemagneticfieldtendstobeconcentratedintheregionswithstrongerfield.Analysisshowsthatananisotropyofturbulencestronglyaffectsthenonlinearturbulentmagneticdiffusion,thenonlineareffectivevelocityandthenonlinearα-effect.Twotypesofnonlinearities(algebraicanddynamic)arealsodiscussed.Thealgebraicnonlinearityimpliesanonlineardependenceofthemeanelectromotiveforceonthemeanmagneticfield.Thedynamicnonlinearityisdeterminedbyadifferentialequationforthemagneticpartoftheα-effect.ItisshownthatfortheαΩaxisymmetricdynamothealgebraicnonlinearityalone(whichincludesthenonlinearα-effect,thenonlinearturbulentmagneticdiffusion,thenonlineareffectivevelocity,etc.)cannotsaturatethedynamogeneratedmeanmagneticfieldwhilethecombinedeffectofthealgebraicanddynamicnonlinearitieslimitsthemeanmagneticfieldgrowth.PACSnumbers:47.65.+a;47.27.-iI.INTRODUCTIONGenerationofmagneticfieldsbyturbulentflowofconductingfluidisafundamentalproblemwhichhasalargenumberofapplicationsinsolarphysicsandas-trophysics,geophysicsandplanetaryphysics,etc.Inrecenttimetheproblemofnonlinearmean-fieldmag-neticdynamoisasubjectofactivediscussions(see,e.g.,[1,2,3,4,5,6,7,8,9,10]).Itwassuggestedin[11]thatthequenchingofthenonlinearα-effectisverystrongandcausesaveryweaksaturatedmeanmagneticfield.However,thelatersuggestionisindisagreementwithob-servationsofgalacticandsolarmagneticfields(see,e.g.,[12,13,14,15,16,17])andwithnumericalsimulations(see,e.g.,[18,19,20]).Saturationofthedynamogeneratedmeanmagneticfieldiscausedbythenonlineareffects,i.e.,bythebackreactionofthemeanmagneticfieldontheα-effect,tur-bulentmagneticdiffusion,differentialrotation,etc.TheevolutionofthemeanmagneticfieldBisdeterminedby∗Electronicaddress:[e-mail];URL:~gary†Electronicaddress:[e-mail]∂B/∂t=∇×(V×B+E−η∇×B),(1)whereVisameanvelocity(e.g.,thedifferentialro-tation),ηisthemagneticdiffusionduetotheelectri-calconductivityoffluid.ThemeanelectromotiveforceE=hu×biinananisotropicturbulenceisgivenbyEi=αijBj+(Veff×B)i−ηij(∇×B)j−κijk(∂ˆB)jk−[δ×(∇×B)]i(2)(see[21,22]),where(∂ˆB)ij=(1/2)(∇iBj+∇jBi),uandbarefluctuationsofthevelocityandmagneticfield,respectively,angularbracketsdenoteaveragingoveranensembleofturbulentfluctuations,thetensorsαijandηijdescribetheα-effectandturbulentmagneticdiffu-sion,respectively,Veffistheeffectivediamagnetic(orparamagnetic)velocity,κijkandδdescribeanontriv-ialbehaviorofthemeanmagneticfieldinananisotropicturbulence.Nonlinearitiesinthemean-fielddynamoim-plydependenciesofthecoefficients(αij,ηij,Veff,etc.)definingthemeanelectromotiveforceonthemeanmag-neticfield.Theα-effectandthedifferentialrotationarethesourcesofthegenerationofthemeanmagneticfield,whiletheturbulentmagneticdiffusionandtheκ-effect(whichisdeterminedbythetensorκijk)contributetothedissipationofthemeanmagneticfield.2Inspitethenonlinearα-effectwasunderactivestudy(see,e.g.,[5,7]),thenonlinearturbulentmagneticdif-fusion,thenonlinearκ-effect,thenonlineardiamagneticandparamagneticeffects,etc.arepoorlyunderstood.Inthepresentpaperwederivedequationsforthenonlinearturbulentmagneticdiffusion,thenonlineareffectivevelocity,thenonlinearκ-effect,etc.forananisotropicturbulence.Theobtainedresultsforthenonlinearmeanelectromotiveforcearespecifiedforananisotropicbackgroundturbulencewithonepreferen-tialdirection.Thebackgroundturbulenceistheturbu-lencewithzeromeanmagneticfield.Wedemonstratedthattoroidalandpoloidalmagneticfieldshavedifferentnonlinearturbulentmagneticdiffusioncoefficients.Itisshownthatevenforahomogeneousturbulencethereisanonlineareffectivevelocitywhichcanbeadiamagneticorparamagneticvelocitydependingonanisotropyoftur-bulenceandlevelofmagneticfluctuationsintheback-groundturbulence.II.THEGOVERNINGEQUATIONSInordertoderiveequationsforthenonlinearturbu-lentmagneticdiffusionandothernonlinearcoefficientsdefiningthemeanelectromotiveforcewewilluseameanfieldapproachinwhichthemagnetic,H,andvelocity,v,fieldsaredividedintothemeanandfluctuatingparts:H=B+b,v=V+u,wherethefluctuatingpartshavezeromeanvalues,V=hvi=const,andB=hHi.Themomentumequationandtheinduct