Modeling space plasma turbulence at the ion gyroscale：空间等离子体湍流模型在离子gyroscale

Modelingspaceplasmaturbulence

attheiongyroscale

P.L.Sulemincollaborationwith

T.Passot,D.Laveder,L.Marradi,P.Hunana,D.BorgognoVlasov-Maxwellkinetiics:theory,simulationsandobservationsinspaceplasmas,WPI,Vienna,March29-April1,2011OUTLINESpaceplasmas:MainfeaturesanddebatedquestionsHowtomodelthesolarwind?FLR-LandaufluidsimulationsSpaceplasmasaremagnetizedandturbulentβparameterisusuallyclosetoorlargerthanunitySonicMachnumberisoforderunityPlasmafluctuationsspanahugerangeofscales,displayingpowerscalinglawsthatextenddowntotheiongyroscaleswherekineticeffectscannotbeignored.Amongthedebatedquestions:

Spectralenergydistributionanditsanisotropy

Dissipationmechanisms

Heatingoftheplasma;temperatureanisotropy

ParticleaccelerationTheconceptsofwavesmakesenseeveninthestrongturbulenceregime.Dispersionisnonnegligibleattheionscales:coherentstructures.Spaceplasmas:MainfeaturesanddebatedquestionsDensityspectrumintheISMAmstrong,Rickett&Spangler,ApJ1995SpectrumofthemagneticfluctuationsinthesolarwindGolstein,Robert&Matthaeus,SpaceplasmassuchasthesolarwindortheEarthmagnetosheath:NaturallaboratoriesforaccurateinsitumeasurementsTurbulentmagnetizedplasmas

withessentiallynocollisions.CascadesextendbeyondtheionLarmorradiusSmall-scalecoherentstructures(filaments,shocklets,magnetosonic

solitons,magneticholes)

withtypicalscalesofafewionLarmorradius.Dispersiveandkineticeffectsplayanimportantrole.Thesolar-terrestrialenvironment

Sahraouietal.PRL102,231102(2009)ExcessofmagneticenergyinthetransversecomponentsSeveralpower-lawranges:Whichwaves?Whichslopes?Importanttoestimatetheheating.(Ngetal.JGR115,A02101;2010)Atwhatscalesdoesdissipationtakeplace?Bywhatmechanism?Mainfeaturesofsolarwindplasmaprotongyrofrequencyperpendicularmagneticspectrumparallelmagneticspectrum~K41electrongyrofrequencyAlexandrovaetal.Planet.SpaceSci.55,2224(2007)k-filtering->

wavevectorsarehighlyobliquerelativelytotheambientfield(θ=86°)SpectralanisotropyNonresonnantheatingoftheprotons,possiblyviaLandaudampingDoesthespectralanisotropypersistatsmallscales?Sahraouietal.PRL102,231102(2009)Alexandrovaetal.JGR111,A12208(2006).DriftkineticAlfvénvorticesalsoobservedinthecuspregion.(Sundkvistetal.Nature,436,825,2005)andofcoherentstructures:-CurrentfilamentsandAlfvénvortices-Mirrorstructures(magneticholesandhumps)MainfeaturesofterrestrialmagnetosheathplasmaImportantroleofthetemperatureanisotropy:leadstomicro-instabilitiesAIC(nearquasi-perpendicularshock)andmirrorinstabilities(furtherinsidemagnetosheath)PresenceofmirrormodesspatialspectrumsteeperthantemporaloneAlfvénvorticesIdentifiedusingk-filteringtechnique(Pinçon&Lefeuvre,JGR96,1789,1991):HaveessentiallyzerofrequencyintheplasmaframeSahraouietal.,PRL96,075002(2006)fastmagnetosonicshocklets

(Stasiewiczetal.GRL2003)Slowmagnetosonicsolitons

(Stasiewiczetal.PRL2003)Mirrorstructuresintheterrestrialmagnetosheath

(Souceketal.JGR2008)Signatureofmagneticfilaments

(Alexandrovaetal.JGR2004)StatisticalstudyoftemperatureanisotropiesinthesolarwindTurbulence(and/orsolarwindexpansion)cangeneratetemparatureanisotropyThisanisotropyislimitedbymirrorandobliquefirehoseinstabilities.Hellingeretal.GRL33,L09101(2006)seealsoBaleetal.PRL103,21101(2009)Howtomodelthesolarwind?Thesolarwindisonlyveryweaklycollisional:thissuggestskineticsimulationsVlasov-Maxwellsimulations:hardlypossibleonthepresentdaycomputersinthreespacedimensions(6variables+time,andabroadrangeoftimescales).Gyrokineticsimulations

(G.Howes,PoP15,055904,2008)

arenowfeasible

and

showthepresenceofcascadesbothinthephysicalandvelocityspacesintherangek┴ρ≥1.Gyrokinetictheory(Howes,ApJ651,2006,

Schekochihinetal.,ApJSupp.,182,310,2009),

concentratesonthequasi-transversedynamicsandaveragesoutthefastwaves.Applicabilitytospaceplasmasofthegyrokinetictheoryisstilltobevalidated.Gyrokineticsimulationsremainchallengingnumericallyanddifficulttointerpret.Oneneedsa

fluidmodelthat(evenifnotrigorouslyjustified)

can

be

integrated

relatively

fast,

allowsforstrong

temperatureanisotropies

doesnotaprioriorderoutthefast

magnetosonic

waves.Simplestapproach:theincompressibleMHDOnlyonetypeofwaves:Alfvénwaves.Interactionbetweencounter-propagative

AlfvénwavesClearconceptofcascades(fortheElsasservariables)bothinthecontextofweakandstrongturbulence.BalanceorunbalanceregimedependingonequalorunequalenergyfluxesassociatedwithAlfvénwavesmodespropagatingalongtheambientmagneticfield.PossibilitytodevelopaphenomenologyclosetoK41(inspiteoflessuniversality)Relativelysimpleequations,whichpermitshigh-resolutionsimulationsFluidapproaches:TheMHDanditsextensionsProvidesareasonabledescriptionatscaleslargecomparedtoplasmamicroscalesObservationalevidenceofsmalldensityfluctuationInfact,compressibilitycannotbetotallyneglectedMatthaeusetal.JGR1991Bavassano&Bruno,JGR1995ForincompressiblehomogeneousisotropicMHDturbulence,existenceofananalogoustothe4/5lawofKarman-Howarthforfluidturbulence,givingstatisticsof3rdordermomentsforvelocityincrementsElsässervariables:Longitudinalcomponents(Politano&Pouquet,GRL25,273,1998).Roleofcompressibility,evenatlargescalesCarboneetal.PRL103,061102(2009)Takecompressibilityintoaccountinaphenomenologicalway:Significanteffectdespitesmalldensityfluctuations.incompressiblecompressibleCompressibleMHD(possiblyretainingHalleffect:Hall-MHD)Richerdynamics:variouskindsofwaves:energyistransferrednotonlybetweenthedifferentscales

butalsobetweenthedifferenttypesofwaves.ExtensionoftheconceptsofinertialcascadeisnotstraightforwardForcedone-dimensionalHall-MHDsimulations

(Lavederetal.2010):EnergytransfertakesplacefromAlfvéntomagnetosonicmodes,

invalidatingthenotionofanAlfvénwaveinertialcascade.Aparallelcascadecoulddevelopatsmalldispersivescales(Yoon&Fang,PPCF50,0985007,2008)ParallelpropagationHMHD1D:parallelpropagationStructuresinpressureequilibriummagnetickineticSpectralbreaksassociatedwithcoherentstructuresthatformatdispersiveionicscales.HMHD1D:quasi-transversepropagation(80°)(kineticdriving)magneticpressurethermalpressuretheirhalfsumvxPressurebalanceperturbedbydispersiveshocksPresenceofashockAbsenceofshocksSpectraofthetransversecomponentsKineticspectrumissteeperthanmagneticone(incontrastwithparallelpropagation)magnetickinetickineticmagneticCompressibleMHDretainswavesthataredampedbyLandauresonanceinVlasov-Maxwelldescriptionofcollisionlessplasmas(Howes,NPG16,219,2009)MHDoverestimatescompressibilityandenergytransferalongtheambientfield.Servidioetal.(PSS55,2239,2007):

ationicscales,spontaneousgenerationofquasi-perpendicularMSwaves:magnetosonicturbulence(anti-correlationdensitymagneticintensity).Contrastswithsolarwindobservations:turbulenceofquasi-transverse(kinetic)Alfvénwaves(Sahraouietal.PRL105,131101,2010).Solarwinddisplaystemperatureanisotropy.Thesimplestmodelretainingtemperature(orpression)anisotropy:doubleadiabaticapproximation:(Chewetal.,Proc.R.Soc.LondonA236,112,1956).Alsocalled“CGL”(forChew,Goldberger&Low).Assumesagyrotropicpressuretensor(neglectsFLRcorrections)UsesasimpleOhm’slawwithnoHalleffectnorelectronpressuregradientNeglectsheatfluxes.Leadsofconservationofalongflowtrajectories.

Whentheplasmaisdriven,temperatureanisotropycandevelop:Beyondthreshold,microinstabilities(i.e.themirrorinstability)takeplace.Fortheproblemtobelinearlywell-posed,instabilitiesshouldbearrestedatsmallscales.FiniteLarmorcorrections(nongyrotropiccontributions)aretoberetainedtoarrestthemirrorinstabilityatsmallscales.andIgnoringLandaudamping,CGLalsopredictsawrongthresholdformirrorinstability.Itisnecessarytoretain

Landaudamping

todepletesonicwaves(andensureaweakercompressibility)tocorrectlycapturethemirrorinstabilitythresholdFLRcorrectionstoarrestthemirrorinstabilityatsmallscalesTheselow-frequencykineticeffectsshouldbeincludedinafluidapproach

(inawaythatdoesnotinducespurioussmall-scaleinstabilities).Gyrofluids:consistsinclosingthehierarchyofmomentequationsderivedfromthegyrokineticequation.Landaufluids:extensionofanisotropicMHDincludinglow-frequencykineticeffects:consistentwiththelinearkinetictheoryevenatsmalltransversescales.•IntroducedbyHammett&Perkins

(PRL64,3019,1990)asaclosureretaininglinearLandaudamping.•Appliedtolarge-scaleMHDbySnyder,Hammett&Dorland(PoP4,3974,1997)

toclosethehierarchyofmomentequationsderivedfromthedriftkineticequation.•ExtendedtodispersiveMHDwithHalleffectandlargescaleFLRcorrections

(Passot&Sulem,PoP10,3906,2003;Goswami,Passot&Sulem,PoP12,102109,2005)

Inclusionofquasi-transversescalesextendingbeyondtheiongyroscale,underthe

gyrokineticscaling

(Passot&Sulem,PoP

14,082502,2007):FLR-Landaufluids.FLRLandau-fluidsarebasedonafulldescriptionofthehydrodynamicnonlinearities,supplementedbyalinear(orsemi-linear)descriptionoflow-frequencykineticeffects(LandaudampingandFLRcorrections),

withinthegyrokineticscaling.Incontrastwithgyrokinetics,Landaufluidsretainfastwavesthatareaccuratelydescribeduptotheiongyroscale.Landaufluids(andalsogyrofluids)neglectwaveparticletrapping,i.e.theeffectofparticlebouncemotiononthedistributionfunctionnearresonance.Fluiddescriptionretaininglow-frequencykineticeffects:LandaufluidmodelsTheclosureisusuallyperformedatthelevelofthefourthordermoments.Theforthordercumulantsareobtainedfromthelinearizedkinetictheory,assumingsmallfrequencieswithrespecttotheiongyrofrequency

andeitherlongwavelengthswithrespecttotheiongyroradiusorquasi-perpendiculardirections.Thenon-gyrotropicpartsofthepressure,heatfluxandfourthordertensorsarealsoexpressedusingthekinetictheory.INPRACTICE:Theabovekineticexpressionstypicallydependonelectromagneticfieldcomponentsandinvolvetheplasmadispersionfunction(whichisnonlocalbothinspaceandtime).Thesevariousexpressionscanbeexpressedintermsofotherfluidmomentsinsuchawayastominimizetheoccurrenceoftheplasmadispersionfunction.ThelatterisotherwisereplacedbysuitablePadéapproximants,thusleadingtolocal-in-timeexpressions.Atsomeplaces,aHilberttransformwithrespecttothelongitudinalspacecoordinateappears,thatmodelizesLandaudamping.BriefdescriptionoftheclosureprocedureLandaufluidsForthesakeofsimplicity,neglectelectroninertia.Iondynamics:derivedbycomputingvelocitymomentsfromVlasovMaxwellequations.=B/|B|.

Electronpressuretensoristakengyrotropic(consideredscales>>electronLarmorradius)andthuscharacterizedbytheparallelandtransversepressuresBFLRcorrectionsForeachparticlespecies,PerpendicularandparallelpressuresheatfluxtensorEquationsfortheparallelandperpendicular(gyrotropic)heatfluxesInvolvethe4thrankgyrotropiccumulants:

standforthenongyrotropiccontributionsofthefourthrankcumulants.workofthenongyrotropicpressureforce2mainproblems:Closurerelationsareneededtoexpressthe4thorder

cumulants

(closure

at

lowest

order

alsopossible,although

usually

less

accurate)(2)FLRcorrections(non-gyrotropic)tothevariousmomentsaretobe

evaluated.Thestartingpointforaddressingthesepointsisthelinearkinetictheoryinthelow-frequencylimit:

Ω:iongyrofrequencyForaunifieddescriptionoffluidandkineticscales,FLRLandau-fluidsretaincontributionsof:•quasi-transversefluctuations•hydrodynamicscaleswith1:ionLarmorradius

replacedbyinstantaneousmeanvaluesinordertotakeintoaccounttheglobalevolutionoftheplasmaHilberttransformSimilarly,thegyroviscoustensoriscomputedbycombiningvariousfluidquantitiesobtainedfromthelinearkinetictheory,allowingtoeliminatemostoccurrencesoftheplasmadispersionfunction.Passot&Sulem,PoP14,082502(2007)Aftersubstitutiononegetaninitialvalueproblem:Thisleadstotheapproximation:Inordertotakeintoaccounttheglobalevolutionoftheplasma,replaceequilibriumquantitiesbyinstantaneousmeanvalues.Themodelconservesthetotalenergy:Conservationofenergyisindependentoftheheatfluxandsubsequentequations,butrequirestheintroductionoftheworkdonebytheFLRstressforces.ImplementationoftheLandaudampingviaHilberttransforms,andalsooftheFLRcoefficientsasBesselfunctionsofk┴ρ,iseasyina

spectralcode.ElectronLandaudampingisanessentialingredientinmanycases(limitingtherangeofvalidityoftheisothermalmodels).Mirrormodesgrowthrate:comparisonofFLR-Landaufluidwithkinetictheory(WHAMPcode)Frequencyanddampingrate

ofAlfvénwaves:

obliquepropagationDoesnotcaptureresonancequasi-transversepropagation(KineticAlfvénwaves)frequencydampingratefrequencydampingrateOnegets(inthelocalreferenceframe):TheinclusionofFLRsintheMHDequationsisconsideredinmanymodelsaimingtomodelizetheslowordriftdynamicsinfusionplasmas.WhenusingtheabovesimplifiedFLR’s,

themirrorinstabilityisstabilizedonlyveryclosetothresholdandspurioussmall-scaleinstabilitiescandevelop(thefunctionsГ0andГ1aretoberetained).Insomeinstances,themodelcansimplifybytakingthelargescalelimitTheabovesimplifiedFLRcanbeinappropriateinthepresenceoftemperatureanisotropy:DecayinstabilityofparallelAlfvénwavesinthelong-wavelengthlimitDrift-kineticanalysis(fromInhester1990)LandaufluidsimulationDecayinstabilityofAlfvénwaveproducesaforwardpropagatingacousticwaveandabackwardAlfvénwavewithawavenumbersmallerthanthatofthepump.ExamplesofuseofthesimplifiedFLRLandaufluid

Nonlinearsimulations:

Athree-dimensionalparallelcodewasdevelopedforthesimplifiedFLR-Landaufluid(D.Borgogno,D.Laveder,P.Hunana).(Borgognoetal.NPG16,275,2009)LandaufluidPICComparisonbetweenLandaufluidandhybridPICsimulationsPropagationofanAlfvénwaveinadensityinhomogeneity:parallelhighdensitychannelofsmallamplitude(10%)alignedwiththeambientfieldDensityhumpoflargeamplitude(100%)DensityfluctuationTransversedirectionDevelopementofstronggradientsandofscalessmallerthantheionLarmorradius.Thespatialsupportofsmall-scalestructuresismoreextendedthanwithoutdispersion(small-scalesconcentratedonlocalizedobliqueschocks).“Dispersivephasemixing”:importanceof3Dgeometry

andofionLandaudampingThissuggeststo

perform3DPICsimulationssimilartothe2DsimulationsofTsiklauriet