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!  ToincludecollisionsweusetheFokker-Plankcollisionoperator,whichisdiscretizedaccordingtothenumericalDGschemeusedintheVlasovsolver

!  Inclusionofacollisionoperatorprovidesregularizationofphasespacestructureandanoutletforenergydissipation

!  Collisionsalsocreateaphysicalsourceofentropyincreaseandhencemakestheevolutionofthedistributionanirreversibleprocess

InvestigationofStochasticHeatinginLowCollisionalityPlasmas

Signsofstochasticmotion(chaos)

Effectofcollisionoperatoronheating

Current,entropy,andpressureinpassivesimulations

Self-consistentsimulations

ReferencesandAcknowledgements

ContinuumVlasov-Maxwellsolver

JackSchroeder1,AmmarHakim2,JimmyJuno3,JasonTenBarge2

1UniversityofWisconsin-Madison,Madison,WI,2PrincetonPlasmaPhysicsLaboratory,Princeton,NJ,3IREAP,UniversityofMaryland,CollegePark,MD

!  The effect of stochastic heating in plasma is important inunderstandingtheheatingofthesolarcoronaandthesolarwind

! WiththerecentlaunchoftheParkerSolarProbePlusthatwill gather unprecedented data of the solar wind andcorona, further theoretical investigation of stochasticheatingprocessesisdesiredtocomparewithexperimentaldata

! What does it take to drive particles into stochastic motion? A time andspatiallyvaryingEfieldperpendiculartoaconstantBfieldissufficient

!  ThisseriesPoincareplotsatvaryingvaluesofE0withB0=1,ω=0.4567,and

k=1/ρe

!  TheVlasov-Maxwell systemprovidesanaccuratekineticmodel forweaklycollisionalplasmas

!  Directly solving a discretized Vlasov system avoids counting noise from

which traditional particle-in-cell methods suffer, especially important forresolvingstochasticityinanon-linearsystem

!  HereaMaxwelliandistributionfunctionisevolved“passively,”meaningevolvingtheVlasovequationwithoutMaxwell’sequationsthatcoupletheplasmacurrentstotheelectromagneticfields

!  Inthefieldconfigurationaboveforthehighlystochasticregime,flatteningofthespatiallyintegrateddistributionisasignatureofstochasticparticlemotions 1D2Vsimulation,Nx=48,Nvx=48,Nvy=48,

polyOrder=3

[1]A.H.Hakim,SimJournalentryJE32.Availablefrom:http://ammar-hakim.org/sj/je/je32/je32-vlasov-test-ptcl.html[2]J.Junoetal.,DiscontinuousGalerkinalgorithmsforfullykineticplasmas,JComput.Phys.353(2018)110-147[3]B.D.G.Chandranetal.,PerpendicularIonHeatingbyLow-frequencyAlfvén-waveTurbulenceintheSolarWind,ApJ.720(2010)503-515! Codecanbeaccessed

!  https://bitbucket.org/ammarhakim/gkyl/src/default/!  condainstall-cgkylgkyl

! HighperformancecomputingwasprovidedbyStampede2,TexasAdvancedComputingCenter(TACC)locatedattheUniversityofTexasatAustin

! ThisprojectwassupportedbytheDOESummerUndergraduateLaboratoryInternshipprogram

!  Inpassivesimulationstheplasmaseesonlytheuserspecifieddrivingfields:

!  Thisresultsinaclearcorrelationbetweentheplasma

current,numericalentropy,andpressureinrelationtheoscillatingelectricfield

!  Aboveshownisaπ/2phaseshiftbetweentheplasma

currentandelectricfieldinacollisionalsimulationwellaftertheresonantheatingperiodends;asimpleFourieranalysisofAmpere’slawyieldsthisexpressionforthecurrent:

Thefactorofimaginaryimathematicallyrepresentsaπ/2phaseshift,affirmingthesimulationoutput!  Asimilarphaseshiftisseenwiththenumericalentropyand

theplasmapressure Summaryandfuturework

!  Drivingfieldsproducestochasticmotionsintheplasmaparticleswhichthenleadstoanenergytransferintotheplasmathermalenergy

!  Lookingatthisenergytransferin“passive”simulationswithvaryingdegreesofcollisionalityrevealshowcollisionscanmediatestochasticheating

!  Inthecollisionlesscaseparticlesresonantwiththeelectricfieldgainenergyupuntilalmost20cyclotronperiods;afterparticlesaredrivenoff-resonancethethermalenergyreachesaquasi-steady-state

!  Additionofcollisionsprovidesanoutletforfurtherenergytransferfromtheelectricwavestotheparticles

!  Thetransitionfromnonlinearresonantheatingtocollisionalheatingcanbeseenwherethecurvesbegintodiverge

!  Keytakeaway:withcollisionalitythethermalenergycanincreasewithoutboundwhenparticlecurrentsdonotfeedbacktofields

!  Line-outsoftheintegrated2Ddistributionfunctiontakenatvy=0andt=100Ωce

-1withsuperimposedMaxwelliandistributionsofthetemperatureindicatedbytheintegratedthermalenergyvalueatthattimeshowthehighlynon-Maxwelliannatureofthedistributions

!  Notethatpresenceofphasespacestructureinthecollisionlesssimulationissmoothedoutwhencollisionsareincluded

CollisionOperator

@f

@t= ⌫

@

@v·(v � u) f + v2t

@f

@v

�

Contact:JackSchroeder,schroeder24@wisc.edu;AmmarHakimahakim@pppl.gov

distributionfunctionsatt=100Ωce-1(elongatedaxisisspace,otheraxesarevxandvy)

left:collisionlesssimulation;rightcollisionalsimulationν/Ωci=0.01

!  Evolvingthedistributionfunctionself-consistentlysuchthattheparticlecurrentsfeedbacktothefieldsbysolvingMaxwell’sequationswiththeVlasovequationprovidesalookatamorephysicalsituation

!  Thesamesimulationsranbeforewithself-consistencyturnedonleadstonegligibleincreasesinthermalenergy

!  Rapidplasmaresponseseverelydampsouttheoscillatingelectricfieldandleadstoveryminimalperturbationtothedistributionfunctionandinhibitsstochasticmotions

!  Keytakeaway:introductionofself-consistencyleadstoaplasmaresponsethatdramaticallycancelsthefieldsandpreventsanystochasticityandsubsequentheating

!  Amorecomplicateddriverthanasingleelectricwavemodemayberequiredtoseestochasticparticlemotions

!  Drivingamagnetizedplasmawithahighamplitudeperturbingelectricfieldcausesstochasticparticleorbitsandnonlinearresonancesthatleadtoparticleheating

!  Collisionsintroducearouteformoreelectromagneticfieldenergytodissipateintotheplasmathermalenergy

!  Self-consistencyleadstocompletedampingofdrivingfield

!  Futureworkwillinclude“priming”theself-consistentsimulationbyinitializingfromadistributionthathasthecurrentsgeneratedfromthedrivingfieldsofapassivesimulation;alsoinvestigatingself-consistentsimulationswithdelta-andtime-correlateddrivers

Abstract:Theeffectofstochasticheatinginplasmaisimportantinunderstandingtheheatingofthesolarcoronaandthesolarwind.WiththecominglaunchoftheParkerSolarProbePlusthatwillgatherunprecedenteddataofthesolarwindandcorona,furthertheoreticalinvestigationofstochasticheatingprocessesisdesiredtocomparewithexperimentaldata.InthisstudyweusethecontinuumcodeGkeylltosolvetheVlasov-Maxwellequationsfortheevolutionofaplasmadistributionfunction.Usingacontinuumsolveravoidsnumericalissuessuchasparticlenoisecharacteristicoftraditionalparticle-in-cellmethods.Weinvestigatetheevolutionofadistributionfunctionindifferentelectromagnetic(EM)fieldconfigurationssuchasthoseobservedinsolarwindconditions.Inadditiontotestparticlesimulations,wewillcomparetodistributionsevolvedself-consistently,suchthattheparticlemotionfeedsbacktotheseedEMfield,andincludingparticlecollisioneffects.Energytransferfromfieldstoparticleswillbestudiedtodeterminetheextenttowhichstochasticmotionofparticlesleadstoparticleheating.