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1 Simulations of the Current Filamentation Instability Thecurrentfilamentationinstabilitywasfirstobservedin3 Dsimulationsof reconnectionwithanambientguidefieldthatisfivetimesthereversedfield.we havecarriedoutavarietyofsimulationstotrytoestablishthephysicsofthe instabilityandtherangeofparameterswhereitisexpectedtobeimportant.we firstcarriedoutareconnectionsimulationidenticaltothecasediscussedindetailin thepaperbutwithaambientguidefieldof2.5timesthereversedfieldina computationaldomainthatwas L x L y L z = 8d i 4d i 4d i.amovieoftheresults ofthesimulationisincludedinthesupplementarymaterial(jez_b025.mov).shown isacutoftheelectronout of planecurrent J ez intheplaneofreconnectionandin they zplaneinacutthroughtheinitialx line(thecoordinatesarethesameasin thepaper,wherezisinthedirectionoftheguidefieldandtheprimarycurrent).the turbulenceismuchstrongerthaninthecaseofaguidefieldoffive,sostrongthat thebroadeningofthecurrentlayernearthex linetriggerstheformationofa secondarymagneticisland.theasymmetricspreadingoftheturbulencealongtwo oftheseparatrices,apossibleobservationalsignatureofthemodel,isalsoevident. Oncetheturbulenceonsets,itremainsstrongforthedurationofthesimulation. Cutsthroughthex linerevealthatthecurrentlayergoesthroughseveralcyclesof thinningandturbulentbroadeninguntilsettlingintoaquasi stationarystate. Inordertopindownthephysicalmechanismdrivingtheinstabilitywehavecarried outaseriesofsimulationsofnarrowelectroncurrentlayersthatareintendedto modelthelate timecurrentlayerthatformsasaresultofreconnection.therighthandpolarizationoftheinstabilityseeninthereconnectionsimulationssuggests thattheinstabilityispartofthewhistlerbranchandthereforethattheionsplay littleroleintheinstability sdevelopment.totestthis,wecompletedsimulations withvaryingvaluesoftheelectron to ionmassratioand,ultimately,thelimiting caseofstationaryions(infinitemassions).asexpected,theinstabilityhaslittle dependenceonmassratio.asafurtherchecweasedacollaborator(dr.paul Cassa)toperformsimilarsimulationswithanelectronMHD(EMHD)codeinwhich theionfluidisstrictlystationaryandcannotactasanythingotherthanapassive bacground.wefindthataninstabilitydevelopsinbothdescriptionswithsimilar growthratesandwavenumbersandwithsimilarstructureatlatetime. ThesimulationdomainforthePICelectroncurrentlayersimulationsisgivenby L x L y L z =10d e 10d e 80d e,wheredeistheelectroninertiallength.herethe normalizationsareintermsofelectronquantities(ratherthanthatoftheions) becausetheelectronsplaythedominantroleasthedriveroftheinstability.the initialstateisacurrentlayerwithwidthequaltoanelectroninertiallengthandno perturbations,asidefromthoseduetoinherentparticlenoise,areimposed.the densityandtemperatureoftheplasmaareinitiallyconstantwhiletheguidefield (herewithanasymptoticvalueequalto2.5timesthefieldsupportingthecurrent 1

2 sheet)variestoensurepressurebalance.thedomainisperiodicinthexandz directionsandhasconductingwallsintheydirection. Figure1showsthedevelopmentoftheinstabilityinthePICsimulationsforthecase ofstationaryionsatthreetimes:t=0,t=80andt=130inverseelectroncyclotron times.eachplotshowsthezcomponentoftheelectroncurrentdensityinthesame yz plane.thetoppanelshowstheinitialquiescentstate.inthesecondpanelthe instabilityhasbecomenoticeable,whilethethirdshowsasnapshotatlatetime whentheentirecurrentsheethasfilamented. InthenextplotweshowequivalentdatafromanEMHDsimulationofthesame system.inthiscaseinitialperturbationsareimposedtoseedtheinstability.the instabilityintheemhddescriptionhasasomewhatlargergrowthrate(~afactorof 1.5)andthewavelengthofthedominantmodeissomewhatsmaller(~afactorof2) soineticeffectsdohavesomeimpact.nevertheless,theoveralldevelopmentofthe instabilityisqualitativelysimilarinbothsimulations. 2

3 WehaveperformedaseriesofPICsimulationstestingthedependenceofthe instabilityonvariousquantitiesofinterest:magnitudeoftheguidefield, temperatureoftheplasma,electron to ionmassratio.thevariationwiththeguide fieldisdiscussedinthearticle(seefigure1c).raisingthetemperaturedrivesthe instabilitytolongerwavelength.wefindthatbelowanelectrontoionmassratioof 1/100theinstabilitydoesnotappreciablyvaryinitsstructureorgrowthrate. FinallywehaveexploredthedynamicsofcurrentlayersinitializedwithHarris equilibria.inthesecaseswehavetaentheguidefieldtobeuniformandequalto 2.5timesthereversedfieldandhaveaddedauniformdensity0.5timestheHarris densityn0.theratiooftheiontoelectrontemperaturewastaenas4.0,whichis representativeoftheearth smagnetosphere.theinterestingresultisthatthe instabilityiscompletelystableiftheionsaretaentobestationary(infinitemass ions)buttheinstabilityisrobustforamassratioof1600.wesuspectthatwith stationaryionstheelectronscannotmoveinthedirectionofthedensityandcurrent gradientsbecauseofthelargechargeseparationthatwouldresult.fortheharris equilibriumthegrowthrateisaroundafactoroftwosmallerthanintheforce free casebutthestructureoftheinstabilityisbasicallyunchanged. Linear Stability Analysis Based on the Electron-MHD Equations Our PIC current simulations reveal that the instability seen in the 3-D reconnection simulations is insensitive to the ion mass. In the frame of our PIC simulation, the timedependence of the transverse electric fields reveal that it is a right-hand circularly polarized electromagnetic wave and hence is part of the whistler/electron-cyclotron branch. Further 3-D electron-mhd simulations of narrow current layers reveal that the nonlinear development is striingly similar to that of the PIC model. The spatial correlation between the current density and transport in Fig. 2 of the paper suggests that it is driven by the current density gradient. A simple fluid description of electromagnetic waves in this regime is given by the electron MHD equation, 3

4 t P B = 1 ne P = 1 d e 2 2. ((PB ) J ); (S1) Linearization of this equation in the presence of a local current density gradient J ez '= dj ez /dy with growth rate γ and wavevectors z and x along and across B yields the dispersion relation, γ γ i v 2 z ez 2 = Ω z d e P ez 2 P z 2 d 2 e + P x J ez ' neω ez ; P = 1+ 2 d 2 e, (S2) where γ = γ + i z v ez, 2 = 2 x + 2 z and v ez = J ez /n e e is the local electron streaming velocity. In the absence of J ez ' this dispersion relation yields whistler/electron-cyclotron waves. The current gradient causes instability. Defining the small parameter ε = J ez '/neω ez, the growth rate peas at γ Ω ez ε /2 for x d e 1 with z / = ε /2. The parallel streaming velocity v ez has a stabilizing influence on long wavelength modes, an effect that increases with decreasing guide field and therefore explains the absence of the instability during anti-parallel reconnection. Defining a normalized streaming velocity by v ˆ = v ez /(2d e Ω ez ), the growth rate in the long wavelength limit is given by γ d e Ω ez ε /(2ˆ v ) for z / ε /(2ˆ v 2 ). As the guide field becomes small v ˆ becomes large, the growth rate drops and the parallel wavelength increases, which is consistent with the general the trends of the PIC simulations. Of course, a inetic treatment will be required to refine this description of the filamentation instability and in particular to properly describe the limit in which the thermal spread of the electron distribution function is comparable to the streaming velocity. ThemagneticviscouscontributiontoOhm slaw Thecorrelationofthefluctuationsinthemagneticfieldwiththoseofthetransverse currentoftheelectronsproducesaviscouscontributiontoohm slaw.acompact formforthiscontributioncanbecalculatedinthelimitofastrongguidefieldb0z andelectromagneticturbulenceforwhich z ismuchsmallerthan,thegradient transversetothezdirection.thus,wedevelopaformalorderinginwhich B /B z ~ z / ~ /Ω e ~ ε and d e ~ 1.Wefurtherconsiderthelimitinwhichthe ioncurrentcanbeneglectedcomparedwiththatoftheelectrons.thisorderingis consistentwiththescalinglawsforthefilamentationinstabilitydiscussed previouslyandwiththeelectromagneticturbulenceseeninthereconnectionand currentlayersimulations.werewriteeq.(s1)as PB 1 J PB = 1 ne ne P B J.(S3) 4

5 Ignoringforthemomenttheconvectivederivativeandrecallingthat P 1andthat ~ B ~ B 0z z ~ ε,wefind B ~ J andδb z = B z B 0z ~ J z.since J = c B /4π, wehave B ~ J z.thus,theaxialandtransversemagneticfieldandcurrent perturbationsareofthesameorder.thescalingoftheseperturbationswithε followsfrombalancingthetimederivativewiththeconvectivenonlinearity: ~ J ~ J ~ J.Thetransverseandaxialperturbationsarethereforefirst orderinε.therepresentationfor B followsfromtherequirementthat B ~ B 0or B = z ˆ sothatthetotalfieldandcurrentcanbewrittenas B = z ˆ (B 0z +δb z ) + z ˆ + O(ε 2 )(S4) J = ˆ z 2 + δb z z ˆ + O(ε 2 ).(S5) Theelectroncurrentinthetransversedirectionisthereforetolowestorder divergence free.withtheseexpressionsforthemagneticfieldandcurrentthez componentofthelorentzforceassumestheform, 1 c ( J ).(S6) Equation(2)ofthemainpaper,wheretheviscoustransporttermiswritteninterms ofthetransportofcanonicalmomentum,followsimmediately. 5