Draft version January 23, 2015 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 ELECTRON HEATING IN A RELATIVISTIC, WEIBEL-UNSTABLE PLASMA Rahul Kumar, David Eichler, and Michael Gedalin PhysicsDepartment,Ben-GurionUniversity,Be’er-Sheba84105,Israel Draft version January 23, 2015 ABSTRACT Thedynamicsoftwoinitiallyunmagnetizedrelativisticcounter-streaminghomogeneousion-electron plasma beams are simulated in two dimensions using the particle-in-cell (PIC) method. It is shown that current filaments, which form due to the Weibel instability, develop a large scale longitudinal 5 electric field in the direction opposite to the current carried by the filaments as predicted by theory. 1 Fast moving ions in the current filaments decelerate due to this longitudinal electric field. The same 0 longitudinal electric field, which is partially inductive and partially electrostatic, is identified as the 2 main source of acceleration of electrons in the current filaments. The transverse electric field, though n larger than the longitudinal one, is shown to play a smaller role in heating electrons, contrary to a previous claims. It is found that, in 1D, the electrons become strongly magnetized and are not J accelerated beyond their initial kinetic energy. Rather, the heating of the electrons is enhanced by 2 the bending and break-up of the filaments, which releases electrons that would otherwise be trapped 2 within a single filament and hence slow the development of the Weibel instability (i.e. the magnetic field growth) via induction as per Lenz’s law. In 2D simulations electrons are heated to about one ] quarter of the initial kinetic energy of ions. The magnetic energy at maximum is about 4 percent, E decaying to less than 1 percent by the end of the simulation. Most of the heating of electrons takes H place while the longitudinal electric field is still growing while only a small portion of the heating is . a result of subsequent magnetic field decay. The ions are found to gradually decelerate until the end h p of the simulation by which time they retain a residual anisotropy less than 10 percent. - o r 1. INTRODUCTION relatively less energetic upstream electrons are signifi- t s Collisionless shocks forming in astrophysical environ- cantlyheatedtoenergycomparabletotheenergyofions a astheycrosstheforeshock,whichappearstobeinagree- mentsarebelievedtobemediatedbyelectromagneticin- [ mentwiththeelectromagneticobservationsofsupernova stabilities(KennelandSagdeev1967;Parker1961;Eich- remnants and gamma-ray bursts afterglows, where the 1 ler 1979; Blandford and Eichler 1987) in which ions are high energy radiation is believed to be synchrotron radi- v scattered by the resulting magnetic field fluctuations. In ation originating from gyration of high energy electrons 6 particlular, the Weibel instability (Weibel 1959), which 6 causesfastgrowthofstrongmagneticfieldatsmall-scale in a magnetic fields significantly higher than the inter- 4 in anisotropic plasma flow, has received much attention stellar magnetic field (Panaitescu and Kumar 2002; Pi- 5 asthemainisotropizationmechanismthatleadstoshock ran 2005; Gehrels and M´esz´aros 2012). It however re- 0 transition in free-streaming ejecta from violent astro- mains unclear how the upstream electrons are energized . intheforeshockregion,bothinnatureandinthenumer- 1 physicalevents(MedvedevandLoeb1999;Wiersmaand ical simulations of collisionless shocks, and whether the 0 Achterberg 2004; Lyubarsky and Eichler 2006; Achter- Weibel instability-induced magnetic field should persist 5 berg and Wiersma 2007; Bret 2009; Yalinewich and over astrophysically significant length scales behind the 1 Gedalin 2010; Shaisultanov et al. 2012). As the domi- shock. : nant modes of the Weibel instability are less than the v Numerical simulations suggest presence of large scale ion gyroradius, which renders them inefficient scatterers i electricfieldinandaroundthecurrentfilamentsforming X of ions, the longstanding question about its role in col- just ahead of the shock and ions are found to decelerate lisions shocks has been how well it competes with other r due to this electric field. The same electric field that de- a mechanisms(e.g.Galeevetal.1964;BlandfordandEich- celeratesionsshouldalsoaccelerateelectrons(Blandford ler 1987; Lyubarsky and Eichler 2006). Clearly mecha- and Eichler 1987; Lyubarsky and Eichler 2006; Gedalin nisms that require a preexisting magnetic field are ques- et al. 2008, 2012). However, the details of the elec- tionable when the magnetic field is weak. On the other tric field and the acceleration mechanism have not been hand, a slow shock mechanism may actually suppress worked out. a faster mechanism because it creates a broader shock Inordertounderstandtheheatingofelectronsinfore- transition, and thus has greater ”reach” upstream of the shock region of relativistic shocks we simulate, using the shock. kineticPICmethod,developmentoftheWeibelinstabil- The role of Weibel instability in forming the shock ity in two relativistic counter-streaming plasma beams, transition in weakly magnetized plasmas has recently whichresemblestheprecursorofarelativisticshock,but been established through several numerical experiments simpler and more idealized and therefore more suitable (Nishikawaetal.2003;Frederiksenetal.2004;Nishikawa for resolving certain fundamental questions. The beams etal.2005;Spitkovsky2008;Keshetetal.2009). Numeri- are taken to be homogeneous and fully interpenetrating calsimulationsofrelativisticshockshaveshownthatthe atthebeginning,sothetimedevelopmentoftheinstabil- 2 ity in our simulation imitates the spatial development of 3. FILAMENTATION AND ELECTRON HEATING theinstabilityintheforeshockregions,wherelatertimes At the very beginning of the simulation the counter- in our simulation correspond the regions closer to the streaming electrons, which are relatively lightweight, as shockfronts. Ourquantitativeanalysissuggeststhatthe compared to ions, and can relatively easily be deflected most of the heating of upstream electrons is due to the bythemagneticperturbations,aresubjecttotheWeibel longitudinal electric field. The transverse electric field, instability. Currentduetothestreamingelectronsgener- thoughmuchstrongerthanthelongitudinal, hasnegligi- ate magnetic field and the electrons moving in the same ble effect on net acceleration of electrons. direction are herded into the filaments by self created We have also inserted virtual test particles into the magnetic field. Growing current in the filaments induces code to act as numerical probes. They assist in the di- an electric field opposite to the direction of current in agnostics of the mechanisms that are at work, and this accordance with Lenz’s law. The induced electric field will be discussed below. slows down the streaming electrons in the filaments and arescatteredbythestrongmagneticfieldaroundthefila- 2. NUMERICAL SIMULATION ments. Duringthisstagerelativelyheavierionscontinue We use parallel version of the PIC code TRISTAN streaming almost unaffected. As the electron Weibel in- (Buneman 1993; Spitkovsky 2005) to simulate relativis- stabilitystageendsinabout20ωp−e1,electronsarenearly tic counter-streamingbeams ofions and electronsin two thermalized to about their initial streaming kinetic en- dimensions. The simulation is initialized by placing ion- ergy n0me(γ0−1)c2, creating a nearly isotropic electron electron pairs at uniformly chosen random locations in background for still streaming ions. a two dimensional square box in x-y plane. Initially ion As the electron filamentation stage ends, relativisti- and electron in each pair are moving in opposite direc- callycounter-streamingionsundergotheWeibelinstabil- tionsalongthex-axiswithLorentzfactorγ . Halfofthe ity and likewise current carrying ions are separated into 0 total number of pairs have ions moving along the posi- long current carrying filaments along the initial stream- tive x-axis while an equal number of another half pairs ing direction. The current filaments are non-stationary have ions moving along the negative x-direction, hence structureswithslightbendinginthetransversedirection creatingtwooppositelystreamingneutralplasmabeams (figure 1), which move along with the current carrying of equal intensity, where each beam has a net current. ions with the strength of magnetic field being small near The initial condition ensures that the simulation is ini- the filaments and large between two adjacent filaments. tiallychargeandcurrentfreeandthatMaxwellequations Again, the inductive electric field develops in the fila- aresatisfiedoverscaleswherethereareequalnumbersof ments opposite to the current due to the streaming ions. forward and backward moving electron-ion pairs, but at In addition to the inductive electric field along the fila- thegridscale,localfluctuationsinthenumberofforward ments an electrostatic electric field develops around the and of backward movers imply a net small scale current. filaments in thedirectiontransverse tothestreamingdi- This noise level, however, is reddened by smoothening rection due to excess of positively charged ions in the the grid scale current. As the electrons and ions in the filaments. The scale of this transverse electrostatic field pairs separate from each other, the electric and mag- is limited by the electrons that cloud around the fila- netic field grow from the noise in the current generated ments and remains ∼ c/ωpe, which grows with time as by streaming electrons and ions. Since we simulate an the electrons are accelerated to higher energies. In Fig- initially unmagnetized plasma, only out-of-plane mag- ure 2 we show the details of the local magnetic field and netic field B and in-plane electric fields E and E are thelocalcurrentduetoionsandelectronsinasmallspa- z x y excited as the Weibel instability set in. We impose pe- tialpartofthesimulationbox. Theapparentcorrelation riodic boundary condition in both x and y directions for betweenthemotionofionsandelectrons(electronsmove both particles and fields. opposite to their current) suggests that the electrons are Here we discuss results mainly from the two largest accelerated in the direction of streaming ions. simulations we have attempted in terms of physical size Thespatialdistributionofelectronscloselyfollowsthe and evolution time. Ion to electron mass ratio m : m distributionofions. Chargeneutralizingelectronsclump i e for the two reported simulations are 16:1 and 64:1, and in rather positively charged filaments and stream along are henceforth referred to as M and M , respectively. with the ions. The electrons in the filaments move along 16 64 All the figures in this paper are for ion to electron the inductive electric field created by the streaming ions mass ratio of 64:1, i.e. simulation M , unless otherwise and consequently get energized. Ions on the other hand 64 stated. The physical sizes of the box for M and M moveagainsttheelectricfieldinducedbythemselvesand 16 64 are 500 c/ω x500 c/ω and 1000 c/ω x1000 c/ω , re- hence lose their energy. Since the current filaments are pe pe pe pe spectively, where c is the speed of light in vacuum and non-stationary and turbulent at very small scale, and ω = (cid:112)4πn e2/m γ is the initial electron plasma fre- so are the field structures, the local details of energy pe 0 e 0 gainandlosscandeviatesignificantlyfromtheonemen- quency,whereeisthechargeofanelectronandn isthe 0 tionedaboveandhenceanstatisticalapproachisneeded initialnumberdensity ofelectrons, or bycharge neutral- to understand the systematic acceleration of electrons. ity, of ions. Both simulations were resolved to 1/10th of We therefore compute spatial average of various quanti- the initial electron skin-depths. Initially there are 32(8) ties and correlates to quantify the presence of systemic particlesperunitcellofthesimulationboxforM (M ) 16 64 heatingofelectronsatscalesmuchlargerthantheirskin and the simulation was evolved for 500 (1000) plasma time ω−1. Initially the ions and electrons are moving depth. pe During the linear stage of the instability not all the along the x-axis with initial Lorentz factor γ = 10 in 0 energy that is lost by ions goes into heating electrons. the both cases. 3 10−1 2Bz 10−2 Bz2 E2 , y 2Ey 2E,x 10−3 E2 x 10−4 0 200 400 600 800 1000 time × ω Figure 1. Left panel: Structure of magnetic fields at 225 ωp−e1. pe The magnetic field is still growing due to the Weibel instability Figure 3. Temporalvariationofthemeansquare(averagedover andformsfilamentarystructuresparalleltothestreamingdirection thephysicaldomainofthesimulation)oftheelectromagneticfield (referredtoasfilamentaryphase). Mostoftheheatingofelectrons components Ex (dashed red), Ey (dotted dashed green), and Bz takesplaceduringthefilamentaryphase. Rightpanel: Structureof (solidblue)arenormalizedto4πn0(mi+me)(γ0−1)c2. Amongst magneticfieldat650ωp−e1. Transversesizeaswellasthebendingof theelectromagneticfields,themagneticfieldgainsmostoftheki- filamentsgrowwithtimeandeventuallyfilamentsgetdisoriented. netic energy lost by the ions to the fields, which holds positively Here,andhenceforth,Bz andallotherelectromagneticfields(i.e., chargesionsinthefilamentstogetheragainsttheelectrostaticelec- ExandEy)arenormalizedto(4πn0(γ0−1)(mi+me)c2)1/2,unless tric field, followed by mostly electrostatic transverse electric field otherwisespecified. Ey andlongitudinalelectricfieldEx. filaments wipe out any preference in the direction of the electric field in the x-y plane (figure 1). 3.1. electron heating estimate Theinductivelongitudinalelectricfieldinthedirection opposite to the current in plasma features in the linear theory of the Weibel instability itself. In fact it is the curl of this longitudinal electric field which accompanies thegrowthofmagneticfieldinaccordancewiththeFara- day’s law of induction. Presence of a large scale electric fieldinaconductingplasmanaturallypredictstransferof kinetic energy from ion to less energetic electrons. Here we estimate the net energy gained by electrons due to Figure 2. Local magnetic field, and the current due to electron and ions are shown in a spatial patch from the simulation M64 ( the inductive electric field in a one dimensional linear mi:me=64:1)attime65ωp−e1. Directionofblackandredarrows theory of the Weibel instability. In the two dimensional indicatedirectionofthelocalcurrentduetotheionsandelectrons, casesimulatedhereelectromagneticfieldfluctuationsdue respectively,atthetailofthearrowsandlengthofthearrowisin to the electrostatic wave modes parallel to the x-axis proportiontothemagnitudeofthecorrespondingvectorquantity. (streamingdirection)andtheobliquemodesalsobecome The local currents shown here are computed by taking the aver- agevelocityofallparticleswithinaboxofsize0.4cωp−e1×0.4cωp−e1 comparabletothetransversemodes,andeventuallylead centered at the tail of arrows. Inductive electric field opposes the tothedisruptionofcurrentfilaments(Shaisultanovetal. current due to fast moving ions in the current filaments and is 2012). Here we consider the heating solely due to the responsible for deceleration of ions as well as acceleration of elec- trons. transverse modes with the wave vector (cid:126)k along the y- axis and electric field E(cid:126) along the x-axis. We assume homogeneity along the x-axis, that is averaging out the A significant part of the total kinetic energy goes into fluctuationsalongthex-axis,hencereducingtheproblem the electromagnetic fields which are responsible for me- to one spatial dimension. diating collective interactions between the charged par- Let p (y) be the x-momentum of an electron at any x ticles. In figure 3 we show how much of initial kinetic given location y. The electric field E (y) at any given x energygoesintodifferentcomponentsoftheelectromag- y determines the rate of change of x-momentum of an netic fields. During the ion filamentation stage, which is individualelectronlocatedaty,i.e. dp (y)/dt=eE (y). x x mainlymagneticinnature, energyinthemagneticfields Averaging the x-momentum of all electrons at a certain reaches few percent of the total initial kinetic energy. y gives, Longitudinal electric field E is the weakest component, x but as shown later, is the most important for the net dp¯ (y)/dt=eE (y) (1) x x transferofkineticenergyfromiontoelectrons. Thelon- gitudinal electric field E has a curl in the direction of where p¯ (y) = (cid:82) p f (y,p(cid:126),t)d3p/(cid:82) f (y,p(cid:126),t)d3p with f x x x e e e magnetic field which predominantly contributes to the being the distribution function of electrons which is as- growth of the magnetic field. In other words, the induc- sumed to depend on y only. The electromagnetic fields tiveelectricfieldE accompaniesthegrowthofthemag- E andB ,aswellasthemeanx-momentumofelectrons x x z netic field and accelerates electrons hence leaving it the p¯ ,canallbewrittenassumofsinusoidalmodesofvari- x weakest amongst all growing electromagnetic field com- ouswavelengths,butwithB quarterawavelengthphase z ponents. As the non-linearity sets in and the filaments shifted from E and p¯ , since B is maximum between x x z are bent and broken, E and E both converge to a sim- twoadjacentcurrentfilamentsbutE andp¯ peakinthe x y x x ilar value since disruption and disorientation of current current filaments. We first consider a single sinusoidal 4 mode of wave number k and write Bz = Bksin(ky), 10−3 10−3 where Bk is time dependent amplitude of the magnetic 250 t t=ω−1 field B in the mode of wave number k. From Maxwell’s pe z equations we relate the electric and magnetic field as 10−4 50 t 10−4 kcE = cos(ky)∂Bk/∂t. Decomposing the linear equa- x kBz 150 t kBz tion1intoFouriercomponentsandthensubstitutingfor the electric field we find 10−5 10−5 ∂p¯k/∂t=(e/kc)∂Bk/∂t (2) x 10−6 10−6 where p¯kx is the amplitude of average x-momentum p¯x 10−2 ck/2πω 100 0 tim20e0 × ω 400 in the mode of wave number k. Equation 2 enables pe pe us to express instantaneous amplitude of average three- Figure 4. Left Panel: average Bk (amplitude of magnetic field z momentumfluctuationintermsoftheamplitudeofmag- inmodeofwavenumberk)asafunctionofwavenumberkattime netic fluctuation as 50, 150, and 250 ωp−e1 are shown by red, blue, and green curves, respectively. The spectrum is obtained by first computing one p¯k =eBk/kc+p (3) dimensionaldiscreteFouriertransformofmagneticfieldinseveral x x0 slicesofthesimulationsboxalongthetransversedirectionandthen Equation3isofcentralimportance: Itimpliesthatthe takingaverageofthemagneticfieldamplitudesinanygivenmode gyroradius of an electron, in the limit that p >> p , koveralltheseslices. Rightpanel: evolutionofaverageamplitude is roughly the wavelength times 1/2π. As wex shall sxeoe oftransversemagneticfluctuationBzk for2π/k=10,2,1,0.5cωp−e1 areshownbyred,green,blue,andblackcurves,respectively. The below, this is confirmed by the simulations. The impli- magnetic field in any given mode grows exponentially and then cationisthatthefieldgrowthisnot stopped bythemag- slowlydecays. netization of the electrons, as conjectured by Lyubarsky andEichler(2006),becausetheelectronsareatalltimes 10−4 10−4 onlymarginallymagnetized. Thattheenergyoftheelec- tronskeepspacewiththefieldgrowth,sothattheynever 250 t t=ωp−e1 250 t t=ω−1 get highly magnetized, is the essential reason the elec- pe 150 t trons get heated as much as they do. 150 t kx ky Itisalsoshownthatthemarginalmagnetizationonthe E E other hand, is enough to suppress the electron heating 10−5 50 t 50 t 10−5 in one dimension, because the filaments remain exactly straight, and the electrons are accelerated exactly along the filaments. By contrast, in more than one dimension, the existence of oblique modes allows the filaments to 10−2ck/2π10ω−1 100 10−2ck/2π10ω−1 100 bend, and it becomes harder for an electron to remain pe pe withinasinglefilament. Thecountercurrentsoftheelec- Figure 5. Transversespectrum,asintheleftpaneloffigure4,of trons are then less likely to cancel those of the ions and longitudinal electric field Ex and transverse electric field Ey as a the field growth can proceed to larger length scales. functionofwavenumberkattime50,150,and250ωp−e1areshown byred,blue,andgreencurves,respectively. At any instant, the rate of change of average kinetic energy of an electron U =m (γ −1)c2 due to the lon- e e e gitudinal electric field E is given by the spatial average the largest transverse scale achieved during the Weibel x (y-average) of eE v¯ . For a single mode, defining Uk as instability by that time. In Figure 4 we show the time x x e y-average of cp¯k we obtain (after squaring the equation evolution of transverse spectrum of the magnetic field x 3 and then taking the derivative with respect to time) structure in the simulation which confirms the exponen- tialgrowthandsaturationofthemagneticfieldatlength UkdUk/dt≈[e2c2/(kc)2]Bk∂Bk/∂t (4) scalessmallerthanafewtimestheskindepthofions. In e e figure 5 we show the transverse spectrum of E and E ThelineartheoryofWeibelinstabilitypredictsexponen- x y as well. It shows the exponential growth and satura- tialgrowthoftheamplitudeofmagneticfieldfluctuation tion of the transverse electric field, and that the trans- Bk atalllengthscales. However, themaximumstrength verseelectricfieldpeaksathalfthewavelengthmagnetic of magnetic field at any given length scale is constrained field peaks, since the transverse electric field is symmet- by the available current in the plasma at that length ric with respect to the current filaments. The transverse scale. Consequently, the growth of magnetic field am- spectrumoflongitudinalelectricfieldinfigure5doesnot plitude Bk at a length scale 1/k reaches a saturation show saturation because of growth of electrostatic par- amplitude Bpkeak, which is inversely proportional to the allel modes, as shown later, which contributes to further wave number k1. Since the amplitude of magnetic field growth of the longitudinal electric field . Bk at smaller scales saturates first, while the magnetic The heating due to the longitudinal electric field E is x field at larger scales are still growing, the net magnetic significant only until the filaments prevail. As the trans- fieldatanyinstantisdominatedbythemagneticfieldin verse scale of the filaments reach ions skin-depth, bend- ing of the filaments in the transverse direction becomes 1 themaximumachievablenetcurrentinthefilamentsoftrans- significant. Thecurrentfilamentsstarttobreak,getran- verse size 1/k is limited by the density of streaming ions in that domly oriented in x-y plane, and the rate of heating of lengthscale,whichis∼n0ec/k,implyingthatBpkeak∝1/k(Kato electrons is substantially reduced. In order to obtain an 2005;Gedalinet al.2010) estimateforthetotalenergyacquiredbytheelectronsby 5 the end of the filamentation it suffices to integrate equa- 101 tion 4 with respect to time for the smallest wavenumber 101 k achievedduringtheWeibelinstabilitywiththeelec- min 2 tronheatinginprogress,andintegratingituptothetime c saturation is achieved in this mode, that is to say until −1pe m e tthheeseheparteisncgripdtuieontsofothritshme noedtehiesastiignngifiecstainmt.ateFowlelowgeitn,g r/cωL E / γ0 100 K Uf2 ≈m γ c2(ω /k c)2(Bkmin2/8πn ) (5) e i 0 pi min peak 0 100 10−1 where Uef is the average energy of an electron towards 0 tim20e0 × ω 400 0 tim20e0 × ω 400 the end of the heating process during the linear phase. pe pe As compared to the gyro-frequency of electrons, the Figure 6. LeftPanel: ThemeanLarmorradiiofelectronsaswell as test particles (charged particles that do not participate in the transverse scale of filaments grow relatively slowly with plasma dynamics but respond to the local electromagnetic field) time. Electrons gain energy in the current filaments and withchargetomassratio0.1thanthatoftheelectronsareshown theirLarmorradiigetenlarged. However,atanyinstant in units of instantaneous skin depth of electrons. The red curve during the linear stage of the Weibel instability the en- shows Larmor radius to electron skin depth ratio for electrons, while black, blue, and green curves show the same for test parti- largedLarmorradiiofelectronsduetoinductiveelectric cles with initial kinetic energy 0.01, 0.1, and 10 times that of the fieldinthecurrentfilamentscannotsubstantiallyexceed electrons. DuringthelinearstageoftheinstabilityLarmorradius the transverse size of the current filaments. This limita- ofelectronsaswellasthetestparticlesgainingenergyfromthelon- tion on electron Larmor radii is due to the fact that for gitudinalelectricfieldremainsabouttheinstantaneousskindepth oftheelectrons. RightPanel: Temporalvariationofmeankinetic anelectronthathasLarmorradiusmuchlargerthanthe energyoftheelectronsandtestparticlesinunitsofγ0mec2. Color spacing between two adjacent filaments, energy gained code for the curves is same as in the left panel. Energetic parti- inonefilamentislostintheneighboringfilamentswhich cles that have Lamor radii much larger than the spacing between carries opposite current and hence E directed in op- thefilaments,suchasthetestparticleswithinitialkineticenergy posite sense. Additionally, as suggestexd by equation 3, (γ0−1)mec2 (shown in blue) do not get efficiently energized by thelongitudinalelectricfield. the energy imparted by the inductive longitudinal elec- tric field in the largest length scale is just enough to m / m = 64 m / m = 16 keep the Larmor radii of electrons at any instant about i e i e 1/2π times the transverse size of the current filaments. Asksint-hdeepetlehctarlosnosignectreaacsceesleirnattehdetosahmigehperroepnoerrtgiioens.thIenir- Ke0 60 ion Ke0 15 ion dooffeettdhh,eeaesLleaocrbtmsreoornrvse,rdawdihniuictshhetiossiatmlhsueolaiantbisootnuasntt(tafihngeeuo1ru/es26π)s,ktitinmheedsreaptthtihoe K , K/e0e 40 K , K/e0e 10 transversesizeofthecurrentfilaments,isnearlyconstant K /i 20 electron K / i 5 electron during the linear stage of the Weibel instability. It sug- 0 0 gests that during the filamentation stage the inductive 0 500 1000 100 200 300 400 500 heating of electrons proceed such that n U (cid:38) B2/8π time × ω time × ω 0 e pe pe remains satisfied. Thegrowthofmagneticfieldandtransversesizeofthe Figure 7. Evolution of mean kinetic energy of electrons Ke (dashed red ) and ions Ki (solid green) are shown for two dif- filaments continue until the transverse scale of the fila- ferentiontoelectronmassratios(normalizedtotheinitialkinetic ments becomes comparable to the ions skin depth c/ωpi energyelectronsKe0). ElectronsheatingduetotheLenzelectric fieldissignificantuntiltheioncurrentfilamentsaredisrupted. By afterwhichthefilamentsaredisorientedandtheheating theendofthesimulationelectronsareheatedtoabout1/3ofthe of electrons due to longitudinal inductive field is sub- ionskineticenergyinbothcasesconsiderhere. stantially reduced(inthe two dimensionalcase electrons continue to gain energy due to decay of magnetic field, While the magnetic field scatter the charged particles but at a much reduced rate). It suggests that for the by altering their trajectories, the only source of acceler- purpose of estimating the total heating due to inductive ation of the plasma particles is electric field in the lab electric field we can take ck ∼ ω . This along with min pi frame. From our PIC simulations we separate out work thecondition thatduringtheheating nUe (cid:38)B2/8π sug- done by the two orthogonal components of the electric gest that Uf (cid:46)m γ c2 (from equation 5), that is to say field, namely E and E , on electrons as well as on pro- e i 0 x y theelectronsareacceleratedtoenergycomparabletothe tons. As evident from Figure 8, in both the simulations energyofionsduringthelinearstageoftheWeibelinsta- reported here the change in total kinetic energy of elec- bility due to the inductive longitudinal electric field. In trons and protons is apparent to be mainly due to the thefollowingwepresentresultsfromthePICsimulation longitudinal electric field E . The net work done by the x which confirms that the heating of electrons due to the transverseelectricfieldE ,whichisratherstrongerthan y inductive longitudinal field is indeed significant, though the longitudinal electric field (Figure 3), is vanishingly thereisadditionalheatingduetotheelectrostaticmodes small 2. which appear in the two dimensional analysis and is es- sential to the heating of electrons. 2 The implication is that the net energization of electrons due toelectronsfallingintothegrowingandmergingfilaments,assug- gested by Hededal et al. (2004) and Spitkovsky (2008), is rather 3.2. Net work done by the electric field smallduringthefilamentationstageoftheWeibelinstability. 6 m / m = 64 m / m = 16 i e i e 2 10−2 Evyy 0.050 − −ioEEnyxvvyx Evyy 0.050 − −ioEEnyxvvyx 22rcr,E,Eyxy 10−3 Eyc2 Eyr2 Exr2 , , E Evxx electron Evxx electron 2cE,x Exc2 −0.05 − −Eyvy −0.05 − −Eyvy Exvx Exvx 0 100 200 300 400 0 500 1000 100 200 300 400 500 time × ω time × ω time × ω pe pe pe Figure 9. Temporal variation of the mean square (as in the Figure 8. Therateofenergygained(lost)byelectrons(protons), figure 3) of the electric field components Er, Ec, Er, and Ec x x y y asafunctionoftime,du√etoEx andEy,i.e. Exvx andEyvy,re- (E(cid:126) = E(cid:126)r +E(cid:126)c, such that ∇·E(cid:126)r = 0, and ∇×E(cid:126)c = 0. Sub- spectively(inunitsofc2 4πγ0me),areshownbysolidanddashed script x and y indicate components along longitudinal (streaming red (black) curves, respectively . The quantities Exvx and Eyvy direction)andtransversedirection,respectively)areshownbythe arethemeanvaluesofExvx andEyvy,respectively,computedfor solid red, solid blue, dashed red, and dashed blue curves, respec- thesampleofparticles(fewpercentofthetotalnumberofparticles) tively. During the filamentation stage the transverse electric field whichwereinitiallyhomogeneouslydistributedoverthesimulation Ey is mostly electrostatic and the longitudinal electric field Ex is box. The left and right panels are for ion to electron mass ratios partiallyinductiveandpartiallyelectrostatic. 16:1and64:1,respectively. ThoughthetransverseelectricfieldEy ismuchlargerinmagnitudethanthelongitudinalelectricfieldEx (figure3),itscontributiontotheelectronaccelerationandionde- Electron Ion celeration(dashedlines)isnegligibleascomparabletotheheating 0.04 0.04 duetoEx. vy 0.02 vy 0.02 In figure 7 we show the time evolution of the total ki- c,ry Eycvy c,ry Eycvy netic energy of electrons and ions. It shows that most E E 0 0 of the energy exchange between ions and electrons takes ,x Eyrvy ,x Eyrvy v v pwlhaecne mduarginnegtitchfieelfidlaismsetniltlagtrioowninstga(gfieguofret3h)eininssttraebniglitthy., c,rEx −0.02 Excvx c,rEx −0.02 Excvx We have run the simulation for long enough to capture −0.04 Exrvx −0.04 Exrvx most of the heating phase and long after the rapid heat- 0 200 400 0 200 400 ing phase ends, electrons are found to have acquired a time × ω time × ω pe pe substantial part of the ions’s kinetic energy, in agree- mentwiththeestimatepresentedabove. Evenlongafter Figure 10. The left (right) panel shows the instantaneous rate of energy gained (lost) by the electrons (protons) due to the ro- the breaking of filaments, ions are not completely ther- tational and compressive components of the longitudinal electric malized and continue to lose energy and the heating of fieldinsolidredandsolidblue, respectively. Thedashedredand electrons continues, though at a much lower rate. bluecurvesshowsthesameduetotherotationalandcompressive Bending of the current filaments is apparent from the components of the transverse electric field Ey, respectively. Nor- malization and computation of the mean shown here are same as very beginning of the Weibel instability. The bending inthefigure8. of current filaments results in mixing of the longitudi- nal and transverse components of the electric field. Two dimensional linear theory of the Weibel instability pre- of oblique and parallel electrostatic waves appearing in dicts the growth of waves with wave vector at oblique the plasma along with the purely transverse waves, the angle with respect to the streaming direction (Shaisul- longitudinalelectricfieldcannotcompletelybedescribed tanov et al. 2012), and we suggest that this can account by a divergence free electric field. for the bending of the filaments. The filaments are also Infigure9weshowtowhatextentthelongitudinaland susceptible to the Buneman instability, which leads to transversecomponentsoftheelectricfieldiselectrostatic the growth of resonant waves parallel to the streaming andinductiveinnature. Itisevidentfromthefigurethat direction. Theobliqueandparallelmodesareratherelec- during the linear stage of the instability the transverse trostaticinnaturewhichimpliesthatthethetruenature electricfieldismostlyelectrostaticisnatureandisdueto ofthelongitudinalandtransverseelectricfield(i.e.,elec- excess of positive charges in the current filaments. The trostatic or induced) is rather mixed and are due to sev- longitudinal electric field on the other hand is partially eral waves growing at the same time, though at different inductive and partially electrostatic, but more of an in- rates (see the section 3.3 ). In order to quantify the ductive in nature. In figure 10 we show the work done roleofdifferenttypesofgrowingwavesintherelativistic by the rotational and compressive part of each compo- counter-steamingplasmainheatingofelectronswesepa- nentsoftheelectricfield. Wefindthattheworkdoneby rate(Helmholtzdecomposition)theelectricfieldintothe the compressive and rotational part of the longitudinal rotational and compressive part, i.e. E(cid:126) =E(cid:126)c+E(cid:126)r, such electric field are comparable, suggesting that the role of resonant heating by the electrostatic waves is compara- that∇·E(cid:126)r =0and∇×E(cid:126)c =0. Iftheonlywavesgrowing ble with the inductive heating due to transverse Weibel in the counter-streaming plasma were the purely trans- modes. verse Weibel modes then the longitudinal electric field would be purely inductive in nature and could solely be 3.3. Parallel and oblique modes described by a rotational electric field. However, in case 7 Figure 11. Power spectrum for Bz, Ex, and Ey at 100 ωp−e1 is shown in the left, middle, and right panel panel, respectively. ThepowerspectrumshowsbroadbandnatureoftheWeibelinsta- bility. The spectrum of Bz shows that the transverse waves are mostly magnetic in nature. The existence of parallel and oblique modes, which are rather electrostatic in nature, is apparant from thespectrumofEx. Figure 13. Thestrengthandsenseofthemagneticfieldisshown x 10−4 x 10−4 inasmallpatchofthesimulationboxfortwovariantofsimulations 1 8 M64 whentransversesizeifthefilamentsisabout30c/ωp−e1 . Top 21/2m c)γi000..68 21/2m c)γi06 pmt:iaosannismessluoe:llfaeticooitnonrlnosynMisnco6aou4rrneditneteorrw-tsbhotericeaahacshmsciuaeinmsvegeeedcilohetnacosrtgrbboeenenassemiatutsrtiern(agailmlistamyot)op.tbhaBierltoeiitnctoiioptrmiaaitnlipnfiloagncniatiene-l n π00.4 n π04 tfihlaempelnatssmiasadbyonuatm4ic0s.c/Tωpime.eAsullcmhotdheastgtrhoewtrfaasntsevreirnsethsiezeabosfenthcee kE / (4x0.2 kB / (4z2 ofelectronswhichresonantlysuppressthegrowthofwaves. mode, are suppressed in the linear regime by the induc- 0 0 0 100 200 300 400 500 0 100 200 300 400 500 tive response of the electrons which are accelerated by, time × ω time × ω pe pe and therefore move along the inductive electric fields. Figure 12. The left (right) panel shows temporal evolution of Note, however, that the purely transverse mode stops the average amplitude of sinusoidal variation in the longitudinal growing and at time of about 250 ω (figure) while the electric(magnetic)fieldEk (Bk)withthewavevectorbeingalong pe x z oblique mode continues to grow and in fact catches up the streaming direction (x-axis). The red, green, blue, and cyan coloredcurveshowthefieldamplitudeofmodeswith2π/k=410, with the purely transverse mode. This can be inter- 90, 16, and 8 c/ωpe, respectively. The average amplitude shown pretedasfollows: Thetransversemode,beingthefastest here are obtained after taking average of amplitude of any given growing, is the first to trap the electrons into the reso- mode over an ensemble of several slices parallel to the y-axis at nance, so that the number of resonant electrons actually equallyspacedlocationsalongthex-axis. increases. This includes those electrons that resonated withtheobliquemodesuntiltrappedbythepurelytrans- We perform two dimensional spectral analysis of the verse mode. We can say that these electrons are in a electromagneticfieldcomponentsforthesimulationM non-linear resonance with the purely trapped mode. At 64 to show the simultaneous growth of several other waves this point, the growth of the purely transverse mode is in addition to the purely transverse Weibel waves. In impeded by the even larger number of non-linearly reso- figure 12, we show the average amplitude of sinusoidal nantelectrons,soitstopsgrowing.Theobliquemodes,in variation in electromagnetic fields along the streaming contrast,sufferlessfromtheinductiveeffectsoftheelec- direction. The amplitude of variation in the magnetic trons so they keep growing. until they have spoiled the fieldservesasaproxyforbendingofthecurrentfilaments alignmentofthefilamentsmadebythepurelytransverse since the parallel modes are mostly electrostatic and do mode. not contribute to the growth of the magnetic field. As The role of electrons in suppressing the oblique modes evident from the figure 12, the growth of electric field can clearly be demonstrated by simulating the Weibel in the same length scale is faster than the growth of the instability with electrons of infinite mass such that elec- magnetic field, hence by comparison a rather stronger trons do not interact with the plasma waves. In the ab- electric field in waves with their wave vector along the senceoflighterresonatingelectronsallmodesgrowfaster longitudinal direction can be attributed to the growth and the Weibel instability give rise to rather stronger of electrostatic modes along the streaming direction in magnetic field (figure 13). In case of infinitely massive addition to the contribution from the bending of current electrons the current filaments which form due to the filaments. Weibel instability are rather short in length along the In figure 14 we show the growth of magnetic and elec- longitudinaldirectionandgetquicklydisorientedascom- tric field in purely transverse, purely longitudinal, and pared to the simulation with electrons interacting with an oblique mode. The general trend is that the purely the waves. transversemodesgrowthefastestandtherateofgrowth The large fluctuation in the amplitudes of the waves decreases with the angle between the wave vector and (figure14)isapparentlyduetointerferenceofwaveswith y-axis. The oblique modes, like the purely transverse samewavenumberbutwithdifferentphasesappearingin 8 10−2 10−1 10−2 B2 1.5 z 2 2 1e k21/E / (4 n m c)πγx0i0 1100−−43 k21/B / (4 n m c)πγz0i0 111000−−−432 222E,E,Bxyz 1100−−43 EEyx22 −ω K/K, r / cee0Lp 0.15 KrLe/c/Kωep−0e1 10−5 0 200 400 600 00 200 400 600 0 200time 4×0 0ω 600 0 200time 4×0 0ω 600 time × ω time × ω pe pe pe pe Figure 14. The left (right) panel shows temporal evolution of Figure 15. Onedimensionalsimulation: A)Leftpanel: Tempo- the amplitude of sinusoidal variation (obtained from two dimen- ral variation of the mean square (averaged over the physical do- sionalFourierspectrumofelectromagneticfields)inthelongitudi- main of the simulation) of the electromagnetic field components nalelectric(magnetic)fieldEk (Bk)withwavevectorbeingalong Ex (red), Ey (green), and Bz (blue) are normalized to the to- x-axis, y-axis, and at an equxal anzgle with respect to the x and tal initial kinetic energy density n(mi+me)(γ0−1)c2. B) Right y axes . Specifically, the red, green, and blue curves correspond Panel : a) Black curve : Temporal variation of Kinetic energy of to (2π/kx,2π/ky) = (0,100),(100,100),and(100,0)c/ωpe, respec- electronKEe (normalizedtotheinitialkineticenergyofelectrons tively. The thick solid line segments in red and blue indicate the KEe0 = (γ0−1)mec2), b) Cyan Curve: Ratio of instantaneous growth rate predicted by the linear theory (taken from Shaisul- Larmor radius to the instantaneous skin depth of electrons as in tanovet al.(2012)forelectrontemperatureme/mi timestheion thefigure6. temperature)forthetransverseandtheobliquewave,respectively. depth level, hence reducing the simulation to effectively theplasmaatdifferentspatiallocationswhichgrowinde- 1.5 dimension. That is to say, though electromagnetic pendently. The phase difference in the purely transverse field and particle’s velocity are allowed to have all three Weibel modes at different locations can also be blamed components, they are allowed to vary along the trans- for the wiggle in the filaments which appears at the very versedirection(y-axis)only. Herewediscussresultsfrom beginning of the simulation and can contribute to the thesimulationwhichhastwodimensionalboxofsize0.3 scattering of the electrons. c/ω × 1600 c/ω and 64 particles per cell. All other pe pe Thepowerspectrumoftheelectromagneticfields(Fig- parameters for this simulation are same as in the case of ure 11) reveals a broadband nature of the instabilities in two dimensional simulation M . 64 the counter-streaming plasma. We have identified the In the one dimensional simulation electrons first un- growth of transverse and oblique Weibel-like modes as dergo the Weibel instability, which essentially is the well as parallel electrostatic modes in our simulation. Weibel instability of one species. The electron Weibel However, there can possibly be several other waves and stage ends with the isotropisation of electrons. As the instabilitiesinthecounter-streamingplasma(Bret2009), ions undergo the Webiel instability size of ion current which make the current filaments unstable, and might filaments and strength of the magnetic field grow. At become more pronounced for plasma parameters which the stage when transverse size of the current filaments are rather more realistic for astrophysical contexts, i.e. becomes∼2πc/ω electrons, whicharemagnetized, get pe for higher ion to electrons mass ratio or larger Lorentz confined in ion current filaments between two adjacent factor of the streaming plasma than simulated here. In peaks of the magnetic field (Wiersma and Achterberg any case, it is evident that whichever wave can bend the 2004;LyubarskyandEichler2006). Electronsinthecur- current filaments of streaming ions can very efficiently rent filaments quiver in transverse direction and drift in scatter electrons out of the filaments since the electrons the same sense as the ions. The countercurrent due to havemuchlessinertialmassascomparedwithions. The electrons efficiently arrest the growth of Weibel instabil- bending of the filaments, which we suggest is due to a ity of the ions. Anisotropic ions continue to stream in broadband nature of the Weibel instability (i.e. growth the current filaments and the growth of magnetic field of the waves with their wave vector at some angle with ceases (Figure 15). The lack of heating of electrons ob- respect to the transverse direction), is a crucial require- served in the one dimensional case can be attributed to ment for the heating of the electrons as demonstrated the lack of scattering of electrons out of the current fil- bytheone-dimensionalsimulationwherepoorscattering aments which is rather efficient in the two-dimensional results in trapped electrons, which efficiently short out caseduetoobliqueswavesgrowingalongwiththepurely theinductiveelectricfieldcreatedbystreamingionsand transverse mode. do not get energized. 4. DISCUSSION AND CONCLUSIONS 3.4. One dimensional simulation: purely transverse We have simulated the development of the Weibel Weibel mode instability in relativistically counter-streaming homoge- In order to illustrate the role of oblique and electro- nous ion-electron beams, using the kinematic PIC static modes parallel to the streaming direction we sim- method. The physical domain of the simulation is taken ulated the counter-streaming plasma in one dimension to be sufficiently large, in both parallel and perpendicu- such that the only modes transverse to the streaming lar tothe plasma streamingdirection, to ensurethat the direction, i.e. purely transverse Weibel models, are al- growth of Weibel unstable modes as well as the longitu- lowed to grow. The growth of oblique as well as longi- dinal electrostatic states are not frustrated due to finite tudinal modes is suppressed by reducing the size of the size of the box. The homogeneous set-up simulated here box along the streaming direction (x-axis) to sub-skin ensuresthatanyeffectoflargescalelongitudinalinhomo- 9 geneitywhichmaybepresentinthecaseofshockissep- Current filamentation and significant electron heating arated out. In the case of a shock transition there might is observed in three dimensional simulations of counter- be some additional heating due to cross-shock potential steaming plasma as well. We expect that the physical that may develop because of longitudinal separation of mechanisms in a more realistic three dimensional simu- ionsfromelectrons,sincethelighterelectronscanberel- lationshouldbethesameasinthetwodimensionalcase ativelyeasilyisotropisedbytheforeshockmagneticfield, discussed here and can be verified in future simulations. hence interrupting the electrons flow before the flow of WethankU.Keshet,Y.Lyubarsky,andA.Spitkovsky ions (Balikhin et al. 1993; Lyubarsky 2006; Lyubarsky for helpful discussions. We thank A. Spitkovsky for a and Eichler 2006). critical reading of the manuscript. We are grateful to Our findings from the plasma simulations concerning U. 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