Simulation study of the formation of a non-relativistic pair shock

Simulation study of the formation of a

non-relativistic pair shock

Mark Eric Dieckmann and Antoine Bret

Journal Article

N.B.: When citing this work, cite the original article.

Original Publication:

Mark Eric Dieckmann and Antoine Bret, Simulation study of the formation of a non-relativistic

pair shock, Journal of Plasma Physics, 2017. 83, pp.1-19.

http://dx.doi.org/10.1017/S0022377816001288

Copyright: Cambridge University Press (CUP): STM Journals

http://www.cambridge.org/uk/

Postprint available at: Linköping University Electronic Press

Simulation study of the formation of a

non-relativisti pair sho k

M. E. Die kmann 1

†

, and A. Bret 2,3

1

DepartmentofS ien eandTe hnology(ITN),LinkopingUniversity,CampusNorrkoping, 60174Norrkoping,Sweden

2

UniversityofCastillaLaMan ha,ETSIInd,E-13071CiudadReal,Spain 3

InstitutodeInvestiga ionesEnergéti asyApli a ionesIndustriales,CampusUniversitariode CiudadReal,13071CiudadReal,Spain

(Re eivedxx;revisedxx;a eptedxx)

We examine with aparti le-in- ell (PIC) simulation the ollision of twoequally dense loudsof oldpairplasma.The loudsinterpenetrateuntilinstabilitiessetin,whi hheat uptheplasmaandtriggertheformationofapairofsho ks.Thefastest-growingwavesat the ollisionspeed /5andlowtemperatureare theele trostati two-streammode and the quasi-ele trostati oblique mode. Both wavesgrowand saturate viathe formation of phase spa e vorti es. The strong ele tri elds of these nonlinear plasma stru tures provideane ientmeansofheatingupand ompressingtheinowingupstreamleptons. Theintera tion of thehotleptons, whi h leak ba kinto theupstream region, withthe inowing oolupstreamleptons ontinuouslydrivesele trostati wavesthatmediatethe sho k. These waves heat up the inowingupstream leptons primarily along the sho k normal, whi h results in an anisotropi velo ity distribution in the post-sho k region. This distributiongivesrisetotheWeibelinstability.Oursimulationshowsthat evenif thesho kismediatedbyquasi-ele trostati waves,strongmagnetowaveswillstilldevelop initsdownstreamregion.

1. Introdu tion

Compa t obje ts like neutron stars or bla k holes that a rete material an emit relativisti jets. Thesejets are omposed of ele trons,positrons andions. Theemission of relativisti jets by mi roquasars (Fabian & Rees 1979; Margon 1984), whi h are stellar-size bla k holes that gather material from a ompanion star, and by some of the supermassive bla k holes in the enters of galaxies has been observed dire tly (Bridle & Perley 1984). The reball model attributes gamma-ray bursts (GRBs) to ultrarelativisti jets, whi h are emittedduring strong supernovae. Adire t observation of the ultrarelativisti jets that trigger the GRBs and o ur at osmologi al distan es isnotpossible.Theirexisten e anthusnotbeestablishedunambiguously(Woosley& Bloom 2006).However,observations of a mildly relativisti plasma outow during the supernova1998bwbyKulkarniet al.(1998)lendsomesupportto thereballmodel.

The e ien y, with whi h the a reting obje t an a elerate the jet plasma, is not onstant in time. A variable plasma a eleration e ien y results in a spatially varying velo ity prole ofthe jet plasma.Internal sho ks anform at lo ations witha largevelo ity hangeand these sho ks an onstitutestrongsour esof ele tromagneti radiation(Rees1978).Thepromptemissionsofgamma-raybursts,whi hareasso iated

withinternalsho ksinultrarelativisti jets,arevisiblea ross osmologi aldistan esand internalsho ksshould thus besour esofintenseele tromagneti radiation.

The relativisti fa tors of theinternal sho ks in GRB jets are probablyof the order ofafew.Awide rangeoftheoreti al andnumeri alstudieshaveaddressedthe ollision of lepton louds at relativisti speeds and the instabilities that sustain the sho k and thermalize the plasma that rossesit (Kazimura et al. 1998; Medvedev& Loeb 1999; Brainerd2000;Sakaietal.2000;Silvaetal.2003;Haruki&Sakai2003;Jaros heketal. 2004; Medvedev et al. 2005; Milosavljevi & Nakar2006; Chang et al. 2008; Sto kem etal.2008;Bretetal.2008,2013;Sironi&Giannios2014;Mar owithetal.2016).Su h sho ks thermalize plasma via the magneti elds that are driven by the lamentation instabilityof ounter-streamingbeamsof hargedparti les, whi h is alsoknownasthe beam-Weibelinstability.

This magneti instability outgrows the ompeting ele trostati instabilities if two equallydenselepton louds ollideorinterpenetrateatarelativisti speed.Ele trostati waves and instabilities an, however, not always be negle ted and they exist even in lepton plasmas where positrons and ele trons are equally dense. An external ele tri elda eleratesele tronsandpositronsintooppositedire tions,whi h reatesa urrent. Even if the initial ele tri eld perturbation is removed, the urrent leads to a self-generatedele tri eldintheplasma.Theele tri eldeventuallybe omesstrongenough to reversetheowdire tions ofele tronsandpositrons. Themotionovershootsthough andele tronsand positronsos illatearoundtheequilibriumposition.Thisos illationis sustainedbyele trostati elds, whi h an oupleresonantlytootherbeams.

Thejetsofmi roquasars ontainasigni antfra tionofpositrons(Trigoetal.2013) alike the jets that trigger GRBs. The high variability of a mi roquasar jet, whi h is emitted by a stellar-size bla k hole that a retes material from a ompanion star, suggests that internal sho ks are present in su h jets (Kaiser et al. 2000; Miller-Jones et al.2005).Mi roquasarjets expandatalowerspeedthanGRB jets.Typi alvelo ity hanges are thus likelyto be smallerand theinternal sho ksof mi roquasarsmaynot alwaysberelativisti ,inwhi h aseele trostati pro essesmaybe omemoreimportant. Nonrelativisti pairsho kshavesofar notre eivedmu hattentionandthestru tureof theirtransitionlayersremainsunknown.

We explore with a parti le-in- ell (PIC) simulation the initial evolution phase of a leptoni sho kthat forms whentwoequallydensepair louds ollideat aspeedthat is onefthof thespeedoflight.The ele tronsandpositronsof ea h loudhaveanequal numberdensityandmeanspeedandthenet hargeand urrentofea h loudvanishes. The thermal spread of the parti le velo ities is small ompared to the loud ollision speedand theinstabilities aninitiallybedes ribedinthe oldplasmalimit.Both,the pair temperature and the ollision speed are probably too low to be realisti for the plasma distribution lose to an internal sho k of the jet of a mi roquasar. We hose theseinitial onditionsbe auseele trostati instabilitiestendtobemoreimportantthan magneti onesfornonrelativisti ollisionsof oldplasma.We anthusstudyasho k,for whi hele trostati ee tsaremaximizedinitstransitionlayerandthatthus onstitutes alowerbound forpairsho kswithrespe ttothedegreeofmagnetization.

Oursimulationshowsthatthetransitionlayerofthepairsho kisindeedmediatedby nonlinearandpredominantlyele trostati waves.However,thein ompletethermalization of the inowing upstream plasma by the sho k results in a downstream plasma with a thermally anisotropi distribution. This anisotropy is strong enough to trigger the growthof theWeibel instability in its original form (Weibel 1959). Even apra ti ally ele trostati pair sho kthus generatesa magneti eld in its downstream region. The

furtherbythepresen eofionsandbytheambipolarele tri eldtheydriveatthesho k front(Sto kemetal.2014).

Ourpaperisstru turedasfollows.Thesho kformationme hanism,thePICsimulation method,ourinitial onditionsandtheexpe tedspe trumofgrowingwavesaredis ussed inSe tion 2.Se tion 3presentsthesimulationresultsandse tion4isthesummary.

2. Sho k formation, the simulation ode and the initial onditions 2.1. The formationme hanismof a ollisionless leptoni sho k

We examinetheformation of sho ksout of the ollisionof two harge-and urrent-neutral loudsofele tronsandpositrons.Theplasmawe onsiderisinitially unmagne-tized,noionsarepresentandallleptonspe ieshavethesametemperature.Theabsent binary ollisionsimplythatbothlepton loudswillmovethroughea hotheruntilplasma instabilitiesstarttogrow.Onlythreewavemodes andevelopforourinitial onditions. The two-stream modes are purely ele trostati and their wave ve tor is aligned with the ollisiondire tion.Thequasi-ele trostati obliquemodeshaveawaveve torthat is orientedobliquelytothe ollisiondire tionandtheybelongtothesamewavebran has thetwo-streammodes.Thethirdmodeisthelamentationmode,whi hisalsoknownas thebeam-Weibelmode(Califanoetal.1998).Thewaveve torsofthesewavesformthe angle

π/2

withthe ollisiondire tion.Thesethreemodesgrowsimultaneouslyduringthe sho k formationstage.Theirgrowthiseventuallyhaltedby nonlinearpro esses,whi h heatuptheplasmain theoverlaplayerand bringit losertoathermalequilibrium.

A leptoni sho k an be reated in a PIC simulation by the ollision of one lepton loud with a ree ting wall. The ree ted leptons move against the inowing leptons that have not yet rea hed the wall and an overlaplayer develops. The instabilities in this overlap layer let waves grow that heat up the plasma when they saturate. The expansionof theheatedplasmaislimitedononesidebythewallandasho kformson theother side. Thesho k evolutionis resolved orre tlyon e adownstream regionhas formed that is thi k enoughtode ouple thesho k from thewall. Theformationphase of the sho k may, however,not be resolved orre tly by this omputationally e ient method. The me hanism that triggers the lamentation orbeam-Weibel instability is thatparti leswithoppositelydire ted urrentve torsrepelea hotherandparti leswith parallel urrentve torsattra tea h other.Theinstabilitysaturatesbyforming urrent hannels that olle t parti les with the same dire tion of the urrent ve tor. Current hannels that ontain parti les with oppositely dire ted urrent ve tors are separated bymagneti elds. Aree tionofaparti lebythewall hangesitsvelo ity omponent alongthewall'snormaldire tion and,thus, thedire tion ofits urrentve tor.Spatially separated urrent hannels an,however,notformatthewallbe ausetheparti leisnot spatially displa edbythe ree tion.Thesuppressionof thelamentation instability at theree tingwallwillae tthespe trumoftheunstablewaves.

Thisspe trumisresolved orre tlyifwelettwoseparatelepton louds ollide.Ifboth louds dier only in their mean speed, then we have to resolve in the simulation two identi al sho ks that en lose the expanding downstream region. It is omputationally expensiveandunne essarytotra kbothsho ksforalongtime. Hereweletalongand ashortlepton loud ollide.Wein reasethetimeintervalduring whi h we anobserve thesho kbetweenthedownstreamregionand thelonglepton loud.These ond sho k movesintotheoppositedire tionanditeventuallyrea hesthesimulationboundary.By

2.2. The parti le-in- ell(PIC)simulationmethod

We model the ollisionof the lepton louds with aparti le-in- ell (PIC) simulation. ThePICsimulation odeisbasedonthekineti plasmamodel,whi happroximatesea h plasma spe ies

i

by a phase spa e density distribution

f

i

(x, v, t)

. The position ve tor

x

and the velo ity ve tor

v

are treated as independent oordinates, whi h allows for arbitraryvelo itydistributionsatanygivenposition.Thenumberdensityofthisspe ies isthe zero'thmomentofthedistribution

n

i

(x, t) =

R f

i

(x, v, t) dv

andthe meanspeed

¯

v

i

(x, t) =

R vf

i

(x, v, t) dv

orresponds to its rst moment. The number density and the mean speed yield the harge density

ρ

i

(x, t) = q

i

n

i

(x, t)

and the urrent density

J

i

(x, t) = q

i

v

¯

i

(x, t)n

i

(x, t)

ofthespe ies

i

.Thetotal hargedensity

ρ(x, t) =

P

i

ρ

i

(x, t)

and urrentdensity

J(x, t) =

P

i

J

i

(x, t)

update the ele tromagneti eldsvia anite dieren eapproximationof Ampere'sandFaraday'slawsonanumeri algrid.

µ

0

ǫ

0

∂E

∂t

= ∇ × B − µ

0

J

,

∂B

∂t

= −∇ × E.

The EPOCH ode (Arber et al. 2015) we use fullls

∇ · B = 0

and

∇ · E = ρ/ǫ

0

to round-opre ision.

Anensembleof omputationalparti les(CPs)withthe harge

q

i

andmass

m

i

approx-imates thephasespa edensitydistribution

f

i

(x, v, t)

. Therelativisti momentum

p

j

of the

j

th

CPofspe ies

i

isupdatedviaadis retizedformoftherelativisti Lorentzfor e equation

∂

∂t

p

j

= q

i

(E(x

j

) + v

j

× B(x

j

))

and its position is updated via

∂

∂t

x

j

= v

j

. Theele tri eld andthemagneti eld areinterpolatedfromthenumeri algridtothe parti le'sposition

x

j

toupdateitsmomentum.The urrentdensityonthegrid,whi his usedtoupdatetheele tromagneti elds,isthesumoverallparti le urrentsafterthey havebeeninterpolatedfromtheparti lepositions tothegridnodes.

2.3. Thesimulation setup

Ourtwo-dimensional simulationboxhasthe length

L

x

along

x

and

L

y

along

y

.The simulation box is subdivided into the two intervals

−0.65 L

x

< x < 0

and

0 6 x <

0.35 L

x

.Theboundaryat

0.35 L

x

isree tingandthatat

−0.65 L

x

isopen.Theboundary onditionsat

y = 0

and

y = L

y

areperiodi .Wepla eele tronsandpositronswithequal densities

n

0

andtemperatures

T

0

=10eVeverywhereintheboxat

t = 0

.Theele trons andpositronsintheintervalwith

x > 0

haveavanishingmeanspeed.Theele tronsand positronsin theinterval

x < 0

havethemeanspeed

v

0

= 0.2c

along

x

.Nonewparti les areintrodu edwhilethesimulationisrunningandthesimulationisstoppedwellbefore theendoftheinowinglepton louden ountersthesho korbeforetheleptonsthatare ree tedbytheboundaryat

x = 0.35 L

x

returntothesho k.

We normalize the position to the ele tron skin depth

λ

s

= c/ω

p

, where

ω

p

=

(n

0

e

2

/ǫ

0

m

e

)

1

/2

istheele tronplasmafrequen yof one loud.Velo itiesare normalized to

c

.Momentaarenormalizedto

cm

e

andwedene

p

0

= v

0

m

e

asthemeanmomentum ofaleptonoftheplasma loudinthehalf-spa e

x < 0

.Theboxsize

L

x

× L

y

= 60 × 2.4

is resolvedby

1.9 × 10

4

grid ellsalong

x

and by 760grid ellsalong

y

. Ele trons and positrons are represented by 25 CPs per ell, respe tively. The time is normalized to

ω

−1

p

.The simulationtime

t

sim

= 120

, whi h issubdividedinto

57200

equaltimesteps. Wenormalizetheele tri eld to

ω

p

cm

e

/e

andthemagneti eld to

ω

p

m

e

/e

.

Figure1.Thesolutionofthelineardispersionrelationfortwobeams,ea hofwhi h onsists ofele tronsandpositronswiththesamenumberdensity,meanspeedandtemperature

T

0

=

10 eV.Thebeamshaveaninniteextentandthey ounterstreamalong

x

withthespeedmodulus

0.1c

.Thegrowthrate

δ

isexpressedinunitsof

ω

p

.

2.4. Thesolution ofthe lineardispersion relation

Wehavetoverifythatourboxislargeenoughtoresolvethe ompetingunstablemodes and we wantto determine the wave mode, whi h growsfastest for the sele ted initial onditions.Wesolveforthispurposethelineardispersionrelationin ordertodetermine the spe trum of the growing waves. The solution is omputed under the assumption thattheoverlaplayerhasaninnitesize.This onditionisapproximatelyfullledifthe olliding louds aninterpenetrateforsometimebeforetheinstabilitiesgrow.

Theinitialvelo ityspreadfor

T

0

= 10

eVisabout

v

th0

= 4.5 × 10

−3

_c

andboth louds drift toward ea h other at

v

0

= 0.2c

. Thermal ee ts an be negle ted for the ratio

v

0

/v

_th0

= 44

and the leptonbeamsare old.We solvethelinear dispersionrelationin theframe ofreferen ein whi h thetotalmomentum vanishes.Thepair louds movein thisreferen eframeintooppositex-dire tionsatthespeedmodulus

β

′

0

≡ v

0

′

/c = 1/10

. Thenon-relativisti dispersionequationforaperturbationoftheform

exp(ik · r − iωt)

andawaveve tor

k

withanarbitraryorientationis(Bretet al.2010)

ω

2

ǫ

xx

− k

y

2

c

2

ω

2

ǫ

yy

− k

x

2

c

2

− ω

2

ǫ

yx

+ k

x

k

y

c

2

2

= 0 .

(2.1) where

δ

αβ

istheKrone kersymboland

ǫ

αβ

(k, ω) = δ

αβ

1 −

ω

2

p

ω

2

!

+

ω

2

p

ω

2

X

j

Z

d

3

p

p

α

p

β

k

·



_∂f

0

j

∂p



mω − k · p

.

(2.2) Theproblemofndingthefastestgrowingmodehasbeensolved(Bret&Deuts h2005; Bretetal.2013)for olddistributions oftheform

f

0

j

(p) = δ(p

y

)δ(p

x

− P

j

)

.

Figure1showsthesolutionofthelineardispersionrelationforourplasmaparameters. Thegrowthratepeaksatthewavenumber

k

x

λ

s

≈ 12

anditsvaluedoesnotdependon

k

y

λ

s

for the onsidered wavenumber interval.The fastest-growingmodes arethus the two-stream/obliquemodes.Theirpeakexponentialgrowthrateis

δ

T S

ω

p

=

√

2

2

.

(2.3)

Thelamentationmodesare hara terizedbyaowaligned omponent

k

x

= 0

.Califano etal.(1998)estimatedtheirgrowthrateas

δ

W

ω

p

= 2β

′

0

.

(2.4)

Figure 1demonstratesthat the growthrateof the lamentationmodeswith

k

x

= 0

is smallerthan that ofthe two-stream/obliquemodes,whi h onrms theaforementoned approximationssin e

δ

W

< δ

T S

for

β

′

0

= 1/10

.

We anestimatewiththehelpofFig.1ifandhowourlimitedboxsizewillae tthe spe trumofgrowingwaves.Thesimulationemploysperiodi boundary onditionsalong

y

andtheboxlengthis

L

y

in thisdire tion.Thesmallestresolvedwavenumberisthus

k

c

= 2π/L

y

or

k

c

λ

s

= 2.6

andwaveswith

k

y

< k

c

annotgrow.Figure1showsthatthe growthrateofthelamentationmodesde reasesbelow

δ

W

for

k

y

< k

c

whilethatofthe two-stream/obliquemodesremains un hanged.The main ee t of the limitedbox size along

L

y

is thus to suppress the wavenumbers where the growthof the lamentation instabilityisnegligible.Ifoursimulationshowsthattheplasmadynami sisgovernedby thetwo-stream/obliquemodes,thenwewouldobtainthesameresultalsoforlarger

L

y

. 3. Simulation results

Wedis ussthesimulationresultsatsele tedtimesandfo usonthesho kthat forms atlowervaluesof

x

.Therstpartaddressesthewavemodesthat triggertheformation of sho ks. The se ond part dis usses the stru ture of the sho k and the ele tri elds thatmediate itandthenalpartexaminesthegrowthofmagneti elds.

3.1. Instabilityandnonlinear saturation

Thetwo loudsofinitiallyunmagnetized ollisionlessleptonplasmawillmovethrough ea h other for some time before plasma instabilities set in. Figure 2(a- ) displays the ele tri

E

x

and

E

y

omponentsaswellasthemagneti

B

z

omponentat thetime

t

1

=

7.6

.Thelepton loud,whi hwasinitiallylo atedin thehalf-spa e

x < 0

,hasmovedby

v

0

t

1

= 1.5

towardsin reasingvaluesofx. Waveshavegrowninthe loudoverlaplayer, whi hspanstheinterval

0 < x < 1.5

atthis time.Thedistribution of

E

x

revealswaves withawavelength

λ ≈ 0.4

.The

E

y

and

B

z

omponentsare losely orrelatedand both os illaterapidly alongy.

Thein-planeele tri eld omponentsandtheout-of-planemagneti eldatthetime

t

2

= 14.7

aredisplayedinFig.2(d-f).Figures2(a)and(d)showthesamedistributionof

E

x

ex ept for thelargeramplitude. Theirspatial onnementdemonstrates that these wavesdonotpropagatealong

y

.Thewavestru turesbelongtoele trostati two-stream modes.Thepatternsin

E

y

resemblethosein

B

z

andtheiramplituderatiois omparable tothatattheearliertime.Thespatial orrelationoftheeldstru turesinthedistribution of

E

y

and

B

z

suggeststhattheybelongtothesamewaves.

We anextra t somepropertiesof the wavesfrom a omparison ofthe amplitudeof

E

y

and

B

z

at thetimes

t

1

or

t

2

. Theratio of theeld energy densities

ǫ

0

(E

2

x

+ E

2

y

)/2

and

B

2

z

/2µ

0

is in the given normalization

(E

2

x

+ E

2

y

)/B

2

z

≈ 100

. The parti les of both louds move at a speed

≈ v

0

/2

relative to the waves, whi h are slow-moving in the referen e frame of the overlaplayer. The ele tri for e imposed on a harged parti le, whi hmoveswith

v

0

/2

=0.1, is 50times largerthanthe magneti for e. We on lude that thewave'smagneti eld neitherhasasigni antenergydensitynordoesitae t theleptondynami s.Thewavesarethus quasi-ele trostati andtheirwavelengthalong the ollisiondire tionis

≈ 0.4

.Theamplitudeofthewaveshasin reasedbyafa tor

≈ 50

Figure 2. The in-plane ele tri eld and the out-of-plane magneti eld lose to the initial ollision boundaryatthe time

t

1

= 7.6

(left olumn)and at

t

2

= 14.7

(right olumn):Panels (a,d)show

E

x

,panels(b,e)show

E

y

andpanels( ,f)show

B

z

.

during thetimeinterval

t

2

− t

1

= 7.1

. Ifweassumethatthewavesgrowexponentially, thentheirgrowthrateis

δ ≈ 0.5

in unitsof

ω

p

,whi hmat hesthatin Fig.1.

Thewavemodesthatyieldtheobservedele tri eld anbeidentiedwithitsspatial powerspe trum.WeFourier-transformthein-planeele tri elddistribution

E

p

(x, y) =

E

x

(x, y) + iE

y

(x, y)

overthespatialinterval

0.2 < x < 2.7

andoverall

y

andmultiplyit withits omplex onjugate.Figures3(a,b) showthepowerspe traatthetimes

t

1

and

t

2

inthequadrant

k

x

> 0

and

k

y

> 0

.Thepowerspe trumat

t

1

= 7.6

showswavepower at

k

x

λ

s

≈ 14

,whi h extendsupto maximumperpendi ularwavenumber

|k

y

λ

s

| ≈ 200

. Thewavenumber

k

x

λ

s

= 14

orrespondstoawavelengthalong

x

ofabout

0.45

.

The ow-aligned wave number

k

x

λ

s

≈ 14

of the fastest-growing waves and the extension of wavepowerto largevalues of

k

y

agree with the numeri alsolutionof the lineardispersionrelationin Fig.1.Thesolutionofthelineardispersionrelationpredi ts a peak growth rate that does not depend on the value of

k

y

for the onsidered wave numbers.Thewavespe trumonFig.3(a) doeshoweversuggest that waveswith alow valueof

k

y

growfaster.Thegrowthrateisproportionaltotheamplitudethewavewould rea h after a given time if its growth would not be limited by nonlinear ee ts. The ele tri eldamplitude,whi h isne essaryto formphasespa evorti es,de reaseswith in reasing values of

k = |k

2

x

+ k

2

y

|

1

/2

0

50

100

150

200

0

50

100

(a) k

_y

λ

S

k

x

λ

S

−2.5

−2

−1.5

−1

−0.5

0

0

50

100

150

200

0

50

100

(b) k

_y

λ

S

k

x

λ

S

−1

0

1

2

3

Figure3.Thespatialpowerspe traofthein-planeele tri eld

E

p

= E

x

(x, y) + iE

y

(x, y)

at thetime

t

1

= 7.6

(a)and

t

2

= 14.7

(b).The olors aleis 10-logarithmi andbothspe traare normalizedtothepeakvaluein(a).

Figure 4. The phase spa e density distribution inthe

x, p

x

-plane at the time

t

2

= 14.7

of ele trons(a)andpositrons(b).Thephasespa edensitydistributionisaveragedoverallother dimensions.Themomentumisnormalizedto

p

0

.The olors aleis10-logarithmi .

3( ) anthus be explained with a strongernonlinear damping that is imposed on the modeswithalargevalueof

k

y

.

Thepowerspe truminFig.3(b)isstill on entratedonthetwo-stream/obliquemode bran h. Itswidth along

k

y

has diminished,whi h suggeststhat thermal dampingis at work;therangeofwavenumbers

k

y

that areunstableto theobliquemodeinstabilityis largein a oldplasma,while thewavegrowthis on entratedat lowvaluesof

k

y

ifthe plasma is hot(Silvaet al. 2002). A rst and se ond harmoni along

k

x

have emerged. Thewaveamplitudes havethusrea hedanon-linearregime (Umedaetal.2003).

Non-linear ee ts in the wave distribution should be tied to hanges in the lepton distribution.Figure4showsthephasespa edensitydistributions

f (x, p

x

)

oftheele trons and positrons. Theoverlaplayerof both louds spans theinterval

−0.2 < x < 3

. The ounterstreaming loudshavenotyetmergedalong

p

x

.However,thesubstantialparti le a elerationdemonstrates that the instability is about to saturate.The density in the overlaplayeristwi e that of asingle loud and thedensity u utations aused bythe wavesare oftheorderof

5% − 10%

(notshown).

Figure5.Panels(a)and(b)showtheele tri

E

x

and

E

y

omponents losetotheinitial ollision boundary.Panels( ) and(d)showthe phasespa e densitydistributions inthe

x, p

x

-planeof ele trons andpositrons,respe tively.Thephase spa edensity distributions are averaged over allother dimensions,theyarenormalizedtothesamevalueanddisplayedona10-logarithmi s ale.Themomentumisnormalizedto

p

0

.Thesimulationtimeis

t

3

= 58.8

.

3.2. Sho kformation

The two-stream instability saturates by forming stable phase spa e vorti es in the ele tron and positron distributions (Berk & Roberts 1967) and the sameholds during the initial saturation stage of the oblique mode instability (Die kmann et al. 2006b). Ele tron phasespa e vorti esare hara terized by strong bipolar pulses in theele tri elddistribution,whi h orrespondtoalo alizedpositiveex ess harge.Positronphase spa evorti es orrespondtoalo alizednegativeex ess harge.

The in-plane ele tri eld omponents at the time

t

3

= 58.8

are displayed in Fig. 5(a,b). The ele tri

E

x

omponent shows su h bipolar eld stru tures. A large quasi-planar eld pulse is lo ated at

x ≈ 2.5

in the interval starting from

y ≈ 1

that goes throughtheperiodi boundaryat

y = 2.4

until

y ≈ 0.5

.Thepolarityof

E

x

indi atesthe presen e of apositive ex ess harge in between bothele tri eld bands. If this quasi-planar bipolar pulse is asso iated with an ele tronphase spa e vortex,then the latter should bedete tablein theele tronphasespa edensitydistribution evenifithasbeen integratedoverallvaluesof

y

.

Figures 5( , d) show the orresponding ele tron and positron distributions. Figure 5( ) onrms theexisten e of aphase spa e vortex in the ele trondistribution at this lo ation.ThevortexinFig.5( )spansthespatialinterval

2 < x < 3

andthemomentum interval

−1 < p

x

/p

0

< 1

. Themean momenta of theupstreamele trons andpositrons aremodulatedbytheele trostati potentialofthevortexwhentheypassit,buttheyare nottrappedbyit.Theupstreamleptons ontinuetomovetoin reasingvaluesof

x

until theyarethermalizeduponenteringthedownstreamregion

x > 4

,whi his hara terized byadensephasespa edensitydistributionbetween

0 < p

x

/p

0

< 1

.Thisthermalization anonlybea omplishedbytheeldstru turesseeninthein-planeele tri eldbetween

x ≈ 3

and

x ≈ 4

inFig.5(a,b).

Thedistribution ofthe positronsshowstwosmallervorti esthat surround thelarge ele tronphase spa e vortex. The positron verti esare entered at

x ≈ 1.8

and

x ≈ 3

. The zero- rossing of the ele tri

E

x

omponent and, thus, the extremal point of the ele trostati potentialat

x ≈ 2.5

in Fig.5(a) orrespondstoastable equilibrium point

Figure6.Panels(a)and(b)showtheele tri

E

x

and

E

y

omponents losetotheinitial ollision boundary.Panels( ) and(d)showthe phasespa e densitydistributions inthe

x, p

x

-planeof ele trons andpositrons,respe tively.Thephase spa edensity distributions are averaged over allother dimensions,theyarenormalizedtothesamevalueanddisplayedona10-logarithmi s ale.Themomentumisnormalizedto

p

0

.Thesimulationtimeis

t

4

= 120

.

for the trapped ele trons. Hen e it is an unstable equilibrium point for the positrons, explainingwhythevorti esofpositronsandele tronsarestaggeredalong

x

.

Asmalllo alized loudofele tronsandpositronsis entredat

x ≈ 0

and

p

x

≈ 0

.The loudisanartifa tfromourinitial onditions.Thenitegrowthtimeoftheele trostati instabilitiesimpliesthatthewavesstarttogrowwellbehindthefrontoftheplasma loud thatwasinitiallylo atedinthehalf-spa e

x > 0

.This harge-and urrentneutral loudis stableagainstele trostati instabilities,be auseitsextentalong

x

isnotsu ientlylarge toallowittointera twiththeinowingupstreamleptonsviaatwo-streaminstability.

Figure6showsthein-plane ele tri elddistribution andtheasso iatedleptonphase spa edensitydistributionsatthetime

t

4

= 120

.Weobservestrongquasi-planarele tri eld stru tures in the

E

x

-distributionin the interval

−3 < x < 0

. Their amplitude is omparabletotheonethatgaverisetophasespa evorti esintheele tron-andpositron distributions at the earlier time. These ele trostati stru tures in

E

x

have propagated well beyond the initial ollisionboundary

x = 0

rea hing aposition

x ≈ −3

. We nd relativelystrongele tri eldos illationsin

E

x

and

E

y

between

0 < x < 8

.Thetransition layerofthissho kthusspansatthis timeanintervalwiththewidth

∆x ≈ 10

.

Thestrongplanarwavesintheinterval

−3 < x < 0

inFig.6are orrelatedwithphase spa evorti esin thehotleptonpopulationatlowspeeds.Thevorti esofele tronsand positronsarestaggeredalong

x

.Theele tronsandpositronsthatgyrateinthesevorti es originatefromthehotplasma omponentandtheyarewell-separatedalong

p

x

fromthe inowing upstreamleptons. The meanspeed of these phase spa e vorti esis lessthan

p

x

= 0

,whi h impliesthat theymovetowardsde reasingvaluesof

x

. Themean speed ofthevorti esde reaseswithanin reasingdistan efromthesho ktransitionlayerand theyarethusa eleratedawayfromthedownstreamplasma.Theleptons,whi hgyrate inthevorti es,rea hapeakmomentum

≈ −p

0

.

The simultaneous presen e in theinterval

−6 < x < 5

of the hotleptons that have leakedfromthedownstreamregionandthe oolerdriftingupstreamleptonsimpliesthat the overall plasma distribution is non-thermal and thus unstable. The ele tri eld of the phase spa e vorti es seeds the instability and we observe momentum os illations along

p

x

inthe oolinowingele tronsandpositronsthatin reasewith

x

intheinterval

−2

−1

0

1

2

3

4

5

6

1

1.5

2

2.5

3

(a) X

N(x)

−4

−2

0

2

4

6

8

10

12

1

1.5

2

2.5

3

(b) X

N(x)

Figure7.Thetotalleptondensity

N(x)

inunitsoftheinitialtotaldensity

2n

0

atthetime

t

3

= 58.8

(a)and

t

4

= 120

(b).

a strong urrent, whi h indu es an ele tri eld. The ele tri eld os illates in spa e anditsos illationamplitudede reasesin unisonwiththenet urrentinthedire tionof de reasingvaluesof

x

.We andes ribethis os illationin termsofaprodu tbetweena sinusoidallyos illatingele tri eldandanenvelopefun tion.

A spatially varying envelope fun tion gives rise to a ponderomotive for e (Kono et al. 1980) that does not depend on the sign of the parti le harge and a elerates ele trons and positrons in the dire tion of de reasing valuesof the envelope fun tion. This ponderomotive for eis ex ertedby themodulated upstreamplasmaonto thehot leptonsthatformthephasespa evorti esandita eleratesthem.

The leptons in the interval

1 < x < 6

in Fig. 6( ,d) are omposed of a hot dilute omponent and the ool denseupstream leptons. Both populationsgradually mix and they merge to a singleone at

x ≈ 6

. We observeele tri elds in this interval in Fig. 6.Theseelds showsomepie ewiseplanarstru tures,whi h orrespondtophasespa e vorti eswithalimited extentalong

y

.The twostrongestlo alizedstru tures at

x ≈ 1

areseparatedbyaperpendi ular

E

y

eldat

y ≈ 0.8

.Theselo alizedstru turesarelikely to betheresult ofan instability of initiallyplanarphase spa evorti esorphasespa e tubes.Indeed,two-dimensionalPICsimulations(Oppenheimetal.1999)ofphasespa e tubes in a stabilizing magneti eld show that the phase spa e tubes gradually break upalong theiraxes. The ollapseof aphasespa e vortex isan ee tivewayto s atter theleptonsin phase spa e,whi h resultsin amixing ofthedownstream andupstream leptonsinthe

x, p

x

plane.Strongsmall-s aleele tri eldsareobservedupto

x ≈ 8

.The absen eofphasespa evorti eswith

x > 8

demonstratesthattheleptondistributionin thisintervalisnolongerunstabletoele trostati instabilities.

The phasespa e density distributionof the leptonswasuniform in the interval

4 <

x < 6

at thetime

t

3

= 144

and in the interval

7 < x < 13

at the time

t

4

= 297

. We on ludethat these intervals orrespond to adownstreamregion that is lose to being in athermal equilibrium, at least with respe t to ele trostati waves and instabilities. The density distribution along

x

sheds lightonto how mu h theplasma is ompressed by thesho k rossing. Figure7 omparesthedensity distributions at thetimes

t

3

and

t

4

. The density onverges at low

x

to the initial density. A density peak is observed loseto

x ≈ 0

in Fig.7(a),whi h orrespondstothedenselepton loudat thisposition showninFig.5(a).ThedensitypeakhasdisappearedinFig.7(b).Theele tri eldsthat grewinthisspatialintervalinresponsetotheinstabilitybetweentheinowingupstream

Figure8.Thetotalleptondistribution

f

t

(v

x

, v

y

)

averagedover

−4.5 < x < −4.2

isshownin panel(a), thataveragedover

1.9 < x < 2.2

inpanel(b)andthat averaged over

8.3 < x < 8.6

isshowninpanel( ).The olors ale is10-logaritmi andnormalizedtothepeakvaluein(a).

ele tronsand positronsinto oppositedire tions.Thedensityrisesfrom about

N (x) = 1

to

N (x) ≈ 3

overafewele tronskindepths. Theplasma ompressionfa torof about3 istheoneexpe ted forastrongnonrelativisti sho k(Zel'Dovi hetal. 1967).

3.3. Se ondaryinstabilitiesandmagneti eldgeneration

The ele tri elds asso iated with the phase spa e vorti esheat up the leptons via Landau damping(Landau 1946;O'Neil1965)and their ollapses attersthemin phase spa e.Theee tsofthisheatingontheleptondistributionisvisualizedbyFig.8,whi h showsthephasespa edensityasafun tion of

p

x

and

p

y

atthreepositionsalong

x

.The distributionhasbeenintegratedover

y

andoveranintervalalong

x

ofwidth 0.3.

ThedistributioninFig.8(a)hasbeensampledfarupstreamofthesho k.Theupstream leptons onstitutethe old densebeamthat is lo ated at

p

x

≈ p

0

. The leakedleptons formahotanddilutebeamthat movesat

p

x

≈ −p

0

.Themeanspeedofthehotlepton beam ex eeds that expe ted from a spe ular ree tion, sin e the sho k is moving to in reasingvaluesof

x

.Figure8(b)revealsthattheinowingupstreamleptonshavebeen heatedupbythetimetheyrea htheposition

x ≈ 2

.Theyaredistributedoverawider velo ity rangeandtheir peak valueofthephasespa e density hasthusde reased.The temperatureisoftheorderof100eV.Theseleptonsareimmersedinahotdilutelepton omponent.Its thermalmomentum spreadisofthe orderof

p

0

and thetemperatureis thusaboutonekeV.Theinowingupstreamleptonsformahotbeamat

p

x

≈ p

0

in8( ) thatis onlyabouttwi e asdenseastheleptonsin thehotpopulation.

Thewavesobserved loseto

x = 2

inFig.6(a,b)suggestthatthevelo itydistribution in Fig.8(b)is stillunstableto anele trostati instability. It annotbe thetwo-stream instabilitybe ause that onerequires twobeams that are well-separatedalong

p

x

. This distribution an,however,stillbeunstabletotheele trona ousti instability.Alikethe well-known ion a ousti instability, whi h is driven by a drift between old ions and hotele trons,theele trona ousti instability andevelopif oldele tronsdriftrelative to a hot ele tronspe ies. Waves grow if the driftspeed between the hotand the old ele tron spe ies ex eeds several times the thermal spread of the old ele tron spe ies (Gary1987).This onditionisfulllledinFig.8(b).Wenoteinthis ontextthatalthough

Figure9.Panels(a)and(b)showtheout-of-plane omponent

B

z

ofthemagneti eldatthe times

t

3

= 58.8

and

t

4

= 120

, respe tively. The olor s ale is the same for both panels. For omparison: Thedownstream regionat

t

3

= 58.8

is en losedby sho ksat

x ≈

3

and

x ≈

8

, whilethe orre tlyresolvedsho kat

t

4

= 120

islo atedat

x ≈

5

.

theinowingupstreamleptonstheirnumberdensity,whi hweobtainbyintegratingthe phasespa edensityalong

p

x

,isof thesameorder.Theintera tionof ounterstreaming leptonbeamswithasimilardensityresultsinrapidly growinginstabilities.Tothebest ofourknowledgethea ousti instabilityin pairplasmahasnotyetbeenexplored.Here we annot unambigously show that it exists in pair plasma, be ausethe ele tri eld mayalsobetheresidualeld ofaphasespa eholethat formedpreviously.

ThedistributioninFig.8( )appearstobestableagainstele trostati instabilitiessin e wedo notobservesigni antele tri eld os illationsin theregion

x > 8

. Thevelo ity distribution of the leptons in this region is, however,not a Maxwellian. Therefore the plasma ontainsfreeenergythat anbereleasedbya ollisionlessinstability.Thethermal anisotropy ontainedinthetotalleptonvelo itydistribution

f

t

(v

x

, v

y

)

andmeasuredin therestframeofthedownstreamplasma anbeestimatedas

A =

R f

t

(v

x

, v

y

)(v

x

− p

0

/2m

e

)

2

dv

x

dv

y

R f

t

(v

x

, v

y

)v

y

2

dv

x

dv

y

− 1.

(3.1)

A value

A = 0

would saythat thethermalenergy in the

x

dire tionequalsthat in the

y

dire tion,whi hwouldimplythat thereisnothermalanisotropy.Weobtainthevalue

A ≈ 6

from the data shown in Fig. 8( ). Su h a large anisotropy valueresults in the Weibelinstabilityin itsoriginalform (Weibel1959;Morse&Nielson1969).

Figure 9 onrmsthat a magneti eld hasgrownin the downstream region. Strong magneti eldswithapproximatelythesamepeakamplitudearepresentatbothtimes. Theamplitudeofthemagneti eldex eedsthatobservedinFig.2( ,f)byafa tor3and itequalsthatoftheele tri eldin thegivennormalizationin Fig.6(a,b).Nevertheless, themagneti for e,whi hisa tingonaleptonthatismovingatthespeedof

0.1

relative to theseeld pat hes,will stillbeanorder ofmagnitude weakerthan theele tri for e inthesho ktransition layer.

Thestrongdownstreammagneti eldsarenot orrelatedwithele tri eldstru tures (SeeFigs.5(a,b)and6(a,b))andtheyarethusnotdrivenbyanobliquemodeinstability. The Weibel instability drives magnetowaves with a negligible ele tri eld and with the samemagneti eld dire tion as theone observedhere and this instability is thus ompatible withthesimulationdata.TheWeibelinstabilityyieldsmagneti elds with

anisotropies (Morse& Nielson1971; Sto kem et al. 2009).The velo ity spreadin Fig. 8( ) orrespondsto aleptontemperatureoftheorderof 1keV or

10

7

K.Themagneti pressureofaeld

B

z

= 0.03

isthusafewper entofthe umulativethermalpressureof ele tronsandpositronsandwithintherangethatisa essibletotheWeibelinstability. We an omparethemaximumsizeofthemagneti pat hesinFig.9tothegyroradius ofanele troninthateld. Themagneti eld

B

˜

z

andthe ollisionspeed

v

˜

0

inphysi al units an be al ulated from the normalized ones via

B

z

= e ˜

B

z

/m

e

ω

p

and

v

˜

0

= v

0

c

. The gyroradius of an ele tronthat movesat the speed

v

0

/2

relative to the stationary perpendi ularmagneti eld,whi hisnormalizedto

λ

s

= c/ω

p

,isthen

r

g

/λ

s

= v

0

/2B

z

. Taking

v

0

= 0.2

and

B

z

= 0.03

gives

r

g

≈ 3λ

s

,whi hisaboutthreetimesthe oheren e s ale of the largest magneti eld pat hes. The magneti eld pat hes are also not stationary on the time s ale needed to perform a gyro-orbit. This time equals for a maximumamplitude

B

max

= 0.03

in ournormalization

2π/B

max

≈ 200

,whi hex eeds thesimulationtime.Theleptons anthusnot omplete afull gyro-orbit.Themagneti eldwillinsteaddee tleptonsbyasmallanglethatdependsonwheretheleptonentered thepat handonhowlongitstayedinsidethepat h.Themagneti eldwillthuss atter theleptonsofthedire tedbeaminFig.8( ).Therepeateds atteringoftheleptonswill eventuallythermalize theirdistribution.

4. Summary

Wehaveexaminedtheformationandtheinitialevolutionofanon-relativisti leptoni sho k. The sho kwas reatedby letting two spatially uniform louds of equallydense ele tronsandpositrons ollideatarelativespeedof0.2 .Theabsen eofbinary ollisions impliedthatboth loudsinitiallyinterpenetratedandformedanoverlaplayer.The two-stream instability grew in this overlap layer. The nonlinear saturation of the growing wavesheateduptheplasmain theoverlaplayerand transformeditinto adownstream regionthatwasen losedbytwosho ks.Wefollowedtheevolutionofoneof them.

Someofthehotdownstreamleptonses apedupstreamandintera tedwiththe inow-ing upstream plasma. Nonlinear and predominantly ele trostati phase spa e vorti es formed,whi hmediatedthesho ktransitionlayer.Thesestru turesareunstable (Whar-tonetal.1968;Berk&Roberts1967;Morse&Nielson1969)andtheir ollapses attered and heated the leptons and gave rise to ele tri eld u utations. The intera tion of ele tri eld u tuations and harged parti les has asimilar ee t as binary ollisions between parti les(Dum 1978;Die kmann et al.2006a; Baleet al. 2002;Baalrud et al. 2009; Bret 2015) with respe t to the thermalization of the inowing plasma and this intera tion ontributedto theparti leheating bythesho k.

A spatially onned region formed, in whi h the plasma density ex eeded the u-mulative density of the olliding louds. We found the ompression fa tor 3, whi h is expe ted forastrongsho k.However,theele trostati sho ktransition layer ouldnot fully thermalize the upstream plasmathat rossed it. Theresidual thermal anisotropy of thedownstream plasma drovemagnetowavesviathe instability proposed by Weibel (1959)in itsoriginalform.Themagneti eldswereweakandspatiallyinhomogeneous. Nevertheless,theleptonsthatwould enterthese eldpat heswouldexperien ea small-angledee tionbythemagneti eld.

The magnitude of the dee tion angle depends on the time the parti le needs to ross themagneti pat h. Repeateddee tions will thus randomize theparti le paths. Thisrandomizationwill resultinathermalizationofthedownstreamplasmaonatime s alethat ex eedsbyfartheonea essibletooursimulation.Wenote thatthe Weibel

Magneti eld pat hes will thus grow and s atter parti lesuntil thelepton population hasrea hedathermalequilibrium.

Themotivationofourworkhasbeentobetterunderstandthepropertiesoftheinternal sho ksofmi roquasarjets.Theinitialtemperature,whi hwegavetotheleptons,andthe sho kspeedare,however,lowerthanthevalueswemaynd losetotheinternalsho ks ofmi roquasarsjets.We hosetheselowvaluestomakethewavesandplasmastru tures inthesho ktransitionlayerquasi-ele trostati .Wewillstudyinfutureworklarger non-relativisti ollisionspeedsand onsider the ee ts of a largerinitial temperature and determineupto whi hvaluesthesho ksresembletheonewehaveexaminedhere.

Wewill also study in moredetailthe spe trum of theunstablewavesthat angrow inapairplasmathat onsistsofa oolbeamimmersedinahotba kgroundpopulation. Oursimulationshowedthepresen eofele trostati wavesinsu haplasma.Weproposed thatsu hwaves ouldbedrivenbyaninstability,whi hissimilartotheele trona ousti instability Gary (1987), and we haveto determine if asimilar instability exists in the pairplasmawe onsidered here.

A knowledgements: Grant ENE2013-45661-C2-1-P from the Ministerio de Edu- a ión y Cien ia, Spain and Grant PEII-2014-008-P from the Junta de Comunidades deCastilla-LaMan ha.ComputertimeandsupportwasprovidedbytheHPC2Nandby theSwedishNationalInfrastru tureforComputing(SNIC)throughthegrant SNIC2015-1-305.

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