astronomy and electron acceleration due to high frequency...

Astron. Astrophys. 356, 377–388 (2000) ASTRONOMYAND

ASTROPHYSICS

Electron acceleration due to high frequency instabilitiesat supernova remnant shocks

M.E. Dieckmann1?, K.G. McClements2, S.C. Chapman1, R.O. Dendy2,1, and L.O’C. Drury 3

1 Space and Astrophysics Group, Department of Physics, University of Warwick, Coventry CV4 7AL, UK2 EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, UK3 Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland

Received 20 August 1999 / Accepted 15 February 2000

Abstract. Observations of synchrotron radiation across a widerange of wavelengths provide clear evidence that electronsare accelerated to relativistic energies in supernova remnants(SNRs). However, a viable mechanism for the pre–accelerationof such electrons to mildly relativistic energies has not yet beenestablished. In this paper an electromagnetic particle–in–cell(PIC) code is used to simulate acceleration of electrons frombackground energies to tens of keV at perpendicular collision-less shocks associated with SNRs. Free energy for electron ener-gization is provided by ions reflected from the shock front, withspeeds greater than the upstream electron thermal speed. ThePIC simulation results contain several new features, including:the acceleration, rather than heating, of electrons via the Bune-man instability; the acceleration of electrons to speeds exceed-ing those of the shock–reflected ions producing the instability;and strong acceleration of electrons perpendicular to the mag-netic field. Electron energization takes place through a varietyof resonant and non–resonant processes, of which the strongestinvolves stochastic wave–particle interactions. In SNRs the dif-fusive shock process could then supply the final step required forthe production of fully relativistic electrons. The mechanismsidentified in this paper thus provide a possible solution to theelectron pre–acceleration problem.

Key words: acceleration of particles – instabilities – shockwaves – stars: supernovae: general

1. Introduction

The detection of radio synchrotron emission from shell–typesupernova remnants (SNRs) is a clear indication that electronsof typically GeV energy are being accelerated in such objects.There is now convincing evidence that synchrotron emissionfrom some remnants extends to X–ray wavelengths (Pohl &Esposito 1998): this implies the presence of electrons with en-

Send offprint requests to: K.G. McClements([email protected])

? Present address:Institutet for teknik och naturvetenskap,Linkopings Universitet, Campus Norrkoping, 601 74 Norrkoping,Sweden

ergies of order 1014eV. The prime example of this is the rem-nant of SN1006, recent observations of which using the ASCA(Koyama et al. 1995) and ROSAT (Willingale et al. 1996) space-craft show that X–ray emission from the bright rim has a hard,approximately power–law spectrum. In contrast, emission fromthe centre is softer, with a strong atomic line component. Thesharp edges and strong limb brightening observed at both X–ray and radio wavelengths indicate that: the acceleration siteis the strong outer shock bounding the remnant; the accelera-tion is continuous; and the local diffusion coefficient of elec-trons near the shock front is substantially reduced relative tothat in the general interstellar medium (Achterberg et al. 1994).The possibility that the X–ray emission from SN1006 is ther-mal bremsstrahlung has been examined by Laming (1998), andfound to be less tenable than the synchrotron interpretation.

There is thus extensive observational evidence that thestrong collisionless shocks bounding shell–type SNRs accel-erate electrons to relativistic energies. The standard interpre-tation of extragalactic radio jet observations is also based onthe premise that the relativistic electrons responsible for ob-served synchrotron emission are produced by shocks, althoughin this case the shock parameters are much less certain. He-liospheric shocks, on the other hand, do not generally appearto be associated with strong electron acceleration, perhaps be-cause the Mach numbers of such shocks are much lower thanthose of SNRs and extragalactic radio jets, although Andersonet al. (1979) have published data showing that keV electrons areproduced in the vicinity of the perpendicular bow shock of theEarth.

While diffusive shock acceleration (Axford et al. 1977;Krymsky 1977; Bell 1978; Blandford & Ostriker 1978) pro-vides an efficient means of generating highly energetic elec-trons from an already mildly relativistic threshold, and can op-erate at oblique shocks as well as parallel ones (Kirk & Heavens1989), the “injection” or “pre–acceleration” question remainsvery open: by what mechanisms can electrons be acceleratedfrom background (sub–relativistic) energies to mildly relativis-tic energies (Levinson 1996)? In this paper, we investigate oneof the possible answers, which has attractive “bootstrap” char-acteristics. Specifically, we suggest that waves are excited bycollective instability of the non–Maxwellian population of ions

378 M.E. Dieckmann et al.: Electron acceleration at supernova remnant shocks

reflected from a perpendicular shock front, and that these wavesdamp on thermal electrons, thereby accelerating them. Such aprocess was first proposed as a candidate acceleration mech-anism for cosmic ray electrons by Galeev (1984). This modelwas developed further by Galeev et al. (1995), with the addedingredient of macroscopic electric fields implied by the need tomaintain quasi–neutrality in a plasma with an escaping popu-lation of electrons. McClements et al. (1997) carried out a pri-marily analytical study of electron acceleration by ion–excitedwaves at quasi–perpendicular shocks, which was necessarilyrestricted to quantifying linear regimes of wave excitation andparticle acceleration, in relation to inferred shock parameters.

Instabilities driven by shock–reflected ions at SNR shockshave also been invoked by Papadopoulos (1988) and Cargill &Papadopoulos (1988) as mechanisms for electron heating, ratherthan electron acceleration. On the basis of a simple analyticalcalculation, Papadopoulos predicted that strong electron heatingwould occur at quasi–perpendicular shocks with “superhigh”Mach numbers (specifically, shocks with fast magnetoacousticMach numbersMF > 30–40) through the combined effectsof Buneman (two–stream) and ion acoustic instabilities. In thismodel the Buneman instability, driven by the relative streamingof shock–reflected ions and upstream electrons, heats the elec-trons to a temperatureTe much greater than the ion temperatureTi: in these circumstances, the ion acoustic instability can bedriven unstable if there is a supersonic streaming between theelectrons and either reflected or non–reflected (“background”)ions. Using a hybrid code, in which ions were treated as parti-cles and electrons as a massless fluid, Cargill & Papadopoulos(1988) found that the electron heating predicted by Papadopou-los (1988) could occur in a self–consistently computed shockstructure. However, as Cargill & Papadopoulos point out in thelast paragraph of their 1988 paper, the use of a fluid model for theelectrons means that hybrid codes cannot be used to investigateelectron acceleration. Recently, Bessho & Ohsawa (1999) haveused a particle–in–cell (PIC) code to investigate accelerationof electrons from tens of keV to highly relativistic energies atoblique shocks in which the electron gyrofrequencyΩe exceedsthe electron plasma frequencyωpe.

An improved theoretical understanding of electron acceler-ation at shocks is desirable not only for intrinsic interest, butalso to enable observations of synchrotron and inverse Comp-ton emission to be related quantitatively to shock parameters.However, almost all work on particle acceleration has concen-trated on ions. There are several reasons for this. First, upstreammomentum and energy fluxes are dominated by ions, and theshock structure problem therefore reduces essentially to that ofisotropizing the ion distribution. Second, much of our under-standing is based on the use of hybrid codes, in which elec-trons are represented as a fluid: such codes cannot provide in-formation on electron acceleration. However, the very fact thatelectron dynamics does not appear to be important for shockstructure allows us to separate the two problems: prescribingthe ion parameters using the results of hybrid code simulations,we can examine in detail physical processes occurring on elec-tron timescales. This is the approach followed in this paper. We

describe the results of a fully nonlinear investigation, carried outby large scale numerical simulation using a PIC code and backedup by analytical and numerical studies, of the underlying plasmaphysics mechanisms. We consider the caseωpe > Ωe, whichis qualitatively distinct from the strongly–magnetized regimeinvestigated by Bessho & Ohsawa (1999). Our primary goalis finding a mechanism capable of producing mildly relativis-tic electrons: once they have attained rigidities comparable tothose of shock–heated protons, they can undergo resonant scat-tering, and subsequent acceleration to relativistic energies canthen proceed via the diffusive shock mechanism. Our approachenables us to test earlier predictions of both electron acceleration(Galeev 1984; Galeev et al. 1995; McClements et al. 1997) andelectron heating (Papadopoulos 1988; Papadopoulos & Cargill1988) at very high Mach number astrophysical shocks. Simu-lation results are presented for a range of reflected ion speedsin Sect. 2; plasma instabilities occurring in the simulations, andother processes likely to play a role in electron acceleration andheating at SNR shocks, are identified in Sect. 3; and the resultsof these investigations are discussed in Sect. 4.

2. Particle–in–cell code simulations

To investigate wave excitation and particle acceleration in thevicinity of a perpendicular SNR shock we use an electromag-netic relativistic PIC code described by Devine (1995). Theparticle–in–cell principle (Denavit & Kruer 1980) relies on self–consistent evolution of electromagnetic fields and macroparti-cles in sequential stages. Relativistic electromagnetic PIC codeshave been used previously to simulate acceleration processes inastrophysical plasmas (e.g. McClements et al. 1993; Bessho &Ohsawa 1999). A distinctive feature of the code used in thepresent study is the fact that the energy density of electromag-netic or electrostatic fluctuations can be readily determined asa function of frequencyω, wavevectork or timet: this greatlyfacilitates the identification of any wave modes excited in a par-ticular simulation.

The code has one space dimension(x) and three velocitydimensions(vx, vy, vz). To model a plasma containing shock–reflected proton beams, we construct a simulation box with 350grid cells in thex–direction and with the local magnetic fieldBoriented in they–direction. McClements et al. (1997) pointedout that, at any given point in the shock foot, there are in facttwo proton beams, one propagating away from the shock, theother towards it. For simplicity, we assume in our PIC modelthat the two beams propagate with equal speedsub⊥ in op-posite directions perpendicular to the magnetic field, and thatboth background ions and electrons have zero net drift: thus,the simulated plasma has zero current. Strictly speaking, thisis unrealistic, since, in self–consistent models of perpendicularshocks, the magnetic field magnitude has a finite gradient alongthe shock normal direction, and a finite current is then requiredby Ampere’s law (Woods 1969). We will discuss this approxi-mation in Sect. 4. The frame of reference in each simulation isthe upstream plasma frame: time evolution in the simulation canthus be interpreted as spatial variation in the shock foot, with

M.E. Dieckmann et al.: Electron acceleration at supernova remnant shocks 379

t = 0 in the simulation corresponding to the interface betweenthe undisturbed upstream plasma and the foot. The size of thefoot Lfoot lies approximately in the range(0.3 − 0.7)vs/Ωi,wherevs is the shock speed andΩi is the upstream ion gyrofre-quency (McClements et al. 1997). Thus, if the simulation is tobe confined to the foot, the duration of the simulation should beno greater than

tmax =Lfoot

vs' (88 − 205)

2π

Ωe, (1)

whereΩe is the electron gyrofrequency (the true proton/electronmass ratio, 1836, was used in the simulations). The simulationsreported here lasted for either 70 or 135 electron cyclotron pe-riods2π/Ωe.

The proton beams were assumed to be initially Maxwellianwith thermal speedδu⊥ = 3 × 105 ms−1 (δu⊥ being definedsuch that the equivalent temperature in energy units ismpδu

2⊥,

wheremp is the proton mass), and a range of perpendicular driftspeedsub⊥ = 3.25ve0, 3.5ve0, 5ve0 and6ve0, whereve0 is theelectron thermal speed, defined in the same way asδu⊥ and ini-tially set equal to3.75× 106 ms−1 (this corresponds to an elec-tron temperatureTe ' 9.3×105 K ' 80 eV). The value chosenfor the total beam number density,0.33ne, is consistent with thehighest values of this parameter found in hybrid simulations ofquasi–perpendicular shocks with Alvenic Mach numbersMA

ranging up to about 60 (Quest 1986). Cargill & Papadopoulos(1988) usedMA = 50 in their hybrid simulation of an SNRshock (it was computationally difficult to simulate shocks withhigherMA). The density of each beamnb is, accordingly, onesixth of the electron densityne, so that the background protondensityni required by charge balance is0.67ne (the backgroundproton thermal speedvi was set equal to1.9 × 105 ms−1). Theelectron plasma frequencyωpe/2π and gyrofrequencyΩe/2πin our simulations were set equal to105Hz and104Hz, respec-tively, corresponding tone ' 1.2×108 m−3 and magnetic fieldB ' 3.6 × 10−7 T. The ratioωpe/Ωe is typically of order102

or higher in HII regions of the interstellar medium. We havechosen a relatively low value of this ratio in order to study andcompare the effects of instabilities occurring on both theω−1

pe

andΩ−1e timescales.

The electrons, background protons and each proton beamwere represented, respectively, by 3200, 800 and 7200 particlesper cell. The use of a relatively small number of background pro-tons per cell is justified by the fact that instabilities driven by theproton beams have much higher frequencies than noise fluctua-tions associated with the background protons: large numbers ofelectrons and beam protons in each cell ensure a level of noiseenergy well below the wave energy produced by the instabilities.In what follows we measure time in electron cyclotron periods,using the notationt = Ωet/2π. We also definek = kve0/Ωe

(only waves propagating in thex–direction are represented), anormalized frequencyω = ω/Ωe, andr = kv⊥/Ωe, v⊥ beingelectron velocity perpendicular to the magnetic field.

In every simulation, transfer of energy from beam protonsto electrons was observed, but the power flux between the twospecies increased dramatically whenub⊥ was raised from3.5ve0

0 20 40 60 80 100 1200

5

10

Normalized time

Nor

m. E

nerg

y

0 10 20 30 40 50 60 700

20

40

60

80

Normalized time

Nor

m. E

nerg

y

3.5ve0

6ve0

ub⊥ = 3.25ve0

5ve0

Fig. 1.Total electron perpendicular kinetic energy, normalized to its ini-tial value, versus simulation time in electron cyclotron periods2π/Ωe,for several values ofub⊥/ve0. Energy transfer to electrons is muchmore rapid atub⊥ = 5ve0 andub⊥ = 6ve0 than it is at lower beamspeeds.

to 5ve0. Fig. 1 is a time evolution plot of perpendicular kineticenergyE⊥e =

∑j mev

2⊥j/2, whereme is electron mass and

the summation is over all electrons in the simulation box. Sincethe total electron number is constant,E⊥e can be regarded asa measure of the effective perpendicular electron temperature(although it should be stressed at the outset that the electronsdo not always have a Maxwellian distribution). The energy isnormalized to its initial value, which was identical in the foursimulations. Whenub⊥ = 3.25ve0 and3.5ve0 (upper plot) theenergy increases by approximately an order of magnitude inaround 60–100 electron cyclotron periods; whenub⊥ = 5ve0and6ve0 (lower plot) the energy increases by a factor of about40 within t ' 15 − 30. The perpendicular energies of the othertwo particle populations, again normalized to the initial electronenergy, are plotted versus time for the case ofub⊥ = 6ve0in Fig. 2. In the case of the beam protons (upper plot), bothbulk motion energy and thermal energy are included. During thesimulation the beam proton energy drops by less than 1%, whilethe background proton energy (lower plot) rises by no more thanabout 10% (in the other simulations the perpendicular energiesof the two ion species changed by even smaller amounts). Inabsolute terms the energy gained by background protons is verysmall compared to that lost by beam protons, with almost allthe energy being transferred to electrons: we will demonstratethat the beam protons excite an instability which couples themefficiently to electrons.

In all the cases studied, electrons were energized in the di-rection perpendicular to the magnetic field. The upper plot inFig. 3 shows, in more detail than Fig. 1, the time evolution ofE⊥e

(once again normalized to its initial value) in the first 25 elec-tron cyclotron periods of the simulation withub⊥ = 6ve0. Thelower plot shows the time evolution of〈ε0Ex(x, t)2/2〉, whereε0 is the permittivity of free space,Ex(x, t) is thex–componentof the electric field, and the brackets〈〉 denote a spatial averageover the simulation box. In general,Ex is the dominant fieldcomponent: since propagation in thex–direction only is rep-


0 10 20 30 40 50 60 70

1.096

1.098

1.1

1.102

1.104x 10

4

Normalized time

Nor

m. E

nerg

y

0 10 20 30 40 50 60 703.1

3.2

3.3

3.4

3.5

Normalized time

Nor

m. E

nerg

y

Fig. 2. Normalized beam proton (upper plot) and background proton(lower plot) perpendicular kinetic energies (both normalized to initialelectron thermal energy) versus simulation time for the simulation withub⊥ = 6ve0. The beam proton energy drops by less than 1%; thebackground proton energy increases by approximately 10%.

resented, it follows that the waves excited are predominatelyelectrostatic. Henceforth, the term “electric field” refers to thex–component. The field has a single value in each simulationbox cell: the electrostatic field energy density〈ε0Ex(x, t)2/2〉is calculated by summingε0Ex(x, t)2/2 over the box and di-viding by the number of cells. The energy density plotted in thelower frame of Fig. 3 is normalized to the perpendicular elec-tron energy density att = 0. The electron energy grows rapidlyin two main phases, att ' 5 and t ' 14, and then continuesto grow at a slower rate. The field energy is greatly enhanced attimes when the particle kinetic energy is growing rapidly: thissuggests strongly that the fields are involved in particle accel-eration. In the case of the wave energy burst att ' 5, the fieldenergy grows to a level comparable to the electron kinetic en-ergy at that time. The energy of the burst occurring att ' 14, onthe other hand, is much lower than that of the electrons. Fig. 4shows the time evolution ofE⊥e and field energy in the simu-lation withub⊥ = 3.25ve0. The upper plot showsE⊥e growingon a timescale comparable to the transit time of the simulationbox through the shock foot. The lower plot shows that electro-static field activity is again correlated with electron acceleration.Fig. 4 resembles the second of the two periods of wave growthin Fig. 3 (att ' 14), in that the wave energy is small comparedto the electron kinetic energy.

We now consider the distribution of wave amplitudes inwavenumber space. Fig. 5 shows the time evolution of this dis-tribution in the simulation withub⊥ = 6ve0. The grey scaleshows the base 10 logarithm of the wave amplitude obtained byFourier transforming in space the electric field of one of twocounter–propagating waves excited by the ion beams. The startof the burst in wave energy in the lower plot of Fig. 3 att ' 3can be identified with the burst atk ' 1.8 in Fig. 5. This reachesan amplitude of 35 Vm−1, generating a harmonic atk ' 3.6.When the peak amplitude is reached there is an increase in waveenergy atk < 1. The frequency of this lowk noise is close tothe upper hybrid frequencyωuh = (ω2

pe + Ω2e)

1/2. Its appear-

0 5 10 15 20 250

10

20

30

40

50

Normalized time

Nor

m. e

nerg

y

0 5 10 15 20 250

1

2

3

4

5

Normalized time

Nor

m. e

nerg

y

Fig. 3.Time evolution of perpendicular electron kinetic energy (upperplot) and electrostatic field energy (lower plot) in the simulation withub⊥ = 6ve0.

0 20 40 60 80 100 1200

5

10

Normalized time

Nor

m. e

nerg

y

0 20 40 60 80 100 1200

0.02

0.04

Normalized time

Nor

m. e

nerg

y

Fig. 4.Time evolution of perpendicular electron kinetic energy (upperplot) and electrostatic field energy (lower plot) in the simulation withub⊥ = 3.25ve0.

ance correlates with the maximum of the first wave burst att ' 5 in the lower plot of Fig. 3, and with the strong increaseof electron kinetic energy in the upper plot, suggesting that itarises from a redistribution of wave energy and changes in theelectron distribution. Aftert ' 8, when the initial wave bursthas disappeared, a more broadband perturbation is generatedat k ' 1.3, the meank decreasing with time. Att = 14 thewave amplitude peaks at about 16 Vm−1: this is considerablylower than the peak electric field of 35 Vm−1 in the first burst,but nevertheless strong enough to generate two harmonics (atk ' 2.6 andk ' 3.9).

The corresponding plot for the simulation withub⊥ =3.25ve0 is shown in Fig. 6. In this case instability occurs at dis-crete, regularly–spaced values ofk. Waves with relatively highk (' 4) are the first to be driven unstable: during the course ofthe simulation, the instability shifts to lower discrete wavenum-bers. Broadband noise develops atk < 1, as in Fig. 5, but at alater time in the simulation (t ' 35). This appears to be associ-ated with a more gradual evolution of the electron distributionthan that which occurs in the simulation withub⊥ = 6ve0. Thedifference in temporal behaviour between Figs. 5 and 6 will be


−0.5

0

0.5

1

1.5

0 1 2 3 40

10

20

30

40

50

60

Normalized wavenumber

Nor

mal

ized

tim

e

Fig. 5. Base 10 logarithm of electric field amplitude (Vm−1) versuskandt in the simulation withub⊥ = 6ve0.

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0 1 2 3 40

20

40

60

80

100

120

Normalized wavenumber

Nor

mal

ized

tim

e

Fig. 6. Base 10 logarithm of electric field amplitude (Vm−1) versuskandt in the simulation withub⊥ = 3.25ve0.

discussed later in this paper. Figs. 5 and 6 show that in both sim-ulations the plasma eventually stabilizes, on a timescale whichdepends on the beam velocity.

The dependence of wave amplitude onk andt whenub⊥ =5ve0 is qualitatively similar to Fig. 5: after an intense burst earlyin the simulation, a wave with slowly–varying amplitude is ob-served to cascade down ink as time progresses. The growthrate of the first wave burst is 20% higher in the simulation withub⊥ = 6ve0 than it is in the simulation withub⊥ = 5ve0. Inthe former case, as mentioned above, the peak amplitude ofthe second burst is 16 Vm−1, at k = 1.25 and t = 14; thecorresponding figures for the simulation withub⊥ = 5ve0 are12 Vm−1, k = 1.88 andt = 12. The wave amplitude distribu-tion in the simulation withub⊥ = 3.5ve0 is similar to Fig. 6:bursts of wave activity occur at discretek, with the highk modesbeing driven unstable first.

In principle, it is also possible to determine the time evo-lution of wave amplitude as a function ofω and k. However,in order to obtain good frequency resolution it is necessary toaverage the amplitude over times longer than the electron accel-eration timescale. Electrostatic waves in the electron cyclotron

range propagating perpendicular to the magnetic field include,for example, electron Bernstein waves, whose dispersion rela-tion depends on the electron distribution. Since this is rapidlyevolving, it can be difficult to interpret observed distributionsof wave amplitude inω and k. However, we have found thatthe most strongly–growing waves in the simulations invariablyhaveω ' kub⊥/ve0: one can thus obtain the frequencies ofthe high intensity modes in Figs. 5 and 6 by multiplyingk byub⊥/ve0. By this means, it is straightforward to verify that themodes excited early in both simulations haveω ' 10, and henceω ' ωpe.

3. Analysis of simulation results

Short–lived bursts of narrowband wave activity, correlated withrapid increases in electron kinetic energy, occur for all four val-ues ofub⊥/ve0 considered above. These bursts appear through-out the simulations withub⊥ = 3.25ve0 andub⊥ = 3.5ve0;in the case ofub⊥ = 5ve0 andub⊥ = 6ve0, they appear onlyat early times. In every case, the instability cascades to longerwavelengths in the course of the simulation. In order to com-pare the simulation results with those given by linear instabilityanalysis (described in the next subsection), we determine growthrates for the first wave that interacts significantly with the elec-trons: in such cases one may assume that the electrons are stillrepresented by a single Maxwellian velocity distribution withthermal speedve0. The simulations provide the wavenumberkof the unstable wave modes and the electric field amplitudeE.The real frequenciesω of the unstable wave modes are assumedto be equal tokub⊥/ve0. The normalized growth rateγ/Ωe isestimated by fitting an exponential to the plot of wave ampli-tude versus time during the period of most rapid growth in eachsimulation.

The results of this analysis are shown in Table 1. The symbolEm denotes the maximum value ofE during each simulation.In three of the four simulations there is a period of wave growthwhich can be described accurately as exponential. In each case,the growth rate falls to zero, and the wave decays: an exampleof this behaviour, for the case ofub⊥ = 6ve0, is shown in Fig. 7,where wave amplitude (defined as in Figs. 5 and 6) atk = 1.8is plotted versus normalized time. A possible reason for wavecollapse (observed in all four simulations) will be discussed laterin this paper. In the case ofub⊥ = 3.25ve0, the mode referredto in Table 1 (k ' 3.6) is the second to be destabilized in thatsimulation. It appears to grow linearly rather than exponentially:for this reason, no figure is given for its growth rate. The firstmode to be destabilized in this simulation, withk ' 3.9, doesnot grow to a large amplitude (compared to the noise level), andso it is difficult to determine its growth rate. Later in the paperwe will present evidence of wave–wave interaction between thesecond mode excited (k ' 3.6) and the third mode excited(k ' 3.3), which may help to explain the linear growth of thelatter.

Let us now examine whether the growth rates derived fromthe PIC simulations in Table 1 and Fig. 7 can be reproducedusing linear stability theory.


0 2 4 6 8 1010

−2

10−1

100

101

102

Normalized time

Wav

e am

plitu

de

Fig. 7. Time evolution of electric field amplitude (Vm−1) at k = 1.8after the onset of instability in the simulation withub⊥ = 6ve0. Thehorizontal line isEi for this wave; the diagonal line shows an expo-nential fit to the growth rate withγ/Ωe = 0.24.

Table 1.Parameters of highest intensity wave mode in each simulation.

ub⊥/ve0 k ω γ/Ωe Em (Vm−1)

6.0 1.8 10.8 0.24 355.0 2.15 10.7 0.2 233.5 3.3 11.6 0.05 2.53.25 3.6 11.7 – 1.6

3.1. Linear stability analysis

The appropriate dispersion relation for electrostatic,perpendicular–propagating waves with frequencies in theelectron cyclotron range and above, excited by an ion beamwith a Maxwellian distribution inv⊥, is (Melrose 1986)

1− ω2pi

ω2 +2ω2

pb [1 + ζbZ(ζb)]k2δu2

⊥− ω2

pe

ω

e−λe

λe

∞∑`=−∞

`2I`

ω − `Ωe=0, (2)

where:ωpi, ωpb are the background and beam ion plasma fre-quencies;Z is the plasma dispersion function, with argumentζb ≡ (ω −kub⊥)/kδu⊥; andI` is the modified Bessel functionof the first kind of order with argumentλe ≡ Tek

2/(meΩ2e).

Both species of ion, having a much longer cyclotron period thanthe electrons, can be treated as unmagnetized particles on thetimescales of interest here. Strictly speaking, there should be aterm in Eq. (2) for each of the two proton beams, but since theyhave mean perpendicular speeds of opposite sign, andω ' kub⊥is a prerequisite for wave–particle interaction, we need only con-sider one of them.

Solutions of Eq. (2) for complexω in terms of realk canbe readily obtained numerically, and compared with the simu-lation results in Table 1. In Figs. 8 and 9γ ≡ Im(ω) is plottedversusk for parameters corresponding to the initial conditionsof the simulations withub⊥ = 6ve0 andub⊥ = 3.25ve0. In theformer case it can be seen that strong instability drive occursat k ' 1.8 with maximum growth rateγ ' 0.25, as observedearly in the simulation (Fig. 5 and Table 1). The unstable real fre-

γ

k

Fig. 8. Predicted linear growth rates of waves withω > Ωe when thebeam speed is6ve0. The other dispersion relation parameters corre-spond to the initial conditions of all four simulations.

γ

k

Fig. 9. Predicted linear growth rates of waves withω > Ωe when thebeam speed is3.25ve0.

quencies range fromω ' 8 to ω ' 10.8, and are thus clusteredaround the dimensionless electron plasma frequency (ω = 10).The main instability appears to be essentially unaffected by cy-clotronic effects: the growth rate does not depend on how closethe frequency is to a cyclotron harmonic. There are, however,two much weaker instabilities atk < 1, which are narrowbandin both k andω, the real frequencies lying just below the sec-ond and third cyclotron harmonics. In Fig. 9 instability occurs atk ∼ 3 − 4, the corresponding real frequencies again clusteringaroundωpe. In this case, however, the instability is modulated bycyclotronic effects, as in the simulation. Instability again occursat k < 1, with real frequencyω ' 1.8.

The mode appearing early in the simulation withub⊥ =6ve0 arises from a two–stream instability (Buneman 1958). Thiscan be driven by ions drifting relative to electrons in an unmag-netized plasma: it can also occur in a magnetized plasma, withions drifting across the field, ifωpe/Ωe is sufficiently large,


and the instability drive is sufficiently strong. Electrons as wellas ions are then effectively unmagnetized and the appropriatedispersion relation is (Melrose 1986)

1 − ω2pi

ω2 +2ω2

pb [1 + ζbZ(ζb)]k2δu2

⊥+

2ω2pe [1 + ζeZ(ζe)]

k2v2e0

= 0 , (3)

whereζe ≡ ω/kve0. In the frequency regime of interest here(ω ' ωpe), it can be shown easily that the background ion termin Eq. (3) can be neglected. Letting the thermal speeds of thetwo remaining species tend to zero, Eq. (3) reduces to

1 − ω2pb

(ω − kub⊥)2− ω2

pe

ω2 = 0 . (4)

This differs slightly from the original two–stream dispersionrelation analysed by Buneman (1958) in that the ions, ratherthan the electrons, have a finite drift speed. Buneman’s equa-tion becomes identical to Eq. (4) under the transformationω →kub⊥ − ω: using this, we can infer from Buneman’s analysisthat Eq. (4) has a rootω = ω0 + iγ, where real frequencyω0and growth rateγ are given approximately by

ω0 ' kub⊥ − ω2/3pb ω1/3

pe cos4/3 θ, (5)

γ ' ω2/3pb ω1/3

pe cos1/3 θ sin θ, (6)

θ being a parameter whose value depends on(kub⊥ −ωpe)/ω

2/3pb ω

1/3pe (Buneman 1958): it is straightforward to ver-

ify that the strongest wave growth occurs whenθ = π/3, whichcorresponds tokub⊥ = ωpe. If kub⊥ ω

2/3pb ω

1/3pe cos4/3 θ, it

follows from Eq. (5) thatω ' kub⊥ and so the strongest driveoccurs atω ' ωpe. However, sinceθ can have a range of values,the instability has finite bandwidth, extending to frequenciessignificantly belowωpe. Solving the full Buneman dispersionrelation [Eq. (4)] withub⊥ = 6ve0, we obtain results which arealmost identical to those obtained from the magnetized disper-sion relation [Eq. (2)], except, of course, that the cyclotronicfeatures atk < 1 in Fig. 8 do not appear. Even in the caseof ub⊥ = 3.25ve0, Eq. (4) yields instability at about the samewavenumbers and frequencies as Eq. (2) [although the growthrates are somewhat higher in the case of Eq. (4)]. The essentialdifference between Figs. 8 and 9 is that the lower beam speed inthe latter yields lower growth rates: whenγ is sufficiently small,the gyromotion of an electron in one wave growth period can-not be neglected, and the instability is modified by cyclotroniceffects. However, the instability remains Buneman–like in char-acter.

Further analysis of Eq. (2) indicates that the instabilitygrowth rate is a slowly–decreasing function ofδu⊥/ub⊥: inthe case ofub⊥ = 6ve0, for example, the maximum growthrate is around0.06Ωe when δu⊥/ub⊥ = 0.3. The Bunemaninstability is thus robust, in the sense that its occurrence is notcritically dependent on the velocity–space width of the reflectedion distribution. In any event, the values ofδu⊥/ub⊥ used inour PIC simulations are broadly consistent with reflected beamion distributions occurring in the hybrid simulations of Cargill& Papadopoulos (1988).

The instabilities atk < 1 in Figs. 8 and 9 arise from theinteraction of a beam mode (ω ' kub⊥) with electron Bern-stein modes. The existence of such instabilities can be inferredanalytically by taking the limit of Eq. (2) for cold beam andbackground protons:

1 − ω2pi

ω2 − ω2pb

(ω − kub⊥)2− ω2

pe

ω

e−λe

λe

∞∑`=−∞

`2I`

ω − `Ωe=0 . (7)

In the absence of the proton beam term, Bernstein mode so-lutions of Eq. (7) have frequencies which approach`Ωe (` =1, 2, 3, ...) ask → ∞ and(` + 1)Ωe ask → 0 (the long wave-length limit is different for frequencies equal to or greater thanthe upper hybrid frequencyωuh, which in the case of the sim-ulations presented in Sect. 2 is about 10Ωe). Whennb /= 0,approximate analytical solutions of Eq. (7) can be obtained bysettingω = kub⊥ + iγ and solving perturbatively forγ in cer-tain limits. For example, lettingλe → 0 and assuming thatωdoes not lie close to a harmonic ofΩe, we obtain

γ

Ωe'

(me

mp

)1/2 (nb

ne

)1/2 (ω2

Ω2e

− 1)1/2

. (8)

Instability, corresponding to realγ, thus requiresω > Ωe. Forω sufficiently close to Ωe, the electron term in Eq. (7) is dom-inated by the –th harmonic, and instead of Eq. (8) we obtain

γ

Ωe' ω

Ωe

(me

mi

)1/2 (nb

ne

)1/2λee

λe

(`I`)1/2

(1 − `Ωe

ω

)1/2

. (9)

Numerical solutions of Eq. (2) forω ∼ Ωe are broadly consis-tent with Eqs. (8) and (9). In both these cases the growth ratescales as(me/mp)1/2: in contrast, the Buneman growth rate[Eq. (6)] scales as(me/mp)1/3. This helps to explain the factthat in Figs. 8 and 9 the Buneman instability is stronger than thelower frequency Bernstein instability. It should also be notedthat most astrophysical plasmas have a higher ratio of electronplasma frequency to gyrofrequency that that assumed in thesimulations (ωpe/Ωe = 10). Normalized toΩe, the Bunemangrowth rate scales asωpe/Ωe, and so the instability is less likelyto be modified by cyclotronic effects whenωpe/Ωe > 10. Theelectron Bernstein modes exist becauseTe is finite: thus, thetransition from Buneman to electron Bernstein instability de-pends on the value ofve0. If the initial electron temperature inthe simulations had been lower than 80 eV, the Buneman in-stability would, again, have been affected to a lesser extent bycyclotronic effects.

3.2. Nonlinear effects

Fig. 1 shows strong increases in electron kinetic energy per-pendicular to the magnetic field in all four simulations. Thereis a strong correlation between acceleration and wave excita-tion via the Buneman instability (Figs. 3 and 4). Although suchwaves can energize electrons via Landau damping (Papadopou-los 1988), one would expect this process to be of limited effec-tiveness when, as in the present case, the waves are propagating


0 10 20 30 40 500

0.5

1

1.5

2

Normalized time

Am

plitu

de V

/m

0 10 20 30 40 500

0.5

1

1.5

2

2.5

Normalized time

Am

plitu

de V

/m

Fig. 10.Time evolution of electric field amplitude (Vm−1) at k = 3.6(upper plot) andk = 3.3 (lower plot) after the onset of instability in thesimulation withub⊥ = 3.25ve0. The vertical lines indicate a period inwhich there is an anti–correlation between the two wave amplitudes.

perpendicular to the magnetic field and have a growth rate whichis comparable to or less thanΩe. It is likely therefore that thevery strong electron acceleration observed in the simulations isdue at least in part to nonlinear processes.

As noted previously, the second mode to be excited in thesimulation withub⊥ = 3.25ve0 does not undergo an exponentialgrowth phase. Fig. 10 shows the time evolving wave amplitudesof this mode, atk = 3.6 (upper plot), and the third mode tobe excited, atk = 3.3 (lower plot). The amplitude atk = 3.6grows linearly up tot ' 27, and then collapses. The amplitudeat k = 3.3 grows exponentially fromt ' 7 to t ' 14, withγ/Ωe ' 0.04: this is close to the growth rate atk = 3.3 found bylinear stability analysis (Fig. 9). The amplitude remains constantuntil t ' 27, and then grows linearly untilt ' 35. The lineargrowth of the wave atk = 3.3 thus correlates strongly withthe collapse of the wave atk = 3.6: this suggests wave–wavecoupling. The linear growth of the wave atk = 3.6 may, in turn,be correlated with the decay of the first mode to be excited, atk ' 3.9 (see Fig. 6): the latter has the highest growth rate ofwaves in this range, according to linear stability analysis (Fig. 9).

We now consider specific nonlinear processes which maybe occurring in the simulations. Karney (1978) examined thenonlinear interaction of large amplitude electrostatic lower hy-brid waves with ions. The particle motion is described by anormalized Hamiltonian

h =12(px + y)2 +

12py

2 − α sin (y − µt), (10)

where:y, zare, respectively, the wave propagation and magneticfield directions; the canonical momentum componentspx, py

are normalized tomΩ/k, wherem andΩ are particle mass andgyrofrequency; the position variabley is normalized to1/k; µis wave frequency in units ofΩ; andα is given by

α =E/B

Ω/k, (11)

E being the wave electric field amplitude. The first two termsin the Hamiltonian describe the motion of the particle in themagnetic field; the third term arises from the electrostatic wave.The system can be regarded as consisting of two harmonic os-cillators: one associated with the particle gyromotion aroundB, the other with the wave. These oscillators are coupled by theparameterα, the value of which determines the extent to whichthe system phase space is regular or stochastic. Karney (1978)solved the Hamiltonian equations corresponding to Eq. (10) fora range of initial conditions, plotting normalized Larmor radiusr = kv⊥/Ω versus wave phase angleφ at the particle’s posi-tion, for successive transits of the particle through a particulargyrophase angle. Particle trajectories were thus represented asdiscrete sets of points in(r, φ) space. For sufficiently small val-ues ofα, all particles have regular orbits, represented by smoothcurves in(r, φ) space, spanning all values ofφ and with onlysmall variations inr. Whenα exceeds a certain thresholdα0,“islands” appear in(r, φ) space, within which particle trajec-tories are confined. Further increases inα lead to the forma-tion of more islands, which cause the phase space to becomestochastic: at sufficiently largeα > αc, the system phase spaceis completely stochastic, with no regular orbits remaining. Theinitial electron distributions in our simulations decrease mono-tonically inv⊥: in such cases stochasticity in phase space tendsto favour particle diffusion to larger velocities, i.e. acceleration.

Karney (1978) obtained the following analytical estimatefor α0:

α0 =∣∣∣∣ r(ω/Ω − `)`(d/dr)J`(r)

∣∣∣∣ , (12)

where`Ω is the cyclotron harmonic closest toω andJ` is theBessel function of order. Karney’s analysis does not explicitlyinvolve a particular type of wave or particle, or a specific particledistribution function. The results can thus be applied to the caseof electrons interacting with electrostatic waves excited by theBuneman instability, in which casem = me and Ω = Ωe.Combining Eqs. (11) and (12) and using the identity

d

drJ`(r) =

`

rJ`(r) − J`+1(r), (13)

we infer that the critical electric fieldE = Ei for island forma-tion in (r, φ) space is

Ei =v⊥B0|µ − `|

`| `r J`(r) − J`+1(r)|

. (14)

In general, it is not possible to determine analytically an ex-pression for the electric field amplitudeE = Ec correspond-ing to α = αc, above which the phase space becomes com-pletely stochastic. Karney obtained an empirical expression forαc, based on numerical calculations with particular values ofµ andr, which may not be applicable to the simulation resultsdiscussed here. However, island formation is a first step in thedestruction of regularity in the system phase space, andEi canbe regarded as an approximate threshold for stochasticity: elec-tric field amplitudes which are significantly higher thanEi willconvert regular orbits at a particularv⊥ into stochastic ones.


Table 2.Values calculated forEi using the wave parameters given inTable 1.

v⊥/ve0 closest (µ − `) kv⊥/Ωe Ei (Vm−1)

6.0 11 0.2 10.8 1.85.0 11 0.3 10.7 2.33.5 12 0.4 11.6 2.13.25 12 0.3 11.7 1.4

In the casesub⊥ = 5ve0 and6ve0, linear stability analysisindicates that wave growth occurs across a range of frequenciesω ∼ ωpe, which includes cyclotron harmonics: in such casesµ = `, and any non–zero wave amplitudeE will cause islands tobe formed. In the case of the lower two beam speeds, the unstablefrequencies lie between cyclotron harmonics, andEi is thusalways finite. Table 2 lists the values ofEi derived from Eq. (14)that are required for comparison with the highest intensity wavemode excited in each simulation. The actual peak electric fieldsof these waves are given in Table 1.

Comparing Tables 1 and 2, we see that waves are excitedwith amplitudes exceedingEi in all four cases. For the simula-tion with ub⊥ = 3.25ve0, E/Ei ' 1.1. This ratio rises to 1.2for ub⊥ = 3.5ve0, 10 forub⊥ = 5ve0, and 19 forub⊥ = 6ve0.In the latter two cases, as we have seen, waves are excited withω = `Ωe, for which island formation occurs regardless of thevalue ofE. The fact thatEm/Ei 1 at higher values ofub⊥ in-dicates that the phase space in these simulations is characterizedby strong stochasticity. The waves rapidly collapse, however,soon after the onset of strong electron acceleration. In the othertwo simulations, the peak amplitudes are only just sufficient forisland formation to occur, and it is likely that little stochasticityoccurs in the system phase space. The waves excited in thesesimulations decay more gradually than those produced at higherub⊥.

We now consider possible explanations for two of the resultsnoted above: the sharp rise in wave amplitude when the beamspeed is raised from3.5ve0 to 5ve0; and wave collapse, whichoccurs in all four simulations but is particularly rapid in the twosimulations with higherub⊥. As far as the dependence of waveamplitude onub⊥ is concerned, the first point to note is thatthe unstable waves all satisfyω ' ub⊥k. In each simulationthe total number of computational particles is, of course, finite,the Maxwellian electron velocity distribution being initializedup tov⊥ ' 5ve0. Thus, the beams withub⊥ = 5ve0 and6ve0excite waves with phase velocities exceeding the velocity of anyelectron in the simulation: this is not so in the simulations withub⊥ = 3.25ve0 and 3.5ve0. The minimum electron velocityrequired for wave–particle interactions is the phase velocity ofthe wave: thus, only the slow beams can excite waves capableof interacting with electrons at the start of the simulations. Thewave–particle interaction results in electron acceleration, theenergy for this being drawn from the wave. This energy lossmay account for the relatively low peak electric field amplitudesof waves excited by the slow beams.

The waves generated by the fast beams, on the other hand,cannot initially interact resonantly with the electron population,and so their amplitudes can grow to levels much higher thanEi.Sufficiently high wave amplitudes can activate a second accel-eration mechanism, which arises from particle trapping in thewave electric field (Karney 1978): electrons with an initiallymonotonic decreasing distribution are re–distributed uniformlywithin the trap, the result being a net increase in kinetic en-ergy. The wave can trap electrons with perpendicular velocitiesdiffering from the wave’s phase velocity by up tovtr, where

vtr =

√eE

mk. (15)

For ub⊥ = 5ve0, the maximum electric field is 23 Vm−1

and the wavenumberk is 5.7 × 10−3 × 2π m−1. In this casevtr = 1.1 × 107 ms−1 ' 2.8ve0. For ub⊥ = 6ve0, the max-imum field is 35 Vm−1 and k = 4.8 × 10−3 × 2π m−1, sothat vtr = 1.4 × 107 ms−1 ' 3.8ve0. The waves excited inthe simulations with higher beam speeds can thus trap electronsdeep within the electron thermal population: a large numberof electrons can then be pre–accelerated to velocities compara-ble to the wave’s phase velocity, with further acceleration tak-ing place via the stochastic mechanism discussed previously. Atwo–stage process of this type was proposed by Karney (1978).Whereas the first burst of wave activity in Fig. 3 contained moreenergy than the electron population, the energy in the secondburst was much lower than the perpendicular electron kineticenergy by that stage of the simulation. This may have been dueto the first burst resulting in trapped electrons populating theregion of phase space atv⊥ ' ub⊥, via the trapping mech-anism. The perpendicular electron velocity distribution wouldthen be considerably broader than it was initially, with an effec-tive thermal speedve > ve0. The beam distribution, on the otherhand, did not change significantly during the simulation (seeFig. 2), and soub⊥/ve < ub⊥/ve0. The situation would thenbe similar to that of the simulations with lower beam speeds,in which electrons can immediately absorb energy from waveswith ω ' kub⊥, and one would expect any subsequent waveburst to have a peak energy much lower than that of the electronpopulation, as observed.

With regard to the second observation, wave collapse, it isinteresting to note that in every case the wave amplitude falls toa level well belowEi: intuitively, one would have expected thewaves to cease interacting with electrons, and hence to reach asteady–state level, when their amplitudes had fallen belowEi.The collapse may be associated with changes in the dispersioncharacteristics of the wave mode resulting from strong particleacceleration. Karney (1978) justified his Hamiltonian approachby considering only stochastic regions of phase space, at par-ticle speeds (and hence wave phase speeds) greatly exceedingve0. The stochastic regions thus lie in the high velocity tail ofthe initial Maxwellian electron distribution, and most electronsare not initially affected by the wave–particle interaction. How-ever, in the simulations withub⊥ = 5ve0, 6ve0 the reduction inub⊥/ve noted above means that perpendicular electron speedsare no longer small compared to the wave phase speed, and we


find that there is a transition from the pure Buneman instabil-ity shown in Fig. 8 to the more complicated instability shownin Fig. 9: the latter, as we have discussed, has a Buneman–likeenvelope, but also has cyclotronic features, and in fact linearstability analysis shows that the variation ofω with k in thiscase is characteristic of the beam/electron Bernstein mode dis-cussed in Sect. 3.1. Asub⊥/ve falls, the maximum growth ratedrops considerably, but remains positive if the electrons retain aMaxwellian distribution. However, as we now demonstrate, theelectron distributions occurring in the simulations are often farfrom Maxwellian.

3.3. Particle distributions

From the simulation results we have evaluated the distributionof perpendicular electron speedsf(v⊥), defined such that∫ ∞

0f(v⊥)dv⊥ = Ne , (16)

whereNe is the total number of electrons in the simulation.With this definition, a Maxwellian velocity distribution is of theform v⊥e−v2

⊥/2v2e , decreasing monotonically forv⊥ > ve. One

advantage of plotting a distribution in this way is that the thermalspeed of a Maxwellian can be readily identified graphically,being the speed at whichdf/dv⊥ = 0. In Fig. 11 f(v⊥) isplotted fort = 0, 45, 90 and135 in the simulations withub⊥ =3.25ve0 (dash–dotted curves) andub⊥ = 3.5ve0 (solid curves).The two curves are identical fort = 0, since the same initialelectron temperature is used in all four simulations. Att = 45the proton beams have generated hot electron tails, peaking atv⊥ ' 4ve0 (ub⊥ = 3.25ve0) andv⊥ ' 6ve0 (ub⊥ = 3.5ve0).The maximum electron speeds in the two cases are7ve0 (ub⊥ =3.25ve0) and 10ve0 (ub⊥ = 3.5ve0). At t = 90 the slowerbeam has produced a local maximum inf(v⊥) at 5ve0, and ahigh velocity cutoff at10ve0. The local maximum has becomeless pronounced att = 135. In the case ofub⊥ = 3.5ve0 alocal maximum can be seen att = 45: by t = 90, however,the distribution is monotonic decreasing above a speed onlyslighly higher than the initial thermal speed. By the end of thissimulationf(v⊥) extends up to 12ve0.

In Fig. 12 f(v⊥) is shown for t = 0, 20, 40 and 70 inthe simulations withub⊥ = 5ve0 (dash–dotted curves) andub⊥ = 6ve0 (solid curves). Att = 20 the two distributions havelocal maxima atv⊥ ve0, as in the second frame of Fig. 11:for ub⊥ = 5ve0 the distribution peaks locally atv⊥ ' 10ve0and falls to zero atv⊥ ' 18ve0. The corresponding figures atthe same stage of the simulation withub⊥ = 6ve0 are12ve0and22ve0. By t = 40, the local maxima still exist and, indeed,the bumps–on–tail containing these maxima actually comprisemost of the electron population in both cases. By this time thehigh velocity tails extend tov⊥ ' 25 − 30ve0. Local maximaclose to the original thermal speedsve0 still exist, but these havedisappeared byt = 70. The strong wave bursts in these simula-tions occur beforet = 20: after this time, a weaker, more broad-band instability occurs at lowerk, but still with ω ' kub⊥/ve0.To model this instability, we can approximate the solid curve at

0 2 4 6 80

0.05

0.1

0.15

0.2

v/ve0

(t=0)

f(v)

0 5 100

0.05

0.1

0.15

0.2

v/ve0

(t=45)

f(v)

0 5 100

0.05

0.1

0.15

0.2

v/ve0

(t=90)

f(v)

0 5 100

0.05

0.1

0.15

v/ve0

(t=135)

f(v)

Fig. 11.Normalized perpendicular electron speed distributions at var-ious times in the simulations withub⊥ = 3.25ve0 (dash–dotted lines)andub⊥ = 3.5ve0 (solid lines).

0 2 4 6 80

0.05

0.1

0.15

0.2

v/ve0

(t=0)

f(v)

0 10 200

0.05

0.1

0.15

v/ve0

(t=20)

f(v)

0 10 20 300

0.01

0.02

0.03

0.04

v/ve0

(t=40)

f(v)

0 10 20 300

0.01

0.02

0.03

0.04

v/ve0

(t=70)f(

v)

Fig. 12.Normalized perpendicular electron speed distributions at vari-ous times in the simulations withub⊥ = 5ve0 (dash–dotted lines) andub⊥ = 6ve0 (solid lines).

t = 20 in Fig. 12 by superposing two Maxwellians, with thermalvelocitiesvec = ve0, veh = 10ve0 and densitiesnec = 0.38ne,neh = 0.62ne. The solid curve in Fig. 13 shows the true distri-bution att = 20; the dashed curve shows the bi–Maxwellian fitto this distribution. The match is not exact, but is close enoughto suggest that we can model wave excitation at this stage of thesimulation by solving a modified version of the dispersion rela-tion [Eq. (2)], with the parameters of the dashed curve definingthe electron distribution. Results indicate an electron Bernsteininstability with maximum growth rateγ ' 0.08Ωe at k ' 1.4,ω ' 8.2Ωe: the wavenumbers are consistent with those of fluc-tuations appearing att ≥ 20 in Fig. 5.

The electron distributions att = 70 in the simulations withub⊥ = 5ve0 andub⊥ = 6ve0 (Fig. 12) can both be approxi-mated by single Maxwellians, respectively withve ' 8ve0 andve ' 12ve0. The proton beam–excited Buneman instability canthus produce electron distributions whose perpendicular thermalspeeds exceed the velocities of the ion beams which producedthem. The fastest–moving electrons havev⊥ ub⊥. This phe-


0 10 20 30 400

0.02

0.04

0.06

0.08

0.1

v / ve0

at t=20

f(v)

Fig. 13.Normalized perpendicular electron speed distribution att = 20in the simulation withub⊥ = 6ve0. The solid curve is the distributionsampled in the simulation; the dashed curve shows a bi–Maxwellianfit.

nomenon, observed in all four simulations, is further strongevidence for nonlinear processes playing a role in electron ac-celeration: in the quasi–linear limit, unmagnetized electrons ofa particular speedv0 can only interact with waves whose phasespeed equalsv0, and the range of wave phase speeds is deter-mined in turn by the ion beam speeds. In the case ofub⊥ = 6ve0,the final electron temperature is about 11.5 keV: this is easilysufficient to account for thermal X–ray emission observed fromSNRs such as Cas A (Papadopoulos 1988). Individual electronenergies up to several tens of keV were obtained in the simula-tions.

4. Conclusions and discussion

Using particle–in–cell (PIC) simulations and linear stabilitytheory, we have shown that electrostatic waves in the electronplasma range, excited by ions reflected from a high Mach num-ber perpendicular shock, can effectively channel the free energyof the shock into electrons. Such shocks are known to be asso-ciated with SNRs, and the processes investigated in this papermay thus help to account for both X–ray thermal bremsstrahlungand the creation of “seed” electron populations for diffusiveshock acceleration to MeV and GeV energies in such objects.The simulation results provide confirmation of a proposal byPapadopoulos (1988) and Cargill & Papadopoulos (1988) thatstreaming between reflected ions and upstream electrons cangive rise to a strong Buneman instability. Whereas these authorsassumed that the sole effect of the Buneman instability wouldbe electron heating, the PIC simulations show acceleration –the development of strongly non–Maxwellian, anisotropic fea-tures in electron velocity distributions. The maximum electronvelocities are considerably higher than those expected on thebasis of quasi–linear theory: this implies that nonlinear wave–particle interactions are contributing to the acceleration. Whenthe beam speed is greater than about four times the initial elec-tron thermal speed, thermalization of the electron population is

observed after saturation of the Buneman instability, the finalelectron temperature being of the order of 10–12 keV.

It is possible that the acceleration process identified inthis paper may be relevant to oblique shocks as well asstrictly perpendicular ones. A necessary requirement is thepresence of reflected ion beams, which have been observed(Sckopke et al. 1983) upstream of both quasi–parallel and quasi–perpendicular regions of the Earth’s bow shock (the term “quasi–perpendicular” is conventionally used to describe a shock atwhich the angleθBn between the shock normal and the upstreammagnetic field is greater than 45). Leroy et al. (1982) used ahybrid code to simulate ion reflection at shocks with a rangeof values ofθBn, finding very similar results forθBn = 80

andθBn = 90. They inferred from this that hybrid simulationswhich use strictly perpendicular geometry may be comparedwith observational data of quasi–perpendicular shocks. Addi-tional necessary requirements for electron acceleration via theBuneman instability are that the projection of the reflected ionbeam velocity onto the plane perpendicular to the upstream fieldexceed the upstream electron thermal speed, and that the plasmabe weakly magnetized, in the sense thatωpe > Ωe. When theseconditions are satisfied locally, the Buneman instability will oc-cur. Whether this instability is sufficient to produce significantelectron energization at oblique shocks as well as perpendicu-lar and nearly perpendicular ones remains to be demonstrated,however. In the simulations presented in this paper, accelerationoccurred on timescales shorter than an ion cyclotron period. Itwas not necessary to represent the shock foot structure, sincethis has dimensions of the order of a reflected ion Larmor ra-dius, and for this reasonθBn is not explicitly a parameter inour model. Leroy et al. (1982) have noted, however, that ex-trapolations of results obtained for nearly perpendicular shocks(80 ≤ θBn ≤ 90) to more oblique shocks should be treatedwith caution, since the physical processes governing the shockstructure can be expected to change asθBn is reduced.

The simulation results and our analysis of them suggestthat the damping of waves in the electron plasma range at per-pendicular SNR shocks could provide a solution to the cosmicray electron injection problem. Although wave–particle inter-actions at such shocks have been invoked previously in this con-text (Galeev 1984; Papadopoulos 1988; Cargill & Papadopoulos1988; Galeev et al. 1995; McClements et al. 1997), the simula-tion results presented here contain several new features. Theseinclude: the acceleration, rather than heating, of electrons viathe Buneman instability; the acceleration of electrons to speedsexceeding those of the shock–reflected ions producing the in-stability; and strong acceleration of electrons perpendicular tothe magnetic field. The wave–particle mechanisms proposed byGaleev (1984) and McClements et al. (1997) gave rise to elec-tron acceleration primarily along the magnetic field. Diffusiveshock acceleration, which is probably essential for the produc-tion of ultra–relativistic electrons, can occur when the electronshave magnetic rigidities comparable to those of ions flowinginto the shock. Since, however, the diffusive shock mechanismrequires electrons to be rapidly scattered, its efficacy does notdepend sensitively on the initial pitch angle distribution. The


geometry of the simulations described here (B perpendicular tothe one space dimension) excludes the possibility of accelerationby electrostatic waves along the parallel direction. The presentmodel is complementary to those of Galeev (1984), Galeev et al.(1995) and McClements et al. (1997), in that it provides an alter-native means of energizing electrons at perpendicular shocks.At a real SNR shock, perpendicular acceleration via the Bune-man instability and parallel acceleration via wave excitation atω < Ωe are both likely to occur. PIC simulations in two spacedimensions would make it possible to assess quantitatively therelative contributions of these two types of instability to electronenergization under a range of conditions.

We discuss finally our neglect of the finite plasma currentpresent in the foot region of perpendicular shocks. This approx-imation does not appear to have introduced unrealistic elementsinto our simulation results, except insofar as the absence of a fi-nite drift between the electrons and background protons removesa possible source of drive for the ion acoustic instability, invokedas one of the electron heating mechanisms in the model of Pa-padopoulos (1988). However, if the background protons andelectrons flowing into the shock foot are approximately isother-mal, it seems unlikely that any instability excited by their relativestreaming could result in a significant transfer of energy fromone species to the other. Another possibility is that ion acous-tic instability could result from the streaming of beam protonsrelative to electrons. This would require, however, the electrontemperature to be extremely large compared to the beam protontemperature (ion Landau damping strongly suppresses the insta-bility when the temperatures are equal): even in the simulationwhich produced an effective electron temperature of 80 timesthe initial temperature, this condition was not satisfied. Thus, thesimulation parameters were such that the ion acoustic instabilitycould not occur. It is interesting to note, however, that the curvecorresponding toub⊥ = 6ve0 in the lower frame of Fig. 1 bears acertain resemblance to a curve in Fig. 5 of Papadopoulos’ 1988paper (Fig. 5), showing schematically the predicted variationof electron temperature with distance in a quasi–perpendicularshock foot: in both cases, there is a very rapid rise in the to-tal electron energy, resulting from strong Buneman instability,followed by a more gradual rise, which coincides in the simula-tions with the excitation of electron Bernstein modes. The lattermay play a role in the simulations which is similar to that of theion acoustic mode in the model of Papadopoulos.

Acknowledgements.This joint work was supported by the Commis-sion of the European Communitities, under TMR Network ContractERB–CHRXCT980168, by the UK Department of Trade and Indus-try, and by EURATOM. S. C. Chapman was supported by a PPARClecturer fellowship.

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astronomy and electron acceleration due to high frequency...

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