STRONG ELECTRON ACCELERATION AT HIGH MACH NUMBER SHOCK WAVES: SIMULATION STUDY OF ELECTRON DYNAMICS N. Shimada and M. Hoshino

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1 The Astrophysical Journal, 543:L67 L71, 000 November The American Astronomical Society. All rights reserved. Printed in U.S.A. STRONG ELECTRON ACCELERATION AT HIGH MACH NUMBER SHOCK WAVES: SIMULATION STUDY OF ELECTRON DYNAMICS N. Shimada and M. Hoshino Department of Earth and Planetary Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113, Japan Received 000 February 8; accepted 000 May 6; published 000 October 11 ABSTRACT Electron-ion dynamics in a perpendicular magnetosonic shock wave in a high Mach number regime is studied by using the particle-in-cell simulation. It is shown that in the shock transition layer nonlinear evolution of twostream instabilities plays an important role on the electron rapid heating and acceleration. As the shock Mach number greatly exceeds the critical Mach number, a series of large-amplitude, coherent electrostatic waves with the electron holes in phase space are excited by the two-stream instability between the reflected ions and the incident electrons in the shock transition layer. As the incident electrons are decelerated by the instability, other electrostatic waves grow in time by another two-stream instability between the incident ions and the decelerated 1 incident electrons. The dynamic timescale of these instabilities is of the order of qpe, where qpe is the plasma frequency. The nonlinear interaction of these waves leads to the strong electron heating as well as the nonthermal high-energy electron acceleration in the shock transition layer. Subject headings: acceleration of particles cosmic rays plasmas shock waves 1. INTRODUCTION The origin of high-energy electrons is still a long-standing problem in many astrophysical applications such as supernova shocks (e.g., Koyama et al. 1995), extragalactic radio sources by jets, the emission from active galactic outflow, interplanetary shocks, etc. Shock acceleration has been discussed as one of the important processes producing the high-energy particles, and there are many theoretical and observational efforts so far at understanding the high-energy ion acceleration/heating in shock waves. The studies of electron acceleration/heating, however, are limited. By taking into account plasma instabilities in detail, Papadopoulos (1988) proposed the electron energization process at high Mach number shocks, in which electron heating is produced through two-step instabilities in the shock transition layer where the reflected ions coexist with the incident ions and electrons (e.g., Leroy et al. 198; Wu et al. 1984). Buneman instability (BI) is first excited by the velocity difference between the reflected ions and the incident electrons, and the electrons are heated up by the instability. As the next step, the ion acoustic (IA) instability is triggered under the preheated electron plasma by BI, and the electrons are further heated up to MA/ be, where M A is the Alfvén Mach number. Cargill & Papadopoulos (1988) also studied this electron heating process for the perpendicular shock waves by using a hybrid code where the electrons are treated as fluid. They assumed phenomenological resistivities to model BI and IA instability and demonstrated the strong electron heating in the shock transition layer. This nonlinear process can successfully explain the origin of high-energy electrons such as in supernova remnants with a high Mach number, although there remain several important unsolved issues. One of them is the physics of the phenomenological resistivity assumed in their study, because the model resistivity may not properly describe the energy exchange between ions and electrons in a highly nonlinear shock wave. It should be noted that in situ observations of high Mach number shocks in our interplanetary space suggest the importance of electron dynamics; at the Uranian bow shock, the Voyager satellite observed the strong electron heating in the L67 quasi-perpendicular shock condition with Mms 0 (Bagenal et al. 1987), where M ms is magnetosonic Mach number. The other observation is in the Earth s foreshock at quasiperpendicular region with Mms 9.5. The Wind satellite measured the strong, localized electrostatic waves with a bipolar signature (Bale et al. 1998), which are thought to be excited by the nonlinear evolution of BI (Davidson et al. 1970; Omura, Kojima, & Matsumoto 1994). We think that we need to reconsider the conjecture of the electron heating process proposed by Papadopoulos (1988) together with the above observations. In this Letter, we study the ion-electron dynamics organizing the electron energization and wave activities in the shock transition layer by using the one-dimensional, particle-in-cell simulation code where both ions and electrons are treated as particles (e.g., Hoshino et al. 199). In the past, Tokar et al. (1986) reported nonadiabatic electron heating at high Mach number shocks in the strongly magnetized condition (namely, the ratio of electron gyrofrequency to plasma frequency, Q ce/qpe p 1) by utilizing a full particle simulation. Contrary to their simulation study, we treat a weakly magnetized condition with Q ce/qpe K 1 and show that the excitation of large-amplitude electrostatic waves due to two-stream instabilities results in strong, rapid electron heating/acceleration.. SIMULATION In our simulation system, a low-entropy, high-speed plasma consisting of electrons and ions is injected from the left boundary region that travels toward positive x. At the injection boundary (x p 0), the plasma carries a uniform magnetic field B z, polarized transverse to the flow. The downstream right boundary condition is a wall where particles and waves are reflected (so-called piston method). The shock wave then propagates backward in the x direction. Initially, each computational cell, which is comparable to the electron Debye length, includes 80 particles for each species (electrons and ions). The plasma parameters are as follows: upstream plasma bep bip / ( bj p 8pnT j/b ), qpe p 0Qce [ qpe p (4pne /m), Qce p eb/mc], and M/m p 0 (where M is the ion mass and m is the

2 L68 STRONG ELECTRON ACCELERATION AT SHOCK WAVES Vol. 543 Fig. 1. Overall shock structure with (a) M p 3.4 and (b) M p (c) Enlarged view of (b). A A electron mass; n, T, and B are, respectively, the density, temperature, and magnetic field strength; e and c are the electric charge and the speed of light). We now discuss the shock properties obtained in two runs with different Mach numbers. Figures 1a (run a) and 1b (run b) show, respectively, snapshots of the nonlinear stages of the shock structures with MA p 3.4 (for run a) and MA p 10.5 (for run b) ( Mms p 3.0 for run a and 9.3 for run b). Figures 1a and 1b show an entire simulation box. After the simulations start, 350T pe and 86T pe have passed for runs a and b, respectively. The time is normalized by T pe { p/qpe defined in the upstream region. From the top, the phase-space diagrams v x for the electrons and ions, the magnetic field Bz, and the electric field Ex are presented versus the x-coordinate. The velocity, magnetic, and electric fields are normalized by the injection plasma flow speed ( u 0 ), the upstream magnetic field ( B 0 ), and the motional electric field ( Ey0 p ub/c 0 0 ), respectively. The x-coordinate is measured by the electron inertia length c/q pe. The shock fronts are located at x 65 for run a and x 35 for run b and propagate with a speed of about 0.5u 0. In Figure 1a, some of the ions flowing into the shock front are simply reflected to keep a rather steady shock potential, while the electrons show adiabatic properties, since the density and magnetic field variations are well correlated. The magnetic field strength varies smooth in the ion dynamics scale without any oscillation of electron inertia scale. This is consistent with previous shock studies (e.g., Lembège & Dawson 1987; Leroy et al. 198). On the other hand, Figure 1b shows a complex shock structure. The electron phase diagram shows several spiky structures at the front side of the reflected ions and the ion phase diagram shows substructures in the shock transition layer. The precursor waves in the magnetic field observed in the upstream region are identified to the X-mode wave (e.g., Tokar et al. 1986; Langdon, Arons, & Max 1988). The enlarged view of Figure 1b is presented in Figure 1c to show the details of the shock transition layer. In the top panel in Figure 1c, we observe several electron vortex (hole) structures at the very front of the shock transition layer corresponding to spikes of the large-amplitude electric field. Figure shows a time series of the instability evolution for 1 run b during several q pe, focusing on vortex dynamics and the other interesting structures. From the left, we show the electron vx, Ex, and ion vxin the shock transition layer. From top to bottom, t p 193.4, 196., 198.0, 0.5, and 05.3 are presented. When the reflected ions are generated to maintain the current continuity, the electrons with the smaller inertia mass are decelerated by emitting the electrostatic wave. (We call this interaction between the incident electrons and the reflected ions process 1, as seen, for example, when x 48 5 at t p The waves excited by process 1 basically propagate to the left in the simulation frame.) The decelerated electron flow and the dense incident ion flow form a velocity difference that is also capable of exciting another instability. (We call this interaction between the decelerated electrons and the incident ions process, as seen, for example, when x 5 at t p 193.4, where electron bulk velocity is decelerated toward zero. The waves excited by process basically propagate to the right in the simulation frame.) These two processes dominate the electron energization. Let us examine details of processes 1 and in the high Mach number regime. Having a large velocity difference between the incident electrons and the reflected ions in mind, we can conclude that process 1 is equivalent to the BI (e.g., Morse & Nielson 1969). In process 1, we observe the formation of coherent electron vortex and wavy structures in the reflected ions. To check this process, we calculate the scale size and the growth rate of the instability. For our regime, the dispersion relation of the electrostatic waves propagating perpendicular to the magnetic field can be reduced to the electrostatic dispersion relation with three plasma populations, i.e., j q j 1 [1 z Z(z j )] p 0, (1) jpe, i0, ir k v j

3 No. 1, 000 SHIMADA & HOSHINO L69 Fig.. Time series of the instability evolution in the shock transition layer where Z(z) is the plasma dispersion function, zj p (q ku j)/kvj, and qj, uj, and vj indicate the plasma frequency, bulk, and thermal velocity for the j-component, respectively. Subscripts e, i0, and ir present the electron, incident ion (proton), and reflected ion components. The ion plasma frequencies are given by qi0 p (m/m)(1 a r)qe and qir p (m/m)arqe, where ar is reflection ratio. We take here ar p 0.5 (in our high Mach number shock simulations a r is about on average, almost independent of M A ). For the same condition of the upstream region of run b, we obtain the unstable wavelength of 0.71c/qe for the maximum growth rate of 0.14 qe. In the top panels of Figure, the newborn vortex at x 48 has a wave- length about 0.75 c/q e, which is in good agreement. At the maximum growth rate, the phase velocity of these vortices is 0.8u 0 in the simulation frame, so that some pop- ulation of the leading edge of the reflected ion (their velocity is u 0 ) can resonate with these waves. Dense wavy structure seen in the reflected ion population is also produced as a result of resonance with the electrostatic field resulting from the twostream instability between the reflected ions and heated electrons. In the nonlinear stage of electron vortices, their deep potential wells deflect even the incident ion population (t 196.), and the energy exchange between the ions and electrons results in the enhancement of the magnitude of the localized, bipolar electric field. During the evolution of the electron vortex, the electrons are decelerated due to the momentum exchange between the incident electrons and the reflected ions through the electrostatic fields. As a result, a velocity difference between the electrons and the dense incident ion ({ Du i0, e ) is formed quickly and destabilizes another strong two-stream instability (process ), which in turn produces wavy structures in the incident ion population. From the dispersion relation, we find that this two-stream instability is not sensitive to properties of the reflected ions but very sensitive to temperature of the incident ions and slightly sensitive to the electron temperature. Large temperatures for both of the incident ions and electrons reduce its growth rate. The rapid growth of this instability is realized in the region where the flow speed of the electrons is reduced (namely, the region with a large Du i0, e ) and where the electrons and incident ions are not yet heated. This explains the wavy structure in the incident ion population propagating to the left in the simulation box (t p ), although their velocity is about 0.6u 0 to the right in the simulation frame, consistent with linear theory. Coexistence of process with 1 tends to destroy vortices of BI origin, which have long-living nature, as stated below. During the nonlinear evolution of the vortex (at x 48), the propagation speed of the vortex is quite reduced (Bujarbarua & Schamel 1981), while the leading edge of the reflected ions propagates further upstream, destabilizing BI and decelerating the electrons after t p (for example, x at t p 0.5). At the same time, process works at the same region as the two-stream instability sets in between newly decelerated electrons upstream of the vortex and the incident ions so that the amplified electric fields propagate toward the vortex and destroy its coherence structure ( x at t p 0.5). In the downstream region of the vortex we also find coexistence of processes 1 and ( x 48 5 at t p 198.0), and they also interact dynamically. These collision and merging processes among the coherent waves are accompanied by the strong thermalization of electrons. This new dynamical process can quickly heat up the electrons. After the vortex decays and merges into the hot electron plasma, a train of new vortices begins to appear at the edge of the newly reflected ions ( t 0.5), and the same dissipation process is repeated again on a timescale of order 10T pe, much less than the typical ion gyroperiod p/q p 380T. The relationship be- ci pe

4 L70 STRONG ELECTRON ACCELERATION AT SHOCK WAVES Vol. 543 Fig. 3. Downstream electron energy spectra for runs a and b. The dotted lines indicate Maxwellian distribution. Fig. 4. Shock-heated electron temperature normalized by T e0 (top) and incident upstream flow energy (bottom) as the function of Mach number. The middle panel is the ratio of T to T. i e tween our strong electron heating process and the shock reformation process (Quest 1986) remains an interesting topic. 3. DISCUSSION We found that the interaction of electrostatic waves under the two-stream instabilities between the electron and two ion components causes rapid, strong electron heating in the shock transition layer. It is also interesting to check whether or not the nonthermal electrons are produced during our strong shockheating processes. Figure 3 shows the downstream electron energy spectra for runs a and b. The energy is normalized by the incident ion bulk energy. The dotted lines indicate a Maxwellian distribution. One can find that the energy spectrum is approximated by a Maxwellian for run a, while run b clearly shows the nonthermal population above the incident ion bulk energy. Another interesting value is the energy conversion rate in our shocks. Figure 4 shows the thermal energy of shock-heated electrons ( T e ) normalized by the upstream electron temperature ( T e0 ; top) and the incident flow energy (bottom), as the function of M A. The middle panel is the ratio of the shock-heated ion temperature ( Ti) to Te. The temperature is calculated using f( p)gm j(v v) d p/ f( p)d p, where p p gmjv, mjis the mass for j species, g is the Lorentz factor, and v is the averaged velocity. Error bars show the maximum and minimum values observed in the shock reformation process. To obtain the temperature, we sample the electron distribution function in the regions with B/B0 p.5 3.5, which is expected as the downstream state of the Rankine-Hugoniot relation with twodimensional plasma heating in the velocity space. At the lowest M A, we can see subcritical feature as a lower ratio of T i /T e. As the Mach number increases, the ion reflection dominates (Leroy et al. 198), and at a higher Mach number regime greater than 10 where a series of the clear electron hole structure appears, more and more of the ion energy goes into electrons, which results in strong electron heating and acceleration. We obtain that about 5% 35% incident bulk flow energy is converted to the electron thermal energy, and the ratio of ion to electron temperature becomes about 6 in the high Mach number regime. Finally, let us scale our simulation to the real mass ratio. 1 1/3 The growth rate of BI ( tg ) is proportional to q pe(m/m) 1/ 1/3 (n/m) (m/m), and the region of a transition layer ( L t ) is 1/ proportional to u /Q (M/n) under the fixed M A condition 0 ci 1/ [ MA u 0(Mn) /B]. The speed for the convection of instability structure is proportional to u 0. Then the growth time normalized by the convection time in the shock transition layer becomes t /(L /u ) (V /c)(m/m) 1/6. Therefore, we find that the time g t 0 A required to reach to the nonlinear stage of electron heating becomes short if we normalize the physical quantity in the ion scale. In a preliminary result of the simulation with M/m p 100, we obtain the similar strong electron heating/acceleration with the formation of electron hole structures. The authors are grateful to T. Terasawa and B. Lembège for fruitful discussions.

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