Electron accelerations at high Mach number shocks:
Two-dimensional Particle-in-Cell simulations on massively parallel
supercomputer systems
Yosuke Matsumoto1 , Takanobu Amano2 , and Masahiro Hoshino2
21

1Graduate School of Science, Chiba University, Japan
Graduate School of Science, University of Tokyo, Japan

Introduction

Plasma kinetic processes at collision-less shocks havebeen investigated and recognized as important for injecting electrons towards so-called the diffusive shock acceleration mechanism. The shock surfing acceleration is oneof the prominent mechanisms that can quickly accelerate
the electrons at the leading edge of the shock foot region
by DC electric fields. The underlying mechanism of theshock surfing acceleration is the plasma kinetic process
between the reflected ions and the incoming electrons that
leads to the excitation of Buneman instability. Numerical investigations of the shock surfing acceleration havebeen reported by Partile-in-Cell (PIC) simulation which
follows particles motions with the electromagnetic fielddevelopment self-consistently. Recently, two-dimensional
PIC simulation studies reported contrary results: [1] reported the shock surfing acceleration is effective in a highMach number perpendicular shock evolution while it is nota dominant process or even not observed in the other PIC
simulation results [2, 3]. In this paper, we report PIC simulation results of the electron acceleration at high Mach
number perpendicular shocks. The dependence of the acceleration mechanism on the ion to electron mass ratio andthe Alfven Mach number is discussed.
2Numerical method

We have examined shock evolutions by a twodimensional Particle-in-Cell simulation code. The codesolves ion and electron motions along with the electric andmagnetic fields developments. The electric and magnetic
fields are solved implicitly for stability. The calculationof the current density is based on a charge conservation
scheme. The code is parallelized via domain decomposition by using Message Passing Interface (MPI) library andOpenMP and runs efficiently on massively parallel supercomputer systems (Figure 1).
The shock is formed by injecting particles from theboundary on the left-hand side (x = 1) and reflecting particles at the boundary on the right-hand side (x = Lx ).
The injected plasma carries a z-component of the magnetic field and thus the convective electric field Ey = Vx Bz(perpendicular shock). The periodic boundary conditions
are applied at y = 1 and Ly . The simulation box sizes inthe x and y directions are Lx = 10V0 /Ωgi and Ly = 5λi ,respectively, where V0 is the upstream speed, Ωgi is theion gyro frequency, and λi is the ion inertia length in theupstream region. The grid size ∆h is set equal to Debye

Fig. 1. (Left) Performance of the PIC code in GFLOPS
vs. number of processor cores. Solid and dashed lines
with symbols are the obtained results for Fujitsu FX1 andFujitsu HX600, respectively. The straight lines indicate
100% efficiency of the parallelization. (Right) Efficiencyof the performance vs. number of processor cores.

length in the upstream region.
We have examined several runs with various ion to electron mass ratios (M/m) and the Alfven Mach numbers(MA ), while the upstream electron βe and the ratio of theelectron plasma to gyro frequencies ωpe /Ωge are fixed toβe = 0.5 and ωpe /Ωge = 10.0 among the simulation runs.
The mass ratio varies from 25 to 100 and the Alfven Mach
number is increased from 15 to 30 for the M/m = 100case. The upstream parameters are summarized in Table 1. The maximum computational resource (Run C) areused with 24001x1024 grid points in which 5 × 109 particles motions are solved with 512 processor cores on Fujitsu
FX1 supercomputer system at JAXA.
3Results

The maximum energy gain of electrons was found in
Run C in which the electrons are accelerated at the edgeof the foot by the shock surfing acceleration. When a particle entered the unstable region of the Buneman instability,
it is accelerated by the convective electric field while being

Table 1. Upstream parameters

Run ARun B
Run CM/m
25100100MA151530V0 /c0.20.10.2

βe0.5
0.50.5

ωpe /Ωge
101010

(a)

(b)

(c)

Fig. 2.
Trajectories of accelerated (red) and nonaccelerated (cyan) electrons superposed on the twodimensional profile of the electrostatic field strength. Opensquare locates the position of the particle at each time step
following the trajectory back for 15 Ω−1ge . A number located beside the open square denotes γ − 1. Snapshots aretaken at times of (a) Ωge T = 513.5, (b) Ωgi T = 522.5, and(c) Ωgi T = 532.5, respectively.

captured by the electrostatic potential. The present twodimensional simulation shows in Figure 2(a) that a sampled electron can be escaped from the potential well at atime after gaining a fraction of energy and drifts toward thedownstream. Since the length of the potential well is limited to the order of the ion inertia length, the particle canescape from the well even though the electric field strengthis sufficiently large. This is the multi-dimensionality effect. Nevertheless it can enter the unstable region fromthe downstream-side again since the gyro radius of the accelerated particle becomes larger than before. Then thesecond surfing acceleration is realized. In the trajectory inFigure 2(b), the energy gain is more effective than in theprevious encounter, and the energy reached γ ∼ 2. The
third encounter with the unstable region is possible in thesame manner, and the energy is increased to γ ∼ 3 (Figure2(c)). For accelerated particles, they typically experience
the shock surfing acceleration three times as shown in thisexample. The present acceleration mechanism can be applied to lucky particles that will constitute the non-thermalpopulation, and most of the particles that will constitutethe thermal population drift toward the downstream without being captured as shown with cyan lines in Figure 2.
Nevertheless the shock surfing acceleration is so effective
as it can energize the electrons from γ ∼ 1 to the relativistic regime of γ ∼ 3 within a time scale of ∼ 20 Ω−1ge , or∼ 0.2 Ω−1inthisparticularcasewithM/m=100.
gi
Figure 3 shows the electron energy spectra obtained in
the downstream for three simulation runs (Table 1). TheFig. 3. Energy spectra of the electron in the downstream
region. The results are compared for (a) Run A (green),Run B (magenta), and Run C (black). Dashed lines are
fitted Maxwell distributions.
energy spectrum in Run C exhibits a high energy tail which
deviates from the thermal population and reaches γ ∼ 9. A
similar non-thermal electron production is observed in Run
A in which the Buneman instability is highly activated atthe edge of the foot as well. A significant reduction of themaximum energy of the electron was observed in Run B
as we increased the mass ratio from 25 to 100 while MA
is fixed to 15. This trend is consistent with the recent
two-dimensional simulation study [2].
4Summary

We have examined electron acceleration mechanisms at
high Mach number shocks by means of two-dimensional
Particle-in-Cell (PIC) simulations with various ion-toelectron mass ratios, Alfvén Mach numbers. We foundelectrons are effectively accelerated at a super-high Mach
number shock (MA = 30) with a mass ratio of M/m = 100
and βe = 0.5. The electron shock surfing acceleration isfound out to be an effective mechanism for accelerating
the particles toward the relativistic regime even in two dimensions with the large mass ratio. Buneman instability
in the super-high Mach number shock resulted in a coherent electrostatic potential structure which enabled an efficient trapping of the electrons. While multi-dimensionality
allows the electrons to escape from the trapping region,
they can interact with the strong electrostatic field severaltimes. This multiple interaction enables effective shocksurfing accelerations as has been found in the small mass
ratio case.
Acknowledgement
Computational resources are provided by Fujitsu FX1
at JAXA and Fujitsu HX600 at Nagoya Univerisity.
References[1] T. Amano and M. Hoshino, ApJ, 690, 244, 2009.
[2] M. A. Riquelme and A. Spitkovsky, ApJ, 733, 63, 2011.
[3] T. N. Kato and H. Takabe, ApJ, 721, 828, 2010.