J. Plasma Fusion Res. SERIES, Vol.2 (1999) 145-148

Acceleration of Energetic lons by Oblique Magnetosonic
Shock Waves
MASAKI Tomohiro. BESSHO Naoki and OHSAWA Yukiharu*
Department of Physics, Nagoya University, Nagoya 464-8602, Japan(Received: 8 December 1998 / Accepted: 2 February 1999)

Abstract
Enhanced acceleration of energetic ions by an oblique shock is studied by means of a fully
relativistic, fully electromagnetic particle code with full ion and electron dynamics. It is found that someof the energetic ions that move with a shock can be accelerated many times. The acceleration takes placewhen the particle velocity becomes parallel to the transverse electric field in the shock.

Keywords:
particle acceleration, energetic particle, oblique shock wave

1. Introduction
Large-amplitude magnetosonic waves canaccelerate some ions to high energies [-9]. Their speeds

acceleration mechanism is quite different from that forthermal ions. Because the speeds of energetic ions are(S l020eV) are often observed and have been one of thehigher than the shock speed, they can go back to theupstream region by the gyro-motion after entering theshock region. They can gain energies when they are in
the shock region and their velocities become nearlyparallel to the transverse electric field formed in themagnetosonic shock. The maximum energy they cangain increases with increasing initial energy (energy
before the encounter with the shock). It is to be noted
that, for a perpendicular shock, they eventually move tomost important issues in plasma physics andthe downstream region after these interactions.

can be comparable to or greater than the Alfv6n speed.
These results have been applied to solar physics toaccount for the prompt ion acceleration to relativisticenergies, lOe - lOroev [0]. In the present paper, wewill examine the question; whether or not shock waves

have mechanisms accelerating particles to higher
energies. In fact, cosmic rays with much higher energies

In this paper, by using a particle simulation code,astrophysics.

In a turbulent plasma where nonlinear

we study interactions of energetic ions with amagnetosonic waves are present, high-energy ions thathave been accelerated by a certain pulse will encountermagnetosonic shock wave propagating obliquely to amagnetic field. It will be found that some of theenergetic ions can be further accelerated by themechanism discussed in the previous paper [11]. A
distinguish feature for the oblique shock is that it cantrap some energetic ions; thus, the acceleration

many different magnetosonic pulses after the firstacceleration. In the previous paper by Maruyama et al.
[11], interactions of such high-energy ions withmagnetosonic waves propagating perpendicular to amagnetic field were studied. They reported that some ofthe energetic ions can be further accelerated. Themechanism can operate many times on them.

@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research145

Masaki T. et al., Acceleration of Energetic Ions by Oblique Magnetosonic Shock Waves

2. Effects of Parallel Motion
In a perpendicular shock, particles cannot movewith a shock for a long time. In an oblique shock,however, some particles could move with a shock, by50. The electron cyclotron frequency in the upstream
region is {D".lo)r, = 1.5, where oo" is the spatiallyaveraged plasma frquency. The Alfv6n speed is volc =0.196, where c is the speed of light. Even though thedrifting along the field line. Let v;1 be the velocity
parallel to the magnetic field B,0the angle between thewave normal and B, v"; the shock speed, then thecode has a small number of He ions as well as H ions,the orbits shown in this paper will be those of H ions.
(The orbits of energetic He ions are similar to those ofparticles withH ions.) More detailed description of the simulation
method can be found in Ref. I l].
Initially we have energetic ions as well thermalions. In the upstream region, the thermal ions andv,, cos0-v,o(1)can move with the shock. We will show by simulations

electrons have Maxwellian distribution functions, which
are uniform in space. The energetic ions have anthat, of the ions that satisfy the above relation, some cangain energies many times from the shock; because they
move with the shock, the acceleration mechanism
discussed in Ref. [11] can work many times on eachisotropic distribution function such that/(7) = n.n6 (y76), where 7 is the Lorentz factor and 76 is the initialvalue. We excite a shock wave in an external magnetic
field Bs = (8,0, 0 , B.o). It propagates in the positive .r
direction; thus the -r component of the magnetic field isconstant, while the y and z components vary with timeand space. In the following, we will show a case withthe propagation angle 0 = 60"; that is, B.s/B,o = tan 60".
Figure I displays profiles of the transverse electricfield E" in a shock at various times. The propagation

particle. They obtain energies when their velocities
become parallel to the transverse electric field in theshock. Obviously, in perpendicular shocks, no particles
satisfy the relation (l).
Even if the shock is oblique, very few ions would
satisfy the relation (l) if there are no energetic ions;because usually the shock speed is much greater than
the ion thermal speed, v,6 ) v1;. However, if there areenergetic ions in a plasma, we will have an appreciable
number of ions that can move with a shock wave,
holding the relation (1). Accordingly, this acceleration
mechanism can take place in oblique shocks in a plasma
having nonthermal energetic ions.
The gyro-radii of energetic ions are supposed to begreater than the shock width (or the pulse width). Thus,
the field quantities can significantly change within thegyro-radii. The quantities I and vn at the position of eachparticle will vary with time as it gyrates. The relation (1)speed is v"n= 7.92 va. We also show there a phase spaceplot (x, y) of energetic ions. Initially, they are uniformly
distributed in space with y6 = 2.0. Here, the ions thatwere in the region 363 < xl(clao) < 447 at t = 0 areshown; they expand in space because of the gyromotion. We find especially high-energy ions near theshock wave. The maximum value of y increases withtime.
We show in Fig. 2 time variations of the position x(upper panel) and energy 7 Qower panel) of an energetic
ion that is accelerated three times. The position of theshock front is represented by the dotted line; theupstream region is above this line. After the encounter
with the shock, the particle moves with the shock. Theenergy Tincreases rapidly when the particle enter theshock region for the first time. It then goes out to theshould therefore be interpreted as an average over agyration.

3. Simulation
We use a one-dimensional (one space coordinate

and three velocity components), relativistic,electromagnetic, particle simulation code with full ion
and electron dynamics. The system length is L, = 4096
A' where A, is the grid spacing. The system is bounded[3,12]; the particles are confined in the region lO0 < x/
Ls<3996, being specularly reflected atxl\,"= 100 and

upstream region; after a while it enters the shock region

again, and the energy increases. The acceleration isrepeated. The simulation was terminated before thisprocess is finished, because the shock wave nearlyreached the plasma boundary. If the system length islonger, then this process would have continued for axlA,, = 3996. The total number of electrons is N. =288,000. As space plasmas, the code contains (majority)hydrogen ions with flr&ss rzs and charge qs and(minority) helium ions with moss z?Hg and charge 4s".
The mass density of He ions is taken to be 40 percent ofthat of H ions. The ion-to-electron mass ratio is mslm" -longer time.

It is important to note that, when the particle is in
the upstream region, the average velocity in the -r
direction is obviously smaller than the shock speed. The
velocitv y- is increased when it encounters the shock. Itt46

Masaki T. et al., Acceleration of Energetic Ions by Oblique Magnetosonic Shock Waves

is caused by the change in the magnetic field, which will
be briefly discussed in the next section.

0.8

4. Summary and Discussion

0.4

We have studied interactions with energetic ionsand an oblique shock wave by means of particle
simulations. It is found that some energetic ions canoect=300014move with a shock and can be accelerated many times
by the transverse electric field in the shock. The present
result indicates that, once particles have high energies,
they can be further accelerated many times by othermagnetosonic waves. Its mechanism is different fromthe one for thermal ions.
Even if the initial parallel velocity is smaller thanthat given by Eq. (1), some particles can move with theshock. It is because v,, of energetic ions can be increased

0.8

9000.4oct[l 0.84y11500
30.4

upon the collision with the shock. To see this, we
assume that, for simplicity, we have magnetic field andparticle orbit shown in Fig. 3. Then, the change in themagnetic field along the particle orbit would be givenal"x"q;..f"e"'^I
40.8dt= (Bt - Bo) t t(/ - t) - 6 G - /ou,) l,Q)

2100o;...";

0.4'"-&;-#"'

where 86 and Br are the magnetic fields in the upstream

0and shock regions, respectively. The particle is supposed

to enter the shock region at t = tin ?nd return to theupstream region at t = tour.Further, in the following

0dllL
=8.
- dt + 'D.dB
dt'800Fig. 1 Profiles of transverse electric field E, and phase(3)

space plots lx, yl of energetic ions. The dots
represents energetic ions.

we may neglect the first term on the right-hand side, if
the Lorentz force is greater than the electric force (thisis the case for energetic particles). Then, integrating Eq.
(3), we find that, after returning to the upstream region,the parallel momentum is increased by an amount

5pr=fpr(r*,)-p-, (ti)l.Bol Bo

400x/ c/ ruDo"

equationd(B'p)

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Masaki T. et al., Acceleration of Energetic Ions by Oblique Magnetosonic Shock Wavesq)

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