Fusion Res.
Res. SERIES,SERIES, Vol.
Vol. 6 (2004) 000-000
60–63
J. Plasma Fusion

Particle Acceleration Mechanisms
in Magnetosonic Shock Waves
OHSAWA Yukiharu
Department of Physics, Nagoya University, Nagoya, 464-8602, Japan(Received: 9 December 2003 / Accepted: 6 July 2004)

Abstract
Theory and simulations of nonstochastic particle acceleration mechanisms occurring in collisionless shock waves
in a magnetized plasma are reviewed. We describe the acceleration mechanisms of hydrogen ions, heavy ions,electrons, nonthermal relativistic ions, and positrons. The first three mechanisms explain the basic properties of solarenergetic particles; i.e., prompt acceleration of ions to energies 10 9 + 1010 eV and electrons to 10 7 + 10 8 eV, withelemental compositions of high-energy heavy ions similar to that of the solar corona. For the last two mechanisms, themaximum energies demonstrated by simulations far exceed the level of solar energetic particles.
Keywords:
particle acceleration, collisionless shock wave, particle simulation, hydrogen, heavy ion, electron, positron

1. Introduction

can become relativistic.
In this paper, shock waves are assumed to propagate in
the x direction with a speed vsh in an external magnetic fieldin the (x, z) plane, B0 = B0 ( cos i, 0, sin i) . Figure 1 shows
ion phase space plots (x, p x ) and (x, p y ) , where p x and p y are
the x and y components of the momentum p, respectively.
Many ions are accelerated to relativistic energies by a shock
wave (the shock front is at x/D g - 1500 at this moment,
where D g is the grid spacing). This result was obtained by arelativistic, electromagnetic particle simulation with full ion
and electron dynamics [12]. In the laboratory frame, thequantities B z - B z0 , E y , and electric potential z have similar
profiles. (In the wave frame, E y is constant.) The simulation
parameters for Fig. 1 are as follows: m i / m e = 100 ,X e /~ pe = 3 , i = 90 % , and vA /c = 0.3 . The observed shock

Acceleration of charged particles in laboratory andastrophysical plasmas has been extensively studied by many
authors [1-8]. To understand particle acceleration, we need tostudy collective plasma behavior as well as individual
particle motions in a self-consistent manner. Particle
simulations are therefore a powerful tool for this research.
Indeed, for instance, they have shown that magnetosonicshock waves (or pulses) can accelerate particles with various
nonstochastic mechanisms; i.e., hydrogen ions [9-16], heavy
ions [17,18], electrons [19,20], nonthermal relativistic ions[20,21], and positrons [23]. The first three mechanisms
explain the basis properties of solar energetic particles;
prompt acceleration of ions to relativistic energies andelectrons to ultrarelativistic energies, with the elemental
compositions of energetic heavy ions quite similar to that of
the solar corona. On the other hand, in the last two
mechanisms (relativistic ions and positrons) particle energies
far exceed the level of the solar energetic particles. This
paper very briefly reviews the theory and simulation resultsof these acceleration processes.

2. Ion acceleration
2.1 Hydrogen ions
In a single-ion-species plasma, a magnetosonic shock
wave can accelerate some ions by reflection; the reflection
occurs owing to the rapid increase in the electric potential or
the magnetic field [9-16]. The maximum speed of reflectedions is v + 2MvA , where M is the Alfvén Mach number andv A is the Alfvén speed. When the electron gyrofrequency isgreater than the plasma frequency, X e /~ pe L 1, these ionsFig. 1 Phase space plots of ions accelerated to relativisticenergies by a shock wave.

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: ohsawa@phys.nagoya-u.ac.jp160

Ohsawa Y., Particle Acceleration Mechanisms in Magnetosonic Shock Waves

speed is v sh = 2.7vA .
In this example, the ions are strongly accelerated once;
both p x and p y monotonically increase to their maximum
values after the reflection. It was suggested [10], however,that multiple reflection of an ion can occur at a shock front.
This can take place when vx is very close to vsh when the ion
encounters the shock wave. If the thermal speed is much
lower than v sh , as shown in Fig. 1, the multiple reflection will
rarely happen [15]., i.e., the number of such particles will besmall in a low beta plasma.

2.2 Heavy ions
In a plasma containing multiple ion species with thehydrogen being the major component, a shock wave
accelerates all the heavy ions [17]. (Some hydrogen ions areaccelerated by the reflection, as discussed in Sec. 2.1.) The
maximum speeds of the heavy ions are nearly the same,independent of particle species,

B
B0v + d Bm + B n v sh ,mFig. 2 Electron guiding center orbit projected on the ( x, y )plane.

(x, y) plane; some electrons are reflected at point D. Point Cand point E are on the peak of the potential z ; hence,x C = x E = x m , where x m designates the x position at which z
and B z take their maximum values. The electric field E x ispositive in the region x > x m and is negative in x < x m .
Substituting the guiding-center velocity(1)0where B m is the peak value of B.
If large-amplitude magnetosonic pulses are excited in
the solar corona, therefore, the elemental compositions ofhigh-energy heavy ions should be quite similar to that of thesolar corona [3]. If they are excited in coronal magnetic
tubes, where the magnetic fields are strong, the particle
energies would be quite high.
This acceleration becomes weak as the wave amplitudeis decreased. It does not vanish, however, as long as theamplitude is finite. Hence, magnetosonic solitary waves aredamped even if they propagate perpendicular to a magnetic
field. Aside from the particle acceleration, the presence ofmultiple ion species introduces various interesting
phenomena in the wave propagation (see refs. [24,25] andreferences therein).

v gx =(5)KBC = e (z C - z B )C+ eE y0 # dx E x /[ E y0 + (v /c)( BB x0 /B z )]B

(6)

C- eE y0 # dt v B y /B.
B

The first term on the right-hand side shows the energy changeby the potential difference. The second and third terms
represent the work done by the constant electric field E y0 .
We have similar expressions for K CD and K DE . The energy
increase from point B to point E is given by
K BE = K BC + K CD + K DE . The magnitudes of K BC and K CD arequite small because the changes in the kinetic energy due tothe electric potential z and due to E y0 almost cancel [19]. In
moving from points D to E, however, electrons gain energy
from both z and E y0 , which makes K DE quite large. Hence,
K BE - K DE . Now, assuming that, shortly after the reflection,
the electrons have relativistic velocities v + c and thatB + B z & B y ( B y is small in the simulations), we obtain theelectron energy at point E as(2)where c is the Lorentz factor, we have the energy
conservation equation in the wave frame,m e c 2 (c - c0 ) = e (z - z 0 ) - eEy0 # vy dt.

BcE x Bz+ v By ,B2in eq. (3), we find the increase in the kinetic energy frompoint B to point C as

If some electrons are reflected near the end of the main
pulse of a shock wave, they can be accelerated toultrarelativistic energies in the shock region [19, 20]. Here,
we theoretically describe this, considering particle
trajectories.
From the relativistic equation of motion for an electron,d (c v)= - e cE + v #c Bm ,dt(4)

vgy =-3. Electron acceleration to ultrarelativisticenergies

mecE y0 B z
B+ v Bx0 ,B2(3)

KBE =Here, we have used the relations E y = E y0 (constant) andE z = 0 , which hold in the wave frame. We show in Fig. 2 aschematic diagram of guiding-center orbit projected on theez E
1 - (v sh /c)( B z0 /B x0 ) .

(7)This indicates that K BE can have extremely great values
when the denominator is close to zero. This condition isidentical to the relation v gx + 0 . In this case, it takes a long261

Ohsawa Y., Particle Acceleration Mechanisms in Magnetosonic Shock Waves

Fig. 3 The maximum electron energy as a function of vsh .
The solid lines and dots show theory and simulation
results, respectively. Here, X e /~ pe = 3 and i = 45 ° in(a), while X e /~ pe = 1 and i = 66 ° in (b).

time to move from point D to E, during which the particle
traverses a long distance in the y direction, gaining a greatamount of energy from E y0 . The magnitude of the potential
in the wave frame is given as [26]RVsin i cos iWez E = m i v A2 csh Ssin 2 i +Scsh (1 + csh2 tan 2 i)1/ 2 WTXVR
1/2W
SJ
2v 2 N# S KK1+ 2 sh2 OO - 1W ,S L vA sin i PW
XT(8)Fig. 4 Stepwise increase in c of a relativistic ion incessantly
accelerated by an oblique shock wave.
Here, X H0 is the nonrelativistic hydrogen
gyrofrequency in the upstream region.

where csh = (1- v sh2 /c 2 ) -1/2 . Figure 3 shows the maximum
electron energy as a function of vsh . The solid line and dots
represent the theory and simulation results, respectively.

4. Acceleration of nonthermal relativisticparticles

dc =(10)TX
Here, q i is the ion charge, and X i is the nonrelativistic ion
gyrofrequency. The subscript 1 refers to quantities in thestrong-field region; p19 is the momentum perpendicular to
the magnetic field. The energy increase rate averaged overthe gyroperiod [22] isIn the previous three mechanisms, particles areaccelerated from thermal to high energies. In this section, wediscuss a mechanism accelerating nonthermal energetic
particles to much higher energies.
Whenvsh + c cos i ,RV
2q i p1 9 E1y
X (t - t )sin SS i1 outc in WW .
2 22m i c X i1

(9)d c g01 v1 9 v sh= π
X i01,dtc2

is satisfied,the interaction time between relativistic particles
and wave fields can become significantly long [21,22,23].
Indeed, since the time-averaged particle velocity in the xdirection, G v x H, is limited by c cos i, particles cannot quickly
outrun the wave if (9) holds. It was shown [21,22] that by
virtue of this effect, nonthermal, fast ions with c + 1 can be
accelerated to ultrarelativistic energies with c > 100 by anoblique shock wave satisfying eq. (9) (see Fig. 4, where c
grows from 4 to 160).
Relativistic ions have gyration speeds comparable to or
faster than v sh and gyroradii much larger than the width of
the shock transition region. In association with thegyromotion, thus, these particles can move back and forth
between the pulse and upstream regions [27,28,29]; when
they are in the shock region, they gain energy from thetransverse electric field E y , because their motions are nearlyparallel to E y there. Hence, while they move with the wave,
their energies increase stepwise, with each energy jump
given bywhere g01 is a numerical factor smaller than unity,
RV
J
B2 N
X (t - t in ) W
B
g01 = d1 - z0 n KK1 - x02 OOsin S i1 out.
B z12cSW
2B z1LPTX

(11)

(12)

5. Positron acceleration
Strong positron acceleration in a shock wave with eq. (9)has also been demonstrated with particle simulations [23].
Although the electric field parallel to the magnetic field, E ,is quite weak in magnetohydrodynamic waves, positrons canbe reflected by E because the positron mass is muchsmaller than ion masses. They gain a large amount of energy
while moving with the wave. (The reflection of hydrogenions is across B.)
If c is large, a slight change in the speed can lead to agreat change in c. Using this fact,

vd c /d t & c dv/d t,362(13)

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Fig. 5 Positron phase space plots (x, c) . The solid line shows
the profile of B z .

one can theoretically find particle motions where the zerothorder velocity is nearly parallel to the external magnetic field.
This theory explains the positron acceleration observed inparticle simulations.
Figure 5 shows phase space plots of positrons
accelerated by a shock wave with v sh = 2.4vA in a plasma
with vA /c = 0.2 and i = 61°. The plasma contains positrons
as well as electrons and ions with the positron-to-electron
density ratio n p /n e = 1/50 . Unlike the above energetic ions,these positrons are in the shock transition region during theacceleration process.

6. Summary
We have very briefly reviewed five different particle
acceleration mechanisms caused by magnetosonic shockwaves. They are demonstrated by simulations and areexplained by a coherent theory. Even though we have notdescribed the nonlinear wave theory [30-35] or simulation
methods, they play an essential role in the understanding ofparticle acceleration.The development of plasma physics isneeded to resolve the mystery of cosmic ray acceleration.

Acknowledgements
The author is grateful to Dr. M. Toida, Dr. N. Bessho,
Dr. S. Irie, Mr. S. Usami, and Mr. H. Hasegawa. This paper
is based on the collaboration with them.

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