J. Plasma Fusion Res. SERIES, Vol.2 (1999) 149-152

Electron Trapping and Acceleration in an Oblique
Magnetosonic Shock Wave
BESSHO Naoki and OHSAWA Yukiharu
Department of Physics, Nagoya University, Nagoya 464-8602, Japan(Received: 8 December 1998 / Accepted: 19 February 1999)

Abstract
Oblique magnetosonic shock waves are studied by means of a fully relativistic, fully electromagnetic
particle code with full ion and electron dynamics, with particular attention to electron motion. It is foundthat some electrons can be trapped and then accelerated to ultra-relativistic energies in a shock wave;
Lorentz factors with 7> 100 have been observed in our simulations. The electron energies can be
extremely high at certain propagation angles and shock speeds.

Keywords:
particle acceleration, ultra-relativistic electron, oblique shock wave

1. Introduction and Overview of Oblique
Shock Waves

related to the shock speed v.6 throughEfl= -v"nBrolc.

Ion acceleration by a shock wave has been shownand studied by a number of particle simulations [1-5](and references therein). Strong electron acceleration inparticle simulations, however, has not been reported,even though high-energy electrons have been observedin laboratory and astrophysical plasmas [6-8]. In this(l)We assume that v.6 > 0l and thus E s < 0.

The nonlinear wave theory tells us the wave
structure. A shock wave will have a positive electric
potential g(x). The quantities g, 8,, and plasma density
have similar profile [9,10]. On the other hand, E, and 8,,(and thus the electric field along the magnetic field E11)
are proportional to the x derivatives of these quantities;
for instance, E, = -)Ql0x.

paper we will demonstrate by using a relativistic,electromagnetic particle simulation code that a large-

amplitude, oblique shock wave can produce ultrarelativistic electrons.
We consider a magnetosonic wave propagating in
the -r direction with speed v,6 in an external magnetic

2. Physical Picture

field in the (x, z) plane. We assume that the field

We outline the mechanism why electrons can gain
great energies if reflection occurs. Figure I shows aschematic diagram of the trajectory of electron guiding
center in the wave frame. Here it is assumed that thereflection takes place at point D; the dotted line shows
the orbit of a passing electron. As an electron moves
from point A to C, it moves to the negative y direction
because of the E, x B. drift. It gains kinetic energy A-E1

quantities depend on -r only; 0/0y = dldz = 0. Then, the-r component of the magnetic field is constant, B, = B,<:.

In the pulse region, B, as well as 8, can have finitevalues.

In the wave frame where the time derivatives are
zero (dldt = 0), the electric field in the y direction isconstant, E, = Ero, and E is zero. Hence we have B =(Bn, By, Br) and E = (E- Ep, 0). The electric field E s is@1999 by The Japan Society of Plasma
C orre sponding author's e -mail : be s sho @ npmi.phy s. nagoya-u.ac.j pt49

Science and Nuclear Fusion ResearchBessho N. et al., Electron Trapping and Acceleration in an Oblique Magnetosonic Shock Wave

from the electric potential; A,E1 - eE@d - etp(xi Q 0).
At the same time, it loses energy AE2 because of theelectric field E o; LE, = -sBrr1 1rc - )n) (< 0). The netchange in the energy is therefore

{---

(2)

A,E=LEr+LEz.

Even though the magnitudes of AE1 and LE2 are quite
large, they almost cancel when an electron moves withdrift approximation; in particular, in a perpendicularpulse they have exactly the same magnitude and AE = 0[1]. However, if an electron is reflected and moves
from D to E, then it would gain energies from both E,
and E s. As a result. the increase in energy isFig. 1 Schematic diagram of electron orbit in the {x, y)plane.

(3)

LE=LEr+LEz,e(F-Fo)+h016

where AE3 is defined &s AE3 = -eErn(ye - yJ (> 0).

3. Estimate of Energy Increase
We now obtain maximum energy of a reflectedelectron. The equation of motion for a relativistic
electron is(4)^"9#=-,(".=")'

where 7is the Lorentz factor. For the stationary,
one-dimensional system being considered, this can beintegrated to give the energy conservation equation

Fig. 2 Schematic diagram of quantity lel,F- Fsl + h.ytl.

m"Tc2 - m"Toc2 =e(Q- qo)- eErol Dydt. (5)Equation (8) indicates that ft is positive if B.e/8,e is oforder unity. If B,olBn is much greater than unity, then ft
can be negative. In addition, Eq. (9) shows that, if theinitial value of ft is positive, ho> O, then /r is always
positive in the region where (F - Fs) is positive.
We show in Fig. 2 a schematic diagram of thefunction le(F - Fi + hoTol.We suppose that F has amaximum value at x = r^ and has a minimum value
right behind the main pulse. If the reflection occurs, itwould be in this dip; rr represents the reflection point.
This function could be negative in the dip. We here,
however, consider the case where it is always positive,
because it can be shown that the strons acceleration
takes place in this case.
We assume that an electron with positive fts isreflected at x = h. After the reflection, parallel electron
velocity v11 increases and takes its maximum value at x =x., where E and F are maximum. Then, Eq. (9)indicates that ft has its minimum (positive) value at thisWe define quantity F asF=-JE,(B /40)

dx.

(6)

It is noted that F has a profile similar to the quantities gand 8,. Using F, we can rewrite Eq. (5) asm"Tc2 - m"Toc2= e(F - Fs) - m"c(Er186) (yv,- Tova). Q)

If we define quantity h as
h = m.c2 - m"v"yvrB6/Bix,(8)then from Eq. (7) we havey= [e(F - Fs) + hsys]lh.

(e)

150

Bessho N. et al., ElectronTrapping and Acceleration in an Oblique Magnetosonic Shock Wave

point; hence, Zhas its maximum value there.
We can estimate the maximum value 7. in thewave frame by substituting the maximum value of F in
Eq. (9).The maximum 7in the laboratory frame can bePx 50

obtained by the Lorentz transformation as

Tn= Tsn(l + v,vs,lc') T^= T"tT^.

(10)

Pv

Now let us consider a case where ft becomes quite
small, ft = 0, at r = rm. There the electron speed would
be close to the speed of light c. Hence, roughly, we havevr- cB,/8. The quantity ft can then be expressed as

h=- m"c2(, - *, ?

+1""d) ,1Pz50(1 1)

where tan 0 = BuolBuo.Here we have used therelations Bro = %rrBvo and B,s = Btro. Because B.
becomes quite large in the shock region, we may put as
BzlB - l. Equation (11) then suggests that the electronenergy can become extremely large when
716 (v,6/c) tan 0 =l.

(r2)(Dpet4. Simulation

Fig. 3 Phase space plots of electrons.

We use a one-dimensional (one space coordinate

and three velocity components), relativistic,
electromagnetic, particle simulation code with full ionand electron dynamics to study the electron acceleration.
The total system length is L, = 4096Le, where A, is thethan py; it is because the parallel velocity is dominant in

the motion of high-energy electrons.

grid spacing. More detailed description about theThe simulation parameters are as follows. The

Next, we study trajectories of electrons; we will
confirm that it is the reflected electrons that gain greatenergies. Figure 4 displays time variations of (.r - v.6r),

number of simulation particles is N, = 19" = 262,144.They, and 7of electrons in a shock with v.6 = 1.75 vt; (x -ion-to-electron mass ratio is rflilm" = 100. The ratio of
the cyclotron to plasma frequencies is {D"rla)r" = 3.0 in
the far upstream region. The electron and ion thermalvelocities a;re v7"lc = 0.38 and vvlc = 0.O1, respectively.
The Alfv6n speed is ralc = 0.3. The time step issufficiently small, oo"A/ = 0.02, so that At is muchsmaller than the plasma and cyclotron periods even in
the shock region. The external magnetic field is in the(r, z) plane, and waves propagate in the -r direction. The
propagation angle is 0 = 45o:' i.e., B"o = B&.
Figure 3 shows phase space plots of electrons in a
shock with propagation speed v,6 = 2.3 yr. Someelectrons are reflected near the end of the main pulse,
and ultra-relativistic electrons are found in the shock
region. The maximum value of their Lorentz factors is 1,= 130. We also see that p, and p. have greater valuesv,5t) is the -r position in the wave frame. The lengths are

simulation code can be found in Ref. [2].

normalized to the electron skin depth c/rrro". (Here thesystem size is a half of that in the previous run; theother parameters such as N/^ are the same.) The thicklines show an electron that was accelerated, and the thinlines represent an electron that was not accelerated. In

all of the three panels, the low- and high-energy
electrons both have small-amplitude, cyclotron
oscillations with short periods, .opet S 20. At the sametime, the accelerated electron has a large-amplitude,
long-period oscillation with period ar"t = 170; it begins
when the particle enters the shock, alo"/ = 180, andcontinues until the end of this simulation run. For

comparison with Fig. l,

we show the times

corresponding to points B, C, D, and E. The top panel
indicates that some electrons are reflected and then151

Bessho N. et al., EleclronTrapping and Acceleration in an Oblique Magnetosonic Shock Wave

References

become high energy in the shock wave. On the other
hand, the quantity (x - v,r,t) of the low-energy electron
keeps decreasing, which means that this particle passes

tll D. Biskamp and H. Welter, Nuclear Fusion 12,through the shock region without strong interactions.
The second panel shows that the y position of thet2l Y. Ohsawa, Phys. Fluids 28,2130 (1985).
t3l B. Lembege and J.M. Dawson, Phys. Fluids 81,663 (1972).

1001 (1989).

accelerated electron also oscillates.

t4l Y.Ohsawa, J. Phys. Soc. Jpn. 59,2782 (1990).
t5l K.Maruyama, N. Bessho and Y. Ohsawa, Phys.
Plasmas 5,3257 (1998).

P

t6l C. Joshi, C.E. Clayton, W.B. Mori, J.M. Dawsonand T. Katsouleas, Comments Plasma Phys.

f50x0

Controlled Fusion 16, 65 (1994).
t7l K. Koyama, R. Petre, E.V. Gotthelf , et al., Nature!

37E,zss (1995).

10t8l T. Tanimori, Y. Hayami, S. Kamei, et al.,y0Astrophys. J. 497, L25 (1998).
t9l T. Kakutani, H. Ono, T. Taniuti and C.C. Wei, J.
Phys. Soc. Jpn.24, I 159 (1968).
ll0l Y. Ohsawa, Phys. Fluids 29, 1844 (1986).

-10tl ll Y. Ohsawa, J. Phys. Soc. Jpn. 58,4445 (1989).

40v,200Fig.4 Time variations of {x- v"6t), y, and yof passing
and trapped electrons.

t52