Cross-scale coupling at a perpendicular collisionless shock

Abstract

A full particle simulation study is carried out on a perpendicular collisionless shock with a relatively low Alfven Mach number (MA = 5). Recent self-consistent hybrid and full particle simulations have demonstrated ion kinetics are essential for the non-stationarity of perpendicular collisionless shocks, which means that physical processes due to ion kinetics modify the shock jump condition for fluid plasmas. This is a cross-scale coupling between fluid dynamics and ion kinetics. On the other hand, it is not easy to study cross-scale coupling of electron kinetics with ion kinetics or fluid dynamics, because it is a heavy task to conduct large-scale full particle simulations of collisionless shocks. In the present study, we have performed a two-dimensional (2D) electromagnetic full particle simulation with a “shock-rest-frame model”. The simulation domain is taken to be larger than the ion inertial length in order to include full kinetics of both electrons and ions. The present simulation result has confirmed the transition of shock structures from the cyclic self-reformation to the quasi-stationary shock front. During the transition, electrons and ions are thermalized in the direction parallel to the shock magnetic field. Ions are thermalized by low-frequency electromagnetic waves (or rippled structures) excited by strong ion temperature anisotropy at the shock foot, while electrons are thermalized by high-frequency electromagnetic waves (or whistler mode waves) excited by electron temperature anisotropy at the shock overshoot. Ion acoustic waves are also excited at the shock overshoot where the electron parallel temperature becomes higher than the ion parallel temperature. We expect that ion acoustic waves are responsible for parallel diffusion of both electrons and ions, and that a cross-scale coupling between an ion-scale mesoscopic instability and an electron-scale microscopic instability is important for structures and dynamics of a collisionless perpendicular shock.

Full text

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arXiv:1104.0187v1 [physics.plasm-ph] 1 Apr 2011
Cross-scalecouplingataperpendicularcollisionlessshock
Takayuki Umeda, Masahiro Yamao
Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya 464-8601, JAPAN
Ryo Yamazaki
Department of Physical Science, Hiroshima University, Higashi-Hiroshima 739-8526, JAPAN
Present address: Department of Physics and Mathematics, Aoyama Gakuin University, 5–10–1, Fuchinobe, Sagamihara,
Kanagawa, 252-5258, JAPAN
Abstract
A full particle simulation study is carried out on a perpendicular collisionless shock with a relatively low Alfven Mach
number (MA = 5). Recent self-consistent hybrid and full particle simulations have demonstrated ion kinetics are
essential for the non-stationarity of perpendicular collisionless shocks, which means that physical processes due to ion
kinetics modify the shock jump condition for fluid plasmas. This is a cross-scale coupling between fluid dynamics and
ion kinetics. On the other hand, it is not easy to study cross-scale coupling of electron kinetics with ion kinetics or
fluid dynamics, because it is a heavy task to conduct large-scale full particle simulations of collisionless shocks. In the
present study, we have performed a two-dimensional (2D) electromagnetic full particle simulation with a “shock-rest-
frame model”. The simulation domain is taken to be larger than the ion inertial length in order to include full kinetics
of both electrons and ions. The present simulation result has confirmed the transition of shock structures from the
cyclic self-reformation to the quasi-stationary shock front. During the transition, electrons and ions are thermalized
in the direction parallel to the shock magnetic field. Ions are thermalized by low-frequency electromagnetic waves (or
rippled structures) excited by strong ion temperature anisotropy at the shock foot, while electrons are thermalized
by high-frequency electromagnetic waves (or whistler mode waves) excited by electron temperature anisotropy at the
shock overshoot. Ion acoustic waves are also excited at the shock overshoot where the electron parallel temperature
becomes higher than the ion parallel temperature. We expect that ion acoustic waves are responsible for parallel
diffusion of both electrons and ions, and that a cross-scale coupling between an ion-scale mesoscopic instability and an
electron-scale microscopic instability is important for structures and dynamics of a collisionless perpendicular shock.
Key words: collisionless shock; particle-in-cell simulation; cross-scale coupling
1. Introduction
Dynamics of shock waves in plasmas are often
discussed by the shock jump conditions (Rankine-
Hugoniot conditions), which describe conservation
laws of mass, momentum, energy, normal magnetic
field and motional electric field for fluid plasmas.
Email address: umeda@stelab.nagoya-u.ac.jp (Takayuki
Umeda).
On the other hand, previous kinetic simulations
revealed that collisionless shocks in plasmas can be
strongly non-stationary in both spatial and tem-
poral scales of ions. In the direction normal to the
shock surface of a quasi-perpendicular collision-
less shock, a new shock front periodically appears
(e.g., Biskamp and Welter, 1972; Quest, 1985; Lem-
bege and Dawson, 1987; Lembege and Savoini,
1992; Hellinger et al., 2002), which is called the
self-reformation. Incoming ions are reflected up-
Article to be published in Planetary and Space Science4 April 2011

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stream at the shock ramp of a supercritical quasi-
perpendicular collisionless shock, and they form a
foot in front of the ramp during their gyration. At
the upstream edge of the foot, ions are accumu-
lated in time and are reflected upstream, which are
responsible for the self-reformation. The cyclic self-
reformation is due to ion dynamics, although this
process has been confirmed in both electromagnetic
hybrid and full particle simulations. In addition,
recent full particle simulations have shown that
electron-scale micro instabilities, such as Buneman
instability (e.g., Shimada and Hoshino, 2000) and
modified two-stream instability (e.g., Scholer et
al., 2003) are excited at the foot during the cyclic
self-reformation. Scholer and Matsukiyo (2004) has
also demonstrated that the modified two-stream in-
stability is also responsible for the self-reformation.
Another mechanism of the self-reformation is steep-
ening of whistler mode waves in upstream regions
of oblique shocks (Krasnoselskikh et al., 2002).
In the shock-tangential direction, on the other
hand, there appear fluctuations in the spatial scale
of ion inertial length in the direction parallel to
the shock magnetic field (Winske and Quest, 1988;
Lowe and Burgess, 2003) or ion gyro radius of re-
flected ions in the direction perpendicular to the
shock magnetic field (Burgess and Scholer, 2007),
which are called the “ripples”. The compression of
incoming ions at collisionless shocks results in their
adiabatic heating in the shock-normal direction. In
quasi-perpendicular shocks, however, the ion heat-
ing in the shock-normaldirection is more dominated
bythegyrationofreflectedions.Thusaniontemper-
ature anisotropy between shock-normal and shock-
tangential directions becomes a common feature in
the transition region of quasi-perpendicular shocks.
The dynamic rippled character of the shock sur-
face is thought to be related to the ion tempera-
ture anisotropy.Although this process has been con-
firmed in two-dimensional (2D) electromagnetic hy-
brid particle simulations, it is difficult to take into
account the dynamic rippled character of the shock
surface in 2D electromagnetic full particle simula-
tions. This is because current computer resources
are not necessarily enough to take such a large sim-
ulation domain of several ion inertial length.
Very recently, however,there are several attempts
of 2D electromagnetic full particle simulations that
take into account ion dynamics in both shock-
normal and shock-surface directions (Hellinger et
al. 2007; Amano and Hoshino, 2009; Lembege et
al., 2009). These results indicate that ion-scale fluc-
tuations at perpendicular collisionless shocks can
dynamically change electron-scale processes such as
wave excitation and electron acceleration. The pur-
pose of this paper is to examine a cross-scale cou-
pling between the dynamic rippled character of the
shock surface and electron-scale micro instabilities.
In order to take into account ion dynamics in both
shock-normal and shock-surface directions, a large-
scale 2D electromagnetic full particle simulation is
carried out by using the “shock-rest-frame model”.
2. Full Particle Simulations
2.1. Shock-Rest-Frame Model
There are several different methods for exciting
collisionless shocks in kinetic simulations of plas-
mas. These include the injection method (or the
reflection/wall method) (e.g., Quest, 1985; Winske
and Quest, 1988; Shimada and Hoshino, 2000;
Hellinger et al. 2002; Lowe and Burgess, 2003;
Scholer et al., 2003; Burgess and Scholer, 2007;
Amano and Hoshino, 2009). the plasma release
method (Ohsawa, 1985), and the magnetic piston
method (e.g., Biskamp and Welter, 1972; Lembege
and Dawson, 1987; Lembege and Savoini, 1992). In
these methods, collisionless shocks are excited by an
interaction between a supersonic plasma flow and
a resting plasma. The simulation domain is taken
in the downstream rest frame with the injection
method, while the simulation domain is taken in
the upstream rest frame withe the plasma release
and magnetic piston methods. Thus an excited
shock wave propagates upstream in these methods.
There is also another method called the flow-flow
method for exciting collisionless shocks (e.g., Omidi
and Winske, 1992). Since collisionless shocks are
excited by an interaction between two supersonic
plasma flows in this method, there exist forward
and reverse shock waves. A big problem in these
methods is that excited collisionless shock waves
propagate at a fast velocity, and it is necessary to
take a very long simulation domain in the propaga-
tion direction of the shock waves in order to follow a
long-time evolution of the shock waves. This makes
it difficult to perform multidimensional simulations
even with current supercomputer systems.
An alternative is to excite collisionless shocks in
the shock rest frame with the “relaxation method”,
whereby collisionless shocks are excited by an inter-
action between a supersonic plasma flow and a sub-
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sonicplasmaflowmovinginthe samedirection.This
method was first used in hybrid particle simulations
in 1980’s (e.g., Leroy et al., 1981, 1982), and then
in full particle simulations in 1990’s (Pantellini et
al., 1992; Krauss-Varban et al., 1995). This method
was not so popular because of several difficulties in
numerical techniques, and its application to long-
term evolution of shock waves was not considered.
In 2000’s, however, long-term 1D simulations with
the relaxation method have been performed by us-
ing Darwin particle code (Muschietti and Lembege,
2006)and full electromagneticparticlecode (Umeda
and Yamazaki, 2006). Very recently, the relaxation
method has also been applied to long-term 2D full
electromagnetic particle simulations (Umeda et al.,
2008, 2009). In general, it is not easy to perform a
large-scale (ion-scale) multidimensional full electro-
magnetic particle simulations of collisionless shocks
even with present-day supercomputers. Hence the
shock-rest-frame model is important to be able to
follow the evolution of shock waves for a long term
with a limited computer resource.
2.2. Simulation Setup
We use a 2D full electromagnetic particle code
(Umeda, 2004), in which the full set of Maxwell’s
equations and the relativistic equation of motion
for individual electrons and ions are solved in a
self-consistent mannar. The continuity equation for
charge is also solved to compute the exact current
density given by the motion of charged particles
(Umeda et al., 2003).
The initial state consists of two uniform regions
separated by a discontinuity. In the upstream re-
gion that is taken in the left hand side of the sim-
ulation domain, electrons and ions are distributed
uniformly in space and are given random velocities
(vx,vy,vz) to approximate shifted Maxwellian mo-
mentum distributions with the drift velocity ux1,
number density n1 ≡ ǫ0meω2
peratures Te1≡ mev2
e, ωpand vtare the mass, charge, plasma frequency
and thermal velocity, respectively. Subscripts “1”
and “2” denote “upstream” and “downstream”,
respectively. The upstream magnetic field By01 ≡
−meωce1/e is also assumed to be uniform, where
ωcis the cyclotron frequency (with sign included).
The downstream region taken in the right-hand
side of the simulation domain is prepared similarly
with the drift velocity ux2, density n2, isotropic
pe1/e2, isotropic tem-
te1and Ti1≡ miv2
ti1, where m,
temperatures Te2and Ti2, and magnetic field By02.
We take the simulation domain in the x-y plane
and assume a perpendicular shock (i.e., Bx0 = 0).
Since the ambient magnetic field is taken in the y
direction, free motion of particles along the ambient
magnetic field is taken into account. As a motional
electric field, a uniform external electric field Ez0=
−ux1By01= −ux2By02is applied in both upstream
and downstream regions, so that both electrons and
ions drift in the x direction. At the left boundary
of the simulation domain in the x direction, we in-
ject plasmas with the same quantities as those in
the upstream region, while plasmas with the same
quantities as those in the downstream region are
also injected from the right boundary in the x direc-
tion. We adopted absorbing boundaries to suppress
non-physical reflection of electromagnetic waves at
both ends of simulation domain in the x direction
(Umeda et al., 2001), while the periodic boundaries
are imposed in the y direction.
In the relaxation method, the initial condition
is given by solving the shock jump conditions
(Rankine-Hugoniot conditions) for a magnetized
two-fluid isotropic plasma consisting of electrons
and ions (Hudson, 1970). In order to determine a
unique initial downstream state, we need given up-
stream quantities ux1, ωpe1, ωce1, vte1, and vti1and
an additional parameter. We assume a low-beta
and weakly-magnetized plasma such that βe1 =
βi1= 0.125 and ωce1/ωpe1= −0.1 in the upstream
region. We also use a reduced ion-to-electron mass
ratio mi/me= 25 for computational efficiency. The
light speed c/vte1= 40.0 and the bulk flow velocity
of the upstream plasma ux1/vte1= 4.0 are also as-
sumed. Then, the Alfv´ en Mach number is calculated
as MA = (ux1/c)|ωpe1/ωce1|?mi/me = 5.0. The
region is given as Ti1/Te1= 1.0. In this study, down-
stream ion-to-electron temperature ratio Ti2/Te2=
8.0 is also assumed as another initial parameter
to obtain the unique downstream quantities by
solving the shock jump conditions, ωpe2/ωpe1 =
1.8372, ωce2/ωpe1= 0.3375, ux2/vte1= 1.1851, and
vte2/vte1= 2.6393.
In this study, we perform two runs with different
sizes of the simulation domain. We use Nx× Ny=
2048×1024 cells for the upstream region and Nx×
Ny= 2048 × 1024 cells for the downstream region,
respectively, in Run A. The grid spacing and time
step of the present simulation are ∆x/λDe1 = 1.0
and ωpe1∆t = 0.0125, respectively. Here λDe1is the
electron Debye length upstream. Thus the total size
ion-to-electron temperature ratio in the upstream
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of the simulation domain is 10.24li×5.12liwhich is
long enough to include the ion-scale rippled struc-
ture, where li= c/ωpi1(= 200λDe1) is the ion iner-
tial length. In Run B, we use Nx×Ny= 2048×128
cells for the upstream region and Nx×Ny= 2048×
128 cells for the downstream region, respectively.
Thus the the total size of the simulation domain is
10.24li× 0.64li, in which ion-scale processes along
the ambient magnetic field is neglected. We used 16
pairs of electrons and ions per cell in the upstream
region and 64 pairs of electrons and ions per cell in
the downstream region, respectively, at the initial
state.
3. Results
Figure 1 shows the tangential component of
magnetic field By as a function of position x and
time t for Runs A and B. The position and time
are renormalized by the ion inertial length li and
the ion cyclotron angular period 1/ωci1, respec-
tively. The magnitude is normalized by the initial
upstream magnetic field By01. In Fig.1, the tan-
gential magnetic fields By are averaged over the
y direction, which means that fluctuations in the
shock-tangential direction are neglected.
In the present shock-rest-frame model, a shock
wave is excited by the relaxation of the two plas-
mas with different quantities. Since the initial state
is given by the shock jump conditions for a “two-
fluid” plasma consisting of electrons and ions, the
kinetic effect is excluded in the initial state and the
excited shock becomes “almost” at rest in the simu-
lation domain. In both runs, the shock front appears
and disappears on a timescale of the downstream
ion gyroperiod, which correspondsto the cyclic self-
reformation of a perpendicular shock. The reforma-
tion takes place for more than ωci1t = 12 in Run B,
while the reformation seems to be less significant af-
ter ωci1t ∼ 8 in Run A. The previous 2D full particle
simulations have demonstrated the transition from
the cyclic self-reformation to a “quasi-stationary”
shock front (Hellinger et al., 2007; Lembege et al.,
2009), which is in agreement with Run A. However,
it should be noted that the self-reformation does
take place even after ωci1t ∼ 8 in Run A, but on
a different timescale, when we focus on a local tan-
gential magnetic field.
When the length of the simulation domain in
the shock-tangential direction is shorter than the
ion inertial length, ion-scale fluctuations along
Fig. 1. Tangential magnetic field By as a function of position
x and time t for Runs A and B. The position and time are
normalized by λi and 1/ωci1, respectively. The magnitude
is normalized by the initial upstream magnetic field By01.
The magnetic fields are averaged over the y direction.
the shock surface (ripples) do not appear and the
profiles of electromagnetic fields become almost
one-dimensional (Umeda et al., 2008, 2009), and
there exists apparent cyclic self-reformation of the
perpendicular shock as seen in Run B. The present
result suggests that the ion-scale fluctuations in the
shock-tangential direction play an important role
in the sequential appearance of non-stationary and
quasi-stationary shock fronts.
Figure 2 shows the electron and ion temperatures
as a function of position x and time t for Run A.
The panels (a), (b) and (c) corresponds to the tem-
perature ratios Te||/Ti||, Ti||/Ti⊥and Te||/Te⊥, re-
spectively. The panels (d) and (e) correspondsto the
parallel temperatures of electrons and ions, Te||and
Ti||, respectively. Note that these temperatures are
averaged over the y direction, and that the paral-
lel temperatures are approximated by the tempera-
tures in the y direction while the perpendicular tem-
peratures are approximated by the average of tem-
peratures in the x and z directions.
From ωci1t ∼ 7, the electron temperature in the
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Fig. 2. Electron and ion temperatures as a function of position x and time t for Run A. (a)Te||/Ti||, (b)Ti||/Ti⊥, (c)Te||/Te⊥,
(d)Te||, and (e)Ti||. The position and time are normalized by λi and 1/ωci1, respectively. The temperatures are normalized
by the initial upstream temperature (Te01 = Ti01). These temperatures are averaged over the y direction. Here, the parallel
temperatures are approximated by the temperatures in the y direction while the perpendicular temperatures are approximated
by the average of temperatures in the x and z directions.
direction parallel to the ambient magnetic field, Te||,
at the shock overshoot becomes twice as large as the
ion parallel temperature, Ti||. At the shock foot, the
electron perpendicular temperature, Te⊥, is higher
than the electron parallel temperature, Te||. On the
other hand, Te||becomes higher than Te⊥ at the
shock overshoot. In the downstream region, Te||is
slightly higher than Te⊥. As seen in Fig.2d, electrons
are strongly thermalized in the direction parallel to
the shockmagneticfieldatthe overshoot,suggesting
thatthereexistsastrongparalleldiffusionprocessat
the overshootfrom ωci1t ∼ 7. On the otherhand, ion
parallel temperature becomes higher at the shock
foot and in the downstream region as seen in Fig.2e.
Fig.2 shows that electrons and ions are thermalized
in different regions, suggesting that electrons and
ions heating takes place on different scales.
Figure 3 shows snapshots of shock magnetic field
By, ion density ni, ion parallel temperature Ti||, ion
parallel temperature Ti⊥, and ion temperature ra-
tio Ti||/Ti⊥. at ωcit = 12 for Runs A and B. Al-
though the Mach number of the present simulation
run is relatively low (MA= 5), the present perpen-
dicular shock is supercritical, and therefore the cy-
clotron motion of reflected ions is dominant for ion
heating in the shock-normal direction. At the shock
overshoot, the ion parallel and perpendicular tem-
peratures are low because of the accumulation of
upstream cold ions. The ion perpendicular temper-
ature becomes higher at the shock foot because of
the non-gyrotropic velocity distribution of reflected
ions. At the shock foot, ions are also thermalized
in the parallel direction because of an anisotropy-
driven ion cyclotron wave in Run A. However, the
ion parallel heating is not responsible for the elec-
tron parallel heating (see Fig.2). As seen in Fig.3,
theiontemperatureanisotropy(Ti⊥/Ti||> 1)would
be a common feature at perpendicular collisionless
shocks. In Run B, there is no ion parallel heating
because the system length does not allow the ex-
istence of an ion cyclotron wave in the parallel di-
rection. Thus the temperature anisotropy becomes
much higher than in Run A.
In order to study mechanisms for parallel heat-
ing of electrons and ions, we take Fourier transfor-
mation of the shock-normal magnetic field compo-
nent Bxand the shock-tangential electric field com-
ponent Eyin the transition region. Figure 4 shows
frequency-wavenumber spectra of the shock-normal
magneticfieldcomponentBxfordifferenttimeinter-
vals: (a)ωci1t = 4 ∼ 6, (b)ωci1t = 6 ∼ 8, (c)ωci1t =
8 ∼ 10, and (d)ωci1t = 10 ∼ 12. The frequency and
wavenumber are normalized by ωpe1and ωpe1/vte1,
respectively. These frequency-wavenumber spectra
are obtained by projection of ω − kx− ky spectra
onto the ω−kyplane. Note that the typical electron
and ion cyclotron frequencies in the transition re-
gion are ωce∼ 0.4ωpe1and ωci∼ 0.016ωpe1, respec-
tively, and their maximum values are ωce∼ 0.7ωpe1
and ωci∼ 0.028ωpe1, respectively, at the overshoot.
In Fig.4, we found a strong enhancement of Bx
component below ωci, which might correspond to
the rippled structures due to the ion temperature
anisotropy. We also found an enhancement of Bx
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Fig. 3. Spatial profiles of shock magnetic field By, ion density
ni, ion parallel temperature Ti||, ion parallel temperature
Ti⊥, and ion temperature ratio Ti||/Ti⊥. at ωcit = 12 for
Runs A and B.
component over ωci, suggesting that electromag-
netic electron cyclotron waves are excited in the
transition region, which might correspond to the
“nonlinear whistler waves” reported by Hellinger et
al. (2007) and Lembege et al. (2009).
For ωci1t = 6 ∼ 8 the high-frequency electromag-
netic electron cyclotron waves are enhanced over
ω/ωpe1= 0.4,whileothertime intervalstheBxcom-
ponent is enhanced up to ω/ωpe1 = 0.4, implying
thatthehigh-frequencywavesareresponsibleforthe
parallel heating of electrons at ωcit ∼ 7 (Fig.2a and
2d).Notethat excitationofelectromagneticwhistler
mode waves due to electron temperature anisotropy
are observed in the previous 2D simulations of per-
pendicular shocks (Umeda et al., 2008). Since the
electron parallel temperature becomes higher than
the electron perpendicular temperature for ωcit >
7, whistler mode waves due to electron temperature
anisotropy.
Ions are thermalized in the direction parallel to
the shock magnetic field by the low-frequency elec-
tromagnetic waves below ωci. However, these waves
are not so much responsible for the parallel heating
of electrons, because there is not significant paral-
lel heating of electrons at the shock foot as seen in
Fig.2d.
Figure 5 shows frequency-wavenumber spectra
of the shock-normal magnetic field component Ey
in the transition region for different time intervals:
(a)ωci1t = 4 ∼ 6, (b)ωci1t = 6 ∼ 8, (c)ωci1t =
8 ∼ 10, and (d)ωci1t = 10 ∼ 12. with the same
format as Fig.4. The typical ion plasma frequency
in the transition region is ωpi ∼ 0.35ωpe1, and its
maximum value is ωce∼ 0.6ωpe1at the overshoot.
For ωci1t = 6 ∼ 12, we found a strong enhance-
ment of the Eycomponent up to ω/ωpe1∼ 0.7,while
there is not any enhancement in a high-frequency
range for ωci1t = 4 ∼ 6. The phase velocity of these
wave are estimated as vp/vte1 = 1.0 ∼ 1.5. From
Fig.2, the typical parallel temperatures of electrons
and ions are estimated as Te||∼ 18Te01and Te||∼
9Ti01. Thus the ion acoustic velocity is obtained as
vs=
?
Te||+ γTi||
mi
∼ 1.3vte1
(1)
with γ = 3, suggesting that the ion acoustic waves
are excited in the transition region. As shown in
Fig.2a, the electron parallel temperature becomes
higher than the ion parallel temperature due to elec-
tron cyclotron waves at the shock overshoot, which
is a suitable condition for excitation of ion acoustic
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Fig. 4. Frequency-wavenumber spectra of the shock-nor-
mal magnetic field component Bx for different time in-
tervals in Run A. (a)ωci1t = 4 ∼ 6, (b)ωci1t = 6 ∼ 8,
(c)ωci1t = 8 ∼ 10, and (d)ωci1t = 10 ∼ 12. The
Fourier transformation is taken for y/li = 0 ∼ 5.12, (a)
x/li = −0.5 ∼ 0.5, (b)-(d) x/li = −1.0 ∼ 0.0. These fre-
quency-wavenumber spectra are obtained by projection of
ω − kx− ky spectra onto the ω − ky plain. The frequency
and wavenumber are normalized by ωpe1 and 1/λDe1, re-
spectively. The magnitude is normalized by the upstream
magnetic field By01.
Fig. 5. Frequency-wavenumber spectra of the shock-normal
magnetic field component Ey for different time intervals in
Run A, with the same format as Fig.4. (a)ωci1t = 4 ∼ 6,
(b)ωci1t = 6 ∼ 8, (c)ωci1t = 8 ∼ 10, and (d)ωci1t = 10 ∼ 12.
The magnitude is normalized by the motional electric field
Ez0.
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waves.
Figs.4 and 5 show active wave phenomena in the
transition region, although the shock front appears
to be “quasi-stationary” when averaged over the y
direction. The shock front becomes turbulent in-
stead of quasi-stationary. We expect that the ion
acoustic waves would play a role in the transition
process from the self-reformation phase to the tur-
bulent phase, because ion acousticwavesarerespon-
sible for diffusion of both electrons and ions along a
magnetic field.
4. Summary
We performed a 2D electromagnetic full particle
simulationofalow-Mach-numberperpendicularcol-
lisionless shock. The results are itemized below.
(i) It has been confirmed that the cyclic self-
reformation of the shock front becomes less
significant as time elapses, which is consistent
with the previous 2D simulations (Hellinger
et al., 2007; Lembege et al., 2009). The shock
front appears to be “quasi-stationary” by
averaging the spatial profiles of electromag-
netic fields over the shock-tangential direc-
tion, although electron-scale microscopic and
ion-scale mesoscopic instabilities are quite dy-
namic and the shock front becomes turbulent.
(ii) During the transition from the cyclic self-
reformation to the turbulent shock front,
electrons and ions are thermalized in the di-
rection parallel to the shock magnetic field in
different regions and by different mechanisms.
The electron parallel temperature is more en-
hanced than the parallel ion temperature at
the shock overshoot.
(iii) Low-frequency electromagnetic (ion cyclotron
ormirrormode) wavesareexcited at the shock
foot, which corresponds to the rippled struc-
ture in the shock-tangential direction. These
waves are excited by the strong temperature
anisotropy of ions. Ions are thermalized in the
direction parallel to the shock magnetic field
by these waves, but electrons are not.
(iv) Electromagnetic electron cyclotron (whistler
mode) waves are excited at the shock over-
shoot. These waves are excited by the tem-
perature anisotropyof electrons. Electrons are
thermalized in the direction parallel to the
shock magnetic field by these waves, but ions
are not.
(v) Strong parallel heating of electrons at the
shock overshoot results in the suitable condi-
tion for excitation of ion acoustic waves. The
rippled structure might be an energy source
of the ion acoustic waves. However, their de-
tailed excitation mechanism is not yet clear.
The ion acoustic waves are responsible for
parallel diffusion of both electrons and ions.
We expect that the ion acoustic waves have a
direct implication with the transition from the
self-reformation phase to the turbulent phase.
(vi) A cross-scale coupling between an ion-scale
mesoscopic instability and an electron-scale
microscopic instability is important for struc-
tures and dynamics of collisionless perpendic-
ular shocks. Hence large-scale full kinetic sim-
ulations are quite important.
It is noted that Yuan et al. (2009) have demon-
strated the cyclic self-reformation in their 2D hy-
brid simulation of a quasi-perpendicular shock with
θ = 85◦, which is different from the results in purely
perpendicular shocks. We are now trying to check
whether the reformation is suppressed or not at a
oblique shock by a large-scale 2D PIC simulation.
However, this is beyond the scope of the present pa-
per.
Finally, the effect of mass ratio is discussed.
In the present simulation parameter (MA
5,ωpe1/ωce1 = 10,β = 0.125), electron cyclotron
(Bernstein) modes is weakly unstable due to cur-
rent driven instability with a reduced mass ratio
(e.g., Muschietti and Lembege, 2006). When we use
the real mass ratio of mi/me= 1836, the thermal
velocity of upstream electrons becomes about 8.6
times as large as the case of mi/me= 25, and the
electron cyclotron modes are stabilized. In contrast
with current driven instability, obliquely propagat-
ing whistler mode waves becomes unstable due to
modified two-stream instability (Matsukiyo and Sc-
holer, 2003, 2006). However, the evolution of modi-
fied two-stream instability at perpendicular shocks
has been studied only in a localized uniform model
(Matsukiyo and Scholer, 2006). The influences of
modified two-stream instability on the reformation
process of perpendicular shocks is an outstanding
issue to be addressed by future 2D PIC simulations
of perpendicular collisionless shocks.
=
8

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Acknowledgements
The authors are grateful to S. Matsukiyo and Y.
Ohira for discussions. The computer simulations
were carried out on Fujitsu HPC2500 at ITC in
Nagoya Univ. and NEC SX-7 at YITP in Kyoto
Univ. as a collaborative computational research
project at STEL in Nagoya Univ. and YITP in
Kyoto Univ. This work was supported by Grant-
in-Aid for Scientific Research on Innovative Areas
No.21200050(T. U.), Grant-in-Aid for Scientific Re-
search on Priority Areas No.19047004 (R. Y.) and
Grant-in-Aid for Young Scientists (B) No.21740184
(R. Y.) from MEXT of Japan.
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