Large-scale two-dimensional (2-D) full particle-in-cell (PIC) simulations are
carried out for studying periodic self-reformation of a supercritical
collisionless perpendicular shock with an Alfvén–Mach number MA∼6.
Previous self-consistent one-dimensional (1-D) hybrid and full PIC
simulations have demonstrated that the periodic reflection of upstream ions
at the shock front is responsible for the formation and vanishing of the
shock-foot region on a timescale of the local ion cyclotron period, which was
defined as the reformation of (quasi-)perpendicular shocks.
The present 2-D full PIC simulations with different ion-to-electron mass
ratios show that the dynamics at the shock front is strongly modified by
large-amplitude ion-scale fluctuations at the shock overshoot, which are
known as ripples.
In the run with a small mass ratio, the simultaneous enhancement of the shock
magnetic field and the reflected ions take place quasi-periodically, which is
identified as the reformation. In the runs with large mass ratios, the
simultaneous enhancement of the shock magnetic field and the reflected ions
occur randomly in time, and the shock magnetic field is enhanced on a
timescale much shorter than the ion cyclotron period.
These results indicate a coupling between the shock-front ripples and
electromagnetic microinstabilities in the foot region in the runs with large
mass ratios.
Space plasma physics (wave–particle interactions)Introduction
It has been well known since one-dimensional (1-D) particle-in-cell (PIC)
simulations in the 1970s that the “shock front” at a supercritical
(quasi-)perpendicular collisionless shock shows a periodic behavior.
first reported the periodic reflection of ions and the
formation of a “shock foot” at a perpendicular shock, which leads to a
non-stationary shock front. Later, this process was described as a
periodically vanishing “leading edge of the shock”
. The term “periodic (self-)reformation” was
first used by and has been widely used after the finding
of the periodic reformation of quasi-parallel shocks
which is generated by a different process from the periodic reformation of
(quasi-)perpendicular shocks. In the past simulation studies, reduced
parameters were used due to the computational cost. However, it has been
confirmed that the reformation of (quasi-)perpendicular shocks is
common in 1-D simulations with various ion-to-electron mass ratios, including
the real one when the Alfvén–Mach number is supercritical but relatively
low (MA<10) and the ion beta is low (βi<0.4)
. The reformation disappears with high
Alfvén–Mach numbers (MA>100) . The existence of the
reformation in the two-dimensional (2-D) system was also confirmed by a full
PIC simulation , although a small simulation system was
used due to the computational cost. In 2-D simulations with a small
simulation system in the shock tangential direction, the development of shock
waves becomes quite similar to that in 1-D simulations. Here, we refer to
such a 2-D system with a small simulation system in the shock tangential
direction as a “quasi-1-D” system.
Recent development of computer technologies allows us to perform larger-scale
full PIC and higher-resolution hybrid PIC simulations in multiple dimensions
with a longer simulation time. These multidimensional simulation studies
sometimes modify previous understanding of the physics of collisionless
shocks. and reported that the
periodic reformation of an exactly perpendicular shock was “suppressed”
with an ion-to-electron mass ratio mi/me=42 where the
shock reformation was evident in an early phase but became less evident in a
later phase, while the reformation was “absent” with
mi/me=400. In their high-resolution 2-D hybrid PIC
simulation, the excitation of electron-scale whistler mode waves at the shock
front was included. In their large-scale 2-D full PIC simulation, the
formation of ion-scale and large-amplitude fluctuations at the shock
overshoot was included, which has been known as the “ripples” that were
first reported by 2-D hybrid PIC simulation studies
.
By contrast, reported that the periodic reformation was
“confirmed” in their 2-D hybrid simulation of a quasi-perpendicular shock
with a shock-normal angle of θBn=85∘. They also
demonstrated that the period of the reformation became longer in the 2-D
simulation than in the 1-D simulation. These results contradict each other,
which brings confusion to shock scientists. Although the simulation
parameters of these previous works are similar, as shown in Table 1, the
obtained results seem to be different from each other. Hence, the existence
of the reformation of (quasi-)perpendicular shocks is still under
debate, even in 2-D kinetic simulations.
In order to study the contradiction between the former
and latter results, a
direct comparison was made between the 2-D full PIC simulation results of
quasi-perpendicular (θBn=80∘) and exactly
perpendicular (θBn=90∘) shocks
. It was confirmed that the shock-normal angle does not
affect the presence or the suppression of the reformation of
(quasi-)perpendicular shocks. It was also shown that the reformation
seems to be suppressed when the shock magnetic field is analyzed by averaging
over the shock-tangential direction, as done by and
. On the other hand, the reformation is present, but its
period is modified by ripples when the shock magnetic field is analyzed at a
local point, as done by .
Simulation parameters used by different authors.
AuthorsCodeMAθBnβi1βe1mi/meωpe1/ωce1Run A (present)Full PIC6.1190∘0.320.32254Run B (present)Full PIC6.4490∘0.320.321004Run C (present)Full PIC6.4790∘0.320.322564Run D (present)Full PIC6.4190∘0.320.326254/Full PIC4.9390∘0.150.244002Full PIC4.9390∘0.150.24422Hybrid PIC5.685∘0.150.2––
Unlike (quasi-)1-D simulations, it is expected that the issue of the
presence, absence, and suppression of the shock reformation of rippled
(quasi-)perpendicular shocks will be related to the ion-to-electron
mass ratio in multidimensional simulations, since the reformation is
suppressed with mi/me=42 and is absent with
mi/me=400 in the previous study .
It is known that the mass ratio controls the types of microinstabilities in
the foot region .
By using the following two dimensionless parameters, i.e., Alfvén–Mach
number
MA=ux1cωpi1ωci1=ux1cωpe1ωce1mime,
and the ratio of the thermal plasma pressure to the magnetic pressure (plasma
beta),
βi=2vti12ωpi12c2ωci12,βe=2vte12ωpe12c2ωce12,
we obtain the ratio of the upstream bulk velocity ux1 to the thermal velocity vt1 as
ux1vti1=2MAβi,ux1vte1=2MAβememi.
Hereafter, the subscripts “1” and “2” denote
“upstream” and “downstream”, respectively.
This equation shows that the ratio of the upstream bulk velocity to the ion
thermal velocity depends only on the Alfvén–Mach number and the ion beta,
which becomes larger with a larger Alfvén–Mach number and a smaller ion
beta. On the other hand, the ratio of the upstream bulk velocity to the
electron thermal velocity depends also on the ion-to-electron mass ratio.
It is known that relative bulk velocities among incoming ions, reflected
ions, and electrons arise in the foot region due to the reflection of a part
of incoming ions at the shock front. The relative velocities become a free
energy source for various types of microinstabilities in the foot region
. The ratio of the relative drift
velocity between ions and electrons to the electron thermal velocity controls
the types of microinstabilities via electron thermal damping. Hence, the mass
ratio plays a role in the full PIC simulation of (quasi-)perpendicular
shocks.
We aim to study the effect of the mass ratio (i.e., microinstabilities in the
foot region) on the periodic self-reformation of perpendicular collisionless
shocks. We make a comparison between 2-D full PIC simulation results with
different mass ratios. In Sect. 2, the simulation setup and the detailed
parameters are presented. In Sect. 3, the periodic self-reformation of
(quasi-)perpendicular shocks is (re)defined in accordance with the
past studies. In Sect. 4, the identification of the reformation is made by
using the 2-D full PIC simulation results with different mass ratios based on
the definition. Section 5 gives the summary of the result and some comments
on the spacecraft in situ observation of the reformation in the geophysical
plasma.
Simulation setup
We use a standard 2-D electromagnetic full PIC code with several improvements
.
A supercritical Alfvén–Mach number (MA∼6), low beta
(βi1=βe1=0.32), and perpendicular
(θBn=90∘) collisionless shock as shown in
Table 1 is excited by using the “relaxation method”
e.g.,, with which
the simulation domain is taken in the shock-rest frame. The shocks excited
with these parameters exhibit the periodic reformation of perpendicular
shocks in (quasi-)1-D systems, as we show later. A reduced
plasma-to-cyclotron frequency ratio
ωpe1/ωce1=4 and a reduced ratio of the speed
of light to the electron thermal velocity c/vte1=10 are chosen
due to the computational cost, which does not play any role in hybrid PIC
simulations but which may modify linear dispersion relations of
microinstabilities due to reflected ions in full PIC simulations.
The simulation domain is taken in the x-y plane with periodic boundaries in
the y direction and absorbing boundaries in the x
direction. An in-plane shock magnetic field is imposed in the y direction
(By0). The plasma flow is directed in the x direction. Hence, a
motional electric field is applied in the z direction.
A detailed initial setup for 2-D simulations of (quasi-)perpendicular
shocks was described in our previous studies
and is not repeated here.
In the present study, we perform four simulation runs, A, B, C, and D, with
different ion-to-electron mass ratios mi/me=25, 100,
256, and 625, respectively.
The grid spacing and the time step of the present simulation runs are common
(Δx=Δy≡Δ=λDe1 and cΔt/Δ=0.5, where λDe is the electron Debye length). For all of Runs
A–D, we perform two types of runs with different sizes of the simulation
domain of Lx×Ly=32di1×6di1 (labeled as “large”)
and 32di1×di1 (labeled as “small”), where di1=c/ωpi1 is the ion inertial length (di1=50λDe1,100λDe1,160λDe1, and 250λDe1 for Runs A, B, C,
and D, respectively). Note that the rippled structures at the shock front are
included in the “large” simulation runs but are excluded in the “small”
simulation runs.
We used 25 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.
The bulk flow velocity of the upstream plasma is
ux1/vte1=3.0, 1.5, 0.9375, and 0.6 for Runs A, B, C, and D,
respectively.
The previous studies showed that
the electron cyclotron drift instability (ECDI) is driven
with the parameters of Run A ,
which excites electrostatic waves at
multiple electron cyclotron harmonic frequencies
.
On the other hand,
the modified two-stream instability (MTSI) is driven
with the parameters of Runs B–D ,
which excites obliquely propagating electromagnetic whistler mode waves
at a frequency
between the electron cyclotron frequency and the lower hybrid resonance frequency
.
Definition of reformation
Development of a perpendicular shock for Run A with the small
simulation domain. Tangential component of the shock magnetic field By at
y/di1=0.5 as a function of position x and time
t(a), and the corresponding ion density (b) and density
of reflected ions (c). The position and time are renormalized by the
ion inertial length di1=c/ωpi1 and the ion
cyclotron angular period 1/ωci1, respectively. The
magnitudes of the magnetic field and the density are normalized by the
initial upstream magnetic field B01 and the initial upstream density
n1. The positions of the shock foot and overshoot used in Figs. 3 and 4
are indicated by the dashed lines.
The left panel of Fig. 1 shows the tangential component of the shock
magnetic field By at y/di1=0.5 for Run A with the small simulation
domain as a function of position x and time t. The position and time are
renormalized by the ion inertial length di1=c/ωpi1 and
the ion cyclotron angular period 1/ωci1, respectively. The
magnitude is normalized by the initial upstream magnetic field B01.
The perpendicular shock generated in the simulation is at rest.
The shock overshoot (at x/di1∼0) shows a periodic
enhancement on a timescale close to the downstream ion cyclotron period
(Tr∼5.9/ωci2=1.7/ωci1∼0.27Tci1). The shock-foot region
at x/di1∼-2) also shows the periodic vanishing, which
indicates the shock reformation of the perpendicular shock.
In the middle panel of Fig. 1, we show the ion density Ni at
y/di1=0.5 as a function of position x and time t in the
same run. The magnitude is normalized by the initial upstream density
n1. We see the periodic behaviors of the ion density both at the
overshoot and in the foot region. In the right panel of Fig. 1, we show the
density of the reflected ion component only. Here, we obtain the density of
the reflected ion by integrating the x-y-vx phase-space (reduced)
distribution function over the velocity as
Nr(t,x,y)≡∫-∞ux2f(t,x,y,vx)dvx,
where ux2 is the downstream bulk velocity. We see the periodic behavior
of the ion density in the foot region (at x/di1∼-2).
Note that we also see these periodic behaviors in the other simulation runs
(B–D) with the small simulation domain, but this is not shown here.
Development of a perpendicular shock at y/di1=3
for Run A with the large simulation domain
with the same format as Fig. 1.
In Fig. 2, we show the tangential component of the shock magnetic field
By, the ion density Ni, and the density of reflected ions
Nr at y/di1=3 with the same format as Fig. 1 for Run A with the
large simulation domain.
The perpendicular shock generated in the simulation is at rest as in Fig. 1
(the other runs are as well, but these are not shown here). The results of
the large-scale simulation run look similar to those of the small-scale
simulation run until ωci1t∼4, but become different
after ωci1t∼4 because of the generation of ripples.
We see quasi-periodic behaviors of the shock magnetic field at the overshoot
(x/di1∼0) and in the foot region (x/di1∼-2).
Although the periodic behaviors of the densities are also seen at the
overshoot, their period seems not to correspond to the period of the shock
magnetic field. The periodic behaviors in the foot region are not so clear in
the ion density and in the density of reflected ions.
In the present simulations, the excited shocks are at rest, as seen in
Figs. 1 and 2. Hence, the development of shock structures (i.e., overshoot
and foot) can be discussed by the temporal variation at fixed positions.
To identify the periodic behaviors more clearly, we first re-define the
position of the shock overshoot from the maximum of the shock magnetic field
averaged for the time interval of ωci1t=4–12. Then, we
estimate the typical gyroradius of reflected ions as
2(ux1-ux2)/(ωci1+ωci2), which is close
to 1.85di1. We define the typical position (i.e., center) of the foot
region as 1.5di1 away upstream from the overshoot by considering the
typical spatial size of the shock ramp and the phase-space trajectory of
reflected ions.
Figure 3 shows the temporal variation of the shock magnetic field By and
the ion density Ni for Run A at the overshoot and in the foot
region in the top and bottom panels, respectively. The result of the
small-scale simulation run is plotted in the left panels, while the result of
the large-scale simulation run is plotted in the right panels. The density of
reflected ions is also plotted in the panels of the foot region.
Temporal variation of the shock magnetic field By and the ion
density Ni at the overshoot and in the foot region for Run A. The
left panels show the result of the small-scale simulation run (at
y/di1=0). The right panels show the result of the large-scale
simulation run (at y/di1=3). The shock overshoot is defined as
a position where the shock magnetic field averaged for the time interval of
ωci1t=4–12 is maximum. The shock foot is defined as a
position 1.5di1 away upstream from the overshoot.
In the small-scale simulation run (left panels), periodic oscillations in
both By and Ni are clearly seen at the overshoot and in the
foot region. There is a positive correlation between By and Ni
both at the overshoot and in the foot region. It is also seen that the shock
magnetic field in the foot region increases when the shock magnetic field at
the overshoot decreases (and vice versa). The result also shows that the
shock magnetic field in the foot region is enhanced when the density of
reflected ions is enhanced, which is identified as the reformation of
(quasi-)perpendicular shocks.
In the large-scale simulation run (right panels), on the other hand, there is
no relationship between the shock magnetic field at the overshoot and in the
foot region. The correlation between By and Ni is positive in
the foot region but not so clear at the overshoot. The result also shows that
the shock magnetic field in the foot region looks enhanced when the density
of reflected ions is enhanced.
Some previous 2-D simulation studies discussed the reformation of
(quasi-)perpendicular shocks based on the temporal variation
(periodicity) of the shock magnetic field at the overshoot. It is possible to
identify the reformation from the shock magnetic field at the overshoot in
1-D simulations and in small-scale 2-D simulations (without ripples), since
there is an inverse relationship between the shock magnetic field in the foot
region and at the overshoot. On the other hand, the present result has
clearly shown that it is not appropriate to identify the shock reformation of
rippled (quasi-)perpendicular shocks only from the shock magnetic
field at the overshoot of rippled shocks.
Hence, we discuss the shock reformation in terms of the quasi-periodic
formation of the foot region, which is consistent with the original
definition of the reformation of (quasi-)perpendicular shocks. We
define the formation of the foot region by the simultaneous enhancement in
the density of the reflected ions and the shock magnetic field.
Identification of reformation in large-scale 2-D simulations
As shown in the previous section, the perpendicular shocks excited in the 2-D
full PIC simulations are at rest. Hence, we analyze the temporal variation of
the “foot region” at a fixed position, which was defined in the previous
section as 1.5di1 away upstream from the overshoot where the
time-averaged By is maximum. As seen in Fig. 3, the typical
characteristics of the shock reformation, i.e., the periodic oscillation of
By and Nr, is well identified at this position in the run with
the small simulation domain.
We detect the formation of the foot, i.e., the simultaneous enhancement of
the reflected ion density and the shock magnetic field, by the following
equation:
If=NrBy-〈By〉.
Here, 〈By〉 represents the averaged shock magnetic field for
the time interval of ωci1t=4–12. This equation
indicates the foot formation by taking a finite and positive value when a
certain number of reflected ions exist in the foot region in association with
the enhancement of the shock magnetic field by exceeding its mean value.
Figure 4 shows the temporal deviation of the shock magnetic field By from
its mean value (〈By〉) together with the density of reflected
ions Nr (defined in Eq. 4) in the foot region for Runs A–D with
the large simulation domain. The circles show the indicator of the foot
formation defined by Eq. (5).
In order to take account of the spatial variation of physical quantities in
the shock tangential (y) direction due to ripples, we plot the temporal
variation of the foot region at the following eight points: y/di1=0,0.75,1.5,2.25,3,3.75,4.5, and 5.25.
Temporal variation of the shock magnetic field By-〈By〉 (solid lines), and density of reflected ions Nr
(dash–dotted lines) in the foot region for Runs A–D with the large
simulation domain. Here, 〈By〉 represents the averaged shock
magnetic field for the time interval of ωci1t=4–12. The
indicator of the foot formation defined in Eq. (5) is shown by circles.
In Run A (mi/me=25), we see the simultaneous
enhancement of the shock magnetic field By and the reflected ions
Nr quasi-periodically, although the periodicity is not as clear
as in the run with a small simulation domain. It is shown that the
enhancement of reflected ions is not necessarily accompanied by the
enhancement of the shock magnetic field. The time period of the simultaneous
enhancement varies in a range of ωci1Tr∼0.5–2.2, but its typical value is Tr∼2.0/ωci1∼0.32Tci1, which is longer than the
period of the reformation in the run with the small simulation domain as
demonstrated by the previous studies .
In Run B (mi/me=100), we also see the simultaneous
enhancement of the shock magnetic field By and the reflected ions
Nr, but its periodicity is not as clear as in Run A. Some of the
enhancement of the shock magnetic field is not related to reflected ions.
In Runs C (mi/me=256) and D
(mi/me=625), the simultaneous enhancement of the shock
magnetic field By and the reflected ions Nr occurs almost randomly.
Some of the enhancement of the shock magnetic field occurs in a timescale
much shorter than the local ion cyclotron period, indicating the existence of
electron-scale electromagnetic waves.
Discussions and summaryDiscussions on the simulation results
Two-dimensional full PIC simulations of rippled
perpendicular collisionless shocks with MA∼6 and
βi=βe=0.32 were performed with different
ion-to-electron mass ratios. The periodic self-reformation of
(quasi-)perpendicular shocks was detected from the simultaneous
enhancement of the shock magnetic field and the reflected ions in the foot
region.
In the runs with the small simulation domain where the shock-front ripples
are absent, the dynamics at the shock front was independent of the mass ratio
and the periodic formation and vanishing of the foot region was clearly
detected at the time period of Tr∼1.7/ωci1∼0.27Tci1. The periodic oscillation of the shock overshoot was also seen at
the same time period.
In the runs with the large simulation domain, the shock-front ripples are
present. In the run with mi/me=25, there was no
correlation between the temporal variation of the shock magnetic field in the
foot region and the overshoot, suggesting that it was not appropriate to
detect the reformation of (quasi-)perpendicular shocks from the
quasi-periodic oscillation of the shock magnetic field at the overshoot as
done by some previous studies. The quasi-periodic formation of the foot
region was detected at the typical time period of Tr∼2.0/ωci1∼0.32Tci1.
In the runs with mi/me=100,256, and 625 and the
large simulation domain, the simultaneous enhancement of the shock magnetic
field and the reflected ions occurred randomly in time at the front of the
shock ramp. The enhancement of reflection ions did not necessarily correspond
to the enhancement of the magnetic field that occurred at a timescale shorter
than the local ion cyclotron period. These results indicate the absence of
the periodic reformation of a rippled perpendicular shock.
The previous studies showed that the ion-to-electron mass ratio controlled
the types of microinstabilities in the foot region
. The MTSI was dominant in the foot region
in the runs with mi/me=100,256, and 625. The excitation of the whistler mode in the
foot region was also indicated by the enhancement of the shock magnetic field
at the timescale much shorter than the local ion cyclotron period. It was
also shown that the wavelength in the shock tangential direction of the
whistler mode excited by the MTSI is close to the wavelength of the ripples
. It is suggested that wave–wave coupling between the
whistler mode and the ripples is possible, which disturbs the periodic
reformation of (quasi-)perpendicular shocks.
The ECDI was driven in the run with mi/me=25. The electron cyclotron harmonic
mode does not have magnetic fluctuations, and does not interact with the
shock-front ripples. It is suggested that the quasi-periodic formation of the
foot region is present when there is no microinstability in the foot region
or when electrostatic instabilities such as ECDI and upper-hybrid drift
instability are driven as seen in the previous 2-D simulations
, but its time period is modified by
the shock-front ripples from the typical time period in 1-D simulations.
The present study does not deny the existence of the reformation of
(quasi-)perpendicular shocks in the real space plasma, although the
reformation is absent in the simulation runs of rippled perpendicular shocks
with larger mass ratios. It is suggested that the existence of the periodic
reformation of (quasi-)perpendicular shocks depends on the physical
parameters indicated by Eq. (3). The reformation of
(quasi-)perpendicular shocks could exist by suppressing the MTSI in
the foot region by other instabilities as in Run A.
Since the present study used the reduced frequency ratio
ωpe1/ωce1=4 (and the reduced speed of light
c/vte1=10), it is worth discussing the effect of the frequency
ratio to microinstabilities at the shock foot. The frequency ratio
ωp/ωc does not affect the structure of shock
on ion scales. The effect of the frequency ratio
ωp/ωc to the velocity distribution functions
at the shock foot is not so large as long as the Alfvén–Mach number and
the plasma beta are the same, as shown by the previous study
.
Let us suppose that velocity distribution functions of electrons and ions at
the shock foot are independent of the frequency ratio as indicated from
Eq. (3). Then, we can use the parameters for the velocity distribution
functions of a three-component plasma at the shock foot
(Vdi1=0.42,Vti1=0.06,ωpi1=0.052,Vdi2=-0.32,Vti2=0.05,ωpi2=0.047,Vde=0.08,Vte=1.94,ωpe=1.75,ωce=0.92), which were obtained from the run with
mi/me=625 (Run A of Table II) in the previous study
. We change ωpe1/ωce1 as
4, 20, and 100, and solve the linear dispersion relation. Note that
(vte1ωpe1)/(cωce1) is constant due
to Eq. (2).
Figure 5 shows the dispersion relation of the MTSI. The wave-normal angle
relative to the ambient magnetic field is 85∘ for kx>0 and
84∘ for kx<0. The growth rate of the MTSI for
ωpe1/ωce1=4 is slightly smaller than that
for ωpe1/ωce1=20 and 100 since the
electron thermal velocity is close to the speed of light. The linear
dispersion relations for ωpe1/ωce1=20 and
100 are almost the same due to c/vte1≫1. The present linear
analysis suggests that the effect of the frequency ratio
ωp/ωc to the linear dispersion relation of
the MTSI is small.
The ECDI does not have a positive growth rate for all of the frequency
ratios, suggesting that the frequency ratio
ωp/ωc does not affect the present simulation
results with MA∼6 and βi=βe=0.32.
It should be noted that the ECDI is dominant in a lower-beta plasma and that
the frequency ratio affects the growth rate of the ECDI
. The ECDI can disturb the generation of the MTSI
through nonlinear saturation as seen in Run A. Further self-consistent full
PIC simulations are needed to study the influence of the frequency ratio on
the competition between the ECDI and the MTSI. However, the size of the
computational domain is proportional to
(ωp/ωc)2 and the time step is proportional
to ωp/ωc in 2-D full PIC simulation. It is a
heavy task to use a large ωp/ωc in
large-scale full PIC simulations. Equation (3) also shows that a larger Mach
number or a lower beta gives a large ux1/vte1 which is
necessary for driving the ECDI with a larger mass ratio.
Linear dispersion relations for a three-component plasma based on
the velocity distribution functions in the foot region of a perpendicular
collisionless shock (Vdi1=0.42,Vti1=0.06,ωpi1=0.052,Vdi2=-0.32,Vti2=0.05,ωpi2=0.047,Vde=0.08,Vte=1.94,ωpe=1.75,ωce=0.92) in Run A of Table II in
the previous study . The frequency ratio
ωpe1/ωce1 is changed as 4 (dotted lines),
20 (dashed lines), and 100 (solid lines) by keeping
vte1ωpe1/cωce1 constant. The
wave-normal angle relative to the ambient magnetic field is 85∘ for
kx>0 and 84∘ for kx<0.
Discussions on the comparison with observations
Since the present simulation study used reduced parameters
(mi/me, ωpe1/ωce1, and
c/vte1), it is not easy to compare the present study with the
spacecraft in situ observations directly.
However, it is worth discussing the in situ
observations of the shock reformation in the geophysical plasma by
extrapolating the present simulation results. There are two references which
explicitly tried to detect the reformation of (quasi-)perpendicular
shocks.
made a statistical analysis of quasi-perpendicular
shocks with parameters ranging for 2≤MA≤6.5, 75∘≤θBn≤90∘, and βi≤0.6
observed by the Cluster spacecraft. As evidence of the reformation of
(quasi-)perpendicular shocks, they showed that the spatial size of the
shock ramp is less than the ion inertial length and has a Gaussian-like
distribution with a standard deviation of 10 and a few electron inertial
lengths.
Figure 6 shows the histogram of the spatial size of the shock ramp for 8
points in position (y/di1=0:0.75:5.25)×251 points in time
(ωci1t=4:0.032:12) in the present simulation run with
mi/me=625 and the large simulation domain where the
periodic reformation of perpendicular shocks is not seen. Here, the spatial
size of the shock ramp is approximated by the gradient of the shock magnetic
field normalized by By,max-By01.
The spatial size of the shock ramp is distributed between
0.04di1(=de1) and 1.1di1, and its typical
size is 0.16di1. The characteristics of the histogram are not the
same as Fig. 6 of , since the result includes the spatial
size of the shock ramp for only an exactly perpendicular shock with a
specific Mach number and plasma beta. However, it is clearly shown that the
spatial size of the shock ramp has a distribution without the periodic
reformation of perpendicular shocks. Hence, the result of
showed the non-stationarity of the quasi-perpendicular
shock, but did not necessarily indicate reformation.
Histogram of the spatial size of the shock ramp for 8 points in
position (y/di1=0:0.75:5.25)×251 points in time
(ωci1t=4:0.032:12) in the run with
mi/me=625 and the large simulation domain. The spatial
size of the shock ramp is approximated by the gradient of the shock magnetic
field normalized by By,max-By01.
reported a quasi-periodic reflection of ions at the front
of a quasi-perpendicular shock with MA=10, θBn=81∘, βi=2.0, and βe=1.7 by the
Cluster spacecraft. The time period for the formation of reflected ion bursts
is ∼8 s that corresponds to 0.5Tci1, which is much longer
than the period of the reformation seen in the present full PIC simulations.
The result seemed to be consistent with the original definition of the
reformation of (quasi-)perpendicular shocks, although the simultaneous
enhancement of the shock magnetic field in the foot region was not shown.
However, the time period of ∼8 s is close to the time resolution of
the instrument (4 s). Also, it is known from the hybrid PIC simulation
results that the shock front at a (quasi-)perpendicular shock with a
high ion beta (βi>1) is stationary
e.g.,. Therefore, it is unclear whether
the periodic behavior identified by really exhibits the
“reformation of quasi-perpendicular shocks” in its original definition.
It is possible that the periodic behavior at the shock front of quasi-perpendicular shocks
is generated by whistler mode waves ,
which is similar to the reformation of quasi-parallel shocks.
This process is, however, outside the scope of the present simulation study.
Further PIC simulation studies with parameters
similar to this event (higher Mach number and higher ion beta) are necessary.
Summary
The present 2-D full PIC simulation study reproduced the results of the
previous 2-D simulations by
using different ion-to-electron mass ratios. It is suggested that the
periodic reformation of (quasi-)perpendicular shocks is not common,
unlike 1-D simulations. The reformation in its original definition exists
with a narrower range of the Alfvén–Mach number and the plasma beta as
well as the electron plasma-to-cyclotron frequency ratio in 2-D than in 1-D.
It is not so easy to identify the reformation in multidimensional simulations
even with a simplified model. It might be quite difficult to identify it from
limited data obtained by in situ observation.
Access to the raw data of the present numerical simulations
may be provided upon reasonable request to one of the authors (Takayuki
Umeda, taka.umeda@nagoya-u.jp).
TU developed the simulation codes, drafted the manuscript, and
approved the final manuscript. YD contributed to the analysis of the
simulation data.
The authors declare that they have no conflict of
interest.
Acknowledgements
One of the authors (Takayuki Umeda) is grateful to Yoshitaka Kidani and
Shuichi Matsukiyo for discussions. This work was supported by MEXT/JSPS under
Grant-In-Aid (KAKENHI) for Scientific Research (B) no. JP26287041. The
computer simulations were performed on the supercomputer systems at the
Institute for Space-Earth Environmental Research and the Solar-Terrestrial
Environment Laboratory in Nagoya University through a joint research
program. The topical editor, Minna
Palmroth, thanks Yann Pfau-Kempf and two anonymous referees for help in
evaluating this paper.
References
Biskamp, D. and Welter, H.:
Numerical studies of magnetosonic collisionless shock waves,
Nucl. Fusion, 12, 663–666, 1972.
Burgess, D.:
Cyclic behavior at quasi-parallel collisionless shocks,
Geophys. Res. Lett., 16, 345–348, 1989.Hellinger, P., Travnicek, P. M., Lembege, B., and Savoini, P.:
Emission of nonlinear whistler waves at the front of perpendicular supercritical shocks:
hybrid versus particle simulations,
Geophys. Res. Lett., 34, L14109, 10.1029/2007GL030239,
2007.Hellinger, P., Travnicek, P., and Matsumoto, H.:
Reformation of perpendicular shocks: Hybrid simulations,
Geophys. Res. Lett., 29, 2234, 10.1029/2002GL015915,
2002.
Lembege, B. and Dawson, J. M.:
Self-consistent study of a perpendicular collisionless and nonresistive shock,
Phys. Fluids, 30, 1767–1788, 1987.
Lembege, B. and Savoini, P.:
Non-stationarity of a two-dimensional
quasi-perpendicular supercritical collisionless shock
by self-reformation,
Phys. Fluids B, 4, 3533–3548, 1992.Lembege, B., Savoini, P., Hellinger, P., and Travnicek, P. M.:
Nonstationarity of a two-dimensional perpendicular shock:
Competing mechanism,
J. Geophys. Res., 114, A03217, 10.1029/2008JA013618,
2009.
Leroy, M. M., Goodrich, C. C., Winske, D., Wu, C. S., and Papadopoulos, K.:
Simulation of a perpendicular bow shock,
Geophys. Res. Lett., 8, 1269–1272, 1981.
Leroy, M. M., Winske, D., Goodrich, C. C., Wu, C. S., and Papadopoulos, K.:
The structure of perpendicular bow shocks,
J. Geophys. Res., 87, 5081–5094, 1982.Lowe, R. E. and Burgess, D.: The properties and causes of rippling in
quasi-perpendicular collisionless shock fronts, Ann. Geophys., 21, 671–679,
10.5194/angeo-21-671-2003, 2003.Lobzin, V. V., Krasnoselskikh, V. V., Bosqued, J.-M.,
Pincon, J.-L., Schwartz, S. J., and Dunlop, M.:
Nonstationarity and reformation of high-Mach-number quasiperpendicular shocks:
Cluster observations,
Geophys. Res. Lett., 34, L05107, 10.1029/2006GL029095,
2007.Matsukiyo, S. and Scholer, M.:
Modified two-stream instability in the foot of high Mach number
quasi-perpendicular shocks,
J. Geophys. Res., 108, 1459, 10.1029/2003JA010080,
2003.Matsukiyo, S. and Scholer, M.:
On microinstabilities in the foot of high Mach number perpendicular shocks,
J. Geophys. Res., 111, A06104, 10.1029/2005JA011409,
2006.Mazelle, C., Lembege, B., Morgenthaler, A., Meziane, A., Horbury, T. S., Genot, V.,
Lucek, E. A., and Dandouras, I.:
Self-reformation of the quasi-perpendicular shock: CLUSTER observations,
in Twelfth International Solar Wind Conference, AIP Conf. Proc. Vol. 1216,
471–474, 10.1063/1.3395905,
2010.
Muschietti, L. and Lembege, B.:
Electron cyclotron microinstability in the foot of a perpendicular shock:
A self-consistent PIC simulation,
Adv. Space Res., 37, 483–493, 2006.
Muschietti, L. and Lembege, B.:
Microturbulence in the electron cyclotron frequency range at
perpendicular supercritical shocks,
J. Geophys. Res., 118, 2267–2285, 2013.Scholer, M. and Burgess, D.:
Whistler waves, core ion heating, and nonstationarity in oblique
collisionless shocks,
Phys. Plasmas, 14, 072103, 10.1063/1.2748391,
2007.Scholer, M. and Matsukiyo, S.: Nonstationarity of quasi-perpendicular shocks:
a comparison of full particle simulations with different ion to electron mass
ratio, Ann. Geophys., 22, 2345–2353,
10.5194/angeo-22-2345-2004, 2004.Scholer, M., Shinohara, I., and Matsukiyo, S.:
Quasi-perpendicular shocks:
length scale of the cross-shock potential,
shock reformation, and implication for shock surfing.
J. Geophys. Res., 108, 1014, 10.1029/2002JA009515,
2003.Shimada, N., Hoshino, M., and Amano, T.: Structure of a strong supernova
shock wave and rapid electron acceleration confined in its transition region,
Phys. Plasmas, 17, 032902, 10.1063/1.3322828,
2010.Sokolov, I. V.:
Alternating-order interpolation in a charge-conserving scheme for particle-in-cell simulations,
Comput. Phys. Commun. 184, 320–328, 2013.
Tokar, R. L., Aldrich, C. H., Forslund, D. W., and Quest, K. B.:
Nonadiabatic electron heating at high-Mach-number perpendicular shocks,
Phys. Rev. Lett., 56, 1059–1062, 1986.Umeda, T., Kidani, Y., Matsukiyo, S., and Yamazaki, R.:
Modified two-stream instability at perpendicular shocks:
Full particle simulations,
J. Geophys. Res., 117, A03206, 10.1029/2011JA017182,
2012a.Umeda, T., Kidani, Y., Matsukiyo, S., and Yamazaki, R.:
Microinstabilities at perpendicular collisionless shocks:
A comparison of full particle simulations with different ion to electron mass ratio,
Phys. Plasmas, 19, 042109, 10.1063/1.3703319, 2012b.Umeda, T., Kidani, Y., Matsukiyo, S., and Yamazaki, R.:
Dynamics and microinstabilities at perpendicular collisionless shock: A
comparison of large-scale two-dimensional full particle simulations with
different ion to electron mass ratio, Phys. Plasmas, 21, 022102,
10.1063/1.4863836, 2014.Umeda, T., Kidani, Y., Yamao, M., Matsukiyo, S., and Yamazaki, R.:
On the reformation at quasi- and exactly perpendicular shocks:
Full particle-in-cell simulations,
J. Geophys. Res. 115, A10250, 10.1029/2010JA015458,
2010.
Umeda, T., Omura, Y., and Matsumoto, H.:
An improved masking method for absorbing boundaries
in electromagnetic particle simulations,
Comput. Phys. Commun., 137, 286–299, 2001.
Umeda, T., Omura, Y., Tominaga, T., and Matsumoto, H.:
A new charge conservation method
for electromagnetic particle simulations,
Comput. Phys. Commun., 156, 73–85, 2003.
Umeda, T., Yamao, M., and Yamazaki, R.:
Two-dimensional full particle simulation of
a perpendicular collisionless shock with a shock-rest-frame model,
Astrophys. J., 681, L85–L88, 2008.
Umeda, T., Yamao, M., and Yamazaki, R.:
Electron acceleration at a low-Mach-number
perpendicular collisionless shock,
Astrophys. J., 695, 574–579, 2009.
Umeda, T., Yamao, M., and Yamazaki, R.:
Cross-scale coupling at a perpendicular collisionless shock,
Planet. Space Sci., 59, 449–455, 2011.
Umeda, T. and Yamazaki, R.:
Particle simulation of a perpendicular collisionless shock:
A shock-rest-frame model,
Earth Planets Space, 58, e41–e44, 2006.Yuan, X., Cairns, I. H., Trichtchenko, L., Rankin, R., and Danskin, D. W.:
Confirmation of quasi-perpendicular shock reformation
in two-dimensional hybrid simulations,
Geophys. Res. Lett., 36, L05103, 10.1029/2008GL036675,
2009.
Winske, D. and Quest, K. B.:
Magnetic-field and density-fluctuations
at perpendicular supercritical collisionless shocks,
J. Geophys. Res., 93, 9681–9693, 1988.