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

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.

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.

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.

By contrast,

In order to study the contradiction between the former

Simulation parameters used by different authors.

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

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

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

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.

We use a standard 2-D electromagnetic full PIC code with several improvements

A supercritical Alfvén–Mach number (

The simulation domain is taken in the

A detailed initial setup for 2-D simulations of (quasi-)perpendicular
shocks was described in our previous studies

In the present study, we perform four simulation runs, A, B, C, and D, with
different ion-to-electron mass ratios

The grid spacing and the time step of the present simulation runs are common
(

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

Development of a perpendicular shock for Run A with the small
simulation domain. Tangential component of the shock magnetic field

The left panel of Fig. 1 shows the tangential component of the shock
magnetic field

The perpendicular shock generated in the simulation is at rest.

The shock overshoot (at

In the middle panel of Fig. 1, we show the ion density

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

In Fig. 2, we show the tangential component of the shock magnetic field

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

We see quasi-periodic behaviors of the shock magnetic field at the overshoot
(

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

Figure 3 shows the temporal variation of the shock magnetic field

Temporal variation of the shock magnetic field

In the small-scale simulation run (left panels), periodic oscillations in
both

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

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.

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

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:

Figure 4 shows the temporal deviation of the shock magnetic field

In order to take account of the spatial variation of physical quantities in
the shock tangential (

Temporal variation of the shock magnetic field

In Run A (

In Run B (

In Runs C (

Two-dimensional full PIC simulations of rippled
perpendicular collisionless shocks with

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

In the runs with the large simulation domain, the shock-front ripples are
present. In the run with

In the runs with

The previous studies showed that the ion-to-electron mass ratio controlled
the types of microinstabilities in the foot region

The ECDI was driven in the run with

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

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
(

Figure 5 shows the dispersion relation of the MTSI. The wave-normal angle
relative to the ambient magnetic field is

The ECDI does not have a positive growth rate for all of the frequency
ratios, suggesting that the frequency ratio

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

Linear dispersion relations for a three-component plasma based on
the velocity distribution functions in the foot region of a perpendicular
collisionless shock (

Since the present simulation study used reduced parameters
(

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.

Figure 6 shows the histogram of the spatial size of the shock ramp for 8
points in position

The spatial size of the shock ramp is distributed between

Histogram of the spatial size of the shock ramp for 8 points in
position

It is possible that the periodic behavior at the shock front of quasi-perpendicular shocks
is generated by whistler mode waves

The present 2-D full PIC simulation study reproduced the results of the
previous 2-D simulations

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.

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.