Particle simulations of re-formation at collisionless perpendicular shocks: Coherent behavior of reflected ions



[1] The properties of reformation in perpendicular collisionless shocks are investigated by means of a one-dimensional particle-in-cell simulation. Reformation is associated with ion reflection at the shock ramp and subsequent ion gyromotion in the upstream region. However, if ions are reflected continuously at the ramp, why does the shock reform intermittently at sufficiently high Mach number? The simulations show that the shock potential changes dramatically through the re-formation cycle, so that it is this potential variation which leads to the intermittent response of the shock.

1. Introduction

[2] A cyclic change of the shock front structure in collisionless plasmas is known as shock re-formation. Both full particle simulations [Biskamp and Welter, 1972; Tokar et al., 1986; Lembege and Savoini, 1992; Krasnoselskikh et al., 2002] and hybrid simulations [Quest, 1985, 1986] have shown cyclic changes in the ramp structure of perpendicular and quasiperpendicular shocks at sufficiently high Mach number. The shock reformation time in these simulations typically scales as the ion gyroperiod, and ion diagnostics demonstrates that reformation is associated with ion reflection at the shock ramp and subsequent ion gyromotion in the upstream region.

[3] The hypothesis that the reflection and the gyromotion of ions causes energy dissipation at high Mach number perpendicular shocks is now widely accepted [Phillips and Robson, 1972; Paschmann et al., 1982], although re-formation has not been confirmed yet by spacecraft observations (e.g., at Earth's bow shock).

[4] However, an essential question still remains to be settled: Why does the shock structure become intermittent and the behavior of reflected ions become coherent, even though plasma particles are continuously injected to the shock ramp from upstream? Little attention has been given to this point.

[5] If the injected particles are continuously reflected with the same rate at the shock front, and if the upstream gyroperiods of the reflected particles are all the same, the gyrophases of the magnetized motion of reflected particles should gradually become random as time elapses. In short, we expect to see that the ion velocity distribution becomes stationary.

[6] This expectation makes it difficult for us to understand the coherent ion behavior seen in the shock re-formation. Here we use the term “coherent ion behavior” to refer to the periodic change of the ion velocity distribution.

[7] A fully random distribution of the ion gyrophases appears to be inconsistent with the macroscopic periodic behavior of the shock. To answer this question, we implement full particle simulations for a long time enough to confirm the periodic behavior of the shock structure. In this report we discuss the detailed mechanism of the shock re-formation of perpendicular shocks in collisionless plasmas and subsequent ion gyromotion in the upstream region.

[8] In our results the most important point to note is the relationship between a periodic change of shock potential and the coherent behavior of reflected ions. No studies have ever discussed this in detail.

[9] Although the simulations of Krasnoselskikh et al. [2002] also show ion reflection, these authors suggest that the transition to a nonstationary quasiperpendicular shock takes place at a sufficiently high Mach number so that the nonlinear whistler wave that represents the shock front can no longer stand in place. Lembege and Savoini [1992] point out a periodic change of the electric field during shock re-formation. However, they make no reference to the coherent ion behavior.

2. Simulation Model

[10] The equations solved in our code are the equations of motion of both ions and electrons and Maxwell's equations. We use a one-dimensional explicit simulation scheme based on the particle-in-cell method for time advancing of plasma particles [Birdsall and Langdon, 1991]. The velocity of a particle is assigned to spatial grids to obtain the spatial distribution of current density. We adopt a quadratic spline as the weighting function for this assignment [Birdsall and Langdon, 1991]. This choice of the weighting function is desirable to reduce numerical noises in particle simulations. By solving Poisson's equation every 500 steps, we correct the electric field during the calculations. The loaded number of superparticles is 2 × 106 for each species (ions and electrons).

[11] The simulation domain Lx is divided into 8192 grids, and the grid separation is uniform, Δx = 2λe, where λe is the electron Debye length defined by the upstream density. A hard wall with perfect conductivity is set at the right-hand boundary, where particles are perfectly reflected. The left-hand boundary is free, where particles with constant flux are injected into the simulation domain. When particles moving in the positive x direction reach the right-hand boundary, they initially reflect at the wall, and the shock is launched ahead of the wall. This typical method to excite shocks in simulation studies is called the “reflecting wall method” [Biskamp and Welter, 1972; Quest, 1985].

[12] We assume that the velocity distributions of both incident ions and electrons are shifted Maxwellians with spatially constant temperature and that the drift speed equals the incident speed of plasma particles V, which is in the positive x direction.

[13] The typical simulation parameters are as follows: The mass ratio is mi/me = 100, the charge ratio is qi/qe = −1, the temperature ratio is Ti/Te = 1, the frequency ratio is ωpee = 4, and the electron plasma beta is βe = 0.0032. The time step for integrating the fundamental equations is Δtωpe = 0.016. Under these parameters the electron thermal speed vTe/c = 0.01, ion-acoustic speed cA/c = 1.0 × 10−3, and the Alfvén speed vA/c = 2.5 × 10−2, where c is speed of light. The Alfvén Mach number of shocks are defined by both the Alfvén speed and the drift speed of incident particles (MA = V/vA).

[14] The upstream electric and magnetic field have components E0 = (0, 0, E0) and B0 = (0, B0, 0), respectively. The speed of incident particles is then V = E0/B0 > 0. Since this simulation is one-dimensional, the shock normal is inevitably in the x direction. Thus the shock normal and the external magnetic field are mutually perpendicular.

3. Two Phases of Shock Re-formation

[15] First, we show a typical shock structure obtained in a calculation with MA = 4. Figure 1 displays the stacked profiles of the magnetic field −By. At tωpe ≃ 1200 we recognize a fully formed shock at x/(cpe) ≃ 110, where the left [x/(cpe) < 110] and right [x/(cpe) > 110] sides of the shock correspond to upstream and downstream, respectively. The position of the shock changes very little [x/(cpe) ≃ 110] during tωpe ≃ 1200–1600. During this period we can see a wavy structure propagating from the shock ramp toward upstream.

Figure 1.

Stacked profiles of the magnetic field −By from tωpe = 1000 to 2400 in the case of MA = 4. The spacing between profiles is Δtωpe = 16, and the color bar indicates the value of −By/B0, where B0 is the initial upstream magnetic field.

[16] This is the shock “foot” associated with ions reflected from the shock ramp. At tωpe ≃ 1600, a new shock begins to grow in the foot region of the old shock. The amplitude of the magnetic field at the new shock front grows as time elapses, and re-formation of the new shock ramp is completed at tωpe ≃ 1900. The development of the foot of the new shock begins after the new shock completely forms (tωpe > 1900).

[17] We define one period of re-formation as the interval between the time when the formation of the old shock is completed and the time when the formation of the new shock is completed. We estimate this period to be Trωpe ≃ 700 (i.e., TrΩi ≃ 1.75, where Ωi is the ion cyclotron frequency defined by the initial magnetic field) from Figure 1.

[18] Figure 2 shows the spatial profiles of the magnetic field −By (top panels), electron phase space (middle panels), and ion phase space (bottom panels) at three different times in the period of the shock re-formation. A dotted line in each top panel represents the downstream asymptotic magnetic field (−By/B0 = 3.19), which is predicted by the Rankine-Hugoniot conditions.

Figure 2.

Spatial profile of the (top panels) magnetic field, (middle panels) xvx phase space of electrons, and (bottom panels) ions at (a) tωpe = 2000, (b) 2240, and (c) 2400 in the case of MA = 4. A dotted line in each top panel represents the downstream magnetic field (−By/B0 = 3.19) predicted by the Rankine-Hugoniot conditions.

[19] We can divide the re-formation period into two different phases. The first phase is defined as the time during which the shock foot region broadens toward upstream. We call this the “broadening phase.” The shock structure at tωpe = 2000 and 2240 shown in Figures 2a and 2b corresponds to this broadening phase. At tωpe = 2000, the beam-like ions which have been reflected at the shock run in the negative x direction. Simultaneously, the electron distribution in the xvx space is disturbed [68 < x/(cpe) < 80] owing to the interaction between the incident electrons and the reflected ions [Shimada and Hoshino, 2000]. The width of the foot at tωpe = 2000 is Δx/(cpe) ≃ 12, from the shock front [x/(cpe) ≃ 80] to the head of the reflected ions [x/(cpe) ≃ 68].

[20] Then, the width of the foot becomes broader [45 < x/(cpe) < 78] at tωpe = 2240 than that at tωpe = 2000, as the distribution of the reflected ions is extended upstream. Note that the x component of the averaged speed of reflected ions at tωpe = 2240 is slower than that at tωpe = 2000. After that, the reflected ions begin to return to the shock because of their gyromotion, so that a beam-like component of reflected ions no longer exists.

[21] At tωpe ≃ 2240, the foot no longer broadens, and a new shock begins to grow in the foot of the old shock. This is the start of the “growing phase.” Figure 2c shows that the amplitude of the new shock ramp becomes comparable to that of the old shock ramp at tωpe = 2400. Then, the amplitude of the old shock ramp gradually decreases as time elapses, and finally, the old shock front disappears into the downstream of the new shock. Thus from Figure 1, the broadening phase covers 1200 < tωpe < 1600 and 1900 < tωpe < 2300, and the growing phase extends over 1000 < tωpe < 1200, 1600 < tωpe < 1900, and 2300 < tωpe < 2400.

4. Influence of Shock Potential on Re-formation

[22] From Figure 2 in the case of MA = 4 the two phases in the re-formation period can be identified clearly. It is also found that the periodic gyromotion of reflected ions becomes coherent even though the incident ions hit the shock front continuously. What makes the reflected ions behave coherently and causes the shock re-formation to be intermittent?

[23] In order to answer the question, we closely look at the shock potential during one period Tr. We call the potential jump across the shock front the “shock potential.” Figure 3 shows the spatial profiles of the potential energy eϕ(x) = −exEx(u)du at three different times in the case of MA = 4. Ions with kinetic energy that exceeds the shock potential can transit the shock front and penetrate downstream. In Figure 3, the spatial profile of the shock potential dramatically varies during one period of the shock re-formation. During the broadening phase, the position of the shock front stays at x/(cpe) ≃ 100 and the amplitude of the shock potential is almost same (compare the profiles at tωpe = 1360 and 1520 in Figure 3), but the potential gradients (i.e., −Ex) in the shock foot are different. Because the gradient of the potential is steeper, the electric field in the negative x direction at tωpe = 1360 is larger than that at tωpe = 1520. However, the ratio of ions reflected at the shock to ions that transit the shock is almost the same at these two times since the shock potential and the kinetic energy of incident ions are almost unchanged.

Figure 3.

Spatial profile of the potential energy at three different times, tωpe = 1360, 1520, and 1760 in the case of MA = 4, where the potential energy is normalized by the flow energy of upstream ions.

[24] On the other hand, the potential jump across the shock front in the growing phase (at tωpe = 1760) is much different from that in the broadening phase. At tωpe = 1760, both the shock potential and the field −Ex become larger than those at tωpe = 1360 and 1520 in Figure 3. This causes the number of reflected ions in the growing phase to become more than that in the broadening phase. Thus we conclude from Figure 3 that the ion reflection rate at the shock front is not steady during the re-formation period primarily due to the temporal change in the shock potential.

[25] Figure 4 displays the time histories of the maximum shock magnetic field −By (Figure 4a), the maximum electric field −Ex (Figure 4b), and the maximum potential energy eϕ, near the shock front in three cases of different Mach number (MA = 2, 4, and 8) (Figure 4c). In these runs we fixed the fundamental parameters and only changed the Mach number. From the temporal evolution of the ion phase space (not shown here), we define the starting times of the broadening phase at which beam-like ions collectively begin to extend toward upstream. The arrows in Figure 4b indicate these starting times for each Mach number. For instance, the broadening phase starts at tωpe ≃ 1200 and 1900 in the case of MA = 4.

Figure 4.

Time histories of (a) maximum magnetic field −By, (b) maximum electric field −Ex, and (c) maximum potential energy eϕ at the shock for three cases with different Mach numbers.

[26] In Figure 4b the period of shock re-formation becomes shorter as the Mach number increases. This is due to the fact that the magnetic field at the shock foot becomes larger (as shown in Figure 4a), and therefore the gyroperiod of reflected ions becomes smaller in the high Mach number cases. The maximum ∣Ex∣ takes relatively large values at the starting times of the broadening phase for each re-formation cycle but then becomes relatively small until it grows rapidly near the end of that cycle. This is qualitatively the same result as obtained by Lembege and Savoini [1992]. However, the maximum potential behaves differently, showing a more gradual increase through the late broadening phase and the full growing phase. This implies that the number of reflected ions is gradually increasing throughout much of the cycle and that this nonstationary property of the ions is what eventually leads to the nonsteady property of the magnetic field.

[27] In addition, the fluctuations of the maximum ∣By∣, ∣Ex∣, and eϕ tend to be smaller as the Mach number decreases. In the case of MA = 2, the maximum eϕ is almost steady during one period. Thus the number of ions reflected at the shock front hardly changes during one period in the case of MA = 2. This can explain why shock re-formation and coherent ion motion are not observed at the low Mach number shocks (MA < 2).

5. Electric Field Due to the Shock Potential

[28] The shock normal component of electric field Ex becomes maximum in the vicinity of the shock front, where the magnetic field also takes a maximum value. It is useful to examine the relationship between the electric field and the magnetic field in our simulations. We obtain the following expression by assuming a fluid description for the plasma and using the electron momentum equation,

display math

where Ve [= (Vex, Vey, Vez)] is the electron flow velocity, Pe is the electron pressure tensor, and ne is the electron density. Considering the one-dimensional property of our simulation, the x component of equation (1) yields

display math

[29] In Figure 5 the amplitude of each term in the right-hand side of equation (2) is displayed as a function of x at tωpe = 2400 in the case of MA = 4. As shown in Figure 2, the position of the shock front is x/(cpe) ≃ 58. The red line in Figure 5 indicates the spatial variation of Ex, which takes a large negative value at the shock ramp. Owing to this electric field some of the incident ions with low kinetic energy are reflected back upstream. The contribution of the term VeyBz is clearly small since Vey is almost zero everywhere in the system. The spatial form of VezBy term (line 1 in Figure 5) apparently matches with that of Ex. Lines 2 and 3 show the electron inertia terms, (me/qe)∂Vex/∂t and (me/qe)VexVex/∂x, respectively. These two terms give no significant contribution to Ex in the vicinity of the shock ramp. The term (1/qene)∂Pe,xx/∂x, represented by the line 4, tends to vary markedly in the downstream region, and its value in the vicinity of the shock front is almost zero. Thus the dominant contribution to Ex is from the term VezBy. Figure 4 shows similar properties between the maximum ∣By∣ and ∣Ex∣ in terms of the temporal evolution and the Mach number dependence. Thus the relation ExVezBy is approximately valid for each of our simulations.

Figure 5.

Spatial profile of the electric field Ex (red curve) in the vicinity of the shock ramp at tωpe = 2400 in the case of MA = 4. The additional curves indicate the contributions to Ex of each term in equation (2), VezBy (line 1), (me/qe)∂Vex/∂t (line 2), (me/qe)VexVex/∂x (line 3), and 1/(qene)∂Pe,xx/∂x (line 4).

6. Conclusions

[30] We used a one-dimensional full particle simulation to study the re-formation of perpendicular shocks. The temporal change of the magnetic field profile exhibits an intermittent behavior in shocks of sufficiently high Mach number. Although upstream particles arrive continuously at the shock front, the ions reflected by the shock potential show coherent collective motions in the upstream. We define two phases (broadening phase and growing phase) in the re-formation period by the spatial profile of the magnetic field. The ion reflection rate at the shock front in the growing phase becomes larger than that in the broadening phase since the shock potentials at two phases are dramatically different. Thus, in the cases of high Mach number shocks, the motion of reflected ions becomes intermittent rather than continuous. When the Mach number is low, we cannot clearly identify the two phases of re-formation, and the temporal evolution of the shock potential tends to be continuous.

[31] Finally, we refer to the fact that the plasma beta in this report (βe = 0.0032) is much smaller than typical values observed in space plasmas (βe ∼ 1). We cannot discuss the difference between shocks with high plasma beta and shocks with low plasma beta since we have no simulation results concerning high beta plasmas. In our present results we observe the shocks exhibit an intermittent behavior when the Mach number is greater than the critical Mach number (MA ≃ 2). We cannot say for certain whether the critical Mach number exists also in the case of high plasma beta. If the critical Mach number is found in the case of high plasma beta, the value may be different from our critical Mach number since it must be dependent on the plasma beta and other initial parameters.


[32] One of the authors (K.N.) would like to thank D. Winske for providing with one of references. This work has been carried out under the COE program granted for one of the authors (K.N.). Computation in the present study was performed with the KDK system of Radio Science Center for Space and Atmosphere (RASC) at Kyoto University as a collaborative research project.

[33] Shadia Rifai Habbal thanks Kevin B. Quest and another referee for their assistance in evaluating this paper.