Nonrelativistic collisionless shocks in weakly magnetized electron--ion plasmas: two-dimensional particle-in-cell simulation of perpendicular shock

A two-dimensional particle-in-cell simulation is performed to investigate weakly magnetized perpendicular shocks with a magnetization parameter of 6 x 10^-5, which is equivalent to a high Alfv'en Mach number M_A of ~130. It is shown that current filaments form in the foot region of the shock due to the ion-beam–Weibel instability (or the ion filamentation instability) and that they generate a strong magnetic field there. In the downstream region, these current filaments also generate a tangled magnetic field that is typically 15 times stronger than the upstream magnetic field. The thermal energies of electrons and ions in the downstream region are not in equipartition and their temperature ratio is T_e / T_i ~ 0.3 - 0.4. Efficient electron acceleration was not observed in our simulation, although a fraction of the ions are accelerated slightly on reflection at the shock. The simulation results agree very well with the Rankine-Hugoniot relations. It is also shown that electrons and ions are heated in the foot region by the Buneman instability (for electrons) and the ion-acoustic instability (for both electrons and ions). However, the growth rate of the Buneman instability is significantly reduced due to the relatively high temperature of the reflected ions. For the same reason, ion-ion streaming instability does not grow in the foot region.

💡 Research Summary

This paper presents a two‑dimensional particle‑in‑cell (PIC) investigation of a perpendicular collisionless shock propagating in a weakly magnetized electron‑ion plasma. The upstream magnetization parameter is σ = 6 × 10⁻⁵, corresponding to an Alfvén Mach number M_A ≈ 130, i.e., an extremely high‑Mach shock. The authors initialize a uniform plasma with realistic ion‑to‑electron mass ratio (m_i/m_e = 1836) and equal electron and ion temperatures, and impose a very weak magnetic field perpendicular to the shock normal. A reflecting wall creates a steady shock that travels into the incoming flow, allowing the formation of a foot region ahead of the main ramp.

The central finding is that the ion‑beam (or ion‑filamentation) Weibel instability dominates the foot dynamics. Reflected ions streaming back into the upstream plasma generate transverse current filaments with a characteristic wavelength of a few ion skin depths. These filaments amplify the magnetic field locally by an order of magnitude, producing B/B₀ ≈ 10–15 in the foot. The amplified, highly tangled field is advected downstream, where it remains the dominant magnetic component and yields a downstream magnetic energy density roughly fifteen times larger than the upstream value.

Downstream, the plasma satisfies the Rankine‑Hugoniot jump conditions: the density compression ratio approaches the theoretical value of ≈4, and the pressure balance is maintained by the turbulent magnetic field. However, electron and ion temperatures are not equilibrated; the downstream electron temperature is only 30–40 % of the ion temperature (T_e/T_i ≈ 0.3–0.4). This non‑equipartition results from the fact that electron acceleration mechanisms are largely absent in the simulation. No significant shock‑drift acceleration, surfing acceleration, or magnetic‑reconnection‑driven processes are observed for electrons. Ions, on the other hand, experience modest reflection at the shock front; a small fraction gains additional energy, but the overall ion energy spectrum remains close to a thermal distribution.

The authors also dissect the micro‑instabilities that heat the plasma in the foot. The relative drift between reflected ions and incoming electrons drives the Buneman instability, which generates electrostatic waves that heat electrons. Because the reflected ion beam already possesses a relatively high temperature, the growth rate of the Buneman mode is reduced compared to the cold‑beam limit. Simultaneously, an ion‑acoustic instability, excited by the same drift, heats both species. The ion‑ion streaming instability, which would otherwise amplify ion density fluctuations, does not develop because the temperature contrast between the two ion populations is insufficient.

Overall, the simulation demonstrates that in a weakly magnetized, high‑Mach number perpendicular shock, the dominant energy conversion channel is the generation of turbulent magnetic fields via the ion‑Weibel instability. This turbulence controls the downstream pressure balance and sets the electron‑to‑ion temperature ratio, while efficient electron acceleration is suppressed by the high temperature of the reflected ion beam. The results are consistent with macroscopic shock theory (Rankine‑Hugoniot) and provide a kinetic‑level explanation for the observed temperature partition.

The study has several implications for astrophysical shocks. First, it suggests that even when the upstream magnetic field is negligible, collisionless shocks can self‑generate strong magnetic turbulence that dominates the downstream dynamics. Second, the lack of electron acceleration under these conditions challenges models that invoke high‑Mach perpendicular shocks as primary sites of cosmic‑ray electron injection, unless additional pre‑conditioning (e.g., pre‑existing turbulence, higher β, or oblique magnetic geometry) is present. Third, the identified heating mechanisms (Buneman and ion‑acoustic instabilities) operate on electron and ion scales, respectively, and could be observable through emitted radiation signatures in supernova remnants or gamma‑ray burst afterglows.

Future work should extend the present 2D study to three dimensions, where filament merging, magnetic reconnection, and additional modes (e.g., kinetic Alfvén waves) may alter both the magnetic amplification and particle acceleration efficiencies. Varying the upstream plasma β, the angle between the magnetic field and shock normal, and the ion‑to‑electron temperature ratio would help map the parameter space where electron acceleration becomes viable. Such extensions will be essential for building a comprehensive kinetic theory of high‑Mach number collisionless shocks in weakly magnetized astrophysical environments.