Magnetic Field Amplification and Particle Acceleration in Weakly Magnetized Trans-relativistic Electron-ion Shocks

We investigate the physics of quasi-parallel trans-relativistic shocks propagating in weakly magnetized plasmas by means of long-duration two-dimensional particle-in-cell simulations. The structure of the shock precursor is shaped by a competition between the Bell instability and the Weibel (filamentation) instability. The Bell instability is dominant at relatively high magnetizations $(σ\gtrsim10^{-3})$, whereas the Weibel instability prevails at lower magnetizations $(σ\lesssim10^{-4})$. Shocks with precursors shaped by Bell modes efficiently accelerate ions, converting a fraction $\varepsilon_{\mathrm{i}}\sim0.2$ of the upstream flow energy into downstream nonthermal ion energy. The maximum energy of nonthermal ions exhibits a Bohm scaling in time, as $E_{\max}\propto t$. A much smaller fraction $\varepsilon_{\mathrm{e}}\ll0.1$ of the upstream flow energy goes into downstream nonthermal electrons in the Bell regime. On the other hand, when the precursor is dominated by Weibel modes, the shock efficiently generates both nonthermal ions and electrons with $\varepsilon_{\mathrm{i}}\sim\varepsilon_{\mathrm{e}}\sim0.1$, albeit with a slower scaling for the maximum energy, $E_{\mathrm{max}}\propto t^{1/2}$. Our results are applicable to a wide range of trans-relativistic shocks, including the termination shocks of extragalactic jets, the late stages of gamma-ray burst afterglows, and shocks in fast blue optical transients.

💡 Research Summary

This paper presents a comprehensive investigation of quasi‑parallel, trans‑relativistic (shock Lorentz factor Γ_sh≈2) electron‑ion shocks propagating in weakly magnetized plasmas, using long‑duration two‑dimensional particle‑in‑cell (PIC) simulations. The authors focus on the structure of the shock precursor, which is shaped by a competition between two plasma instabilities driven by the streaming of reflected, non‑thermal particles (cosmic rays): the Bell (current‑driven) instability and the Weibel (filamentation) instability. By varying the upstream magnetization σ from 10⁻³ down to 10⁻⁴ (and an unmagnetized case in the appendix), they identify a clear transition. For σ≳10⁻³ the Bell instability dominates; its modes have wavevectors nearly parallel to the background field, grow on scales of several ion skin depths, and reach near‑equipartition magnetic energy (δB≈10 B₀) in the nonlinear stage. The resulting precursor contains large‑scale cavities and dense filaments, and it efficiently reflects and re‑accelerates ions. Consequently, about 20 % of the upstream bulk kinetic energy is transferred to non‑thermal ions (ε_i≈0.2), while electrons receive a negligible fraction (ε_e≪0.1). The maximum ion energy follows a Bohm‑like scaling, E_max∝t, indicating rapid acceleration.

When σ≲10⁻⁴ the Weibel instability takes over. Its wavevectors are perpendicular to the background field, the dominant magnetic component is B_z, and the characteristic scale shrinks to ∼ d_i. The magnetic turbulence is generated mainly by electron currents, leading to δB/B₀≈10 as well, but the structure is filamentary rather than cavity‑dominated. In this regime both ions and electrons are accelerated with comparable efficiencies, ε_i≈ε_e≈0.1, but the energy growth is slower, E_max∝t¹/², reflecting the rapid saturation of the small‑scale Weibel modes.

The authors also distinguish a high‑current Bell regime (where the cosmic‑ray current exceeds a critical value) from the classical low‑current regime. In the high‑current case the magnetic energy density scales with the upstream field (δB∝B₀), whereas in the low‑current case it scales with the cosmic‑ray momentum flux. This distinction is crucial for understanding the competition with the Weibel mode, especially at intermediate magnetizations (σ≈10⁻³·⁵) where Bell modes are present but with reduced amplitude.

Technical details of the simulations include a reduced ion‑to‑electron mass ratio (m_i/m_e=100), 16 particles per cell per species, a cell size of Δx=0.4 d_e, and a time step Δt=0.5 Δx/c. Numerical artifacts are minimized using an alternative Yee stencil, extensive filtering, and a moving injector that supplies fresh upstream plasma. The runs reach unprecedented durations (ω_pi t≈7 000–12 000), allowing the authors to capture the fully nonlinear evolution of both instabilities.

The results have direct astrophysical implications. In the termination shocks of extragalactic jets (σ∼10⁻⁴–10⁻³) the Bell‑dominated regime predicts efficient ion acceleration, potentially contributing to ultra‑high‑energy cosmic rays. In the late stages of gamma‑ray burst afterglows, where the ambient interstellar medium is essentially unmagnetized (σ∼10⁻⁹), the Weibel‑dominated regime naturally yields comparable ion and electron acceleration efficiencies, offering a plausible explanation for the bright X‑ray and radio emission observed in events like GW170817. Fast blue optical transients (FBOTs) also require strong electron acceleration; the Weibel‑driven precursor provides the necessary electron energy budget.

In summary, the paper demonstrates that the dominant precursor instability in weakly magnetized, trans‑relativistic shocks is set by the upstream magnetization: Bell modes at higher σ produce fast, ion‑dominated acceleration with Bohm scaling, while Weibel modes at lower σ generate balanced ion‑electron acceleration with slower, diffusive scaling. This unified picture bridges the gap between previous high‑σ studies and the low‑σ environments relevant to many observed high‑energy transients, offering quantitative predictions for particle spectra, acceleration efficiencies, and magnetic field amplification that can be tested against multi‑wavelength observations.