Weibel Instability in Collisionless Plasmas Across Astrophysical and Laboratory Shocks

TL;DR

This paper provides a comprehensive cold-fluid analysis of the Weibel instability across various plasma regimes, deriving dispersion relations, scaling laws, and validating predictions with laboratory and space plasma observations.

Contribution

It introduces a unified linear fluid framework for the Weibel instability applicable to multiple plasma regimes and compares predictions with experimental and space data.

Findings

Relativistic effects reduce growth rates by up to 40%.

Cold-fluid predictions match experimental filament spacing within 2%.

Space observations confirm theoretical wavenumber scaling across diverse environments.

Abstract

We present a cold-fluid analysis of the purely transverse Weibel (current-filamentation) instability across four regimes: non-relativistic (NR) single-species, NR multi-species, relativistic single-species, and relativistic multi-species (electron--positron and electron--proton). Beginning from linearized fluid equations, we derive the dispersion relations in each regime and extract scaling laws for the maximum growth rate $\gamma_{\rm max}$ and characteristic unstable wavenumber $k_{\rm max} = \omega_{pi}/c$. Relativistic corrections suppress $\gamma_{\rm max}$ by up to 40 per cent above $v_0 \approx 0.2c$, peaking near $v_0 \approx 0.9c$. Multi-species effects are significant only for $m_e/m_i \gtrsim 1/500$. For the tabletop laser experiment of Bai et al., Nat.Commun., 16, 3770 (2025), the cold-fluid prediction gives $d_i = c/\omega_{pi} \approx 31.7\,\mu{\rm m}$, within 2 per cent of the measured filament spacing $\lambda_F \approx 31\,\mu{\rm m}$. The saturation field estimate $B_{\rm sat} \approx 2.3\times10^4$ T is an upper bound, consistent with the measured $\approx 5000$ T under kinetic suppression. Two MMS burst-mode bow shock crossings (October 16, 2015 and November 25, 2017) confirm $k_{\rm max} d_i = 1$ from FGM/FPI data. A multi-environment scatter plot spans 21 orders of magnitude in $n_i$, with all points within a factor of 3 of the 1:1 line.