Weibel Instability in Collisionless Plasmas Across Astrophysical and Laboratory Shocks

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 $γ_{\rm max}$ and characteristic unstable wavenumber $k_{\rm max} = ω_{pi}/c$. Relativistic corrections suppress $γ_{\rm max}$ 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 $d_i = c/ω_{pi} \approx 31.7\,μ{\rm m}$, within 2 per cent of the measured filament spacing $λ_F \approx 31\,μ{\rm m}$. 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.

Figures (10)

• Figure 1: MMS1 burst-mode timeseries for the 2015 October 16 quasi-perpendicular bow shock crossing (Event 1). Panel 1: magnetic field magnitude $|B|$. Panel 2: GSM components $B_x$ (green), $B_y$ (blue), $B_z$ (orange); the pink band marks the shock ramp interval. Panel 3: ion number density $n_i$ (green, left axis) and derived ion skin depth $d_i = c/\omega_{pi}$ (purple dashed, right axis). Panel 4: ion temperature anisotropy $T_\perp/T_\parallel$ (blue); contextual plasma characterisation only (see text). Upstream parameters: $n_i^{\rm up} = 20.2$ cm$^{-3}$, $M_A = 2.4$, $\beta_i = 9.0$, $d_i = 50.8$ km. Data source: MMS FGM level-2 (128 S s$^{-1}$) and FPI-DIS (150 ms) via PySPEDAS grimes2022pyspedas.
• Figure 2: MMS1 burst-mode timeseries for the 2017 November 25 high-$M_A$ bow shock crossing (Event 2; Kropotina et al. kropotina2023weibel). Panel layout as in Fig. \ref{['fig:mmstsA']}. Upstream parameters: $n_i^{\rm up} = 13.6$ cm$^{-3}$, $M_A = 2.3$, $\beta_i = 2.4$, $d_i = 61.8$ km. The shock ramp (pink band) is confined to the first $\approx 5$ s of the interval, after which the spacecraft resides in the magnetosheath.
• Figure 3: Perpendicular magnetic power spectral density $P_{B_\perp}(kd_i) = P(f)\cdot V_{\rm sw,\,up}/(2\pi)$ for the two MMS bow shock events, plotted against $kd_i$ via the Taylor hypothesis ($k = 2\pi f/V_{\rm sw,\,up}$). Left: Event 1 (2015-10-16). Black dashed line: power-law fit $k^{-2.17}$ in the MHD inertial range. Red dashed line: $k^{-2.77}$ in the sub-ion range. Right: Event 2 (2017-11-25). Red dashed line: $k^{-2.65}$ in the sub-ion range. In both panels the orange dash-dot vertical line marks the cold-fluid prediction $k_{\rm max}d_i = 1$; the shaded orange band shows $\pm 40$ per cent Taylor-hypothesis uncertainty. The $y$-axis is in units of nT$^2$ Hz$^{-1}$; multiplying by $V_{\rm sw}/(2\pi)$ converts to nT$^2$ km rad$^{-1}$ in wavenumber space.
• Figure 4: Multi-environment test of the cold-fluid Weibel scale prediction $k_{\rm max} = \omega_{pi}/c$. Abscissa: predicted $d_i = c/\omega_{pi}$ (metres); ordinate: observed filament or spectral-break scale (metres). The dashed line is the 1:1 prediction; the grey band is the factor-of-3 envelope. Orange star: Bai et al. (2025) Al$^{+8}$ laser plasma ($d_i \approx 32\,\mu{\rm m}$, $\lambda_F = 31\,\mu{\rm m}$). Yellow pentagon: Kropotina et al. (2023) MMS 2017 event ($d_i \approx 68$ km). Green square: SNR G1.9+0.3 ($d_i \approx 1600$ km, estimated). Blue circle: MMS Event 1 from this work ($d_i \approx 51$ km). Purple circle: MMS Event 2 from this work ($d_i \approx 62$ km). All five environments lie within a factor of 3 of the 1:1 line across 21 orders of magnitude in $n_i$.
• Figure 5: Growth rate $\gamma(k)$ for relativistic (solid) and non-relativistic (dashed) single-species Weibel instability. The three panels cover low ($v_0 = 0.1$--$0.3c$), intermediate ($v_0 = 0.4$--$0.7c$), and relativistic ($v_0 = 0.8$--$0.95c$) beam velocities. Relativistic corrections are negligible below $v_0 \approx 0.2c$, grow progressively through the trans-relativistic regime, and reach up to $\sim 40$ per cent near $v_0 \approx 0.9c$. For $v_0 \leq 0.07c$ (the conditions of the Bai et al. experiment), the two curves are indistinguishable on this scale.