Weibel Instability in Collisionless Plasmas Across Astrophysical and Laboratory Shocks

We present a systematic 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 the linearized fluid equations, we derive the dispersion relations in each regime and extract scaling laws for ${\gamma }_{\max }$ and $k_{\max }={\omega }_{\mathrm{pi}}/c$. The main contributions are: (i) explicit regime-transition criteria in the $\left(v_0/c,,,m_i/m_e\right)$ parameter space; (ii) contour maps of relativistic suppression and mass-ratio dependence serving as diagnostic charts and (iii) an assessment of where the purely transverse mode dominates over competing instabilities. Relativistic corrections suppress ${\gamma }_{\max }$ by up to 40% above $v_0\approx 0.2c$, peaking near $v_0\approx 0.9c$. For the tabletop laser experiment of Bai et al., the predicted ion skin depth $d_i\approx 31.7\mathrm{\mu m}$ agrees with the observed filament spacing to within 2%, and $B_{\mathrm{sat}}\approx 2.3\times {10}^{4}T$ is consistent with the measured $\approx 5000T$ under kinetic suppression. Two MMS burst-mode bow shock crossings confirm $k_{\max }{d}_i=1$ from FGM/FPI PySPEDAS data. A multi-environment scatter plot spanning 21 orders of magnitude in $n_i$, from laser-plasma to young SNR scales, shows all points within a factor of 3 of the 1:1 line.