# Quasi-nonlinear approach to the Weibel instability

## TL;DR

This paper investigates how small-scale magnetic fluctuations in collisionless plasmas influence the growth and evolution of the Weibel instability, revealing that quasi-collisional effects can modify but not suppress the instability.

## Contribution

It introduces a quasi-nonlinear approach showing how plasma quasi-collisionality affects the Weibel instability's development in nonlinear regimes.

## Key findings

- Quasi-collisions slightly suppress the Weibel instability.
- The instability cannot be completely shut down by quasi-collisional effects.
- Results support the self-similar model of the collisionless foreshock.

## Abstract

Astrophysical and high-energy-density laboratory plasmas often have large-amplitude, sub-Larmor-scale electromagnetic fluctuations excited by various kinetic-streaming or anisotropy-driven instabilities. The Weibel (or the filamentation) instability is particularly important because it can rapidly generate strong magnetic fields, even in the absence of seed fields. Particles propagating in collisionless plasmas with such small-scale magnetic fields undergo stochastic deflections similar to Coulomb collisions, with the magnetic pitch-angle diffusion coefficient representing the effective "collision" frequency. We show that this effect of the plasma "quasi-collisionality" can strongly affect the growth rate and evolution of the Weibel instability in the deeply nonlinear regime. This result is especially important for understanding cosmic-ray-driven turbulence in an upstream region of a collisionless shock of a gamma-ray burst or a supernova. We demonstrate that the quasi-collisions caused by the fields generated in the upstream suppress the instability slightly but can never shut it down completely. This confirms the assumptions made in the self-similar model of the collisionless foreshock.