Quasi-collisional approach to Weibel turbulence in GRB shocks

Weibel instability is known to control collisionless shocks. These shocks are expected to form in high-energy-density astrophysical plasma sources such as gamma-ray bursts (GRB) as well as in laboratory laser plasmas. The energetic particles escaping upstream of the shock form the outgoing population of supra-thermal particles which magnetize the shock precursor and further affect particle diffusion and cosmic ray propagation in the upstream. The self-induced pitch-angle diffusion in the Weibel sub-larmor-scale fields induce particle stochastic deflections similar to Coulomb collisions. We show that this effect of the plasma "quasi-collisionality" affects the growth rate and evolution of the Weibel instability in the deeply nonlinear regime. We demonstrate that the quasi-collisions suppress the instability, yet can never shut it down completely. This confirms the assumptions made in the self-similar model of the collisionless foreshock.