  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.
