Simulations of Ion Acceleration at Non-relativistic Shocks. III.   Particle Diffusion
  We use large hybrid (kinetic protons-fluid electrons) simulations to investigate the transport of energetic particles in self-consistent electromagnetic configurations of collisionless shocks. In previous papers of this series, we showed that ion acceleration may be very efficient (up to $10-20\%$ in energy), and outlined how the streaming of energetic particles amplifies the upstream magnetic field. Here, we measure particle diffusion around shocks with different strengths, finding that the mean free path for pitch-angle scattering of energetic ions is comparable with their gyroradii calculated in the self-generated turbulence. For moderately-strong shocks, magnetic field amplification proceeds in the quasi-linear regime, and particles diffuse according to the self-generated diffusion coefficient, i.e., the scattering rate depends only on the amount of energy in modes with wavelengths comparable with the particle gyroradius. For very strong shocks, instead, the magnetic field is amplified up to non-linear levels, with most of the energy in modes with wavelengths comparable to the gyroradii of highest-energy ions, and energetic particles experience Bohm-like diffusion in the amplified field. We also show how enhanced diffusion facilitates the return of energetic particles to the shock, thereby determining the maximum energy that can be achieved in a given time via diffusive shock acceleration. The parametrization of the diffusion coefficient that we derive can be used to introduce self-consistent microphysics into large-scale models of cosmic ray acceleration in astrophysical sources, such as supernova remnants and clusters of galaxies. 
Damiano Caprioli, Anatoly Spitkovsky (Princeton University)
astro-ph.HE