Large-scale hybrid simulations of particle acceleration at a parallel shock

Using one-dimensional hybrid simulations with very large spatial domains, we study the acceleration of protons by a parallel collisionless shock to energies higher than have been obtained previously with self-consistent plasma simulations of this type. Energy spectra and energetic particle fluxes are determined for four simulations with different-sized spatial domains. We find that the density of energetic particles upstream decays with distance from the shock and approaches a constant. The energetic particles also excite magnetic fluctuations. We find that the variance of these transverse fluctuations decreases with distance from the shock into the upstream region. This implies that the mean free path, λ, of the energetic particles increases with distance from the shock. Since our simulations are spatially limited, the downstream energy spectra are expected to deviate from a power law and become exponential at a characteristic energy E_c. However, we find that because λ increases with distance upstream, E_c is smaller than expected from a simple application of the diffusive theory (which assumes a constant mean free path and a free-escape boundary). Our results are qualitatively consistent with diffusive theories of the coupling of the particles and self-generated waves. However, in contrast to these theories, the hybrid-simulated energetic particle flux approaches a constant at some point far upstream, rather than vanishing, as assumed in the diffusion theory.