Diffusive Shock Acceleration Simulations: Comparison with Particle Methods and Bow Shock Measurements

Abstract

Direct comparisons of diffusive particle acceleration numerical simulations have been made against Monte Carlo and hybrid plasma simulations by Ellison et al. (1993) and against observations at the Earth's bow shock presented by Ellison et al. (1990). Toward this end we have introduced a new numerical scheme for injection of cosmic-ray particles out of the thermal plasma, modeled by way of the diffusive scattering process itself; that is, the diffusion and acceleration across the shock front of particles out of the suprathermal tail of the Maxwellian distribution. Our simulations take two forms. First, we have solved numerically the time-dependent diffusion-advection equation for the high-energy (cosmic-ray) protons in one-dimensional quasiparallel shocks. Dynamical feedback between the particles and thermal plasma is included. The proton fluxes on both sides of the shock derived from our method are consistent with those calculated by Ellison et al. (1993). A similar test has compared our methods to published measurements at the Earth's bow shock when the interplanetary magnetic field was almost parallel to the solar wind velocity (Ellison et al. 1990). Again our results are in good agreement. Second, the same shock conditions have been simulated with the two-fluid version of diffusive shock acceleration theory by adopting injection rates and the closure parameters inferred from the diffusion-advection equation calculations. The acceleration efficiency and the shock structure calculated with the two-fluid method are in good agreement with those computed with the diffusion-advection method. Thus, we find that all of these computational methods (diffusion-advection, two-fluid, Monte Carlo, and hybrid) are in substantial agreement on the issues they can simultaneously address, so that the essential physics of diffusive particle acceleration is adequately contained within each. This is despite the fact that each makes what appear to be very different assumptions or approximations.