Numerical studies of cosmic-ray injection and acceleration

A numerical scheme that incorporates a thermal leakage injection model into a combined gasdynamics and cosmic-ray (CR) diffusion-convection code has been developed. The hydro/CR code can follow in a very cost-effective way the evolution of CR-modified planar quasi-parallel shocks by adopting subzone shock tracking and multilevel adaptive mesh refinement techniques. An additional conservative quantity, S = P_g/ρ^γg-1, is introduced to follow the adiabatic compression accurately in the precursor region, especially in front of strong, highly modified shocks. The "thermal leakage" injection model is based on the nonlinear interactions of the suprathermal particles with self-generated MHD waves in quasi-parallel shocks. The particle injection is followed numerically by filtering the diffusive flux of suprathermal particles across the shock to the upstream region according to a velocity-dependent transparency function that controls the fraction of leaking particles. This function is determined by a single parameter, ε, which should depend on the strength of postshock wave turbulence but is modeled as a constant parameter in our simulations. We have studied CR injection and acceleration efficiencies during the evolution of CR-modified planar shocks for a wide range of initial shock Mach numbers, M ₀, assuming a Bohm-like diffusion coefficient. For expected values of ε the injection process is very efficient when the subshock is strong, leading to fast and significant modification of the shock structure. As the CR pressure increases, the subshock weakens and the injection rate decreases accordingly so that the subshock does not disappear. Although some fraction of the particles injected early in the evolution continue to be accelerated to ever higher energies, the postshock CR pressure reaches an approximate time-asymptotic value because of a balance between fresh injection/acceleration and advection/diffusion of the CR particles away from the shock. In the strong shock limit of M₀ ≳ 30, the injection and acceleration processes are largely independent of the initial shock Mach number for a given ε, while they are sensitively dependent on M₀ for M₀ < 30. We conclude that the injection rates in strong parallel shocks are sufficient to lead to rapid nonlinear modifications to the shock structures and that self-consistent injection and time-dependent simulations are crucial to understanding the nonlinear evolution of CR-modified shocks.