The Evolution and Efficiency of Oblique MHD Cosmic-Ray Shocks: Two-Fluid   Simulations
  Using a new, second-order accurate numerical method we present dynamical simulations of oblique MHD cosmic ray (CR) modified plane shock evolution using the two-fluid model for diffusive particle acceleration. The numerical shocks evolve to published analytical steady state properties. In order to probe the dynamical role of magnetic fields we have explored for these time asymptotic states the parameter space of upstream fast mode Mach number, $M_f$, and plasma $\beta$, compiling the results into maps of dynamical steady state CR acceleration efficiency, $\epsilon_c$. These maps, along with additional numerical experiments, show that $\epsilon_c$ is reduced through the action of compressive work on tangential magnetic fields in CR-MHD shocks. Thus $\epsilon_c$ in low $\beta$, moderate $M_f$ shocks tends to be smaller in quasi perpendicular shocks than it would be high $\beta$ shocks of the same $M_f$. This result supports earlier conclusions that strong, oblique magnetic fields inhibit diffusive shock acceleration. For quasi parallel shocks with $\beta < 1$, on the other hand, $\epsilon_c$ seems to be increased at a given $M_f$ when compared to high $\beta$ shocks. The apparent contradiction to the first conclusion results, however, from the fact that for small $\beta$ quasi parallel shocks, the fast mode Mach number is not a good measure of compression through the shock. That is better reflected in the sonic Mach number, which is greater. Acceleration efficiencies for high and low $\beta$ having comparable sonic Mach numbers are more similar. Time evolution of CR-MHD shocks is qualitatively similar to CR-gasdynamical shocks. However, several potentially interesting differences are apparent. 
Adam Frank, T. W. Jones, and Dongsu Ryu