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, Mf , and plasma β, compiling the results into maps of dynamical steady state CR acceleration efficiency, ǫc. These maps, along with additional numerical experiments, show that ǫc is reduced through the action of compressive work on tangential magnetic fields in CR-MHD shocks. Thus ǫc in low β, moderate Mf shocks tends to be smaller in quasi perpendicular shocks than it would be high β shocks of the same Mf . This result supports earlier conclusions that strong, oblique magnetic fields inhibit diffusive shock acceleration. For quasi parallel shocks with β < 1, on the other hand, ǫc seems to be increased at a given Mf when compared to high β shocks. The apparent contradiction to the first conclusion results, however, from the fact that for small β 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 β 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. We have run simulations using constant, and non-isotropic, obliquity dependent forms of the diffusion coefficient κ. Comparison of the results show that while the final steady states achieved are the same in each case, the history of CR-MHD shocks can be strongly modified by variations in κ and, therefore, in the acceleration timescale. Also, the coupling of CR and MHD in low β, oblique shocks substantially influences the transient density spike that forms in strongly CR-modified shocks. We find that inside the density spike a MHD slow mode wave is generated that eventually steepens into a shock. A strong shear layer develops within the density spike, driven by MHD stresses. We conjecture that currents in the shear layer could, in non-planar flows, result in enhanced particle acceleration through drift acceleration.