Time-dependent simulation of oblique MHD cosmic-ray shocks using the two-fluid model

Adam Frank, Thomas W Jones, Dongsu Ryu

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Using a new, second-order accurate numerical method we present dynamical simulations of oblique MHD cosmic-ray (CR)-modified plane shock evolution. Most of the calculations are done with a two-fluid model for diffusive shock acceleration, but we provide also comparisons between a typical shock computed that way against calculations carried out using the more complete, momentum-dependent, diffusion-advection equation. We also illustrate a test showing that these simulations evolve to dynamical equilibra consistent with previously published steady state analytic calculations for such shocks. In order to improve understanding of the dynamical role of magnetic fields in shocks modified by CR pressure we have explored for time asymptotic states the parameter space of upstream fast mode Mach number, Mf, and plasma β. We compile the results into maps of dynamical steady state CR acceleration efficiency, ∈c. Since the models are simplifications, such maps should not be used to predict quantitatively ∈c in real shocks; however, they are internally consistent, so that they can enable us to compare various competing dynamical effects. The maps, along with additional numerical experiments, show that ∈c is reduced through the action of compressive work on tangential magnetic fields in CR-MHD shocks. Thus ∈c in low β, moderate Mf shocks tends to be smaller in quasi-perpendicular shocks than it would be for 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, ∈c 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 in these instances. 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 nonisotropic, obliquity (and hence spatially) dependent forms of the diffusion coefficient κ. Comparison of the results shows 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 can be 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 nonplanar flows, result in enhanced particle accretion through drift acceleration.

Original languageEnglish (US)
Pages (from-to)629-643
Number of pages15
JournalAstrophysical Journal
Issue number2
StatePublished - Mar 10 1995


  • Acceleration of particles
  • Cosmic rays
  • MHD
  • Shock waves


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