VISCOUS AND INERTIAL EFFECTS AT COSMIC-RAY SHOCKS

To illustrate the effect of the viscous and inertial terms on the structure of cosmic-ray-modified shocks, we present an analysis of a time-independent one-dimensional shock, in which the pressure of the background fluid and the magnetic-field energy density are negligible compared with the cosmic-ray pressure. We retain the effect of the magnetic field on cosmic-ray transport coefficients. The diffusion, viscous and inertial terms work to "thicken" the shock. When the average magnetic field is negligible and the cosmic-ray transport is isotropic, diffusion is the dominant effect unless the parameters are such that the scattering time becomes comparable to the characteristic times for velocity change in the fluid frame. For a perpendicular shock, where the diffusion is strongly reduced by the magnetic field, we find that the relative magnitude of the viscous effect is increased by a factor approximately (OMEGA-tau)2, leading to the conclusion that cosmic-ray viscosity can be important to the structure of collisionless MHD shocks. In both cases considered, the viscous and inertial terms become important for those parameter ranges where the diffusion approximation begins to break down.