Solution for jump conditions at fast shocks in an anisotropic magnetized plasma

Abstract

We study the magnetic field and plasma parameters downstream of a fast shock as functions of normalized upstream parameters and the rate of pressure anisotropy (defined as the ratio of perpendicular to parallel pressure). We analyse two cases: with the shock (i) perpendicular and (ii) inclined with respect to the magnetic field. The relations on the fast shock in a magnetized anisotropic plasma are solved taking into account the criteria for the mirror instability and firehose instability bounding the pressure anisotropy downstream of the shock. Our analysis shows that the parallel pressure and the parallel temperature as well as the tangential component of the velocity are the parameters that are most sensitive to the rate of pressure anisotropy. The variations of the other parameters, namely density, normal velocity, tangential component of the magnetic field, perpendicular pressure, and perpendicular temperature are much less pronounced, in particular when the perpendicular pressure exceeds the parallel pressure. The variations of all parameters increase substantially for a very low rate of anisotropy, which is bounded by the firehose instability in the case of inclined shocks. Using the criterion for mirror instability as a closure relation for the jump conditions at the fast shock, we obtain the plasma parameters and the magnetic field downstream of the shock as functions of the Alfvén Mach number. For each Alfvén Mach number, the criterion for mirror instability determines the minimum jumps in such parameters as density, tangential magnetic field component, parallel pressure, and temperature, and determines the maximum values of the velocity components and the perpendicular temperature. Ideal anisotropic magnetohydrodynamics (MHD) has wide applications for space plasma physics. Observations of the field and plasma behaviour in the solar wind as well as in the Earth's magnetosheath have highlighted the need for an MHD model where the plasma pressure is treated as a tensor.