Characteristic analysis of a complex two-dimensional magnetohydrodynamic bow shock flow with steady compound shocks

A simple, compact, and systematic derivation is given of the characteristic properties of the magnetohydrodynamic (MHD) equations with two independent variables (time-dependent MHD in the $\mathrm{xt}$ plane and steady MHD in the $\mathrm{xy}$ plane), based on the symmetrizable Galilean invariant form of the equations and using a matrix approach. A numerically obtained stationary planar field-aligned MHD bow shock flow with interacting shocks is then analyzed in terms of hyperbolic and elliptic regions, steady $\mathrm{xy}$ characteristics, limiting lines, and allowed shock transitions. With the help of this analysis, a wave structure present in the bow shock flow can be interpreted as a double steady compound shock. This interpretation is based on the complete analogy demonstrated in our analysis, between the $\mathrm{xy}$ characteristic structure of this novel steady compound shock and the $\mathrm{xt}$ characteristic structure of the well-known time-dependent MHD compound shock.