A general theory of self-similar expansion waves in magnetohydrodynamic flows

The general theory of self-similar magnetohydrodynamic (MHD) expansion waves is presented. Building on the familiar hydrodynamic results, a complete range of possible field-flow and wave-mode orientations are explored. When the magnetic field and flow are parallel, only the fast-mode wave can undergo an expansion to vacuum conditions: the self-similar slow-mode wave has a density that increases monotonically. For fast-mode waves with the field at all arbitrary angle with respect to the flow, the MHD equations have a critical point. There is a unique solution that passes through the critical point that has 1/2gammabeta = 1 and B-r = 0 there, where gamma is the polytropic index, beta the local plasma beta and B-r the radial component, of the magnetic field. The critical point is an umbilical point, where sound and Alfven speeds are equal, and the transcritical solution undergoes a change from a fast-mode to a slow-mode expansion at the critical point. Slow-mode expansions exist for field-flow orientations where the angle between field and flow lies either between 90degrees and 180degrees or between 270degrees and 360degrees. There is also an umbilic point in these solutions when the initial plasma beta beta(0) exceeds a critical value beta(crit). When beta(o) greater than or equal to beta(crit), the solutions require a transition through a critical point. When beta(o) beta(crit), there is a smooth solution involving all inflection in the density and angular velocity. For other angles between field and flow, all the slow-mode waves are compressive. An analytic solution for the case of a magnetic field everywhere perpendicular to the flow with gamma = 2 is presented.