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

Abstract

Abstract. 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 an 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 ½γβ = 1 and B_r = 0 there, where γ is the polytropic index, β the local plasma beta and B_r the radial component of the magnetic field. The critical point is an umbilical point, where sound and Alfvén 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 90° and 180° or between 270° and 360°. There is also an umbilic point in these solutions when the initial plasma beta β₀ exceeds a critical value β_crit. When β₀ [ges ] β_crit, the solutions require a transition through a critical point. When β₀ < β_crit, there is a smooth solution involving an 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 γ = 2 is presented.