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 Br = 0 there,
where γ is the polytropic index, β the local plasma beta and Br 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 βcrit.
When β0 [ges ] βcrit, the solutions require a transition through a critical point. When
β0 < βcrit, 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.
Mathematical Theory of One-dimensional Isothermal Blast Waves in a Magnetic Field
