Magnetohydrodynamic shock wave formation: effect of area and density variation

Abstract

The nonlinear steepening of finite amplitude Magnetohydrodynamic (MHD) waves propagating perpendicular to the magnetic field is investigated. The nonlinear evolution of a planar fast magnetosonic wave in a homentropic flow field is understood well through simple waves. However, in situations where the wave is moving through a variable area duct or when the flow field is nonhomentropic, the concept of simple waves cannot be used. In the present paper, the quasi-one-dimensional MHDequations that include the effect of area variation and density gradients are solved using the wave front expansion technique. The analysis is performed for a perfectly conducting fluid and also for a weakly conducting fluid. Closed form solutions are obtained for the nonlinear evolution of the slope of the wave front in the limits of infinitely large and small conductivity. A general criterion for a compression wave to steepen into a shock is obtained. An analytical expression for the location of shock formation is derived. The effect of area variation and density gradient on shock formation is studied and examples highlighting the same are presented.