Speaker: Vincent Wheatley, Graduate Student, California Institute of Technology
We consider the problem of regular refraction (where regular implies all waves meet at a single point) of a shock at an oblique, planar contact discontinuity separating conducting fluids of different densities in the presence of a magnetic field. Ideal MHD simulations indicate that the presence of a magnetic field inhibits the Kelvin-Helmholtz instability of the shocked contact. We show that the shock refraction process produces a system of from five to seven plane waves that may include fast, intermediate, and slow MHD shocks, slow compound waves, $180^o$ rotational discontinuities, and slow-mode expansion fans, that intersect at a point. These solutions are not always unique; up to four different entropy-satisfying solutions exist for certain parameter ranges. These non-unique solutions differ in the type of discontinuous waves that participate. Thus the physical solution may be selected based on which types of discontinuous waves are considered realistic. The set of equations governing the structure of these multiple-wave solutions are obtained in which fluid property variation is allowed only in the azimuthal direction about the wave-intersection point. Corresponding solutions are referred to as either quintuple-points, sextuple-points, or septuple-points, depending on the number of participating discontinuous waves. A numerical method of solution is described and examples are compared to the results of numerical simulations for moderate magnetic field strengths. The limit of vanishing magnetic field at fixed permeability and pressure is studied for two solution types. The relevant solutions correspond to the hydrodynamic triple-point with the shocked contact replaced by a singular structure consisting of a wedge, whose angle scales with the field strength, bounded by either two slow compound waves or two $180^o$ rotational discontinuities, each followed by a slow-mode expansion fan. These bracket the MHD contact which itself cannot support a tangential velocity jump in the presence of a non-parallel magnetic field. The magnetic field strength within the singular wedge is finite and the shock-induced change in tangential velocity across the wedge is supported by the expansion fans that form part of the compound waves or follow the rotational discontinuities. To verify these findings, an approximate leading order asymptotic solution appropriate for both flow structures was computed. The full and asymptotic solutions are compared quantitatively.