Speaker: Vincent Wheatley, Graduate Student, California Institute of Technology

Abstract:

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.

Last modified: Tue Nov 11 15:35:09 EST 2003