On the Richtmyer-Meshkov Instability in Magnetohydrodynamics

Citation

Wheatley, Vincent (2005) On the Richtmyer-Meshkov Instability in Magnetohydrodynamics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/N407-2B54. https://resolver.caltech.edu/CaltechETD:etd-05272005-145538

Abstract

The Richtmyer-Meshkov instability is important in a wide variety of applications including inertial confinement fusion and astrophysical phenomena. In some of these applications, the fluids involved may be plasmas and hence be affected by magnetic fields. For one configuration, it has been numerically demonstrated that the growth of the instability in magnetohydrodynamics is suppressed in the presence of a magnetic field. Here, the nature of this suppression is theoretically and numerically investigated.

In the framework of ideal incompressible magnetohydrodynamics, we examine the stability of an impulsively accelerated, sinusoidally perturbed density interface in the presence of a magnetic field that is parallel to the acceleration. This is accomplished by analytically solving the linearized initial value problem, which is a model for the Richtmyer-Meshkov instability. We find that the initial growth rate of the interface is unaffected by the presence of a magnetic field, but for a finite magnetic field the interface amplitude asymptotes to a constant value. Thus the instability of the interface is suppressed. The interface behavior from the analytical solution is compared to the results of both linearized and non-linear compressible numerical simulations for a wide variety of conditions.

We then consider the problem of the regular refraction of a shock at an oblique, planar contact discontinuity separating conducting fluids of different densities in the presence of a magnetic field aligned with the incident shock velocity. Planar ideal MHD simulations indicate that the presence of a magnetic field inhibits the deposition of vorticity on the shocked contact, which leads to the suppression of the Richtmyer-Meshkov instability. We show that the shock refraction process produces a system of five to seven plane waves that may include fast, intermediate, and slow MHD shocks, slow compound waves, 180° rotational discontinuities, and slow-mode expansion fans that intersect at a point. In all solutions, the shocked contact is vorticity free and hence stable. These solutions are not unique, but differ in the type of waves that participate. The set of equations governing the structure of these multiple-wave solutions is 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 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 applied field magnitude, bounded by either two slow compound waves or two 180° 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 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 and there is shown to be excellent agreement between the two.

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