Part I: The Equations of Plasma Physics and the Richtmyer-Meshkov Instability in Magnetohydrodynamics. Part II: Evolution of Perturbed Planar Shockwaves.

Abstract

Part I: Mitigating the Richtmyer-Meshkov instability (RMI) is critical for energy production in inertial confinement fusion. Suitable plasma models are required to study the hydrodynamic and electromagnetic interactions associated with the RMI in a conducting medium. First, a sequence of asymptotic expansions in several small parameters, as formal limits of the non-dissipative and non-resistive two-fluid plasma equations, leads to five simplified plasma/magnetohydrodynamics (MHD) systems. Each system is characterized by its own physical range of validity and dispersion relations, and includes the widely used magnetohydrodynamic (MHD) and Hall-MHD equations. Next we focus on the RMI in MHD. Using ideal MHD, it has been shown that the RMI is suppressed by the presence of an external magnetic field. We utilize the incompressible, Hall-MHD model to investigate the stabilization mechanism when the plasma ion skin depth and Larmor radius are nonzero. The evolution of an impulsively accelerated, sinusoidally perturbed density interface between two conducting fluids is solved as a linearized initial-value problem. An initially uniform background magnetic field of arbitrary orientation is applied. The incipient RMI is found suppressed through oscillatory motions of the interface due to the ion cyclotron effect. This suppression is most effective for near tangential magnetic fields but becomes less effective with increasing plasma length scales. The vorticity dynamics that facilitates the stabilization is discussed. Part II: We consider the evolution of a planar gas-dynamic shock wave subject to smooth initial perturbations in both Mach number and shock shape profile. A complex variable formulation for the general shock motion is developed based on an expansion of the Euler equations proposed by Best [Shock Waves, {1}: 251-273, (1991)]. The zeroth-order truncation of Best s system is related to the well-known geometrical shock dynamics (GSD) equations while higher-order corrections provide a hierarchy of closed systems, as detailed initial flow conditions immediately behind the shock are prescribed. Solutions to Best s generalized GSD system for the evolution of two-dimensional perturbations are explored numerically up to second order in the weak and strong shock limits. Two specific problems are investigated: a shock generated by an impulsively accelerated piston with a corrugated surface, and a shock traversing a density gradient. For the piston-driven flow, it is shown that this approach allows full determination of derivative jump conditions across the shock required to specify initial conditions for the retained, higher-order correction equations. In both cases, spontaneous development of curvature singularity in the shock shape is detected. The critical time at which a singularity occurs follows a scaling inversely proportional to the initial perturbation size. This result agrees with the weakly nonlinear GSD analysis of Mostert et al. [J. Fluid Mech., {846}: 536-562, (2018)].