Stability of an Impulsively Accelerated Density Interface in Magnetohydrodynamics

Abstract

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 nonlinear compressible numerical simulations. In these applications, the fluids involved may be ionized and hence be affected by magnetic fields. Samtaney [4] has demonstrated, via numerical simulations, that the growth of the RMI is suppressed in the presence of a magnetic field. The particular flow studied was that of a shock interacting with an oblique planar contact discontinuity (CD) separating conducting fluids of different densities within the framework of planar ideal magnetohydrodynamics (MHD). It was shown that the suppression of the instability is caused by changes in the shock refraction process at the CD with the application of a magnetic field [5]. These changes prevent the deposition of circulation on the CD.A more widely studied flow results from a shock wave accelerating a density interface with a single-mode sinusoidal perturbation in amplitude. Our goal is to understand the effect of a magnetic field on this flow when conducting fluids are involved. The magnetic field is again aligned with the motion of the shock. The initial condition for this flow is illustrated in Fig. 1(a). It is characterized by the incident shock sonic Mach number, M, the density ratio across the CD, 2 = 1 , the ratio of the CD's initial amplitude to its wavelength, 0 =, the ratio of specific heats, , and the nondimensional strength of the applied magnetic field, ÿ1 B 2 =2p 0 . Here B is the magnitude of the applied magnetic field and p 0 is the initial pressure in the unshocked regions of the flow. As a model for this flow, we will examine the growth of a sinusoidally perturbed interface separating incompressible conducting fluids that is impulsively accelerated at t 0. The setup for the model problem is illustrated in Fig. 1(b). This problem is characterized by 1 = , 2 = , 0 =, , and the normalized magnitude of the impulse, V =p 0 p . We choose to be 1 from the corresponding shock driven flow. In this investigation, it is convenient to consider solutions to the linearized equations of ideal, incompressible MHD in a noninertial reference frame that has acceleration Vt in the z direction. Here, t is the Dirac delta function and V c. The equations are linearized about a base flow that results from the impulsive acceleration of an unperturbed interface. This flow has no x dependence and zero vertical velocity (u). ...

Richtmyer–Meshkov instability: theory of linear and nonlinear evolution

“…This interaction is mediated by sound waves. Besides compressibility, other interesting physical phenomena, for example shocks propagating into viscous fluids (Miller & Ahrens 1991), the influence of elastic/plastic properties of solid materials Piriz et al 2008) and the effect of magnetic fields on the dynamics of corrugated shocks in plasmas (Wheatley et al 2005), are certainly worth studying. In fact, when dealing with more complex properties and states of matter, as well as the inclusion of magnetic fields for shocks propagating in plasmas, the interaction between the shock fronts and boundaries may be sustained not only by the longitudinal adiabatic sonic waves, but with another family of wave fronts that propagate at speeds other than the speed of sound in ordinary gases.…”

A method to simulate linear stability of impulsively accelerated density interfaces in ideal-MHD and gas dynamics

“…We will appeal to such an interpretation in discussing results from our numerical simulations. Recently, Samtaney [5] has shown that the RM instability is suppressed by the presence of a magnetic field, which was also confirmed analytically by Wheatley et al [6]. The magneto-hydrodynamics (MHD) RM instability includes nonlinear MHD waves such as a slow-mode shock [5,7].…”

“…The amplitude of the oscillations are larger for stronger incident shocks as expected. In Figure 1(d) the amplitude of the density interface for each shock strength are plotted and for comparison the saturation amplitudes predicted by the incompressible theory of Wheatley et al [6] are shown as horizontal lines. Predictably, the numerical simulations agree well with the incompressible theory at low Mach numbers, with the differences growing with increasing Mach number.…”

Linear Simulations of the Cylindrical Richtmyer-Meshkov Instability in Hydrodynamics and MHD

“…This was further reconfirmed by analytical incompressible Magnetohydrodynamics (MHD) theory of an impulsively accelerated interface developed by Wheatley et al [7,8,9]. In Samtaney's work, the magnetic field was aligned to the motion of the shock, the suppression of the RMI was caused by the bifurcation of the vortex sheet which transported the baroclinically generated vorticity away from the contact discontinuity by a pair of MHD shocks.…”