Linear Analysis of Converging Richtmyer–Meshkov Instability in the Presence of an Azimuthal Magnetic Field

Abstract

We investigate the linear stability of both positive and negative Atwood ratio interfaces accelerated either by a fast magnetosonic or hydrodynamic shock in cylindrical geometry. For the magnetohydrodynamic (MHD) case, we examine the role of an initial seed azimuthal magnetic field on the growth rate of the perturbation. In the absence of a magnetic field, the Richtmyer-Meshkov growth is followed by an exponentially increasing growth associated with the Rayleigh-Taylor instability. In the MHD case, the growth rate of the instability reduces in proportion to the strength of the applied magnetic field. The suppression mechanism is associated with the interference of two waves running parallel and anti-parallel to the interface that transport vorticity and cause the growth rate to oscillate in time with nearly a zero mean value. IntroductionThe Richtmyer-Meshkov instability (RMI) occurs when a shock wave impulsively accelerates a perturbed density interface separating two different fluids [1,2]. RMI may not be classified as a classical hydrodynamic instability with modal analysis because the perturbations grow algebraically (linearly in time) rather than exponentially. The RMI occurs in a variety of physical applications. It is important in astrophysical phenomena such as supernova explosions, supersonic combustion in hypersonic air-breathing engines, and inertial confinement fusion (ICF) [3,4,5]. In direct-drive ICF, small capsules containing a deuterium and tritium fuel are compressed by laser-generated shock waves, where achieving sufficiently high pressures and temperatures inside the compressed ICF targets to ignite the fuel is the primary goal of this process. On the other hand, in indirect drive approach to ICF, the driver energy (from laser beams or ion beams) is first absorbed in a high-Z enclosure (a hohlraum), which surrounds the capsule. The material heated by the driver releases X-rays, which drive the capsule implosion [6]. The mixing between the outer shell and the inner fuel inhibits the fusion reaction due to RMI and RTI (Rayleigh Taylor instability) which limits the energy production [7,8,9].The RMI in the presence of a magnetic field initially oriented parallel to the incident shock direction has been investigated numerically and theoretically. Nonlinear ideal magnetohydrodynamics (MHD) simulations of a shock interaction with an oblique planar contact discontinuity were discussed by Samtaney [10]. It was demonstrated that the RMI is suppressed by the existence of such a magnetic field. The primary suppression mechanism may be attributed to transport of baroclinic vorticity away from the density interface by slow-or intermediate mode shocks reducing the vorticity at the interface. This results were reconfirmed in the context of incompressible MHD linear stability analysis by Wheatley et al. [11]. They demonstrated that vorticity is transported away from the interface by Alfvén fronts. More detailed nonlinear MHD simulations extending the parameter ranges of earlier investigations [12] demo...

Incompressible models of magnetohydrodynamic Richtmyer-Meshkov instability in cylindrical geometry

“…Then, we present the numerical solution of the incompressible models in the cylindrical geometry, and we compare the results with compressible linear simulations of MHD RMI for both cases of normal and azimuthal magnetic fields. These compressible simulations have been introduced and investigated by Bakhsh et al [20,23]. A uniform grid with 1024 cells is utilized for the numerical solution.…”

“…Next, the results of the incompressible model in cylindrical geometry are presented and compared with the numerical simulations of the compressible MHD RMI which have been previously discussed in detail in Refs. [20,23].…”

“…For large m, the model shows better results because it reproduces the right frequency and magnitude of the amplitude of the interface and matches the linear simulation results at early scaled time. The investigation of compressible MHD RMI in the presence of an azimuthal magnetic field in cylindrical geometry [23] explained the dominant physical effect was the transport of vorticity by waves propagating at the local Alfvén speed, parallel and antiparallel to the interface. The interference of these waves, with alternating phase causes constructive and destructive interference and vorticity patterns alternate with changing signs.…”

“…This can be explained as follows: the vorticity on either side of the interface is transported by the effect of waves propagating at the local Alfvén speed, parallel and antiparallel to the interface. The interference of these waves alternating phase causes constructive and destructive interference, and vorticity patterns alternate with changing sign [23].…”

Linear Stability of an Impulsively Accelerated Density Interface in an Ideal Two-Fluid Plasma

“…A purely numerical approach for linear analysis of the RM instability was developed for hydrodynamics and MHD 6 . This numerical approach was then used to investigate RM and Rayleigh-Taylor instabilities in cylindrical geometry by Bakhsh et al 7 and Baksh & Samtaney 8 .…”