The Richtmyer–Meshkov instability in magnetohydrodynamics

In ideal magnetohydrodynamics (MHD), the Richtmyer–Meshkov instability can be suppressed by the presence of a magnetic field. The interface still undergoes some growth, but this is bounded for a finite magnetic field. A model for this flow has been developed by considering the stability of an impulsively accelerated, sinusoidally perturbed density interface in the presence of a magnetic field that is parallel to the acceleration. This was accomplished by analytically solving the linearized initial value problem in the framework of ideal incompressible MHD. To assess the performance of the model, its predictions are compared to results obtained from numerical simulation of impulse driven linearized, shock driven linearized, and nonlinear compressible MHD for a variety of cases. It is shown that the analytical linear model collapses the data from the simulations well. The predicted interface behavior well approximates that seen in compressible linearized simulations when the shock strength, magnetic field strength, and perturbation amplitude are small. For such cases, the agreement with interface behavior that occurs in nonlinear simulations is also reasonable. The effects of increasing shock strength, magnetic field strength, and perturbation amplitude on both the flow and the performance of the model are investigated. This results in a detailed exposition of the features and behavior of the MHD Richtmyer–Meshkov flow. For strong shocks, large initial perturbation amplitudes, and strong magnetic fields, the linear model may give a rough estimate of the interface behavior, but it is not quantitatively accurate. In all cases examined the accuracy of the model is quantified and the flow physics underlying any discrepancies is examine

Effects of magnetic fields on magnetohydrodynamic cylindrical and spherical Richtmyer-Meshkov instability

The effects of seed magnetic fields on the Richtmyer-Meshkov instability driven by converging cylindrical and spherical implosions in ideal magnetohydrodynamics are investigated. Two different seed field configurations at various strengths are applied over a cylindrical or spherical density interface which has a single-dominant-mode perturbation. The shocks that excite the instability are generated with appropriate Riemann problems in a numerical formulation and the effect of the seed field on the growth rate and symmetry of the perturbations on the density interface is examined. We find reduced perturbation growth for both field configurations and all tested strengths. The extent of growth suppression increases with seed field strength but varies with the angle of the field to interface. The seed field configuration does not significantly affect extent of suppression of the instability, allowing it to be chosen to minimize its effect on implosion distortion. However, stronger seed fields are required in three dimensions to suppress the instability effectively

Converging cylindrical magnetohydrodynamic shock collapse onto a power-law-varying line current

We investigate the convergence behaviour of a cylindrical, fast magnetohydrodynamic (MHD) shock wave in a neutrally ionized gas collapsing onto an axial line current that generates a power law in time, azimuthal magnetic field. The analysis is done within the framework of a modified version of ideal MHD for an inviscid, non-dissipative, neutrally ionized compressible gas. The time variation of the magnetic field is tuned such that it approaches zero at the instant that the shock reaches the axis. This configuration is motivated by the desire to produce a finite magnetic field at finite shock radius but a singular gas pressure and temperature at the instant of shock impact. Our main focus is on the variation with shock radius r, as r→0, of the shock Mach number M(r) and pressure behind the shock p(r) as a function of the magnetic field power-law exponent μ ⩾ 0, where μ = 0 gives a constant-in-time line current. The flow problem is first formulated using an extension of geometrical shock dynamics (GSD) into the time domain to take account of the time-varying conditions ahead of the converging shock, coupled with appropriate shock-jump conditions for a fast, symmetric MHD shock. This provides a pair of ordinary differential equations describing both M(r) and the time evolution on the shock, as a function of r, constrained by a collapse condition required to achieve tuned shock convergence. Asymptotic, analytical results for M(r) and p(r) are obtained over a range of μ for general γ, and for both small and large r. In addition, numerical solutions of the GSD equations are performed over a large range of r, for selected parameters using γ=5/3. The accuracy of the GSD model is verified for some cases using direct numerical solution of the full, radially symmetric MHD equations using a shock-capturing method. For the GSD solutions, it is found that the physical character of the shock convergence to the axis is a strong function of μ. For 0 ⩽ μ < 4/13, M and p both approach unity at shock impact r=0 owing to the dominance of the strong magnetic field over the amplifying effects of geometrical convergence. When μ⩾0.816 (for γ=5/3), geometrical convergence is dominant and the shock behaves similarly to a converging gas dynamic shock with singular M(r) and p(r), r→0. For 4/13 < μ ⩽ 0.816 three distinct regions of M(r) variation are identified. For each of these p(r) is singular at the axis

Impulse-driven Richtmyer-Meshkov instability in Hall-magnetohydrodynamics

We utilize the incompressible, Hall-magnetohydrodynamics (MHD) model for conducting fluids to investigate the effect of Hall current on the stability of an impulsively accelerated, perturbed density interface, or contact discontinuity (CD) separating two fluids in the presence of a background magnetic field. This is used as a simple model, in a conducting fluid, of a Richtmyer-Meshkov type flow that is characterized in a neutral fluid by a shock-wave-density-interface interaction. Two versions of the Hall-MHD equations are explored. In the first, the ions are treated as an incompressible fluid but the electron gas retains its compressibility, while for the second version the incompressible limit for both species is invoked. The linearized equations of motion are first formulated for a sinusoidal interface perturbation and then solved as an initial-value problem using a Laplace transform method with general numerical inversion but with analytical inversion for some limiting-parameter cases. While the field equations are identical for both Hall-MHD models, the CD-jump conditions differ leading to qualitatively similar but quantitatively different CD dynamics. For both models, the presence of the magnetic field is found to suppress the incipient interfacial growth associated with neutral-gas, Richtmyer-Meshkov instability (RMI). When the ion skin depth is finite, the vorticity dynamics that drive the suppression of the RMI differ markedly from the ideal MHD, RMI flow. On the interface, the Hall-MHD description allows the presence of a tangential slip velocity which leads to finite circulation deposition. Away from the interface, vorticity is produced by the perturbed magnetic fields and transported to infinity by a dispersive wave system. This leads to decay of the velocity slip at the interface with the effect that interface growth remains bounded but distorted by damped oscillations associated with the ion cyclotron effect. The flow behavior for several limits of the ion skin depth and the Larmor radius is explored. Specific comparisons with the results from both models against ideal MHD are presented

Geometrical shock dynamics for magnetohydrodynamic fast shocks

We describe a formulation of two-dimensional geometrical shock dynamics (GSD) suitable for ideal magnetohydrodynamic (MHD) fast shocks under magnetic fields of general strength and orientation. The resulting area–Mach-number–shock-angle relation is then incorporated into a numerical method using pseudospectral differentiation. The MHD-GSD model is verified by comparison with results from nonlinear finite-volume solution of the complete ideal MHD equations applied to a shock implosion flow in the presence of an oblique and spatially varying magnetic field ahead of the shock. Results from application of the MHD-GSD equations to the stability of fast MHD shocks in two dimensions are presented. It is shown that the time to formation of triple points for both perturbed MHD and gas-dynamic shocks increases as ϵ^(-1), where ϵ is a measure of the initial Mach-number perturbation. Symmetry breaking in the MHD case is demonstrated. In cylindrical converging geometry, in the presence of an azimuthal field produced by a line current, the MHD shock behaves in the mean as in Pullin et al. (Phys. Fluids, vol. 26, 2014, 097103), but suffers a greater relative pressure fluctuation along the shock than the gas-dynamic shock

Magnetohydrodynamic implosion symmetry and suppression of Richtmyer-Meshkov instability in an octahedrally symmetric field

We present numerical simulations of ideal magnetohydrodynamics showing suppression of the Richtmyer-Meshkov instability in spherical implosions in the presence of an octahedrally symmetric magnetic field. This field configuration is of interest owing to its high degree of spherical symmetry in comparison with previously considered dihedrally symmetric fields. The simulations indicate that the octahedral field suppresses the instability comparably to the other previously considered candidate fields for light-heavy interface accelerations while retaining a highly symmetric underlying flow even at high field strengths. With this field, there is a reduction in the root-mean-square perturbation amplitude of up to approximately 50% at representative time under the strongest field tested while maintaining a homogeneous suppression pattern compared to the other candidate fields

Converging cylindrical shocks in ideal magnetohydrodynamics

We consider a cylindrically symmetrical shock converging onto an axis within the framework of ideal, compressible-gas non-dissipative magnetohydrodynamics (MHD). In cylindrical polar co-ordinates we restrict attention to either constant axial magnetic field or to the azimuthal but singular magnetic field produced by a line current on the axis. Under the constraint of zero normal magnetic field and zero tangential fluid speed at the shock, a set of restricted shock-jump conditions are obtained as functions of the shock Mach number, defined as the ratio of the local shock speed to the unique magnetohydrodynamic wave speed ahead of the shock, and also of a parameter measuring the local strength of the magnetic field. For the line current case, two approaches are explored and the results compared in detail. The first is geometrical shock-dynamics where the restricted shock-jump conditions are applied directly to the equation on the characteristic entering the shock from behind. This gives an ordinary-differential equation for the shock Mach number as a function of radius which is integrated numerically to provide profiles of the shock implosion. Also, analytic, asymptotic results are obtained for the shock trajectory at small radius. The second approach is direct numerical solution of the radially symmetric MHD equations using a shock-capturing method. For the axial magnetic field case the shock implosion is of the Guderley power-law type with exponent that is not affected by the presence of a finite magnetic field. For the axial current case, however, the presence of a tangential magnetic field ahead of the shock with strength inversely proportional to radius introduces a length scale R = √μ_0/p_0I/(2π) where I is the current, μ_0 is the permeability, and p_0 is the pressure ahead of the shock. For shocks initiated at r ≫ R, shock convergence is first accompanied by shock strengthening as for the strictly gas-dynamic implosion. The diverging magnetic field then slows the shock Mach number growth producing a maximum followed by monotonic reduction towards magnetosonic conditions, even as the shock accelerates toward the axis. A parameter space of initial shock Mach number at a given radius is explored and the implications of the present results for inertial confinement fusion are discussed

Effects of seed magnetic fields on magnetohydrodynamic implosion structure and dynamics

The effects of various seed magnetic fields on the dynamics of cylindrical and spherical implosions in ideal magnetohydrodynamics are investigated. Here, we present a fundamental investigation of this problem utilizing cylindrical and spherical Riemann problems under three seed field configurations to initialize the implosions. The resulting flows are simulated numerically, revealing rich flow structures, including multiple families of magnetohydrodynamic shocks and rarefactions that interact non-linearly. We fully characterize these flow structures, examine their axi- and spherisymmetry-breaking behaviour, and provide data on asymmetry evolution for different field strengths and driving pressures for each seed field configuration. We find that out of the configurations investigated, a seed field for which the implosion centre is a saddle point in at least one plane exhibits the least degree of asymmetry during implosion