Dynamics and hydrodynamic instability of magnetohydrodynamic implosions

Abstract

Inertial confinement fusion (ICF) is a promising candidate for renewable energy generation. It involves the compression of a spherical capsule of deuterium-tritium fuel by the action of an imploding shock wave. In concept, the shock wave provides thermonuclear ignition of the fuel on convergence to the capsule centre, and the inward inertial motion of the surrounding fuel provides confinement long enough for the fusion burn to be sustained. The process, however, suffers from the presence of hydrodynamic instabilities in the confinement process. As the fluids involved in such processes are plasmas, they may be affected by magnetic fields. There is some evidence that application of a magnetic field to a planar flow in ideal magnetohydrodynamics (MHD) suppresses the Richtmyer-Meshkov instability, and the possibility exists of it suppressing this instability in converging flows as well. There is also experimental evidence suggesting increased performance of ICF due to electron and alpha particle confinement by the magnetic field. However, the application of a magnetic field to a converging plasma may disrupt the ability of the imploding shock to produce thermonuclear ignition. Here, the dynamic effects of the external application of magnetic fields (referred to as seed magnetic fields) to converging plasma flows such as those seen in ICF are investigated numerically and analytically under the framework of ideal magnetohydrodynamics (MHD). In the first study, cylindrical and spherical implosions in MHD are examined. We formulate Riemann problems to generate a full set of imploding waves and determine the symmetry features in the flow. Three configurations of seed magnetic fields are applied: a uniform, unidirectional field, an azimuthal, axisymmetric field which is zero in part of the domain, and a field with a stagnation point at the centre of the domain (referred to as a “saddle”-field). The application of a given field results in the breaking of the axi- or spherisymmetry (for cylindrical or spherical geometry respectively) of the implosion into a symmetry type which matches that of the applied field. The cylindrical geometry with the azimuthal, axisymmetric field therefore shows an axisymmetric implosion, unlike its spherical counterpart which shows a profound disruption of spherisymmetry. The least distorted shock wave implosions are provided by “saddle”-fields, in both cylindrical and spherical problems. Second, we consider the cylindrical collapse of a shock wave onto a constant current which runs along the cylinder axis. This current produces an axisymmetric magnetic field which varies with radius r as 1/r and which, unlike the similar field considered in the first study, is singular on the axis. The shock Mach number and pressure ratio are both found to weaken as M(r)-1∼r and p(r)-1∼r respectively, making the use of a constant current such as this unsuitable since it would inhibit thermonuclear fusion. By varying the current to zero precisely as the shock collapses onto it in a power-law decrease with decay exponent μ, as is done in the third study, the picture becomes more complicated. A strong dependency of shock behaviour on the choice of m shows up to five distinct behaviours. However, a strong-shock singular pressure of, at weakest, p(r)∼r^((4-13μ)/(4(μ+1))) could be assured by choosing μ>4/13. A choice of μ>0.816 could further ensure strong-shock collapse comparable to that of gas-dynamic shocks, with M(r)∼r^(-1/n) and p(r)∼r^(-2/n), where n≃0.225425 for the plasma of interest. Finally, we investigate the effect of the magnetic field on the Richtmyer-Meshkov instability in cylindrical and spherical numerical formulations similarly to the first study. The uniform, unidirectional and “saddle”-field configurations are considered only. A perturbed density interface is initialized on the interior of the initial Riemann interface, and the Richtmyer-Meshkov and Rayleigh-Taylor instabilities are provoked by the action of the incoming MHD shocks. Both instabilities are suppressed by the seed magnetic field. Suppression extent is largely insensitive to the choice of field configuration, but does show different behaviour depending on the local orientation of the magnetic field to the interface, as the vorticity which causes perturbation growth is transported along field lines. Three-dimensional magnetic fields show weaker suppression of the instabilities for a given field strength. The studies suggest that a seed magnetic field could be used to increase performance of ICF by suppression of hydrodynamic instabilities such as, in particular, the Richtmyer-Meshkov instability. However, the field should be judiciously chosen in both its configuration and strength in order to avoid the effects of distortion or weakening of imploding shock waves.