Numerical Experiments of Magnetohydrodynamic Shocks and the Stability of Switch-On Shocks

Abstract

Numerical experiments are performed on a variety of magnetohydrodynamic shock waves to determine their behavior in a dissipative fluid. The Lundquist equations are integrated with Lax's finite difference scheme in which physical dissipation is imitiated by numerical dissipation. Fast and slow magnetohydrodynamic shocks respond to small disturbances just as predicted by linear nondissipative theory. The disintegration of an intermediate (nonevolutionary) shock is followed in time to its final (stable) flow pattern. In this case, the disturbance builds up inside the shock structure to a maximum, and the shock splits into two shocks (e.g., a switch-on and a switch-off shock) separated by a uniform region. This analysis gives the qualitatively correct time behavior of the splitting which, for small times, agrees with previous linearized dissipative results. The magnetohydrodynamic switch-on shock is examined using the same techniques. Small transverse disturbances, which might split the shock according to a linear analysis, merely rotate the magnetic field in the plane of the shock (repolarize the shock) and leave an Alfvén wave tail behind it. The fluid dynamic properties of the shock are not changed, the shock does not split, and therefore the switch-on shock is stable. In short, this approach overcomes the insufficiencies of linear, nondissipative analyses and gives physically realistic and reasonable answers to the long unresolved questions of magnetohydrodynamic shock stability.