Similarity solution for cylindrical magnetohydrodynamic blast waves

Abstract

A similarity solution is obtained for the flow behind a very strong (in the hydrodynamic sense) cylindrical magnetohydrodynamic shock wave produced by the sudden release of energy along a line of infinite extent in a plasma. The plasma is assumed to be an ideal gas with infinite electrical conductivity, and to be permeated by the azimuthal magnetic field of a line current. It is shown that it is of critical importance to take into account the ambient magnetic pressure, no matter how small. It is found that, to preserve similarity, the external circuit is required to maintain a constant axial current; this result also appears in the related problem, treated by Greenspan, where the ambient plasma is nonconducting. This boundary condition is shown to have some interesting consequences, especially with regard to the energy content of the system. The dependence of the shock speed on the explosive energy is determined as a function of the ambient magnetic field both for the present case and for Greenspan's case, and interesting differences are noted. Other differences between the two cases are also discussed.