On cylindrical magnetohydrodynamic shock waves

Abstract

If an axial rod is surrounded by an ionized gas, an expanding cylindrical shock wave can be produced by passing through the gas a current which returns along the rod. The azimuthal magnetic field of the current acts like a piston, pushing the plasma away from the rod and leaving behind a cylindrical vacuum region. The case considered is that where a uniform magnetic field parallel to the axis is initially present in the gas; in this case a transverse magnetohydrodynamic shock wave results from the current discharge. The flow is analyzed under the assumptions that the plasma is a nonviscous, nonheat-conducting, ideal gas of infinite electrical conductivity, and that the discharge current increases linearly with time. The analysis is made first on the basis of the "snowplow" theory of Rosenbluth, and then from a similarity solution of the full magnetohydrodynamic equations. The results of the two solutions are compared for the case = 7/5. It is found that the speed predicted by the snowplow theory is in very good agreement with the speed of the contact front obtained from the solution of the full equations over the entire range of shock strength, but that the snowplow speed is a good approximation to the shock speed only in the limit of strong shocks. The effect on the flow of varying the axial field is discussed.