Interactions of Hydromagnetic Waves

Abstract

A generalized Rankine-Hugoniot relation for hydromagnetic shock waves propagating perpendicularly to the magnetic field is discussed. Then the interaction of the shock or rarefaction waves with the contact surface on which the magnetic field has a discontinuity is, analyzed by a graphical method. The motion of the high temperature part of a plasma surrounded by the strong magnetic field and the electric current is investigated applying the result of the above analysis. It is shown that under some initial condition the contracting motion of the plasma occurs succeeding its expansion. The results obtained by them show that the propagation of shocks is generally complicated for the arbitrary configuration of the magnetic field. However, it was shown by Taniuti G ) that the analysis of the fluid with infinite conductivity can be comparatively simplified, and that the propagation of hydromagnetic waves may be described in the way analogous to that of the ordinary hydrodynamics so far as the one-dimensional propagation of the hydromagnetic wave perpendicular to the magnetic field is concerned. In the present article, we deal with these perpendicular shock and rarefaction waves and their interaction. In § 2, the various shock relations, involving an equation which corresponds to the Rankine-Hugoniot equation in the hydrodynamics, will be presented. 4 ) The analysis of the interaction between the shock, rarefaction wave and the contact discontinuity will be given in § 3. In § 4 we shall discuss, on the basis of these results, an initial value problem in which the fluid is initially at rest and the high density part of the fluid associated with a weak magnetic field is surrounded by the low density part with a strong magnetic field. The subsequent b~haviour of the fluid and the magnetic field will be calculated in detail and it will be pointed out that under certain conditions the contact surface can oscillate due to its interaction with the rarefaction and shock waves. We hope that this result concerning the motion of the contact surface may be of some use for the qualitative explanations of some experimental facts, resulting from the pinched discharge in the cylindrical tube, although it may be difficult to find any actual phenomenon exactly corresponding with this kind of initial value problem.