High-Field Steady-State Shock Wave in a Plasma

The high magnetic field limit, of a shock wave in a steady-state model plasma moving along the x axis, is studied transverse to the magnetic field, in which the only spatial dependence is that on the x coordinate. The collisions of the model plasma are described by (Krook-type) modified Boltzmann collision terms with constant relaxation times. What occurs is that drift velocity of both electrons and ions behind the transition is converted to an increase in temperature of both species ahead of the transition through ion momentum exchange resulting in Joule heating. First formulation of the equations describing this phenomenon is presented, and then an exact asymptotic development of the integro-differential transport equations, together with the Maxwell equations and subsidiary conditions. It is shown analytically, in the case for which the relaxation times of the ions and electrons are equal, that a shock wave is possible, provided the plasma is sufficiently dense, which means that the shock is bounded away from the low-temperature plasma limit. The case in which the relaxation times are unequal reduces to the investigation of a nonlinear ordinary differential equation which is not carried out here, but it is conjectured that a shock wave is still possible.