The structure of resistive‐dispersive intermediate shocks

Abstract

The structure of intermediate shocks is studied on the basis of the resistive, nonviscous two‐fluid equations. Electron inertia effects are neglected so that the generalized Ohm's law contains only the Hall current and the electron pressure terms in addition to the usual resistive term and the electric field. As for the case of purely resistive MHD, reported recently by Hau and Sonnerup (Journal of Geophysical Research, 94, 6539, 1989), fixed‐point analysis is performed to examine the nature of the magnetic structure near the upstream and downstream states of the intermediate shock. The one‐dimensional, steady state, resistive Hall MHD equations are then integrated numerically to generate complete shock structures which are presented in the form of magnetic hodograms. These hodograms describe fast and slow shocks in addition to intermediate shocks. As expected, the calculations show that the main effect of Hall currents is to remove the symmetry between left‐hand and right‐hand polarized shock structures found in the purely resistive case and sometimes to convert the smooth shock transitions obtained from the resistive MHD model into transitions that incorporate oscillatory standing wave train structures at their upstream and/or downstream edge. The magnetic structure in the plane of the shock near the possible upstream and downstream states of the intermediate shock, which in the case of purely resistive MHD is either a node or a saddle, can be either a node, a saddle or a spiral point, the latter corresponding to a standing wave train, when the Hall term is included. As a result, the number of possible types of magnetic hodogram topology increases from 3 in the resistive case, to a total of 20. However, it appears that the constraints provided by the shock jump conditions make certain of these topologies unattainable: only 13 of the 20 cases have been found and are reported in the paper. The relationship between small‐amplitude dispersive waves in the flow upstream or downstream of a shock and the nature of the corresponding fixed point is also discussed.