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Structure of resistive magnetohydrodynamic shocks
thesisposted on 2022-03-28, 18:08 authored by Stephen George McAndrew
This thesis presents results of a study of the steady state structure of slow, intermediate and fast magnetohydrodynamic (MHD) shocks in the case where energy dissipation is due to resistive heating of the fluid and the upstream Alfven speed is greater than the sound speed. A new parametric solution of the jump conditions is given and the various shock families are shown to be determined by the presence and number of stationary points in the magnetic field phase plane and the speed of sound. It is shown that there are 14 types of MHD shock structure contained in 12 families with the "switch-on" shock being the limiting case of a fast and intermediate shock combination. The thesis proceeds as follows. First, the background to the MHD shock structure problem is given. The equations for resistive MHD are then presented in the normal incidence frame for the steady state and the jump conditions relating the upstream and downstream states are then given in conventional and parametric form. The classification of shocks in terms of the MHD signal speeds is then described. The MHD equations yield a pair of ODEs in the transverse components of B, allowing the shock transition to be represented as a trajectory in the By-Bz plane. The magnetic field phase plane is then used to show the emergence of families of shocks, with different families of intermediate shocks found to have 0, 1 or 2 degrees of freedom in their internal structure. The multiplicity arises due to the possibility of an entropy increasing gas dynamic jump inside them from a supersonic to subsonic state across which the magnetic field components do not change. An original analysis is presented of the smooth passage of an intermediate shock from a supersonic to a subsonic state at a special point known as the transonic transition point. The final chapter gives a table of the shocks possible in resistive MHD, presents a calculation of the effect of fluid viscosity on the structure of a shock and gives details of a preliminary study of shock stability using an eigenvalue analysis. The calculations in this thesis confirm that intermediate shocks have the requisite number of degrees of freedom in their structure to be stable according to the hypothesis of Wu (J. Geophys. Res., 95: 8149, 1990) which had cast doubt on the original view that intermediate shocks are structurally unstable.