A Hall-MHD model for dipolarization fronts

In recent studies of transient reconnection, dipolarization fronts which are characterized by a steep increase or jump in the z component of the earth's tail magnetic field, have been reported in kinetic simulations with open boundaries as well as in many satellite observation, including the most recent five-probe THEMIS mission. One possible interpretation of these jumps is the formation of supersonic shocks using the standard MHD Rankine-Hugoniot relations. The observed jumps in the measured quantities like the density, magnetic field and velocity components are compared with the different kinds of oblique shocks that can occur in MHD. Using these jump conditions obtained from MHD, an initial value code for the 1D Hall-MHD system of equations is then solved to investigate the role of Hall physics on these shocks. Because of the strong coupling between the z component and an out-of-plane y component of the magnetic field in Hall-MHD, a discontinuous y component develops when the initial condition does not have a B$_{y}$ component. The sharp shock discontinuity leads to the generation of whistler waves which have different propagation characteristics on either side of the shock boundary.