Using self-similar solutions to understand magnetised transport in high-energy-density plasmas relevant to magneto-inertial fusion

The transport of magnetic fields and thermal energy are important processes in magneto-inertial fusion schemes. Accurate modelling of these effects is crucial for the success of future experiments. To this end, we develop semi-analytic self-similar solutions to subsonic magnetised transport. We solve the extended MHD equations in 1D planar geometry under assumption of pressure balance, for arbitrary plasma beta, using a Newton-Raphson steered shooting code. Agreement between the self-similar solutions and simulations using the MHD code Chimera is good, validating the use of these semi-analytic solutions as flexible test problems for extended MHD algorithms.  We then use these self-similar solutions to assess the relative roles of the Nernst and resistive diffusion effects in magnetic field evolution. We find that the Ettingshausen and Ohmic heating terms can be significant even in an order unity beta plasma and dominate in low beta regimes, but their effect depends strongly on the plasma profiles. We also discuss the breakdown of these self-similar solutions as the Ohmic heating drives strong heating of the electrons, finding that the Ettingshausen effect prevents electron-ion temperature separation in this low beta limit. This suggests that the exclusion of the Ettingshausen effect may cause resistive MHD simulations to overestimate the electron temperature at a plasma-vacuum interface, such as at the edge of a z pinch.