Self-Similar Solutions With Electrothermal Processes for Plasmas of Arbitrary Beta

Abstract

Self-similar magnetohydrodynamic (MHD) solutions are developed for a cold planar wall next to a hot plasma with an embedded magnetic field parallel to the wall, including the relevant electrothermal terms. Velikovich et al. (Phys. Plasmas, vol. 22, no. 4, 042702, 2015) studied the Nernst electrothermal effect in the induction equation for such a problem under the assumption that the ratio of thermal to magnetic pressure ( $\beta$ ) was large. Other electrothermal processes, such as the Ettingshausen and plasma Thomson effects, vary inversely with $\beta$ , as does Joule heating, and may impact the plasma evolution at low $\beta$ . To study all these processes, we have extended the self-similar formulation to allow for an arbitrary $\beta$ . The self-similar ansatz requires constant total pressure, i.e., thermal plus magnetic. Self-similar solutions are presented with the density and temperature boundary conditions characteristic of the plasma immediately following laser preheat in the Magnetized Liner Inertial Fusion experiment, and with the far-field magnetic field equal to that at the wall. Four cases are examined: two with a large $\beta$ and two with a moderate $\beta$ , and for each of these with and without electrothermal terms. Solutions for the density, temperature, and velocity depend on the choice of $\beta$ but are largely insensitive to the electrothermal effects. However, for the same value of $\beta$ , whether large or moderate, the profile of the magnetic field changes significantly when the electrothermal effects are included. In the case of moderate $\beta$ and electrothermal effects, the magnetic field profile has two extrema. This interesting case is further simulated with a 1-D MHD code that allows for total pressure changes through the momentum equation. Comparison with the self-similar solution shows that the simulation results approach the self-similar solution between the wall and an outward propagating rarefaction wave. The solutions are proposed as verification tests for advanced, multiphysics MHD codes.