Self-similar solutions for resistive diffusion, Ohmic heating and Ettingshausen effects in plasmas of arbitrary $β$

Abstract: MIF approaches, such as the MagLIF experiment, use magnetic fields in dense plasma to suppress cross-field thermal conduction, attempting to reduce heat losses and trap alpha particles to achieve ignition. However, the magnetic field can introduce other transport effects, some of which are deleterious. An understanding of these processes is thus crucial for accurate modelling of MIF. We generalise past work exploiting self-similar solutions to describe transport processes in planar geometry and compare the model to the radiation-magnetohydrodynamics code Chimera. We solve the 1D extended MHD equations under pressure balance, making no assumptions about the ratio of magnetic and thermal pressures in the plasma. The resulting ODE boundary value problem is solved using a shooting method, combining an implicit ODE solver and a Newton-Raphson root finder. We show that the Nernst effect dominates over resistive diffusion in high $\beta$ plasma, but its significance is reduced as the $\beta$ decreases. On the other hand, we find that Ettingshausen and Ohmic heating effects are dominant in low $\beta$ plasma, and can be observable in even order unity $\beta$ plasma, though in the presence of a strong temperature gradient heat conduction remains dominant. We then present a test problem for the Ohmic heating and Ettingshausen effects which will be useful to validate codes modelling these effects. We also observe that the Ettingshausen effect plays a role in preventing temperature separation when Ohmic heating is strong. Neglecting this term may lead to overestimates for the electron temperature at a vacuum-plasma interface, such as at the edge of a z-pinch. The model developed can be used to provide test problems with arbitrary boundary conditions for magnetohydrodynamics codes, with the ability to freely switch on terms to compare their individual implementations.