Self-Similar Solutions with Electro-Thermal Processes for Plasmas of Arbitrary Beta

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 electro-thermal terms. Velikovich et al., [1] studied the Nernst effect in the induction equation for such a problem under the assumption that the ratio of thermal to magnetic pressure
$(\beta)$
was large. Other electro-thermal processes, such as Ettingshausen, Joule heating, and Nernst-thermal heating vary inversely with
$\beta$
. To study all these we have extended the self-similar formulation to allow for an arbitrary
$\beta$
. The self-similar ansatz requires constant total pressure, thermal plus magnetic. Four self-similar solutions are presented with the same density and temperature boundary conditions: two of high
$\beta$
and two of low
$\beta,$
and for each of these with and without electro-thermal terms. At high
$\beta$
only the N ernst term
$i$
n the induction equation affects the solution. At low
$\beta$
The Ettingshausen term is larger than thermal conduction at the wall, and the thermal N ernst term is negative. The results are proposed as verification tests for advanced, multi-physics, MHD codes. An example calculation is presented from a 1D MHD simulation code that allows for total pressure changes through the momentum equation. Comparison with the self-similar solution for one case shows that the simulation results approach the self-similar solution.