The magnetized Noh problem with both axial and azimuthal magnetic fields

The classic Noh problem [1] describes in simple analytic formulae the divergent hydrodynamic shock resulting from the stagnation of a supersonic flow against a wall, axis, or point. It has been used for years as a verification test for simulation codes. In a recently published work the cylindrical version of the Noh problem was generalized to the magnetohydrodynamic (MHD) realm by including an azimuthal magnetic field and it presented exact solutions for the MHD flow [2]. In the present work the analysis is further developed to include both azimuthal and axial magnetic fields in the inflow. Physical constraints on the azimuthal current density at the axis limit the acceptable solutions to those with 1 ≤ γ <; 3/2 and γ ≥ 2, where γ is the ratio of specific heats. We present self-similar, exact solutions in radius and time for the case γ = 2 with a radial inflow initially at zero pressure, constant velocity, and with equal amplitude for the azimuthal and axial magnetic components. The flow departs from these conditions even in the pre-shock region due to the magnetic stresses. In the post-shock region the density and azimuthal magnetic field vanish at the axis but, surprisingly, the axial magnetic field is nearly uniform. These exact solutions are compared with simulations using the FLASH (Eulerian Riemann solver) and MACH2 (ALE) MHD codes. The numerical techniques employed to match the proper physics are discussed. This magnetized Noh problem with azimuthal and axial fields presents a verification test for advanced codes intended to model Z-pinch flux compression experiments, such as in MagLIF on the Z generator, or gas puffs on university scale machines.