Exact self-similar solutions for the magnetized Noh Z pinch problem

Abstract

A self-similar solution is derived for a radially imploding cylindrical plasma with an embedded, azimuthal magnetic field. The plasma stagnates through a strong, outward propagating shock wave of constant velocity. This analysis is an extension of the classic Noh gasdynamics problem to its ideal magnetohydrodynamics (MHD) counterpart. The present exact solution is especially suitable as a test for MHD codes designed to simulate linear Z pinches. To demonstrate the application of the new solution to code verification, simulation results from the cylindrical RÀZ version of Mach2 and the 3D Cartesian code Athena are compared against the analytic solution. Alternative routines from the default ones in Athena lead to significant improvement of the results, thereby demonstrating the utility of the self-similar solution for verification.

Stability of stagnation via an expanding accretion shock wave

“…[34][35][36] Purely gasdynamic stability analysis, like that done here and in Ref. 37, is, in turn, a pre-requisite for a more complete analysis accounting for additional physics, including MHD effects, 32,33 radiative energy losses from the stagnated cylindrical precursor plasma, 12,13 or both radiative losses and gravity in astrophysical accretion-shock flows. 17 Such study is obviously necessary for the analysis of the laboratory-astrophysics experiments proposed in Ref.…”

“…This code had originally been developed for astrophysical studies and is actively supported by a growing community. In fact, our choice of a Cartesian mesh rather than a cylindrical one was dictated in part by the confidence that we had developed in the Cartesian version of Athena during our previous work on the MHD version of the Noh problem, 32 and in part out of our curiosity about whether such a high-performance code could properly resolve decaying perturbations even on a non-aligned mesh. Our original intent had been to subsequently exercise the newly-added cylindrical mesh capabilities of Athena for this test, but we encountered some numerical difficulties with that effort which we have not yet been able to resolve.…”

“…37, where such analysis has been done for a converging shock wave. Stability analysis of other hydrodynamic 31 and MHD 32,33 generalizations of the classic Noh solution could shed light on the stability of hot-spot formation and other cases of shock-wave stagnation in laser-fusion and Z-pinch implosions, in laboratory astrophysics experiments, 16 as well as guide verification of hydrodynamic and MHD codes.…”

“…31) and in MHD. 32,33 Stability analysis of the classic Noh solution is of interest for several reasons. First, the stability problem is solved analytically, so its explicit solution could be used for verification of hydrocodes in two and three dimensions.…”

Anomalous wave structure in magnetized materials described by non-convex equations of state

“…Magnetic acceleration is the basis of magnetic pinch facilities where large magnetic pressure is used to compress material samples [21][22][23][24][25] . Magnetic field is also commonly used to process materials in industry.…”