Stable and unstable supersonic stagnation of an axisymmetric rotating magnetized plasma

TL;DR

This paper extends the classic Noh problem to magnetized, rotating plasma flows, providing self-similar solutions that include stability analysis, useful for testing numerical MHD codes and understanding magnetically driven implosions.

Contribution

It introduces a five-parameter family of self-similar MHD solutions with rotation and magnetic fields, including a special case where magnetic tension creates shocks without initial flow.

Findings

Identified stable and unstable regions in parameter space.

Validated solutions with the Athena MHD code.

Provided solutions for testing numerical code accuracy and stability.

Abstract

The Naval Research Laboratory "Mag Noh problem", described in this paper, is a self-similar magnetized implosion flow, which contains a fast MHD outward propagating shock of constant velocity. We generalize the classic Noh (1983) problem to include azimuthal and axial magnetic fields as well as rotation. Our family of ideal MHD solutions is five-parametric, each solution having its own self-similarity index, gas gamma, magnetization, the ratio of axial to the azimuthal field, and rotation. While the classic Noh problem must have a supersonic implosion velocity to create a shock, our solutions have an interesting three-parametric special case with zero initial velocity in which magnetic tension, instead of implosion flow, creates the shock at $t=0+$. Our self-similar solutions are indeed realized when we solve the initial value MHD problem with finite volume MHD code Athena. We numerically investigated the stability of these solutions and found both stable and unstable regions in parameter space. Stable solutions can be used to test the accuracy of numerical codes. Unstable solutions have also been widely used to test how codes reproduce linear growth, transition to turbulence, and the practically important effects of mixing. Now we offer a family of unstable solutions featuring all three elements relevant to magnetically driven implosions: convergent flow, magnetic field, and a shock wave.