Stable and Unstable Solutions of the Mag Noh Problem

NRL Mag Noh problem is a semi-analytic self-similar implosion flow, which contains a fast MHD outward propagating shock of constant velocity. This is a generalization of the classic zero-parametric Noh gasdynamics problem to include magnetic field and rotation. The main application of this series of ideal MHD solutions is verification of numerical codes that simulate magnetically driven implosions, such as Z-pinches. Self-similarity means that the solution at any given time can be trivially rescaled to obtain a solution at some other time. In this work we generalized previous results by adding axial magnetic field and rotation, making our family of ideal exact Mag Noh solutions five-parametric, each solution having its own self-similarity index, gas gamma, magnetization, ratio of axial to azimuthal field and rotation. While classic Noh problem must have a supersonic implosion velocity to create a shock, our solution has an interesting three-parametric special case with zero initial velocity, which creates the shock instantaneously. We verified that our self-similar solutions are indeed realized when we solve initial condition MHD problem with finite volume MHD code Athena. We numerically investigated stability of our solutions and found both stable and unstable regions in parameter space. While testing accuracy of numerical codes on exact stable solutions has always been important, commonly we also test how numerical codes handle instabilities, such as Kelvin-Helmholtz or Raleigh-Taylor. The way codes reproduce linear growth, transition to turbulence and practically important effects of mixing have always been interesting to high energy density physicists. Now we add another physically unstable test, which, for the first time, features all three elements relevant to magnetically driven implosions: convergent flow, magnetic field and the shock. A subset of our solutions has singular azimuthal current at the origin. Engineering such singularity in real implosions may help to create a hot spot in magnetically driven fusion experiments.