Stable and Unstable Solutions of the Mag Noh Problem

Recently, there had been an increased interest in rotating implosions. Such rotation may appear spontaneously in Z-pinches with the imposed Bz field. The conservation of angular momentum in the imploding flow will naturally create rotation profiles, which are unstable to the so-called magnetorotational instability (MRI). In this article, we will use a convenient theoretical framework that we developed for the implosion stagnated on a shock: the NRL Mag Noh problem. This is a self-similar implosion flow with a shock propagating outward with constant velocity. Such solutions, obtained in ideal MHD, can be used for various purposes, including verifying numerical schemes and codes, studying instabilities that may develop in shocked flows, and many others. Our Mag Noh solutions are five-parametric and include both B-phi and B-z magnetic fields as well as rotation. There are many curious physical limiting cases, including the family of solutions with zero initial velocity, which develops a shock at t = 0+, very unlike the original classic Noh that requires a supersonic velocity. All our theoretical self-similar solutions are verified by solving an initial value MHD problem with finite volume MHD code Athena. In this article, we explain how Mag Noh is derived and demonstrate a few classes of solutions, and then, we study a rotating unstable solution and show that in the postshock flow, the perturbations can become large and significantly modify the flow profile.