Magnetohydrodynamic Equilibrium and Stability of Spheromak with Spheroidal Plasma-Vacuum Interface

TLDR: In this article, an analytic solution to the Grad-Shafranov equation was obtained for a prolate and an oblate spheroidal plasma by using Hill's vortex model.

Abstract: The analytic solutions to the Grad-Shafranov equation are obtained for a prolate and an oblate spheroidal plasma by using Hill's vortex model. Effects of a toroidal magnetic field B ϕ on the MHD equilibrium configurations are investigated by using these analytic solutions. When B ϕ is stronger than that of the force-free configuration, the spheroidal plasmas in a vacuum magnetic field are shown to be unable in the MHD equilibrium. The several physical quantities on the equilibrium configuration are evaluated. The spheromak plasma is proved to be unstable if d p /dψ≠0 and d 2 F /dψ 2 ≥0 on the magnetic axis. Here p is the pressure and V (ψ) the volume surrounded by a magnetic surface of ψ=const. The equilibrium configurations of the spheroidal plasmas by using Hill's vortex model are shown to satisfy the above conditions, i.e., to be unstable.