Equilibria and Stability in Partially Relaxed Plasma-Vacuum Systems

Abstract We develop a multiple interface variational model, comprising multiple Taylor-relaxed plasma regions, each of which are separated by an ideal MHD barrier. A principle motivation is the development of a mathematically rigorous ideal MHD model to describe intrinsically 3D equilibria, with nonzero internal pressure. A second application is the description of transport barriers as constrained minimum energy states. As a first example, we calculate the plasma solution in a periodic cylinder, generalizing the analysis of the treatment of Kaiser and Uecker, Q. Jl. Mech. Appl. Math.,57(1), 2004, who treated the single interface in cylindrical geometry. Expressions for the equilibrium field are generated, and equilib-rium states computed. Unlike other Taylor relaxed equilibria, for the equilibria investigated here, only the plasma core necessarily has reverse magnetic shear. We show the existence of tokamak like equilibria, with increasing safety factor and stepped-pressure profiles. A stability treatment of the multiple barrier configuration reduces to an eigenvalue problem, where the eigenvectors are the normal displacements of the ideal barriers, and the eigen-matrix has tridiagonal structure. Next, marginal stability thresholds are explored in parameter space. For a single interface, results are benchmarked to Kaiser and Uecker. For multiple interfaces, we check our working via convergence tests, which reveal that the system approaches the single barrier case in the limit of vanishing interface width. The analysis provides a foundation upon which to study the stability of systems with a single internal barrier, placed at the reverse shear point. 1