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 equilibrium 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.

Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets

“…These "stepped-pressure" states are extrema of the multi-region, relaxed energy functional [20] and resolve the singularities by allowing local relaxation, and thus are not globally ideal.…”

“…The SPEC code [25] does allow for discontinuities. SPEC formally finds extrema of the multi-region, relaxed, MHD (MRxMHD) energy functional, as proposed by Hole, Hudson and Dewar [20,26]. While in globally-ideal equilibria the topology of the magnetic field is continuously constrained, in MRxMHD the topology is discretely constrained at a finite number, N , of so-called, "ideal" interfaces, where discontinuities in the pressure and tangential magnetic field are allowed.…”

MHD memes

“…This has developed in various ways that other contributors to this Proceedings will no doubt touch on in much more detail. However, sticking to the MHD theme, it could be said that the theory of flux-minimizing surfaces [22,23,24,25,26] oscillation centre theory, and this has close links with our current research on three-dimensional MHD equilibrium theory [5,6]. Another intersecton concerns the theory of quantum chaos, a field in which Allan made an important contribution [27], which is better known outside the field of plasma physics than within it.…”

Verification of the SPEC code in stellarator geometries

“…SPEC [3] was developed to calculate MHD equilibria as extrema of the multi-region, relaxed MHD (MRxMHD) energy functional proposed by Hole, Hudson and Dewar [8,9]. While in ideal MHD the magnetic topology is continuously constrained, in MRxMHD the topology is only discretely constrained, thus allowing for partial relaxation.…”