The spectrum of multi-region-relaxed magnetohydrodynamic modes in topologically toroidal geometry

A general formulation of the problem of calculating the spectrum of stable and unstable eigenmodes of linearized perturbations about a magnetically confined toroidal plasma is presented. The analysis is based on a new hydromagnetic dynamical model, multi-region relaxed magnetohydrodynamics, which models the plasma magnetic-field system as consisting of multiple regions, containing compressible Euler fluid and Taylor-relaxed magnetic field, separated by interfaces in the form of flexible MHD current sheets. This is illustrated using a first-principles analysis of a two-region slab geometry, with periodic boundary conditions to model the outer regions of typical tokamak or reversed-field pinch plasmas. The lowest and second-lowest eigenvalues in plasmas unstable to tearing and kink-tearing modes are calculated. Near-marginal stability, the lowest mode obtained using the incompressible approximation to the kinetic energy normalization of the present study is shown to correspond to the eigenvalues found in previous studies where all mass was artificially loaded onto the interfaces.