Relaxed MHD states of a multiple region plasma

We calculate the stability of a multiple relaxation region MHD (MRXMHD) plasma, or stepped-Beltrami plasma, using both variational and tearing mode treatments. The configuration studied is a periodic cylinder. In the variational treatment, the problem reduces to an eigenvalue problem for the interface displacements. For the tearing mode treatment, analytic expressions for the tearing mode stability parameter Δ′, being the jump in the logarithmic derivative in the helical flux across the resonant surface, are found. The stability of these treatments is compared for m = 1 displacements of an illustrative reverse field pinch-like configuration, comprising two distinct plasma regions. For pressureless configurations, we find the marginal stability conclusions of each treatment to be identical, confirming the analytical results in the literature. The tearing mode treatment also resolves ideal MHD unstable solutions for which Δ′ → ∞: these correspond to displacement of a resonant interface. Wall stabilization scans resolve the internal and external ideal kink. Scans with increasing pressure are also performed: these indicate that both variational and tearing mode treatments have the same stability trends with β, and show destabilization in configurations with increasing core pressure. Combined, our results suggest that variational stability of MRXMHD configurations is sufficient for both ideal and tearing (Δ′ < 0) stability. Such configurations, and their stability properties, are of emerging importance in the quest to find mathematically rigorous solutions of ideal MHD force balance in 3D geometry.