Magnetohydrodynamic Stability of Plasmas with Ideal and Relaxed Regions

A unified energy principle approach is presented for analysing the magnetohydrodynamic (MHD) stability of plasmas consisting of multiple ideal and relaxed regions. By choosing an appropriate gauge, we show that the plasma displacement satisfies the same Euler-Lagrange equation in ideal and relaxed regions, except in the neighbourhood of magnetic surfaces. The difference at singular surfaces is analysed in cylindrical geometry: in ideal MHD only Newcomb's [W. A. Newcomb (2006) Ann. Phys., 10, 232] small solutions are allowed, whereas in relaxed MHD only the odd-parity large solution and even-parity small solution are allowed. A procedure for constructing global multi-region solutions in cylindrical geometry is presented. Focussing on the limit where the two interfaces approach each other arbitrarily closely, it is shown that the singular-limit problem encountered previously [M.J. Hole et al. (2006) J. Plasma Phys., 77, 1167] in multi-region relaxed MHD is stabilised if the relaxed-MHD region between the coalescing interfaces is replaced by an ideal-MHD region. We then present a stable (k, pressure) phase space plot, which allows us to determine the form a stable pressure and field profile must take in the region between the interfaces. From this knowledge, we conclude that there exists a class of single interface plasmas that were found stable by Kaiser and Uecker [R. Kaiser et al (2004) Q. Jl Mech. Appl. Math., 57, 1], but are shown to be unstable when the interface is resolved.