Magnetohydrodynamic Stability of Plasmas with Ideal and Relaxed Regions

摘要

A unified energy principle approach is presented for analysing themagnetohydrodynamic (MHD) stability of plasmas consisting of multiple ideal andrelaxed regions. By choosing an appropriate gauge, we show that the plasmadisplacement satisfies the same Euler-Lagrange equation in ideal and relaxedregions, except in the neighbourhood of magnetic surfaces. The difference atsingular surfaces is analysed in cylindrical geometry: in ideal MHD onlyNewcomb's [W. A. Newcomb (2006) Ann. Phys., 10, 232] small solutions areallowed, whereas in relaxed MHD only the odd-parity large solution andeven-parity small solution are allowed. A procedure for constructing globalmulti-region solutions in cylindrical geometry is presented. Focussing on thelimit where the two interfaces approach each other arbitrarily closely, it isshown 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 ifthe relaxed-MHD region between the coalescing interfaces is replaced by anideal-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 musttake in the region between the interfaces. From this knowledge, we concludethat there exists a class of single interface plasmas that were found stable byKaiser and Uecker [R. Kaiser et al (2004) Q. Jl Mech. Appl. Math., 57, 1], butare shown to be unstable when the interface is resolved.