Magnetohydrodynamic spectrum in a class of toroidal equilibria

A normal mode analysis is performed for small amplitude magnetohydrodynamic motions about arbitrary, axially symmetric, static equilibria with purely poloidal magnetic fields. It is shown that the maximum growth rate pertaining to a fixed azimuthal mode number m increases with ‖m‖, and that the point eigenvalues become densely spaced in the limit ‖m‖→∞, thus forming an unstable continuum. For large ‖m‖, the determination of the normal modes is reduced to a problem in ordinary differential equations, and the dependence of the point eigenvalues on mode numbers is studied. As a practical consequence, it is found that the criterion derivable from the energy principle through trial functions localized at a pressure surface is necessary for gross stability (i.e., for stability of modes with small mode numbers). This conclusion does not apply to equilibria with shear.