Non-ideal stability: variational method for the determination of the outer-region matching data

Within the framework of studies of the stability of magneto-plasmas to non-ideal modes, such as resistive modes, the problem of determining the asymptotic matching data arising from the outer (ideal) region is considered. Modes possessing both tearing and interchange (ballooning) parity are considered in finite-pressure plasmas. The matching data, which form a matrix whose elements represent the small solution response to forcing by a big solution, are shown to derive from a variational (energy) principle. The variational principle, as presented, applies to both cylindrical and two-dimensional (toroidal) geometries. Allowing for the presence of multiple rational surfaces, a reciprocity relation between off-diagonal elements of the matching data matrix is obtained. The variational principle is suitable for numerical approximation, and, in particular, for the finite-element method, for which convergence rates are estimated. By packing nodes near the rational surface, maximum convergence, proportional to the inverse square of the number of mesh nodes for tent functions, is achieved.