RESONANT MAGNETOHYDRODYNAMIC MODES WITH TOROIDAL COUPLING .1. TEARING MODES

In a cylindrical plasma, tearing modes can be calculated by asymptotic matching of ideal magnetohydrodynamic (MHD) solutions across a critical layer. This requires a quantity DELTA' that represents the "discontinuity" in the ideal solution across the layer. In a torus, poloidal harmonics are coupled and there are many critical surfaces for each toroidal mode number, and correspondingly many discontinuities DELTA'-m. The ideal MHD solutions do not then determine the DELTA'-m but only a relation between them-described by an "E matrix." The calculation of the E matrix for a large-aspect-ratio tokamak is discussed. In a weak-coupling approximation, it is tridiagonal and can be computed from integrals over the uncoupled eigenfunctions or from simple "basis functions" comprising triplets of coupled poloidal harmonics. This weak-coupling approximation fails if DELTA'-m is already small for an uncoupled harmonic. An alternative strong-coupling approximation is developed for this case.