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 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. The ideal MHD solutions do not then determine the many discontinuities, 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.
