Resonant magnetohydrodynamic modes with toroidal coupling. Part II: Ballooning-twisting modes

This is Part II of a study of resonant perturbations, such as resistive tearing and ballooning modes, in a torus. These are described by marginal ideal magnetohydrodynamic (MHD) equations in the regions between resonant surfaces; matching across these surfaces provides the dispersion relation. Part I [Phys. Fluids B 3, 1532 (1991)] described how all the necessary information from the ideal MHD calculations could be represented by a so-called E matrix. The calculation of this E matrix for tearing modes (even parity in perturbed magnetic field) in a large-aspect-ratio torus was also described. There the toroidal modes comprise coupled cylinder tearing modes and the E matrix is a generalization of the familiar Δ' quantity in a cylinder. In the present paper, resistive ballooning, or twisting modes, which have odd parity in perturbed magnetic field, are discussed. Unlike the tearing modes, these odd-parity modes are intrinsically toroidal and are not directly related to the odd-parity modes in a cylinder. This is evident from the analysis of the high-n limit in ballooning space, where the twisting mode exhibits a singular transition at large aspect ratio when the interchange effect is small (as in a tokamak). Analysis of the high-n limit in coordinate space, rather than ballooning space, clarifies this singular behavior. It also yields a prescription for treating low-n twisting modes and a method for calculating an E matrix for resistive ballooning modes in a large-aspect-ratio tokamak in the limit the interchange term vanishes. The elements of this matrix are given in terms of cylindrical tearing mode solutions