The guiding-center kinetic equation with Fokker-Planck collision term is used to study, in cylindrical geometry, a class of dissipative instabilities of which the classical tearing mode is an archetype. Variational solution of the kinetic equation obviates the use of an approximate Ohm’s law or adiabatic assumption, as used in previous studies, and it provides a dispersive relation which is uniformly valid for any ratio of wave frequency to collision frequency. One result of using the rigorous collision operator is the prediction of a new instability. This instability, driven by the electron temperature gradient, is predicted to occur under the long mean-free path conditions of present tokamak experiments, and has significant features in common with the kink-like oscillations observed in such experiments.

We studied ballooning instabilities in tokamaks of arbitrary cross section and finite shear. These azimuthally localized, ideal magnetohydrodynamic modes have large toroidal-mode numbers, but finite variation along the field and across the flux surfaces. Stability is determined by solving a second-order ordinary differential equation on each Qux surface, subject to the proper boundary conditions. Qualitative agreement is achieved with the Princeton p E $T stability code.The economics of conceptual tokamak reactors improves significantly for P as large as 10/o, where P is the ratio of plasma pressure to magnetic pressure The a. chievable P is likely to be determined by "ballooning modes. " These are magnetoydrodynamic (MHD) instabilities, analogous to the aneurisms which develop at weak spots in a pressurized elastic container. In plasmas, these modes are driven by the interaction

Although resistive diffusion is much slower than the growth of resistive instabilities, the conventional neglect of diffusion in tearing mode calculations is incorrect. The proper criterion for neglect of diffusion in resistive instability calculations, which is not satisfied for tearing modes, is ω≫v/δ, where v is the resistive diffusion velocity and δ is the resistive layer thickness. The effect of diffusion is calculated in the limit of large and small ωδ/v for the plane slab model, and new expressions for growth rate and stability boundary are obtained. The diffusion appears to have a stabilizing effect.