Kinetic-ballooning-mode theory in general geometry

A systematic procedure for studying the influence of kinetic effects on the stability of MHD ballooning modes is presented. The ballooning mode formalism, which is particularly effective for analysing high-mode-number perturbations of a plasma in toroidal systems, is used to solve the Vlasov-Maxwell equations for modes with perpendicular wavelengths on the scale of the ion gyroradius. The local stability on each flux surface is determined by the solution of three coupled integro-differential equations which include effects due to finite gyroradius, trapped particles, and wave-particle resonances. More tractable forms of these equations are then obtained in the low (ω < ω_bi, ω_ti) and intermediate- (ω_bi, ω_ti < ω < ω_be, ω_te) frequency regimes with ω_bj and ω_tj being the average bounce and transit frequencies of each species. After further simplifying approximations, the kinetic results here are shown to be reducible to the MHD-ballooning-mode equations in the analogous limits, ω ≶ ω_s where ω_s = c_s/L_c, with c_s being the acoustic speed and L_c the connection length.