The kinetic theory of high-n electromagnetic low-frequency modes in a finite-β toroidal plasma is investigated in the collisionless limit. The high-n electromagnetic ballooning mode is identified and compared with the MHD ballooning mode. This mode is found to be always unstable, with the growth rate of the order of the drift frequency ω*. As the β-value increases and approaches the critical β predicted by the MHD theory, the growth rate becomes large and a real-frequency downshift appears. The toroidal shift of the magnetic surface and the magnetic well have a stabilizing effect, but the mode is unstable, because of the wave-particle interactions taking place even in the 'second stability region' of the MHD theory. The growth rate normalized to ω* has a peak in the MHD-unstable parameter region, but the peak is low. In the zero-β limit, this mode turns out to be the electrostatic ballooning mode. The drift branch and the drift-Alfvén branch are also identified in a toroidal plasma but remain stable, because of magnetic shear.
Resistive MHD ballooning mode in tokamaks