Structure of the compressible MHD equations for the internal kink mode in a toroidal plasma with large aspect ratio

Abstract

The equations for the ideal, internal m = n = 1 kink mode in a toroidal plasma are derived from a direct, large-aspect-ratio perturbation expansion of the compressible magnetohydrodynamic (MHD) equations. The derivation complements earlier investigations of the internal kink mode based either on the energy principle or on direct expansions of the incompressible MHD equations. It is shown that five poloidal harmonics (m = −1, 0, 1, 2 and 3) have to be retained in a direct expansion of the compressible MHD equations, as compared with the three poloidal harmonics m = 0, 1 and 2 needed in the case of an incompressible plasma, or when working from the energy principle. Furthermore, the sound velocity is found to replace the Alfvén velocity in the generalized Pfirsch–Schlüter factor (the kinetic energy enhancement factor in a toroidal plasma) previously derived for an incompressible plasma. Taking this factor fully into account in the calculation of the growth rate of the m = n = 1 mode, it is shown that, while the Bussac result γ_B is recovered near marginal stability, growth rates of the order of 30% larger than γ_B are obtained when γ_B becomes of the order of the sound frequency.