Comprehensive linear model for the n = m = 1 fishbone kinetic-MHD instability

A comprehensive linear model for the interaction between the MHD n = m = 1 internal kink and fast particles, based on the Hamiltonian angle-action formalism, is derived. On the basis of this model, a linear code, MHD-K, that solves partially analytically and numerically the kinetic-MHD dispersion relation non-perturbatively is presented. The impact of passing fast particles on the fishbone is shown to be an essential drive of the instability, where previous models highlighted only trapped particles as the driver of the kinetic-MHD instability. Resonant planes in phase space are presented, showing multiple resonant branches for both trapped and passing particles.