Vertical displacement oscillatory modes in tokamak plasmas

Vertical displacement normal modes in shaped tokamak plasmas are studied analytically, based on the reduced ideal-magnetohydrodynamic model. With help of quadratic forms, and using appropriate eigenfunction for vertical displacements with toroidal mode number $n=0$ dominant elliptical-angle $m=1$ , a dispersion relation is derived, including effects ideal or resistive walls through single parameter, $D_w(\gamma )$ which is, general, function complex eigenfrequency $\gamma = -{\rm i}\omega$ . For resistive-wall case, cubic $\gamma$ One root corresponds to well-known, non-rotating mode, growing time scale. The other two roots weakly damped by wall resistivity, but oscillate frequency below poloidal Alfvén frequency, makes them immune continuum damping, subject possible instability due resonant interaction fast ions.