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