An analytic derivation of the relevant dispersion relation for vertical displacements in shaped tokamak plasmas is presented, valid for arbitrary values of the ellipticity parameter. The theory is developed within the framework of the reduced ideal-MHD model. A nearby, perfectly conducting wall can provide passive feedback stabilization of vertical displacements on the ideal-MHD timescale. The mechanism for passive stabilization relies on image currents induced on the metallic wall. However, if the plasma extends to the magnetic separatrix, where magnetic X-points are located, as in the case of a divertor tokamak configuration, perturbed axisymmetric currents carried by the plasma in the vicinity of the X-points are triggered. It is shown that these X-point currents can provide passive feedback stabilization, even in the absence of a nearby wall. X-point currents are excited due to the resonant nature of magnetic X-points with respect to toroidal axisymmetric perturbations. An intermediate case, where the plasma boundary is located just inside the magnetic separatrix, is also analyzed, providing additional insight into the stabilization mechanism.
Transport-driven Scrape-Off-Layer flows and the boundary conditions imposed at the magnetic separatrix in a tokamak plasma
Multimode observations and 3D magnetic control of the boundary of a tokamak plasma