Resistivity and sheared rotation effects on the toroidal external kink mode

We present PEST-III analysis of the toroidal external kink with plasma resistivity and sheared rotation for a range of equilibria varying elongation, driven unstable by increasing $\beta$. The results show that the typical ordering for marginally stable $\beta$ values is $\beta_{\textrm{rp,rw}}<\beta_{\textrm{rp,iw}}<\beta_{\textrm{ip,rw}}<\beta_{\textrm{ip,iw}}$, where rp,ip signify resistive or ideal plasma, and iw,rw indicate ideal wall or resistive wall (no-wall). The two resistive plasma $\beta$ limits are significantly lower than the two ideal plasma values. We vary aspects of the tearing layer physics by means of a variational principle with Pad\'e approximants, and compare with a general computational solution for the layers to gain insight. We also include pressure gradient and local velocity shear within the layers. Global rotation shear $\Omega'$ is included in the form of a relative rotation of the $q=2,~3,\cdots$ surfaces and we investigate the resultant effect on the poloidal mode number spectrum. We then present a model for active feedback control, which is the toroidal generalization building on recent results in cylindrical mode control theory [D.P. Brennan and J.M. Finn and submitted to Physics of Plasmas (2014)].