A general MHD stability formulation for plasmas with flow and resistive walls

Plasma rotation, induced either by means of neutral beams (e.g. in NSTX and DIII-D) or appearing spontaneously (e.g. in Alcator C-Mod, JET and Tore Supra) is routinely observed in modern tokamak experiments. Plasma rotation has a major effect on plasma stability. In particular, flow and flow shear stabilize external modes such as the resistive wall mode (as observed e.g. in DIII-D), and also have a significant influence on turbulence, internal kinks and ballooning modes. A self-consistent analysis of the effect of rotation requires the use of numerical tools. In this work, we extend our previous analysis and present a general \textit{variational} eigenvalue formulation of the stability problem. The analysis includes arbitrary (both toroidal and poloidal) plasma rotation and a thin resistive wall of arbitrary shape and resistivity. It is shown the problem can always be reduced to a classic eigenvalue formulation of the kind i$\omega $\textbf{A}$\cdot $\textbf{$\zeta $=B}$\cdot $\textbf{$\zeta $}, where \textbf{$\zeta $} is the unknown eigenvector related to the plasma displacement, and $\omega $ the (complex) evolution frequency of the perturbation. The formulation is well suited for a finite element analysis.