Control of resistive wall modes in a cylindrical tokamak with plasma rotation and complex gain

Feedback stabilization of magnetohydrodynamic (MHD) modes is studied in a cylindrical model for a tokamak with resistivity, viscosity and toroidal rotation. The control is based on a linear combination of the normal and tangential components of the magnetic field just inside the resistive wall. The feedback includes complex gain, for both the normal and for the tangential components, and the imaginary part of the feedback for the former is equivalent to plasma rotation. The work includes (1) analysis with a reduced resistive MHD model for a tokamak with finite \beta and with stepfunction current density and pressure profiles, and (2) computations with full compressible visco-resistive MHD and smooth decreasing profiles of current density and pressure. The equilibria are stable for \beta=0 and the marginal stability values $\beta_{rp,rw}<\beta_{rp,iw}<\beta_{ip,rw}<\beta_{ip,iw}$ (resistive plasma, resistive wall; resistive plasma, ideal wall; ideal plasma, resistive wall; ideal plasma, ideal wall) are computed for both cases. The main results are: (a) imaginary gain with normal sensors or plasma rotation stabilizes below $\beta_{rp,iw}$ because rotation supresses the diffusion of flux from the plasma out through the wall and, more surprisingly, (b) rotation or imaginary gain with normal sensors destabilizes above $\beta_{rp,iw}$ because it prevents the feedback flux from entering the plasma through the resistive wall to form a virtual wall. The effect of imaginary gain with tangential sensors is more complicated but essentially destabilizes above and below $\beta_{rp,iw}$. A method of using complex gain to optimize in the presence of rotation in the $\beta>\beta_{rp,iw}$ regime is presented.