Resistive wall mode feedback stabilization studies using a lumped-parameter circuit equation formulation

The role of the resistive wall mode in limiting tokamak plasma performance is well chronicled and is a central topic of the Feedback Stabilization Initiative (FSI). It is believed that stabilization of this mode, which is a converted branch of the ideal-MHD external kink mode, may lead to the design of devices capable of accessing higher performance advanced operating regimes. We have developed a formulation of the resistive wail mode, for the limiting case of infinite aspect ratio, using the elementary physical concepts of self and mutual inductance. This results in a set of coupled lumped-parameter circuit equations with the variables being the perturbed plasma current, the helical component of induced current in the resistive shell, and (with feedback) the current in the active coil. These equations, which describe plasma perturbations of n greater than or equal to 1, have a one to one correspondence with plasma vertical positional n = 0 control. Comparisons between the dispersion relations for the two cases show that the quantity that carries the strength of the instability for the resistive wall mode, equivalent to the negative decay index in vertical position control, is L-pl(1 - f) where L-pl is the helical inductance of the perturbed plasma current and (1 - f) is related to the helicity of the ideal-MHD kink mode. This method has been applied successfully to describe the resistive wall mode in general terms and to describe analytically resistive wall mode feedback stabilization schemes. Formulation in this manner should facilitate numerical simulation of resistive wall mode feedback schemes. In this paper, we will describe the formulation in detail, show how the resulting circuit equations compare to the equations arrived at using traditional MHD analysis methods (particularly with the inclusion of feedback), and compare the resistive wall mode equations to those that describe the (n = 0) vertical instability.