Control of linear and nonlinear resistive wall modes

Abstract

Control of linear and nonlinear resistive wall modes (RWMs) is studied with a model that assumes: (1) a single Fourier harmonic of the normal component of the magnetic field is measured at the resistive wall; and (2) the control applied consists of that same harmonic at a larger radius, the control surface. For this model, it is shown that real gain Gr (zero phase shift) is exactly equivalent to having a perfectly conducting wall between the actual resistive wall and the control surface. It is also shown that imaginary gain Gi (π/2 phase shift) is exactly equivalent to the rotation of the resistive wall, which is in turn equivalent to plasma rotation. If there are two resistive walls separated by an insulator, Gi is equivalent to the rotation of the outer wall, and this effective differential rotation of the resistive walls can stabilize the modes for arbitrary plasma rotation. Complex gain Gr+iGi is equivalent to a closer conducting wall with rotation of the resistive wall. These equivalences are exact in two-dimensional linear theory (single Fourier harmonic k), and are good approximations when there is a spectrum of k. It is also shown in this slab model that “mode control,” used in DIII–D [J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)], is equivalent to higher Gr. Two-dimensional nonlinear simulations of control of RWM driven by current and pressure are presented. Investigations are shown of the validity of the feedback equivalences in nonlinear theory, showing that the equivalences hold to a good approximation even when a spectrum of k is present. It is found that the real gain required to give benign saturation of the nonlinear RWMs can be much less than that required for linear stabilization, particularly near the threshold for instability with a perfectly conducting wall.