Feedback Stabilization of a High-β, Sharp-Boundaried Plasma Column with Helical Fields

A sharp-boundaried, high-β stellarator of l-fold helical symmetry (plasma radius a, periodicity length 2π/h) is magnetohydrodynamically unstable, particularly to the m = 1, long-wavelength mode. For this mode the instability is driven by l ± 1 fields introduced by the unstable displacement ξ. Compensating fields (or, equivalently, plasma distortions) to counteract the instability can be introduced externally and controlled by a feedback arrangement. Estimates of the required external currents are given. The self-adjointness property of the magnetohydrodynamic energy-principle operator is used to derive the general relationship between the external fields and the unstable motion. Since perturbation theory is used, the question arises whether the normal mode with feedback differs appreciably from that without feedback for large feedback. The mode may be driven to a pure interchange, which is not subject to feedback control. In the stellarator this mode is stable above a critical β [of order (ha)2 for l = 1], and linear feedback stabilization is apparently possible in this case.