2.1 Feedback Stabilization of Hydromagnetic Continua: Review and Prospects

Abstract

Three types of analytical models for determining adequate feedback spatial and temporal resolution are distinguished for representing dynamics with finite sampling: piecewise continuous, (‘spliced’) discrete coupled modes, and coupled wavetrains. The stabilization of the z‐θ pinch is used for comparing the various representations, particularly as they represent a quasi‐one‐dimensional model for the $\mathrm{m}\text{\hspace{0.17em}=\hspace{0.17em}1}$ modes, and specific stability regimes are given. The advantages of the modal and wavetrain approaches in describing three‐dimensional effects are illustrated, with the wavetrain approach shown as particularly convenient for systems having many potentially unstable wavelengths within the system boundaries, hence necessitating many sampling stations. A discussion of difficulties in stabilizing interchange modes is given, and nonlinear forms of ‘bang‐bang’ feedback obviating these difficulties suggested.