A general long wavelength instability for Z-pinches and for Extrap within the Hall model

The stability of long wavelength perturbations is analysed within the framework of the Hall model. Free boundary modes with m = 1 and ksub(z) → O are shown to be unstable for all pressure profiles which go to zero at the plasma surface, if feedback from the wall can be neglected. The growth rate of the instability increases with decreasing plasma radius. Similar results are found for Extrap. Nonlinearities in combination with losses at the X-points are possibly responsible for the gross stability of free boundary modes in some Extrap discharges. In recent Extrap experiments, where an axial field (Bsub(T) = 0.5 Bsub(p)) is added, the improved stability might instead be due to passive feedback. (author)