Theory of the Slow Pinch Discharge. I. Magnetohydrodynamic Stability of the Discharge Core

Among the unexplained properties of the slow pinch discharge observed in Zeta, Sceptre, and other experiments is the gross magnetohydrodynamic stability of the discharge. Since the pinch of the magnetic lines of force is approximately constant within the main core of these discharges, the energy principle is applied to such a configuration. The stability condition for interchange modes is $\frac{\mathrm{dp}}{\mathrm{dr}}\ge -\frac{2\gamma p{B_{\theta }}^2}{r(B^2+\gamma p)}$ and, for kink modes, $\frac{\mathrm{dp}}{\mathrm{dr}}$ should be greater than an undetermined positive quantity. Comparison with experimental results suggests that interchange instabilities are occurring and limiting the negative pressure gradient to the above value, whereas the kink modes can be occurring with, at most, only small amplitude. Possible reasons are given to explain why the kink instability is not observed.