Nonlinear kink mode dynamics in circular and noncircular pinches

Abstract

It is shown that the global (free-boundary) m=1 kink instability of the ideal, magnetohydrodynamic (MHD) sharp boundary (surface current) pinch is stabilized by nonlinear effects, provided Be≲1 and βp<1, where βp=1+B2e−B2i and Bi and Be denote, respectively, the internal and external axial magnetic fields of the pinch, normalized to the poloidal magnetic field. The stabilization has to do with the bending of the interior, ‘‘frozen’’ field lines and associated volume currents induced in the pinch, and does not occur in a pure surface current model, which neglects these currents and only conserves the total magnetic flux through the pinch. It is suggested that the global, helical m=1 structures observed in various pinch experiments may have to do with the stabilizing mechanism above. The nonlinear stability has been calculated by means of a new approach to the bifurcated equilibria of the helical m=1 mode, and the method should also be useful in connection with other nonlinear, ideal MHD phenomena. The regime of nonlinear stability above corresponds to intermediate or short wavelengths of the marginal mode (ka≳1). In the opposite, long-wavelength regime, the ideal MHD model and the pure surface current model give similar results, predicting nonlinear instability for, e.g., the nearly marginal Kruskal–Shafranov mode in tokamaks, in agreement with previous theories. Effects of mode rotation as well as of a noncircular cross section of the pinch, modeling the Extrap [Fusion Technol. 16, 7 (1989)] configuration, have also been considered, extending the results of a previous, linear investigation [Phys. Fluids B 2, 1601 (1990)] to the nonlinear regime.