Nonlinear properties of the internal m = 1 kink instability in the cylindrical tokamak

An analysis is presented, in a cylindrical approximation, of the nonlinear behavior of the m = 1 magnetohydrodynamic kink instability that occurs in a tokamak when the “safety factor” q(r) = rBz / RB θ(r) falls below unity on axis. A kinked neighboring equilibrium is found, which is accessible from the initial straight equilibrium in the sense of satisfying the flux-conservation constraints. Owing to the singular nature of the fundamental, all harmonics are excited in a singular region near where q (r) = 1. The nonlinear amplitude is moderate. It is shown that growing modes of this type should produce negative voltage spikes and inward shifts in major radius, as are seen in the experiments. The predicted magnitudes of these two effects are, however, much smaller than those observed.