Stability of straight tokamak equilibria without wall stabilization

A specific straight tokamak equilibrium surrounded by vacuum and without a conducting wall is proved to be stable to all ideal magnetohydrodynamic (MHD) modes by analytically evaluating linear stability. The analysis starts with a circular-cylindrically symmetric equilibrium which has a piecewise constant current profile. The effect of corrugation, which is of great importance to the stability of axisymmetric modes, is taken into account by using perturbation methods. Stability of axisymmetric modes is proved by solving the eigenvalue problem up to second order in the corrugation amplitude. Whereas elliptical corrugation (N=2) leads to instability for arbitrary current density, an equilibrium with N≥3 may be stabilized to axisymmetric modes by current reversal. To treat nonaxisymmetric global modes, the potential energy is evaluated using tokamak scaling. A sufficient stability criterion is derived according to which the equilibrium is stable to nonaxisymmetric modes if the current density in the outer plasma area is reversed and in the center sufficiently peaked, and if the safety factor q at the magnetic axis is greater than unity, increasing monotonically toward the plasma edge