Nonlinear magnetohydrodynamic stability and plasma containment with closed field lines

Previous formulations of the nonlinear energy principle of ideal magnetohydrodynamics in toroidal geometry cease to be relevant for stability if all magnetic field lines are closed. On the other hand, nonlinear stability is more vital in this degenerate case because the conservation laws cease to prevent plasma from touching the wall. The appropriate formulation of the nonlinear energy principle is given for this case, and from this the necessary and sufficient condition is derived for stability to finite line-preserving perturbations. Apart from the well known requirement of stability to infinitesimal interchanges and flutes deducible from the linear theory, this condition contains the additional requirement that the level surfaces of the quantities q = ∮ dl/ B, I = ∮ dl B, and the pressure p, coincide globally. It is further shown that nonlinear stability is compatible with containment only if the (purely geometrical) quantity Λ = Iq attains its minimum at the wall. The entire analysis is completely general and fully nonlinear.