The potential energy of an ideal static MHD plasma is minimized using the invariants of motion as variational constraints and assuming a general symmetry (dependence on two space variables only). For simplicity only the plasma-on- the-wall case is considered. The first variation yields a generalized Shafranov equation, the second the desired stability criterion. It is found that equilibria with a longitudinal current increasing monotonicaily towards the boundary are always stable with respect to symmetric modes. For equilibria with an outwardly decreasing current a sufficient criterion (for symmetric modes) is derived, which only requires the solution of a linear eigenvalue problem. The theory is applied to the straight circular cylinder and to the axisymmetric torus.
Grad-Shafranov equation for non-axisymmetric MHD equilibria
