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The ideal-magnetohydrodynamics (MHD) energy principle is used to derive a necessary stability criterion for high-toroidal-number (n) external modes in axisymmetric equilibria. The corresponding trial functions are expressed in the ballooning representation, but have a finite amplitude at the plasma boundary and can apply to equilibria where the conventional, high-n internal ballooning criterion predicts stability. These trial functions are constructed by solving the standard local ballooning equation at the plasma boundary flux surface with the radial wave number parameter as a complex eigenvalue, such that the radial envelope of the mode is an exponential decaying into the plasma. The resulting stability criterion includes the surface and vacuum contributions to the MHD potential energy associated with the mode finite edge amplitude, and provides a framework for analyzing free-boundary ballooning and peeling modes.

Stability criterion for edge-localized, high-n external modes in tokamaks