Free boundary ballooning mode representation
A new type of ballooning mode invariance is found in this paper. Application of this invariance is shown to be able to reduce the two-dimensional problem of free boundary high n modes, such as the peeling-ballooning modes, to a one-dimensional problem. Here, n is toroidal mode number. In contrast to the conventional ballooning representation, which requires the translational invariance of the Fourier components of the perturbations, the new invariance reflects that the independent solutions of the high n mode equations are translationally invariant from one radial interval surrounding a single singular surface to the other intervals. The conventional ballooning mode invariance breaks down at the vicinity of plasma edge, since the Fourier components with rational surfaces in vacuum region are completely different from those with rational surfaces in plasma region. But, the new type of invariance remains valid. This overcomes the limitation of the conventional ballooning mode representation for studying free boundary modes.