Gap modes in Arnold tongues: the topological origin of toroidal Alfven eigenmodes

The topological origin of the Toroidal Alfven eigenmodes (TAEs) is demonstrated in the limit of high toroidal mode number using the ballooning representation, in which the governing equation reduces to a modified Mathieu equation with a delta function potential perturbation. We prove that the modified Mathieu equation admits isolated gap modes with topological origins in the unstable regions of the Mathieu equation, which are known as Arnold tongues. The modes are identified as TAE modes. A generalization of this argument shows that gap modes can be induced in regimes of instability by localized potential perturbations for a large class of periodic Hamiltonians.