The effect of nonlinear mode coupling on the stability of toroidal Alfvén eigenmodes

A cylindrical model of a tokamak is used to describe the nonlinear coupling between Alfven waves and sound waves. The model is applied to the case of a toroidal Alfven eigenmode (TAE) driven unstable by fusion alpha particles to explore its relevance to the nonlinear saturation of such modes, which can produce a loss of alpha particles in a burning plasma. The mechanism is the modulational instability of a finite-amplitude wave, in this case a TAE mode, in which a density fluctuation and Alfven sidebands are spontaneously excited when the finite-amplitude wave exceeds a threshold amplitude. In general, such a nonlinear calculation in a bounded inhomogeneous plasma would require a numerical solution. However, in the present case, an approximate analytic solution is derived by making use of the fact that the sound wave is logarithmically singular on a particular magnetic surface. The calculation is therefore carried out on the magnetic surface on which the TAE instability is centered and a nonlinear dispersion relation is derived. The threshold for the modulational instability yields a value for the ratio of the perturbed radial component of the magnetic field to the equilibrium magnetic field, which is comparable to the saturated amplitude of a TAE instability obtained by other workers. In addition, the nonlinear dispersion relation also includes a resonant case for which the threshold can be significantly lower. The present, heuristic calculation suggests that the modulational instability may indeed be relevant to the evolution and eventual saturation of a TAE instability.