On non-linear frequency chirping in connection with collisions

The non-linear behaviour of toroidicity-induced Alfvén eigenmodes, destabilized by fast ions, is investigated in tokamak geometry and for a Wendelstein 7-X high-mirror equilibrium. Both cases show frequency chirping in the non-linear phase. The focus of this paper is on how particle collisions influence the non-linear dynamics and the associated frequency chirping. Pitch-angle scattering and fast-ion drag, which together are described by the fast-ion collision operator, are considered. We study the effect of a Krook operator, relaxing the distribution function to its unperturbed value, on the non-linear dynamics. The Krook operator leads to a periodic re-appearance of the chirping. This is also observed in experiments in which a fast-particle source is usually present.The simulations are carried out using the non-linear and fully three-dimensional CKA-EUTERPE model. The model is perturbative in the sense that a fixed mode structure is used. Since such an investigation is undertaken for the first time for the stellarator Wendelstein 7-X, the tokamak case as well as analytical theory are used for comparison. The parameters of the fast-ion distribution function in Wendelstein 7-X are inspired by the 2018 experimental campaign which, for the first time, includes neutral beam injection to supply fast ions.