Alfvenic frequency chirping in tokamaks: Theoretical modelling and simulation using the MEGA code*

In this work, we firstly develop a theoretical framework to study long range frequency chirping waves associated with unstable Alfven eigenmodes. Using a Lagrangian formalism and an adiabatic approximation, we show how the nonlinear contribution of energetic particles modifies the radial structure of a global Alfven eigenmode (GAE). Also, a new conservation law is introduced for energetic particles in resonance with chirping waves. This conservation law does not depend on the frequency of the perturbation and remains conserved even during the frequency chirping. For the second step, self-consistent simulations are performed using the hybrid model of the MEGA code. Using the new conservation law, we propose a new phase-space analysis method which enables an appropriate study of energetic particles phase space during frequency chirping of Alfvenic perturbations. For chirping waves associated with an n=6 toroidicity-induced Alfven eigenmode (TAE), coherent structures (holes and clumps) are formed and evolved in phase space. We demonstrate that these structures cause convective transport in phase-space leading to radial drifts of the particles. It is also shown that the rate of frequency chirping increases with the damping rate of the TAE. Our observations of the nonlinear behaviour of the TAE and the phase space dynamics of the energetic particles are consistent with our theoretical model and the Berk-Breizman (BB) theory.