Analysis of the nonlinear dynamics of a chirping-frequency Alfv\'en mode in a Tokamak equilibrium

Chirping Alfv\'{e}n modes are considered as potentially harmful in burning Tokamak plasmas. In this paper, the nonlinear evolution of a single-toroidal-number chirping mode is analysed by numerical particle simulation. This analysis can be simplified if the different resonant phase-space structures can be investigated as isolated ones. This can be done adopting a coordinate system that includes two constants of motion. In our simulations, we adopt as constants of motion, the magnetic momentum and the initial particle coordinates. For each resonant structure, a density-flattening region is formed around the respective resonance radius, with radial width that increases as the mode amplitude grows. It is delimited by two large negative density gradients, drifting inward and outward. With constant mode frequency, this density flattening would be responsible for the exhausting of the drive when large negative density gradients leave the resonance region. The frequency chirping, however, causes the resonance radius and the resonance region to drift inward. This drift delays the moment in which the inner density gradient reaches the inner boundary of the resonance region. On the other side, the island reconstitutes around the new resonance radius; as a consequence, the large negative density gradient further moves inward. This process continues as long as it allows to keep the large gradient within the resonance region. When this is no longer possible, the resonant structure ceases to be effective in driving the mode. To further grow, the mode has to tap a different resonant structure, possibly making use of additional frequency variations.