Analysis of the nonlinear dynamics of a chirping-frequency Alfvén mode in a Tokamak equilibrium

Abstract

Chirping Alfvén modes are considered as potentially harmful for the confinement of energetic particles in burning Tokamak plasmas because of their capability, by modifying their frequency, of extracting as much power as possible from such particles and, in turn, enhancing their transport.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; that is, if the phase space can be cut into slices that do not exchange particles and interact with the mode independently of each other. This can be done adopting a coordinate system that includes two global invariants of the system or, if it is not possible to identify these invariants (this is generally the case for chirping modes), two constants of motion. In the frame of a numerical simulation approach, we adopt as constants of motion, the magnetic momentum and a suited function of the initial particle coordinates. The relevant resonant structures are then identified. The analysis is focused on the dynamics of two of them: namely, those yielding the largest drive during, respectively, the linear phase and the nonlinear one. It is shown that, 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. If the mode frequency were constant, this density flattening would be responsible for the exhausting of the drive yielded by the resonant structure, which would occur as the large negative density gradients leave the resonance region. causes the resonance radius and the resonance region to drift inward. This drift, along with a relevant resonance broadening, delays the moment in which the inner density gradient reaches the inner boundary of the resonance region, leaving it. 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.