The guiding centre dynamics of fast particles can alter the behaviour of energetic particle (EP) driven modes with chirping frequencies. In this paper, the applicability of an earlier trapped/passing locus model (Hezaveh et al 2017 Nucl. Fusion57 126010) has been extended to regimes where the wave trapping region can expand and trap ambient particles. This extension allows the study of waves with up-ward and down-ward frequency chirping across the full range of EP orbits. Under the adiabatic approximation, the phase-space of EPs is analysed by a Lagrangian contour approach where the islands are discretised using phase-space waterbags. In order to resolve the dynamics during the fast formation of phase-space islands and find an appropriate initialisation for the system, full-scale modelling is implemented using the bump-on-tail code. In addition to investigating the evolution of chirping waves with deepening potentials in a single resonance, we choose specific pitch-angle ranges in which higher resonances can have a relatively considerable contribution to the wave-particle interaction. Hence, the model is also solved in a double-resonance scenario where we report on the significant modifications to the behaviour of the chirping waves due to the 2nd resonance. The model presented in this paper gives a comprehensive 1D paradigm of long range frequency chirping signals observed in experiments with both up-ward and down-ward chirping and multiple resonances.
Frequency chirping in the early stage of a near-threshold bump-on-tail instability