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The guiding centre dynamics of fast particles can alter the behaviour ofenergetic particle driven modes with chirping frequencies. In this paper, theapplicability of an earlier trapped/passing locus model [H. Hezaveh et al 2017Nucl. Fusion 57 126010] has been extended to regimes where the wave trappingregion can expand and trap ambient particles. This extension allows the studyof waves with up-ward and down-ward frequency chirping across the full range ofenergetic particle orbits. Under the adiabatic approximation, the phase-spaceof energetic particles is analysed by a Lagrangian contour approach where theislands are discretised using phase-space waterbags. In order to resolve thedynamics during the fast formation of phase-space islands and find anappropriate initialisation for the system, full-scale modelling is implementedusing the bump-on-tail (BOT) code. In addition to investigating the evolutionof chirping waves with deepening potentials in a single resonance, we choosespecific pitch-angle ranges in which higher resonances can have a relativelyconsiderable contribution to the wave-particle interaction. Hence, the model isalso solved in a double-resonance scenario where we report on the significantmodifications to the behaviour of the chirping waves due to the $2^{\text{nd}}$resonance. The model presented in this paper gives a comprehensive 1D paradigmof long range frequency chirping signals observed in experiments with bothup-ward and down-ward chirping and multiple resonances.

Theoretical description of chirping waves using phase-space waterbags