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Description
Frequency chirping of Alfvénic modes is a ubiquitous signature of the nonlinear phase-space dynamics of energetic particles (EPs) in tokamaks [1], and the chirping rate carries direct information on the resonant EP redistribution that limits fast-ion confinement [1]. Recent work shows that the nonlinear dynamics of the thermal plasma — beat-driven zonal fields and zonal currents together with the phase-space zonal structures of the thermal species — plays an essential role in the chirping and saturation of Alfvén eigenmodes, in some regimes dominating over the EP nonlinearity [2, 3]. We present a systematic global gyrokinetic study of chirping toroidal Alfvén eigenmodes (TAEs) with the particle-in-cell code ORB5, addressing how the chirping rate is set by the EP drive and how it is modified when several toroidal mode numbers are allowed to interact.
The simulations use a circular, large-aspect-ratio equilibrium with a monotonic safety-factor profile, a flat bulk-plasma density and a core-peaked energetic-ion population whose concentration is scanned to vary the linear drive. A scan over toroidal mode number identifies a most unstable intermediate-n TAE. The saturation amplitude nevertheless decreases monotonically with toroidal mode number, indicating that the radial width of the mode envelope, rather than the linear growth rate, sets the attainable mode level.
In the nonlinear phase all cases chirp downward, the sense expected both from the evolution of resonant EP phase-space structures [1] and from zonal-current-mediated frequency shifts [2]. Tracking the instantaneous frequency [4], the chirping rate increases with the EP drive, but distinctly more slowly than the adiabatic phase-space-structure scaling near marginal stability [1, 4]; the saturation level likewise grows only weakly with the linear growth rate. The central result is that, within each simulation family, the chirping rate is a linear function of the saturated mode amplitude, consistent with the chirp being governed by the wave-trapping frequency of the resonant EPs rather than by the linear drive itself [1, 4].
Multi-mode simulations show that this coupling is strong. Retaining the axisymmetric zonal component alone raises both the saturation level and the chirping rate of the driven TAE severalfold relative to the single-n case at the same drive, in line with the zonal-field-mediated enhancement of TAE saturation found when thermal-plasma nonlinearities are retained [3]. Adding further toroidal harmonics then reduces both quantities monotonically, as the modes compete for the same EP free energy, while all cases remain on a common chirp-rate–amplitude line. Single-n predictions of chirping rates, and hence of the inferred EP transport, can therefore be substantially in error: zonal flows, zonal currents, mode–mode coupling and the nonlinear thermal-plasma response [2, 3] must be retained in quantitative modelling of chirping AEs [1].
References
[1] L. Chen and F. Zonca, “Physics of Alfvén waves and energetic particles in burning plasmas”, Rev. Mod. Phys. 88, 015008 (2016).
[2] R. Ma, P. Liu, L. Chen, F. Zonca and Z. Qiu, “How zonal fields suppress reversed shear Alfvén eigenmode in tokamak plasmas”, Phys. Rev. Lett. 137, 045101 (2026).
[3] N. Chen, T. Hayward-Schneider, F. Zonca et al., “On nonlinear saturation of toroidal Alfvén eigenmode due to thermal plasma nonlinearities”, arXiv:2604.15024 (2026).
[4] R. Wu, A. Biancalani, M. V. Falessi et al., “Frequency chirping of energetic-particle-driven geodesic acoustic modes in tokamaks”, arXiv:2603.23430 (2026).