On the Effect of Beating during Nonlinear Frequency Chirping

Abstract

Spectroscopic analyses of energetic particle (EP) driven bursts of MHD fluctuations in magnetically confined plasmas often exhibit chirps that occur simultaneously in groups of two or more. While the superposition of oscillations at multiple frequencies necessarily causes beating in the signal acquired by a localized external probe, self-consistent hybrid simulations of chirping EP modes in a JT-60U tokamak plasma have demonstrated the possibility of global beating, where the mode's electromagnetic field vanishes globally between beats and reappears with opposite phase [Bierwage et al., Nucl. Fusion 57, 016036 (2017)]. This implies that there can be a single coherent field mode that oscillates at multiple frequencies simultaneously when it is resonantly driven by multiple density waves in EP phase space. Conversely, this means that the EP density waves are mutually coupled and interfere with each other via the jointly driven field, a mechanism ignored in some theories of chirping. In this thesis-style treatise, we study the role of field pulsations in general and beating in particular using the Hamiltonian guiding center orbit-following code ORBIT with a reduced wave-particle interaction model in realistic geometry. Beating is found to drive the evolution of EP phase space structures. A key mechanism is the pulsation of effective phase space islands combined with the alternation of their effective O-and X-points due to phase jumps between each beat. Observations: (1) Beating causes density wave fronts to advance radially in a pulsed manner and the resulting chirps become staircase-like. (2) The pulsations facilitate convective transfer of material between neighboring layers of phase space density waves. On the one hand, this may inhibit the early detachment of solitary phase space vortices. On the other hand, it facilitates the accumulation of hole and clump fragments into larger structures. (3) Long-range chirping is observed when massive holes or clumps detach and drift away from the turbulent belt around the seed resonance. It is remarkable that the detached vortices remain robust and, on average, maintain their concentric nested layers while being visibly perturbed by the field's continued beating.

Fast-ion transport in quasisymmetric equilibria in the presence of a resonant Alfvénic perturbation

“…In the future, we plan to build on this work to obtain a more complete picture of the evolution of AEs in stellarator reactor scenarios. For example, the analysis tools described here, such as kinetic Poincaré plots, could be used to study the nonlinear evolution of phase-space structures such as hole-clump pairs (Bierwage, White & Duarte 2021) and zonal structures (Zonca et al 2015) in quasisymmetric configurations. by a factor of the magnetic drifts.…”

NSTX-U theory, modeling and analysis results

“…This flow of particles across the resonance provides an energy source and local gradients for sustained, repeated chirping [48,49]. Detailed spectroscopic analyses [50] clarified the role of violent amplitude pulsations and phase jumps of the simulated chirping Alfvén mode. These phenomena are attributed to the beating of multiple non-normal modes associated with the presence of holes and clumps, and consequent half-period shift of mode phase.…”

Perturbative analysis of low-frequency instabilities in high-field ST40 experiments

“…To better understand the underlying physics of oscillation bifurcation, we use two selected discharges in which clear transitions between constant frequency and chirping phases were observed. The results are then interpreted in terms of reduced nonlinear analytic theory of driven marginally-unstable waves in the presence of background dissipation [27,28], specifically regarding the sustainment of phase space structures that are thought to be connected with chirping [27,29,30].…”