Weighted Birkhoff Averaging for Measuring Energetic Particle Integrability*

In a burning plasma, energetic particles (EPs) maintain the fusion reaction by heating the plasma core. Alfvén waves can cause EP transport due to islands and island overlap in phase space [1]. Poincaré plots are frequently used to measure EP transport in stellarators and tokamaks [3]. In the case of multiple Alfven waves or perturbation to quasi-symmetry, Poincaré maps have higher dimensionalities and are difficult to interpret. The Weighted Birkhoff Average (WBA) involves taking the average of a quantity over a trajectory weighted by a Gaussian function, which is super-convergent for smooth dynamic flows, giving it utility in distinguishing integrable from chaotic flows [2]. The accuracy with which the WBA converges and the relative time to convergence distinguishes between integrable and chaotic flows. We use Weighted Birkhoff Averaging to measure integrability of EPs and to distinguish between periodic, quasi-periodic, or chaotic orbits. These orbit types can be indicative of EP transport due to an Alfvén wave. We apply WBA to map stochastic regions of phase space. We apply WBA to passing EP momenta with varying magnetic perturbations and initial conditions. 

We acknowledge funding through the US Department of Energy, under contract No. DE-SC0024630.

[1] Paul, Elizabeth J., et al. Fast-ion transport in quasisymmetric equilibria in the presence of a resonant alfvénic perturbation. Journal of Plasma Physics , 2023.

[2] Nathan Duignan, James D. Meiss. Distinguishing between regular and chaotic orbits of flows by the weighted Birkhoff average, Physica D: Nonlinear Phenomena, 2023.

[3] White, R B. Modification of particle distributions by magnetohydrodynamic instabilities II. Plasma Physics and Controlled Fusion.