Resonance frequency broadening of wave-particle interaction in tokamaks due to Alfv\'{e}nic eigenmode

We use a guiding center code ORBIT to study the broadening of resonances and the parametric dependence of the resonance frequency broadening width $\Delta\Omega$ on the nonlinear particle trapping frequency $\omega_b$ of wave-particle interaction with specific examples using realistic equilibrium DIII-D shot 159243 (Collins et al. 2016 Phys. Rev. Lett. 116 095001). When the mode amplitude is small, the pendulum approximation for energetic particle dynamics near the resonance is found to be applicable and the ratio of the resonance frequency width to the deeply trapped bounce frequency $\Delta\Omega/\omega_b$ equals 4, as predicted by theory. This factor 4 is demonstrated for the first time in realistic instability conditions. It is found that as the mode amplitude increases, the coefficient $a=\Delta\Omega/\omega_b$ becomes increasingly smaller because of the breaking down of the nonlinear pendulum approximation for the wave-particle interaction.