Gyrokinetic theory of toroidal Alfv\'en eigenmode saturation via nonlinear wave-wave coupling

Nonlinear wave-wave coupling constitutes an important route for the turbulence spectrum evolution in both space and laboratory plasmas. For example, in a reactor relevant fusion plasma, a rich spectrum of symmetry breaking shear Alfv\'en wave (SAW) instabilities are expected to be excited by energetic fusion alpha particles, and self-consistently determine the anomalous alpha particle transport rate by the saturated electromagnetic perturbations. In this work, we will show that the nonlinear gyrokinetic theory is a necessary and powerful tool in qualitatively and quantitatively investigating the nonlinear wave-wave coupling processes. More specifically, one needs to employ the gyrokinetic approach in order to account for the breaking of the ``pure Alfv\'enic state" in the short wavelength kinetic regime, due to the short wavelength structures associated with nonuniformity intrinsic to magnetically confined plasmas. Using well-known toroidal Alfv\'en eigenmode (TAE) as a paradigm case, three nonlinear wave-wave coupling channels expected to significantly influence the TAE nonlinear dynamics are investigated to demonstrate the strength and necessity of nonlinear gyrokinetic theory in predicting crucial processes in a future reactor burning plasma. These are: 1. the nonlinear excitation of meso-scale zonal field structures via modulational instability and TAE scattering into short-wavelength stable domain; 2. the TAE frequency cascading due to nonlinear ion induced scattering and the resulting saturated TAE spectrum; and 3. the cross-scale coupling of TAE with micro-scale ambient drift wave turbulence and its effect on TAE regulation and anomalous electron heating.