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Nonlinear wave-wave coupling constitutes an important route for theturbulence spectrum evolution in both space and laboratory plasmas. Forexample, in a reactor relevant fusion plasma, a rich spectrum of symmetrybreaking shear Alfv\'en wave (SAW) instabilities are expected to be excited byenergetic fusion alpha particles, and self-consistently determine the anomalousalpha particle transport rate by the saturated electromagnetic perturbations.In this work, we will show that the nonlinear gyrokinetic theory is a necessaryand powerful tool in qualitatively and quantitatively investigating thenonlinear wave-wave coupling processes. More specifically, one needs to employthe gyrokinetic approach in order to account for the breaking of the ``pureAlfv\'enic state" in the short wavelength kinetic regime, due to the shortwavelength structures associated with nonuniformity intrinsic to magneticallyconfined plasmas.  Using well-known toroidal Alfv\'en eigenmode (TAE) as a paradigm case, threenonlinear wave-wave coupling channels expected to significantly influence theTAE nonlinear dynamics are investigated to demonstrate the strength andnecessity of nonlinear gyrokinetic theory in predicting crucial processes in afuture reactor burning plasma. These are: 1. the nonlinear excitation ofmeso-scale zonal field structures via modulational instability and TAEscattering into short-wavelength stable domain; 2. the TAE frequency cascadingdue to nonlinear ion induced scattering and the resulting saturated TAEspectrum; and 3. the cross-scale coupling of TAE with micro-scale ambient driftwave turbulence and its effect on TAE regulation and anomalous electronheating.

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