  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.
