Statistical methodology [1] for the calculation of the nonlinear growth rate γ_q for long-wavelength convective cells (including zonal flows) driven by short-wavelength drift-wave (DW) interactions is reviewed. The physical amd mathematical relations of γ_q to eddy viscosity, modulational instability, and ray propagation are described. Previous results for scalar fields are generalized to a wide class of systems of coupled partial differential equations by introducing a covariant Hamiltonian formalism for the gyrofluid nonlinearities and employing a tensor wave kinetic equation. A statistical energy theorem is proven that relates γ_q to a tensor second functional derivative of the DW energy, and a general formula for γ_q is obtained in terms of the structure constants of the noncanonical Poisson bracket. The roles of Casimir invariants and random Galilean invariance in constraining the results are explained. Applications are made to (i) electrostatic ion-temperature-gradient-driven modes at small ion temperature, and (ii) weakly electromagnetic collisional DW's. However, it is stressed that complete results cannot be obtained until a fully self-consistent analysis of the steady-state turbulence is performed.

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