Turbulent generation of large-scale flows and nonlinear dynamics of flute modes

Generation of large-scale flows (zonal flows and streamers) by flute mode turbulence is examined. The evolution equations for mean flow generation are obtained by averaging the model equations over fast small scales. For a system containing both drift-type waves and large-scale structures, small scales are modulated by larger scale shear flows so that energy in the small-scale component is not conserved. A WKB-type wave kinetic equation that describes the conservation (along the rays) of an action-like invariant of the flute mode turbulence with slowly varying parameters due to the mean sheared flow is formulated. The relevant action-like integral is shown to correspond to the quantity conserved for the small-scale component alone. The structure of the action integral is determined by the structure of the matrix element describing the interaction of the small-scale and large-scale component. The k-space diffusion coefficient for the zonal flows and streamers are calculated.