A Hybrid Statistics/Amplitude Approach to the Theory of Interacting Drift Waves and Zonal Flows

An approach to the theory of drift-wave--zonal-flow systems is adopted in which only the DW statistics but the full ZF amplitude are kept. Any statistical description of turbulence must inevitably face the closure problem. A particular closure, the Stochastic Structural Stability Theory (SSST), has been recently studied in plasma\footnote{B.~F.~Farrell and P.~J.~Ioannou, Phys.\ Plasmas \textbf{16}, 112903 (2009).} as well as atmospheric-science contexts. First, the predictions of the SSST are examined in the weakly inhomogeneous limit, using the generalized Hasegawa--Mima model as a simple example. It is found that the equations do not admit a complete solution, as the characteristic ZF scale cannot be calculated. To address that deficiency, an analysis is performed of a bifurcation from a DW-only state to a DW--ZF state in the Hasegawa--Wakatani model in order to gain analytical insight into a nonlinear DW--ZF equilibrium, including prediction of the charactistic scale. The calculation permits discussion of the relative importance of eddy shearing and coupling to damped eigenmodes for the saturation of the self-consistently regulated turbulence level.