Interactions of disparate scales in drift-wave turbulence

Renormalized statistical theory is used to calculate the interactions between short scales (wave vector $\mathit{k})$ and long scales (wave vector $\mathit{q}\ll \mathit{k})$ in the Hasegawa-Mima model of drift-wave turbulence (generalized to include proper nonadiabatic response for $k_{\Vert }=0\mathrm{}$ fluctuations). The calculations include the zonal-flow growth rate as a special case, but also describe long-wavelength fluctuations with $\mathit{q}$ oriented at an arbitrary angle to the background gradient. The results are fully renormalized. They are subtly different from those of previous authors, in both mathematical form and physical interpretation. A term arising in previous treatments that is related to the propagation of short-scale wave packets is shown to be a higher-order effect that must consistently be neglected to lowest order in a systematic expansion in $q/k.\mathrm{}$ Rigorous functional methods are used to show that the long-wavelength growth rate ${\gamma }_{\mathit{q}}$ is related to second-order functional variations of the short-wavelength energy and to derive a heuristic algorithm. The principal results are recovered from simple estimates involving the first-order wave-number distension rate ${\tilde{\gamma }}_{\mathit{k}}^{\mathrm{}\left(1\right)\mathrm{}}\doteq \mathit{k}\cdot \mathbf{\nabla }{\tilde{\Omega }}_{\mathit{k}}{/k}^2,$ where ${\tilde{\Omega }}_{\mathit{k}}$ is a nonlinear random advection frequency. Fokker-Planck analysis involving ${\tilde{\gamma }}_{\mathit{k}}^{\mathrm{}\left(1\right)\mathrm{}}$ is used to heuristically recover the evolution equation for the small scales, and a random-walk flux argument that relates ${\tilde{\gamma }}_{\mathit{k}}^{\mathrm{}\left(1\right)\mathrm{}}$ to an effective autocorrelation time is used to give an independent calculation of ${\gamma }_{\mathit{q}}.$ Both the rigorous and heuristic derivations demonstrate that the results do not depend on, and cannot be derived from, properties of linear normal modes; they are intrinsically nonlinear. The importance of random-Galilean-invariant renormalization is stressed.