Interactions of disparate scales in drift-wave turbulence

Abstract

Renormalized statistical theory is used to calculate the interactions between short scales (wave vector k) and long scales (wave vector q<<k) in the Hasegawa-Mima model of drift-wave turbulence (generalized to include proper nonadiabatic response for k( ||)=0 fluctuations). The calculations include the zonal-flow growth rate as a special case, but also describe long-wavelength fluctuations with 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. Rigorous functional methods are used to show that the long-wavelength growth rate gamma(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 gamma;((1))(k)equals, single dot abovek small middle dotnablaOmega;(k)/k(2), where Omega;(k) is a nonlinear random advection frequency. Fokker-Planck analysis involving gamma;((1))(k) is used to heuristically recover the evolution equation for the small scales, and a random-walk flux argument that relates gamma;((1))(k) to an effective autocorrelation time is used to give an independent calculation of gamma(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.