Within the context of statistical homogeneity, we examine geostrophic turbulence, employing a simple,eddy-damped closure. We find for decaying turbulence that high wavenumbers tend toward three-dimensional isotropy, as predicted by Charney, but wavenumbers smaller than the energy peak tend toward anapproximate two-dimensional state, with the crossover wavenumber near the peak energy wavenumber.The high-wavenumber energy spectrum is found to be log-modified k3, where k is the three-dimensionalwavenumber. Analytic information for the isotropization rate at small scales as well as for the large-scale"barotropization" is proposed. Finally, we describe the relation of these results to the more familiarlayered approximations of the equations of motion.
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Jackson R. Herring (1980) studied this question.