In 1941, Kolmogorov proposed a theory for three-dimensional homogeneous isotropic turbulence, where an exact relation between the third-order velocity correlation function and the energy flux is provided in the inertial range. This theory was generalized to two-dimensional turbulence in around 60 years, and because it builds a connection between measurable quantities (velocity correlation function) and physical concepts (energy flux) it was applied to atmospheric and oceanic flows. For example, the two-dimensional theory is used to explain the atmospheric Nastrom-Gage spectrum, which has been observed for 30 years but without a satisfactory explanation.

This talk first systematically explains the difference between three- and two-dimensional turbulence in the perspective of the relation between third-order correlation function and energy flux based on their different energy flux directions. Then we derive and justify an exact extension of the law for two-dimensional turbulence that also covers the forcing scales. Next, considering the rotation and stratification in the atmospheric and oceanic flows, we apply our formulation to a two-dimensional rotating stratified turbulence. Finally, for real atmospheric and oceanic turbulence where energy transfers to both large and small scales, we propose that based on our forcing-scale resolving theory a constraint on the third-order velocity correlation functions of different scales can be obtained.

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