On the Arrest of Inverse Energy Cascade and the Rhines Scale

The notion of the cascade arrest in a β-plane turbulence in the context of continuously forced flows is revised in this paper using both theoretical analysis and numerical simulations. It is demonstrated that the upscale energy propagation cannot be stopped by a β effect and can only be absorbed by friction. A fundamental dimensional parameter in flows with a β effect, the Rhines scale, LR, has traditionally been associated with the cascade arrest or with the scale that separates turbulence and Rossby wave–dominated spectral ranges. It is shown that rather than being a measure of the inverse cascade arrest, LR is a characteristic of different processes in different flow regimes. In unsteady flows, LR can be identified with the moving energy front propagating toward the decreasing wavenumbers. When large-scale energy sink is present, β-plane turbulence may attain several steady-state regimes. Two of these regimes are highlighted: friction-dominated and zonostrophic. In the former, LR does not have any particular significance, while in the latter, the Rhines scale nearly coincides with the characteristic length associated with the large-scale friction. Spectral analysis in the frequency domain demonstrates that Rossby waves coexist with turbulence on scales smaller than LR thus indicating that the Rhines scale cannot be viewed as a crossover between turbulence and Rossby wave ranges.