Quantifying resonant and near-resonant interactions in rotating turbulence

Abstract

(Received ?; revised ?; accepted ?. -To be entered by editorial office)Nonlinear triadic interactions are at the heart of our understanding of turbulence. In flows where waves are present modes must not only be in a triad to interact, but their frequencies must also satisfy an extra condition: the interactions that dominate the energy transfer are expected to be resonant. We derive equations that allow direct measurement of the actual degree of resonance of each triad in a turbulent flow. We then apply the method to the case of rotating turbulence, where eddies coexist with inertial waves. We show that for a range of wave numbers, resonant and near-resonant triads are dominant, the latter allowing a transfer of net energy towards two-dimensional modes that would be inaccessible otherwise. The results are in good agreement with approximations often done in theories of rotating turbulence, and with the mechanism of parametric instability proposed to explain the development of anisotropy in such flows. We also observe that, at least for the moderate Rossby numbers studied here, marginally near-resonant and non-resonant triads play a non-negligible role in the coupling of modes.

On the inverse energy transfer in rotating turbulence

“…It is important to note that in simulation FULL, no energy is being injected directly into the 2D modes. So while resonant interactions make the spectra evolution anisotropic, quasi-resonant interactions must be coming into play in order to couple the 3D and 2D modes [42,38,39,34].…”

“…known as the resonance condition. Resonant interactions are very important to the evolution of a rotating turbulent flow, but they do not encompass all of the interactions that happen in it [32,34]. The Reynolds and the Rossby numbers are the two non-dimensional parameters which control the dynamic evolution of the flow.…”

Inverse cascades and resonant triads in rotating and stratified turbulence

“…His "instability assumption" states that the statistical direction of the energy transfer is determined by the stability of the corresponding elemental triadic interaction. In practice, this results in a preferential transfer of energy towards the modes with zero frequency, although wave turbulence theories cannot explain how energy ultimately reaches these modes [84,85] (as shown in [86], near-resonances are needed to explain the efficient transfer of energy to these modes, while nonresonant interactions act reducing this net energy transfer). In spite of these limitations, the argument is succesful in explaining how anisotropy develops.…”

Flows driven by libration, precession, and tides in planetary cores

“…[33]). Explaining these complex behaviors is very challenging and reflects in several aspects the intense research activity in rotating turbulence [51,52]: it is clearly beyond the scope of this paper. But without claiming exhaustivity or mathematical rigor, one can suggest here, from very simple considerations, two plausible physical mechanisms in connection with the two observed types of saturation.…”