Nonlinear energy transfers and phase diagrams for geostrophically balanced rotating-stratified flows

Abstract

Equilibrium statistical mechanics tools have been developed to obtain indications about the natural tendencies of nonlinear energy transfers in two-dimensional and quasi two-dimensional flows like rotating and stratified flows in geostrophic balance. In this article, we consider a simple model of such flows with a non-trivial vertical structure, namely two-layer quasi-geostrophic flows, which remain amenable to analytical study. We obtain the statistical equilibria of the system in the case of a linear vorticity-stream function relation, build the corresponding phase diagram, and discuss the most probable outcome of nonlinear energy transfers, both on the horizontal and on the vertical, in the presence of stratification and rotation.

Disentangling the triadic interactions in Navier-Stokes equations

“…Hence, when the small-scales are strongly helically-signed the forward energy transfer is depleted. The existence of inverse cascade even when helicity is not positive-definite contradicts the predictions based only on the absolute equilibrium in the inviscid and unforced limit [27,28]. By concentrating the negative helical modes at small scales (high wavenumbers) we showed that as soon as triads of the other two classes (Class III and Class IV) become competitive, they take the leadership in the energy transfer mechanisms and the energy flux is reversed, reaching a more standard forward-cascade regime.…”

“…E 38, 114 (2015) helical case, is highly fragile [26]: it is enough to have a tiny number of helical modes with the opposite sign distributed uniformly on the Fourier space to revert the system to a forward cascade regime. Such a conclusion is also supported by arguments based on absolute equilibrium [27,28]. In this paper we explore the case when all Fourier modes have the same helicity (say positive) except for a small subset possessing also the opposite (negative) helicity.…”

Role of helicity for large- and small-scale turbulent fluctuations

“…On the other hand, it is well known that the external mechanisms such as rotation [22,23], confinement [24,25], shear [26] or coupling with the magnetic field [27] might revert the direction of the energy cascade. Strikingly enough, such a reversal of the flux has been predicted and observed also in 3D HIT with explicit breaking of parity invariance, i.e., by restricting the dynamics to a subset of Fourier modes such that the helicity becomes sign definite [28][29][30], suggesting that inverse energy transfer events are much broader than previously thought and they are potentially present in all flows in nature. Another not understood crucial aspect of fully developed turbulence is intermittency, the tendency of the flow to develop more and more non-Gaussian velocity fluctuation at smaller and smaller scales.…”

Mathematical and physical ideas for climate science

“…In addition to the 2‐D and quasi‐2‐D cases mentioned above, the theory has also been applied to stratified fluids (essentially in the quasi‐geostrophic regime). Herbert [] has obtained and classified the statistical equilibria of the two‐layer QG model in the framework of the Robert‐Miller‐Sommeria theory, and updated the discussion of the vertical distribution of energy at statistical equilibrium (see section 3.2.2): in particular, it is shown that even at statistical equilibrium, there will remain some residual energy in the baroclinic mode, unless the initial vertical profile of fine‐grained enstrophy is uniform. In the context of continuously stratified flows, Venaille [] has taken up the thread initiated by Merryfield [] (see section 3.2.2) and shown that bottom‐trapped currents are indeed statistical equilibria of the Robert‐Miller‐Sommeria theory.…”