Restricted Equilibrium and the Energy Cascade in Rotating and Stratified Flows

Abstract

Most of the turbulent flows appearing in nature (e.g. geophysical and astrophysical flows) are subjected to strong rotation and stratification. These effects break the symmetries of classical, homogenous isotropic turbulence. In doing so, they introduce a natural decomposition of phase space in terms of wave modes and potential vorticity modes. The appearance of a new time scale associated to the propagation of waves, in addition to the eddy turnover time, increases the complexity of the energy transfers between the various scales; nonlinearly interacting waves may dominate at some scales while balanced motion may prevail at others. In the end, it is difficult to predict \emph{a priori} if the energy cascades downscale as in homogeneous isotropic turbulence, upscale as expected from balanced dynamics, or follows yet another phenomenology. In this paper, we suggest a theoretical approach based on equilibrium statistical mechanics for the ideal system, inspired from the restricted partition function formalism introduced in metastability studies. In this framework, we show analytically that in the presence of rotation, when the dynamics is restricted to the slow modes, the equilibrium energy spectrum features an infrared divergence characteristic of an inverse cascade regime, whereas this is not the case for purely stratified flows.

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• Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation
• Interplay of waves and eddies in rotating stratified turbulence and the link with kinetic-potential energy partition
• Inverse cascades and resonant triads in rotating and stratified turbulence
• Absorption of waves by large scale winds in stratified turbulence
• A statistical mechanics approach to mixing in stratified fluids
• Energy cascades in rapidly rotating and stratified turbulence within elongated domains
• Equilibrium statistical mechanics and energy partition for the shallow water model
• An Introduction to Large Deviations and Equilibrium Statistical Mechanics for Turbulent Flows