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Most of the turbulent flows appearing in nature (e.g. geophysical andastrophysical flows) are subjected to strong rotation and stratification. Theseeffects break the symmetries of classical, homogenous isotropic turbulence. Indoing so, they introduce a natural decomposition of phase space in terms ofwave modes and potential vorticity modes. The appearance of a new time scaleassociated 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 motionmay 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 anotherphenomenology.  In this paper, we suggest a theoretical approach based on equilibriumstatistical mechanics for the ideal system, inspired from the restrictedpartition function formalism introduced in metastability studies. In thisframework, we show analytically that in the presence of rotation, when thedynamics is restricted to the slow modes, the equilibrium energy spectrumfeatures an infrared divergence characteristic of an inverse cascade regime,whereas this is not the case for purely stratified flows.

Restricted Equilibrium and the Energy Cascade in Rotating and Stratified Flows