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Kraichnan seminal ideas on inverse cascades yielded new tools to study commonphenomena in geophysical turbulent flows. In the atmosphere and the oceans,rotation and stratification result in a flow that can be approximated astwo-dimensional at very large scales, but which requires consideringthree-dimensional effects to fully describe turbulent transport processes andnon-linear phenomena. Motions can thus be classified into two classes: fastmodes consisting of inertia-gravity waves, and slow quasi-geostrophic modes forwhich the Coriolis force and horizontal pressure gradients are close tobalance. In this paper we review previous results on the strength of theinverse cascade in rotating and stratified flows, and then present new resultson the effect of varying the strength of rotation and stratification (measuredby the ratio $N/f$ of the Brunt-V\"ais\"ala frequency to the Coriolisfrequency) on the amplitude of the waves and on the flow quasi-geostrophicbehavior. We show that the inverse cascade is more efficient in the range of$N/f$ for which resonant triads do not exist, $1/2 \le N/f \le 2$. We then usethe spatio-temporal spectrum, and characterization of the flow temporal andspatial scales, to show that in this range slow modes dominate the dynamics,while the strength of the waves (and their relevance in the flow dynamics) isweaker.

Inverse cascades and resonant triads in rotating and stratified turbulence