# Inverse cascades and resonant triads in rotating and stratified turbulence

## TL;DR

This paper reviews and presents new findings on how rotation and stratification influence inverse cascades and wave interactions in geophysical turbulence, highlighting the conditions under which slow modes dominate and inverse cascade efficiency varies.

## Contribution

It provides new insights into the effect of the N/f ratio on inverse cascade strength and wave dynamics in rotating and stratified turbulence.

## Key findings

- Inverse cascade is more efficient for 1/2 ≤ N/f ≤ 2.
- Slow quasi-geostrophic modes dominate in this N/f range.
- Wave strength is weaker when resonant triads do not exist.

## Abstract

Kraichnan seminal ideas on inverse cascades yielded new tools to study common phenomena in geophysical turbulent flows. In the atmosphere and the oceans, rotation and stratification result in a flow that can be approximated as two-dimensional at very large scales, but which requires considering three-dimensional effects to fully describe turbulent transport processes and non-linear phenomena. Motions can thus be classified into two classes: fast modes consisting of inertia-gravity waves, and slow quasi-geostrophic modes for which the Coriolis force and horizontal pressure gradients are close to balance. In this paper we review previous results on the strength of the inverse cascade in rotating and stratified flows, and then present new results on the effect of varying the strength of rotation and stratification (measured by the ratio $N/f$ of the Brunt-V\"ais\"ala frequency to the Coriolis frequency) on the amplitude of the waves and on the flow quasi-geostrophic behavior. 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 use the spatio-temporal spectrum, and characterization of the flow temporal and spatial 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) is weaker.