Parameterisation of the interactions of waves and zonal flows taking the full Coriolis acceleration into account

The necessity of going beyond the traditional approximation in the sub-inertial regime and in weakly stratified regions

Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM F-91191 Gif-sur-Yvette, France

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Received: 22 July 2025
Accepted: 23 October 2025

Abstract

Context. From the Earth’s atmosphere and oceans to stellar radiation zones, inertia-gravity waves, which are called gravito-inertial waves (GIWs) in astrophysics, transport momentum and mix matter when they are damped through heat and viscous diffusions and when they break. Their short-timescale dynamics is governed by the combined action of the buoyancy force and of the Coriolis acceleration. Through the transport they trigger, they modify the long-term evolution of the large-scale planetary atmospheric (oceanic) circulation and of the structure and rotation of stars. Many current models assume the so-called traditional approximation of rotation (TAR), where the local projection of the rotation vector along the horizontal direction is neglected, is assumed.

Aims. The TAR can be adopted for very thin fluid layers or when the projection of the Coriolis acceleration along the direction of the entropy and chemical stratifications can be neglected when compared to the buoyancy force. It is often assumed when momentum fluxes, heat, and matter transported by GIWs are evaluated. We identify the applicability regime of this approximation and propose a non-traditional parameterisation of the interactions of the waves and zonal flows in planetary atmospheres (oceans) and stellar interiors, in which the full Coriolis acceleration is taken into account.

Methods. We built a prototype local non-traditional Cartesian model in which we took the full Coriolis acceleration, buoyancy, and heat and viscous diffusions into account. We studied the two channels through which GIWs exchange momentum with mean flows while transporting heat and mixing matter: their linear damping and their non-linear breaking. We did not assume any hierarchy between stratification and rotation to allow us to explore the possible large parameter space in geophysical and astrophysical flows, in particular, the cases of weak stable stratification and rapid rotation.

Results. On the one hand, the radiative and viscous dampings of GIWs increase with a decreasing ratio of the wave frequency (ω) with the inertial frequency (2Ω; Ω is the angular rotation frequency). In the sub-inertial regime (ω < 2Ω), the TAR strongly underestimates GIWs damping, especially when the buoyancy and the Coriolis acceleration are equally strong. In this regime, a non-traditional modelling must be adopted. It predicts the correct altitude at which momentum is deposited, which is closer to the excitation region of waves than the altitude predicted using the TAR. On the other hand, non-traditional modelling of GIWs convective and shear-induced breakings were proposed, and we demonstrate that the TAR overestimates the momentum flux deposited by GIWs through these channels. When the full Coriolis acceleration is taken into account, the efficiency of the convective and shear-induced overturning is reduced and the transport is weaker than predicted when the TAR is assumed. Finally, we derived a fully non-traditional parameterisation of the interaction of GIWs with mean zonal flows. This can be implemented in numerical models of the long-term evolution of the atmospheric (oceanic) general circulation and of the structure of rotating stars.