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A direct derivation of the Gent–McWilliams/Redi diffusion tensor from quasi-geostrophic dynamics

Published online by Cambridge University Press:  19 May 2023

Julie Meunier ,
Benjamin Miquel Open the ORCID record for Benjamin Miquel [Opens in a new window]  and
Basile Gallet Open the ORCID record for Basile Gallet [Opens in a new window]
Julie Meunier
Affiliation:
Service de Physique de l'Etat Condensé, Université Paris-Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France
Benjamin Miquel
Affiliation:
Univ Lyon, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, LMFA, UMR5509, 69130, Ecully, France
Basile Gallet*
Affiliation:
Service de Physique de l'Etat Condensé, Université Paris-Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France
*
†Email address for correspondence: basile.gallet@cea.fr
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Abstract

The transport induced by ocean mesoscale eddies remains unresolved in most state-of-the-art climate models and needs to be parametrized instead. The natural scale separation between the forcing and the emergent turbulent flow calls for a diffusive parametrization, where the eddy-induced fluxes are related to the large-scale gradients by a diffusion tensor. The standard parametrization scheme in climate modelling consists in adopting the Gent–McWilliams/Redi (GM/R) form for the diffusion tensor, initially put forward based on physical intuition and educated guesses before being put on firm analytical footing using a thickness-weighted average (TWA). In the present contribution, we provide a direct derivation of this diffusion tensor from the quasi-geostrophic (QG) dynamics of a horizontally homogeneous three-dimensional patch of ocean hosting a large-scale vertically sheared zonal flow on the $\beta$ plane. The derivation hinges on the identification of a useful cross-invariant defined as the product of the buoyancy and QG potential vorticity fluctuations. While less general than the TWA approach, the present QG framework leads to rigorous constraints on the diffusion tensor. First, there is no diapycnal diffusivity arising in the QG GM/R tensor for low viscosity and small-scale diffusivities. The diffusion tensor then involves only two vertically dependent coefficients, namely the GM transport coefficient $K_{GM}(z)$ and the Redi diffusivity $K_R(z)$. Second, as identified already by previous authors, the vertical structures of the two coefficients are related by the so-called Taylor–Bretherton relation. Finally, while the two coefficients differ generically in the interior of the water column, we show that they are equal to one another near the surface and near the bottom of the domain for low-enough dissipative coefficients. We illustrate these findings by simulating numerically the QG dynamics of a horizontally homogeneous patch of ocean hosting a vertically sheared zonal current resembling the Antarctic Circumpolar Current.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Geostrophic turbulence Geophysical and Geological Flows: Ocean processes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 963 , 25 May 2023 , A22
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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