  The transport induced by ocean mesoscale eddies remains unresolved in most
state-of-the-art climate models and needs to be parameterized instead. The
natural scale separation between the forcing and the emergent turbulent flow
calls for a diffusive parameterization, where the eddy-induced fluxes are
related to the large-scale gradients by a diffusion tensor. The standard
parameterization scheme in climate modeling 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 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. 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 $K_R(z)$. Secondly, as already identified by previous authors the
vertical structures of the two coefficients are related by the so-called
Taylor-Bretherton relation. Finally, while the two coefficients generically
differ 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 numerically
simulating the QG dynamics of a horizontally homogeneous patch of ocean hosting
a vertically sheared zonal current resembling the Antarctic Circumpolar
Current.
