Planetary-geometric constraints on isopycnal slope in the Southern Ocean

On planetary scales, surface wind stress and differential buoyancy forcing act together to produce isopycnal surfaces that are relatively flat in the tropics/subtropics and steep near the poles, where they tend to outcrop. Tilted isopycnals in a rapidly rotating fluid are subject to baroclinic instability. The turbulent, mesoscale eddies generated by this instability have a tendency to homogenize potential vorticity (PV) along density surfaces. In the Southern Ocean (SO), the tilt of isopycnals is largely maintained by competition between the steepening effect of surface forcing and the flattening effect of turbulent, spatially inhomogeneous eddy fluxes of PV. Here we use quasi-geostrophic theory to investigate the influence of a planetary-geometric constraint on the equilibrium slope of tilted density/buoyancy surfaces in the SO. If the meridional gradients of relative vorticity and PV are small relative to β, then quasi-geostrophic theory predicts ds/dz = β/ f0 = cot(ϕ0)/a, or equivalently r ≡ |∂zs/(β/ f0)| = 1, where s is the isopycnal slope, ϕ0 is a reference latitude, a is the planetary radius, and r is the depth-averaged criticality parameter. We find that the strict r = 1 condition holds over specific averaging volumes in a large-scale climatology. A weaker r = O(1) condition for depth-averaged quantities is generally satisfied away from large bathymetric features. We employ the r = O(1) constraint to derive a depth scale to characterize large-scale interior stratification, and we use an idealized sector model to test the sensitivity of this relationship to surface wind forcing. Finally, we discuss the possible implications for eddy flux parameterization and for the sensitivity of SO circulation/stratification to changes in forcing.