Southern Ocean Overturning Transport Closure

For the Southern Ocean, the meridional overturning circulation is assumed to be maintained by a balance between the wind-driven Ekman transport and the eddy transport (following Marshall & Radko, 2003 and Marshall & Zanna, 2014):

\[\Psi_{SO}=-\frac{\tau L_x}{\rho_of}+L_x K s\]

where \(\tau\) is the wind stress magnitude, \(L_x\) is the zonal extent of the channel, \(\rho_o\) is the reference density, and \(f\) is the Coriolis frequency. \(K\) is an eddy “diffusivity”, following the Gent & McWilliams (1990) (GM) parameterization.

\(s=s(b)\) is the isopycnal slope in the Southern Ocean:

\[\begin{split}\begin{aligned} s\left(b\right)&=\begin{cases} \frac{z_B\left(b\right)}{L_y-y_{SO}\left(b\right)} & b >= b_{min} \\ \frac{\tau}{K\rho_o f} & b < b_{min} \end{cases} \end{aligned}\end{split}\]

where \(L_y\) is the meridional extent of the Southern Ocean channel, \(z_B\left(b\right)\) is the depth of the isopycnal with density \(b\) in the adjoining basin, and \(y_{SO}(b)\) is the latitude at which the isopycnal of density \(b\) outcrops in the channel. \(b_{min}\) is the minimum surface buoyancy in the channel. Isopycnals with density less than \(b_{min}\) do not outcrop at the surface in the Southern Ocean. Since we approximate the interior of the Southern Ocean to be adiabatic, the residual circulation has to vanish on isopycnals that do not outcrop, and their slope is set to be such that the eddy-driven transport exactly cancels the wind-driven Ekman transport.