Thermal Wind ClosureΒΆ
The circulation between basins or basin regions is represented by a Thermal wind closure (c.f. Nikurashin & Vallis (2012), Jansen & Nadeau (2019), Nadeau & Jansen (2020) )
\[\partial_{zz}\Psi^z\left(z\right)=\frac{1}{f}\left[b_2\left(z\right)-b_1\left(z\right)\right]\]
Where \(b_1(z)\) and \(b_2(z)\) are the buoyancy profiles in the adjacent regions, \(f\) is the coriolis parameter, and \(\Psi^z(z)\) is the overturning streamfunction in depth space.
Vanishing net mass flux between the columns yields the boundary conditions:
\[\begin{split}\begin{aligned}
\Psi^z\left(z=0\right)&=0 \\
\Psi^z\left(z=-H\right)&=0
\end{aligned}\end{split}\]
The isopycnal overturning streamfunction is then obtained by mapping the z-coordinate streamfunction obtained from the thermal wind relation onto the buoyancy in the respective up-stream column:
\[\Psi^b\left(b\right) = \int_{-H}^0 \partial_z\Psi^z\left(z\right)\mathcal{H}\left[b - b_{up}\left(z\right)\right]\]
where \(\mathcal{H}\) is the Heaviside step function and
\[\begin{split}\begin{aligned}
b_{up}\left(z\right) =
\begin{cases}
b_1\left(z\right), & \partial_z\Psi^z\left(z\right) > 0 \\
b_2\left(z\right), & \partial_z\Psi^z\left(z\right) < 0 \,.
\end{cases}
\end{aligned}\end{split}\]