Sverdrup balance

Remarks

Integrating the momentum equations over depth gives $-f V = -\rho_0^{-1} \partial_x P + \rho_0^{-1} \tau^x$ and $f U = -\rho_0^{-1} \partial_y P + \rho_0^{-1} \tau^y$, where $U$ is the zonal transport and $P = \int_{-H}^{0} p \, dz$. Taking the curl eliminates $P$ and leaves $f (\partial_x U + \partial_y V) + \beta V = \rho_0^{-1} (\partial_x \tau^y - \partial_y \tau^x)$. The depth-integrated continuity equation with $w = 0$ at top and bottom gives $\partial_x U + \partial_y V = 0$, and the balance follows.

Example: in the northern hemisphere the westerlies blow eastward near latitude $45^\circ$, and the trade winds blow westward near latitude $15^\circ$. So $\tau^x$ increases with $y$ in between, and the wind-stress curl is negative. The balance then gives $V < 0$: the interior of the subtropical gyre carries water toward the equator. The return flow cannot obey the balance, and it occurs in a narrow western boundary current such as the Gulf Stream.

The assumption $\beta \ne 0$ is essential: on an $f$-plane the left side vanishes and the equation gives no information on the transport.

Going deeper

For a student

The balance is the vorticity equation $\beta v = f \, \partial w / \partial z$ of geostrophic flow, integrated over the depth, with the vertical velocity at the base of the Ekman layer given by Ekman pumping. Given $V$, the zonal transport $U$ follows from continuity by integrating $\partial_x U = -\partial_y V$ westward from the eastern boundary, where $U = 0$. The interior solution cannot also satisfy the condition at the western boundary, and this mismatch is resolved by the frictional boundary layers of Stommel and Munk.