Approximate Solution of the Stommel Model: Sverdrup Balance
The Stommel model
$$
r \nabla^2 \psi + \beta \frac{\partial \psi}{\partial x} = F_\tau(x,y)
$$
is linear and it is possible to obtain an exact, analytic solution.
However, it is more insightful to approach the problem perturbatively, by supposing that the frictional term is small, meaning there is an approximate balance between wind stress and the $\beta$-effect
For Friction is small if $|r\zeta| \ll |\beta v|$ or
$$
\frac{\overbrace{r}^{=\,f_0 \delta_B/H}}{\underbrace{L}_{\text{horizontal scale}}}
= \frac{f_0 \delta_B}{HL} \ll \beta
$$
generally speaking this
inequality is well satisfied for large-scale flow
The vorticity equation becomes
$$
\beta v \approx \text{curl}_z \tau_T
$$
which is known as Sverdrup balance, a balance between the beta effect and wind stress curl
Sverdrup balance provides a useful, if not impregnable, foundation on which to build. However, the observational support for Sverdrup balance is rather mixed, with discrepancies arising from the presence of small-scale eddying motion with concomitantly large nonlinear terms, as well as from non-negligible vertical velocities induced by interaction with bottom topography.
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