In a narrow western layer, $x$ derivatives dominate $y$ derivatives. Subtracting the forced interior balance leaves
$$
\frac{\beta}{f_0}\psi_{{\rm bl},x}
\simeq-\gamma\psi_{{\rm bl},xx}.
$$
If the layer has width $\delta$, then $\psi_x=O(\psi/\delta)$ and $\psi_{xx}=O(\psi/\delta^2)$. Balancing the beta effect with linear bottom drag gives
$$
\frac{\beta}{f_0\delta}
\sim\frac{\gamma}{\delta^2},
$$
and therefore the <Stommel boundary layer> width is
$$
\boxed{\delta=\frac{\gamma f_0}{\beta}}.
$$
This scaling assumes $\delta\ll L$ and that along-boundary variations occur on the basin scale.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-333/2/c/solution