Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 2 / a / Solution

Assume $f>0$; replacing it by $|f|$ gives the thickness for either hemisphere. With $W=(u-U)+iv$, the steady anomaly equations reduce toThus the Bottom Ekman layer hasIts integrated anomalous transport isIt is the transport required by the vertically integrated momentum balance between Coriolis acceleration and bottom stress. For slowly varying geostrophic flow, $\nabla \cdot {\mathbf{U}}_g=0$ and $\nabla \cdot (\hat{z}\times {\mathbf{U}}_g)=-{\zeta }_g$, soMass conservation therefore gives the interior Ekman pumping condition