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

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

In the quasi-geostrophic approximation, buoyancy is proportional to $f{\psi }_z$ and its material equation givesThe governing interior and boundary equations are consequentlyAfter linearization about rest, put $m=Nk/|f|$ and $r=Nk\delta /2$. The initial PV is $-k^2{\psi }_0\sin kx$, and the decaying homogeneous vertical solution givesThe boundary current decays on time $r^{-1}=2/(Nk\delta )$, while the original current survives aloft. The boundary influence penetrates onlyStronger stratification or shorter horizontal scale confines the adjustment more tightly; rotation communicates it farther upward. This is quasi-geostrophic Ekman spin-down.