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

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.