In the <quasi-geostrophic approximation>, buoyancy is proportional to $f\psi_z$ and its material equation gives
$$
w=-\frac f{N^2}\frac{D_g\psi_z}{Dt}.
$$
The governing interior and boundary equations are consequently
$$
\boxed{\frac{D_g}{Dt}\left(\nabla_h^2\psi+\frac{f^2}{N^2}\psi_{zz}\right)=0\quad(z>0),}
$$
$$
\boxed{\frac{D_g\psi_z}{Dt}=-\frac{N^2\delta}{2f}\nabla_h^2\psi\quad(z=0).}
$$
After 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 gives
$$
\boxed{psi=\psi_0\left[1-(1-e^{-rt})e^{-mz}\right]\sin kx.}
$$
The boundary current decays on time $r^{-1}=2/(Nk\delta)$, while the original current survives aloft. The boundary influence penetrates only
$$
\boxed{m^{-1}=\frac{|f|}{Nk}.}
$$
Stronger stratification or shorter horizontal scale confines the adjustment more tightly; rotation communicates it farther upward. This is <quasi-geostrophic Ekman spin-down>.

Solution

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

Bottom <Ekman pumping> damps a horizontal quasi-geostrophic Fourier mode while its influence penetrates vertically only over $|f|/(Nk)$. For a half-space with a vertically uniform initial mode, the boundary amplitude decays exponentially while the remote interior remains unchanged.

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