Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 333 / 3 / Solution

The interior equation materially conserves three-dimensional quasi-geostrophic potential vorticity. At each rigid horizontal boundary, ${\psi }_z$ is proportional to the boundary buoyancy anomaly, so the other two equations express material conservation of that buoyancy.

A background streamfunction $\overline{\psi }=-\Lambda zy$ gives ${\overline{\mathbf{u}}}_g=(\Lambda z,0,0)$. Its interior PV is uniform. Writing $\psi =\overline{\psi }+{\psi }^{\prime }$ and retaining first-order terms givesin the interior. Since ${\overline{\psi }}_z=-\Lambda y$, perturbation advection of the boundary buoyancy givesat $z=0,H$. At the lower boundary this is replaced by

For a Fourier component $\hat{\psi }(z,t)e^{ikx}$ with zero interior PV,Solving this boundary-value problem in terms ofgiveswhere, with $\mu =NHk/f$,The boundary equations consequently reduce toFor $q_0,q_H\propto e^{-ikct}$, defineThe determinant gives the Damped Eady-wave dispersion relation