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

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

For $ϵ=0$, the quantity under the square root is real:It is negative for small $\mu$ and positive for large $\mu$. Therefore the Eady instability occurs for $\mu <{\mu }_c$, whereand modes are neutral for $\mu >{\mu }_c$.

For $\mu \gg 1$, $\coth \mu =1+O(e^{-2\mu })$ and $\mathrm{csch}\mu$ is exponentially small. HenceThese are two decoupled Boundary Rossby waves, one localized near each boundary. Their laboratory phase speeds lie just inside the basic velocities $\Lambda H$ and $0$ because each propagates intrinsically against the local flow.

At $\mu ={\mu }_c$, the real undamped part $F$ vanishes. PutThe neutral equation implies $d=1/4-1/{\mu }_c^2>0$. The radicand is thereforeIf $\sqrt{z}=u+iv$ is chosen with $v>0$, thenThus $v>ϵ/2$, and the plus branch hasFor $k\Lambda >0$ this is exponential growth. Lower-boundary damping therefore destabilizes the formerly neutral cutoff mode, an example of dissipation-induced instability.

When $\mu \gg 1$, the boundary coupling $B/H=\mathrm{csch}\mu /\mu$ is exponentially small. The two eigenvalues are therefore the diagonal boundary-wave speeds through order $1/\mu$:The lower wave is damped at the imposed rate, while the upper wave is exponentially isolated from the lower boundary and remains neutral to algebraic order. Damping destroys the phase-locked counterpropagating-wave interaction at short horizontal wavelength rather than damping both waves equally.