Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 333 / 3

The quasi-geostrophic approximation requires small Rossby number, nearly horizontal geostrophic balance, hydrostatic vertical balance, small interface or density displacements, stable background stratification, and horizontal scales much larger than the vertical scale. The Boussinesq approximation and a beta plane are used, ageostrophic motion enters only at the order needed to evolve potential vorticity, and here the buoyancy frequency $N$ is constant. Under these assumptions the materially conserved three-dimensional quasi-geostrophic potential vorticity is

The QG thermodynamic relation makes vertical velocity proportional to the material derivative of ${\psi }_z$. At a flat rigid boundary, no normal flow therefore givesor ${\hat{\psi }}^{\prime }(0)=0$ for a nonzero-frequency mode. Let $K^2=k^2+l^2$ and take the vertical structure $\hat{\psi }={\psi }_0\cos (mz)$, which satisfies this condition. Linearizing $q_t+\beta {\psi }_x=0$ gives the Baroclinic Rossby wave dispersion relationFor long horizontal waves, $K{R}_d\ll 1$, so $\omega \simeq -\beta k{R}_d^2$: the vertical-mode deformation term controls the response. For short waves, $K{R}_d\gg 1$, so $\omega \simeq -\beta k/K^2$, the Barotropic Rossby wave form. In both limits the zonal phase propagation is westward relative to the mean flow.

Linearization about $U\hat{\mathbf{x}}$ changes the frequency to the intrinsic value $\omega -Uk$. ThusA mountain-fixed disturbance has $\omega =0$, so its vertical wavenumber satisfiesThe quasi-geostrophic mountain wave propagates vertically only when $m^2>0$. For $U>0$, this requiresOtherwise the disturbance is vertically evanescent.

At the topographic boundary, no normal flow gives $w^{\prime }=U{h}_x$. Since QG buoyancy is $b^{\prime }=f_0{\psi }_z^{\prime }$ and $U{b}_x^{\prime }+N^2{w}^{\prime }=0$, the linearized lower condition isFor the specified ridge $h=h_0\cos kx$, matching its horizontal structure sets $l=0$; retaining $K^2=k^2+l^2$ also covers a sinusoidal ridge with that horizontal wavevector. When $U>\beta /K^2$, defineThe decaying solution and its boundary amplitude areIncreasing $N$ raises the streamfunction amplitude in proportion to $N$ while reducing the vertical e-folding length ${\gamma }^{-1}$ in proportion to $N^{-1}$: stronger stratification produces a stronger but more shallowly trapped disturbance.