Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 80 / 3 / Solution

Balanced variables and PV derivation. Write the disturbance reduced pressure as $p^{\prime }/{\rho }_0=f_0\psi$. Leading geostrophic balance gives $(u_g,v_g)=(-{\psi }_y,{\psi }_x)$, and leading hydrostatic balance gives buoyancy anomaly ${\sigma }^{\prime }=p_z^{\prime }/{\rho }_0=f_0{\psi }_z$. The geostrophic material derivative isIt includes horizontal advection, not an independent prognostic vertical velocity. The appropriate regime has small Rossby number, small stratified Froude number, thin aspect ratio and slow evolution; a common scaling has $[NH/(f_{0}L){]}^2$ of order one and $\beta L/|f_0|\ll 1$.

Only the vertical rotation component is retained in the traditional approximation. In the horizontal momentum balance, the horizontal part of the rotation vector multiplies the small vertical velocity; in vertical momentum, its contribution is small compared with leading hydrostatic pressure and buoyancy in the thin-layer scaling. Away from the equator these corrections are smaller by a factor of order $(H/L)\cot \lambda$. They need not be small in equatorial or strongly nonhydrostatic settings, so the approximation has a definite regime of validity.

Put $A(z)=f_0^2/N^2(z)$. Eliminating $w$ from buoyancy givesThe last step needs justification because $D_g$ and ${\partial }_z$ do not commute on arbitrary fields. Their extra terms here areSubstitution into the supplied vorticity equation givesSince $D_{g}f=\beta v_g=\beta {\psi }_x$,This is three-dimensional quasi-geostrophic potential vorticity conservation; the height-dependent coefficient is retained inside the derivative.

At a small steady lower elevation $b(x,y)$, impermeability gives $w=u_g{b}_x+v_g{b}_y=D_{g}b$, evaluated at $z=0$ to leading order. Substitution into buoyancy and division by $N^2$ gives the topographic buoyancy boundary condition for quasi-geostrophic flow,

Rossby waves at rest. For constant $N$ let $\alpha =f_0^2/N^2$ and $K^2=k^2+\alpha m^2$. Linearization about relative rest gives ${\partial }_t({\psi }_{xx}+\alpha {\psi }_{zz})+\beta {\psi }_x=0$. A Fourier wave givesFor $\beta >0$, the zonal phase speed $\omega /k=-\beta /K^2$ is westward. At fixed altitude the PV perturbation is $q^{\prime }=-K^2{\psi }^{\prime }$; its total field is the background $f_0+\beta y$ with sinusoidally displaced contours, as sketched below.

A northward parcel displacement has $q^{\prime }=-\beta {\xi }_y$: it retains the smaller background PV of its origin. The inverted geostrophic velocity associated with a positive PV crest is northward on its eastern flank and southward on its western flank. Advection of the background PV then decreases the anomaly on the eastern flank and increases it on the western flank, shifting the crest westward. This is a PV-restoring mechanism, not a westward mean current.

Differentiating the dispersion relation gives the group velocityFor nonzero $k,m$, the vertical phase speed is $\omega /m=-\beta k/[m(k^2+\alpha m^2)]$, soThus upward energy propagation accompanies downward vertical phase propagation, and conversely. For $m=0$ the vertical group speed is zero and a vertical phase speed is not defined.

Stationary topographic response. Assume $f_0\ne 0$, $\beta >0$ and nonzero corrugation wavenumber $k$. In a uniform current the laboratory dispersion is ${\omega }_{\mathrm{lab}}=Uk-\beta k/K^2$. For a stationary quasi-geostrophic mountain wave, $K^2=\beta /U$, givingFor $U\ne 0$, the linear lower-boundary condition becomes $U{\partial }_x[(f_0/N^2){\psi }_z^{\prime }+b]=0$. Its forced nonzero-$k$ component therefore hasIf $0<U<\beta /k^2$, the vertical wavenumber is real. Both signs would be bounded aloft, so boundedness alone is insufficient for uniqueness. Impose an upward radiation condition, excluding waves incident from infinity; equivalently switch on the terrain forcing causally with arbitrarily weak absorption aloft. Since $c_{gz}$ has the sign of $km$, chooseThe unique outgoing forced component is thenThe bottom derivative and $w^{\prime }(0)=U{b}_x$ verify the phase and amplitude. In this stationary pattern it is the intrinsic frequency ${\omega }_{\mathrm{int}}=-Uk$ that describes downward vertical phase propagation relative to the current; the laboratory frequency is zero.

If $U<0$ or $U>\beta /k^2$, defineExclude exponential growth at infinity. The unique decaying forced component isAt the excluded value $U=\beta /k^2$, a nonzero prescribed bottom derivative cannot be matched by a bounded constant/linear vertical solution; the steady idealization is singular. At $U=0$ the boundary is not crossed and there is no forced response in the linear nonzero-$k$ sector: ${\psi }^{\prime }=0$, rather than an expression obtained by dividing by $U$. Uniqueness throughout refers to the prescribed forced Fourier component, with the reference state fixed and no independently incident waves or added mean-state changes.

The propagation criterion explains why topographically forced planetary Rossby waves can carry energy into a moderately westerly stratosphere, while easterlies and sufficiently strong westerlies block their vertical propagation. Larger horizontal scales more readily meet the criterion. Real dissipation, height-dependent winds and critical levels modify the simple solution and allow wave–mean-flow interaction; the ideal constant-coefficient calculation establishes the propagation filter, not a complete theory of stratospheric evolution.