Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2019 / 6

The underlying principle is material conservation of potential vorticity for adiabatic, inviscid, rotating, stably stratified motion. In the quasi-geostrophic approximation, the leading horizontal velocity is nondivergent and geostrophic:Here $f_0=2\Omega \sin {\varphi }_0$ is the reference Coriolis parameter, $\beta =(df/dy{)}_0=2\Omega \cos {\varphi }_0/a$ on a spherical planet, and $f=f_0+\beta y$. The positive buoyancy frequency squared is $N^2$; in a Boussinesq fluid it is $-(g/{\rho }_0)d\bar{\rho }/dz$. To quasi-geostrophic order the potential vorticity is the background $f_0+\beta y$ plus the stated perturbation $q^{\prime }$.

Linearizing its material derivative about uniform zonal velocity $U$ giveswhich is the supplied equation. The uniform basic flow contributes no relative-vorticity gradient or vertical shear term.

The approximation needs small Rossby number, time scales long compared with $|f_0{|}^{-1}$, vertical hydrostatic balance, stable stratification, small departures from the reference state and meridional extent small enough for a local beta plane. Buoyancy and rotation must balance at the relevant aspect ratio. Friction, diabatic heating and strong nonlinear advection are neglected here. It applies to balanced midlatitude ocean and atmospheric motions; an atmospheric version also requires a consistent reference-density or pressure-coordinate treatment. It fails in a sufficiently near-equatorial region where $f_0$ vanishes.

The quasi-geostrophic potential vorticity perturbation combines relative vorticity, ${\nabla }_h^2{\psi }^{\prime }$, with the stretching contribution $f_0^2{\partial }_z(N^{-2}{\psi }_z^{\prime })$. The latter measures how buoyancy surfaces and rotating fluid columns are displaced in a stratified fluid. Even a flow with no vertical dependence can have nonzero relative vorticity, while its explicit stretching term vanishes.

For a vertically uniform plane wave with horizontal wavenumbers $(k,l)$, $q^{\prime }=-(k^2+l^2){\psi }^{\prime }$. Substitution givesThus phase propagates westward relative to the current. Physically, northward parcel displacement carries lower background planetary vorticity into a region with larger $f$, requiring negative relative vorticity to conserve total potential vorticity; southward displacement requires positive relative vorticity. The velocity induced by that vorticity pattern shifts the displacement wave westward. This is a statement about phase motion, not a claim that every Rossby-wave packet has westward group velocity.

For constant $N^2$, substitution of the vertical cosine and horizontal plane wave givesThe top rigid-lid condition is satisfied by $\cos (mz)$. To select a discrete vertical mode one must also impose the corresponding no-flow, adiabatic condition at the flat bottom, ${\psi }_z^{\prime }(-H)=0$. It requires $mH=n\pi$. The mode $n=0$ is barotropic; the gravest nonzero baroclinic mode has $m=\pi /H$. HenceThe lid condition alone is insufficient: $\cos [\pi z/(2H)]$ also has zero derivative at the lid but violates the bottom condition. Constant depth motivates the additional physical boundary, which must be used to obtain the printed denominator.

Let $a_d=\pi |f_0|/(HN)=L_d^{-1}$ and set $l=0$. The zonal phase velocity and group velocity areThe phase curve is even in $k$, with limiting value $U-\beta /a_d^2$ at zero and approach to $U$ from below at large $|k|$. The group curve is also even: it starts at the same long-wave limit, equals $U$ at $|k|=a_d$, reaches $U+\beta /(8a_d^2)$ at $|k|=\sqrt{3}{a}_d$, and approaches $U$ from above at large $|k|$. The normalized curves appear in the bottom panels of the preceding figure. For $k=0$, the phase speed means the limit $\omega /k$; there is no zonal crest propagation to define directly at exactly zero wavenumber.

Use the standard Earth radius $a_E=6.371\times 10^6\mathrm{m}$ and rotation rate $\Omega =7.292\times 10^{-5}{\mathrm{s}}^{-1}$. At $10^{\circ }$S,Take a predominantly zonal wave train, $l\simeq 0$, and the gravest baroclinic mode from part (c). ThenThe background current is westward, so $U=-0.025\mathrm{m}{\mathrm{s}}^{-1}$. The pulse uses the baroclinic Rossby-wave group velocity, not the crest speed:ThereforeFor comparison $c_{px}\simeq -0.373\mathrm{m}{\mathrm{s}}^{-1}$ would give about $0.85$ years, but that is a crest transit time and not the requested pulse delay.

The approximately three-year group delay is comparable to the stated three-to-five-year El Niño–Southern Oscillation period. Wind-driven thermocline disturbances can store and convey oceanic adjustment westward; interaction with the western boundary and subsequent faster signals can contribute a delayed feedback on the tropical coupled system. This calculation supports a relevant time scale, not a complete ENSO oscillator: realistic meridional structure, variable stratification and currents, damping, boundary conversion and equatorial coupling are omitted. A nonzero unspecified meridional wavenumber would also change the numerical group speed, so the $l\simeq 0$ assumption is material.