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

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.