Past exam of the geophysical fluid dynamics course of the University of Oxford / 2023 / 2 / a

Use eastward $x$, northward $y$, pressure coordinate $p$ increasing downward, and geopotential $\Phi =gz$. Horizontal derivatives are taken on an isobaric surface. The horizontal geostrophic balance and vertical hydrostatic balance arewhere $R$ is the specific gas constant of dry air and the final equality uses the ideal gas law. At fixed latitude $f$ is independent of $p$. Cross-differentiating givesConsequently the pressure-coordinate thermal wind equations areEquivalently, $f{\mathbf{e}}_z\times {\partial }_{\ln p}{\mathbf{u}}_g=R{\nabla }_{p}T$. Notice the sign: increasing eastward velocity upward means decreasing eastward velocity with $\ln p$, and hence poleward-decreasing temperature in the Northern Hemisphere.

For the stratospheric polar vortex estimate, the original diagram must be read on its logarithmic pressure axis. Near $60^{\circ }$ in the winter hemisphere, adjacent $10\mathrm{m}{\mathrm{s}}^{-1}$ wind contours suggest an increase of roughly $20\mathrm{m}{\mathrm{s}}^{-1}$ through a pressure decade surrounding $10\mathrm{mb}$: for example, about $10\mathrm{m}{\mathrm{s}}^{-1}$ near $30\mathrm{mb}$ and $30\mathrm{m}{\mathrm{s}}^{-1}$ near $3\mathrm{mb}$. These are coarse graphical readings, not a sounding. ThusWith $|f|=2\Omega \sin 60^{\circ }\simeq 1.26\times 10^{-4}{\mathrm{s}}^{-1}$ and $R\simeq 287\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, the poleward temperature gradient is approximatelyIf the vortex edge is taken to span roughly $50^{\circ }$ to $70^{\circ }$, its width is $a(20\pi /180)\simeq 2.2\times 10^6\mathrm{m}$, givingThe estimate depends on the width assigned to the edge and the graphical vertical shear; a range of several to roughly fifteen kelvin is more justified than a precise value. The sign remains a cold winter pole in either hemisphere. The diagram does not by itself prescribe a unique pair of edge latitudes.

At low latitudes, horizontal free-atmospheric temperature gradients are usually much smaller than at the winter vortex edge. For a comparable finite wind vertical shear, the rearranged thermal-wind balance, $T_y=(f/R)u_{\ln p}$, is consistent with weaker temperature gradients as $|f|$ decreases. However, geostrophic balance loses validity near the equator because the Coriolis parameter vanishes; division by $f$ cannot be used there to derive a prediction. The weak temperature gradient approximation is better motivated by rapid horizontal internal gravity wave adjustment, together with tropical convection and the Hadley cell. Thermal-wind balance describes the balanced extratropical relation and its tendency approaching the tropics; it is not a complete causal explanation of equatorial thermal uniformity.