Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 333 / 3 / i

Assume inviscid, unforced evolution, so shallow-water potential vorticity is conserved. If a column has initial depth $H_1$ and relative vorticity ${\zeta }_1$, moving to $H_2>H_1$ at fixed Coriolis parameter givesIts absolute vorticity is amplified by vortex stretching. A column initially without relative vorticity develops cyclonic spin: positive in the Northern Hemisphere and negative in the Southern Hemisphere. More generally the increase is in the direction of its original absolute vorticity, not necessarily the direction of $f$ if the initial spin is unusually strong.

Conservation of shallow-water potential vorticity with fixed relative vorticity givesFor ordinary planetary-vorticity-dominated columns, moving towards the equator decreases the magnitude of absolute vorticity, so the column must become shallower and spread horizontally. This is the compression counterpart of vortex stretching. For $\zeta =0$, the depth ratio is $f_2/f_1$, in either hemisphere before reaching the equator. The displayed general formula is essential if the fixed relative vorticity is large: the sign of the depth change then depends on the original absolute vorticity, and conservation cannot continue through zero absolute vorticity with positive finite depth and nonzero conserved potential vorticity.