= Solution
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> gives
$$
\boxed{\zeta_2=\frac{H_2}{H_1}(f+\zeta_1)-f,\qquad \zeta_2-\zeta_1=\left(\frac{H_2}{H_1}-1\right)(f+\zeta_1).}
$$
Its <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.
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