= Solution
Conservation of <shallow-water potential vorticity> with fixed <relative vorticity> gives
$$
\boxed{\frac{H_2}{H_1}=\frac{f_2+\zeta}{f_1+\zeta}.}
$$
For ordinary planetary-vorticity-dominated columns, moving towards the equator decreases the magnitude of <absolute vorticity>, so the column must \b[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.
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