= Solution
For both $y>0$ and $y<0$, the velocity $v=-W_0y/(2f_0)$ carries parcels toward $y=0$, where $H=y^2$ decreases. Their leading potential vorticity $f_0/H$ therefore increases along the path, exactly as required by the positive source $W_0/H$. The explicit interior velocity has zero relative vorticity, so the leading change comes from vortex-column stretching rather than from $\zeta/H$. Relative vorticity and friction become important only where boundary layers close the circulation or where the singular depth profile near $y=0$ invalidates the approximation.
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