 Solution

For both $y>0$ and $y<0$, the velocity $v=-W_{0}y/(2f_0)$ 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.