Vorticity dilution in a shallow draining layer ($D(\zeta /h)/Dt=0$)

For inviscid nonrotating shallow water equations, columnar horizontal flow has material vorticity equation $D\zeta /Dt=-\zeta {\nabla }_h\cdot {\mathbf{u}}_h$ and depth equation $Dh/Dt=-h{\nabla }_h\cdot {\mathbf{u}}_h$. Thus $\zeta /h$ is conserved, not $\zeta$ itself. Squashing a column to much smaller depth decreases its vertical vorticity. This permits a late draining flow to approximate an irrotational $u_{\theta }=C/r$ profile outside the drain, despite nonzero initial solid-body vorticity. At finite nonzero depth the residual vorticity need not be exactly zero. In a rotating frame, replace $\zeta$ by absolute vorticity $\zeta +f$ in the conserved ratio. Kelvin's circulation theorem remains compatible with these changes because it follows a material loop, not a fixed spatial circle.