= Vorticity dilution in a shallow draining layer
{title2=$D(\zeta/h)/Dt=0$}
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.
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