Slow-drain angular-momentum amplification ($r^2\omega \simeq {\omega }_0{R}^2$)

Axisymmetric inviscid flow has no azimuthal pressure torque, so each parcel conserves specific axial angular momentum $j=r{u}_{\theta }=r^2\omega$. If late draining fluid originally lay near the rim $R$ of a uniformly rotating pool, then $j\simeq {\omega }_0{R}^2$ and $\omega \simeq {\omega }_0{R}^2/r^2$. The late-origin assumption is essential; torque conservation alone gives $j={\omega }_0{r}_0^2$ for each parcel's own initial radius $r_0$. Quasistatic radial and vertical pressure balance then gives free-surface depression $h(r)={\omega }_0^2{R}^4(r^{-2}-R^{-2})/(2g)$. The singular profile applies outside a finite drain or vortex core, not at the rotation axis.