= Slow-drain angular-momentum amplification
{title2=$r^2\omega\simeq\omega_0R^2$}
Axisymmetric <inviscid flow> has no azimuthal pressure torque, so each parcel conserves specific axial <angular momentum> $j=ru_\theta=r^2\omega$. If late draining fluid originally lay near the rim $R$ of a uniformly rotating pool, then $j\simeq\omega_0R^2$ and $\omega\simeq\omega_0R^2/r^2$. The late-origin assumption is essential; torque conservation alone gives $j=\omega_0r_0^2$ for each parcel's own initial radius $r_0$. Quasistatic radial and vertical pressure balance then gives free-surface depression $h(r)=\omega_0^2R^4(r^{-2}-R^{-2})/(2g)$. The singular profile applies outside a finite drain or vortex core, not at the rotation axis.
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