= Solution
Take a singly <quantized vortex> of positive <circulation>. The <superfluid velocity> is $\mathbf u=(\hbar/m)\nabla S$, where $S$ is the <complex argument> of the condensate. Since $S=\theta$, <quantized circulation> gives $u_\theta=\hbar/(mr)$. Insert the radial <Thomas–Fermi approximation for a condensate> into the cylindrical volume <integral>, excluding the core $r<a_0$:
$$
E_{\rm kin}=\frac{\pi H_0\hbar^2n_0}{m}\int_{a_0}^{R_r}\left(1-\frac{r^2}{R_r^2}\right)\frac{dr}{r}.
$$
Integrating the logarithmic and quadratic terms separately gives
$$
\boxed{E_{\rm kin}=\frac{\pi H_0\hbar^2n_0}{m}\left[\log\frac{R_r}{a_0}-\frac12+\frac{a_0^2}{2R_r^2}\right].}
$$
The <logarithm> is the familiar long-range flow contribution of a <quantum vortex>; the $-1/2$ comes from the declining trapped <number density>. If the bucket truncates the cloud, put $R=R_0<R_r$ and the same integration gives $E_{\rm kin}=\pi H_0\hbar^2n_0[\log(R/a_0)-(R^2-a_0^2)/(2R_r^2)]/m$, provided $a_0<R$. A vortex of integer <winding number> $q$ has $u_\theta=q\hbar/(mr)$ and multiplies this flow <energy> by $q^2$. The core radius acts as a cutoff of order the local <healing length>; its internal <gradient> and interaction <energies> are not computed by this shell estimate.
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