= Radial flow near a driven vortex core
{title2=$u_r'(0)=-\operatorname{Im}\xi/2$}
For $[\nabla^2+\xi(1-|\psi|^2)]\psi=0$ and a regular unit-charge core $\psi=f(r)e^{i[\theta+\chi(r)]}$, write current <velocity> $u_r=2\chi'$ when the time-dependent kinetic operator is $-\nabla^2$. Separating the imaginary part gives
$$
(rf^2u_r)'=-2\operatorname{Im}\xi\;rf^2(1-f^2).
$$
With $f=ar+O(r^3)$ and no singular core flux, integration gives $u_r=-\operatorname{Im}\xi\,r/2+O(r^3)$. If $\xi=gn_\infty-i\alpha$, the slope is $\alpha/2$. The phase-gradient convention $k=\chi'$ has half this slope, $\alpha/4$. These are local regularity results, not a global vortex-existence assertion.
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