  Vorticity distributions in axisymmetric vortex rings produced by a
piston-pipe apparatus are numerically studied over a range of Reynolds numbers,
$\mathrm{Re}$, and stroke-to-diameter ratios, $L/D$. It is found that a state
of advective balance, such that $\zeta \equiv \omega_\phi/r \approx F(\psi,
t)$, is achieved within the region (called the vortex ring bubble) enclosed by
the dividing streamline. Here $\zeta \equiv\omega_\phi/r$ is the ratio of
azimuthal vorticity to cylindrical radius, and $\psi$ is the Stokes
streamfunction in the frame of the ring. Some but not all of the $\mathrm{Re}$
dependence in the time evolution of $F(\psi, t)$ can be captured by introducing
a scaled time $\tau = \nu t$, where $\nu$ is the kinematic viscosity. When $\nu
t/D^2 \gtrsim 0.02$, the shape of $F(\psi)$ is dominated by the
linear-in-$\psi$ component, the coefficient of the quadratic term being an
order of magnitude smaller. An important feature is that as the dividing
streamline ($\psi = 0$) is approached, $F(\psi)$ tends to a non-zero intercept
which exhibits an extra $\mathrm{Re}$ dependence. This and other features are
explained by a simple toy model consisting of the one-dimensional cylindrical
diffusion equation. The key ingredient in the model responsible for the extra
$\mathrm{Re}$ dependence is a Robin-type boundary condition, similar to
Newton's law of cooling, that accounts for the edge layer at the dividing
streamline.
