  The macroscale structure and microscale fluctuation statistics of late-time
asymptotic steady state flows in cylindrical geometries is studied using the
methods of equilibrium statistical mechanics. The axisymmetric assumption
permits an effective two-dimensional description in terms of the (toroidal)
flow field $\sigma$ about the cylinder axis and the vorticity field $\xi$ that
generates mixing within the (poloidal) planes of fixed azimuth. As for a number
of other 2D fluid systems, extending the classic 2D Euler equation, the flow is
constrained by an infinite number of conservation laws, beyond the usual
kinetic energy and angular momentum. All must be accounted for in a consistent
equilibrium description. It is shown that the most directly observable impact
of the conservation laws is on $\sigma$, which displays interesting large-scale
radius-dependent flow structure. However, unlike in some previous treatments,
we find that the thermodynamic temperature is always positive. As a
consequence, except for an infinitesimal boundary layer that maintains the
correct (conserved) value of the overall poloidal circulation, the impact on
$\xi$ resides in the statistics of the strongly fluctuating, fine-scale mixing,
where it is sensitive to `antiferromagnetic' microscale correlations that help
maintain the analogue of local charge neutrality. The poloidal flow is
macroscopically featureless, displaying no large scale circulating jet- or
eddy-like features (which typically emerge as negative temperature states in
analogous Euler and quasigeostrophic equilibria).
