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The macroscale structure and microscale fluctuation statistics of late-timeasymptotic steady state flows in cylindrical geometries is studied using themethods of equilibrium statistical mechanics. The axisymmetric assumptionpermits an effective two-dimensional description in terms of the (toroidal)flow field $\sigma$ about the cylinder axis and the vorticity field $\xi$ thatgenerates mixing within the (poloidal) planes of fixed azimuth. As for a numberof other 2D fluid systems, extending the classic 2D Euler equation, the flow isconstrained by an infinite number of conservation laws, beyond the usualkinetic energy and angular momentum. All must be accounted for in a consistentequilibrium description. It is shown that the most directly observable impactof the conservation laws is on $\sigma$, which displays interesting large-scaleradius-dependent flow structure. However, unlike in some previous treatments,we find that the thermodynamic temperature is always positive. As aconsequence, except for an infinitesimal boundary layer that maintains thecorrect (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 helpmaintain the analogue of local charge neutrality. The poloidal flow ismacroscopically featureless, displaying no large scale circulating jet- oreddy-like features (which typically emerge as negative temperature states inanalogous Euler and quasigeostrophic equilibria).

Strong vorticity fluctuations and antiferromagnetic correlations in axisymmetric fluid equilibria