  In the present paper, microcanonical measures for the dynamics of three
dimensional (3D) axially symmetric turbulent flows with swirl in a
Taylor-Couette geometry are defined, using an analogy with a long-range lattice
model. We compute the relevant physical quantities and argue that two kinds of
equilibrium regimes exist, depending on the value of the total kinetic energy.
For low energies, the equilibrium flow consists of a purely swirling flow whose
toroidal profile depends on the radial coordinate only. For high energies, the
typical toroidal field is uniform, while the typical poloidal field is
organized into either a single vertical jet or a large scale dipole, and
exhibits infinite fluctuations. This unusual phase diagram comes from the
poloidal fluctuations not being bounded for the axi-symmetric Euler dynamics,
even though the latter conserve infinitely many ''Casimir invariants''. This
shows that 3D axially symmetric flows can be considered as intermediate between
2D and 3D flows.
