Statistical mechanics of the 3D axisymmetric Euler equations in a Taylor-Couette geometry

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 regime 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 axisymmetric 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.