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

Using an analogy with an Ising-like spin model, we define microcanonical measures for the dynamics of three dimensional (3D) axisymmetric turbulent flow in a Taylor-Couette geometry. We compute the relevant physical quantities and argue that axisymmetry induces a large scale organization in turbulent flows. We show that there exists a low energy, low temperature regime, for which the orthoradial velocity field is organized into vertical stripes, as well as a high energy, infinite temperature regime where the typical orthoradial vorticity field gets organized into either a single vertical jet or a large scale dipole, and exhibits infinite fluctuations. The mechanisms yielding the large scale organizations are argued to be different from the ones involved in two dimensional (2D) turbulence. This shows that the 3D axisymmetric case is truly an intermediate case between 2D and 3D turbulence.