  We characterize the thermodynamical equilibrium states of axisymmetric
Euler-Beltrami flows. They have the form of coherent structures presenting one
or several cells. We find the relevant control parameters and derive the
corresponding equations of state. We prove the coexistence of several
equilibrium states for a given value of the control parameter like in 2D
turbulence [Chavanis and Sommeria, J. Fluid Mech. 314, 267 (1996)]. We explore
the stability of these equilibrium states and show that all states are saddle
points of entropy and can, in principle, be destabilized by a perturbation with
a larger wavenumber, resulting in a structure at the smallest available scale.
This mechanism is therefore reminiscent of the 3D Richardson energy cascade
towards smaller and smaller scales. Therefore, our system is truly intermediate
between 2D turbulence (coherent structures) and 3D turbulence (energy cascade).
We further explore numerically the robustness of the equilibrium states with
respect to random perturbations using a relaxation algorithm in both canonical
and microcanonical ensembles. We show that saddle points of entropy can be very
robust and therefore play a role in the dynamics. We evidence differences in
the robustness of the solutions in the canonical and microcanonical ensembles.
A scenario of bifurcation between two different equilibria (with one or two
cells) is proposed and discussed in connection with a recent observation of a
turbulent bifurcation in a von Karman experiment [Ravelet et al., Phys. Rev.
Lett. 93, 164501 (2004)].
