  A simplified thermodynamic approach of the incompressible axisymmetric Euler
equations is considered based on the conservation of helicity, angular momentum
and microscopic energy. Statistical equilibrium states are obtained by
maximizing the Boltzmann entropy under these sole constraints. We assume that
these constraints are selected by the properties of forcing and dissipation.
The fluctuations are found to be Gaussian while the mean flow is in a Beltrami
state. Furthermore, we show that the maximization of entropy at fixed helicity,
angular momentum and microscopic energy is equivalent to the minimization of
macroscopic energy at fixed helicity and angular momentum. This provides a
justification of this selective decay principle from statistical mechanics.
These theoretical predictions are in good agreement with experiments of a von
Karman turbulent flow and provide a way to measure the temperature of
turbulence and check Fluctuation-Dissipation Relations (FDR). Relaxation
equations are derived that could provide an effective description of the
dynamics towards the Beltrami state and the progressive emergence of a Gaussian
distribution. They can also provide a numerical algorithm to determine maximum
entropy states or minimum energy states.
