Thermodynamics of magnetohydrodynamic flows with axial symmetry

We present strategies based upon optimization principles in the case of the axisymmetric equations of magnetohydrodynamics (MHD). We derive the equilibrium state by using a minimum energy principle under the constraints of the MHD axisymmetric equations. We also propose a numerical algorithm based on a maximum energy dissipation principle to compute in a consistent way the nonlinearly dynamically stable equilibrium states. Then, we develop the statistical mechanics of such flows and recover the same equilibrium states giving a justification of the minimum energy principle. We find that fluctuations obey a Gaussian shape and we make the link between the conservation of the Casimirs on the coarse-grained scale and the process of energy dissipation. We contrast these results with those of two-dimensional hydrodynamical turbulence where the equilibrium state maximizes a H function at fixed energy and circulation and where the fluctuations are nonuniversal.