Dynamics and thermodynamics of axisymmetric flows: Theory

We develop variational principles to study the structure and the stability of equilibrium states of axisymmetric flows. We show that the axisymmetric Euler equations for inviscid flows admit an infinite number of steady state solutions. We find their general form and provide analytical solutions in some special cases. The system can be trapped in one of these steady states as a result of an inviscid violent relaxation. We show that the stable steady states maximize a (nonuniversal) function while conserving energy, helicity, circulation, and angular momentum (robust constraints). This can be viewed as a form of generalized selective decay principle. We derive relaxation equations which can be used as numerical algorithm to construct nonlinearly dynamically stable stationary solutions of axisymmetric flows. We also develop a thermodynamical approach to predict the equilibrium state at some fixed coarse-grained scale. We show that the resulting distribution can be divided in two parts: one universal coming from the conservation of robust invariants and one non-universal determined by the initial conditions through the fragile invariants (for freely evolving systems) or by a prior distribution encoding nonideal effects such as viscosity, small-scale forcing, and dissipation (for forced systems). Finally, we derive a parametrization of inviscid mixing to describe the dynamics of the system at the coarse-grained scale. A conceptual interest of this axisymmetric model is to be intermediate between two-dimensional (2D) and 3D turbulence.