  We develop new variational principles to study stability and equilibrium of
axisymmetric flows. We show that there is an infinite number of steady state
solutions. We show that these steady states maximize a (non-universal)
$H$-function. We derive relaxation equations which can be used as numerical
algorithm to construct stable stationary solutions of axisymmetric flows. In a
second part, we develop a thermodynamical approach to the equilibrium states at
some fixed coarse-grained scale. We show that the resulting distribution can be
divided in a universal part 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
non-ideal effects such as viscosity, small-scale forcing and dissipation (for
forced systems). Finally, we derive a parameterization of inviscid mixing to
describe the dynamics of the system at the coarse-grained scale.
