One of the most important problems in turbulence is the prediction of
large-scale structures of very high Reynolds number flows. The class of
two-dimensional and geostrophic flows is relevant for geophysical
applications (ocean and atmosphere).
We consider the two-dimensional Navier-Stokes equation with weak stochastic
forcing and dissipation in the inertial limit. This is an example of a
dynamical system in which an out-of-equilibrium stationary state is reached,
without detailed balance. The existence of an invariant measure has been
mathematically proved recently, together with mixing and ergodic properties.
This problem has however never been considered from a physical point of
view. We thus address the following
issues: when is the measure concentrated on an inertial equilibrium; how are
the large scales selected by the forcing and what is the level of the
fluctuations ?
The most striking result is the existence of out-of-equilibrium phase
transitions. One observes transitions from one type of flow (unidirectional)
to another one (dipolar), at random instants of time. This is similar to the
classical two-wheel potential with noise. By contrast, in our case, no such
potential exists and the turbulent nature of the flow (infinite number of
degrees of freedom) renders the phenomena much richer.
Analogies with the Earth magnetic field reversals and similar phenomena in
experiments on two- dimensional and geophysical flows will be discussed.