Non-equilibrium statistical mechanics of the stochastic Navier–Stokes equations and geostrophic turbulence

International audience; Two-dimensional and geophysical turbulent flows have the property to self organize and create large scale coherent jets and vortices. This is for instance one of the major processes for the dynamics of Earth's atmosphere. Following On-sager initial insight, based on conjugated works by mathematicians and physicists, this fundamental physical process has found some explanations in the framework of statistical mechanics. An important step, initiated twenty years ago, has been the study of the equilibrium statistical mechanics for the 2D Euler and the related quasi-geostrophic models (the Miller-Robert-Sommeria theory). Real geophysical and experimental flows are however dissipative and maintained by external forces. These lectures focus on recent theoretical development of the statistical mechanics of those non-equilibrium situations. Those progresses have been achieved using tools from field theory (path integrals and instantons), non-equilibrium statistical mechanics (large deviations, stochastic averaging). The aim of these lectures is to briefly introduce the theoretical aspects of this program in the simplest context: the 2D stochastic Euler or Navier-Stokes equations and the quasi-geostrophic equations. We review path integral representations of stochastic processes, large deviations for transition probabilities, action minimization, instanton theory, for general mechanical systems forced by random forces. We will apply this framework in order to predict equilibrium and non-equilibrium phase transitions for the 2D Euler, Navier-Stokes, and quasi-geostrophic dynamics, and to predict the rates of rare transitions between two attractors in situations of first order phase transitions. Kinetic theory of systems with long range interactions, both with and without stochastic external forces, are explained. Based on this kinetic theory, we predict non-equilibrium phase transitions, and discuss their recent experimental observations and numerical simulations. Even if the model we have considered so far are too simple academic models, the expected relevance of those approaches in the future for Earth atmosphere and climate dynamics is briefly discussed.