Freddy BOUCHET - ENS de Lyon and CNRS

CLIMATE AND STATISTICAL MECHANICS



TURBULENCE, STATISTICAL MECHANICS, AND LARGE DEVIATION THEORY

Statistical mechanics approaches to turbulence proved fruitful in order to explain many phenomena. One of the key theoretical tools are large deviation theory and kinetic theory. Using large deviation theory we studied equilibrium and out of equilibrium statistical mechanics of geophysical flows, non-equilibrium phase transitions in turbulence, and large deviation of Reynolds stresses. Instantons derived from large deviation theory predict most probable transitions between attractors. Kinetic theory approaches have been used to study the formation of atmosphere jets and the self organization of two dimensional turbulence. We have also studied rare transitions and instantons for turbulent atmospheres, for instance related to abrupt changes of Jupiter's climate.


Turbulence and large deviation theory: Seminar given during the Workshop on Instantons and Extreme Events in Turbulence and Dynamical Systems, held at Rio de Janeiro in December 2015 [movie] [slides]

Large deviation theory for atmosphere jets: Seminar given during the Workshop on Instantons and Extreme Events in Turbulence and Dynamical Systems, held at Rio de Janeiro in December 2015 [movie] [slides]

Large deviation theory and instanton predict transition paths between turbulent attractors


The instanton of the 2D Navier-Stokes equations

In equilibrium statistical mechanics transition rates between attractors are given by Arrhenius law (chemistry, phase transitions in magnetic systems, condensed matter, bio-molecules). For complex dynamical systems like turbulent flows, large deviation theory extend the equilibrium theory. Transitions rates between attractors can be computed. Moreover transition paths are predictable as action minima (instantons). For the first time we have computed instantons related to non-equilibrium phase transitions between dipoles and parallel flows in two dimensional turbulence.

2D Navier Stokes Instanton
The 2D Navier-Stokes instanton for the transition between dipoles and parallel flows

J. LAURIE and F. BOUCHET, 2014, Computation of rare transitions in the barotropic quasi-geostrophic equations, New J. Phys. 17 (2015) 015009,  [.pdf]


Instantons for quasi-geostrophic dynamics

For quasigeostrophic models that describe atmosphere jets, and for a special set of forces, phase transitions and instantons can be computed explicitly.

F. BOUCHET, J. LAURIE, and O. ZABORONSKI, 2014, Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations, J Stat Phys (2014) 156:1066Ė1092, and  arXiv:1403.0216 [.pdf]

Equilibrium and out of equilibrium statistical mechanics of geophysical flows


In many applications of fluid dynamics, one of the most important problem is the prediction of the very high Reynolds large-scale flows. The highly turbulent nature of such flows, for instance ocean circulation or atmosphere, renders a probabilistic description desirable. We describe theoretical progresses in order to use statistical mechanics ideas for these problems and some applications to real geophysical flows.

Review paper on the statistical mechanics of geophysical flows:
F. BOUCHET, and A. VENAILLE, Statistical mechanics of two-dimensional and geophysical flows, Physics Reports, 2012 [.pdf]


Recent publications on the equilibrium statistical mechanics of turbulent flows:

1) A. RENAUD, A. VENAILLE, and F BOUCHET, 2016, Equilibrium statistical mechanics and energy partition for the shallow water model, arXiv:1505.01356,  [.pdf]
2) 
S. THALABARD, B. DUBRULLE, and F. BOUCHET, 2014, Statistical mechanics of the 3D axisymmetric Euler equations in a Taylor-Couette geometry. J. Stat. Mech.: Theory and Experiment, 1, P01005. [.pdf]



Applications of Stat. Mech. to real geophysical flows


Jupiter-Equilibrium

Jupiter's Great
              Red Spot
The Jupiter's Great Red Spot

An equilibrium statistical mechanics explanation of the self-organization of geophysical flows has been proposed by Robert-Sommeria and Miller (RSM). The RSM theory has been successfully applied to the Jupiterís troposphere: cyclones, anticyclones and jets have been quantitatively described by this theory

(see: A statistical equilibrium model of the Great Red Spot and other Jovian vortices and the publications Bouchet-Sommeria-2002 and Bouchet-Dumont-2002).

Ocean-Equilibrium

Jupiter's Great
              Red Spot
The see surface height of the North Atlantic.

A natural question is to know if some aspects of the large scale organization of ocean dynamics, the map of currents for instance, can be understood in the context of equilibrium statistical mechanics.
Ocean rings are the most common vortices in oceans. They explain most of the kinetic energy variability of ocean dynamics. We have explained the structures, shapes, and velocity fields of ocean rings using equilibrium statistical mechanics. We have also explained from statistical mechanics why the complex mixing resulting from turbulence leads to the formation of strong mid-basin jets like the Gulf Stream in the Atlantic Ocean and the Kuroshio in the Pacific Ocean.

(see Equilibrium Statistical Mechanics of Mid Basin Eastward Jets and of Ocean Vortices)



Theoretical progresses in the Stat. Mech. of 2D and geophysical flows


Out of equilibrium phase transitions

Out of
              equilibrium phase transitions
Prediction of the bistability of turbulent flows for the 2D Navier Stokes Equations

We have studied the two dimensional Navier Stokes equation with stochastic forces (SNS Eq.). We predict theoretically and observe numerically out of equilibrium phase transitions: the turbulent flow switches randomly from a state with a large scale dipole to a state with a unidirectional flow. Similar theoretical considerations lead to the prediction of out of equilibrium phase transitions in a large class of other geometries, and also for geostrophic, high rotating flow experiments or 2D magnetic flows. We thus infer that such phase transitions exist in possible experimental situations and in geophysical flows.

(please see Out of Equilibrium Phase Transitions in 2D and Geophysical Flows and Bouchet-Simonnet-PRL-2009)


Classification of phase transitions

Classification

Equilibrium phase diagram of an ensemble of self graviating stars, illustrating ensemble inequivalence: microcanonical phase transitions (black lines) do not coincide with canonical phase transitions (magenta lines) (see Bouchet-Barre-JStatPhys-2005)


The 2D Euler equations are an example of systems with long range interactions. Systems with long range interactions are not additive, which can lead to inequivalence between the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative heat capacities (the temperature decrease when the energy is increased) and other non-common behaviors like negative temperature jumps when the energy is increased, at a microcanonical critical point.
We have proposed a generalization of Landau classification for systems with long range interactions that describes all the possible phase transitions associated with situations of ensemble inequivalence. The phenomenology for such phase transitions is richer than the classical one. We have then predicted new ensemble inequivalence situations that have never been observed yet and others than have been observed only after our work

(see Bouchet-Barre-JStatPhys-2005).


Simpler variational problems for the RSM statistical equilibria

The Robert-Sommeria-Miller equilibrium statistical mechanics predicts the final organization of two dimensional flows. This powerful theory is difficult to handle practically, due to the complexity associated with an infinite number of constraints. Several alternative simpler variational problems, based on Casimirís or stream function functionals, have been considered recently. We have established the relations between all these variational problems, justifying the use of simpler formulations. This provides a drastic mathematical simplifications for the study of equilibria, and increases our physical insight by justifying new physical analogies.

(see Bouchet-PhysicaD-2008).



Previously unobserved phase transitions and ensemble inequivalence for academic ocean models


Vortices in two dimensional flows interact with a logarithmic range. 2D flows are thus examples of systems with non integrable interaction. Systems with such long range interactions are known to have extremely peculiar thermodynamical properties, for instance negative heat capacities (the temperature decreases when the energy is increased), or energy ranges where the microcanonical and canonical ensembles of statistical physics are non equivalent.
We propose a theoretical description for the equilibrium states of a large class of models of two-dimensional and geophysical flows. Statistical ensemble inequivalence is found to exist generically in those models, related with the occurrence of peculiar phase transitions in the flow topology. The first examples of a bicritical point in the context of systems with long range interactions is reported. Steady states of academic ocean models, the Fofonoff flows, are studied in the perspective of those results. A more detailed description is proposed in and will be the subject of the forthcoming publication.

(see Venaille-Bouchet-PRL-2009).