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]
theory and instanton predict transition paths between
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.
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
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
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
progresses in order to use statistical mechanics
ideas for these problems and some applications
to real 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:
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]
The Jupiter's Great Red Spot
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
The see surface height of the North Atlantic.
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.
Prediction of the bistability of turbulent flows for the 2D Navier Stokes Equations
Classification of phase transitions
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
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.
Previously unobserved phase transitions and ensemble inequivalence for academic ocean models
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.