Projet ANR-06-JCJC-0037 – STATFLOW : Out of equilibrium statistical mechanics of two dimensionnal and geophysical flows
Programme : Programme "Jeunes chercheuses et jeunes chercheurs" (JCJC) 2006
Année de financement : 2006
Coordinateur du projet : CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION REGIONALE COTE D'AZUR (CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION REGIONALE COTE D'AZUR)
Partenaires : Centre national de la recherche scientifique
Montant : 112000.0 euros
Date de début : 2006
Durée : 36 mois
Résumé : SCIENTIFIC DESCRIPTION OF THE PROJECT STATFLOW   1 Introduction  The description of the large-scale structures of  turbulent flows is of wide practical and fundamental  interest, in many domains of fluid mechanics. For  instance, the knowledge of the evolution of the  large-scale atmosphere and ocean, is both a fascinating  scientific issue and a very important problem for  understanding climate change. A statistical description  of the large-scale organization of conservative  two-dimensionnal and geophysical flows has been  described by the equilibrium Robert-Sommeria-Miller  (RSM) theory [Miller-90, Robert-Sommeria-91]. This theory does not consider any  forcing. However, when a weak forcing and dissipation  is added, a situation relevant for any physical  applications, the forcing is known to select the  large-scale structures. The aim of this project is to  consider a probabilistic description of the large  scales for the 2-D Navier-Stokes with weak forcing. In  particular, we want to systematically relate the  forcing to the observed large-scale structures, and the  possible links with the RSM equilibria.  Ideally, one would like to obtain a probabilistic  description of the flow, starting from the equations of  fluid mechanics. This represents one of the most  complex and fascinating unsolved problem faced by  science, and it still appears out of reach in general.  For instance, the description of the velocity  increments statistics in 3-D turbulent flows is one of  the greatest problem of physics [Frisch], still to be  understood theoretically. For simpler problems in  turbulence, progress have been achieved, in the  direction of a probabilistic description, starting from  the fluid equation, for instance using the Kraichnan  model [Gawedski] of a passive scalar advected by a  delta-correlated velocity. Another example is the  computation of the energy spectrum of waves when  nonlinear interactions are weak. In this last example,  a kinetic theory is indeed possible thanks to the  separation between the time scale associated with the  linear wave dynamics and the typical time scale for  nonlinear wave interactions. This provides a decisive  simplification, allowing a natural closure for the  probability distribution functions [Zakharov, Newell].  Our main point is that 2-D turbulence, in the limit of  weak forcing and dissipation, should be another  instance where a kinetic theory approach, to a  probabilistic description of the large-scale dynamics,  is possible. Our first argument is that the statistical  mechanics of conservative dynamics is well understood,  it is the Robert-Sommeria-Miller theory. The second  argument is that, when both forcing and dissipation  vanish (the Eulerian limit) the 2-D Navier-Stokes  equation is well behaved. The small parameter necessary  to obtain a relevant kinetic description is then the  small parameter characterizing the amplitude of the  forcing and the dissipation.  The possibility to use kinetic theory in order to  predict the statistics of the large scales in 2-D  Navier-Stokes flows has never been considered before.  We will derive the prediction of kinetic theory and  analyze its physical implications. By contrast with the  spectral behavior of the inertial scales, we do not  expect some universal behavior for the large scales.  Their behavior will be (strongly) dependent on the  boundary conditions, on the type of forcing as well as  the dissipation mechanisms. Because we do not  underestimate the difficulty of the theoretical  problem, our approach will also rely on very precise  numerical computations. For instance, we will assess  the range of validity for the kinetic approach  hypothesis, and we will test its predictions using  intensive numerical simulations.  Kinetic theory describes the evolution of the  macroscopic dynamics (the large scale flow) on time  scale which are much longer than the time scale typical  of the microscopic dynamics (the small-scale turbulence