Statistical approach of Turbulence R. Monchaux N. Leprovost, F. Ravelet, P-H. Chavanis*, B. Dubrulle, F. Daviaud and A. Chiffaudel GIT-SPEC, Gif sur Yvette France *Laboratoire de Physique Théorique, Toulouse France

Out-of-equilibrium systems vs. Classical equilibrium systems Degrees of freedom:

Statistical approach of turbulence: Steady states, equation of state, distributions 2D: Robert and Sommeria 91’, Chavanis 03’ Quasi-2D: shallow water, β-plane Bouchet 02’’ 3D: still unanswered question (vortex stretching) Axisymmetric flows: intermediate situation 2D and vortex stretching Theoretical developments by Leprovost, Dubrulle and Chavanis 05’

2D and quasi-2D results Statistical equilibrium state of 2D Euler equation (Chavanis): - Classification of isolated vortices: monopoles and dipoles - Stability diagram of these structures: dependence on a single control parameter Quasi 2D statistical mechanics (Bouchet): – Intense jets – Great Red Spot

Approach Principle Basic equation: Euler equation –Forcing is neglected –Viscosity is neglected Variable of interest: Probability to observe the conserved quantity at Maximization of a mixing entropy at conserved quantities constraints

2D vs axisymmetric (1) 2D axisymmetric Vorticity conservation Angular momentum conservation No vortex stretchingVortex stretching 2D experiment Coherent structures Bracco et al. Torino

2D versus axisymmetric (2) Von Karman Taylor-Couette Presentation of Laboratory experiments 2D turbulence in a Ferro Magnetic fluid Jullien et al., LPS, ENS Paris Daviaud et al. GIT, Saclay, France

2D versus axisymmetric (3) Basic equations Vertical vorticity : 2D: Azimuthal vorticity : AXI: azimuthal vorticity: angular momentum: poloidal velocity: Variables of interest:

2D versus axisymmetric (4) Inviscid stationary states Inviscid Conservation laws (Casimirs) F and G are arbitrary functions in infinite number infinite number of steady states Casimirs (F) Generalized helicity (G)

Statistical description (1) Mixing occurs at smaller and smaller scales More and more degrees of freedom Meta-equilibrium at a coarse-grained scale Use of coarse-grained fields Coarse-graining affects some constraints Casimirs are fragile invariant

Statistical description (2) Probability distribution to observe at point r Mixing Entropy: Coarse-grained A. M. Coarse-grained constraints: Robust constraints Fragile constraints

Statistical description (3bis) Maximisation of S under conservation constraints Equilibrium state Equation for most probable fields The Gibbs State Steady solutions of Euler equation

Steady States (1)  What happens when the flow is mechanically stirred and viscous? T1T2 Two thermostats T1>T2 F

Working hypothesis (Leprovost et al. 05’): NS: Steady States (2)

Steady states of turbulent axisymmetric flow F and G are arbitrary functions in infinite number infinite number of steady states - How are F and G selected? - Role of dissipation and forcing in this selection? Steady States (3)

Von Kármán Flow - LDV measurement

Data Processing (1)

Data Processing (2) Time-averaged fmpv

Test: Beltrami Flow with 60% noise A steady solution of Euler equation:

Data Processing (3) F is fitted from the windowed plot F is used to fit G Whole flow50% of the flow Distance to center <0.7 >0.85 intermediate Flow Bulk

Comparison to numerical study Simulation: Piotr Boronski (Limsi, Orsay, France) Re=3000 “inertial” stirring Re=5000 viscous stirring

Dependence on viscosity (1) (+)(-) F function: Legend

Dependence on viscosity (2) (+)(-) G function: Legend

(+) 92.5mm Re = Re = Re = mm Dependence on forcing

Conclusions

Perspectives