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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.

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Statistical approach of turbulence
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
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
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)



No vortex stretching

Vortex stretching

Angular momentum conservation

Vorticity conservation

2D experiment



Bracco et al. Torino

2d versus axisymmetric 2

2D turbulence in a

Ferro Magnetic fluid

2D versus axisymmetric (2)


Von Karman

Jullien et al., LPS, ENS Paris

Daviaud et al. GIT, Saclay, France

Presentation of Laboratory experiments

2d versus axisymmetric 3

Vertical vorticity:

Azimuthal vorticity:

angular momentum:

poloidal velocity:

azimuthal vorticity:

2D versus axisymmetric (3)

Basic equations




of interest:


F and G are arbitrary

functions in infinite


infinite number of

steady states

2D versus axisymmetric (4)

Inviscid Conservation laws

Casimirs (F)

Generalized helicity


Inviscid stationary states

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
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
Statistical description (3bis)

Maximisation of S under conservation constraints

The Gibbs State

Equilibrium state

Equation for most

probable fields

Steady solutions of Euler equation

Steady states 1




Two thermostats T1>T2

Steady States (1)

  • What happens when the flow is mechanically stirred and viscous?

Steady States (2)


Working hypothesis(Leprovost et al. 05’):

F and G are arbitrary

functions in infinite


infinite number of

steady states

Steady States (3)

Steady states of turbulent axisymmetric flow

- How are F and G selected?

- Role of dissipation and forcing in this selection?

Data processing 2
Data Processing (2)



Test beltrami flow with 60 noise

A steady solution of Euler equation:

Test: Beltrami Flow with 60% noise

Data processing 3




Flow Bulk

Data Processing (3)

Whole flow

50% of the flow

Distance to center

  • F is fitted from the windowed plot

  • F is used to fit G

Comparison to numerical study
Comparison to numerical study


viscous stirring


“inertial” stirring

Simulation: Piotr Boronski (Limsi, Orsay, France)

F function

Dependence on viscosity (1)




F function:

G function

Dependence on viscosity (2)

G function:




Dependence on forcing




Dependence on forcing

Re = 190 000

Re = 250 000

Re = 500 000