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EquilibriumOut of equilibrium
Stochastic Orr mechanism
Phase transitions in the 2D Navier-Stokes equationand other systems with long range interactions
At equilibrium and out of equilibrium (NESS)
F. BOUCHET INLN (Nice) CNRS
February 2009
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
Collaborators
Equilibrium statistical mechanics of two dimensional and
geophysical ows : A. Venaille (Grenoble).
Stability of equilibrium states : F. Rousset (Lab. Dieudonné
Nice) (ANR Statow)
Random change of ow topology (out of equilibrium) : H.
Morita and E. Simonnet (INLN-Nice) (ANR Statow)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The Physical Phenomena
Theoretical ideas :
Self organisation processes. Large number of degrees of
freedom (turbulence).
This has to be explained using statistical physics !!!
Mainly out of equilibrium statistical mechanics. We have to work
out new theoretical concepts with such phenomena in mind.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The Physical Phenomena
Theoretical ideas :
Self organisation processes. Large number of degrees of
freedom (turbulence).
This has to be explained using statistical physics !!!
Mainly out of equilibrium statistical mechanics. We have to work
out new theoretical concepts with such phenomena in mind.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The Physical Phenomena
Theoretical ideas :
Self organisation processes. Large number of degrees of
freedom (turbulence).
This has to be explained using statistical physics !!!
Mainly out of equilibrium statistical mechanics. We have to work
out new theoretical concepts with such phenomena in mind.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The Physical Phenomena
Theoretical ideas :
Self organisation processes. Large number of degrees of
freedom (turbulence).
This has to be explained using statistical physics !!!
Mainly out of equilibrium statistical mechanics. We have to work
out new theoretical concepts with such phenomena in mind.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The 2D Stochastic-Navier-Stokes (SNS) Equations
The simplest model for two dimensional turbulence
Navier Stokes equation with a random force
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs (1)
where ω = (∇∧u) .ez is the vorticity, fs is a random force, α is the
Rayleigh friction coecient.
An academic model with experimental realizations (Sommeria
and Tabeling experiments, rotating tanks, magnetic ows, and
so on). Analogies with geophysical ows (Quasi Geostrophic
and Shallow Water layer models).
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The 2D Stochastic-Navier-Stokes (SNS) Equations
The simplest model for two dimensional turbulence
Navier Stokes equation with a random force
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs (1)
where ω = (∇∧u) .ez is the vorticity, fs is a random force, α is the
Rayleigh friction coecient.
An academic model with experimental realizations (Sommeria
and Tabeling experiments, rotating tanks, magnetic ows, and
so on). Analogies with geophysical ows (Quasi Geostrophic
and Shallow Water layer models).
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The 2D Stochastic-Navier-Stokes (SNS) Equations
The simplest model for two dimensional turbulence
Navier Stokes equation with a random force
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs (2)
This is an example of Non Equilibrium Steady State
(NESS).Fluxes of energy and enstrophy
Vorticity patches interact via long range interactions (log(r))(by contrast with usual NESS)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
Equilibrium : the 2D Euler Equations
Conservative dynamics
2D Euler equation∂ω
∂ t+u.∇ω = 0 (3)
Vorticity ω = (∇∧u) .ez . Stream function ψ : u = ez ×∇ψ ,
ω = ∆ψ
Steady solutions to the Euler Eq. :
u.∇ω = 0 or equivalently ω = f (ψ)
Steady states of the Euler equation will play a crucial role.
Degeneracy : what does select f ?
f can be predicted using classical equilibrium statistical
mechanics
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
Equilibrium : the 2D Euler Equations
Conservative dynamics
2D Euler equation∂ω
∂ t+u.∇ω = 0 (3)
Vorticity ω = (∇∧u) .ez . Stream function ψ : u = ez ×∇ψ ,
ω = ∆ψ
Steady solutions to the Euler Eq. :
u.∇ω = 0 or equivalently ω = f (ψ)
Steady states of the Euler equation will play a crucial role.
Degeneracy : what does select f ?
f can be predicted using classical equilibrium statistical
mechanics
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
Equilibrium : the 2D Euler Equations
Conservative dynamics
2D Euler equation∂ω
∂ t+u.∇ω = 0 (3)
Vorticity ω = (∇∧u) .ez . Stream function ψ : u = ez ×∇ψ ,
ω = ∆ψ
Steady solutions to the Euler Eq. :
u.∇ω = 0 or equivalently ω = f (ψ)
Steady states of the Euler equation will play a crucial role.
Degeneracy : what does select f ?
f can be predicted using classical equilibrium statistical
mechanics
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
Outline
1 Equilibrium statistical mechanics
Equilibrium
2 Out of equilibrium statistical mechanics
The 2D Navier Stokes equation with random forces
Experiments
Random changes of ow topology in the 2D S-Navier-Stokes
Eq. (E. Simonnet, H. Morita and F. B.)
3 The 2D linearized Euler Eq. with stochastic forces (F.B.)
General considerations
Constant shear
Shear ows : the general case. Theorems and illustrations.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Outline
1 Equilibrium statistical mechanics
Equilibrium
2 Out of equilibrium statistical mechanics
The 2D Navier Stokes equation with random forces
Experiments
Random changes of ow topology in the 2D S-Navier-Stokes
Eq. (E. Simonnet, H. Morita and F. B.)
3 The 2D linearized Euler Eq. with stochastic forces (F.B.)
General considerations
Constant shear
Shear ows : the general case. Theorems and illustrations.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Statistical Mechanics for 2D and Geophysical Flows
Statistical hydrodynamics ? Very complex problems
Example : Intermittency in 3D turbulence : phenomenological
approach , simplied models (Kraichnan)
It may be much simpler for 2D or geophysical ows :
conservative systems
Statistical equilibrium : A very old idea, some famous contributions
Onsager (1949), Joyce and Montgomerry (1970), Caglioti
Marchioro Pulvirenti Lions (1990), Robert Sommeria (1991),
Miller (1991), Lebowitz (1994)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Statistical Mechanics for 2D and Geophysical Flows
Statistical hydrodynamics ? Very complex problems
Example : Intermittency in 3D turbulence : phenomenological
approach , simplied models (Kraichnan)
It may be much simpler for 2D or geophysical ows :
conservative systems
Statistical equilibrium : A very old idea, some famous contributions
Onsager (1949), Joyce and Montgomerry (1970), Caglioti
Marchioro Pulvirenti Lions (1990), Robert Sommeria (1991),
Miller (1991), Lebowitz (1994)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Statistical Mechanics for 2D and Geophysical Flows
Statistical hydrodynamics ? Very complex problems
Example : Intermittency in 3D turbulence : phenomenological
approach , simplied models (Kraichnan)
It may be much simpler for 2D or geophysical ows :
conservative systems
Statistical equilibrium : A very old idea, some famous contributions
Onsager (1949), Joyce and Montgomerry (1970), Caglioti
Marchioro Pulvirenti Lions (1990), Robert Sommeria (1991),
Miller (1991), Lebowitz (1994)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Statistical Mechanics for 2D and Geophysical Flows
Statistical hydrodynamics ? Very complex problems
Example : Intermittency in 3D turbulence : phenomenological
approach , simplied models (Kraichnan)
It may be much simpler for 2D or geophysical ows :
conservative systems
Statistical equilibrium : A very old idea, some famous contributions
Onsager (1949), Joyce and Montgomerry (1970), Caglioti
Marchioro Pulvirenti Lions (1990), Robert Sommeria (1991),
Miller (1991), Lebowitz (1994)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Statistical Mechanics for 2D and Geophysical Flows
Statistical hydrodynamics ? Very complex problems
Example : Intermittency in 3D turbulence : phenomenological
approach , simplied models (Kraichnan)
It may be much simpler for 2D or geophysical ows :
conservative systems
Statistical equilibrium : A very old idea, some famous contributions
Onsager (1949), Joyce and Montgomerry (1970), Caglioti
Marchioro Pulvirenti Lions (1990), Robert Sommeria (1991),
Miller (1991), Lebowitz (1994)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Robert-Sommeria-Miller (RSM) TheoryEquilibrium statistical mechanics : the most probable vorticity eld
A probabilistic description of the vorticity eld ω : ρ (x,σ) is
the local probability to have ω (x) = σ at point x
A measure of the number of microscopic eld ω corresponding
to a probability ρ :
Maxwell-Boltzmann Entropy: S [ρ]≡−∫
Ddx
∫ +∞
−∞
dσ ρ logρ
The microcanonical RSM variational problem (MVP) :
S(E0,d) = supρ|N[ρ]=1
S [ρ] | E [ω] = E0 ,D [ρ] = d (MVP).
Critical points are stationary ows of Euler's equations :
ω = f (ψ)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Robert-Sommeria-Miller (RSM) TheoryEquilibrium statistical mechanics : the most probable vorticity eld
A probabilistic description of the vorticity eld ω : ρ (x,σ) is
the local probability to have ω (x) = σ at point x
A measure of the number of microscopic eld ω corresponding
to a probability ρ :
Maxwell-Boltzmann Entropy: S [ρ]≡−∫
Ddx
∫ +∞
−∞
dσ ρ logρ
The microcanonical RSM variational problem (MVP) :
S(E0,d) = supρ|N[ρ]=1
S [ρ] | E [ω] = E0 ,D [ρ] = d (MVP).
Critical points are stationary ows of Euler's equations :
ω = f (ψ)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Robert-Sommeria-Miller (RSM) TheoryEquilibrium statistical mechanics : the most probable vorticity eld
A probabilistic description of the vorticity eld ω : ρ (x,σ) is
the local probability to have ω (x) = σ at point x
A measure of the number of microscopic eld ω corresponding
to a probability ρ :
Maxwell-Boltzmann Entropy: S [ρ]≡−∫
Ddx
∫ +∞
−∞
dσ ρ logρ
The microcanonical RSM variational problem (MVP) :
S(E0,d) = supρ|N[ρ]=1
S [ρ] | E [ω] = E0 ,D [ρ] = d (MVP).
Critical points are stationary ows of Euler's equations :
ω = f (ψ)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Robert-Sommeria-Miller (RSM) TheoryEquilibrium statistical mechanics : the most probable vorticity eld
A probabilistic description of the vorticity eld ω : ρ (x,σ) is
the local probability to have ω (x) = σ at point x
A measure of the number of microscopic eld ω corresponding
to a probability ρ :
Maxwell-Boltzmann Entropy: S [ρ]≡−∫
Ddx
∫ +∞
−∞
dσ ρ logρ
The microcanonical RSM variational problem (MVP) :
S(E0,d) = supρ|N[ρ]=1
S [ρ] | E [ω] = E0 ,D [ρ] = d (MVP).
Critical points are stationary ows of Euler's equations :
ω = f (ψ)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Robert-Sommeria-Miller (RSM) TheoryEquilibrium statistical mechanics : the most probable vorticity eld
A probabilistic description of the vorticity eld ω : ρ (x,σ) is
the local probability to have ω (x) = σ at point x
A measure of the number of microscopic eld ω corresponding
to a probability ρ :
Maxwell-Boltzmann Entropy: S [ρ]≡−∫
Ddx
∫ +∞
−∞
dσ ρ logρ
The microcanonical RSM variational problem (MVP) :
S(E0,d) = supρ|N[ρ]=1
S [ρ] | E [ω] = E0 ,D [ρ] = d (MVP).
Critical points are stationary ows of Euler's equations :
ω = f (ψ)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
The Great Red Spot of JupiterThe Real Flow and the Statistical Mechanics Predictions
Observation data (Voyager) Statistical equilibrium
A very good agreement. A simple model, analytic description,
from theory to observation + New predictions.
F. BOUCHET and J. SOMMERIA 2002 JFM (QG model) F. BOUCHET,P.H. CHAVANIS and J. SOMMERIA preprint (Shallow Water model)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
The Great Red Spot of JupiterThe Real Flow and the Statistical Mechanics Predictions
Observation data (Voyager) Statistical equilibrium
A very good agreement. A simple model, analytic description,
from theory to observation + New predictions.
F. BOUCHET and J. SOMMERIA 2002 JFM (QG model) F. BOUCHET,P.H. CHAVANIS and J. SOMMERIA preprint (Shallow Water model)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
The Great Red Spot of JupiterThe Real Flow and the Statistical Mechanics Predictions
Observation data (Voyager) Statistical equilibrium
A very good agreement. A simple model, analytic description,
from theory to observation + New predictions.
F. BOUCHET and J. SOMMERIA 2002 JFM (QG model) F. BOUCHET,P.H. CHAVANIS and J. SOMMERIA preprint (Shallow Water model)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Jovian Vortex Shape : Phase CoexistenceAn isoperimetrical problem balanced by the eect of the deep ow
Left : analytic results
Below left : the Great Red Spot and
a White Ovals
Below right : Brown Barges
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanismEquilibrium
Other Recent Results for the Equilibrium RSM Theory
Simplied variational problems for the statistical equilibria of 2Dows. F. Bouchet, Physica D, 2008
Stability of a new class of equilibrium states (generalisation ofArnold's theorems). E. Caglioti, M. Pulvirenti and F. Rousset,(submitted to Comm. Math. Phys., preprint)
Phase transitions, ensemble inequivalenceand Fofono ows. A. Venaille and F.Bouchet, Phys. Rev. Lett.
Are strong mid-basin eastward jets
(Gulf Stream, Kuroshio) statistical
equilibria ? A. Venaille and F.
Bouchet, in preparationAntoine Venaille
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Outline
1 Equilibrium statistical mechanics
Equilibrium
2 Out of equilibrium statistical mechanics
The 2D Navier Stokes equation with random forces
Experiments
Random changes of ow topology in the 2D S-Navier-Stokes
Eq. (E. Simonnet, H. Morita and F. B.)
3 The 2D linearized Euler Eq. with stochastic forces (F.B.)
General considerations
Constant shear
Shear ows : the general case. Theorems and illustrations.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
The 2D Stochastic-Navier-Stokes (SNS) Equations
The simplest model for two dimensional turbulence
Navier Stokes equation with a random force
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs (4)
where ω = (∇∧u) .ez is the vorticity, fs is a random force, α is the
Rayleigh friction coecient.
An academic model with experimental realizations (Sommeria
and Tabeling experiments, rotating tanks, magnetic ows, and
so on). Analogies with geophysical ows.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
The 2D Stochastic-Navier-Stokes (SNS) Equations
The simplest model for two dimensional turbulence
Navier Stokes equation with a random force
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs (4)
where ω = (∇∧u) .ez is the vorticity, fs is a random force, α is the
Rayleigh friction coecient.
An academic model with experimental realizations (Sommeria
and Tabeling experiments, rotating tanks, magnetic ows, and
so on). Analogies with geophysical ows.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Numerical Simulation of the 2D Stochastic-NS Eq.
Self similar growth of a dipole
structure, for the 2D S-NS Eq.
Left : vorticity eld
Bottom : vorticity proles
M Chertkov, C Connaughton, I Kolokolov, V Lebedev (PRL 2007)
Also M. G. Shats, H. Xia, H. Punzmann and G. Falkovich (PRL 2007)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
The 2D Stochastic Navier-Stokes Equation
∂ω
∂ t+u.∇ω = ν∆ω +
√νfs
Some recent mathematical results : Kuksin, Sinai, Shirikyan,
Bricmont, Kupianen, etc
Existence of a stationary measure µν . Existence of limν→0 µν
In this limit, almost all trajectories are solutions of the Eulerequation
We would like to obtain more physical results :
What is the link of this limit ν → 0 with the RSM theory ?Will we stay close to some steady solutions of the Eulerequation ?Can we describe these statistically stationary states and theirproperties ?
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Outline
1 Equilibrium statistical mechanics
Equilibrium
2 Out of equilibrium statistical mechanics
The 2D Navier Stokes equation with random forces
Experiments
Random changes of ow topology in the 2D S-Navier-Stokes
Eq. (E. Simonnet, H. Morita and F. B.)
3 The 2D linearized Euler Eq. with stochastic forces (F.B.)
General considerations
Constant shear
Shear ows : the general case. Theorems and illustrations.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Out of Equilibrium Phase Transitions in Real Flows2D MHD experiments (2D Navier Stokes dynamics)
J. Sommeria, J. Fluid. Mech. (1986)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Out of Equilibrium Phase Transitions in Real FlowsRotating tank experiments (Quasi Geostrophic dynamics)
Transitions between blocked and zonal states
Y. Tian and col, J. Fluid. Mech. (2001) (groups of H. Swinney and
M. Ghil)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Random Transitions in Other Turbulence ProblemsMagnetic Field Reversal (Turbulent Dynamo, MHD Dynamics)
VKS experiment Earth
(VKS experiment)
Other examples :
Turbulent convection, Van Karman and Couette turbulence
Random changes of paths for the Kuroshio current, weather
regimes, and so on.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Outline
1 Equilibrium statistical mechanics
Equilibrium
2 Out of equilibrium statistical mechanics
The 2D Navier Stokes equation with random forces
Experiments
Random changes of ow topology in the 2D S-Navier-Stokes
Eq. (E. Simonnet, H. Morita and F. B.)
3 The 2D linearized Euler Eq. with stochastic forces (F.B.)
General considerations
Constant shear
Shear ows : the general case. Theorems and illustrations.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
The 2D Stochastic-Navier-Stokes (SNS) Equations
In 2D and geostrophic turbulence, such random transitions can
be predicted and observed
Navier Stokes equation with a random force
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs (5)
where ω = (∇∧u) .ez , fs is a random force, α is the Rayleigh
friction coecient.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
The Stochastic Forces
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs (6)
fS(x, t) = ∑k
fkηk(t)ek(x) (7)
where the ek's are the Fourier modes (Laplacian eigenmodes) and< ηk(t)ηk′(t
′) >= δk,k′δ (t− t ′)
(white in time)
For instance fk = Aexp− (|k|−m)2
2σ2 with 1
2∑|fk|2|k|2 = 1 (smooth in space).
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
The Stochastic Forces
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs (6)
fS(x, t) = ∑k
fkηk(t)ek(x) (7)
where the ek's are the Fourier modes (Laplacian eigenmodes) and< ηk(t)ηk′(t
′) >= δk,k′δ (t− t ′)
(white in time)
For instance fk = Aexp− (|k|−m)2
2σ2 with 1
2∑|fk|2|k|2 = 1 (smooth in space).
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Balance Relations (Energy Conservation)
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√σ fs
Energy conservation :
d 〈E 〉dt
=−2α 〈E 〉−ν 〈Ω2〉+ σ
In a statistically stationary regime :
〈E 〉S =σ
2α− ν
2α〈Ω2〉S
Time unit change in order to x an energy of order one (the
turnover time will be of order one) :
t ′ =√
σ/2αt ; ω ′ =√2α/σω ; α ′ = (2α)3/2 /
(2σ1/2
)and ν ′ = ν (2α/σ)1/2
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
The 2D Stochastic Navier-Stokes Equation
The 2D Stochastic Navier Stokes equation :
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√2αfs (8)
where fs is a random force (white in time, smooth in space).
We use very small Rayleigh friction, to observe large scale
energy condensation (this is not the inverse cascade regime).
We study the limit : limα→0 limν→0 (ν α) (Re Rα 1)
(Weak forces and dissipation).
We have time scale separations :
turnover time = 11/α = forcing or dissipation time
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Large Scale Structures and Euler Eq. Steady States
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√2αfs (9)
Time scale separation : Magenta terms are small.
At rst order, the dynamics is nearly a 2D Euler dynamics.
The ow self organizes and converges towards steady solutions
for the Euler Eq. :
u.∇ω = 0 or equivalently ω = f (ψ)
where the Stream Function ψ is given by : u = ez ×∇ψ
Steady states of the Euler equation will play a crucial role.
Degeneracy : what does select f ?
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Large Scale Structures and Euler Eq. Steady States
∂ω
∂ t+u.∇ω = ν∆ω−αω +
√2αfs (9)
Time scale separation : Magenta terms are small.
At rst order, the dynamics is nearly a 2D Euler dynamics.
The ow self organizes and converges towards steady solutions
for the Euler Eq. :
u.∇ω = 0 or equivalently ω = f (ψ)
where the Stream Function ψ is given by : u = ez ×∇ψ
Steady states of the Euler equation will play a crucial role.
Degeneracy : what does select f ?
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Steady States of Euler Eq. as Maxima of VariationalProblemsEnergy-Casimir Variational Problems
S(E ) = maxω
∫Ddr s (ω)
∣∣∣12
∫Ddr
v2
2= E
Maxima : ω = ∆ψ =(s′)−1
(βψ) (stable steady states of the
Euler Eq.)
In the following, normal form analysis with
s (ω) =−ω2
2+a4
ω4
4+ ...
Geometry parameter g = E (λ1−λ2) ∝ (Lx −Ly )
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Steady States for the 2D-Euler Eq. (doubly periodic)
Bifurcation analysis : degeneracy removal, either by the domain
geometry (g) or by the nonlinearity of the vorticity-stream function
relation (f , parameter a4)
Derivation : normal form for an Energy-Casimir variational problem.
A general degeneracy removal mechanism.F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Numerical Simulation of the 2D Stochastic NS Eq.
Very long relaxation times. 105 turnover times
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Numerical Simulation of the 2D Stochastic NS Eq.
Very long relaxation times. 105 turnover times
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Out of Equilibrium Stationary States : Dipoles
Are we close to some steady states of the Euler Eq. ?
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Out of Equilibrium Stationary States : Dipoles
Are we close to some steady states of the Euler Eq. ?
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Vorticity-Streamfunction Relation
Conclusion : we are close to steady states of the Euler Eq.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Vorticity-Streamfunction Relation
Conclusion : we are close to steady states of the Euler Eq.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Steady States for the 2D-Euler Eq. (doubly periodic)
A second order phase transition
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Out of Equilibrium Phase TransitionThe time series and PDF of the Order Parameter
Order parameter : z1 =∫dxdy exp(iy)ω (x ,y).
For unidirectional ows |z1| ' 0, for dipoles |z1| ' 0.6−0.7 .
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Out of Equilibrium Phase Transitions in Real FlowsRotating tank experiments (Quasi Geostrophic dynamics)
Transitions between blocked and zonal states
Y. Tian and col, J. Fluid. Mech. (2001) (groups of H. Swinney and
M. Ghil)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Experimental Applications
Using the equilibrium theory, we can predict the existence of
out of equilibrium phase transitions
Or phase transitions governed by the domain geometry, by the
topography, by the energy
Prediction of ow topology change in Quasi-Geostrophic and
Shallow Water dynamics (rotating tank experiments)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
The SNS EquationsRandom transitions in real owsRandom change of ow topology (E.S., H.M. and F.B.)
Experimental Applications
Using the equilibrium theory, we can predict the existence of
out of equilibrium phase transitions
Or phase transitions governed by the domain geometry, by the
topography, by the energy
Prediction of ow topology change in Quasi-Geostrophic and
Shallow Water dynamics (rotating tank experiments)
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Outline
1 Equilibrium statistical mechanics
Equilibrium
2 Out of equilibrium statistical mechanics
The 2D Navier Stokes equation with random forces
Experiments
Random changes of ow topology in the 2D S-Navier-Stokes
Eq. (E. Simonnet, H. Morita and F. B.)
3 The 2D linearized Euler Eq. with stochastic forces (F.B.)
General considerations
Constant shear
Shear ows : the general case. Theorems and illustrations.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D stochastic linearized Euler equation
2D Euler equations
∂ Ω
∂ t+V.∇Ω = 0, (10)
Base state : a steady state v0, with vorticity ω0 : v0.∇ω0 = 0.
The 2D Euler equation, linearized close to v0, Ω = ω0 + ω and
V = v+v0, with stochastic forces :
dω +v.∇ω0dt +v0.∇ωdt =√
σ ∑kl
fkl ekldWkl (t) ω(t = 0) = 0
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Why study the 2D stochastic linearized Euler Eq.Fluctuations Statistics - Kinetic theory
It appears naturally to describe uctuations close to statistical
equilibria :
1) Generally speaking (see for instance H. Spohn, Large scale
dynamics of interacting particles)
Fluctuating Vlasov equation
Fluctuating Boltzmann equation
2) More specically in the kinetic theory of 2D turbulence and
other systems with long range interactions
Lenard Balescu Eq., see for instance recent works in the HMF
model (P.H. Chavanis, T. Dauxois, D. Mukamel, S. Ruo, ...)
3) It also appears for any attempt to a kinetic theory or quasi-linear
approach for the 2D Stochastic Navier Stokes (SNS) equation.
Understanding the 2D stochastic linearized Euler Eq. is essential in
understanding the uctuation statistics in the SNS. Eq.F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D stochastic linearized Euler EquationsGeneral considerations
dω +v.∇ω0dt +v0.∇ωdt =√
σ ∑kl
fkl ekldWkl (t) ω(t = 0) = 0
Linear : Gaussian process (two point correlations, Lyapounov
equation)
Theoretical diculty : the deterministic linearized operator is
non normal (no mode decomposition)
Innite dimensional linear operator : non trivial
non-exponential behaviours
Landau damping or Orr mechanism
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Linear SDE : Brownian Motion and Orstein UlhenbeckProcessSome trivial remarks
First case : without dissipation. Ex : Brownian motion
dx =√
σdWt with x(0) = 0⟨x2⟩
(t) = σ t
Second case : with dissipation. A simple 1-d Orstein
Ulhenbeck process (with dissipation)
dx =−αxdt +√
σdWt with x(0) = 0⟨x2⟩S
=σ
2α
A linear stochastic dierential equation with dissipation leads to a
statistically stationary process.F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler EquationLinearized close to a stable steady state
dω +v.∇ω0dt +v0.∇ωdt =√
σ ∑kl
fkl ekldWkl (t) and ω(t = 0) = 0
In which case are we ? A linear variance growth or the
stabilisation to a stationary process ?
There is no dissipation (reversible conservative dynamics), we
should be in the rst case. We expect a linear growth.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler EquationLinearized close to a stable steady state
dω +v.∇ω0dt +v0.∇ωdt =√
σ ∑kl
fkl ekldWkl (t) and ω(t = 0) = 0
In which case are we ? A linear variance growth or the
stabilisation to a stationary process ?
There is no dissipation (reversible conservative dynamics), we
should be in the rst case. We expect a linear growth.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler Eq.Stochastic Orr mechanism
〈v(r , t + τ)v(r ′, t)〉 → 〈v(r ,τ)v(r ′,0)〉S 〈v(r , t)v(r ′, t)〉= O (σ).
(The velocity autocorrelation function has a nite limit)
The velocity vx variance
(here : the base ow is a constant shear v0 (y) = σy)
Even without dissipation, the velocity process converges
towards a stationary process (Stochastic Orr mechanism)F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler Eq.Stochastic Orr mechanism
〈v(r , t + τ)v(r ′, t)〉 → 〈v(r ,τ)v(r ′,0)〉S 〈v(r , t)v(r ′, t)〉= O (σ).
(The velocity autocorrelation function has a nite limit)
The velocity vx variance
(here : the base ow is a constant shear v0 (y) = σy)
Even without dissipation, the velocity process converges
towards a stationary process (Stochastic Orr mechanism)F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
A simple explanation
Linear process :
dx = L[x ]dt +√
σdWt with x(0) = 0⟨x2(t)
⟩S
= σ
∫ t
0
[exp(Lu)]2 du
Usually, without dissipation exp(Lu) does no converge to zero
and the integral diverge⟨x2(t)
⟩S
∝t→∞
σ t
Here, for the 2D linearized Euler Eq., even without dissipation,
[exp(Lu)] →u→∞
0 and the integral converge.
Why and how the evolution operator for the velocity converge
to zero even without dissipation or friction ?
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
A simple explanation
Linear process :
dx = L[x ]dt +√
σdWt with x(0) = 0⟨x2(t)
⟩S
= σ
∫ t
0
[exp(Lu)]2 du
Usually, without dissipation exp(Lu) does no converge to zero
and the integral diverge⟨x2(t)
⟩S
∝t→∞
σ t
Here, for the 2D linearized Euler Eq., even without dissipation,
[exp(Lu)] →u→∞
0 and the integral converge.
Why and how the evolution operator for the velocity converge
to zero even without dissipation or friction ?
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Outline
1 Equilibrium statistical mechanics
Equilibrium
2 Out of equilibrium statistical mechanics
The 2D Navier Stokes equation with random forces
Experiments
Random changes of ow topology in the 2D S-Navier-Stokes
Eq. (E. Simonnet, H. Morita and F. B.)
3 The 2D linearized Euler Eq. with stochastic forces (F.B.)
General considerations
Constant shear
Shear ows : the general case. Theorems and illustrations.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The case of a constant shear in a channelEasily solvable For pedagogical purpose
dω+v.∇ω0dt +v0.∇ωdt =√
σe(r)dW
v0 = syex . −l ≤ y ≤ l . Then ω0 = 0. A drastic simplication.
dω + sy∂ω
∂xdt = 0
Fourier series for the spatial variableω(x ,y , t) = ω (y , t)exp(ikx) :
dω + iksyωdt = 0 then ω(y , t) = ω(y ,0)exp(−iksyt)
The solution for the vorticity is trivial in that case.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The case of a constant shear in a channelEasily solvable For pedagogical purpose
dω+v.∇ω0dt +v0.∇ωdt =√
σe(r)dW
v0 = syex . −l ≤ y ≤ l . Then ω0 = 0. A drastic simplication.
dω + sy∂ω
∂xdt = 0
Fourier series for the spatial variableω(x ,y , t) = ω (y , t)exp(ikx) :
dω + iksyωdt = 0 then ω(y , t) = ω(y ,0)exp(−iksyt)
The solution for the vorticity is trivial in that case.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The case of a constant shear in a channelThe Orr mechanism - Base ow v0 = syex
ω(y , t) = ω(y ,0)exp(−iksyt)
We look at the solution for the velocity v(y , t) :
v(y ,t) =∫
dy G(y ,y ′)ω(y ′,0)exp(−iksy ′t
)We have an oscillating integral. For large times :
vx (y ,t)∼ ∼t→∞
vx ,∞ (y)
texp(−iksyt) and vy (y ,t) ∼
t→∞
vy ,∞ (y)
t2exp(−iksyt)
The velocity decreases algebraically
Orr mechanism-Case (1969)
For the stochastic process :
⟨v(y ,t)v∗(y ′,t)
⟩= σ
∫ t
0
du [exp(Lu)]2→ σ
∫∞
0
du |exp(Lu)|2
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The case of a constant shear in a channelThe Orr mechanism - Base ow v0 = syex
ω(y , t) = ω(y ,0)exp(−iksyt)
We look at the solution for the velocity v(y , t) :
v(y ,t) =∫
dy G(y ,y ′)ω(y ′,0)exp(−iksy ′t
)We have an oscillating integral. For large times :
vx (y ,t)∼ ∼t→∞
vx ,∞ (y)
texp(−iksyt) and vy (y ,t) ∼
t→∞
vy ,∞ (y)
t2exp(−iksyt)
The velocity decreases algebraically
Orr mechanism-Case (1969)
For the stochastic process :
⟨v(y ,t)v∗(y ′,t)
⟩= σ
∫ t
0
du [exp(Lu)]2→ σ
∫∞
0
du |exp(Lu)|2
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The case of a constant shear in a channelDeterministic evolution - The Orr mechanism - Base ow v0 = syex
Evolution of the perturbation vorticity ω(t), advected by a constant
shear s
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The case of a constant shear in a channelDeterministic evolution - The Orr mechanism - Base ow v0 = syex
Evolution of the perturbation kinetic energy for the transverse and
longitudinal components of the velocity v(t)
The shear acts as an eective dissipation (Phase mixing)F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Outline
1 Equilibrium statistical mechanics
Equilibrium
2 Out of equilibrium statistical mechanics
The 2D Navier Stokes equation with random forces
Experiments
Random changes of ow topology in the 2D S-Navier-Stokes
Eq. (E. Simonnet, H. Morita and F. B.)
3 The 2D linearized Euler Eq. with stochastic forces (F.B.)
General considerations
Constant shear
Shear ows : the general case. Theorems and illustrations.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The Linearized Euler Eq. close to Shear ows
Base ow : v0 (r) = U (y)ex . The linearized Euler equation :∂ω
∂ t+ ikU (y)ω− ikψU ′′ (y) = 0, (11)
with ω(x ,y , t) = ω (y , t)exp(ikx) and ω = d2ψ
dy2−k2ψ .
Laplace transform : φ (y ,c ,ε) =∫
∞
0dtΨ(y , t)exp(ik(c + iε)t)(
d2
dy2−k2
)φ − U ′′(y)
U(y)− c− iεφ =
ω (y ,0)
ik (U(y)− c− iε)(12)
This is the celebrated Rayleigh equation. A one century old
classical problem in uid mechanics, applied mathematics and
mathematics. Rayleigh (1842-1919)
Large time asymptotic is related to the limit ε → 0
Singularity of the equation : critical layer U(yc) = c
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The Linearized Euler Eq. close to Shear ows
Base ow : v0 (r) = U (y)ex . The linearized Euler equation :∂ω
∂ t+ ikU (y)ω− ikψU ′′ (y) = 0, (11)
with ω(x ,y , t) = ω (y , t)exp(ikx) and ω = d2ψ
dy2−k2ψ .
Laplace transform : φ (y ,c ,ε) =∫
∞
0dtΨ(y , t)exp(ik(c + iε)t)(
d2
dy2−k2
)φ − U ′′(y)
U(y)− c− iεφ =
ω (y ,0)
ik (U(y)− c− iε)(12)
This is the celebrated Rayleigh equation. A one century old
classical problem in uid mechanics, applied mathematics and
mathematics. Rayleigh (1842-1919)
Large time asymptotic is related to the limit ε → 0
Singularity of the equation : critical layer U(yc) = c
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The Linearized Euler Eq. close to Shear ows
Base ow : v0 (r) = U (y)ex . The linearized Euler equation :∂ω
∂ t+ ikU (y)ω− ikψU ′′ (y) = 0, (11)
with ω(x ,y , t) = ω (y , t)exp(ikx) and ω = d2ψ
dy2−k2ψ .
Laplace transform : φ (y ,c ,ε) =∫
∞
0dtΨ(y , t)exp(ik(c + iε)t)(
d2
dy2−k2
)φ − U ′′(y)
U(y)− c− iεφ =
ω (y ,0)
ik (U(y)− c− iε)(12)
This is the celebrated Rayleigh equation. A one century old
classical problem in uid mechanics, applied mathematics and
mathematics. Rayleigh (1842-1919)
Large time asymptotic is related to the limit ε → 0
Singularity of the equation : critical layer U(yc) = c
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Historical results
1)Base ows without stationary points : for any y , U ′(y) 6= 0
(monotonic prole)
Rayleigh (1880) Mode equation.
Case (Phys. Fluid. 1960) Algebraic laws for the velocity eld
in the case of constant shear (wrong).
Rosencrans and Sattinger (J. Math. Phys 1966) v =t→0
O (1/t)
(Laplace transform tools).
Brown,Stewartson (JFM 1980) Ansatz for the asymptotic
expansion, for large time.
Friedlander Howard (Comm. Math. Phys. 2004)
2) Base ows with one or several stationary points : U ′(y0) = 0
No results !
This the case of interest for the Stochastic Navier Stokes equation !F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Historical results
1)Base ows without stationary points : for any y , U ′(y) 6= 0
(monotonic prole)
Rayleigh (1880) Mode equation.
Case (Phys. Fluid. 1960) Algebraic laws for the velocity eld
in the case of constant shear (wrong).
Rosencrans and Sattinger (J. Math. Phys 1966) v =t→0
O (1/t)
(Laplace transform tools).
Brown,Stewartson (JFM 1980) Ansatz for the asymptotic
expansion, for large time.
Friedlander Howard (Comm. Math. Phys. 2004)
2) Base ows with one or several stationary points : U ′(y0) = 0
No results !
This the case of interest for the Stochastic Navier Stokes equation !F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Base ow with stationary points : U ′(y0) = 0
Mathematical methods : Laplace transform and detailed
analysis of the singularities due to the critical layers and
stationary points.
By contrast with what was previously believed, we can deal
with the diculty related to the stationary points.
Theorem : a) Asymptotic oscillatory vorticity eld
ω (y , t) ∼t→∞
ω∞ (y)exp(ikU(y)t) +O
(1
tα
)b) With a cancellation of the vorticity at stationary points :
For any stationary point of the ow (y0 such that U ′′ (y0) = 0) then
ω∞ (y0) = 0
+ Prediction of the asymptotic vorticity ω∞ (y).F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Base ow with stationary points : U ′(y0) = 0
Mathematical methods : Laplace transform and detailed
analysis of the singularities due to the critical layers and
stationary points.
By contrast with what was previously believed, we can deal
with the diculty related to the stationary points.
Theorem : a) Asymptotic oscillatory vorticity eld
ω (y , t) ∼t→∞
ω∞ (y)exp(ikU(y)t) +O
(1
tα
)b) With a cancellation of the vorticity at stationary points :
For any stationary point of the ow (y0 such that U ′′ (y0) = 0) then
ω∞ (y0) = 0
+ Prediction of the asymptotic vorticity ω∞ (y).F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Base ow with stationary points : the vorticity eld evolution
Evolution of the perturbation vorticity ω(t), advected by a shear ow
U(y) with stationary points
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Base ow with stationary points : the velocity eld
Theorem : algebraically decaying asymptotic velocity eld
vx(y , t) ∼t→∞
vx ,∞ (y)
texp(−ikU(y)t) (13)
vy (y , t) ∼t→∞
vy ,∞ (y)
t2exp(−ikU(y)t) (14)
What about stationary points ? They should give
contributions of order 1/t1/2 !
No contribution from the stationary points thanks to the
cancellation of the vorticity at stationary points.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Base ow with stationary points : the velocity eld
Theorem : algebraically decaying asymptotic velocity eld
vx(y , t) ∼t→∞
vx ,∞ (y)
texp(−ikU(y)t) (13)
vy (y , t) ∼t→∞
vy ,∞ (y)
t2exp(−ikU(y)t) (14)
What about stationary points ? They should give
contributions of order 1/t1/2 !
No contribution from the stationary points thanks to the
cancellation of the vorticity at stationary points.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Base ow with stationary points : the velocity eld
Evolution of the perturbation velocity, components vx(t) and vy (t),
advected by a constant shear ow U(y) with stationary points
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq.Base ow with stationary points : the velocity eld
Evolution of the perturbation velocity, components vx(t) and vy (t),
advected by a constant shear ow U(y) with stationary points
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Asymptotic behaviour of the linearized Euler Eq. :Conclusions
Asymptotically oscillating vorticity elds
Algebraic decay of the velocity eld with 1/t or 1/t2 laws,
whatever the case.
All cases of base ow with any type of shear or monopolar
vortices have been treated.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler Eq. : ConclusionsThe results for shear ows also hold for axisymmetric vortex
Scalings are the same. Quantitative results are dierent.
〈ω(r , t)ω(r , t)〉= O (σt) ∝ δ (r − r ′) (Resonance over streamlines
for the vorticity autocorrelation function)
〈v(r , t + τ)v(r ′, t)〉 → 〈v(r ,τ)v(r ′,0)〉S 〈v(r , t)v(r ′, t)〉= O (σ).
(The velocity autocorrelation function has a nite limit)
Even without dissipation, the velocity stochastic process
converges towards a stationary process (Stochastic Orr
mechanism)
Proof :
Laplace transform (frequency representation), detailed study of the
resolvent operator and its singularities, dierential equation
techniques.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler Eq. : ConclusionsThe results for shear ows also hold for axisymmetric vortex
Scalings are the same. Quantitative results are dierent.
〈ω(r , t)ω(r , t)〉= O (σt) ∝ δ (r − r ′) (Resonance over streamlines
for the vorticity autocorrelation function)
〈v(r , t + τ)v(r ′, t)〉 → 〈v(r ,τ)v(r ′,0)〉S 〈v(r , t)v(r ′, t)〉= O (σ).
(The velocity autocorrelation function has a nite limit)
Even without dissipation, the velocity stochastic process
converges towards a stationary process (Stochastic Orr
mechanism)
Proof :
Laplace transform (frequency representation), detailed study of the
resolvent operator and its singularities, dierential equation
techniques.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler Eq.The Reynolds tensor
Can we evaluate the Reynolds tensor 〈v(r, t).∇ω (r, t)〉 :
〈v(r, t).∇ω (r, t)〉 ∼t→∞
σC (r) log (t)
And the rms value of the nonlinear term :√⟨[v(r, t).∇ω (r, t)]2
⟩∝ σC ′ (r) t3/2
Conclusion : the stochastic linearized Euler Eq. can be valid
only for nite time. No quasi-linear theory can be
self-consistent in the inertial regime.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler Eq.The Reynolds tensor
Can we evaluate the Reynolds tensor 〈v(r, t).∇ω (r, t)〉 :
〈v(r, t).∇ω (r, t)〉 ∼t→∞
σC (r) log (t)
And the rms value of the nonlinear term :√⟨[v(r, t).∇ω (r, t)]2
⟩∝ σC ′ (r) t3/2
Conclusion : the stochastic linearized Euler Eq. can be valid
only for nite time. No quasi-linear theory can be
self-consistent in the inertial regime.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
The 2D Linearized Stochastic Euler Eq.The Reynolds tensor
Can we evaluate the Reynolds tensor 〈v(r, t).∇ω (r, t)〉 :
〈v(r, t).∇ω (r, t)〉 ∼t→∞
σC (r) log (t)
And the rms value of the nonlinear term :√⟨[v(r, t).∇ω (r, t)]2
⟩∝ σC ′ (r) t3/2
Conclusion : the stochastic linearized Euler Eq. can be valid
only for nite time. No quasi-linear theory can be
self-consistent in the inertial regime.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Publications
1 F. Bouchet, Physica D, 2008 Simplied variational problems for thestatistical equilibria of 2D ows.
2 F. Bouchet and E. Simmonet, PRL (January 2009), Randomchanges of ow topology in 2D and geophysical turbulence.
3 A. Venaille and F. Bouchet, PRL (January 2009), Phase transitions,ensemble inequivalence and Fofono ows.
4 A. Venaille and F. Bouchet, to be submitted to J. Phys.Oceanography. Are strong mid-basin eastward jets (Gulf Stream,Kuroshio) statistical equilibria ?
5 F. Bouchet and H. Morita, to be submitted to Physica D,Asymptotic stability of the 2D Euler and of the 2D linearized Eulerequations
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Summary
Messages :
We can predict and observe out of equilibrium phase
transitions for the 2D-Stochastic Navier Stokes equation
We propose experiments to observe such phenomena (Navier
Stokes, Quasi Geostrophic, or Shallow Water dynamics)
Theory for the 2D stochastic linearized Euler equation. The
velocity process converges towards a stationary one even
without dissipation.
F. Bouchet INLN-CNRS-UNSA Phase transitions
EquilibriumOut of equilibrium
Stochastic Orr mechanism
General considerationsConstant shearShear ows : the general case. Theorems and illustrations.
Perspectives
Messages :
Towards statistical mechanics models of the bistability of the
Kuroshio or of other bistable ocean currents.
Analytical and numerical computations oflarge deviations for
the statistics of the large scales of the 2D Stochastic Navier
Stokes Eq.
F. Bouchet INLN-CNRS-UNSA Phase transitions