Out of equilibrium phase transitions in the 2D Navier ... · Motivations Equilibrium Out of...

MotivationsEquilibrium

Out of equilibriumThe 2D linearized Euler Eq. with stochastic forces (F.B.)

Out of equilibrium phase transitions in the 2DNavier-Stokes equation

Random change of flow topology in 2D and geophysicalturbulence

F. BOUCHET – Institut Non Linéaire de Nice

GDR Phénix, Lyon, June 2008

F. Bouchet INLN-CNRS-UNSA Phase transitions



Outline

1 Motivations

2 Recent results in equilibrium statistical mechanicsIs the map of ocean currents a statistical equilibria ?Other recent results for the equilibrium RSM theory

3 Out of equilibrium statistical mechanicsThe 2D Navier Stokes equation with random forcesClassical views for 2D turbulence, inverse energycascade or equilibrium stat. mech. ?MotivationsRandom changes of flow topology in the 2DS-Navier-Stokes Eq. (E. Simonnet, H. Morita and F. B.)

4 The 2D linearized Euler Eq. with stochastic forces (F.B.)




Collaborators

Equilibrium statistical mechanics of two dimensional andgeophysical flows : N. Sauvage, A. Venaille (Grenoble).Stability of equilibrium states : F. Rousset (Lab. DieudonnéNice) (ANR Statflow)Random change of flow topology (out of equilibrium) : H.Morita and E. Simonnet (INLN-Nice) (ANR Statflow)




The Physical Phenomena

Theoretical ideas :

Self organisation processes. Large number of degrees offreedom (turbulence).This has to be explained using statistical physics !!!

Mainly out of equilibrium statistical mechanics. We have to workout new theoretical concepts with such phenomena in mind.





Theoretical ideas :







Theoretical ideas :







Theoretical ideas :






Is the map of ocean currents a statistical equilibria ?Other recent results for the equilibrium RSM theory

Outline

1 Motivations








Robert-Sommeria-Miller (RSM) TheoryEquilibrium statistical mechanics : the most probable vorticity field

A probabilistic description of the vorticity field ω : ρ (x, σ) isthe local probability to have ω (x) = σ at point xA measure of the number of microscopic field ωcorresponding to a probability ρ :

Maxwell-Boltzmann Entropy: S [ρ] ≡ −∫D

dx∫ +∞

−∞dσ ρ log ρ

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 flows of Euler’s equations :

ω = f (ψ)








dx∫ +∞

−∞dσ ρ log ρ



S[ρ] | E [ω] = E0 ,D [ρ] = d (MVP).


ω = f (ψ)








dx∫ +∞

−∞dσ ρ log ρ



S[ρ] | E [ω] = E0 ,D [ρ] = d (MVP).


ω = f (ψ)








dx∫ +∞

−∞dσ ρ log ρ



S[ρ] | E [ω] = E0 ,D [ρ] = d (MVP).


ω = f (ψ)








dx∫ +∞

−∞dσ ρ log ρ



S[ρ] | E [ω] = E0 ,D [ρ] = d (MVP).


ω = f (ψ)





The Map of Ocean Currents

North Atlantic height

Map of ocean currents





The Simplest Model

We describe the upper layer of an ocean by the QuasiGeostrophic model (one and half layer) :

∂q∂t

+ v ·∇ q = 0 ; v = ez ×∇ψ ; q = ∆ψ − ψ

R2 + βy

An extremely rough model of an ocean. The simplest one.A simple question : does it exist strong eastward midbassin jets, which are statistical equilibria of this model ?





The Simplest Model

We describe the upper layer of an ocean by the QuasiGeostrophic model (one and half layer) :

∂q∂t

+ v ·∇ q = 0 ; v = ez ×∇ψ ; q = ∆ψ − ψ

R2 + βy

An extremely rough model of an ocean. The simplest one.A simple question : does it exist strong eastward midbassin jets, which are statistical equilibria of this model ?





Strong Eastward Jets as Equilibria of the QG Model ?Antoine Venaille

These flows are not equilibrium states, but are dynamicallystable

N. Sauvage and F. Bouchet, en preparationA. Venaille, F. Bouchet and E. Simonnet, en preparation





Outline

1 Motivations








Other Recent Results for the Equilibrium RSM Theory

Simplified variational problems for the statistical equilibria of 2Dflows. F. Bouchet, Physica D, 2008Stability of a new class of equilibrium states (generalization ofArnold’s theorems). E. Caglioti, M. Pulvirenti and F. Rousset,(submitted to Comm. Math. Phys., preprint)Phase transitions, ensemble inequivalence and Fofonoff flows.A. Venaille and F. Bouchet, sub. to Phys. Rev. Lett.




The SNS EquationsCascade or Eq. Statistical Mechanics ?Random transitions in real flowsRandom change of flow topology (E.S., H.M. and F.B.)

Outline

1 Motivations








The 2D Stochastic-Navier-Stokes (SNS) Equations

The simplest model for two dimensional turbulenceNavier Stokes equation with a random force

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√σfs (1)

where ω = (∇∧ u) .ez is the vorticity, fs is a random force, α isthe Rayleigh friction coefficient.An academic model with experimental realizations(Sommeria and Tabeling experiments, rotating tanks,magnetic flows, and so on). Analogies with geophysicalflows.






The simplest model for two dimensional turbulenceNavier Stokes equation with a random force

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√σfs (1)

where ω = (∇∧ u) .ez is the vorticity, fs is a random force, α isthe Rayleigh friction coefficient.An academic model with experimental realizations(Sommeria and Tabeling experiments, rotating tanks,magnetic flows, and so on). Analogies with geophysicalflows.





Numerical Simulation of the 2D Stochastic-NS Eq.

Self similar growth of a dipolestructure, for the 2D S-NSequationLeft : vorticity fieldBottom : vorticity profiles

M Chertkov, C Connaughton, I Kolokolov, V Lebedev(nlin.CD/0612052, PRL 2007) (Los Alamos)F. Bouchet INLN-CNRS-UNSA Phase transitions




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 oflimν→0 µνIn 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 stationary solutions of the Eulerequation ?Can we describe these stationary states and theirproperties ?





Outline

1 Motivations








Inverse Energy CascadeSelf similar statistics of inertial scales of 2D flows

Sketch of the doublecascade in 2D turbulence

Self similar statistics - Energy spectrum and velocityincrements.





What are Real Flow RegimesInverse energy cascade or equilibrium statistical mechanics ?











Equilibrium statistical predicts stationary flows. It does not takeinto account forces and dissipation.











Real flows : out of equilibrium statistical mechanics or inversecascade governed by large scales (and not self similar).





Outline

1 Motivations








Out of Equilibrium Phase Transitions in Real Flows2D MHD experiments (2D Navier Stokes dynamics)

J. Sommeria, J. Fluid. Mech. (1986)





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. Swinneyand M. Ghil)





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 turbulenceRandom changes of paths for the Kurushio current,weather regimes, and so on.





Outline

1 Motivations









In 2D and geostrophic turbulence, such random transitionscan be predicted and observedNavier Stokes equation with a random force

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√σfs (2)

where ω = (∇∧ u) .ez , fs is a random force, α is the Rayleighfriction coefficient.





The Stochastic Forces

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√σfs (3)

fS(x, t) =∑

k

fkηk(t)ek(x) (4)

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 = A exp− (|k|−m)2

2σ2 with 12

∑ |fk|2|k|2 = 1 (smooth in

space).





The Stochastic Forces

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√σfs (3)

fS(x, t) =∑

k

fkηk(t)ek(x) (4)

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 = A exp− (|k|−m)2

2σ2 with 12

∑ |fk|2|k|2 = 1 (smooth in

space).





Balance Relations (Energy Conservation)

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√σfs

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 fix an energy of order one (theturnover 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





The 2D Stochastic Navier-Stokes Equation

The 2D Stochastic Navier Stokes equation :

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√2αfs (5)

where fs is a random force (white in time, smooth in space).

We use very small Rayleigh friction, to observe large scaleenergy condensation (this is not the inverse cascaderegime).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





Large Scales Structures and Euler Eq. Steady States

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√2αfs (6)

Time scale separation : Magenta terms are small.At first order, the dynamics is nearly a 2D Euler dynamics.The flow self organizes and converges towards steadysolutions for the Euler Eq. :

u.∇ω = 0 or equivalently ω = f (ψ)

where the Stream Function ψ is given by : u = ez ×∇ψSteady states of the Euler equation will play a crucial role.Degeneracy : what does select f ?





Large Scales Structures and Euler Eq. Steady States

∂ω

∂t+ u.∇ω = ν∆ω − αω +

√2αfs (6)

Time scale separation : Magenta terms are small.At first order, the dynamics is nearly a 2D Euler dynamics.The flow self organizes and converges towards steadysolutions for the Euler Eq. :

u.∇ω = 0 or equivalently ω = f (ψ)

where the Stream Function ψ is given by : u = ez ×∇ψSteady states of the Euler equation will play a crucial role.Degeneracy : what does select f ?





Steady States for the 2D-Euler Eq. (doubly periodic)

Bifurcation analysis : degeneracy removal, either by the domaingeometry (g) or by the nonlinearity of the vorticity-stream

function relation (f , parameter a4)

Derivation : normal form for an Energy-Casimir variationalproblem. A general degeneracy removal mechanism.





Numerical Simulation of the 2D Stochastic NS Eq.

Very long relaxation times. 105 turnover times





Numerical Simulation of the 2D Stochastic NS Eq.

Very long relaxation times. 105 turnover times





Out of Equilibrium Stationary States : Dipoles

Are we close to some steady states of the Euler Eq. ?





Out of Equilibrium Stationary States : Dipoles

Are we close to some steady states of the Euler Eq. ?





Vorticity-Streamfunction Relation

Conclusion : we are close to steady states of the Euler Eq.





Vorticity-Streamfunction Relation

Conclusion : we are close to steady states of the Euler Eq.





Steady States for the 2D-Euler Eq. (doubly periodic)

A second order phase transition





Out of Equilibrium Phase TransitionThe time series and PDF of the Order Parameter

Order parameter : z1 =∫

dxdy exp(iy)ω (x , y).

For unidirectional flows |z1| ' 0, for dipoles |z1| ' 0.6− 0.7 .





An Analogy with a Subcritical Bifurcation ?

A subcritical bifurcation perturbed by an additive noise :

dx = x(µ+ x2 − x4

)dt +

√σdWt

Deterministic bifurcation diagram PDFs





Incompatible with the Steady States Phase Transition?

At equilibrium we observe a second order phase transition(symmetry breaking)

Pitchfork bifurcation diagram(equilibrium)

Bifurcation diagram for theSDE model





Experimental Applications

Using the equilibrium theory, we can predict the existenceof out of equilibrium phase transitionsOr phase transitions governed by the domain geometry, bythe topography, by the energyPrediction of flow topology change in Quasi-Geostrophicand Shallow Water dynamics (rotating tank experiments)





Experimental Applications

Using the equilibrium theory, we can predict the existenceof out of equilibrium phase transitionsOr phase transitions governed by the domain geometry, bythe topography, by the energyPrediction of flow topology change in Quasi-Geostrophicand Shallow Water dynamics (rotating tank experiments)





A Theory for the Large Scales of the 2D Navier-StokesEq. ?An Adiabatic Reduction ?

Euler’s equations have an infinity of stationary solutions :ω = f (ψ) for any f (Euler’s equations have degenerateequilibria.)For the conservative dynamics, equilibrium statisticalmechanics selects fFor Navier-Stokes with weak forces, what does select f ?Time scale separation. An adiabatic reduction ?























The A-B ModelA pedagogical model in order to illustrate adiabatic reduction

dAdt = −ABdBdt = A2

A quadratic nonlinearityConservation of energy

E = A2 + B2

Degenerate equilibrium states : for any B, A = 0 is anequilibrium








E = A2 + B2









E = A2 + B2






Phase Space of the A-B Model





The Stochastic AB ModelThe limit of weak forces and dissipation

dA = (−AB − νA) dt +√νσ1dW1

dB =(A2 − νB

)dt +

√νσ2dW2

Stationary measure in the limit ν →∞






dA = (−AB − νA) dt +√νσ1dW1

dB =(A2 − νB

)dt +

√νσ2dW2

Stationary measure in the limit ν →∞






First step of the adiabatic treatment : understand theevolution of the rapid variable A, for a fixed value of theslow variable B.At first order, for small v , A is a Orstein-Ulhenbeck process.dA = (−AB − νA) dt +

√νσ1dW1. Locally Gaussian :

P(A) = C(B) exp

(−BA2

νσ21

)

P(A,B) = C1 exp

(−

BA2

νσ21

)B

σ21

σ22

+ 12

exp

(−

B2

σ22

); P(E) = C1E

σ21

σ22 exp

(−

E2

σ22

)

A non trivial distribution. The PDF is not concentrated. Theweak forces and dissipation do not select a singleequilibrium E .






First step of the adiabatic treatment : understand theevolution of the rapid variable A, for a fixed value of theslow variable B.At first order, for small v , A is a Orstein-Ulhenbeck process.dA = (−AB − νA) dt +

√νσ1dW1. Locally Gaussian :

P(A) = C(B) exp

(−BA2

νσ21

)

P(A,B) = C1 exp

(−

BA2

νσ21

)B

σ21

σ22

+ 12

exp

(−

B2

σ22

); P(E) = C1E

σ21

σ22 exp

(−

E2

σ22

)

A non trivial distribution. The PDF is not concentrated. Theweak forces and dissipation do not select a singleequilibrium E .





Adiabatic Reduction for the Stochastic-NS Eq.First step (current work) : the stochastic linearised Navier Stokes equation

The first step : linearised Navier Stokes equation close toan Euler equilibrium, with random forcesThe linear operator is non normal (no modedecomposition) ! Further difficultiesThe second step : An equation that describes only thelarge scale evolution ? Is this approach self consistent ?
















The 2D Linearized Stochastic Euler Equation

A stable equilibria for the 2D Euler equation v0, withvorticity ω0 : v0.∇ω0 = 0.The 2D Euler equation, linearized close to u0, withstochastic forces :

dω + v.∇ω0dt + v0.∇ωdt = −αωdt +√σ∑

kl

fkl ekldWkl (t)

An infinite dimensional Orstein-Ulhenbeck process(Gaussian, two point correlations, Lyapounov equation)Theoretical difficulty : the deterministic linearized operatoris non normal (no mode decomposition)Landau damping or Orr mechanism




The 2D Linearized Stochastic Euler Eq.Base flow : an axisymmetric vortex

Fourier decomposition on azimuthal wave numbers :ω = ωk (r , t)eikθ, v = vk (r , t)eikθ.Stochastic linearized Euler equation, in the k th sector :

dωk−ikω′0 (r)

rψk dt+ikΩ0 (r)ωk dt = −αωk dt+

√σ∑

l

fkl ekldWkl (t)

angular frequency Ω0 and stream function ψk :

Ω0 (r) ≡ v0 (r)

rand ∆kψk ≡

d2ψk

dr2 +1r

dψk

dr− k2

r2 ψk = ωk .






dωk−ikω′0 (r)


√σ∑

l

fkl ekldWkl (t)


Ω0 (r) ≡ v0 (r)

rand ∆kψk ≡

d2ψk

dr2 +1r

dψk

dr− k2

r2 ψk = ωk .




Usual Force-Dissipation Balance for an OrsteinUlhenbeck Process

A simple 1-d Orstein Ulhenbeck process

dx = −αxdt +√σdWt⟨

x2⟩

S=

σ

2α




The 2D Linearized Stochastic Euler Eq.Stochastic Landau damping (F. B.)

Resonance for the vorticity autocorrelation function forsmall α : 〈ω(r ,0)ω(r ′,0)〉S = O (σ/α) ∝α→0 δ (r − r ′)

No resonances for the stream function and velocity :〈ψ(r ,0)ψ(r ′,0)〉S = O (σ) and 〈v(r ,0)v(r ′,0)〉 = O (σ)

In the small dissipation limit, the velocity stochasticprocess has a definite limit (Stochastic Landau damping)

Proof :

1 Very technical. Long and detailed analysis and estimatesfor the resolvent operator.

1 Is an adiabatic reduction possible ?




The 2D Linearized Stochastic Euler Eq.Stochastic Landau damping (F. B.)

Resonance for the vorticity autocorrelation function forsmall α : 〈ω(r ,0)ω(r ′,0)〉S = O (σ/α) ∝α→0 δ (r − r ′)


In the small dissipation limit, the velocity stochasticprocess has a definite limit (Stochastic Landau damping)

Proof :

1 Very technical. Long and detailed analysis and estimatesfor the resolvent operator.

1 Is an adiabatic reduction possible ?







kl

fkl ekldWkl (t)

Hypothesis : The base state v0 has at most one critical layer

Monotonous flows, No stagnation point, no separatrix. (anessential technical hypothesis)Ex1 : v0 = v0 (y) ex a shear flow in a 2D channel. v0 (y) ismonotonous.Ex2 : v0 = v0 (r) eθ an axisymmetric vortex in a 2D disk.Ω0 (r) is strictly decreasing.







kl

fkl ekldWkl (t)

Hypothesis : The base state v0 has at most one critical layer

Monotonous flows, No stagnation point, no separatrix. (anessential technical hypothesis)Ex1 : v0 = v0 (y) ex a shear flow in a 2D channel. v0 (y) ismonotonous.Ex2 : v0 = v0 (r) eθ an axisymmetric vortex in a 2D disk.Ω0 (r) is strictly decreasing.






dωk−ikω′0 (r)


√σ∑

l

fkl ekldWkl (t)


Ω0 (r) ≡ v0 (r)

rand ∆kψk ≡

d2ψk

dr2 +1r

dψk

dr− k2

r2 ψk = ωk .






dωk−ikω′0 (r)


√σ∑

l

fkl ekldWkl (t)


Ω0 (r) ≡ v0 (r)

rand ∆kψk ≡

d2ψk

dr2 +1r

dψk

dr− k2

r2 ψk = ωk .




The 2D Linearized Stochastic Euler Eq.An Orstein Ulhenbeck process in infinite dimension

dω−iω′0 (r)

rψdt + ikΩ0 (r)ωdt = −αωdt +

√σe(r)dW (t)

Gaussian stochastic process. Autocorrelation functions

c(r , r ′, t) =⟨ω (r ,0)ω∗

(r ′, t)⟩

S

The bare vorticity equation (for pedagogical purposes)

dω + ikΩ0 (r)ωdt = −αωdt +√σe(r)dW (t)




The 2D Linearized Stochastic Euler Eq.An Orstein Ulhenbeck process in infinite dimension

dω−iω′0 (r)

rψdt + ikΩ0 (r)ωdt = −αωdt +

√σe(r)dW (t)

Gaussian stochastic process. Autocorrelation functions

c(r , r ′, t) =⟨ω (r ,0)ω∗

(r ′, t)⟩

S

The bare vorticity equation (for pedagogical purposes)





The bare vorticity Eq.Easily solvable


Statistically stationary solution :

ω(r , t) =√σel (r)

∫ t

−∞exp

[−(iΩ0 (r) + α)(t − t ′)

]dW (t ′)

For t>0⟨ω(r , t)ω∗(r ′,0)

⟩S =

σe (r) e∗ (r ′)2α + i(Ω0 (r)− Ω0 (r ′))

e−(iΩ0(r)+α)t

Decorrelation time 1/α. (Very long)Resonance of 〈ω(r , t)ω∗(r ′,0)〉S for r = r ′ for small α.Study the limit α→ 0







ω(r , t) =√σel (r)

∫ t

−∞exp

[−(iΩ0 (r) + α)(t − t ′)

]dW (t ′)

For t>0⟨ω(r , t)ω∗(r ′,0)

⟩S =

σe (r) e∗ (r ′)2α + i(Ω0 (r)− Ω0 (r ′))

e−(iΩ0(r)+α)t








ω(r , t) =√σel (r)

∫ t

−∞exp

[−(iΩ0 (r) + α)(t − t ′)

]dW (t ′)

For t>0⟨ω(r , t)ω∗(r ′,0)

⟩S =

σe (r) e∗ (r ′)2α + i(Ω0 (r)− Ω0 (r ′))

e−(iΩ0(r)+α)t





The bare vorticity Eq.The limit α→ 0.

Resonance for the vorticity autocorrelation function.〈ω(r ,0)ω∗(r ,0)〉S = O (σ/α)

〈ω(r ,0)ω∗(r ′,0)〉S ∼√σe (r) e∗ (r ′)

[iP(

1Ω0 (r)− Ω0 (r ′)

)− π

Ω′0 (r)δ(r − r ′)

]

No resonance for the velocity ψ(r) =∫

dr ′G(r , r ′)ω(r ′).

〈ψ(r ,0)ψ(r ′,0)〉S = O (σ) and 〈v(r ,0)v(r ′,0)〉 = O (σ)

Order of magnitude of the nonlinear terms :

〈〈v.∇ω〉1l〉S (r) = O( σ

α3/2

)Conclusion (loosely speaking) : When α is small : very roughvorticity field but smooth velocity field




The bare vorticity Eq.The limit α→ 0.

Resonance for the vorticity autocorrelation function.〈ω(r ,0)ω∗(r ,0)〉S = O (σ/α)

〈ω(r ,0)ω∗(r ′,0)〉S ∼√σe (r) e∗ (r ′)

[iP(

1Ω0 (r)− Ω0 (r ′)

)− π

Ω′0 (r)δ(r − r ′)

]

No resonance for the velocity ψ(r) =∫

dr ′G(r , r ′)ω(r ′).

〈ψ(r ,0)ψ(r ′,0)〉S = O (σ) and 〈v(r ,0)v(r ′,0)〉 = O (σ)

Order of magnitude of the nonlinear terms :

〈〈v.∇ω〉1l〉S (r) = O( σ

α3/2

)Conclusion (loosely speaking) : When α is small : very roughvorticity field but smooth velocity field




The 2D Linearized Stochastic Euler Eq.The same qualitative results as for the bare vorticity Eq. hold.

dω = −A [ω] dt−αωdt+√σ∑

l

fleldWl with A[ω] = −iω′0 (r)

rψ + ikΩ0 (r)ω

Scalings will be the same. Quantitative results will be different.A is the Rayleigh operator (linear stability of Euler’s flows)Lyapounov Equation (LE) for c(r , r ′) = 〈c(r)c∗(r ′)〉S

(A + α) c + c(A+ + α

)= σBB+

(B is diagonal in the basis el and is defined by Bel = flel )Solution for LE using the resolvent operatorR (ω) = (A− iω)−1 : c(r , r ′) = σ

∑l|fl |22π Fl (r , r ′) with

Fl(r , r ′) =

∫ +∞

−∞dΩ R (Ω + iα, r) [el ]R (Ω + iα, r)+ [el ]




The 2D Linearized Stochastic Euler Eq.The same qualitative results as for the bare vorticity Eq. hold.

dω = −A [ω] dt−αωdt+√σ∑

l

fleldWl with A[ω] = −iω′0 (r)

rψ + ikΩ0 (r)ω

Scalings will be the same. Quantitative results will be different.A is the Rayleigh operator (linear stability of Euler’s flows)Lyapounov Equation (LE) for c(r , r ′) = 〈c(r)c∗(r ′)〉S

(A + α) c + c(A+ + α

)= σBB+

(B is diagonal in the basis el and is defined by Bel = flel )Solution for LE using the resolvent operatorR (ω) = (A− iω)−1 : c(r , r ′) = σ

∑l|fl |22π Fl (r , r ′) with

Fl(r , r ′) =

∫ +∞

−∞dΩ R (Ω + iα, r) [el ]R (Ω + iα, r)+ [el ]




The Resolvent of the 2D Rayleigh OperatorAdapted from Briggs Daugherty and Levy (1970)

We define the stream function resolvent S (Ω) [u] by

d2Sdr2 +

1r

dSdr− k2

r2 S = R (Ω) [u],

then S (Ω) [u] verifies an inhomogeneous Rayleigh equation

d2Sdr2 +

1r

dSdr− k2

r2 S +kω′0 (r)

r (Ω− kΩ0 (r))S =

iuΩ− kΩ0 (r)

This equation is singular for r = rc (ω) with Ω = kΩ0 (rC)(critical layer).




The Resolvent of the 2D Rayleigh OperatorProperties

We have the following results :1 S (Ω) [u] is analytic for all Ω except for real Ω with

Ω ∈ [min(kΩ0(r)); min(kΩ0(r))] (BDL)2 The spectrum of the Rayleigh operator is

[min(kΩ0(r)); min(kΩ0(r))] (singular spectrum). No linearmode exist. (BDL)

3 For each Ω ∈ [min(kΩ0(r)); min(kΩ0(r))], S (Ω + iα) [u]has a finite limit S (Ω + i0+) [u] for α→ 0+.

4 For each Ω ∈ [min(kΩ0(r)); min(kΩ0(r))],

R (Ω + iα) [u] = Rr (Ω + iα) [u] +P(Ω + iα)[u]

Ω− kΩ0 (r)

where the limits Rr (Ω + i0+) [u] and P(Ω + i0+)[u] exist.











Ω− kΩ0 (r)












Ω− kΩ0 (r)





The 2D Linearized Stochastic Euler Eq.Conclusions for the case of a single critical layer

Resonance for the vorticity autocorrelation function forsmall α : 〈ω(r ,0)ω∗(r ,0)〉S = O (σ/α)


Order of magnitude of the nonlinear terms :〈〈v.∇q〉1l〉S (r) = O

(σ/α3/2

)Further issues :

1 Numerical resolution of the Lyapounov equation (morephysical issues) (F. Gallaire)

2 Treat the case of multiple critical layers.3 Treat the case with of a separatrix in the base flow.4 Add a small viscosity.5 Is an adiabatic reduction possible ?




The 2D Linearized Stochastic Euler Eq.Conclusions for the case of a single critical layer

Resonance for the vorticity autocorrelation function forsmall α : 〈ω(r ,0)ω∗(r ,0)〉S = O (σ/α)


Order of magnitude of the nonlinear terms :〈〈v.∇q〉1l〉S (r) = O

(σ/α3/2

)Further issues :

1 Numerical resolution of the Lyapounov equation (morephysical issues) (F. Gallaire)

2 Treat the case of multiple critical layers.3 Treat the case with of a separatrix in the base flow.4 Add a small viscosity.5 Is an adiabatic reduction possible ?




Summary

Messages :

We can predict and observe out of equilibrium phasetransitions for the 2D-Stochastic Navier Stokes equationWe propose experiments to observe such phenomena(Navier Stokes, Quasi Geostrophic, or Shallow Waterdynamics)Theory for the 2D stochastic linearized Euler equation.Stochastic Orr mechanism or Landau damping.


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