OUT-OF-EQUILIBRIUM PHASE TRANSITIONS FOR THE LARGE...
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OUT-OF-EQUILIBRIUM PHASE TRANSITIONS FOR THE LARGE SCALES OF 2D TURBULENCE Freddy BOUCHET, Francois GALLAIRE, Frédéric ROUSSET and Eric SIMONNET INLN-CNRS. ANR JC STATFLOW. Contact : [email protected] Statistical Mechanics of Large Scale Geophysical Flows Large scale statistics of turbulent flows In many applications of fluid dynamics, one of the most important prob- lem is the prediction of the very high Reynolds’ large-scale flows. The highly turbulent nature of such flows, for instance ocean circulation or at- mosphere dynamics, renders a probabilistic description desirable, if not nec- essary. A statistical mechanics explanation of the self-organization of geo- physical flows has been proposed by Robert-Sommeria and Miller (RSM). The RSM theory has been success- fully applied to the Jupiter’s tro- posphere : cyclones, anticyclones and jets have been quantitatively described by this theory (F. Bouchet and J. Sommeria) Observation (Voyager) Statistical Equilibrium Velocity field of Jupiter’s Great Red Spot For applications : statistical mechanics beyond equilibrium ? The RSM theory starts from the conservative dynamics and parametrizes equilibria by the energy and other dynamical invariants. However, the theory does not predict the long- term effects of the forcing, which is a relevant issue for any application. It is a practical and fundamental problem to understand how the invariants are selected by the presence of a weak forcing and dissipation, what are the associated fluctuations, are all forcings compatible with RSM equilibria ? The relaxation towards equilibrium of 2-D flows has been considered in the past, however the out-of-equilibrium statistical mechanics has never been considered yet. From a statistical mechanics point of view, this problem is a logical continuation of the RSM theory. Out-of-Equilibrium Phase Transitions in Geophysics In many turbulent geophysical flows, one can see transitions, at random time, between two states with different large scale flow. The most famous example are probably the time reversal of the earth magnetic field, or the Milankovitch cylcle for the earth ice age cycles. Such general phenomena correspond to systems with a large number of degrees of freedom. The case of simple turbulent flows may be studied in much details theoreticaly and numericaly. Velocity signal Rotating tank experiment, where a transition from a blocking state to a zonal state is observed. Y. Tian and col, J. Fluid. Mech. (2001) Stream function for both states J. Sommeria, JFM (1986) Left : Amplitude of the first mode vs time for an experi- mental 2D flow, in a square box. Right : Earth (top) and experimental (bottom) mag- netic field reversal (VKS ex- periment). M. Berhanu and col. arxiv:physics/0701076 The Stochastic Navier-Stokes Equation We consider the 2-D Navier-Stokes equation with weak stochastic forcing and dissipation (Euler limit). ∂w ∂t + u.∇w = ν ∆w − αw + f d + f s where f d is deterministic, f s is a random force. This is the usual framework of turbulence studies. How- ever, we are interested on large scales : • We don’t care with self-similar behavior • Because it is relevant for applications, our forcing is not localized in Fourier space • We use very small Rayleigh friction, to observe the large scale energy condensation We study the out-of-equilibrium invariant measure resulting from the statistical balance between forcing and dissipation. Some recent mathematical results : S. Kuksin (Sinai, Shirikyan, Bricmont, Kupianen, ...) : • Existence of a stationary measure μ ν . Existence of lim ν →0 μ ν • In this limit, almost all trajectories are solutions of the Euler equation We would like to obtain more physical results : what determines the large scale flow, is the measure con- centrated close to RSM equilibria, what is the fluctua- tions level, ... ? The forcing : f S (x,t)= ∑ k f k η k (t)e k (x) where e k ’s are the Fourier modes and <η k (t)η k ′ (t ′ ) >= δ k,k ′ δ (t − t ′ ) (white in time). For instance f k = A exp − (|k|−m) 2 2σ 2 with 1 2 ∑ |f k | 2 |k| 2 =1 (smooth in space). Dipole stationary state (vorticity) Out-of-Equilibrium Flows and the RSM Theory 0 0.5 1 1.5 2 2.5 x 10 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Energy/Enstrophy time Stochastic Navier Stokes 2-D periodic (L x /L y = 1.1, ν=5.10 -5 ) Vorticity Vorticity Enstrophy ν=5.10 -5 Stochastic Navier-Stokes 2-D periodic vorticity vorticity Out-of-equilibrium (DNS of Navier Stokes) These two figures help to compare the predictions of equilibrium statis- tical mechanics (conservative) to the Quasi-Stationary large scale flows ob- tained in the Navier-Stokes equation (dissipative) with random forcing. It illustrates that the qualitative prop- erties of the later can be understood using the former. 0 1 2 3 4 5 6 x 10 -3 -2 -1 0 1 2 3 4 Energy (L x /L y = 1.1) Third derivative of Omega(Psi) DIPOLE D x ZONAL Z x DIPOLE D y ZONAL Z y Dipole Dx y Zonal Zx y DIPOLE Dx y Dipole Dx y Equilibrium Phase Diagram The convergence towards equilibrium occurs on a time scale scaling like ν . Vorticity-Streamfunction Relation Are out-of-equilibrium flow close to steady states of the Euler equation ? In order to addrees this, we plot the vorticity-streamfunction relation, which is a curve for Euler equilibria. -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 ω-ψ Streamfunction (ψ) Vorticity ( ω) Dipole (snapshot) -2 -1 0 1 2 -3 -2 -1 0 1 2 ω-ψ Streamfunction (ψ) Vorticity ( ω) Dipole (Temporal average) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 ω-ψ Streamfunction (ψ) Vorticity ( ω) Zonal (unidirectionnal) flow We are close to some conservative steady states. The zonal case seems to show a non- monotonic vorticity-streamfunction relationship. Such states would be uncompatible with RSM equilibria (To be confirmed by computation at lower ν ), Out-of-Equilibrium Phase Transition 4 Γ 0 50 100 150 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Energy/Enstrophy/Γ 4 time*ν Stochastic Navier Stokes 2-D periodic (L x /L y = 1.03, ν=10 -3 ) streamfunction 20 40 60 80 100 120 20 40 60 80 100 120 vorticity 20 40 60 80 100 120 20 40 60 80 100 120 streamfunction 20 40 60 80 100 120 20 40 60 80 100 120 vorticity 20 40 60 80 100 120 20 40 60 80 100 120 Energy Enstrophy Left : Energy (black), enstro- phy (red) and fourth order moment of the vorticity (blue), versus time. This clearly illustrates an out- of-equilibrium dipole-zonal phase transition. 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Energy/Enstrophy/Γ 4 time*ν Stochastic Navier Stokes 2-D periodic (L x /L y = 1.03, ν=10 -3 ) Idem, in a stationary situation In order to study this transition, we study the time evolution of the modulus of the first Fourrier coefficient for the vorticity, |z 1 | (|z 1 | is close to zero for zonal states), for different values of the control parameter (here the aspect ratio of the domain). 0 5 10 15 20 25 30 35 40 45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time*ν |z 1 | Stochastic Navier Stokes 2-D periodic (L x /L y = 1.02, ν=10 -3 ) 0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time*ν |z 1 | Stochastic Navier Stokes 2-D periodic (L x /L y = 1.03, ν=10 -3 ) 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time*ν |z 1 | Stochastic Navier Stokes 2-D periodic (L x /L y = 1.04, ν=10 -3 ) The following are the pdf of |z 1 |, for the threes same values of the control parameter. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 Re(z 1 ) z 1 PDF, ratio=1.02 Im(z 1 ) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 Re(z 1 ) z 1 PDF, ratio=1.03 Im(z 1 ) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 Re(z 1 ) z 1 PDF, ratio=1.04 Im(z 1 ) Conclusion • The large scales of out-of-equilibrium flows are close to Euler steady states. • What is the link between the control parameters (forcing) and the observed flows ? • This requires a theory (the RSM theory only gives a qualitative understanding) • Letting the energy pile up to larger scales may lead to very interesting phenomena • Poorly studied by experimentalists, and poorly studied by numericians • Probably very relevant for geophysical applications (with other models) ANR Statflow This project is funded by the french agency ANR. It is led by a physicist of the statistical mechanics of geophysical flows (F. Bouchet, sec. 02), by a specialist of the numerical stability of 2-D and 3-D flows (F. Gallaire, sec. 10), by a mathematician of geophysi- cal flows and kinetic theory (F. Rousset, sec. 01) and a specialist of the low frequency variability of ocean dynamics (E. Simonnet, sec. 19) F. Bouchet and E. Simonnet are based at INLN (Nice/Sophia Antipolis) and F. Gallaire and F. Rousset at the Nice university.
Transcript of OUT-OF-EQUILIBRIUM PHASE TRANSITIONS FOR THE LARGE...
OUT-OF-EQUILIBRIUM PHASE TRANSITIONS FOR THELARGE SCALES OF 2D TURBULENCE
Freddy BOUCHET, Francois GALLAIRE, Frédéric ROUSSET and Eric SIMONNETINLN-CNRS. ANR JC STATFLOW. Contact : [email protected]
Statistical Mechanics of Large Scale Geophysical Flows
Large scale statistics of turbulent flows
In many applications of fluid dynamics, one of the most important prob-lem is the prediction of the very high Reynolds’ large-scale flows. Thehighly turbulent nature of such flows, for instance ocean circulation or at-mosphere dynamics, renders a probabilistic description desirable, if not nec-essary. A statistical mechanics explanation of the self-organization of geo-physical flows has been proposed by Robert-Sommeria and Miller (RSM).
The RSM theory has been success-fully applied to the Jupiter’s tro-posphere : cyclones, anticyclonesand jets have been quantitativelydescribed by this theory (F. Bouchetand J. Sommeria)
Observation (Voyager) Statistical EquilibriumVelocity field of Jupiter’s Great Red Spot
For applications : statistical mechanics beyond equilibri um ?
The RSM theory starts from the conservative dynamics and parametrizes equilibria by theenergy and other dynamical invariants. However, the theory does not predict the long-term effects of the forcing, which is a relevant issue for any application. It is a practicaland fundamental problem to understand how the invariants are selected by the presenceof a weak forcing and dissipation, what are the associated fluctuations, are all forcingscompatible with RSM equilibria ? The relaxation towards equilibrium of 2-D flows hasbeen considered in the past, however the out-of-equilibrium statistical mechanics hasnever been considered yet. From a statistical mechanics point of view, this problem is alogical continuation of the RSM theory.
Out-of-Equilibrium Phase Transitions in GeophysicsIn many turbulent geophysical flows, one can see transitions, at random time, betweentwo states with different large scale flow. The most famous example are probably thetime reversal of the earth magnetic field, or the Milankovitch cylcle for the earth ice agecycles. Such general phenomena correspond to systemswith a large number of degrees offreedom. The case of simple turbulent flows may be studied in much details theoreticalyand numericaly.
Velocity signal
Rotating tank experiment,where a transition from ablocking state to a zonalstate is observed.Y. Tian and col, J. Fluid. Mech.
(2001)Stream function for both states
J. Sommeria, JFM (1986)
Left : Amplitude of the firstmode vs time for an experi-mental 2D flow, in a squarebox.Right : Earth (top) andexperimental (bottom) mag-netic field reversal (VKS ex-periment). M. Berhanu and col. arxiv:physics/0701076
The Stochastic Navier-Stokes Equation
We consider the 2-D Navier-Stokes equation with weak stochastic forcing and dissipation(Euler limit).
∂w
∂t+ u.∇w = ν∆w − αw + fd + fs
where fd is deterministic, fs is a random force. This is the usual framework of turbulence studies. How-ever, we are interested on large scales :
• We don’t care with self-similar behavior
• Because it is relevant for applications, our forcing is not localized in Fourier space
• We use very small Rayleigh friction, to observe the large scale energy condensation
We study the out-of-equilibrium invariant measure resulting from the statistical balancebetween forcing and dissipation. Some recent mathematical results : S. Kuksin (Sinai,Shirikyan, Bricmont, Kupianen, ...) :
• Existence of a stationary measure µν. Existence of limν→0 µν
• In this limit, almost all trajectories are solutions of the Euler equation
We would like to obtain more physical results : whatdetermines the large scale flow, is the measure con-centrated close to RSM equilibria, what is the fluctua-tions level, ... ?The forcing : fS(x, t) =
∑kfkηk(t)ek(x) where ek’s are the
Fourier modes 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).Dipole stationary state (vorticity)
Out-of-Equilibrium Flows and the RSM Theory
0 0.5 1 1.5 2 2.5
x 105
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Ene
rgy/
Ens
trop
hy
time
Stochastic Navier Stokes 2−D periodic (Lx/L
y = 1.1, ν=5.10−5)
Vorticity
Vorticity
Enstrophy
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Stochastic Navier−Stokes 2−D periodicvorticity
vorticity
Out-of-equilibrium(DNS of Navier Stokes)
These two figures help to comparethe predictions of equilibrium statis-tical mechanics (conservative) to theQuasi-Stationary large scale flows ob-tained in the Navier-Stokes equation(dissipative) with random forcing.It illustrates that the qualitative prop-erties of the later can be understoodusing the former.
0 1 2 3 4 5 6
x 10−3
−2
−1
0
1
2
3
4
Energy (Lx/L
y = 1.1)
Thi
rd d
eriv
ativ
e of
Om
ega(
Psi
)
DIPOLE Dx
ZONAL Zx
DIPOLE Dy
ZONAL Zy
Dipole Dx
x
y
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
Zonal Zy
x
y
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
DIPOLE Dy
x
y
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
Dipole Dx
x
y
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
EquilibriumPhase Diagram
The convergence towards equilibrium occurs on a time scale scaling like ν.
Vorticity-Streamfunction Relation
Are out-of-equilibrium flow close to steady states of the Euler equation ? In order toaddrees this, we plot the vorticity-streamfunction relation, which is a curve for Eulerequilibria.
−2 −1 0 1 2 3
−4
−3
−2
−1
0
1
2
ω−ψ
Streamfunction ( ψ)
Vor
ticity
(ω
)
Dipole (snapshot)
−2 −1 0 1 2
−3
−2
−1
0
1
2
ω−ψ
Streamfunction ( ψ)
Vor
ticity
(ω
)
Dipole (Temporal average)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−1.5
−1
−0.5
0
0.5
1
1.5
ω−ψ
Streamfunction ( ψ)
Vor
ticity
(ω
)
Zonal (unidirectionnal) flowWe are close to some conservative steady states. The zonal case seems to show a non-monotonic vorticity-streamfunction relationship. Such states would be uncompatiblewith RSM equilibria (To be confirmed by computation at lower ν),
Out-of-Equilibrium Phase Transition
4Γ0 50 100 150
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Ene
rgy/
Ens
trop
hy/
Γ 4
time* ν
Stochastic Navier Stokes 2−D periodic (Lx/L
y = 1.03, ν=10−3)
streamfunction20 40 60 80 100 120
20
40
60
80
100
120
vorticity20 40 60 80 100 120
20
40
60
80
100
120
streamfunction20 40 60 80 100 120
20
40
60
80
100
120
vorticity20 40 60 80 100 120
20
40
60
80
100
120
Energy
Enstrophy
Left : Energy (black), enstro-phy (red) and fourth ordermoment of the vorticity(blue), versus time. Thisclearly illustrates an out-of-equilibrium dipole-zonalphase transition.
0 10 20 30 40 50 60 70 800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Ene
rgy/
Ens
trop
hy/
Γ 4
time* ν
Stochastic Navier Stokes 2−D periodic (Lx/L
y = 1.03, ν=10−3)
Idem, in a stationarysituation
In order to study this transition, we study the time evolution of the modulus of thefirst Fourrier coefficient for the vorticity, |z1| (|z1| is close to zero for zonal states),for different values of the control parameter (here the aspect ratio of the domain).
0 5 10 15 20 25 30 35 40 450
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time* ν
|z1|
Stochastic Navier Stokes 2−D periodic (Lx/L
y = 1.02, ν=10−3)
0 10 20 30 40 50 60 70 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time* ν
|z1|
Stochastic Navier Stokes 2−D periodic (Lx/L
y = 1.03, ν=10−3)
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time* ν
|z1|
Stochastic Navier Stokes 2−D periodic (Lx/L
y = 1.04, ν=10−3)
The following are the pdf of |z1|, for the threes same values of the control parameter.
−0.6−0.4
−0.20
0.20.4
0.6
−0.6−0.4
−0.20
0.20.4
0.6
0
0.2
0.4
0.6
0.8
1
Re(z1)
z1 PDF, ratio=1.02
Im(z1)
−0.6−0.4
−0.20
0.20.4
0.6
−0.6−0.4
−0.20
0.20.4
0.6
0
0.2
0.4
0.6
0.8
1
Re(z1)
z1 PDF, ratio=1.03
Im(z1)
−0.6−0.4
−0.20
0.20.4
0.6
−0.6−0.4
−0.20
0.20.4
0.6
0
0.2
0.4
0.6
0.8
1
Re(z1)
z1 PDF, ratio=1.04
Im(z1)
Conclusion
• The large scales of out-of-equilibrium flows are close to Euler steady states.
• What is the link between the control parameters (forcing) and the observed flows ?
• This requires a theory (the RSM theory only gives a qualitative understanding)
• Letting the energy pile up to larger scales may lead to very interesting phenomena
• Poorly studied by experimentalists, and poorly studied by numericians
• Probably very relevant for geophysical applications (with other models)
ANR Statflow
This project is funded by the french agency ANR. It is led by a physicist of the statisticalmechanics of geophysical flows (F. Bouchet, sec. 02), by a specialist of the numericalstability of 2-D and 3-D flows (F. Gallaire, sec. 10), by a mathematician of geophysi-cal flows and kinetic theory (F. Rousset, sec. 01) and a specialist of the low frequencyvariability of ocean dynamics (E. Simonnet, sec. 19)F. Bouchet and E. Simonnet are based at INLN (Nice/Sophia Antipolis) and F. Gallaireand F. Rousset at the Nice university.
