pattern formation in rayleigh-bénard convectionphysbio/activities/lectures/cross/...b) dislocations...

Back Forward

Benasque PHYSBIO 2003:Pattern Formation in Rayleigh-Bénard Convection - Lecture 3 1

Pattern Formation inRayleigh-Bénard Convection

Lecture 3

Chaos

Back Forward


Outline

• Low dimensional chaos

• Pattern Chaos

• Spatiotemporal chaos

Back Forward


Outline

• Low dimensional chaos

• Pattern Chaos

• Spatiotemporal chaos

Collaborators: Marc Bourzutschky, Keng-Hwee Chiam, Henry Greenside,

Roman Grigoriev, Pierre Hohenberg, Michael Louie, Dan Meiron, Mark

Paul, Yuhai Tu.

Support: NSF and DOE

Back Forward


Lorenz Chaos

H o t

C o l d

x

z

T (x, z, t) ' −rz

Back Forward


Lorenz Chaos

H o t

C o l d

x

z

T (x, z, t) ' −rz+ 9π3√

3Y (t) cos(πz) cos

(π√2x

)

Back Forward


Lorenz Chaos

H o t

C o l d

x

z

T (x, z, t) ' −rz+ 9π3√

3Y (t) cos(πz) cos

(π√2x

)+ 27π3

4Z(t) sin(2πz)

Back Forward


Lorenz Chaos

H o t

C o l d

x

z

T (x, z, t) ' −rz+ 9π3√

3Y (t) cos(πz) cos

(π√2x

)+ 27π3

4Z(t) sin(2πz)

ψ(x, z, t) = 2√

6X(t) cos(πz) sin

(π√2x

)

Back Forward


Lorenz Chaos

H o t

C o l d

x

z

T (x, z, t) ' −rz+ 9π3√

3Y (t) cos(πz) cos

(π√2x

)+ 27π3

4Z(t) sin(2πz)

ψ(x, z, t) = 2√

6X(t) cos(πz) sin

(π√2x

)

(u = −∂ψ/∂z, v = ∂ψ/∂x, r = R/Rc)

Back Forward


Lorenz Model

X = −σ(X − Y )Y = rX − Y −XZZ = b (XY − Z)

b = 8/3 andσ is the Prandtl number.

“Classic” values areσ = 10 andr = 27.

Back Forward


-10-5

05

10

X

-10

0

10

Y

0

10

20

30

40

Z

-50

510

X

Back Forward


The Butterfly Effect

The “sensitive dependence on initial conditions” found by Lorenz is often

called the “butterfly effect”.

In fact Lorenz first said (Transactions of the New York Academy of

Sciences, 1963)

One meteorologist remarked that if the theory were correct, one

flap of the sea gull’s wings would be enough to alter the course

of the weather forever.

By the time of Lorenz’s talk at the December 1972 meeting of the

American Association for the Advancement of Science in Washington,

D.C. the sea gull had evolved into the more poetic butterfly - the title of

his talk was

Predictability: Does the Flap of a Butterfly’s Wings in Brazil set

off a Tornado in Texas?

Back Forward


The Lorenz model does not describe Rayleigh-Bénard convection!

Thermosyphon Rayle igh-Benard Convect ion

Also: Rititake homopolar dynamo

Back Forward


Small system chaos

Back Forward


Theoretical Context

Landau (1944) Turbulence develops by infinite sequence of transitions

adding additional temporal modes and spatial complexity

Ruelle and Takens (1971)Suggested the onset of aperiodic dynamics

from a low dimensional torus (quasiperiodic motion with a small

numberN frequencies)

Feigenbaum (1978)Quantitative universality for period doubling route

to chaos

Back Forward


Some Experimental Highlights

Ahlers (1974) Transition from time independent flow to aperiodic flow at

R/Rc ∼ 2 (aspect ratio 5)

Gollub and Swinney (1975)Onset of aperiodic flow from time-periodic

flow in Taylor-Couette

Maurer and Libchaber, Ahlers and Behringer (1978) Transition from

quasiperiodic flow to aperiodic flow in small aspect ratio convection

Lichaber, Laroche, and Fauve (1982)Quantitative demonstration of the

Fiegenbaum period doubling route to chaos

Back Forward


Larger System Chaos

In “large” systems presumably the chaos becomes higher dimensional.

We might divide up the phenomena into two classes:

Pattern chaos global properties of spatial patterns are important, perhaps

induced by boundary effects

Spatiotemporal chaosstatistical aspects of local units are more

important (0→∞)

In either case we might be interested in

• the mechanism of the dynamics

• characterizing the dynamics

Back Forward


Pattern chaos: convection in intermediate aspect ratiocylinders

G. Ahlers, Phys. Rev. Lett.30, 1185 (1974)

G. Ahlers and R.P. Behringer, Phys. Rev. Lett.40, 712 (1978)

H. Gao and R.P. Behringer, Phys. Rev. A30, 2837 (1984)

V. Croquette, P. Le Gal, and A. Pocheau, Phys. Scr. T13, 135 (1986)

Back Forward


• First experiments:0 = 5.27 cell, cryogenic (normal) liquidHe4 as

fluid. High precision heat flow measurements (no flow visualization).

• Onset of aperiodic time dependence in low Reynolds number flow:

relevance of chaos to “real” (continuum) systems.

• Power law decrease of power spectrumP(f ) ∼ f−4

• Aspect ratio dependence of the onset of time dependence

0 2 5 57

Rt 10Rc 2Rc 1.1Rc

• Flow visualization (Argon,0 = 7.66)

Back Forward


(from Ahlers 1974)

Back Forward


(from Ahlers and Behringer 1978)

Back Forward


Gao and Behringer (1984)

Back Forward


0

0.005

0.010

0.015

0 1 2 3Reduced Rayleigh Number ε

Noi

se A

mpl

itude

(from Ahlers and Behringer 1978)

Back Forward


Croquette, Le Gal, and Pocheau (1986)

Back Forward


Numerical Simulations

(Paul, MCC, Fischer and Greenside)

• 0 = 4.72,σ = 0.78, 2600. R . 7000

• Conducting sidewalls

• Random thermal perturbation initial conditions

• Simulation time∼ 100τh

– Simulation time∼ 12 hours on 32 processors

– Experiment time∼ 172 hours or∼ 1 week

Back Forward


0 500 1000 1500 2000time

1.4

1.5

1.6

1.7

1.8

1.9

2

Nu

R = 6949R = 4343R = 3474R = 3127R = 2804R = 2606

50 τh

Γ = 4.72σ = 0.78 (Helium)Random Initial Conditions

Back Forward


R = 3127(?) R = 6949

Back Forward


Power Spectrum

• Simulations oflow dimensional chaos(e.g. Lorenz model) show

exponential decaying power spectrum

• Power law power spectrum easily obtained fromstochasticmodels

(white-noise driven oscillator, etc.)

Deterministic Chaos ?⇒? Exponential decay

Stochastic Noise ?⇒? Power law decay

Back Forward


Simulation Results

Simulations reproduce experimental results....

Back Forward


Simulation yields a power law over the range accessible to experiment

10-2 10-1 100 101 102

ω10-12

10-11

10-10

10-9

10-8

10-7

10-6

P(ω

)

R = 6949ω-4

Γ = 4.72, σ = 0.78, R = 6949Conducting sidewalls

R/Rc = 4.0

Back Forward


When larger frequencies are included an exponential tail is found

10-2 10-1 100 101 102

ω10-21

10-19

10-17

10-15

10-13

10-11

10-9

10-7

10-5P

(ω)

R = 6949ω-4

Γ = 4.72, σ = 0.78, R = 6949Conducting sidewalls

R/Rc = 4.0

Back Forward


1 2 3 4 5 6 7ν (1/τv)

10-20

10-18

10-16

10-14

10-12

10-10

P(ν

)

R = 2804, σ = 0.78, Γ = 4.72Conducting sidewalls

Exponential region, slope = -3.8

Exponential tail not seen in experiment because of instrumental noise floor

Back Forward


Where does the power law come from?

Power law arises fromquasi-discontinuous changes in the slopeof N(t)

on at = 0.1− 1 time scale associated with roll pinch-off events.

This is clearest to see for the low Rayleigh number where the motion is

periodic , but again the power spectrum has a power law fall off:

• Spectrogram

• Details of time seriesN(t)

• Windowed power spectrum

• Compare periodic and chaotic events

Back Forward


Periodic dynamicsR = 2804

Back Forward


Power law spectra

10-3 10-2 10-1 100 101

ν (1/τv)10-25

10-23

10-21

10-19

10-17

10-15

10-13

10-11

10-9

10-7

P(ν

)

R = 2804ν-4

0 0.025 0.05 0.075 0.1ν (1/τv)

0

2E-07

4E-07

6E-07

8E-07

1E-06

1.2E-06

1.4E-06

P(ν

)

Γ = 4.72, R = 2804, σ = 0.78Conducting sidewallsNo overlappingLinear detrending

Back Forward


Spectrogram

Back Forward


Time seriesN(t)

600 700time

1.4

1.405

1.41

1.415

1.42

1.425N

u

a) dislocation nucleation

b) dislocations climb

c) both dislocations areat the lateral walls

one dislocation quickly glidesinto a wall foci e) the other dislocation slowly glides

into the same wall focisecondannihilation

Note: Dislocation glide alternates left and right,the portion highlighted here goes right.

d) first annihilation

R = 2804, Γ = 4.72, σ = 0.78Conducting

f) process repeats

t = 187 = 8.4 τh

Back Forward


Windowed power spectra

⟨P(ν)⟩

10-1 100 101

ν

10-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

P(ν

)t = 594, Nucleationt = 639, Annihilationt = 700, Glideν-4

Back Forward


Compare periodic and chaotic events

770 775 780 785 790time

1.395

1.4

1.405

1.41

1.415

1.42

1.425

Nu


270.5 271 271.5 272time

1.85

1.86

1.87

1.88

1.89

1.9

1.91

1.92

Nu


Back Forward


Mechanism of Dynamics

• Pan-Am texture with roll pinch-off events creating dislocation pairs

(Flow visualization, Pocheau, Le Gal, and Croquette 1985)

• Mean flow compresses rolls outside of stable band (Model equations,

Greenside, MCC, Coughran 1985)

• Theoretical analysis of mean flow (Pocheau and Davidaud 1997)

• Numerical simulation of importance of mean flow (Paul, MCC,

Fischer, and Greenside 2001)

Back Forward


Generalized Swift-Hohenberg Simulations

Back Forward


Convection Simulations

3 convection cells with different side wall conditions: (a) rigid; (b) finned;

and (c) ramped. Case (a) is dynamic, the others static.

Back Forward


Spatiotemporal chaos

Back Forward



Might be defined as the dynamics, disordered in time and space, of a large

aspect ratio system (i.e. one that is large compared to the size of a basic

chaotic element)

Natural examples

• The atmosphere and ocean (weather, climate etc.)

• Heart fibrillation

Spatiotemporal chaos is a new paradigm of unpredictable dynamics.

(What effect does the butterfly really have?)

Back Forward


Spatiotemporal chaos and turbulence both correspond toL→∞

Define length scales:

• Energy injection scaleLI

• Energy dissipation scaleLD

• System sizeL

Spatiotemporal chaosL >> LD ∼ LITurbulence L ∼ LI >> LD

“Spatiotemporal turbulence” L >> LI >> LD

Back Forward


Spatiotemporal Chaos v. Turbulence

EnergyInjection

EnergyDissipation

ÿ�

ÿ�

Turbulence

Spatiotemporal Chaos

Spatiotemporal Turbulence

ÿ

Back Forward


Systems

• Coupled Maps

x(n+1)i = f (x(n)i )+ g × (x(n)i−1− 2x(n)i + x(n)i+1)

• PDE simulations

– Complex Ginzburg-Landau Equation

∂tA = A+ (1+ ic1)∇2A− (1− ic3) |A|2A

– Kuramoto-Sivashinsky equation

∂tu = −∂2xu− ∂4

xu− u∂xu

• Physical systems (experiment and numerics)

Back Forward


Examples from convection

T+∆T

T

• Spiral Defect Chaos ( experiment , simulations )

• Domain Chaos (model simulations: stripes , orientations , walls ;

convection simulations )

Back Forward


Issues

• System-specific questions

• Definition and characterization

– How to narrow the phenomena

– How to decide if theory, simulation, and experiment match

• Transitions to spatiotemporal chaos and between different chaotic

states

• Thermodynamics and hydrodynamics (cf. Granular Flows)

• Control

Back Forward


Issues

• System-specific questions

• Definition and characterization

– How to narrow the phenomena

– How to decide if theory, simulation, and experiment match

• Transitions to spatiotemporal chaos and between different chaotic

states

• Thermodynamics and hydrodynamics (cf. Granular Flows)

• Control

Ideas and methods from dynamical systems, statistical mechanics, phase

transition theory …

Back Forward


Spatiotemporal Chaos in RotatingRayleigh-Bénard Convection

Back Forward


Ray

leig

h N

umbe

r

stripes(convect ion)

R C

Back Forward


Rotat ion Rate

Ray

leig

h N

umbe

r

no pattern(conduction)


R C

Back Forward


Rotat ion Rate

Ray

leig

h N

umbe

r



KL

R C

Back Forward


Rotat ion Rate

Ray

leig

h N

umbe

r



KL

domainchaos

R C

Back Forward


Rotat ion Rate

Ray

leig

h N

umbe

r



KL

spiralchaos

domainchaos

R C

Back Forward


Rotat ion Rate

Ray

leig

h N

umbe

r



KL

spiralchaos

lockeddomainchaos

(?)

domainchaos

R C

Back Forward


Onset of Domain Chaos(Tu and MCC)

Back Forward



Amplitudes of rolls at 3 orientationsAi(r , t), i = 1 . . .3

Back Forward




∂tA1 = εA1

Back Forward




∂tA1 = εA1+ ∂2x1A1

Back Forward




∂tA1 = εA1+ ∂2x1A1− A1(A

21

Back Forward




∂tA1 = εA1+ ∂2x1A1− A1(A

21+ g+A2

2+ g−A23)

Back Forward




∂tA1 = εA1+ ∂2x1A1− A1(A

21+ g+A2

2+ g−A23)

∂tA2 = εA2+ ∂2x2A2− A2(A

22+ g+A2

3+ g−A21)

∂tA3 = εA3+ ∂2x3A3− A3(A

23+ g+A2

1+ g−A22)

whereε = (R − Rc(�)/Rc(�)

Back Forward




∂tA1 = εA1+ ∂2x1A1− A1(A

21+ g+A2

2+ g−A23)

∂tA2 = εA2+ ∂2x2A2− A2(A

22+ g+A2

3+ g−A21)

∂tA3 = εA3+ ∂2x3A3− A3(A

23+ g+A2

1+ g−A22)

whereε = (R − Rc(�)/Rc(�)Rescale space, time, and amplitudes:

Back Forward


RescaleX = ε1/2x, T = εt, A = ε−1/2A

∂T A1 = A1+ ∂2X1A1− A1(A

21+ g+A2

2+ g−A23)

∂T A2 = A2+ ∂2X2A2− A2(A

22+ g+A2

3+ g−A21)

∂T A3 = A3+ ∂2X3A3− A3(A

23+ g+A2

1+ g−A22)

Back Forward


RescaleX = ε1/2x, T = εt, A = ε−1/2A

∂T A1 = A1+ ∂2X1A1− A1(A

21+ g+A2

2+ g−A23)

∂T A2 = A2+ ∂2X2A2− A2(A

22+ g+A2

3+ g−A21)

∂T A3 = A3+ ∂2X3A3− A3(A

23+ g+A2

1+ g−A22)

Numerical simulations show chaotic dynamics

Therefore in unscaled (physical) units

Length scale ξ ∼ ε−1/2

Time scale τ ∼ ε−1

Back Forward


Experimental results

Hu et al. Phys. Rev. Lett.74, 5040 (1995)

Back Forward


0 0.1 0.2ε

100/

ξ2

0

1

2

3

4

5

Back Forward


Effect of finite size

Hypothesis: deviations from the predicted scaling behavior comes from

the finite size of the experimental system (MCC, Louie, and Meiron 2001)

Use numerical simulations to test a finite size scaling ansatz

ξM = ξf (0/ξ), ξ = ξ0ε−1/2

Generalized Swift-Hohenberg equation (real field of two spatial

dimensionsψ(x, y; t))∂ψ

∂t= εψ + (∇2+ 1)2ψ − g1ψ

3

+ g2z·∇ × [(∇ψ)2∇ψ ] + g3∇·[(∇ψ)2∇ψ ]

Back Forward


Variation withε

Back Forward


Variation with system size

Back Forward


Calculation of domain size

qx

qy

ξΜ

Back Forward


Domain size v.ε

ε

1/ξ Μ

00

Γ=30πΓ=40πΓ=50πΓ=80π

0.1 0.2 0.3

0.02

0.04

0.06

Back Forward


Finite size scaling

0

Γ=30πΓ=40πΓ=50πΓ=80π

ε−1/2/Γ

0 0.05 0.10

0.1

0.2

0.3

ε−1/2

/ξΜ

ξM = ξf (0/ξ), ξ = ξ0ε−1/2

Back Forward


Characterizing Spatiotemporal Chaos

Methods from statistical physics: Correlation lengths and times, etc.

Methods from dynamical systems:Lyapunov exponents and attractor

dimensions.

Quantifies sensitive dependence on initial conditions.

Back Forward


Lyapunov Exponent

t0

t1 t2

t3

tf

δu0

δufX

Y

Z

λ = lim tf→∞1

tf−t0 ln∣∣∣δufδu0

∣∣∣

Back Forward


Lyapunov vector for spiral defect chaos

(from Keng-Hwee Chiam, Caltech thesis 2003)

Back Forward


Lyapunov spectrum for spiral defect chaos

(from Egolf et al. 2000)

Back Forward


Dimensions

Most definitions of fractal dimension involve the geometry of the chaotic

attractor in phase space, which is inaccessible in spatiotemporal chaos.

TheLyapunov dimension:

• is accessible to simulations (but probably not to experiment)

• is usually equal to the information dimension

• is expected to be extensive for large system sizes (Ruelle, 1982)

Back Forward


Lyapunov dimension

Defineµ(n) =∑ni=1 λi (λ1 ≥ λ2 · · · ) with λi theith Lyapunov

exponent.

DL is the interpolated value ofn givingµ = 0 (the dimension of the

volume that neither grows nor shrinks under the evolution)

Back Forward


Lyapunov dimension for spiral defect chaos

(Egolf et al. 2000)

Back Forward


Microextensive scaling for the 1d Kuramoto-Sivashinskyequation

78 83 88 93L

15

16

17

18

Lyap

unov

Dim

ensi

on

(from Tajima and Greenside 2000)

Back Forward


Microextensive scaling

Interesting questions:

• Are there (tiny) windows of periodic orbits or chaotic orbits of

non-scaling dimension so that smooth variation is only in theL→∞limit, or is the variation smooth at finiteL

• Can we use this to define spatiotemporal chaos for finiteL?

• Does spatiotemporal chaos in Rayleigh-Bénard Convection show

microextensive scaling of the Lyapunov dimension?

Back Forward


Conclusions

Small system chaos

• Lorenz model

• experimental demonstration of chaos in continuum systems

Pattern chaos

• aperiodic flow due to pattern dynamics

• power law tail in spectrum due to defect nucleation events


• new paradigm for unpredictable dynamics (cf. chaos, turbulence)

• quantitative predictions for domain chaos in rotating convection, but discrepancies

with experiment remain

• Lyapunov exponent, vector and dimension

Back Forward


References

Small system chaos

E. Lorenz J. Atmos. Sciences,20, 130 (1963)

M. Gorman, P. J. Widmann, and K. A. Robbins (thermosyphon) Physica19D, 255 (1986);36D, 157(1989)

L. D. Landau Akad. Nauk. Dok.44 339 (1944) (Also in early editions ofFluid Mechanicsby Landau andLifshitz.)

D. Ruelle and F. TakensCommun. Math. Phys.20 167 (1971)

M. J. Feigenbaum J. Stat. Phys.19 25 (1978)

G. Ahlers Phys. Rev. Lett.30, 1185 (1974)

J. P. Gollub and H. L. Swinney Phys. Rev. Lett.35927 (1975)

A. Libchaber and J. Maurer J. Phys. Lett. Paris39 L369 (1978)

G. Ahlers and R.P. Behringer Phys. Rev. Lett.40, 712 (1978)

A. Libchaber, C. Laroche, and S. FauveJ. Phys. Lett. Paris43L211 (1982)

Pattern Chaos

G. Ahlers Phys. Rev. Lett.30, 1185 (1974)

Back Forward


G. Ahlers and R.P. Behringer Phys. Rev. Lett.40, 712 (1978)

H. Gao and R.P. Behringer Phys. Rev. A30, 2837 (1984)

V. Croquette, P. Le Gal, and A. PocheauPhys. Scr. T13, 135 (1986)

M. R. Paul, M. C. Cross, P. F. Fischer, and H.§. GreensidePhys. Rev. Lett.87 154501 (2001)

H. S. Greenside, M. C. Cross, and W. M. CoughranPhys. Rev. Lett.60 2269 (1988)

A. Pocheau and F. DaviaudPhys. Rev.E55353 (1997)


Domain Chaos

Y. Tu and M. C. Cross Phys. Rev. Lett.69 2515 (1992)

Y. Hu, R. E. Ecke, and G. Ahlers Phys. Rev. Lett.74 5040 (1995)

M. C. Cross, M. Louie, and D. Meiron Phys. Rev.E6345201 (2001)

Lyapunov exponents and dimension

D. A. Egolf, I. V. Melnikov, W. Pesch, and R. E. EckeNature404736 (2000)

S. Tajima and H. S. GreensidePhys. Rev.E66 017205 (2002)

pattern formation in rayleigh-bénard convectionphysbio/activities/lectures/cross/...b) dislocations...

Documents