Pattern Formation and Spatiotemporal Chaos inSystems Far from Equilibrium

Michael Cross

California Institute of TechnologyBeijing Normal University

May 2006

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 1 / 57

Outline

1 Introduction

2 Rayleigh-Bénard Convection

3 Pattern FormationBasic QuestionNonlinearity

4 Spatiotemporal ChaosWhat is it?Spatiotemporal Chaos in Rayleigh-Bénard ConvectionTheory, Experiment, and Simulation

5 Conclusions

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 2 / 57

An Open System

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 3 / 57

Patterns in Geophysics

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 4 / 57

Patterns in Biology

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 5 / 57

Pattern Formation

The spontaneous formation of spatial structure in opensystems driven far from equilibrium

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 6 / 57

Origins of Pattern Formation

Fluid Instabilities

1900 Bénard’s experimentson convection in a dish of fluidheated from below and with a free surface

1916 Rayleigh’s theoryexplaining the formation of convectionrolls and cells in a layer of fluid with rigid top and bottomplates and heated from below

…

Chemical Instabilities

1952 Turing’s suggestionthat instabilities in chemical reactionand diffusion equations might explain morphogenesis

1950+ Belousov and Zhabotinskii workon chemical reactionsshowing oscillations and waves

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 7 / 57

Bénard’s Experiments

(From the website of Carsten Jäger)Movie

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 8 / 57

Ideal Hexagonal Pattern

From the website of Michael Schatz

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 9 / 57

Rayleigh’s Stability Analysis

Rayleigh made two simplifications:

In the present problem the case is much more complicated, unless wearbitrarily limit it to two dimensions. The cells of Bénard are then reducedto infinitely long strips, and when there is instability we may ask for whatwavelength (width of strip) the instability is greatest.

…and we have to consider boundary conditions. Those have been chosenwhich aresimplest from the mathematical point of view, and they deviatefrom those obtaining in Bénard’s experiment, where, indeed, the conditionsare different at the two boundaries.

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 10 / 57

Rayleigh’s Stability Analysis

Rayleigh made two simplifications:

In the present problem the case is much more complicated, unless wearbitrarily limit it to two dimensions. The cells of Bénard are then reducedto infinitely long strips, and when there is instability we may ask for whatwavelength (width of strip) the instability is greatest.

…and we have to consider boundary conditions. Those have been chosenwhich aresimplest from the mathematical point of view, and they deviatefrom those obtaining in Bénard’s experiment, where, indeed, the conditionsare different at the two boundaries.

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 10 / 57

Rayleigh and his Solution

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 11 / 57

Schematic of Instability

Fluid

Rigid plate

Rigid plate

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 12 / 57

Schematic of Instability

Fluid

Rigid plate

Rigid plate

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 13 / 57

Schematic of Instability

HOT

COLD

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 14 / 57

Schematic of Instability

HOT

COLD

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 15 / 57

Schematic of Instability

HOT

COLD

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 16 / 57

Schematic of Instability

HOT

COLD

2π/q

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 17 / 57

Rayleigh’s Solution

Linear stability analysis:

linear instability towards two dimensional mode with wave numberq

exponential time dependence with growth/decay rateσ(q)

Two important parameters:

Rayleigh numberR ∝ 1T (ratio of buoyancy to dissipative forces)

Prandtl numberP, a property of the fluid (ratio of viscous and thermaldiffusivities). For a gasP ∼ 1, for oil P ∼102, for mercuryP ∼ 10−2.

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 18 / 57

Rayleigh’s Solution

Linear stability analysis:

linear instability towards two dimensional mode with wave numberq

exponential time dependence with growth/decay rateσ(q)

Two important parameters:

Rayleigh numberR ∝ 1T (ratio of buoyancy to dissipative forces)

Prandtl numberP, a property of the fluid (ratio of viscous and thermaldiffusivities). For a gasP ∼ 1, for oil P ∼102, for mercuryP ∼ 10−2.

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 18 / 57

Rayleigh’s Growth Rate

σq/π

21

R=0.5 Rc

Rc = 27π4

4, qc = π√

2

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 19 / 57

Rayleigh’s Growth Rate

σq/π

21

R=Rc q = qc

Rc = 27π4

4, qc = π√

2

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 20 / 57

Rayleigh’s Growth Rate

σq/π

21

R=1.5 Rc

q = qc

Rc = 27π4

4, qc = π√

2

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 21 / 57

Rayleigh’s Growth Rate

σq/π

21

R=1.5 Rc

q = qc

q+q−

Rc = 27π4

4, qc = π√

2

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 22 / 57

Outline

1 Introduction

2 Rayleigh-Bénard Convection

3 Pattern FormationBasic QuestionNonlinearity

4 Spatiotemporal ChaosWhat is it?Spatiotemporal Chaos in Rayleigh-Bénard ConvectionTheory, Experiment, and Simulation

5 Conclusions

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 23 / 57

Pattern Formation: the Question

qx

qy q+

q−

σ > 0 qc

What spatial structures can be formed from the growth and saturation of theunstable modes?

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 24 / 57

Ideal Patterns from Experiment

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 25 / 57

More Patterns From Experiment

From the website of EberhardBodenschatz

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 26 / 57

What is Hard about Pattern Formation?

Why are we still working on this 50 or 100 years later?

The analysis of Rayleigh, Taylor, and Turing was largelylinear,and gaveinteresting, but in the end unphysical solutions.

To understand the resulting patterns we need to understandnonlinearity.

One would like to be able to follow this more general [nonlinear]process mathematically also. The difficulties are, however, such thatone cannot hope to have any very embracing theory of such processes,beyond the statement of the equations. It might be possible, however,to treat a few particular cases in detail with the aid of adigital computer. [Turing]

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 27 / 57

What is Hard about Pattern Formation?

Why are we still working on this 50 or 100 years later?

The analysis of Rayleigh, Taylor, and Turing was largelylinear,and gaveinteresting, but in the end unphysical solutions.

To understand the resulting patterns we need to understandnonlinearity.

One would like to be able to follow this more general [nonlinear]process mathematically also. The difficulties are, however, such thatone cannot hope to have any very embracing theory of such processes,beyond the statement of the equations. It might be possible, however,to treat a few particular cases in detail with the aid of adigital computer. [Turing]

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 27 / 57

What is Hard about Pattern Formation?

Why are we still working on this 50 or 100 years later?

The analysis of Rayleigh, Taylor, and Turing was largelylinear,and gaveinteresting, but in the end unphysical solutions.

To understand the resulting patterns we need to understandnonlinearity.

One would like to be able to follow this more general [nonlinear]process mathematically also. The difficulties are, however, such thatone cannot hope to have any very embracing theory of such processes,beyond the statement of the equations. It might be possible, however,to treat a few particular cases in detail with the aid of adigital computer. [Turing]

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 27 / 57

Outline

1 Introduction

2 Rayleigh-Bénard Convection

3 Pattern FormationBasic QuestionNonlinearity

4 Spatiotemporal ChaosWhat is it?Spatiotemporal Chaos in Rayleigh-Bénard ConvectionTheory, Experiment, and Simulation

5 Conclusions

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 28 / 57

Lorenz Model (1963)

HOT

COLD

X

Z

Y

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 29 / 57

Equations of Motion

X = −P(X − Y)

Y = r X − Y − X ZZ = b (XY − Z)

(whereX = d X/dt, etc.).

The equations give the “velocity”V = (X, Y, Z) of the pointX = (X, Y, Z) inthephase space

r = R/Rc, b = 8/3 andP is the Prandtl number.Lorenz usedP = 10 andr = 27.

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 30 / 57

Linear Equations

X = −P(X − Y)

Y = r X − YZ = −bZ

Solution (P = 1)

X = a1e(√

r −1)t + a2e−(√

r +1)t

Y = a1√

re(√

r −1)t − a2√

re−(√

r +1)t

Z = a3e−bt

with a1, a2, a3 determined by initial conditions.

This is exactly Rayleigh’s solution with an instability atr = 1

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 31 / 57

Nonlinear Equations

-10-5

05

10

X

-10

0

10

Y

0

10

20

30

40

Z

-50

510

X

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 32 / 57

Outline

1 Introduction

2 Rayleigh-Bénard Convection

3 Pattern FormationBasic QuestionNonlinearity

4 Spatiotemporal ChaosWhat is it?Spatiotemporal Chaos in Rayleigh-Bénard ConvectionTheory, Experiment, and Simulation

5 Conclusions

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 33 / 57

Spatiotemporal Chaos

Definitionsdynamics, disordered in time and space, of a large, uniform systemcollective motion of many chaotic elementsbreakdown of pattern to dynamics

Natural examples:atmosphere and ocean (weather, climate etc.)arrays of nanomechanical oscillatorsheart fibrillation

Cultured monolayers of cardiac tissue (from Gil Bub, McGill)

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 34 / 57

Outline

1 Introduction

2 Rayleigh-Bénard Convection

3 Pattern FormationBasic QuestionNonlinearity

4 Spatiotemporal ChaosWhat is it?Spatiotemporal Chaos in Rayleigh-Bénard ConvectionTheory, Experiment, and Simulation

5 Conclusions

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 35 / 57

Spiral Chaos in Rayleigh-Bénard Convection

…and from experiment

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 36 / 57

Domain Chaos in Rayleigh-Bénard Convection

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 37 / 57

Rotating Rayleigh-Bénard Convection

Ω

Rotate

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 38 / 57

Spiral and Domain Chaos in Rayleigh-Bénard Convection

Rotation Rate

Ray

leig

h N

umbe

r

no pattern(conduction)

stripes(convection)

KL Instability

ÿ

stripes unstable

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 39 / 57

Spiral and Domain Chaos in Rayleigh-Bénard Convection

Rotation Rate

Ray

leig

h N

umbe

r

no pattern(conduction)

stripes(convection)

KL

spiralchaos

domainchaos

ÿ

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 40 / 57

Outline

1 Introduction

2 Rayleigh-Bénard Convection

3 Pattern FormationBasic QuestionNonlinearity

4 Spatiotemporal ChaosWhat is it?Spatiotemporal Chaos in Rayleigh-Bénard ConvectionTheory, Experiment, and Simulation

5 Conclusions

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 41 / 57

Theory, Experiment, and Simulation

ComplexSystem

Experiment

Theory

Simulations

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 42 / 57

Summary of Results

Theory predicts

Length scale ξ ∼ ε−1/2

Time scale τ ∼ ε−1

Velocity scale v ∼ ε1/2

with ε = (R − Rc(Ä))/Rc(Ä)

Numerical Tests

model equationsXfull fluid dynamic simulationsX

Experiment× (but now we understand why)

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 43 / 57

Theory for Domain Chaos: Amplitudes

Rayleigh, Turing etc., found solutions

u = u0eσ t cos(qx) . . . = A(t) cos(qx) . . .

so that in the linear approximation and forR nearRc

d A

dt= σ A with σ ∝ ε = R − Rc

Rc

Useε as small parameter in expansion about threshold

Nonlinear saturationd A

dt= (ε − A2)A

Spatial variation∂ A

∂t= εA − A3 + ∂2A

∂x2

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 44 / 57

Theory for Domain Chaos: Amplitudes

Rayleigh, Turing etc., found solutions

u = u0eσ t cos(qx) . . . = A(t) cos(qx) . . .

so that in the linear approximation and forR nearRc

d A

dt= σ A with σ ∝ ε = R − Rc

Rc

Useε as small parameter in expansion about threshold

Nonlinear saturationd A

dt= (ε − A2)A

Spatial variation∂ A

∂t= εA − A3 + ∂2A

∂x2

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 44 / 57

Theory for Domain Chaos: Amplitudes

Rayleigh, Turing etc., found solutions

u = u0eσ t cos(qx) . . . = A(t) cos(qx) . . .

so that in the linear approximation and forR nearRc

d A

dt= σ A with σ ∝ ε = R − Rc

Rc

Useε as small parameter in expansion about threshold

Nonlinear saturationd A

dt= (ε − A2)A

Spatial variation∂ A

∂t= εA − A3 + ∂2A

∂x2

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 44 / 57

Theory for Domain Chaos: Amplitudes

Rayleigh, Turing etc., found solutions

u = u0eσ t cos(qx) . . . = A(t) cos(qx) . . .

so that in the linear approximation and forR nearRc

d A

dt= σ A with σ ∝ ε = R − Rc

Rc

Useε as small parameter in expansion about threshold

Nonlinear saturationd A

dt= (ε − A2)A

Spatial variation∂ A

∂t= εA − A3 + ∂2A

∂x2

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 44 / 57

Three Amplitudes + Rotation + Spatial VariationTu and MCC (1992)

A1 A2 A3

KLKL

KL

∂ A1/∂t = εA1 − A1(A21 + g+ A2

2 + g− A23) + ∂2A1/∂x2

1

∂ A2/∂t = εA2 − A2(A22 + g+ A2

3 + g− A21) + ∂2A2/∂x2

2

∂ A3/∂t = εA3 − A3(A23 + g+ A2

1 + g− A22) + ∂2A3/∂x2

3

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 47 / 57

Three Amplitudes + Rotation + Spatial VariationTu and MCC (1992)

A1 A2 A3

KLKL

KL

∂ A1/∂t = εA1 − A1(A21 + g+ A2

2 + g− A23) + ∂2A1/∂x2

1

∂ A2/∂t = εA2 − A2(A22 + g+ A2

3 + g− A21) + ∂2A2/∂x2

2

∂ A3/∂t = εA3 − A3(A23 + g+ A2

1 + g− A22) + ∂2A3/∂x2

3

gives chaos!

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 47 / 57

Simulations of Amplitude EquationsTu and MCC (1992)

Length scale ξ ∼ ε−1/2

Time scale τ ∼ ε−1

Velocity scale v ∼ ε1/2

Grey: A1 largest; White:A2 largest; Black:A3 largest

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 48 / 57

Model EquationsMCC, Meiron, and Tu (1994)

Real field of two spatial dimensionsψ(x, y; t)

∂ψ

∂t= εψ + (∇2 + 1)2ψ − ψ3 gives stripes

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 49 / 57

Model EquationsMCC, Meiron, and Tu (1994)

Real field of two spatial dimensionsψ(x, y; t)

∂ψ

∂t= εψ + (∇2 + 1)2ψ − ψ3

+ g2z·∇ × [(∇ψ)2∇ψ] + g3∇·[(∇ψ)2∇ψ]gives chaos!

Stripes Orientations Domain Walls

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 49 / 57

Full Fluid SimulationsMCC, Greenside, Fischer et al.

With modern supercomputers we can now simulate actual experiments

Spectral Element Method

Accurate simulation of long-time dynamics

Exponential convergence in space, third order in time

Efficient parallel algorithm, unstructured mesh

Arbitrary geometries, realistic boundary conditions

Conducting

ÿþýüû

Insulating

dT/dx=0

"Fin" Ramp

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 50 / 57

Fluid Simulations Complement Experiments

Knowledge of full flow field and other diagnostics (e.g. total heat flow)

No experimental/measurement noise (roundoff “noise” very small)

Measure quantities inaccessible to experiment e.g. Lyapunov exponentsand vectors

Readily tune parameters

Turn on and off particular features of the physics (e.g. centrifugal effects,mean flow, realistic v. periodic boundary conditions)

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 52 / 57

Full Fluid Dynamic SimulationsScheel, Caltech thesis (2006)

Realistic Boundaries

Periodic Boundaries

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 53 / 57

Lyapunov ExponentJayaraman et al. (2005)

Temperature Temperature Perturbation

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 54 / 57

Lyapunov ExponentJayaraman et al. (2005)

Aspect ratio0 = 40, Prandtl numberσ = 0.93, rotation rateÄ = 40

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 55 / 57

Importance of Centrifugal Force…Becker, Scheels, MCC, Ahlers (2006)

Aspect ratio0 = 20,ε ' 1.05,Ä = 17.6

Centrifugal force 0 Centrifugal force x4 Centrifugal force x10

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 56 / 57

Conclusions

I have described the study of pattern formation and spatiotemporal chaos inopen systems driven far from equilibrium.

Linear stability analysis gives an understanding of the origin of the patternand the basic length scale

Nonlinearity is a vital ingredient, and makes the problem difficult

There has been significant progress in understanding patterns, althoughmany questions remain

I illustrated the approaches we use by describing attempts to reach aquantitativeunderstanding of spatiotemporal chaos in rotating convectionexperiments

Close interaction between experiment, theory, and numerical simulation isimportant to understand complex systems

Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 57 / 57