Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
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