Pattern Formation and Spatiotemporal Chaos in Systems Far ...mcc/Talks/Shaanxi_06.pdf · Outline 1...
Transcript of Pattern Formation and Spatiotemporal Chaos in Systems Far ...mcc/Talks/Shaanxi_06.pdf · Outline 1...
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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
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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
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An Open System
Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 3 / 57
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Patterns in Geophysics
Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 4 / 57
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Patterns in Biology
Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 5 / 57
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Pattern Formation
The spontaneous formation of spatial structure in opensystems driven far from equilibrium
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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
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Bénard’s Experiments
(From the website of Carsten Jäger)Movie
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Ideal Hexagonal Pattern
From the website of Michael Schatz
Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 9 / 57
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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
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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
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Rayleigh and his Solution
Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 11 / 57
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Schematic of Instability
Fluid
Rigid plate
Rigid plate
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Schematic of Instability
Fluid
Rigid plate
Rigid plate
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Schematic of Instability
HOT
COLD
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Schematic of Instability
HOT
COLD
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Schematic of Instability
HOT
COLD
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Schematic of Instability
HOT
COLD
2π/q
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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.
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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
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Rayleigh’s Growth Rate
σq/π
21
R=0.5 Rc
Rc = 27π4
4, qc = π√
2
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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
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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
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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
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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
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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
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Ideal Patterns from Experiment
Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 25 / 57
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More Patterns From Experiment
From the website of EberhardBodenschatz
Michael Cross (Caltech, BNU) Pattern Formation and Spatiotemporal Chaos May 2006 26 / 57
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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
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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
![Page 31: Pattern Formation and Spatiotemporal Chaos in Systems Far ...mcc/Talks/Shaanxi_06.pdf · Outline 1 Introduction 2 Rayleigh-Bénard Convection 3 Pattern Formation Basic Question Nonlinearity](https://reader033.fdocuments.us/reader033/viewer/2022050109/5f47015b6ba16d75366b6ce5/html5/thumbnails/31.jpg)
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
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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
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Lorenz Model (1963)
HOT
COLD
X
Z
Y
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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.
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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
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Nonlinear Equations
-10-5
05
10
X
-10
0
10
Y
0
10
20
30
40
Z
-50
510
X
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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
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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
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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
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Spiral Chaos in Rayleigh-Bénard Convection
…and from experiment
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Domain Chaos in Rayleigh-Bénard Convection
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Rotating Rayleigh-Bénard Convection
Ω
Rotate
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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
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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
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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
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Theory, Experiment, and Simulation
ComplexSystem
Experiment
Theory
Simulations
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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)
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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
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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
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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
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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
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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
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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!
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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
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Model EquationsMCC, Meiron, and Tu (1994)
Real field of two spatial dimensionsψ(x, y; t)
∂ψ
∂t= εψ + (∇2 + 1)2ψ − ψ3 gives stripes
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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
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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
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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)
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Full Fluid Dynamic SimulationsScheel, Caltech thesis (2006)
Realistic Boundaries
Periodic Boundaries
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Lyapunov ExponentJayaraman et al. (2005)
Temperature Temperature Perturbation
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Lyapunov ExponentJayaraman et al. (2005)
Aspect ratio0 = 40, Prandtl numberσ = 0.93, rotation rateÄ = 40
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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
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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
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