Spatio-Temporal Chaos in Pattern-Forming Systems: Defects ...people.esam.northwestern.edu/~riecke/research/Talks/dpg06.pdf · Defect Chaos Phase Chaos Frozen Vortices L BF T Stable

Spatio-Temporal Chaos

in Pattern-Forming Systems:

Defects and Bursts

with

Santiago Madruga, MPIPKS Dresden

Werner Pesch, U. Bayreuth

Yuan-Nan Young, New Jersey Inst. Techn.

DPG Frühjahrstagung 31.3.2006 supported by NASA and DOE

Pattern Formation

Symmetry-breaking instabilities ⇒ patterns

Rayleigh-Bénard convection of a

fluid layer heated from below

T+dT

T

(Plapp & Bodenschatz, 1996) (Bodenschatz, de Bruyn, Ahlers, Cannell, 1991)

Spatio-Temporal Chaos

Spiral-Defect Chaos

(Morris, Bodenschatz, Cannell, Ahlers, 1993)

• Small Prandtl number:

large-scale flows

Küppers-Lortz Chaos

(Hu, Ecke, Ahlers, 1997) ··

• Rotation:

Küppers-Lortz Instability

Goals:

• Quantitative characterization

of spatio-temporally chaotic states

and of transitions between them

(HR & Madruga, 2006)

• Origin of temporal and spatial chaos

and the mechanisms maintaining it

Microscopic

equations

Reduction

⇐⇒

Order parameters

Symmetries

Macroscopic

equations

Non-Boussinesq Convection

(Bodenschatz, de Bruyn, Ahlers, Cannell, 1991)

T+dT

T

Hexagons in Rotating Systems

• Weakly nonlinear description of hexagons

without rotation

γ=0Rolls

µ

Steady Hexagons

with rotation

γ>0

µ

Steady Hexagons

Rolls

Whirling Hexagons

(J.W. Swift, 1984; Soward, 1985)

• Hopf bifurcation to whirling hexagons

• weak non-Boussinesq effects:

coupled Ginzburg-Landau equations ⇒ CGL defect chaos(Echebarria & HR, 2000)

Rotating Non-Boussinesq Convection

- Navier-Stokes equation for momentum Vi

1

Pr

(

∂tVi + Vj∂j

(

Vi

ρ

))

= −∂ip+ δi3

(

1 + γ1(−2z +Θ

R)

)

Θ +

+∂j

[

νρ

(

∂i(Vj

ρ) + ∂j(

Vi

ρ)

)]

+ 2Ωǫij3Vj

- Heat equation and continuity equation

- Weakly temperature-dependent fluid parameters

ρ(T ) = 1 − γ0T − T0R

(1 + γ1T − T0R

) ν(T ) = 1 + γ2T − T0R

.....

Rotating Non-Boussinesq Convection

- Navier-Stokes equation for momentum Vi

1

Pr

(

∂tVi + Vj∂j

(

Vi

ρ

))

= −∂ip+ δi3

(

1 + γ1(−2z +Θ

R)

)

Θ +

+∂j

[

νρ

(

∂i(Vj

ρ) + ∂j(

Vi

ρ)

)]

+ 2Ωǫij3Vj

- Heat equation and continuity equation

- Weakly temperature-dependent fluid parameters

ρ(T ) = 1 − γ0T − T0R

(1 + γ1T − T0R

) ν(T ) = 1 + γ2T − T0R

.....

• Fully nonlinear hexagon

solution

linear stability analysis

• Direct numerical simulation of

chaotic states

Interpretation:

Reduction to complex

Ginzburg-Landau equation

Reentrant Hexagons in Non-Rotating Convection

• Stability of hexagons with respect to amplitude perturbations

20 30 40 50 60Mean Temperature T

0

0.00

0.20

0.40

0.60

0.80

1.00

Red

. Ray

leig

h N

umbe

r ε

h=1.8 mm

Unstable Rolls

Stable Rolls

Unstable Hexagons

Stable HexagonsReentrant Hexagons

• Non-Boussinesq effects increase

with decreasing mean temperature T0




0

0.00

0.20

0.40

0.60

0.80

1.00

Red

. Ray

leig

h N

umbe

r ε

h=1.8 mm

Unstable Rolls

Stable Rolls

Unstable Hexagons


• Strong non-Boussinesq effects (low mean temperature):

- no instability of hexagons to rolls

- coexistence of stable hexagons and rolls




0

0.00

0.20

0.40

0.60

0.80

1.00

Red

. Ray

leig

h N

umbe

r ε

h=1.8 mm

Unstable Rolls

Stable Rolls

Unstable Hexagons


• Weak non-Boussinesq effects




0

0.00

0.20

0.40

0.60

0.80

1.00

Red

. Ray

leig

h N

umbe

r ε

h=1.8 mm

Unstable Rolls

Stable Rolls

Unstable Hexagons



• Intermediate non-Boussinesq effects




0

0.00

0.20

0.40

0.60

0.80

1.00

Red

. Ray

leig

h N

umbe

r ε

h=1.8 mm

Unstable Rolls

Stable Rolls

Unstable Hexagons



• Intermediate non-Boussinesq effects

• Strong non-Boussinesq effects

Weak Non-Boussinesq Effects

water: thickness h = 4.92mm

mean temperature T0 = 12oC

critical temperature difference

∆Tc = 6.4oC

rotation rate Ω = 65 (∼ 1 Hz)

Hopf bifurcation at ǫ = 0.07

0 0.05 0.1 0.15 0.2 0.25

Red. Rayleigh Number ε

0

200

400

600

Am

plit

ude

(a.u

.)

OscillationAmplitude

steady oscillating

ǫ = 0.2

Description within CGL Framework

~

~

~

q1

q

q

3

2

• Extract oscillation amplitude H(X,T )

– wavevectors near q̃n – frequencies near ωH

vx(x, t, z = 0) =3

∑

n=1

(

R+[

e2πni/3H(X,T ) eiωH t + c.c.])

exp (iq̃n · x) + . . .

∂tH = µH + (1 + ib1)∇2H− (b3 − i)H|H|

2

• Complex Ginzburg-Landau equation:

the universal description of weakly nonlinear oscillations


~

~

~

q1

q

q

3

2



vx(x, t, z = 0) =3

∑

n=1

(

R+[


exp (iq̃n · x) + . . .

∂tH = µH + (1 + ib1)∇2H− (b3 − i)H|H|

2

• CGL Defect Chaos (Hr)

0 0.5 1 1.5 2 2.5b3

-2

0

2

4

b1

Defect Chaos Phase Chaos

Frozen

Vortices

L

BF

T

StablePlane Waves

S2

(Chaté & Manneville, 1996)


~

~

~

q1

q

q

3

2



vx(x, t, z = 0) =3

∑

n=1

(

R+[


exp (iq̃n · x) + . . .

∂tH = µH + (1 + ib1)∇2H− (b3 − i)H|H|

2

• Extract coefficients b1 and b3from direct Navier-Stokes

simulations

• Bistability:

Plane waves ↔ defect chaos

0 0.5 1 1.5b3

-1

0

1

b1

Defect Chaos

L BF

T

StablePlane Waves



~

~

~

q1

q

q

3

2



vx(x, t, z = 0) =3

∑

n=1

(

R+[


exp (iq̃n · x) + . . .

∂tH = µH + (1 + ib1)∇2H− (b3 − i)H|H|

2

• Navier-Stokes Simulation (|H|)

0 0.5 1 1.5b3

-1

0

1

b1

Defect Chaos

L BF

T

StablePlane Waves


Intermediate Non-Boussinesq Effects

h = 4.92mm Ω = 65 Pr = 8.7

T0 = 14oC ∆Tc = 8.3

oC Ω = 65

• Hopf bifurcation backward

• Hysteresis and bistability

of steady and oscillating hexagons

ǫ = 0.5

• restabilization of steady hexagons

at larger ǫ

• fluctuating localized domains

of whirling hexagons

0 0.2 0.4 0.6

Red. Rayleigh ε

0

0.5

1

1.5

Am

plit

ude

(a.u

.)

εH

steady hexagonsoscillating hex.

Quintic Complex Ginzburg-Landau Equations

• Quintic Ginzburg-Landau equation

∂tH = µH + (dr + idi)∇2H− (cr + ici)H|H|

2 − (gr + igi)H|H|4

• Extract coefficients

d = 1.90 + 0.033i, c = −1.1 + 7.2i, g = 3.6 + 1.5i

• Demodulation

Navier-Stokes quintic Ginzburg-Landau

Localization Mechanism

• CGL coupled to phase modes

∂tH = (µ−Qδ∇ · ~ϕ)H + d∇2H− cH|H|2 − gH|H|4

Two contributions:

1. Wavenumber selection by front

H = Reiψ ~k = ∇ψ

• large ci, gi:

gradients in the oscillation

magnitude R induce

wavevector ~k

• diffusion dr:

oscillation amplitude damped

|H|

|~k|

(Bretherton and Spiegel, 1983;...,Coullet and Kramer, 2004)

Localization Mechanism

• CGL coupled to phase modes

∂tH = (µ−Qδ∇ · ~ϕ)H + d∇2H− cH|H|2 − gH|H|4

Two contributions:

1. Wavenumber selection by front

2. Compression ∇ · ϕ of underlying hexagon pattern

Pattern Compression ∇ · ϕ

4.4 4.6 4.8 5 5.2 5.4Wavenumber q

0

0.2

0.4

0.6

Gro

wth

rate

σ

A ε=0.2 B ε=0.5

Strong Non-Boussinesq Effects

• h = 4.6mm Ω = 65 Pr = 8.7 T0 = 12◦C ∆Tc = 10

◦C

• no Hopf bifurcation to whirling hexagons

• only side-band instabilities

4.4 4.6 4.8 5 5.2 5.4 5.6Wavenumber q

0

0.5

1

Red

uced

Ray

leig

h (R

-Rc)

/Rc

steady hexagonslinearly stable

ǫ = 1.0

• whirling destroys order of hexagonal lattice

Strong Non-Boussinesq Effects

• h = 4.6mm Ω = 65 Pr = 8.7 T0 = 12◦C ∆Tc = 10

◦C

• no Hopf bifurcation to whirling hexagons

• only side-band instabilities

4.4 4.6 4.8 5 5.2 5.4 5.6Wavenumber q

0

0.5

1

Red

uced

Ray

leig

h (R

-Rc)

/Rc

steady hexagonslinearly stable

ǫ = 0.87

• whirling destroys order of hexagonal lattice

Defect Statistics

• ǫ = 1

• Triangulation

• Heptagons

• Pentagons

• Defect Statistics

Broader than

squared Poisson:

Correlations

20 30 40 50 60 70 80 90N

0

0.02

0.04

0.06

0.08

Rel

ativ

e F

requ

ency

Conclusions

Defects and bursts in rotating non-Boussinesq convection

• Whirling hexagons

- reduction to CGL

- 2d CGL defect chaos

• Localized bursts of oscillations

– quintic CGL: ‘retracting fronts’, collapse

– compression of lattice

• Whirling chaos

– whirling ⇔ penta-hepta defects

Phys. Rev. Lett. 96 (2006) 074501, J. Fluid Mech. 548 (2006) 341, New J. Phys. 5 (2003) 135.

www.esam.northwestern.edu/riecke

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Spatio-Temporal Chaos in Pattern-Forming Systems: Defects ...people.esam.northwestern.edu/~riecke/research/Talks/dpg06.pdf · Defect Chaos Phase Chaos Frozen Vortices L BF T Stable