Complex Patterns: Diagnostics and...

Complex Patterns:

Diagnostics and Excitation

with

Santiago MadrugaMPI Complex Systems, Germany

Werner PeschU. Bayreuth, Germany

Jessica ConwayNorthwestern University

supported by DOE and NSF

Spatio-Temporal Chaos

Undulation Chaos

(Daniels and Bodenschatz, 2002)

T+dT

T

up

down

Kuppers-Lortz Chaos

Rotation rate Ω = 12.1, ǫ = 0.2

(Hu, Ecke, and Ahlers, 1997)

Spatio-Temporal Chaos

Undulation Chaos

(Daniels and Bodenschatz, 2002)

T+dT

T

up

down

Kuppers-Lortz Chaos

Rotation rate Ω = 19.8, ǫ = 0.18

(Hu, Ecke, and Ahlers, 1997)

Spiral Defect Chaos

Convection at low Prandtl numbers

(Morris, Bodenschatz, Cannell, Ahlers, 1993)

Pr = 1.5

Boussinesq

Pr = 0.3

Non-Boussinesq

Defect Chaos in Belousov-Zhabotinsky Reaction

• Transition from spiral waves to defect chaos

(Ouyang and Flesselles, 1996)

Periodically Forced Belousov-Zhabotinsky Reaction

Near-resonant forcing of chemical oscillations

(Lin et al., 2004)

⇐ 1:2 Labyrinth

3:2 Cellular ⇒

(Petrov, Ouyang, & Swinney, 1997)

(Lin et al., 2004)

1:3 Spirals

⇐ weaker forcing

stronger forcing ⇒

(Lin et al., 2004)

Characterization of Complex Patterns

• Correlation functions, spectral entropy

• Local wavevector (Egolf, Melnikov, and Bodenschatz, 1998)

• Disorder function (Gunaratne, Hoffman, and Kouri, 1998)

• Statistics for number of dislocations

- complex Ginzburg-Landau equation (Gil, Lega, and Meunier, 1990)

- electroconvection (Rehberg et al., 1989)

- reaction-diffusion system (Hildebrand, Bär, and Eiswirth, 1995)

- undulation chaos (Daniels and Bodenschatz, 2002)

- penta-hepta defect chaos (Young and HR, 2003)

• Statistics of defect trajectories

- ‘unbinding’ transition of defect pairs (Granzow and HR, 2001)

• Homology: components and holes (Krishan et al., 2007)

Geometric Diagnostics of Spiral Defect Chaos

• automated analysis of experimentally accessible data

• mid-level contour

• smoothing of

contours

• segments:

tip → vertex

• number of closed contours

(cf. Schatz, Mischaikow; Krishan et al., 2007)

• contour length

• compactness of contour

• arclength of segments

• winding number of segments

• statistics based on

1000-2000 snapshots

Number of Components

Pr = 0.3 Pr = 1.5Topological measure

(Betti numbers)

(cf. Krishan et al., 2007)

0.5 1 1.5Red. Rayleigh Number ε

0

10

20

30

40

50

# C

lose

d C

onto

urs

Pr=1.5Pr=1.0Pr=0.3

Number of closed contours

• increases with increasing ǫ

• decreases with increasing Pr

Mean Arclength

Pr = 0.3 Pr = 1.0 Pr = 1.5

ǫ = 1


0

100

200

300

Arc

leng

th

Pr=1.5Pr=1.0Pr=0.3

• arclength decreases

with increasing ǫ

• arclength increases

with increasing Pr

Standard Deviation of Winding Number

Pr = 0.3 Pr = 1.0 Pr = 1.5

ǫ = 1


0

0.2

0.4

0.6

0.8

1

SD

Win

ding

Num

ber

Pr=1.5Pr=1.0Pr=0.3

• winding number decreases

with increasing ǫ

• winding number increases

with increasing Pr

• dependence not only.

on ǫ − ǫSDC

Compactness: Targets and Bubbles

Pr = 0.3 Pr = 1.5Compactness:

C = 4πA

P2

A = area

P = perimeter

Compactness: Targets and Bubbles

Pr = 0.3 Pr = 1.5Compactness:

C = 4πA

P2

circular: C ∼ 1

elongated:

C ∼ 4π λP/2P2 ∝ P−1

−6−5

−4−3

−2−1

012345678910

0

0.01

0.02

0.03

0.04

Ln (Compactness)Ln (Contourlength)

−6−5

−4−3

−2−1

012345678910

0

0.02

0.04

0.06

Ln (Compactness)Ln (Contourlength)

Winding Number and Arclength of Contours

• Large Pr:

large segments are

spirals

• Small Pr:

few clear spirals

• Winding number has

exponential distribution

(cf. Hu & Ecke, 1997)

0 1 2 3 4 5|Winding Number|

0.0001

0.001

0.01

0.1

1R

el. F

requ

ency

ε=0.7ε=1.0ε=1.4

Winding Number and Arclength of Contours

• Large Pr:

large segments are

spirals

• Small Pr:

few clear spirals

• Winding number has

exponential distribution

(cf. Hu & Ecke, 1997)

• Decay rate depends

on Pr

0 2 4 6 8|Winding Number|

0.0001

0.001

0.01

0.1

1R

el. F

requ

ency

Pr=1.5Pr=1.0Pr=0.3

Spiral Statistics

Distribution function for spiral number

• for ǫ = 1 and Pr = 1 narrower than Poisson distribution

(cf. Ecke et al., 1995; Ecke & Hu, 1997)

• collapse for different spiral thresholds

0 10 20 30 40 50Number of Spirals

0

0.05

0.1

0.15

0.2

Rel

. Fre

quen

cy

quarter-turn spiralshalf-turn spiralsfull-turn spiralsPoisson

-3 -2 -1 0 1 2 3

(# Spirals - Mean) /(Mean)1/2

0

0.2

0.4

0.6

Rel

. Fre

quen

cy

quarter turnhalf turnfull turn

Dependence on Vertical Position

So far all measurements were at the mid-plane z = 0

• Distribution function for

the winding number

0 1 2|Winding Number|

0.001

0.01

0.1

Rel

. Fre

quen

cy

z=-0.125z=-0.25z=0

• Mean quantities

-0.2 -0.1 0 0.1 0.2 0.3

Vertical Position z

0.6

0.8

1

1.2

1.4

Enh

ance

men

t ove

r z=

0

# `White’ Contours# `Black’ Contours# All Contours# SpiralsArc LengthSD Winding Number

Non-Boussinesq Effects

Temperature-dependent

fluid properties

• break up-down symmetry

• introduce

quadratic triad interaction

CO2: h = 0.080cm, T0 = 40C

3 4 5 6Wave number q

0

0.2

0.4

0.6

0.8

1

Red

uced

Ray

leig

h N

umbe

r ε

Stab

le

Am

plitu

de u

nsta

ble



fluid properties


• introduce


CO2: h = 0.052cm, T0 = 27C

2.4 2.6 2.8 3 3.2Wave number q

0

0.2

0.4

0.6

0.8

1

Red

uced

Ray

leig

h N

umbe

r ε

Stable hexagons

• Longitudinal

phase mode

ǫ = 0.195, q = 2.5

• Transverse

phase mode

ǫ = 0.16, q = 3.1



fluid properties


• introduce


CO2: h = 0.052cm, T0 = 27C

2.4 2.6 2.8 3 3.2Wave number q

0

0.2

0.4

0.6

0.8

1

Red

uced

Ray

leig

h N

umbe

r ε

Stable hexagons

• Transient

rolls

ǫ = 0.3

q = 3.12

Non-Boussinesq Spiral Defect Chaos

• Working fluid SF6

h = 0.0542cm

p = 140psi

T0 = 80C

Boussinesq Non-Boussinesq

• Number of components

strongly enhanced

0 0.2 0.4 0.6 0.8 1 1.2 1.4Reduced Rayleigh Number ε

0

10

20

30

40N

o. o

f Clo

sed

Con

tour

sNBOB

Non-Boussinesq Spiral Defect Chaos

• Working fluid SF6

h = 0.0542cm

p = 140psi

T0 = 80C

Boussinesq Non-Boussinesq

• Many small components

−4−3

−2−1

012345678910

0

0.01

0.02

0.03

0.04

0.05

0.06

Ln(Contourlength)Ln(Compactness)

(a)

Rel

ativ

e F

requ

ency

−4−3

−2−1

012345678910

0

0.02

0.04

0.06

Ln(Compactness)Ln(Contourlength)

(b)

Rel

ativ

e F

requ

ency

Complex Patterns in the Faraday System

(Kudrolli, Pier, & Gollub, 1992)

• Control of mode interaction by

choice of forcing function

(Arbell & Fineberg, 2002)

Complex Patterns in Resonantly ForcedComplex Ginzburg-Landau Equation

Weakly nonlinear oscillations forced at multiple resonances

∂tA = (1 + iβ)∇2A + (µ + iσ)A − (1 + iα)|A|2A

+(

cos χ + sinχeiνt)

γA∗ + ηA∗2

• 1:2 resonance: spatial

patterns through dispersion

• 1:3 resonance: quadratic

resonant triad interaction

• time-dependent coefficient:

subharmonic patterns ⇒

no transcritical hexagons

• Floquet ansatz (µ < 0)

0 0.5 1 1.5 2 2.5 3Wave Number k

5

10

15

20

25

1:2

Forc

ing

Stre

ngth

γ0.4 0.6 0.8 1 1.2

k

2.4

2.5

2.6

γ

Pattern Selection through Weakly Damped Modes

Weakly nonlinear theory (µ < 0)

A = ǫ(

A1eik1·r + A2e

ik2·r + . . .)

F (t) + ǫ2B(r, t) + O(ǫ3)

• Quadratic order:

strong excitation of

weakly damped

harmonic modes

• Cubic order:

feed-back into

amplitude equation

for fundamentals

dA1

dt= λA1 − b0A1|A1|

2 −

−b(θ)|A1A2|2 + . . .

k1

θr

k2+k 21k

2k2



A = ǫ(

A1eik1·r + A2e

ik2·r + . . .)

F (t) + ǫ2B(r, t) + O(ǫ3)

η = ρ eiπ/4 k(H)c = 2 k

(SH)c

0 20 40 60 80Rhomb Angle θ

0

0.5

1

1.5

2

Cro

ss C

oupl

ing

b(θ)

/b0

ρ=1ρ=1.5

dA1

dt= λA1 − b0A1|A1|

2 −

−b(θ)|A1A2|2 + . . .

k1

θr

k2+k 21k

2k2

(Viñals et al.)



A = ǫ(

A1eik1·r + A2e

ik2·r + . . .)

F (t) + ǫ2B(r, t) + O(ǫ3)

|η| = 1 k(H)c = 1.86 k

(sh)c

0 20 40 60 80Rhomb Angle θ

-30

-20

-10

0

Cro

ss C

oupl

ing

b(θ)

/b0

η=eπi/4

η=e3πi/4

dA1

dt= λA1 − b0A1|A1|

2 −

−b(θ)|A1A2|2 + . . .

k1

+k 21k

θr

k2

22k

(Silber et al.)

Stability of Super-Lattice and Quasi-Patterns

• Cross-coupling coefficient b(θ)/b0: stability of multi-mode patterns

• Conditions for relative stability of patterns on the same Fourier grid

• Overview of ‘preferred’ patterns: Lyapunov functional

η = ρ eiπ/4 k(H)c = 2 k

(SH)c

0 0.5 1 1.5 21:3 Forcing Strength ρ

-0.08

-0.06

-0.04

-0.02

0

Lyap

unov

Ene

rgy

StripesSquaresHexagons4-mode Quasipattern5-mode Quasipattern6-mode Quasipattern

0.6 0.8 1 1.2ρ

-0.05

-0.04

-0.03

Numerical Simulations

Starting from random initial conditions

η = 0.8 eiπ/4



η = 0.9 eiπ/4



η = 1.2 eiπ/4



η = 2 eiπ/4



η = 3 eiπ/4



Spectral Entropy

5000 10000 15000 20000Period T

5.5

6

6.5

7

7.5

8

8.5

Ent

ropy

S

ρ=0.8ρ=0.9ρ=1ρ=1.1ρ=1.2ρ=1.3ρ=1.4ρ=1.5ρ=1.8ρ=2ρ=3

Conclusions

Geometric Analysis of Patterns

• based on experimentally accessible data

• here applied to spiral defect chaos

• sub-Poisson distribution for number of spirals

• exponential distribution for winding number

• many small components for small Prandtl and non-Boussinesq

• patterns depend not only on distance from SDC onset

Quasi-Patterns in Multi-Resonance Complex Ginzburg-Land au Equation

• 3-mode lattices, 4-mode and 5-mode quasipatterns

• additional complexity from Hopf bifurcation?

www.esam.northwestern.edu/riecke

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