spatio-temporal chaos in pattern-forming systems: defects...
TRANSCRIPT
-
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
-
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
• Strong non-Boussinesq effects (low mean temperature):
- no instability of hexagons to rolls
- coexistence of stable hexagons and rolls
-
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
• Weak non-Boussinesq effects
-
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
• Weak non-Boussinesq effects
• Intermediate non-Boussinesq effects
-
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
• Weak non-Boussinesq effects
• 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
-
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
• 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)
-
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
• 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
(Chaté & Manneville, 1996)
-
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
• Navier-Stokes Simulation (|H|)
0 0.5 1 1.5b3
-1
0
1
b1
Defect Chaos
L BF
T
StablePlane Waves
(Chaté & Manneville, 1996)
-
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