Spatio-Temporal Chaos in Pattern-Forming Systems: Defects and Bursts

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

Spatio-Temporal Chaos Spiral-Defect Chaos Küppers-Lortz Chaos (Morris, Bodenschatz, Cannell, Ahlers, 1993) (Hu, Ecke, Ahlers, 1997) Small Prandtl number: large-scale flows 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 T T+dT (Bodenschatz, de Bruyn, Ahlers, Cannell, 1991)

Hexagons in Rotating Systems Weakly nonlinear description of hexagons without rotation with rotation γ=0 Rolls γ>0 Rolls Steady Hexagons Steady Hexagons Whirling Hexagons µ µ (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 V i ( ( )) 1 Vi t V i + V j j Pr ρ = i p + δ i3 ( 1 + γ 1 ( 2z + Θ R ) ) Θ + + j [νρ - Heat equation and continuity equation - Weakly temperature-dependent fluid parameters ( i ( V j ρ ) + j( V )] i ρ ) + 2Ωǫ ij3 V j T T 0 ρ(t) = 1 γ 0 R (1 + γ T T 0 1 R ) ν(t) = 1 + γ T T 0 2 R... 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 Red. Rayleigh Number ε 1.00 0.80 0.60 0.40 0.20 Stable Hexagons Stable Rolls h=1.8 mm Unstable Rolls Reentrant Hexagons Unstable Hexagons 0.00 20 30 40 50 60 Mean Temperature T 0 Non-Boussinesq effects increase with decreasing mean temperature T 0

Reentrant Hexagons in Non-Rotating Convection Stability of hexagons with respect to amplitude perturbations Red. Rayleigh Number ε 1.00 0.80 0.60 0.40 0.20 Stable Hexagons Stable Rolls h=1.8 mm Unstable Rolls Reentrant Hexagons Unstable Hexagons 0.00 20 30 40 50 60 Mean Temperature T 0 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 Red. Rayleigh Number ε 1.00 0.80 0.60 0.40 0.20 Stable Hexagons Stable Rolls h=1.8 mm Unstable Rolls Reentrant Hexagons Unstable Hexagons 0.00 20 30 40 50 60 Mean Temperature T 0 Weak non-boussinesq effects Intermediate non-boussinesq effects

Reentrant Hexagons in Non-Rotating Convection Stability of hexagons with respect to amplitude perturbations Red. Rayleigh Number ε 1.00 0.80 0.60 0.40 0.20 Stable Hexagons Stable Rolls h=1.8 mm Unstable Rolls Reentrant Hexagons Unstable Hexagons 0.00 20 30 40 50 60 Mean Temperature T 0 Weak non-boussinesq effects Intermediate non-boussinesq effects Strong non-boussinesq effects

Weak Non-Boussinesq Effects water: thickness h = 4.92mm mean temperature T 0 = 12 o C critical temperature difference T c = 6.4 o C rotation rate Ω = 65 ( 1 Hz) Hopf bifurcation at ǫ = 0.07 Amplitude (a.u.) 600 400 200 steady oscillating Oscillation Amplitude 0 0 0.05 0.1 0.15 0.2 0.25 Red. Rayleigh Number ε ǫ = 0.2

Description within CGL Framework Extract oscillation amplitude H(X, T) wavevectors near q n frequencies near ω H v x (x, t, z = 0) = 3 n=1 ( R + [ ]) e 2πni/3 H(X, T)e iωht + c.c. t H = µh + (1 + ib 1 ) 2 H (b 3 i) H H 2 ~ q 2 ~ q 3 ~ q 1 exp (i q n x) +... Complex Ginzburg-Landau equation: the universal description of weakly nonlinear oscillations

Description within CGL Framework Extract oscillation amplitude H(X, T) wavevectors near q n frequencies near ω H v x (x, t, z = 0) = 3 n=1 ( R + [ ]) e 2πni/3 H(X, T)e iωht + c.c. t H = µh + (1 + ib 1 ) 2 H (b 3 i) H H 2 ~ q 2 ~ q 3 ~ q 1 exp (i q n x) +... CGL Defect Chaos (H r ) b 1 4 2 0-2 S 2 Defect Chaos Frozen Vortices Phase Chaos 0 0.5 1 1.5 2 2.5 b 3 T L BF Stable Plane Waves (Chaté & Manneville, 1996)

Description within CGL Framework Extract oscillation amplitude H(X, T) wavevectors near q n frequencies near ω H v x (x, t, z = 0) = 3 n=1 ( R + [ ]) e 2πni/3 H(X, T)e iωht + c.c. t H = µh + (1 + ib 1 ) 2 H (b 3 i) H H 2 ~ q 2 ~ q 3 ~ q 1 exp (i q n x) +... Extract coefficients b 1 and b 3 from direct Navier-Stokes simulations Bistability: Plane waves defect chaos b 1 1 0-1 L Defect Chaos Stable Plane Waves BF 0 0.5 1 1.5 b 3 T (Chaté & Manneville, 1996)

Description within CGL Framework Extract oscillation amplitude H(X, T) wavevectors near q n frequencies near ω H v x (x, t, z = 0) = 3 n=1 ( R + [ ]) e 2πni/3 H(X, T)e iωht + c.c. t H = µh + (1 + ib 1 ) 2 H (b 3 i) H H 2 ~ q 2 ~ q 3 ~ q 1 exp (i q n x) +... Navier-Stokes Simulation ( H ) 1 L Defect Chaos BF T b 1 0-1 Stable Plane Waves 0 0.5 1 1.5 b 3 (Chaté & Manneville, 1996)

Intermediate Non-Boussinesq Effects h = 4.92mm Ω = 65 Pr = 8.7 T 0 = 14 o C T c = 8.3 o C Ω = 65 Hopf bifurcation backward Hysteresis and bistability ǫ = 0.5 of steady and oscillating hexagons Amplitude (a.u.) 1.5 1 0.5 steady hexagons oscillating hex. 0 0 0.2 ε H 0.4 0.6 Red. Rayleigh ε restabilization of steady hexagons at larger ǫ fluctuating localized domains of whirling hexagons

Quintic Complex Ginzburg-Landau Equations Quintic Ginzburg-Landau equation t H = µh + (d r + id i ) 2 H (c r + ic i )H H 2 (g r + ig i )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 t H = (µ Qδ ϕ)h + d 2 H ch H 2 gh H 4 Two contributions: 1. Wavenumber selection by front H = Re iψ k = ψ large c i, g i : gradients in the oscillation magnitude R induce wavevector k diffusion d r : oscillation amplitude damped H k (Bretherton and Spiegel, 1983;...,Coullet and Kramer, 2004)

Localization Mechanism CGL coupled to phase modes t H = (µ Qδ ϕ)h + d 2 H ch H 2 gh H 4 Two contributions: 1. Wavenumber selection by front 2. Compression ϕ of underlying hexagon pattern Pattern Compression ϕ Growthrate σ 0.6 0.4 0.2 A ε=0.2 B ε=0.5 0 4.4 4.6 4.8 5 5.2 5.4 Wavenumber q

Strong Non-Boussinesq Effects h = 4.6mm Ω = 65 Pr = 8.7 T 0 = 12 C T c = 10 C no Hopf bifurcation to whirling hexagons only side-band instabilities Reduced Rayleigh (R-R c )/R c 1 0.5 steady hexagons linearly stable 0 4.4 4.6 4.8 5 5.2 5.4 5.6 Wavenumber q ǫ = 1.0 whirling destroys order of hexagonal lattice

Defect Statistics ǫ = 1 Triangulation Heptagons Pentagons Defect Statistics Broader than squared Poisson: Correlations Relative Frequency 0.08 0.06 0.04 0.02 0 20 30 40 50 60 70 80 90 N

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