Pattern Formation and Spatiotemporal Chaos

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 1 Pattern Formation and Spatiotemporal Chaos Insights from Large Scale Numerical Simulations of Rayleigh-Bénard convection Collaborators: Mark Paul, Keng-Hwee Chiam, Janet Scheel, Henry Greenside, Anand Jayaraman, Paul Fischer Support: DOE; Computer time: NSF, NERSC, NCSA, ANL

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 2 Outline Use numerical simulation of Rayleigh-Bénard convection in realistic geometries to learn about complex spatial patterns and dynamics in spatially extended systems. Examples: Pattern chaos: power spectrum Lyapunov exponents Coarsening and wavenumber selection

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 3 Rayleigh-Bénard Convection RBC allows a quantitative comparison to be made between theory and experiment.

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 4 Nondimensional Boussinesq Equations Momentum Conservation [ 1 u ( ) ] σ t + u u = p + R T ê z + 2 u + 2 ê z u Energy Conservation T t + ( ) u T = 2 T Mass Conservation u = 0 Aspect Ratio: Ɣ = r h BC: no-slip, insulating or conducting, and constant T

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 5 Spectral Element Numerical Solution Accurate simulation of long-time dynamics Exponential convergence in space, third order in time Efficient parallel algorithm, unstructured mesh Arbitrary geometries, realistic boundary conditions

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 6 Convection in an elliptical container cf. Ercolani, Indik, and Newell, Physica D (2003)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 7 Pattern chaos: convection in small cylindrical geometries First experiments: Ɣ = 5.27 cell, cryogenic (normal) liquid He 4 as fluid. High precision heat flow measurements (no flow visualization). Onset of aperiodic time dependence in low Reynolds number flow: relevance of chaos to real (continuum) systems. Power law decrease of power spectrum P(f) f 4 G. Ahlers, Phys. Rev. Lett. 30, 1185 (1974) G. Ahlers and R.P. Behringer, Phys. Rev. Lett. 40, 712 (1978) H. Gao and R.P. Behringer, Phys. Rev. A30, 2837 (1984) V. Croquette, P. Le Gal, and A. Pocheau, Phys. Scr. T13, 135 (1986)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 8 (from Ahlers and Behringer 1978)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 9 Numerical Simulations Ɣ = 4.72, σ = 0.78, 2600 R 7000 Conducting sidewalls Random thermal perturbation initial conditions Simulation time 100τ h Simulation time 12 hours on 32 processors Experiment time 172 hours or 1 week

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 10 Γ = 4.72 σ = 0.78 (Helium) Random Initial Conditions 2 1.9 R = 6949 R = 4343 R = 3474 R = 3127 R = 2804 R = 2606 1.8 Nu 1.7 1.6 1.5 1.4 50 τ h 0 500 1000 1500 2000 time

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 11 R = 3127 R = 6949

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 12 Power Spectrum Simulations of low dimensional chaos (e.g. Lorenz model) show exponential decaying power spectrum Power law power spectrum easily obtained from stochastic models (white-noise driven oscillator, etc.)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 13 Simulation yields a power law over the range accessible to experiment 10-6 R = 6949 ω -4 10-7 P(ω) 10-8 10-9 Γ = 4.72, σ = 0.78, R = 6949 Conducting sidewalls R/R c = 4.0 10-10 10-11 10-12 10-2 10-1 10 0 10 1 10 2 ω

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 14 When larger frequencies are included an exponential tail is found 10-5 R = 6949 ω -4 10-7 P(ω) 10-9 10-11 10-13 Γ = 4.72, σ = 0.78, R = 6949 Conducting sidewalls R/R c = 4.0 10-15 10-17 10-19 10-21 10-2 10-1 10 0 10 1 10 2 ω Exponential tail not seen in experiment because of instrumental noise floor

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 15 Where does the power law come from? Power law arises from quasi-discontinuous changes in the slope of N(t) on a t = 0.1 1 time scale associated with roll pinch-off events. This is clearest to see for the low Rayleigh number where the motion is periodic, but again the power spectrum has a power law fall off. Sharp events similar in chaotic and periodic signals

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 16 Spectrogram

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 17

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 18 Role of mean flow 3 convection cells with different side wall conditions: (a) rigid; (b) finned; and (c) ramped. Case (a) is dynamic, the others static.

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 19 Sensitive dependence on initial conditions Lyapunov exponents: Quantify the sensitivity to initial conditions Define chaos Lyapunov vectors: Associate sensitivity with specific events (defect creation, etc.) Propagation of disturbances (Lorenz s question!) Lyapunov dimension: Quantifies the number of active degrees of freedom Scaling with system size may perhaps be used to define spatiotemporal chaos (microextensive chaos: Tajima and Greenside, 2002)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 20 Lyapunov exponents δu 0 δu 1 δu 2 δu = δu 0 e λ 1t, λ 1 = lim t 1 t ln δu δu 0 Line lengths e λ 1t, Areas e (λ 1+λ 2 )t, Volumes e (λ 1+λ 2 +λ 3 )t,... Lyapunov Dimension D L = ν + 1 λ ν+1 ν i=1 where ν is the largest index such that the sum is positive. λ i

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 21 Numerical Approach Chaotic Boussinesq driving solution: [ 1 u ( ) ] σ t + u u T t + ( ) u T = 2 T u = 0 = p + RT ê z + 2 u Linearized equations (tangent space equations): ( u, p, T ) ( u + δ u k,p+ δp k,t + δt k ), for k = 1,...,n [ 1 δ uk ( ) ( ) ] + u δ u k + δ u k u = δp k + RδT ê z + 2 δ u k σ t δt ( ) ( ) k + u δt k + δ u k T = 2 δt k t δ u k = 0

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 22 Simulations in an experimental geometry Ɣ = 4.72, σ = 0.78, R = 6950

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 23 30 data λ = 0.6 20 log Norm 10 0 10 20 30 40 50 t Experimentally realistic system is truly chaotic.

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 24 Lyapunov vector for spiral defect chaos (from Keng-Hwee Chiam, Caltech thesis 2003)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 25 Coarsening Development of ordered state from random initial conditions Much studied in statistical mechanics for relaxation to an ordered state in thermodynamic equilibrium. Questions: Nature of asymptotic state Ideal pattern or frozen disordered state? Coarsening dynamics Often find power law growth of characteristic length scale L t p with p often, but not always 1 2 Phase diffusion or defect dynamics may be important

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 26 Coarsening to a stripe state Coarsening in stripe systems is a hard problem Many types of defects: dislocations, disclinations, grain boundaries: hard to identify important dynamical processes A number of different arguments give slow growth p 1 4, roughly consistent with numerical simulations, Recent experiments on very large relaxational systems suggest a dominant mechanism that is consistent with the observed scaling [Harrison et al. (2002)]

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 27 Coarsening to a state far from equilibrium has additional difficulties No energetic arguments to simplify discussion Asymptotic pattern not a priori known (wave number selection) Progress so far Few experiments, none on rotationally invariant systems. Previous results are mainly based on numerical simulations of model equations. Much of the work is on relaxational models [Elder, Vinals, and Grant (1992) ] Two dimensional numerics on generalized Swift-Hohenberg models [MCC and Meiron (1995)] both relaxational and nonrelaxational models.

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 28 Rayleigh-Bénard simulations Not very large! Limited to aspect ratios Ɣ 50 to 100 and therefore not very long times before finite size affects the dynamics. Main focus of work is on asymptotic state, and gross features of transient (e.g. whether γ = 1 4 or 1 2 not whether γ = 1 4 or 1 5 )

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 29 R = 2169, σ = 1.4, and Ɣ = 57

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 30 Structure factor S(q,t) / t γ (arb. units) 0.3 0.25 0.2 0.15 0.1 0.05 t=4 t=8 t=16 t=32 t=64 t = 128 t = 256 0-1.5-1 -0.5 0 0.5 1 1.5 (q - q ) t γ

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 31 Orientation field correlation length R = 2169, σ = 1.4, Ɣ = 57, t = 203 ) C 2 ( r r,t = e i2(θ( r,t) θ( r,t)) C 2 e r/ξ o

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 32 Mean wave number evolution 3.4 3.3 Initial condition 1 Initial condition 2 3.2 q 3.1 3 q f 2.9 2.8 q d 2.7 0 100 200 [q d from Bodenschatz et al., Ann. Rev. Fluid Mech. 32 2000] Wave number appears to approach dislocation selected value, not q f. t

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 33 Correlation lengths

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 34 Defect density (defined as regions of large curvature) R = 2169, σ = 1.4, Ɣ = 57, t = 8, 128 Defect lines (grain boundaries) clearly evident

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 35 Defect density 0.3 0.2 Initial condition 1 Initial condition 2 t -0.45 ρ d 0.1 10 1 10 2 For isolated defects L D ρ 1/2 D so that L D t 1/4. For defect lines ρ d is the length of line, and the domain size scales as t 1/2 t

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 36 Conclusions Numerical simulations on realistic experimental geometries complement experimental work and yield new insights Pattern chaos Lower noise flows gives consistency of power spectrum with expectation based on deterministic chaos Visualization of dynamics explains observed power law observed in spectrum Confirmation of role of mean flow Lyapunov exponents and eigenvectors Confirms early experiments were chaotic Promising tools for studying spatiotemporal chaos Coarsening Experiments needed! Results largely consistent with previous nonrelaxational model simulations Future work will investigate specific dynamics

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 37 THE END

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 38 Power law spectra 1.4E-06 Γ = 4.72, R = 2804, σ = 0.78 Conducting sidewalls No overlapping Linear detrending 1.2E-06 1E-06 P(ν) 8E-07 6E-07 4E-07 2E-07 0 0 0.025 0.05 0.075 0.1 ν (1/τ v ) 10-7 10-9 10-11 R = 2804 ν -4 10-13 P(ν) 10-15 10-17 10-19 10-21 10-23 10-25 10-3 10-2 10-1 10 0 10 1 ν (1/τ v )

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 39 Time series N(t) 1.425 R = 2804, Γ = 4.72, σ = 0.78 Conducting one dislocation quickly glides into a wall foci d) first annihilation e) the other dislocation slowly glides into the same wall foci second annihilation 1.42 1.415 Nu 1.41 c) both dislocations are at the lateral walls b) dislocations climb 1.405 t = 187 = 8.4 τ h 1.4 a) dislocation nucleation f) process repeats 600 700 time Note: Dislocation glide alternates left and right, the portion highlighted here goes right.

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 40 R = 2804, σ = 0.78, Γ = 4.72 Conducting sidewalls 10-10 P(ν) 10-12 10-14 Exponential region, slope = -3.8 10-16 10-18 10-20 1 2 3 4 5 6 7 ν (1/τ v )

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 41 Compare periodic and chaotic events 1.425 R = 2804, σ = 0.78, Γ = 4.72 Conducting sidewalls 1.92 R = 6949, σ = 0.78, Γ = 4.72 Conducting sidewalls 1.42 1.91 1.415 1.9 1.89 Nu 1.41 Nu 1.88 1.405 1.87 1.4 1.86 1.395 770 775 780 785 790 time 1.85 270.5 271 271.5 272 time

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 42 Mechanism of Dynamics Pan-Am texture with roll pinch-off events creating dislocation pairs (Flow visualization, Pocheau, Le Gal, and Croquette 1985) Mean flow compresses rolls outside of stable band (Model equations, Greenside, MCC, Coughran 1985) Theoretical analysis of mean flow (Pocheau and Davidaud 1997) Numerical simulation of importance of mean flow (Paul, MCC, Fischer, and Greenside 2001)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 43 Lyapunov dimension Define µ(n) = n i=1 λ i exponent. (λ 1 λ 2 ) with λ i the ith Lyapunov D L is the interpolated value of n giving µ = 0 (the dimension of the volume that neither grows nor shrinks under the evolution)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 44 Lyapunov dimension for spiral defect chaos in a periodic geometry (Egolf et al. 2000)

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 45 Previous Results Phase diffusion + focus wavenumber selection D 0 [MCC and Newell (1984)] L t p, p = 1 4 One dimensional numerics on Swift-Hohenberg model [Schober et al. (1986)] p = 1 4 Consistent with 1d phase diffusion with conserved phase winding [Rutenberg and Bray (1995)] Two dimensional numerics on Swift-Hohenberg model [Elder et al. (1992) ] p 1 4 with noise; p 1 5 without noise

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 46 Summary of results For all cases without noise the growth of the correlation length measured from the width of the structure factor is consistent with L t 1/5 (γ 0.2) For the non-relaxational models the correlation length measured from the orientational correlation function appears to follow a different scaling L o t 1/2 The morphology and defect structure of the patterns appeared to be different in the relaxational and non-relaxational cases Relaxational: predominance of patches of straight stripes with sharp boundaries showing up as defect lines Non-relaxational: predominance of smoothly curved stripes with many isolated dislocation defects For the non-relaxational models the asymptotic wave number is not consistent with D (q) = 0 but appears to be the wave number q d for which dislocation climb does not occur.

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 47 Morphology

Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 48 Defects (a),(b) relaxational; (c),(d) non-relaxational

Pattern Formation and Spatiotemporal Chaos - Chennai, 200448-1 R = 2169, σ = 1.4, Ɣ = 57, t = 16, 32, 64, and 128