Some unresolved issues in pattern-forming non-equilibrium systems Guenter Ahlers Department of Physics University of California Santa Barbara CA Kapil Bajaj, Nathan Becker, Eberhard Bodenschatz Robert Ecke, Yuchou Hu, Jun Liu, Worawat Meevasana Stephen Morris, Jaechul Oh, Xin-Liang Qiu Sheng-Qi Zhou, and others Supported by the National Science Foundation And the US Department of Energy

Some unresolved issues in pattern-forming non-equilibrium systems or Some questions to which I really would like to know the answer Guenter Ahlers Department of Physics University of California Santa Barbara CA Kapil Bajaj, Nathan Becker, Eberhard Bodenschatz Robert Ecke, Yuchou Hu, Jun Liu, Worawat Meevasana Stephen Morris, Jaechul Oh, Xin-Liang Qiu Sheng-Qi Zhou, and others Supported by the National Science Foundation And the US Department of Energy

Rayleigh-Benard convection Q d DT e = DT/DTc - 1 s = n / k Prandtl number n = kinematic viscosity k = thermal diffusivity z x

DT = 0.1320 K e = 0.003 0.1311 K -0.003 0.1248 K -0.051 1.28x1.28x0.0343 mm3 J. Swift, P. C. Hohenberg, Phys. Rev. A 15, 319 (1977). P. C. Hohenberg, J. Swift, Phys. Rev. A 46, 4 773 (1992) J. Oh and G.A., Phys. Rev. Lett. 91, 094501 (2003).

ec = 6x10-3 Fexp = 7.1x10-4 Fth = 5.1x10-4 d = 34.3mm

p q q 621.999.bmp e = 0.009 021212a/621/*.999*

p q q Director Angle 621.999.bmp e = 0.009 J. Toner and D.R. Nelson, Phys. Rev. B 23, 316 (1981). 021212a/621/*.999*

Do we know why the fluctuations are symmetric About onset? Can we understand the Phonons quantitatively (e.g. Derive their corelation functions)? The dislocation density?

Electroconvection in a nematic liquid crystal z Planar Alignment y x Director V = V0 cos( wt ) Convection for V0 > Vc e = (V0 / Vc) 2 - 1 Anisotropic !

Oblique rolls zig zag Director Normal rolls

e = -0.0013 X.-L. Qiu + G.A., Phys. Rev. Lett. 94, 087802 (2005)

Oblique -roll Normal-roll fluctuations fluctuations Director f = 4000 Hz e = -1.3x103 f = 25 Hz e = -1.3x103

Time series of S(k0, e, t) at e = -1.0x10-2 for (a) f = 25 Hz, k0 = (3.936, 1.968) and (b) f = 4000 Hz, k0 = (4.830, 0)

f = 25 Hz, k0 = (3.936, 1.968)

p direction (all f): q direction (f = 25 Hz): q direction (f = 4 kHz): f = 25 Hz f = 4000 Hz

Normal rolls f = 4000 Hz

Normal rolls Oblique rolls f = 4000 Hz f = 25 Hz

Alternatively: fits of S(k) Oblique: Normal: Oblique: Normal: X.-L. Qiu and G.A., unpublished.

X.-L. Qiu and G.A., unpublished.

Open symbols: from S(k) Solid symbols: from growth rates 4 kHz 25 Hz

Summary At large frequency (i.e. for normal rolls), the results are consistent with the prediction of linear theory. At low frequency (i.e. for oblique rolls), there were deviations from linear theory. S0, s0, and x tended toward a finite limit as e vanished.

What is the problem? At low frequency (i.e. for oblique rolls), there were deviations from linear theory. S0, s0, and x tended toward a finite limit as e vanished. What is the critical behavior expected from the two coupled stochastic GL equations (E. Bodenschatz, W. Zimmermann, and L. Kramer, J. Phys. (Paris) 49, 1875 (1988) ?

? But some skeptics may still believe that the finite limits of What is the problem? At low frequency (i.e. for oblique rolls), there were deviations from linear theory. S0, s0, and x tended toward a finite limit as e vanished. What is the critical behavior expected from the two coupled stochastic GL equations (E. Bodenschatz, W. Zimmermann, and L. Kramer, J. Phys. (Paris) 49, 1875 (1988) ? But some skeptics may still believe that the finite limits of S0 , s0, and x are due to some kind of undetermined experimental problem. So let us do one more thing: MODULATE e

X.-L. Qiu and G.A., unpublished.

Vary d W/2p = 0.1 Hz Black: d = 0.000 Red: d = 0.014 Blue: d = 0.021 5x10-10 / |e| squares: zag circles: zig W/2p = 0.1 Hz Black: d = 0.000 Red: d = 0.014 Blue: d = 0.021 And similarly for the correlation lengths X.-L. Qiu and G.A., unpublished.

Vary W And similarly for the correlation lengths squares: zag circles: zig 5x10-10 / |e| d = 0.021 Black: W/2p = 1.0 Hz Red: W/2p = 0.2 Hz Blue: W/2p = 0.1 Hz And similarly for the correlation lengths X.-L. Qiu and G.A., unpublished.

How about more complicated cases where A and B are amplitudes of travelling waves? Oblique and normal rolls, Lifshitz points? Etc?

Back to Rayleigh-Benard ! Shadowgraph image of the pattern. The sample is viewed from the top. In essence, the method shows the temperature field. Wavenumber Selection by Domain wall

SDC Crosses: Aspect ratio < 60 Solid circles: Aspect ratio > 60 S.W. Morris, E. Bodenschatz, D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026 (1993). Y.-C. Hu, R. Ecke, and G.A., Phys. Rev. E 51, 3263 (1995). SDC Crosses: Aspect ratio < 60 Solid circles: Aspect ratio > 60 Movie by N. Becker

S. W. Morris, E. Bodenschatz, D. S. Cannell, and G. A. , Phys. Rev S.W. Morris, E. Bodenschatz, D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026 (1993).

Can we understand the wavenumber selection by domiain walls? In large samples?

0.14 +/- 0.03 Gamma = 30 sigma = 0.32 J. Liu and G.A., Phys. Rev. Lett. 77, 3126 (1996)

s = 0.8 SDC Crosses: Aspect ratio < 60 S.W. Morris, E. Bodenschatz, D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026 (1993). Y.-C. Hu, R. Ecke, and G.A., Phys. Rev. E 51, 3263 (1995). Movie by N. Becker SDC s = 0.8 Crosses: Aspect ratio < 60 Solid circles: Aspect ratio > 60

s = 0.8 Data from several sources:, Bodenschatz, Ecke, Hu, SB group

Circles: G = 30 triangles: G = 70 J. Liu and G.A., Phys. Rev. Lett. 77, 3126 (1996); and unpub.

Can we understand the onset of spiral-defect chaos as a function of aspect ratio and Prandtl number?

Gamma = 28, sigma = 1.0 J. Liu and G.A., unpublished.

Large Scale Circulation (“Wind of Turbulence”) R. Krishnamurty and L.N. Howard, Proc. Nat. Acad. Sci. 78, 1981 (1981): Large Scale Circulation (“Wind of Turbulence”) R = 6.8x108 s = 596 = 1 cylindrical slightly tilted in real time Movie from the group of K.-Q Xia, Chinese Univ., Hong Kong Why up one side and down the other (rather than in the middle)?

Do we have anything to contribute to the understanding of the formation of relatively “simple” patterns at very large Rayleigh numbers?

n = kinematic viscosity Q d DT e = DT/DTc - 1 W = 2p f d2/ n s = n / k Prandtl number n = kinematic viscosity k = thermal diffusivity

Prandtl = 0.9 CO2 Omega = 17 Movies by N. Becker

Prandtl = 0.51 He-SF6 Omega = 260 X300 K. M. S. Bajaj, J. Liu, B. Naberhuis, and G. A, Phys. Rev. Lett.81, 806 (1998).

Prandtl = 0.17 H2-Xe K.M.S. Bajaj, G. A., and W. Pesch, Phys. Rev. E 65 , 056309 (2002).

X10 Omega = 42

Summary I. Fluctuations 1. Rayleigh-Benard a. Fluctuations above onset b. "Phonons" and dislocations 2. Electro-Convection Finite correlation lengths etc. at onset II. Deterministic patterns, RB 1. Wavenumber selection by domain walls 2. Wavenumber selection in large samples 3. Onset of spiral-defect chaos a.) as a function od aspect ratio b.) as a function of Prandtl number 4. "Simple" patterns at very large Rayleigh a.) R = 20,000 and large aspect ratio b.) R = 108 III. Deterinistic patterns, RB with rotation 1. Squares at onset 2. First-order transition without hysteresis