1. 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

2. 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

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

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

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

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

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

8. 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?

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

10. Oblique rolls
zig
zag
Director
Normal rolls

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

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

13. 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)

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

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

16. Normal rolls
f = 4000 Hz

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

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X.-L. Qiu and G.A., unpublished.

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 X.-L. Qiu and G.A., unpublished.

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

21. 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.

22. 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) ?

23. ? 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

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 X.-L. Qiu and G.A., unpublished.

25. 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.

26. 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.

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

28. 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

29. 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

30. 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).

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

32. 0.14 +/- 0.03
Gamma = 30 sigma = 0.32
J. Liu and G.A., Phys. Rev. Lett. 77, 3126 (1996)

33. 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

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

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

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

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

38. 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)?

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

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

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

42. 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).

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

44. X10 Omega = 42

45. 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