1. Solitary States in Spatially Forced Rayleigh-Bénard Convection Cornell University (Ithaca, NY) and MPI for Dynamics and Self- Organization (Göttingen, Germany) Jonathan McCoy, Will Brunner EB Supported by NSF-DMR, MPI-DS Werner Pesch University of Bayreuth (Bayreuth, Germany)

2. Striped Patterns in Nature ZebrasSand dunes (photo by Ansel Adams) Spatially extended, complex systems form patterns with characterisitc length scales

3. Convection Patterns Cloud streets over Ithaca (photo by J. McCoy)

4. forcing of patterns How does forcing affect the dynamics? Time periodic forcing is studied in a number of low- dimensional nonlinear systems (van der Pol, Mathieu, etc) å Resonance tongues, Phase-locking, Chaos

5. Spatially extended pattern forming systems offer many spatial and temporal variations on these themes. Examples: Parametric surface waves, Frequency-locking in reaction-diffusion systems, Commensurate/Incommensurate transitions in EC Lowe and Gollub (1983-6); Hartung, Busse, and Rehberg (1991); Ismagilov et al (2002); Semwogerere and Schatz (2002)

6. Commensurate- Incommensurate Transitions Phase solitons (Lowe and Gollub, 1985)

7. Rayleigh-Bénard Convection Horizontal layer of fluid, heated from below Buoyancy instability leads to onset of convection at a critical temp difference Control parameter:  T = T 2 - T 1 Reduced control parameter:  =  T/  T c - 1

10.  f luid: compressed SF 6  pressure: 1.72 ± 0.03 MPa  p. regulation: ±0.3 kPa  mean T: 21.00 ± 0.02 °C  T regulation: ±0.0004 °C  cell height: (0.616 ± 0.015) mm  Prandtl #: 0.86   T c : (1.14 ± 0.02) °C

11. Periodic Forcing of RBC  some parameter of the system: Cell height (geometric parameter) Temperature difference (external control parameter) Gravitational constant (intrinsic parameter) Time periodic forcing (frequency,  ):     1 +  cos(  t)  Spatially periodic forcing (wavenumber, k):     1 +  cos(kx) 

12.  Time-periodic forcing at onset thoroughly investigated  Earlier work on spatial forcing has focused on anisotropic or quasi-1d systems ==> What changes in a 2-dim isotropic system?

13. 1-d forcing in a 2-d system Striped forcing in a large aspect ratio convection cell  One continuous translation symmetry unbroken here: Periodic modulation of cell height by microfabricating an array of polymer stripes on cell bottom

14. 1:1 Resonance

15. Forcing Parameters Cell height: 0.616 ± 0.015 mm Polymer ridges: 0.050 mm high, 0.100 mm wide Modulation wavelength: 1 mm  k f - k c = 0.242 k c  k f close enough to k c for resonance at onset (Kelly and Pal, 1978)

16. Forcing Parameters k f = 1.24 k c

17. I. Resonance at Onset Imperfect Bifurcation (Kelly and Pal, 1978)

18. two predictions imperfect bifurcation (Kelly & Pal 1978) amplitude equations (Kelly and Pal, 1978; Coullet et al., 1986):

19. Cells: Circular cell, with forcing (diameter: 106d) Square reference cell, without forcing (side length: 32d)

20. Forced cell Reference cell

21. Experiment Theory Coherence Length, c1 1.749 (±0.450)1.435 Imperfection Parameter, c2 0.0287 (±0.0004)  = height 0.0423 0.144  Order of magnitude agreement, despite non-sinusoidal forcing: Choosing  d equal to the modulation height or one half this height gives c2 = 0.0423 or 0.0214

22. II. Nonlinear regime How does STC respond to spatially periodic forcing?

23. bulk instability of the forced roll pattern start pattern of forced rolls (recall: wavenumber lies outside of the Busse balloon) Abruptly increase temperature difference, moving system beyond the stability regime of straight rolls Instability modes of the forced rolls are observed before other characteristics emerge

25. Subharmonic resonant structure 3-mode resonance of mode inside the balloon

26. going up 

27. going up 

28. going down 

29. going down 

30. solitary arrays of beaded kinks

31. solitary horizontal beaded array

32. Invasive Structures  = 0.83

33. Invasive Structures II  = 0.91

35. Dynamics of the Kink Arrays Motion preserves zig- and zag- orientation The arrays travel horizontally, climbing along the forced rolls No vertical motion, except for creation and annihilation events Intermittent locking events and reversals of motion

36. Dynamics of the Kink Arrays The diagonal arrays often lock together side-by-side, aligning the kinks to form oblique rolls The oblique roll structures can have defects, curvature, etc.

37. bound kink arrays 3 Mode Resonance

38. 2:1 resonance

39.  = 1.19  = 1.62 SDC ?

40. Summary Part 1 How does a pattern forming system respond when forced spatially outside of the stability region. Observed imperfect bifurcation in agreement with existing theory. Resonances above onset: use modes from inside the stability balloon. Variety of localized states - kinks, beads, …?

41. Part 2 He Hexachaos of inclined layer convection 0.001<  < 0.074 downhill ===>

42. Part 2 He Hexachaos of inclined layer convection 0.001<  < 0.074 drift uphill <===

43. θ = 5° d = 0.3 mm region: 142d x 95d  10 6 images over 35 t h

44. x 78 0.2 t h

45. Isotropic system Penta Hepta Defects (PHD) De Bruyn et al 1996

46. reactions isotropic system

47. anisotropic system:

48. Same Mode Complexes (SMC)

49. Same Mode Complexes (SMC)

50. reactions ==>

51. reactions rates as function of number N of defects

52. reactions rates as function of number N of defects

53. Summary Part 2 complicated state of hexachaos in NOB ILC. earlier theory shows linear in N annihilation. here defect turbulence explainable by two types of defect structures.