Solitary States in Spatially Forced Rayleigh-B

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Solitary States in Spatially Forced
Rayleigh-Bénard Convection
Jonathan McCoy, Will Brunner EB
Cornell University (Ithaca, NY) and MPI for
Dynamics and Self-Organization (Göttingen,
Germany)
Werner Pesch
University of Bayreuth (Bayreuth, Germany)
Supported by NSF-DMR, MPI-DS
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Striped Patterns in Nature
Zebras
Sand dunes (photo by Ansel Adams)

• Spatially extended, complex systems form
  patterns
• with characterisitc length scales

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Convection Patterns
Cloud streets over Ithaca (photo by J. McCoy)
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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)
• Resonance tongues, Phase-locking, Chaos

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• 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)
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Commensurate-Incommensurate Transitions
Phase solitons (Lowe and Gollub, 1985)
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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 T2 - T1
Reduced control parameter ? ?T/ ?Tc - 1
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• fluid compressed SF6
• 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
• ?Tc (1.14 0.02) C

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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)?

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

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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

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11 Resonance
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Forcing Parameters

• Cell height 0.616 0.015 mm
• Polymer ridges 0.050 mm high, 0.100 mm wide
• Modulation wavelength 1 mm
• ? kf - kc 0.242 kc
• ? kf close enough to kc for resonance at onset
• (Kelly and Pal, 1978)

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Forcing Parameters
kf 1.24 kc
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I. Resonance at Onset
Imperfect Bifurcation (Kelly and Pal, 1978)
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two predictions

• imperfect bifurcation (Kelly Pal 1978)
• amplitude equations (Kelly and Pal, 1978
• Coullet et
  al., 1986)

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Cells

• Circular cell, with forcing (diameter 106d)
• Square reference cell, without forcing (side
  length 32d)

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Forced cell
Reference cell
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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
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II. Nonlinear regime
How does STC respond to spatially periodic
forcing?
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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

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Subharmonic resonant structure

• 3-mode resonance of mode inside the balloon

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going up???????
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going up??????
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going down???????
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going down???????
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solitary arrays of beaded kinks
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solitary horizontal beaded array
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Invasive Structures
? 0.83
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Invasive Structures II
? 0.91
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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

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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.

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bound kink arrays
3 Mode Resonance
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21 resonance
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? 1.19
SDC ?
? 1.62
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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, ?

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Part 2
HeHexachaos of inclined layer convection
0.001lt ? lt 0.074
downhill gt
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Part 2
HeHexachaos of inclined layer convection
0.001lt ? lt 0.074
drift uphill lt
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? 5 d 0.3 mm region 142d x
95d ????????? 106 images over 35 th
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x 78 0.2 th
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Isotropic system Penta Hepta Defects (PHD)

De Bruyn et al 1996
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reactions isotropic system

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anisotropic system

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Same Mode Complexes (SMC)

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Same Mode Complexes (SMC)

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reactions
gt
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reactions rates as function of number N of
defects
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reactions rates as function of number N of
defects
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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.