solitary states in spatially forced rayleigh-bénard convection cornell university (ithaca, ny) and...
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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 PeschUniversity of Bayreuth (Bayreuth, Germany)
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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
==> 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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1:1 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 Parameterskf = 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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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
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= 0.83
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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 ModeResonance
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2:1 resonance
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= 1.19 = 1.62
SDC ?
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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 2HeHexachaos of inclined layer convection0.001< < 0.074
downhill ===>
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Part 2HeHexachaos of inclined layer convection0.001< < 0.074
drift uphill <===
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θ = 5°d = 0.3 mmregion: 142d x 95d106 images over 35 th
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x 780.2 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
==>
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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.