Post on 17-Jan-2016
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)
Convection Patterns
Cloud streets over Ithaca (photo by J. McCoy)
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
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)
Commensurate-Incommensurate Transitions
Phase solitons (Lowe and Gollub, 1985)
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
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
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)
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?
•
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
1:1 Resonance
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)
Forcing Parameterskf = 1.24 kc
I. Resonance at Onset
Imperfect Bifurcation (Kelly and Pal, 1978)
two predictions
• imperfect bifurcation (Kelly & Pal 1978)
• amplitude equations (Kelly and Pal, 1978; Coullet et al., 1986):
Cells:
• Circular cell, with forcing (diameter: 106d) • Square reference cell, without forcing (side length: 32d)
Forced cell
Reference cell
II. Nonlinear regime
How does STC respond to spatially periodic forcing?
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
Subharmonic resonant structure
• 3-mode resonance of mode inside the balloon
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
solitary horizontal
beaded array
Invasive Structures
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= 0.83
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
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.
bound kink arrays
3 ModeResonance
2:1 resonance
= 1.19 = 1.62
SDC ?
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, …?
Part 2HeHexachaos of inclined layer convection0.001< < 0.074
downhill ===>
Part 2HeHexachaos of inclined layer convection0.001< < 0.074
drift uphill <===
θ = 5°d = 0.3 mmregion: 142d x 95d106 images over 35 th
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x 780.2 th
Isotropic system Penta Hepta Defects
(PHD)
De Bruyn et al 1996
reactions isotropic system
anisotropic system:
Same Mode Complexes (SMC)
Same Mode Complexes (SMC)
reactions
==>
reactions rates as function of
number N of defects
reactions rates as function of
number N of defects
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.