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.