Solitary States in Spatially Forced RayleighBnard Convection Jonathan

Solitary States in Spatially Forced Rayleigh-Bénard Convection Jonathan Mc. Coy, 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

Convection Patterns Cloud streets over Ithaca (photo by J. Mc. Coy)

forcing of patterns How does forcing affect the dynamics? Time periodic forcing is studied in a number of lowdimensional 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 = T 2 - T 1 Reduced control parameter: = T/ Tc - 1

Ø Ø Ø Ø fluid: pressure: compressed SF 6 1. 72 ± 0. 03 MPa p. regulation: ± 0. 3 k. Pa 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)

Ø Time-periodic forcing at onset thoroughly investigated Ø Earlier work on spatial forcing has focused on anisotropic or quasi-1 d 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 Parameters kf = 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: 106 d) • Square reference cell, without forcing (side length: 32 d)

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

going down

solitary arrays of beaded kinks

solitary horizontal beaded array

Invasive Structures = 0. 83

Dynamics of the Kink Arrays • Motion preserves zig- and zagorientation • 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 Mode Resonance

2: 1 resonance

SDC ? = 1. 19 = 1. 62

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 2 He. Hexachaos of inclined layer convection 0. 001< < 0. 074 downhill ===>

Part 2 He. Hexachaos of inclined layer convection 0. 001< < 0. 074 drift uphill <===

θ = 5° d = 0. 3 mm region: 142 d x 95 d 106 images over 35 th

x 78 0. 2 th

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

reactions isotropic system

anisotropic system:

Same Mode Complexes (SMC)

reactions ==>

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.