Hexagonal Patterns in a Model for Rotating Convection

Madruga, Santiago; Pérez–García, C.

doi:10.1142/s0218127404009107

Cited by 3 publications

(5 citation statements)

References 25 publications

(37 reference statements)

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“…The merging of the lower stability limit of hexagons with the restabilization line also provides an explanation for the large contiguous stability range that was found in rotating non-Boussinesq convection using water (Young et al 2003). An interesting question is how the restabilization interacts with the Hopf bifurcation of the hexagons to oscillating hexagons, which is induced by rotation (Swift 1984;Soward 1985;Echebarria & Riecke 2000;Madruga & Pérez-García 2004).…”

Section: Discussionmentioning

confidence: 86%

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Re-entrant hexagons in non-Boussinesq convection

2006

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While non-Boussinesq hexagonal convection patterns are well known to be stable close to threshold (i.e. for Rayleigh numbers R ≈ R c ), it has often been assumed that they are always unstable to rolls already for slightly higher Rayleigh numbers. Using the incompressible Navier-Stokes equations for parameters corresponding to water as a working fluid, we perform full numerical stability analyses of hexagons in the strongly nonlinear regime (ǫ ≡ R − R c /R c = O(1)). We find 'reentrant' behavior of the hexagons, i.e. as ǫ is increased they can lose and regain stability. This can occur for values of ǫ as low as ǫ = 0.2. We identify two factors contributing to the reentrance: i) the hexagons can make contact with a hexagon attractor that has been identified recently in the nonlinear regime even in Boussinesq convection (Assenheimer & Steinberg (1996);Clever & Busse (1996)) and ii) the non-Boussinesq effects increase with ǫ. Using direct simulations for circular containers we show that the reentrant hexagons can prevail even for side-wall conditions that favor convection in the form of the competing stable rolls. For sufficiently strong non-Boussinesq effects hexagons become stable even over the whole ǫ-range considered, 0 ≤ ǫ ≤ 1.5.

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Section: Discussionmentioning

confidence: 86%

“…An interesting question is how the restabilization interacts with the Hopf bifurcation of the hexagons to oscillating hexagons, which is induced by rotation (Swift (1984);Soward (1985); Echebarria & Riecke (2000); Madruga & Pérez-García (2004)). …”

Section: Discussionmentioning

confidence: 99%

Re-entrant hexagons in non-Boussinesq convection

2006

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“…So far, strongly non-linear NB convection has been studied mostly for large Prandtl numbers [16,24,25], but little is known for small ( P r ≃ 1) or very small Prandtl numbers (P r ≪ 1). In particular, whether reentrant hexagons exist at large ǫ in the presence of the large-scale flows that arise at low P r, and how NB effects impact spiral defect chaos are interesting questions, which we address in this paper.…”

Section: Introductionmentioning

confidence: 99%

Hexagons and spiral defect chaos in non-Boussinesq convection at low Prandtl numbers

Madruga

Riecke

2007

Phys. Rev. E

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We study the stability and dynamics of non-Boussinesq convection in pure gases ͑CO 2 and SF 6 ͒ with Prandtl numbers near PrӍ 1 and in a H 2 -Xe mixture with Pr= 0.17. Focusing on the strongly nonlinear regime we employ Galerkin stability analyses and direct numerical simulations of the Navier-Stokes equations. For Pr Ӎ 1 and intermediate non-Boussinesq effects we find reentrance of stable hexagons as the Rayleigh number is increased. For stronger non-Boussinesq effects the usual, transverse side-band instability is superseded by a longitudinal side-band instability. Moreover, the hexagons do not exhibit any amplitude instability to rolls. Seemingly, this result contradicts the experimentally observed transition from hexagons to rolls. We resolve this discrepancy by including the effect of the lateral walls. Non-Boussinesq effects modify the spiral defect chaos observed for larger Rayleigh numbers. For convection in SF 6 we find that non-Boussinesq effects strongly increase the number of small, compact convection cells and with it enhance the cellular character of the patterns. In H 2 -Xe, closer to threshold, we find instead an enhanced tendency toward roll-like structures. In both cases the number of spirals and of targetlike components is reduced. We quantify these effects using recently developed diagnostics of the geometric properties of the patterns.

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“…The merging of the lower stability limit of hexagons with the restabilization line provides also an explanation for the large contiguous stability range that was found in rotating non-Boussinesq convection using water (Young et al (2003)). An interesting question is how the restabilization interacts with the Hopf bifurcation of the hexagons to oscillating hexagons, which is induced by rotation (Swift (1984);Soward (1985); Echebarria & Riecke (2000); Madruga & Pérez-García (2004)).…”

Section: Discussionmentioning

confidence: 99%

“…However, since the Hopf bifurcation is subcritical in this regime solutions with whirling hexagons still exist. In fact, the whirling activity, which does deform the underlying hexagon pattern [30,31,33], is so strong that it breaks up the hexagonal lattice, introducing many defects. In weakly disordered hexagon patterns the dominant defects are typically penta-hepta defects which consist of two adjacent convection cells that have 5 and 7 immediate neighbors, respectively.…”

Section: Convection With Rotationmentioning

confidence: 99%

Reentrant and whirling hexagons in non-Boussinesq convection

Madruga

Riecke

2007

Eur. Phys. J. Spec. Top.

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We review recent computational results for hexagon patterns in non-Boussinesq convection. For sufficiently strong dependence of the fluid parameters on the temperature we find reentrance of steady hexagons, i.e. while near onset the hexagon patterns become unstable to rolls as usually, they become again stable in the strongly nonlinear regime. If the convection apparatus is rotated about a vertical axis the transition from hexagons to rolls is replaced by a Hopf bifurcation to whirling hexagons. For weak non-Boussinesq effects they display defect chaos of the type described by the two-dimensional complex Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and localized bursting of the whirling amplitude is found. In this regime the coupling of the whirling amplitude to (small) deformations of the hexagon lattice becomes important. For yet stronger non-Boussinesq effects this coupling breaks up the hexagon lattice and strongly disordered states characterized by whirling and lattice defects are obtained.

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Hexagonal Patterns in a Model for Rotating Convection

Cited by 3 publications

References 25 publications

Re-entrant hexagons in non-Boussinesq convection

Re-entrant hexagons in non-Boussinesq convection

Hexagons and spiral defect chaos in non-Boussinesq convection at low Prandtl numbers

Reentrant and whirling hexagons in non-Boussinesq convection

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