Symmetries and pattern selection in Rayleigh-Bénard convection

Golubitsky, Martin; Swift, James W.; Knobloch, Edgar

doi:10.1016/0167-2789(84)90179-9

Cited by 140 publications

(151 citation statements)

References 22 publications

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“…(39) and, for Φ = 0, π, solutions with pentagonal rather than decagonal symmetry can bifurcate from the trivial state. The situation is analogous to that of the triangle solutions in the case of hexagonal symmetry [25]. The global phase gets fixed by the higher-order resonance term…”

Section: Ginzburg-landau Equationsmentioning

confidence: 86%

Sideband instabilities and defects of quasipatterns

Echebarria

Riecke

2001

Physica D: Nonlinear Phenomena

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Quasipatterns have been found in dissipative systems ranging from Faraday waves in vertically vibrated fluid layers to nonlinear optics. We describe the dynamics of octagonal, decagonal and dodecagonal quasipatterns by means of coupled GinzburgLandau equations and study their stability to sideband perturbations analytically using long-wave equations as well as by direct numerical simulation. Of particular interest is the influence of the phason modes, which are associated with the quasiperiodicity, on the stability of the patterns. In the dodecagonal case, in contrast to the octagonal and the decagonal case, the phase modes and the phason modes decouple and there are parameter regimes in which the quasipattern first becomes unstable with respect to phason modes rather than phase modes. We also discuss the different types of defects that can arise in each kind of quasipattern as well as their dynamics and interactions. Particularly interesting is the decagonal quasipattern, which allows two different types of defects. Their mutual interaction can be extremely weak even at small distances.

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Section: Ginzburg-landau Equationsmentioning

confidence: 86%

Sideband instabilities and defects of quasipatterns

Echebarria

Riecke

2001

Physica D: Nonlinear Phenomena

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“…The expressions given in Bees et al (1998b) can be used for the terms 1p2 and D. To proceed further we should have to consider particular forms for n and & F; for example, we may introduce hexagonal or square planforms (see Buzano andGolubitsky 1983 andGolubitsky et al 1984). These equations may be used in future analysis to predict the three-dimensional patterns in gyrotactic bioconvection and to analyse their stability.…”

Section: Extension To a Three-dimensional Flow Fieldmentioning

confidence: 99%

“…Subcritical bifurcations may imply the existence of stable bioconvecting solutions below the critical parameter value and, hence, below the neutral curve. See, for example, Coullet and Fauve (1985) and Fauve (1985) for discussions on amplitude equations, and Buzano and Golubitsky (1983) and Golubitsky et al (1984) for the general form of amplitude equations subject to spatial symmetrical constraints. It is possible, in most systems, to generate a long wavelength theory of the evolution of initial disturbances close to the critical point (see Childress and Spiegel 1978;Chapman andProctor 1980 andKnobloch 1990).…”

Section: Amplitude Equations For Long Vertical Wavelength Instabilitiesmentioning

confidence: 99%

Non-linear bioconvection in a deep suspension of gyrotactic swimming micro-organisms

Bees

Hill

1999

Journal of Mathematical Biology

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Abstract. The non-linear structure of deep, stochastic, gyrotactic bioconvection is explored. A linear analysis is reviewed and a weakly non-linear analysis justifies its application by revealing the supercritical nature of the bifurcation. An asymptotic expansion is used to derive systems of partial differential equations for long plume structures which vary slowly with depth. Steady state and travelling wave solutions are found for the first order system of partial differential equations and the second order system is manipulated to calculate the speed of vertically travelling pulses. Implications of the results and possibilities of experimental validation are discussed.

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“…29 In both fields, the onset of pattern formation can be studied by weakly nonlinear stability theory, for instance, on pre-imposed lattices. 24,25 The transformation presented in the proof of Lemma 3 that lifts the 1D results from Sec. II A to 2D, has a counterpart in fluid mechanics: the "Squire's transformation."…”

Section: Introductionmentioning

confidence: 99%

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Siero¹,

Doelman²,

Eppinga³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

124

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For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in an arid ecosystem model. We numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations. This is corroborated by numerical simulations with a slowly decreasing water input parameter. Here, long wavelength striped patterns are unstable against transverse perturbations, which we also rigorously prove on flat ground through an Evans function approach. In addition, we prove a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes our model, showing that the onset of pattern formation in 2D is due to 1D instabilities in the direction of advection, which naturally leads to striped patterns. V C 2015 AIP Publishing LLC. This paper has been motivated by studies in one space dimension of a scaled phenomenological model for vegetation on possibly sloped planes in arid ecosystems. 53,60 One-dimensional patterns ideally represent striped patterns in two space dimensions by trivially extending them into a transversal direction. Such patterns are referred to as banded vegetation and have received considerable attention after reports of widespread observations. 8,58 Understanding the appearance and disappearance of vegetation bands may ultimately help prevent land degradation. The restriction to one space dimension may overestimate stability: patterns that are stable against 1D perturbations are not necessarily stable against all 2D perturbations. Natural questions to pose are:• Which of the 1D stable patterns extend to 2D stable striped patterns? • In case of destabilization by 2D perturbations, which mechanisms are responsible?In this paper, we answer these questions for the arid ecosystem model and determine the impact of slope induced advection of water. The influence of advection on striped pattern formation is studied in a more general setting. This approach provides a clear argumentation that is unobscured by model-specific details. Equally important, the results will be applicable to a wide range of models. Applicability to the arid ecosystem model is carefully checked though, assuring that the abstract requirements can in fact be met in practice.

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Symmetries and pattern selection in Rayleigh-Bénard convection

Cited by 140 publications

References 22 publications

Sideband instabilities and defects of quasipatterns

Sideband instabilities and defects of quasipatterns

Non-linear bioconvection in a deep suspension of gyrotactic swimming micro-organisms

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

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