Non-linear properties of thermal convection

Busse, Friedrich H.

doi:10.1088/0034-4885/41/12/003

Cited by 914 publications

(552 citation statements)

References 91 publications

(79 reference statements)

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“…This structure is quite similar to the so-called « skewed-varicose » observed in Rayleigh-B6nard convection [4,6]. However, it is a stationary state in our system, contrary to Rayleigh-B6nard convection.…”

Section: The Defects In the Zig-zag Structure -supporting

confidence: 86%

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Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

1986

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Abstract. -We present the first spontaneous sequence of hydrodynamic instabilities due to the evolution of the convective flow from the rest state toward chaos in an anisotropic fluid : a nematic liquid crystal subjected to an AC electric field. The resulting structures are of decreasing symmetry as the control parameter is gradually increased. The study of the streamlines allows the interpretation of the entire sequence by the evolution of the convective flow up to the onset of chaos. We show that the first important step in the flow evolution is the build up of a second mode of rotation which is associated with a pinching instability of the vortex lines. A simple picture is then proposed in order to describe the essential steps of a possible route to chaos, and which could be extended to other convective systems.

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Section: The Defects In the Zig-zag Structure -supporting

confidence: 86%

“…Physique 47 (1986) 595 [1,2]. On the other hand spatial structures of decreasing symmetry are observed, depending on both the Rayleigh number (the constraint or control parameter related to the temperature gradient) and the Prandtl number [3][4][5][6]. However [7,8].…”

mentioning

confidence: 99%

Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

1986

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“…The representation of stable patterns by Busse balloons originates from this field. 4 Although fundamentally different, similar patterns exist, e.g., context striped patterns are commonly called roll-waves and the transverse instabilities of striped patterns we find correspond to certain "oblique-roll" instabilities. 29 In both fields, the onset of pattern formation can be studied by weakly nonlinear stability theory, for instance, on pre-imposed lattices.…”

Section: Introductionmentioning

confidence: 66%

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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“…In pure fluids, convection in the form of parallel rolls is a stable form of convection directly above onset for all values of P r [13], although they compete with spiral defect chaos at low Prandtl numbers where the latter structure has a larger basin of attraction [14]. In binary mixtures, they exist also as stable patterns above Ra c for all positive ψ if Le is large enough to equilibrate the concentration sufficiently.…”

Section: Rollsmentioning

confidence: 95%

Convective Patterns in Binary Fluid Mixtures with Positive Separation Ratios

Huke

Lücke

2002

Thermal Nonequilibrium Phenomena in Fluid Mixtures

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Abstract. We summarize our findings about laterally periodic convection structures in binary mixtures in the Rayleigh-Bénard system for positive Soret effect. Stationary roll, square, and crossroll solutions and their stability are determined with a multimode Galerkin expansion. The oscillatory competition of squares and rolls in the form of crossroll oscillations is reviewed. They undergo a subharmonic bifurcation cascade where the oscillation period grows in integer steps as a consequence of an entrainment process.

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Non-linear properties of thermal convection

Cited by 914 publications

References 91 publications

Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

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

Convective Patterns in Binary Fluid Mixtures with Positive Separation Ratios

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