Domain walls in nonequilibrium systems and the emergence of persistent patterns

Hagberg, Aric; Meron, Ehud

doi:10.1103/physreve.48.705

Cited by 47 publications

(26 citation statements)

References 33 publications

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“…[19][20][21][27][28][29] The velocity of a chemical front in a two-dimensional system can be written as v n ϭ v 0 Ϫ D e , to lowest order in , where v n is the normal velocity of the local front, v 0 is the velocity of a planar front, is the curvature of the front, and D e is a constant, an effective diffusivity. 20,21 If D e Ͼ 0, the local velocity of the front is smaller in a region of large curvature than it is in a region of small curvature.…”

Section: Discussionmentioning

confidence: 99%

“…Our study focuses on the bistable regime where two homogeneous regions can coexist with a sharp chemical front separating the two regions. Analyses of a bistable reaction-diffusion model by Hagberg and Meron [19][20][21][22] show that a variety of patterns can form near a parity breaking bifurcation ͑nonequilibrium Ising-Bloch bifurcation͒, 19,23 where a single chemical front bifurcates into two counter propagating fronts. Figure 1 presents some patterns observed in the neighborhood of a transition that we interpret to be a nonequilibrium Ising-Bloch bifurcation.…”

Section: Introductionmentioning

confidence: 99%

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Transitions in two-dimensional patterns in a ferrocyanide–iodate–sulfite reaction

Li¹,

Ouyang²,

Swinney³

1996

The Journal of Chemical Physics

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Transitions in two-dimensional ͑2D͒ spatial patterns were investigated in a ferrocyanide-iodatesulfite ͑FIS͒ reaction in a circular thin gel reactor. The state of the gel reactor was maintained by contact of one side of the gel with a continuously refreshed well-stirred reservoir. For long residence times of the chemicals in the reservoir, the gel reactor was in a spatially uniform state of low pH ͑about 4͒, while at short reservoir residence times the reactor was in a uniform state of high pH ͑about 7͒. At intermediate residence times the spatiotemporal 2D structures observed include a large low pH oscillating spot, small metastable high pH oscillating spots, shrinking rings, spirals that formed when the axisymmetry of shrinking rings was broken, self-replicating spots that either grew and divided or died from overcrowding, and highly irregular, stationary lamellae. Transitions among the different patterns were examined as a function of gel thickness ͑0.2-0.6 mm͒, reservoir residence time ͑0.6-4 min͒, and ferrocyanide concentration ͑12-80 mM͒. Iodate and sulfite concentrations were held fixed at 75.0 and 89.0 mM, respectively. Several transitions were examined in detail: from a stationary spot to an oscillating spot; from an oscillating spot to a shrinking ring or spirals; the onset of replicating spots; and the transition from a homogeneous state to lamellar patterns. The observed phenomena can all be described in terms of a parity-breaking front bifurcation ͑nonequilibrium Ising-Bloch bifurcation͒.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Transitions in two-dimensional patterns in a ferrocyanide–iodate–sulfite reaction

Li¹,

Ouyang²,

Swinney³

1996

The Journal of Chemical Physics

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“…Equation (3.11) is the time-dependent Ginzburg-Landau equation for the real field, f (e.g. Hagberg & Meron 1993, 1994, and has three steady uniform solutions f ± , f 0 corresponding to (3.7) for γ = 0. Note that (3.11) is a 'gradient system' given by…”

Section: Stability Of the Ice-stream Shear Marginsmentioning

confidence: 99%

Spatiotemporal dynamics of ice streams due to a triple-valued sliding law

Sayag

Tziperman

2009

J. Fluid Mech.

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We show that a triple-valued sliding law can be heuristically motivated by the transverse spatial structure of an ice-stream velocity field using a simple one-dimensional model. We then demonstrate that such a sliding law can lead to some interesting stream-like patterns and time-oscillatory solutions. We find a generation of rapid stream-like solutions within a slow ice-sheet flow, separated by narrow internal boundary layers (shear margins), and analyse numerical simulations in two horizontal dimensions over a homogeneous bed and including longitudinal shear stresses. Different qualitative behaviours are obtained by changing a single physical parameter, a mass source magnitude, leading to changes from a slow creeping flow to a relaxation oscillation of the stream pattern, and to steady ice-stream-like solution. We show that the adjustment of the ice-flow shear margins to changes in the driving stress in the one-dimensional approximation is governed by a form of the Ginzburg–Landau equation and use stability analysis to understand this adjustment. In the model analysed here, the width scale of the stream is not set spontaneously by the ice flow dynamics, but rather, it is related to the mass source intensity and spatial distribution.

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“…The very recent observation of patterns induced by interacting fronts [102], in a different reaction [103], is probably related to Turing-saddle-node interactions, still little documented even from the theoretical point of view. Such front patterns as well as more classical Turing patterns resulting from a subcritical bifurcation have large amplitUde at onset and can exhibit amazing pattern growth dynamics, some reminiscent of fingering [104], others of cell division [31,68].…”

Section: Resultsmentioning

confidence: 99%