Intermittency through modulational instability

Bretherton, C. S.; Spiegel, E. A.

doi:10.1016/0375-9601(83)90491-7

Cited by 138 publications

(76 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Higher order derivatives enter with higher powers of (; (<< 1) and are neglected. See [11,[13][14][15][16] for details. Numerical simulations of these equations have revealed behavior that qualitatively resembles some of that found in the experiments,' including traveling waves and blinking states.…”

Section: [V = A(x T)e Iwt W = B(x T)e-iwt ]mentioning

confidence: 99%

Oscillatory bifurcation with broken translation symmetry

Landsberg

Knobloch

1996

Phys. Rev. E

View full text Add to dashboard Cite

The effect of distant end walls on the bifurcation to traveling waves is considered. Previous approaches have treated the problem by assuming that.it is a weak perturbation of the translation invariant problem. When the problem is formulated instead in a finite box of length L and the limit L --+ 00 is taken, one obtains amplitude equations that differ from the usual Ginzburg-Landau description by the presence of an additional nonlinear term. This formulation leads to a description in terms of the amplitudes of the primary box modes, which are odd and even parity standing waves. For large L, the equations that result take the form of a Hopf bifurcation with approximate D4 symmetry. These equations are able to describe, qualitatively, not only traveling and "blinking" states, but also asymmetrical blinking states and "repeated transients," all of which have been observed in binary fluid convection experiments. PACS number(s): 47.20. Bp, 47.20.Ky, 03.40.Kf

show abstract

Section: [V = A(x T)e Iwt W = B(x T)e-iwt ]mentioning

confidence: 99%

Oscillatory bifurcation with broken translation symmetry

Landsberg

Knobloch

1996

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…This bifurcation is indicated in Fig. 1(a) by a solid dot, and in domains of sufficiently large horizontal extent leads to a spatio-temporally chaotic state known as dispersive chaos [11,22]. The time-independent convectons discussed here emerge from this state via relaxation oscillations as described in [5].…”

mentioning

confidence: 83%

Dissipative solitons in binary fluid convection

Mercader¹,

Batiste²,

Alonso³

et al. 2011

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

Abstract.A horizontal layer containing a miscible mixture of two fluids can produce dissipative solitons when heated from below. The physics of the system is described, and dissipative solitons are computed using numerical continuation for three distinct sets of experimentally realizable parameter values. The stability of the solutions is investigated using direct numerical integration in time and related to the stability properties of the competing periodic state.1. Introduction. Many fluid systems exhibit spatially localized structures in both two [2,4,5,7,9,14,16] and three [10,23] dimensions. Of these the localized structures or convectons arising in binary fluid convection are perhaps the best studied. These states are similar to localized structures studied in other areas of physics [1] despite the fact that fluid systems must always be confined between boundaries. On the other hand in fluid systems the length scale is typically set by the layer depth or the distance between any confining boundaries instead of being an intrinsic length scale selected by a Turing or modulational instability. As a result when we speak of localized states in binary fluid convection we mean states that are localized in the horizontal direction only. In this sense the problem resembles laser systems in short cavities in which the standing wave structure in the longitudinal direction remains of paramount importance [15].In fluids dissipation, whether through viscosity or thermal diffusion, is generally of great importance. For example, it is responsible for the presence of a finite threshold value of the Rayleigh number, a dimensionless measure of thermal forcing, for convection to occur. As a result the solitons of interest in the present article are strongly dissipative and hence require strong forcing for their maintenance. States of this type cannot therefore be understood in terms of (an infinite-dimensional) Hamiltonian system with small forcing and dissipation.In this paper we study localized states in binary fluid convection in a horizontal layer of depth h heated from below. Binary liquids, such as water-ethanol and watersalt mixtures or mixtures of He 3 -He 4 at cryogenic temperatures, are characterized by a cross-diffusion effect called the Soret effect that describes the diffusive separation

show abstract

“…dx Equation (26) is the same as that governing the first transition which takes place at wave number q0 given by (5). The lowest order eigenvalue, A0, also plays a role in the present calculation.…”

Section: Paul K Newton and Lawrence Sirovichmentioning

confidence: 99%

“…Introduction. The Ginzburg-Landau (G-L) amplitude equation governing the modulation of quasi-monochromatic waves in fluid systems with supercritical dimensionless parameter (e.g., Reynolds number) has been the focus of many recent studies on transition to chaos [1][2][3][4][5][6][7][8][9]. The general form of this equation is A, ~(Xr + i^i)Axx = arA -(ft + iPi)\A\2A (1) where Ar, ar, fir, fit are real quantities and under a suitable renormalization it may be written as [8] iA, +(1 -ic0)Axx = ipA -(1 + ip)\A\2A (2) where 0 < c20 < q, p = co/cj.…”

mentioning

confidence: 99%