Large-scale thermal convection in a horizontal porous layer

Goldobin, Denis S.; Shklyaeva, Elizaveta V.

doi:10.1103/physreve.78.027301

Cited by 12 publications

(23 citation statements)

References 14 publications

(26 reference statements)

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“…Note, in the latter equations, the amplitude degree of freedom, which was disregarded in previous works, impacts the instantaneous growth rate of perturbations, but averages out to zero. Thus, on the one hand, our results demonstrate the importance of amplitude degrees of freedom for the stability of response of a general limit cycle oscillator even in the limit of vanishing noise; on the other hand, its average impact turns out to be zero up to the leading order of accuracy for general noise, proving that analytical calculations and conclusions presented in [13,16] are valid for real situations. Notice, the negative Lyapunov exponent and its decrease with increase of the noise strength are related to the stability of the noisy system response in sense that it attracts trajectories (the phenomenon is known as noise-induced synchronization), but this does not mean that the response is regular due to the nonzero phase diffusion.…”

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confidence: 59%

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Dynamics of Limit-Cycle Oscillators Subject to General Noise

Goldobin¹,

Teramae²,

Nakao³

et al. 2010

Phys. Rev. Lett.

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109

110

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The phase description is a powerful tool for analyzing noisy limit cycle oscillators. The method, however, has found only limited applications so far, because the present theory is applicable only to the Gaussian noise while noise in the real world often has non-Gaussian statistics. Here, we provide the phase reduction method for limit cycle oscillators subject to general, colored and non-Gaussian, noise including heavy-tailed one. We derive quantifiers like mean frequency, diffusion constant, and the Lyapunov exponent to confirm consistency of the results. Applying our results, we additionally study a resonance between the phase and noise. Limit cycle oscillators effectively model various sustained oscillations in many fields of science and technology including chemical reactions, biology, electric circuits, and lasers [1][2][3][4]. The phase reduction method is a powerful analytical tool which approximates highdimensional dynamics of limit cycle oscillators with single phase variable that characterizes timing of oscillation [1,5]. Since the phase is neutrally stable, phase perturbations persist in time and result in various remarkable phenomena where weak action leads to significant effects, such as those addressed in the theory of synchronization [6,7]. While the theory of the phase reduction had been developed for deterministic oscillators, recent studies successfully extended the theory to limit cycle oscillators subject to noise [4,8,9] and revealed that interplay between nonlinearity and noise results in fascinating noise-induced phenomena including frequency modulation and noise-induced synchronization [12,13]. This extended phase reduction method, however, has found limited applications so far, since the method is applicable only to Gaussian noise. While the noise in the real world often has non-Gaussian statistics, few theories have considered nonlinear systems subject to general non-Gaussian noise, which has forced people to use the Gaussian approximation. In particular, whether the phase description is still valid for oscillators subject to non-Gaussian noise and how quantifiers of the phase dynamics should be amended remains unknown. In this paper, we develop the phase reduction method for limit cycle oscillators subject to general, colored and nonGaussian noise. By correctly evaluating the influence of amplitude perturbations up to second order in the noise strength, we derive the stochastic differential equation of phase, which allows us to study nonlinear oscillations in the real world without the Gaussian approximation. To confirm consistency of the result, we derive closed expressions of quantifiers of the phase dynamics such as mean frequency, phase diffusion constant, and the Lyapunov exponent. The only limitation we impose is the weakness of the noise. Thus, the obtained results are applicable even when higher order moments of the noise diverge as long as the second order moment is finite and we confirm this fact numerically. As an application of the results, we study a limit cycle oscillato...

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confidence: 59%

“…Thus, for instance, the analytical results and important conclusions of Refs. [15,16] for limit cycle oscillators subject to weak noise and delayed feedback control remain correct. For the leading Lyapunov exponent, the situation is more subtle.…”

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confidence: 89%

Dynamics of Limit-Cycle Oscillators Subject to General Noise

Goldobin¹,

Teramae²,

Nakao³

et al. 2010

Phys. Rev. Lett.

Self Cite

109

110

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“…The properties (7)- (8) are demonstrated in figure 4a with the spectrum of LEs for q 0 = −1. Thus, due to (7) and 9, it is enough to calculate the largest LE γ 1 as a function of u.…”

Section: Spatial Lyapunov Exponentsmentioning

confidence: 99%

“…Frozen-disorder-induced effects in the system can be observed also when the heating is uniform but macroscopic properties of the porous matrix are weakly inhomogeneous (inhomogeneity of porosity, permeability, and heat conductivity is inevitable in real systems). Moreover, the system will be governed by the same equation (1), with the only difference in the relationship between q(x) and physical parameter inhomogeneities [8]. Nonetheless, we consider an inhomogeneous heating in order to make it more obvious that our findings can be observed for convection without a porous matrix as well.…”

Section: Physical Problem Formulationmentioning

confidence: 99%

Localization and advectional spreading of convective currents under parametric disorder

Goldobin

Shklyaeva

2013

J. Stat. Mech.

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We address a problem which is mathematically reminiscent of the one of Anderson localization, although it is related to a strongly dissipative dynamics. Specifically, we study thermal convection in a horizontal porous layer heated from below in the presence of a parametric disorder; physical parameters of the layer are time-independent and randomly inhomogeneous in one of the horizontal directions. Under such a frozen parametric disorder, spatially localized flow patterns appear. We focus our study on their localization properties and the effect of an imposed advection along the layer on these properties. Our interpretation of the results of the linear theory is underpinned by numerical simulation for the nonlinear problem. Weak advection is found to lead to an upstream delocalization of localized current patterns. Due to this delocalization, the transition from a set of localized patterns to an almost everywhere intense "global" flow can be observed under conditions where the disorder-free system would be not far below the instability threshold. The results presented are derived for a physical system which is mathematically described by a modified Kuramoto-Sivashinsky equation and therefore they are expected to be relevant for a broad variety of dissipative media where pattern selection occurs.PACS numbers: 44.25.+f, 72.15.Rn Submitted to: J. Stat. Mech.: Theory Exp.

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“…u is not presented in expression (2) owing to its smallness in comparison to the excited convective currents v. The impact of a weak imposed advective flow on the evolution of temperature perturbations is caused by its symmetry properties: the gross advective flux through the vertical cross-section is u, while the convective flow v possesses zero gross flux and, therefore, yields a less effective heat transfer along the layer [17]. Although equation (1) is valid for a large-scale inhomogeneity q(x), which means h|q x |/|q| ≪ 1, one can set such a hierarchy of small parameters, namely h ≪ (h|q x |/|q|) 2 ≪ 1, that a frozen random inhomogeneity may be represented by white Gaussian noise ξ(x):…”

Section: Problem Formulation and Current State Of Researchmentioning

confidence: 99%

Advectional enhancement of eddy diffusivity under parametric disorder

Goldobin¹

2010

Phys. Scr.

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Frozen parametric disorder can lead to appearance of sets of localized convective currents in an otherwise stable (quiescent) fluid layer heated from below. These currents significantly influence the transport of an admixture (or any other passive scalar) along the layer. When the molecular diffusivity of the admixture is small in comparison to the thermal one, which is quite typical in nature, disorder can enhance the effective (eddy) diffusivity by several orders of magnitude in comparison to the molecular diffusivity. In this paper we study the effect of an imposed longitudinal advection on delocalization of convective currents, both numerically and analytically; and report subsequent drastic boost of the effective diffusivity for weak advection.

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Large-scale thermal convection in a horizontal porous layer

Cited by 12 publications

References 14 publications

Dynamics of Limit-Cycle Oscillators Subject to General Noise

Dynamics of Limit-Cycle Oscillators Subject to General Noise

Localization and advectional spreading of convective currents under parametric disorder

Advectional enhancement of eddy diffusivity under parametric disorder

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