On the relation between Rayleigh–Bénard convection and Lorenz system

Dedicated to Prof. Mohamed Salah El Naschie on his 60th birthday.
https://doi.org/10.1016/j.chaos.2005.08.010Get rights and content

Abstract

Based on an extension of the Lorenz truncation scheme, a chaotic mathematical model is developed to provide a profile of the chaotic attractor associated with the Rayleigh–Bénard convection problem in a plane fluid motion. The attractor of the Lorenz system is a cross-section of the attractor of the proposed model, in which solutions always exist in circles mirroring those appearing in the convection problem.

Introduction

Since the seminal study of Lorenz [1], chaos occurs in the real world as observed in a wide variety of nonlinear dynamical systems. For example, Ruelle and Takens [2] show low-dimensional chaos in a fluid motion, Li and Yorke [3] display chaos in a one-dimensional discrete dynamical system, El Naschie and Elnashaie [4], El Naschie [5] and El Naschie et al. [6] characterize a turbulent fluid flow as soliton loop chaos. In this study we examine the Rayleigh–Bénard problem [7], [8] from the viewpoint of an analysis based on the investigation of Lorenz [1].

Convection in a horizontal layer heated from below is known as the Rayleigh–Bénard problem, which stems from the laboratory experiments of Bénard [9], [10] and the theoretical study of Rayleigh [11]. In a study of atmospheric fluid motion, Lorenz [1] derived a three-mode truncation model of the Rayleigh–Bénard convection problem and suggested that difficulties of accuracy are inherent in very-long-range forecasting because this model gives rise to chaotic behaviour.

However, a comparison of results derived for the Rayleigh–Bénard convection problem in a plane domain and the Lorenz system shows significant differences. For example, if the Rayleigh number R is less than its first critical value, a state of zero convection of the Rayleigh–Bénard or the basic steady-state solution of the Lorenz model is stable and attracts globally. When R increases across the first critical Rayleigh number value, the state of zero convection bifurcates into a stable attractor consisting of a circle of steady-state solutions describing the Rayleigh–Bénard convection behaviour, whereas in the Lorenz model the stable attractor bifurcating away from the basic steady-state solution is a combination of a pair of steady-state solutions. In fact, solutions of the Rayleigh–Bénard convection always exist in circles, but solutions of the Lorenz system always exist in pairs. The topological structures of the attractors of these two systems are always different when the Rayleigh number R increases across its first critical value.

It is the purpose of this study to reconcile these differences by introducing a truncation model in an extended Lorenz description to display a profile of the chaotic attractor of the convection problem. This model encapsulates the property that solutions always exist in circles, which is inherent in Rayleigh–Bénard convection. Furthermore, this system retains the same attractor to describe Rayleigh–Bénard convection when the Rayleigh number is just above its first critical value, whereas the attractor of the Lorenz system is always a cross-section of the attractor of the derived new system for any value of Rayleigh number.

Section snippets

Rayleigh–Bénard convection problem in a plane domain and Lorenz system

The mathematical model described herein is derived by developing the Lorenz truncation scheme. Therefore, in this section, to aid understanding, we present a derivation of the Lorenz system to describe the Rayleigh–Bénard convection problem.

The Rayleigh–Bénard convection problem (see, for example, [7], [8], [9]) describing a fluid motion in a layer of fluid of uniform depth H heated from below is based on the Boussinesq approximation applied to a coupled system involving the Navier–Stokes

Five-mode truncation model of the Rayleigh–Bénard convection problem

As an extension to the Lorenz [1] mathematical model, we develop a truncation scheme to include five modes, of which three modes are used by Lorenz [1] to ensure the occurrence of chaotic behaviour and the other two modes are chosen to retain symmetry of the Rayleigh–Bénard convection in the truncation model. This model is a generalized form of the Lorenz system.

It is readily seen that the following property is valid: (ψ(t, x, z), θ(t, x, z)) is a solution of (4), (5), (6) if and only if (ψ(t, x + ϕ, z),

Conclusion

Based on the previously proposed Lorenz truncation scheme to describe Rayleigh–Bénard convection in a plane domain, we use the two modessinaπxsinπzandcosaπxsinπz,in the truncation of the stream function ψ and the three modessinaπxsinπz,cosaπxsinπzandsin2πz,in the truncation of the thermal function θ to derive a five-mode truncation model of the Rayleigh–Bénard convection problem. This mathematical model shows that solutions always exist in circles reflecting the similar property inherent to the

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