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Abstract

The classic Lorenz equations were originally derived from the two-dimensional Rayleigh–Bénard convection system considering an idealized case with the lowest order of harmonics. Although the low-order Lorenz equations have traditionally served as a minimal model for chaotic and intermittent atmospheric motions, even the dynamics of the two-dimensional Rayleigh–Bénard convection system is not fully represented by the Lorenz equations, and such differences have yet to be clearly identified in a systematic manner. In this paper, the convection problem is revisited through an investigation of various dynamical behaviors exhibited by a two-dimensional direct numerical simulation (DNS) and the generalized expansion of the Lorenz equations (GELE) derived by considering additional higher-order harmonics in the spectral expansions of periodic solutions. Notably, GELE allows us to understand how nonlinear interactions among high-order modes alter the dynamical features of the Lorenz equations including fixed points, chaotic attractors, and periodic solutions. It is verified that numerical solutions of the DNS can be recovered from the solutions of GELE when we consider the system with sufficiently high-order harmonics. At the lowest order, the classic Lorenz equations are recovered from GELE. Unlike in the Lorenz equations, we observe limit tori, which are the multi-dimensional analog of limit cycles, in the solutions of the DNS and GELE at high orders. Initial condition dependency in the DNS and Lorenz equations is also discussed.




The Lorenz equations are a simplified nonlinear dynamical system derived from the two-dimensional Rayleigh–Bénard (RB) convection problem. They have been one of the best-known examples in chaos theory due to the peculiar bifurcation and chaos behaviors. They are often regarded as the minimal chaotic model for describing the convection system and, by extension, weather. Such an interpretation is sometimes challenged due to the simplifying restriction of considering only a few harmonics in the derivation. This study loosens this restriction by considering additional high-order harmonics and derives a system we call the generalized expansion of the Lorenz equations (GELE). GELE allows us to study how solutions transition from the classic Lorenz equations to high-order systems comparable to a two-dimensional direct numerical simulation (DNS). This study also proposes mathematical formulations for a direct comparison between the Lorenz equations, GELE, and two-dimensional DNS as the system’s order increases. This work advances our understanding of the convection system by bridging the gap between the classic model of Lorenz and a more realistic convection system