Abstract
In this study, we first present a generalized Lorenz model (LM) with [Formula: see text] modes, where [Formula: see text] is an odd number that is greater than three. The generalized LM (GLM) is derived based on a successive extension of the nonlinear feedback loop (NFL) with additional high wavenumber modes. By performing a linear stability analysis with [Formula: see text] and [Formula: see text], we illustrate that: (1) within the 3D, 5D, and 7D LMs, the appearance of unstable nontrivial critical points requires a larger Rayleigh parameter in a higher-dimensional LM and (2) within the 9DLM, nontrivial critical points are stable. By comparing the GLM with various numbers of modes, we discuss the aggregated negative feedback enabled by the extended NFL and its role in stabilizing solutions in high-dimensional LMs. Our analysis indicates that the 9DLM is the lowest order generalized LM with stable nontrivial critical points for all Rayleigh parameters greater than one. As shown by calculations of the ensemble Lyapunov exponent, the 9DLM still produces chaotic solutions. Within the 9DLM, a larger critical value for the Rayleigh parameter, [Formula: see text], is required for the onset of chaos as compared to a [Formula: see text] for the 3DLM, a [Formula: see text] for the 5DLM, and a [Formula: see text] for the 7DLM. In association with stable nontrivial critical points that may lead to steady-state solutions, the appearance of chaotic orbits indicates the important role of a saddle point at the origin in producing the sensitive dependence of solutions on initial conditions. The 9DLM displays the coexistence of chaotic and steady-state solutions at moderate Rayleigh parameters and the coexistence of limit cycle and steady-state solutions at large Rayleigh parameters. The first kind of coexistence appears within a smaller range of Rayleigh parameters in lower-dimensional LMs (i.e. [Formula: see text] within the 3DLM) but in a wider range of Rayleigh parameters within the 9DLM (i.e. [Formula: see text]). The second kind of coexistence has never been reported in high-dimensional Lorenz systems.
Highlights
Our current view on the predictability of weather and climate has been significantly influenced by the model of Prof
We suggest that the aggregated negative feedback associated with successive extension of the nonlinear feedback loop may eventually change the stability of the nontrivial critical point, turning an unstable critical point in a lower-dimensional Lorenz model (LM) into a stable critical point in a higher-dimensional LM
We presented a generalized Lorenz model (GLM) with M modes, where M is an odd number that is greater than three
Summary
As discussed below, numerous studies have been performed for examining the role of nonlinearity in producing chaotic responses and for revealing the source of chaos as well as the source of recurrence To achieve these goals, high-dimensional Lorenz models (LMs) have been developed [Shen, 2014, 2016, 2017]) in order to generalize the Lorenz 1963 model using many modes to study the impact of an increased degree of nonlinearity, as well as its collective effect with heating and dissipations, on solution stability. Using 3D and 5D nondissipative Lorenz models (NLMs), we have recently studied the role of nonlinear terms in producing recurrence [Shen, 2018; Shen & Faghih-Naini, 2017; Faghih-Naini & Shen, 2018]. In Appendix A, we provide a mathematical analogy between the linearized 9D nondissipative LM and a coupled system with four masses and four springs
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