Affiliation:
1. Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182, USA
Abstract
A recent study suggested that the nonlinear feedback loop (NFL) of the three-dimensional nondissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a nontrivial critical point. This study discusses the role of the extension of the NFL in producing quasi-periodic trajectories using a five-dimensional nondissipative Lorenz model (5D-NLM). An analytical solution to the locally linear 5D-NLM is first obtained to illustrate the association of the extended NFL and two incommensurate frequencies whose ratio is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself. While the NFL of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended NFL within the 5D-NLM additionally introduces two new pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. One pair of downscaling and upscaling processes provides a two-way interaction between the original (primary) Fourier modes of the 3D-NLM and the newly-added (secondary) Fourier modes of the 5D-NLM. The other pair of downscaling and upscaling processes involves interactions amongst the secondary modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that the two-way interaction is crucial for producing the quasi-periodic solution. A follow-up study using a 7D nondissipative LM shows that a further extension of NFL, which may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection, is capable of producing one more incommensurate frequency.
Publisher
World Scientific Pub Co Pte Lt
Subject
Applied Mathematics,Modeling and Simulation,Engineering (miscellaneous)
Cited by
22 articles.
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