By analyzing the mode-mode interaction in the 3DLM, we previously pointed out that the two nonlinear terms of the 3DLM form a nonlinear feedback loop (NFL) representing a pair of downscaling and upscaling processes. The NFL may be viewed as the main trunk of a tree. By extending the NFL that can provide negative nonlinear feedback, we derived high-dimensional LMs that are more predictable. The extension part can be viewed as a branch of a tree. By performing mathematical analysis and calculating the numerical solutions of the non-dissipative models, we discuss the role of the extended feedback loop in producing quasi-periodic solutions with two or more incommensurate frequencies. Compared to the 3D-NLM, the 5D-NLM additionally introduces two 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 disabling one of the two pairs of the downscaling and upscaling processes in the models, we can reveal the importance of two-way interactions in producing quasi-periodic solutions. A system with a one-way interaction always produces periodic solutions with two commensurate frequencies. This analysis is also applied to analyze higher-dimensional LMs.
The 3DLM has three different kinds of solutions that are dependent on its normalized Rayleigh parameter, (or the heating term), including steady state, chaotic, and limit cycle solutions for small, moderate, and large Rayleigh parameters, respectively. As compared to the 3DLM (or 3D-NLM), the high-dimensional LMs (or NLMs) may have solutions that display similar or different features. For example, a bifurcation near the saddle point of the 3D-NLM and 5D-NLM can be very different. To effectively compare the solutions in the models, we develop Python codes for multi-panel animations that are synchronized in time. Using 3D animations, we can clearly show that a quasi-periodic solution trajectory in the high-dimensional NLMs or limit torus in the LMs moves endlessly on a torus but never intersects itself, supporting the statement “weather never repeats itself”. As a much higher-dimensional LM can be derived based on the further extension of the NFL, the simplicity of the Python code style provides great potential for analyzing such a higher-dimensional LM (e.g., with hundreds of Fourier modes), which is discussed at the end.