Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations

A two-dimensional and dissipative Rayleigh–Bénard convection can be approximated by the Lorenz model, which was originally derived by taking into account only three Fourier modes. Numerous attempts have been made to generalize this 3D model to higher dimensions and several different methods of selecting Fourier modes have been proposed. In this paper, generalized Lorenz models with dimensions ranging from four to nine are constructed by selecting vertical modes that conserve energy in the dissipationless limit and lead to systems that have bounded solutions. An interesting result is that the lowest-order generalized Lorenz model, which satisfies these criteria, is a 9D model and that its route to chaos is the same as that observed in the original 3D Lorenz model. The latter is in contradiction to some previous results that imply different routes to chaos in several generalized Lorenz systems. This discrepancy is explained by the fact that previously obtained generalized Lorenz models were derived by using different methods of selecting Fourier modes and, as a result, most of them did not obey the principle of conservation of energy in dissipationless limit.