Energy-conserving low-order models for three-dimensional Rayleigh-Bénard convection

Constructing hydrodynamic low-order models in the form of coupled gyrostats eliminates the possibility of certain unphysical behaviors, such as solutions diverging to infinity, that often appear in models resulting from ad hoc truncations of Galerkin approximations. In this paper, a simple low-order model in a gyrostatic form that conserves energy in the dissipationless limit (Model I) is constructed for three-dimensional (3D) Rayleigh-Bénard convection. It can be considered an energy-conserving extension of the model by Das et al. [Phys. Rev. E 62, R3051 (2000)] (Model II) that does not conserve energy and possesses solutions diverging to infinity. Also studied here is a smaller but energy-conserving subsystem of Model I that has the form of two coupled gyrostats (Model III). This new system is the 3D analog of the celebrated Lorenz model [J. Atmos. Sci. 20, 130 (1963)]. Stability diagrams and heat transport behavior are calculated and compared for the three models. Model I has improved qualitative agreement with experimental observations compared to that of Model II and Model III.