Energy-Conserving Truncations for Convection with Shear Flow

A method is presented for making finite Fourier mode truncations of the Rayleigh--Benard convection system that preserve invariants of the full partial differential equations in the dissipationless limit. These truncations are shown to have no unbounded solutions and provide a description of the thermal flux that has the correct limiting behavior in a steady-state. A particular low-order truncation (containing 7 modes) is selected and compared with the 6 mode truncation of Howard and Krishnamurti (1986), which does not conserve the total energy in the dissipationless limit. A numerical example is presented to compare the two truncations and study the effect of shear flow on thermal transport.