  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.
