  Upper bounds on time-averaged heat transport are obtained for an eight-mode
Galerkin truncation of Rayleigh's 1916 model of natural thermal convection.
Bounds for the ODE model---an extension of Lorenz's three-ODE system---are
derived by constructing auxiliary functions that satisfy sufficient conditions
wherein certain polynomial expressions must be nonnegative. Such conditions are
enforced by requiring the polynomial expressions to admit sum-of-squares
representations, allowing the resulting bounds to be minimized using
semidefinite programming. Sharp or nearly sharp bounds on mean heat transport
are computed numerically for numerous values of the model parameters: the
Rayleigh and Prandtl numbers and the domain aspect ratio. In all cases where
the Rayleigh number is small enough for the ODE model to be quantitatively
close to the PDE model, mean heat transport is maximized by steady states. In
some cases at larger Rayleigh number, time-periodic states maximize heat
transport in the truncated model. Analytical parameter-dependent bounds are
derived using quadratic auxiliary functions, and they are sharp for
sufficiently small Rayleigh numbers.
