Optimal transport in truncated models of Rayleigh-B\'enard convection

We investigate absolute limits on heat transport in a truncated model of Rayleigh-B\'enard convection. Two complementary analyses are used to derive upper bounds in an eight model: a background method analysis and an optimal control approach. In the optimal control formulation the flow no longer obeys an equation of motion, but is instead a control variable. The background method and the optimal control approach produce the same estimate. However, in contrast to a simpler system (i.e., the Lorenz equations) the optimizing flow field---which is observed to be time independent---does not correspond to an exact solution of the equations of motion.