Bounding heat transport for a model of Rayleigh-Benard convection using sum-of-squares optimization

We determine bounds on the maximum rate of heat transport (Nusselt number) for an 8-ODE model of Rayleigh-Benard convection developed by Gluhovsky et al. (2002). This truncated model is distinguished in the sense that it obeys many desired conservation laws and physical properties of the PDE. We use a general framework for bounding infinite-time averages in dynamical systems, which is similar to the use of Lyapunov functions in stability theory. Applying this framework, the maximal heat transport problem is computed numerically using sum-of-squares optimization. New upper bounds are established for the truncated system that are sharper than previously known bounds derived by Souza and Doering (2015). Additionally, the numerical results are used to inform the construction of new analytical bounds for the truncated model.