
RayleighBénard convection is the buoyancydriven flow of a fluid heated from below and cooled from above. Heat transport by convection an important physical process for applications in engineering, atmosphere and ocean science, and astrophysics, and it serves as a fundamental paradigm of modern nonlinear dynamics, pattern formation, chaos, and turbulence. Determining the bulk transport properties of high Rayleigh number convection turbulent convection remains a grand challenge for experiment, simulation, theory, and analysis. In this talk, after a general survey of the theory and applications of RayleighBénard convection we describe recent results for mathematically rigorous upper limits on the vertical heat transport in two dimensional RayleighBénard convection between stressfree isothermal boundaries derived from the Boussinesq approximation of the NavierStokes equations. These bounds challenge some popular theoretical arguments regarding the asymptotic high Rayleigh number heat transport scaling.

