Convection is buoyancy-driven flow resulting from unstable density stratification in the presence of a gravitational field. Beyond its central role in myriad engineering heat transfer applications, convection underlies many of nature’s dynamical designs on larger-than-human scales. Indeed, solar heating of Earth’s surface generates buoyancy forces that cause the winds to blow, which in turn drive the oceans’ flow. Convection in Earth’s mantle on geological timescales makes the continents drift, and thermal and compositional density differences induce buoyancy forces that drive a dynamo in Earth’s liquid metal core—the dynamo that generates the magnetic field protecting us from solar wind that would otherwise extinguish life as we know it on the surface. The structure of the Sun itself relies on convection in the outer layers to transfer heat from the interior to radiate away from the surface.
The key feature of convection is transport: thermal convection actively transports the heat that generates the density variations that produce the buoyancy forces. Determining the rate at which “heat rises” in turbulent convection is one of the most important open problems in fluid dynamics. In this presentation the confounding question of asymptotically high Rayleigh number heat transport in Rayleigh-Bénard convection – the buoyancy-driven flow in a horizontal layer of fluid heated from below modeled by the Boussinesq approximation to the Navier-Stokes equations – is reviewed from viewpoints of theory (models of the model), computation (direct numerical simulations), experiment (laboratory tests), and mathematical analysis (theorems).