Rayleigh Bénard convection: bounds on the Nusselt number

Rayleigh-Bénard convection is the buoyancy-driven flow of a fluid heated from below and cooled from above. This model of thermal convection is a paradigm for pattern formation and turbulence [1] and it plays an important role in a large range of phenomena in geophysics, astrophysics, meteorology, oceanography and engineering. The problem under investigation is: given an incompressible fluid enclosed in a rectangular container heated from below and cooled from above, what are the flow dynamics? In particular, what is the heat transfer from the bottom to the top? The two main mechanisms associated to the temperature variations are conduction, which tends to remove local temperature differences, and convection, which transports fluid at macroscopic scales. In the specific, convection is generated by the interplay of the buoyancy force that arise from density variations and the limiting effect of inner friction, especially near the plates. If the driving forces, as measures by the Rayleigh number defined below, are small the fluid is at rest in the bulk (i.e the pure conduction state is the global attractor). Above an explicitly known critical Rayleigh number the global attractor consists of stationary convection rolls which appear due to the density differences between layers: hot parcels move upwards and cold parcels, denser, move downwards. As the Rayleigh number increases further the convection pattern is destabilized by finger-like structures that detach from the boundary layers, called plumes. This ultimate stage is classified as turbulent.