Nonlinear two-dimensional Rayleigh-Bénard convection

Two dimensional Rayleigh-Bénard convection in a Boussinesq fluid is the simplest possible system that exhibits convective instability. Moreover it contains the same basic physics as occurring in many geophysical and astrophysical systems, such as the interiors of the Earth and the Sun. We study this ubiquitous system with and without the effect of rotation, for stress free boundary conditions. We review the linear stability theory of two dimensional Rayleigh-Bénard convection, deriving conditions on the dimensionless parameters of the system, under which we expect convection to occur. Building on this we solve the equations governing the dynamics of the nonlinear system using a pseudospectral numerical method. This is done for a range of different values of the Rayleigh, Prandtl and Taylor numbers. We analyse the results of these simulations using a variety of applied mathematical techniques. Paying particular attention to the manner in which the flow becomes unstable and looking at global properties of the system such as the heat transport, we concur with previous work conducted in this area. For a particular subset of parameters studied, we find that motion is always steady. Motivated by this we develop an asymptotic theory to describe these nonlinear, steady state solutions, in the limit of large Rayleigh number. This asymptotic theory provides analytical expressions for the governing hydrodynamical variables as well as predictions about the heat transport. With only a few terms we find excellent agreement with the results of our numerical simulations.