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.