Nonlinear Thermal Convection

Abstract

The topic of this. review article is thermal convection in thin horizontal fluid layers, uniformly heated from below and/or cooled from above. If the temperature difference between the two horizontal boundaries is sufficiently small, the heat will be transferred through the fluid by conduction alone. For greater temperature differences the conduction state becomes unstable and a convective motion is set up. The first experimental investigations on thermal convection date back to Thomson (1881) and Benard ( 1900). The experiments by Benard in particular have attracted great attention and are today considered classical in fluid mechanics. This is essentially due to the surprising and fascinating pattern of very regular hexagonal cells obtained for large values of time in his experiments. Benard studied convective motion in a very shallow layer of viscous fluid (molten spermaceti) and made the motion visible by graphite or aluminum powder. The theory of thermal convection was initiated by Rayleigh ( 1916). He assumed that the amplitude of the motion was infinitesimal such that the equations could be linearized. He thus derived the critical temperature gradient (or in modern language, the critical Rayleigh number) for the onset of convection together with the wavenumber for the marginal stable mode. The resemblance of the cell pattern in the Rayleigh theory to certain cloud forms was early noticed by meteorologists. For example, it was pointed out that often the observed ratio between the vertical height and the lateral extent in cumulus clouds is the same as found in the theory for hexagonal cells. The theory on cloud formation is perhaps best applied to the convective motion set up in an altostratus layer when it breaks up into altocumulus clouds because of radiative cooling at the top of the layer. This application is, however, obscured by the effect of the release of latent heat, which is not accounted for in the theory. The theory on thermal convection has become very important in the study of motion in the earth's interior (see Turcotte & Oxburgh 1972) and also in some branches of astrophysics (Spiegel 1971 , 1972). The great interest shown to the theory for the last 10--15 years is partly due to these and other applications. The main