The smooth transition to a convective régime in a two-dimensional box

Abstract
The fluid motion in a two-dimensional box heated from below is considered. The horizontal surfaces are taken to be free and isothermal while the sidewalls are first taken to be rigid and perfect insulators. Linear stability theory shows that the critical Rayleigh number for the onset of convection is higher than that when no side walls are present and the eigenvalue spectrum is discrete. Finite amplitude theory shows that the onset of convection is sudden, that is, bifurcation occurs. The effect of allowing the sidewalls to be slightly imperfect insulators is also investigated. It is found that if the boundary conditions of the sidewalls depart only slightly from those given above, there is a significant change in the response of the fluid. In the most general circumstances a resonance of the free mode is excited as the Rayleigh number approaches its critical value and finite amplitude effects become important. Then it is shown that the onset of convection is quite smooth and the concept of a sharp bifurcation at a critical Rayleigh number is no longer tenable. For a particular class of imperfections it is shown that a 'transcritical' bifurcation as described by Benjamin (1976) is possible. The limiting case of a very long box is given special consideration.

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