Scale selection in low-order spectral models of twodimensional thermal convection

Abstract

The non-linear dynamics of two-dimensional thermal convection is investigated with the help of two low-order spectral models.The first model is the three-coefficient Lorenz model, in which the aspect ratio has not been used in scaling the variables. This means that the aspect ratio is a free parameter in addition to the Rayleigh number and the Prandtl number. The stability and the energetics of steady-state convection as a function of the three free parameters are investigated.The second model is a ten-coefficient model which describes the evolution and non-linear interaction of two horizontal scales of motion (with vertical wave number equal to one). It is found that the circulation mostly chooses a steady-state equivalent to the steady-state convective solution of the Lorenz model, i.e., a roll with a definite aspect ratio, even when one or more of the seven other coefficients would grow according to linear theory. It is also found that in most cases, the larger scale is favoured over the smaller scale, as the Rayleigh number is increased, especially at low Prandtl number. This agrees qualitatively with laboratory experiments.The paper is, however, principally meant as a contribution to the physical and mathematical insight into, as well as an illustration of, the non-linear dynamics of two-dimensional thermal convection.