Modelisation and simulation of convection in extended geometry

Abstract

Models of convection such as the one of $2 give an interesting starting point for a theoretical analysis devoted to the elucidation of concepts which seem important for an understanding of weak turbulence at large aspect ratios.In particular one can study in nearly realistic conditions the problems of pattern selection and defect motion using the standard amplitude equation formalism. As a matter of fact one can easily recover from (3a,b) the modified amplitude equations derived by Siggia and Zippelius for nearly straight rolls in 3-D convection with stress free BC. However it should be kept in mind that, due to the approximations made in order to have models sufficiently simple to remain tractable, one should not expect more than semi-quantitative results at best.

Another and perhaps more promising use of these models is for numerical simulations in domains of lateral extent incomparably larger than those accessible to 3-D simulations. This study is at its very beginning. The above example of a turbulent transient observed at low Prandtl number (Pr=1.6) shows what one can obtain from such simulations. First it seems that they are able to reproduce the often quoted laboratory observation of alternating agitated and calm periods. A careful examination of the evolution allows to interpret this phenomenon in terms of “dynamical frustration“ linked to the compatibility between the large scale drift flow which tends to advect the rolls and the overall curvature of the current structure which generates the vertical vorticity. Calm periods then correspond to the slow distortion of the structure by the flow and agitated periods to the nucleation and motion of dislocations responding to distortions getting too large.Whether this feed-back mechanism is responsible for the occurrence of weak turbulence at large aspect ratios is not yet known but this is a very tempting conjecture in view of our preliminary results.