Nonlinear dynamics of traveling waves in rotating Rayleigh-Bénard convection: effects of the boundary conditions and of the topology.

Motivated by the experimental results of Liu and Ecke (1997, 1999), different models are developed to analyze the weakly nonlinear dynamics of the traveling-wave sidewall modes appearing in rotating Rayleigh-Bénard convection. These models assume fully rigid boundary conditions for the velocity field. At the linear level, this influences most strongly the critical frequencies: they appear to be proportional to the logarithm of the Coriolis number, which is twice the inverse of the Ekman number. An annular flow domain is considered. This multiply connected geometry is shown to lead generally to the existence of a global mean-flow mode proportional to the average, over the azimuthal coordinate, of the square of the modulus of the envelope of the waves. Because this mode feeds back on the active wave modes at cubic order, the resulting Ginzburg-Landau envelope equation contains a nonlocal term. This new term, however, vanishes in the large-gap limit relevant to the experiments of Liu and Ecke. As compared with previous theoretical work, the present models lead to reduced discrepancies with the results of these experiments concerning the coefficients of the envelope equation. It is also shown that the new nonlocal effects may be realized experimentally in a small-gap annular geometry if a small-Prandtl-number fluid is used, despite the fact that no regime of Benjamin-Feir instability is predicted to occur.