Amplitude equations for Rayleigh-Benard convective rolls far from threshold

An extension of the amplitude method is proposed. An iterative algorithm is developed to build an amplitude equation model that is shown to provide precise quantitative results even far from the linear instability threshold. The method is applied to the study of stationary Rayleigh-Benard thermoconvective rolls in the nonlinear regime. In particular, the generation of second and third spatial harmonics is analyzed. Comparison with experimental results and direct numerical calculations is also made and a very good agreement is found.Peer reviewe

Optimum fields and bounds on heat transport for nonlinear convection in rapidly rotating fluid layer

By means of the Howard-Busse method of the optimum theory of turbulence we investigate numerically the effect of strong rotation on the upper bound on the convective heat transport in a horizontal fluid layer of infinite Prandtl number Pr. We discuss the case of fields with one wave number for regions of Rayleigh and Taylor numbers R and Ta where no analytical asymptotic bounds on the Nusselt number Nu can be derived by the Howard-Busse method. Nevertheless we observe that when R > 108 and Ta is large enough the wave number of the optimum fields comes close to the analytical asymptotic result α1 = (R/5)1/4. We detect formation of a nonlinear structure similar to the nonlinear vortex discussed by Bassom and Chang [Geophys. Astrophys. Fluid Dyn. 76, 223 (1994)]. In addition we obtain evidence for a reshaping of the horizontal structure of the optimum fields for large values of Rayleigh and Taylor numbers