A low dimensional model for Rayleigh-Benard convection in rectangular domains

A low dimensional model ({\bf LDM}) for Rayleigh-B\'enard ({\bf R-B}) convection in rectangular boxes, based on the Galerkin projection of the Boussinesq equations onto a finite set of empirical eigenfunctions, is presented. The empirical eigenfunctions are obtained from Proper Orthogonal Decomposition ({\bf POD}) of the field using the Snapshot Method. The most energetic {\bf POD} modes give us a hint on the dynamic dominance of coherent flow patterns, and how well the original inhomogeneous flow can be modeled with a reduced number of modes. A quadratic non-homogeneous {\bf ODE} system is obtained for the evolution of the modal amplitudes. A solution which considers the additional dissipation due to the neglected less energetic modes is considered in terms of a parameter $e \ge 0$, fixed at a value where the ensemble average of the total viscous and thermal dissipation in the model is the same as in the full simulation ({\bf DNS}). We discuss first results of the evolution of the {\bf LDM} and compare it with the {\bf DNS} data of the {\bf R-B} problem.