  A low-dimensional model (LDM) for turbulent Rayleigh-Benard convection in a
Cartesian cell with square domain, based on the Galerkin projection of the
Boussinesq equations onto a finite set of empirical eigenfunctions, is
presented. The empirical eigenfunctions are obtained from a joint Proper
Orthogonal Decomposition (POD) of the velocity and temperature fields using the
Snapshot Method on the basis of a direct numerical simulation (DNS). The
resulting LDM is a quadratic inhomogeneous system of coupled ordinary
differential equations which we use to describe the long-time temporal
evolution of the large-scale mode amplitudes for a Rayleigh number of 1e5 and a
Prandtl number of 0.7. The truncation to a finite number of degrees of freedom,
that does not exceed a number of 310 for the present, requires the additional
implementation of an eddy viscosity-diffusivity to capture the missing
dissipation of the small-scale modes. The magnitude of this additional
dissipation mechanism is determined by requiring statistical stationarity and a
total dissipation that corresponds with the original DNS data. We compare the
performance of two models, a constant so-called Heisenberg
viscosity--diffusivity and a mode-dependent or modal one. The latter
viscosity--diffusivity model turns out to reproduce the large-scale properties
of the turbulent convection qualitatively well, even for a model with only a
few hundred POD modes.
