An improved dynamical approximation to Boussinesq equation using Karhunen-Loève basis

Karhunen-Loève (K-L) basis is used to provide an efficient relatively low dimensional dynamical approximation to Boussinesq equation governing thermal convection phenomena. K-L basis is empirically generated from an ensemble of numerically obtained realizations of turbulent thermal convection flow field. Fourier collocation spectral method is used to numerically integrate Boussinesq equation with stress-free boundary conditions at Pr = 0.72 and Ra ≈ 10 000. An algorithm, which incorporates the lost dissipative effects of the truncation into the dynamical approximation without increasing the number of degrees of freedom involved, is proposed and numerically tested.