A Fourier-Hermite pseudospectral method is developed to study numerically the three-dimensional penetrative convection problem under the Boussinesq approximation. An S-shaped temperature profile in the absence of motion is prescribed in the vertical direction. All variables are expanded in terms of Fourier-Hermite basis functions. The Hermite functions are scaled to adjust the length of the computational domain in the vertical. A semi-implicit scheme is used for time marching with the third-order Adam-Bashforth and Crank-Nicholson scheme for the nonlinear and linear terms, respectively. An implementation of the numerical method on a parallel computer is also described. Numerical simulation results of resolution 643are presented for low-to-moderate Rayleigh numbers with a Prandtl number of unity. The highly stable outer regions are seen to act as effective lids and all penetrative flow are contained within the computational box. Variances, heat fluxes, and their budgets are reported for several Rayleigh numbers to demonstrate the efficacy of the numerical method. Copyright 1998 Academic Press.
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