The Oberbeck-Boussinesq equations are reduced to a two-dimensional form governing “roll” convection between two free surfaces maintained at a constant temperature difference. These equations are then transformed to a set of ordinary differential equations governing the time variations of the double-Fourier coefficients for the motion and temperature fields. Non-linear transfer processes are retained and appear as quadratic interactions between the Fourier coefficients. Energy and heat transfer relations appropriate to this Fourier resolution, and a numerical method for solution from arbitrary initial conditions are given. As examples of the method, numerical solutions for a highly truncated Fourier representation are presented. These solutions, which are for a fixed Prandtl number and variable Rayleigh numbers, show the transient growth of convection from small perturbations, and in all cases studied approach steady states. The steady states obtained agree favorably with steady-state solutions ob... Abstract The Oberbeck-Boussinesq equations are reduced to a two-dimensional form governing “roll” convection between two free surfaces maintained at a constant temperature difference. These equations are then transformed to a set of ordinary differential equations governing the time variations of the double-Fourier coefficients for the motion and temperature fields. Non-linear transfer processes are retained and appear as quadratic interactions between the Fourier coefficients. Energy and heat transfer relations appropriate to this Fourier resolution, and a numerical method for solution from arbitrary initial conditions are given. As examples of the method, numerical solutions for a highly truncated Fourier representation are presented. These solutions, which are for a fixed Prandtl number and variable Rayleigh numbers, show the transient growth of convection from small perturbations, and in all cases studied approach steady states. The steady states obtained agree favorably with steady-state solutions ob...