Finite Amplitude Free Convection as an Initial Value ProblemI.
Abstract
The OberbeckBoussinesq equations are reduced to a twodimensional 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 doubleFourier coefficients for the motion and temperature fields. Nonlinear 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 steadystate solutions obtained by previous investigators.
 Publication:

Journal of Atmospheric Sciences
 Pub Date:
 July 1962
 DOI:
 10.1175/15200469(1962)019<0329:FAFCAA>2.0.CO;2
 Bibcode:
 1962JAtS...19..329S