Pergamon, New York, 1959). ^These equations were applied to the Benard problem before in K. M. Zaitsev and M. I. Shliomis, Zh. Eksp. Teor. Fiz. 59, 1583 (1970 [Sov. Phys. JETP^, 866 (1971)], where a linearized theory of the transition has been given. ^A. Schliiter, D. LortZj and F. Busse, J. Fluid Mech. 23, 129 (1965).^The well-known hexagonal convection cells would require non-Boussinesq terms in the equations of motion; cf. R. E. Krishnamurti, J. Fluid Mech. ^, 445 (1968).^A. C. Newell and J. A. Whitehead, J. Fluid Mech. ^, 279 (1969).^Length, time, velocity, and temperature are measured in units of I, f/v, v/l, and (ATv^/g^Kl^)^^^, respectively. I, V, AT, )8, g^ and K-are the cell thickness, the kinematic viscosity, the externally maintained temperature difference between bottom and top of the layer, the fluid's volume expansion coefficient, the gravitational acceleration, and the thermometric conductivity, respectively. The Rayleigh number R =g^AT l^/vK and the Prandtl number P = V/K are formed from these constants. Later we will have to use the fluid density p, the average fluid temperature T, its specific heat C^, the cell length in the y direction (I^y), and the total area F of the layer. K is Boltzmann*s con-^^The second-order functional derivatives taken at the same point, which appear in Eq.(2), are not well defined if operating on a functional containing \w\^^ as, e.g., the functional
