Convection in binary fluid mixtures: A model of four coupled amplitudes

The Rayleigh–Bénard system [1, 2] is one of the most popular model systems to study spontaneous structure formation emerging via instabilities. The system consists of a fluid layer confined between two parallel plates perpendicular to the direction of gravity, and exposed to a vertical temperature gradient that is established by keeping the upper and lower plate at two different temperatures T0−∆T/2 and T0 +∆T/2 respectively. Rayleigh–Bénard convection in one–component fluids at small and moderate temperature differences is well understood, mainly thanks to the work of Busse and co–workers in the 70s and 80s [3, 4, 5, 6]. The bifurcation behavior becomes more complex in binary mixtures. Here, the concentration enters as an additional relevant dynamic field. Concentration differences are created in the presence of a non–vanishing Soret effect, a nonequilibrium effect that describes the driving of concentration currents by temperature differences. The concentration couples back into the system’s dynamics via variations that determine the buoyancy force density. When the lighter component of the mixture is driven into the direction of higher temperature, thereby enhancing the density gradient and further destabilizing the layer one speaks of a positive Soret effect, and of a negative Soret effect in the opposite case. Convection in binary mixtures shows a very rich bifurcation behavior. For negative separation ratios convection rolls bifurcate backwards and time– dependent patterns, namely traveling waves and standing waves appear. The bistability of convective patterns and the ground state leads to the existence of localized structures and fronts. In the case of positive separation ratios on which we will focus in this paper the two–dimensional roll structures are often replaced by squares at onset, but become the preferred pattern at higher temperature differences. For intermediate ∆T two types of crossroll structures can exist, a stationary and an oscillatory type that transfer stability from squares to rolls. There are few–mode models that satisfactorily explain the bifurcation properties for negative Soret effects [7]. However, for positive Soret effects a model with only a few degrees of freedom, that is able to properly describe rolls, squares and crossrolls seems to be lacking. In this paper we will present a system of four equations that describe all important properties of the square – crossroll – roll transition for the case of a positive Soret effect. The paper is organized as follows: after this in-