Thermosolutal and binary fluid convection as a 2 x 2 matrix problem

We describe an interpretation of convection in binary fluid mixtures as a s uperposition of thermal and solutal problems, with coupling due to advectio n and proportional to the separation parameter S. Many of the properties of binary fluid convection are then consequences of generic properties of 2 x 2 matrices. The eigenvalues of 2 x 2 matrices varying continuously with a parameter r undergo either avoided crossing or complex coalescence, dependi ng on the sign of the coupling (product of off-diagonal terms). We first co nsider the matrix governing the stability of the conductive state. When the thermal and solutal gradients act in concert (S > 0, avoided crossing), th e growth rates of perturbations remain real and of either thermal or soluta l type. In contrast, when the thermal and solutal gradients are of opposite signs (S < 0, complex coalescence), the growth rates become complex and ar e of mixed type. Surprisingly, the kinetic energy of nonlinear steady state s is also governed by an eigenvalue problem very similar to that governing the growth rates. More precisely, there is a quantitative analogy between t he growth rates of the linear stability problem for infinite Prandtl number and the amplitudes of steady states of the minimal five-variable Veronis m odel for arbitrary Prandtl number. For positive S, avoided crossing leads t o a distinction between low-amplitude solutal and high-amplitude thermal re gimes. For negative S, the transition between real and complex eigenvalues leads to the creation of branches of finite amplitude, i.e. to saddle-node bifurcations. The codimension-two point at which the saddle-node bifurcatio ns disappear, leading to a transition from subcritical to supercritical pit chfork bifurcations, is exactly analogous to the Bogdanov codimension-two p oint at which the Hopf bifurcations disappear in the linear problem. (C) 20 01 Elsevier Science B.V. All rights reserved.