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

Abstract

We describe an interpretation of convection in binary fluid mixtures as a superposition of thermal and solutal problems, with coupling due to advection and proportional to the separation parameter S. Many of the properties of binary fluid convection are then consequences of generic properties of 2×2 matrices. The eigenvalues of 2×2 matrices varying continuously with a parameter r undergo either avoided crossing or complex coalescence, depending on the sign of the coupling (product of off-diagonal terms). We first consider the matrix governing the stability of the conductive state. When the thermal and solutal gradients act in concert ( S>0, avoided crossing), the growth rates of perturbations remain real and of either thermal or solutal type. In contrast, when the thermal and solutal gradients are of opposite signs ( S<0, complex coalescence), the growth rates become complex and are of mixed type. Surprisingly, the kinetic energy of nonlinear steady states is also governed by an eigenvalue problem very similar to that governing the growth rates. More precisely, there is a quantitative analogy between the growth rates of the linear stability problem for infinite Prandtl number and the amplitudes of steady states of the minimal five-variable Veronis model for arbitrary Prandtl number. For positive S, avoided crossing leads to a distinction between low-amplitude solutal and high-amplitude thermal regimes. 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 bifurcations disappear, leading to a transition from subcritical to supercritical pitchfork bifurcations, is exactly analogous to the Bogdanov codimension-two point at which the Hopf bifurcations disappear in the linear problem.