root 2:1 Resonance in non-Boussinesq convection

Numerical and laboratory experiments on non-Boussinesq convection have revealed a possibly stable pattern with square symmetry but with upflow in isolated plumes and downflow in connected sheets, a topology normally associated with a hexagonal pattern. The appearance of this pattern can be interpreted as the sum of two different pairs of orthogonal rolls aligned at 45 degrees and with wave numbers in the ratio root 2:1. We examine the interaction between these modes by expanding about the multiple bifurcation point at which they are both marginal. The relevant amplitude equations are written down and are shown to account for several of the features found in the experiments: the new pattern can be the only stable solution near onset and can occur subcritically. These features of the solution depend crucially on the lack of Boussinesq symmetry, which induces quadratic resonance between the two sets of rolls, in close analogy with the 2:1 resonance problem in non-Boussinesq convection with O(2) symmetry.