Strong spatial interactions with 1:1 resonance: a three-layer convection problem

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Abstract
The resonant interactions between two spatial structures are investigated, using a three-layer thermal convection model as a reference system with translation and reflection symmetry. In the asymptotic limit of a vanishingly small central fluid layer it is shown that when the depth ratio of the outermost fluid layers are equal, two spatially distinct modes of convection can occur with preferred horizontal wavenumber in the ratio 1:1. The normal form equations are derived and the nonlinear temporal evolution of a disturbance consisting of both these modes is considered. It is shown that the multiple bifurcation is a codimension-4 phenomenon and that nonlinear convection resulting from steady state interactions involve both steady mixed mode states, and oscillatory states in the form of standing, travelling and modulated waves. A detailed description of the possible pattern transitions between these states is given.