Interacting convection modes in a saturated porous medium of nearly square planform: a special case

A multitude of convection modes may occur within a confined rectangular box of saturated porous medium when the associated dimensionless Rayleigh number R is above some critical value. For particular sizes of box, however, it is possible for multiple modes (typically three) to share a common critical Rayleigh number. For box shapes close to these geometries, modes can interact and compete nonlinearly near the onset of convection. The generic examples of this phenomenon can be conveniently classified as belonging to one of two distinctive classes distinguished by whether or not fixed points are possible with all three modes having a non-zero amplitude. It transpires that this classification is not quite exhaustive for there is one particular case which falls outside this pattern. This last case, which is described by a system of evolution equations that is structurally different from that applying in the generic situation, is explored in this paper. Some rich dynamical behaviours are uncovered and, in particular, it is shown that at sufficiently large R stable states arise in which all three modes persist. This contrasts to the typical generic behaviour where a single mode is preferred. The bifurcation sequence that develops as the Rayleigh number grows is mapped out from primary through to tertiary stages. In addition, it is found that a cusp bifurcation is present and that the details of the ordering of the various bifurcations are shown to be sensitive to the precise geometry of the box.