The nonlinear interactions of convection modes in a box of a saturated porous medium

A plethora of convection modes may occur within a confined box of porous medium when the associated dimensionless Rayleigh number R is above some critical value dependent on the geometry. In many cases the crucial Rayleigh number R_c for onset is different for each mode, and in practice the mode with the lowest associated R_c is likely to be the dominant one. For particular sizes of box, however, it is possible for multiple modes (typically three) to share a common R_c. For box shapes close to these special geometries the modes interact and compete nonlinearly near the onset of convection. Here this mechanism is explored and it is shown that generically the dynamics of the competition takes on one of two possible structures. A specific example of each is described, while the general properties of the system enables us to compare our results with some previous calculations for particular box dimensions.