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

• We model convection in a finite box of porous media where three modes are viable.
• Each box is classified into one of two classes, with one exception.
• The bifurcation behaviour for an example from each class is studied.
• The behaviour for each box is inferred from these examples.
• The results qualitatively agree with examples in the literature.

A plethora of convection modes may occur within a confined box of porous medium when the associated dimensionless Rayleigh number RR is above some critical value dependent on the geometry. In many cases the crucial Rayleigh number RcRc for onset is different for each mode, and in practice the mode with the lowest associated RcRc 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 RcRc. 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.