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

Abstract

Convection in an infinite layer of a porous medium occurs if the dimensionless Rayleigh number exceeds a critical value. This is also true for a box of a porous medium; however, each discrete modal solution has its own associated critical Rayleigh number. Usually just one mode will be generated at the onset of convection, but there are many critical box dimensions for which up to four modes share the same critical Rayleigh number, and all may be generated at the onset of convection. In such circumstances there will be a slow interchange of energy between the preferred modes. A perturbation method is applied to a system where three modes are generated at onset to yield a system of ordinary differential equations which govern the evolution of the amplitudes of the viable modes. Three unique cases arise, each with a different phase space structure. Critical boxes with ‘moderate’ aspect ratios are systematically categorised into these cases. While two of the examples represent the usual case where just one mode survives in the final state, the third example is a special case where it is possible for the three modes to coexist. The initial conditions determine which mode(s) will survive.