Penetrative convection in a fluid overlying a porous layer

An accurate numerical calculation is presented for the onset of thermal convection in a two layer system which is comprised of a layer of porous material described by Darcy’s law, over which lies a layer of water. The porous layer is also saturated with water. The two layer system is maintained with the lower (porous) surface at 0 °C and the upper (fluid) surface is stress free with the temperature being above 0 °C. This physical picture is capable of encompassing water at the density maximum of 4 °C in the layer and is thus capable of describing a model for patterned ground formation under water. To account for the fact that the density may have a maximum in the layer we adopt an equation of state which expresses the density in the buoyancy force as a quadratic function of temperature. The onset of convection may have a bi-modal nature in which convection may be dominated by the porous medium or by the fluid depending on parameters which appear in the problem. Here, the important parameters are the ratio of fluid to porous medium depth, the upper surface temperature, and the Darcy number δ=K/dm, a parameter representing non-dimensional permeability of the porous matrix. The coefficient K is the permeability and dm is the depth of the porous layer. A surprising array of streamline patterns is found at the onset of convection as one varies the appropriate parameters.