Single-mode solutions for convection and double-diffusive convection in porous media*

This work employs single-mode equations to study convection and double-diffusive convection in a porous medium. The single-mode solutions resembling steady convection rolls reproduce the qualitative behavior of root-mean-square and mean temperature profiles of time-dependent states at high Rayleigh numbers from direct numerical simulations (DNS). We show that the single-mode solutions are consistent with the heat-exchanger model that describes well the mean temperature gradient in the interior. The Nusselt number predicted from the single-mode solutions exhibits a scaling law with Rayleigh number close to that followed by exact 2D steady convection rolls, and the single-mode solutions at a high wavenumber predict Nusselt numbers close to the DNS results in narrow domains. We also employ single-mode equations to analyze the influence of active salinity, introducing a salinity contribution to the buoyancy, but with a smaller diffusivity than the temperature. The single-mode solutions capture the stabilizing effect of an imposed salinity gradient and describe the standing and traveling waves observed in DNS. The Sherwood numbers obtained from single-mode solutions show a scaling law with the Lewis number that is close to the DNS computations with passive or active salinity.