Double-diffusive convection in porous media: The Darcy and Brinkman models

The nonlinear stability of a horizontal layer of a binary fluid mixture in an isotropic and homogeneous porous medium heated and salted from below is studied, for the Oberbeck-Boussinesq - Darcy and Oberbeck-Boussinesq - Brinkman-Forchheimer models, through the Lyapunov direct method. This is an interesting geophysical case because the solute concentration gradient is stabilizing while heating from below provides a destabilizing effect. Unconditional nonlinear stability is found for any value of the porosity and the Lewis number. In particular, if the normalised porosity c is equal to 1, a necessary and sufficient condition for nonlinear stability is proved: in this case the critical linear and nonlinear Rayleigh numbers coincide. For other values of E a conditional stability theorem is shown and the coincidence of the critical parameters holds whenever the Principle of Exchange of Stabilities is valid.