Onset of Convection in Magnetohydrodynamic porous media: absence of subcritical instabilities and global stability

Magnetohydrodynamics (MHD) concerns with a fluid which is immersed and moves in a magnetic field. Accounting for the Lorentz force rise up actions between the fluid and the magnetic field: the fluid velocity influences the magnetic field and viceversa. The equations governing this interaction are different, depending on whether you are in the Galileian or relativistic scheme. We will get in the Galileian scheme. If the fluid fills a porous medium one has to choose a more appropriate momentuum equation (depending on the case at stake). On accounting the Maxwell equations and Boussinesq approximation, we will consider the Darcy momentuum equation