Brinkmann viscosity action in porous MHD convection

•The paper is devoted to MHD convection in a porous horizontal layer with large pores.•The stabilizing action of the Brinkmann viscosity is analyzed.•Steady and oscillatory instability are analyzed.•Unconditional asymptotic nonlinear stability is obtained by linear stability.•The definitive boundedness of solutions is obtained by looking for L2-absorbing sets.

The stabilizing effect of Brinkmann viscosity (BV) in MHD convection in a horizontal porous layer L – filled by an electrically conducting fluid, heated from below and imbedded in a transverse magnetic field – is analyzed. The critical Rayleigh number of linear stability is found and – in closed forms – conditions for the onset of steady or oscillatory convection are obtained. Via the linearization principle given in [16] it is shown that unconditional nonlinear stability of thermal magnetic conduction solution is guaranteed by linear stability. The long-time behavior is characterized via the existence of L2-absorbingL2-absorbing sets.