Stabilisation by magnetic field of the convection in a cylindrical cavity heated from below

We consider the convection in a cylindrical cavity with aspect ratio A (A=height/diameter), heated from below (Rayleigh-Benard situation) and submitted to a horizontal, uniform and constant magnetic field. The three-dimensional Navier-Stokes equations are discretized using the spectral element method, and solved either by time stepping or by Newton method. By a continuation technique, we have obtained stability diagrams giving the thresholds for the onset of convection, and bifurcation diagrams giving the evolution of solutions beyond the thresholds. This study gives information about the stabilisation by a magnetic field, but also affords an example of dynamical system interesting by its symmetry properties and by the selection and evolution of the convective modes.