Oscillatory and steady convection in a magnetic field

Abstract

Two-dimensional convection in a Boussinesq fluid in the presence of an imposed magnetic field is described in terms of a simplified model, which is exact to second order in the amplitude of the motion and appears to be qualitatively correct for larger amplitudes. If the ratio of the magnetic diffusivity to the thermal diffusivity is sufficiently small and the imposed magnetic field is sufficiently large, convection sets in when r = r(o) as overstable oscillations, which grow in amplitude as the normalized Rayleigh number r is increased. There is also a branch of steady solutions that bifurcates from the static equilibrium at r = r(e) < r(o) and stable steady solutions exist for r > rmin. For certain choices of parameters subcritical steady convection, with rmin < r(e), is found and the oscillatory branch ends on the unstable portion of the steady branch, where the period of the oscillations becomes infinite. In some circumstances there may be a bifurcation from symmetrical to asymmetrical oscillations, followed by a sequence of bifurcations at each of which the period doubles. Other choices of parameters allow only supercritical convection with r increasing monotonically on the steady branch; if convection first appears as overstable oscillations the steady branch is then unstable for r(e) < r < rmin and there is a Hopf bifurcation at r = rmin. This complicated pattern of behaviour is consistent with the results of numerical experiments on the full two-dimensional problem.