Influence of an axial magnetic field on the stability of convective flows between non-isothermal concentric spheres

This paper deals with the linear stability analysis of the convective flow of an electrically conducting fluid in a spherical gap in the presence of an axial magnetic field which is parallel to the vector of gravitational acceleration. The inner shell is warmer then the outer one (T1 > T2). The numerical investigations are performed for the radii ratios η=R1R2=0.4-0.8. The corresponding stability diagrams, i.e. the critical values of the Grashof number Grc and the wave number mc, are presented in dependence on the Hartmann number Ha. We show that the critical Grashof number decreases with increasing η in the case of absence of the magnetic field. A steady axial magnetic field stabilizes the flow, i.e. Grc increases with the Hartmann number Ha for each η investigated. The instability sets in either as a Hopf bifurcation or a steady pitchfork bifurcation in dependence on η and Ha. The stability analysis is accompanied by simulation of the three-dimensional states. We found that according to the bifurcation type there are two classes of 3D states: steady and oscillatory. Consequences of the symmetry with respect to ±ϕ direction are discussed.