Linear stability analysis of three-dimensional natural convection at low Prandtl number in an annular enclosure in the presence of a toroidal magnetic field

The natural convection of a liquid metal in an annular enclosure with a square cross section in the presence of a toroidal static magnetic field was investigated by linear stability analysis. Three-dimensional steady disturbances were obtained in an annular enclosure where the walls parallel to the gravitational field were heated and cooled. The Prandtl number Pr was set to 0.025 and the radius ratio of the enclosure κ was set to 0.5, while the Rayleigh number Ra, the Hartmann number Ha, and the angular wavenumber m were considered as parameters. The dimensionless governing equations were discretized by the finite difference method. Since the newly developed dual staggered grid was employed, the interpolation in the outer product terms was not required. The linear growth rate for a standing wave mode was amplified by increasing Ra and attenuated by increasing Ha. For any integer m, the neutral Rayleigh number Ran at Ha = 0 and the neutral Hartmann number Han for Ran ≤ Ra ≤ 200 000 were identified. Based on these, the neutral lines for a given m were obtained as Han ∼ (Ra − Ran)1/3. Furthermore, the critical values Rac, Hac, and mc were obtained, where mc was not necessarily limited to an integer. The distribution of mc was approximated by the power of Ra. The disturbances had symmetry in the azimuthal direction and constituted a pair of vortices rotating in opposite directions. These vortices were elongated along the main stream.