Semi-Ellipse Theorem for the Heterogeneous Swirling Flow in an Azimuthal Magnetic Field with Respect to Axisymmetric Disturbances

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The instability of the density-stratified swirling flow in an azimuthal magnetic field is investigated for the axisymmetric disturbance. An assumption of positive Rayleigh-Synge-Michael’s discriminant is made. It is shown that the complex phase-velocity of any unstable axisymmetric mode must lie within the semi-ellipse which has the range of axial flow-velocity for major axis and (Remark: Graphics omitted.) times the range for minor axis. Here J represents an extended local Richardson number to the present flow and the suffix m means the minimum value in the flow field. J_m must be less than 1/4 by the necessary condition for instability.