The effects of an AXIAL MAGNETIC FIELD on both the vortex breakdown process and fluid layers development in a cylindrical container filled with a conducting viscous fluid are numerically analyzed by using the Generalized Integral Transform Technique (GITT) with a stream function-only formulation. A temperature gradient is imposed in the AXIAL direction on the swirling flow which is advanced by the rotation of the bottom disk under the stabilizing effect of the external MAGNETIC FIELD. Flows are studied for a range of parameters: the Richardson number, Ri, 0 ≤Ri ≤2. 0, and three values of the Prandtl number are investigated, Pr = 0. 025 (liquid Mercury), 0. 032 ( PbLi 17 alloy), and 0. 065 (the molten lithium). Three combinations of aspect ratios (H/R) and Reynolds numbers are compared: (case A: Re=1500, H/R=1. 5), (case B: Re=1855, H/R=2. 0) and (case C: Re=2400, H/R=2. 5). The results reveal that the increase in the values of Hartmann number, Ha suppresses the vortex breakdown in the isothermal case and reduces the number of fluid layers in the layering case. The stability diagram (Hacr–Ri) corresponding to the transition from the multiple fluid layers zone to the one fluid layer zone for increasing Prandtl number is obtained.