We study the flow field during the radial solidification of an Al-7wt pct Si melt inside a cylindrical cavity with cooled side walls which is placed in a rotating magnetic field (RMF). The solidification process is simulated by means of a continuum model which is implemented in a code which solves the axisymmetric Navier-Stokes equations. We first analyze the start-up of the buoyancy-driven convection driven by the radial heat flux in absence of the RMF. We show that the initial large vortex breaks up into three smaller one. The resulting radial temperature profiles are compared with experimental data. Second, we study how the vortex structure is modified by the application of small RMF's.