Numerical stability analysis and general flow characterization were performed for the flow induced by transverse rotating magnetic fields in cylindrical containers. Starting from Richardson’s theoretical critical value for the generation of Taylor vortices in infinite cylinders, overstability in finite cylinders with different aspect ratios was investigated. Decreasing aspect ratio leads to a more stable configuration. For small cylinders and reasonable frequencies of the magnetic field, instabilities arise at low magnetic inductions, e.g.,Bc≈0.5 mT[gallium as model liquid, with radius of the cylinderR=12.5 mm,aspect ratio(h/d)=2and frequencyf=50 Hz]. As a source of instabilities Taylor vortices have been identified. They are transported by the secondary flow, emerging in finite cylinders due to the imbalance of pressure gradient and centrifugal force close to the top and bottom, thus generating time-dependent flow. Their generation is statistical and manifests itself in, e.g., nonperiodic temperature oscillations with small amplitudes in a classical Be´nard configuration. Furthermore the intensity of the secondary flow was investigated for the isothermal case to give hints on its general influence on the overall mass transport. ©1998 American Institute of Physics.