A cylindrical cavity with an aspect ratio of unity is filled with liquid metal and suddenly exposed to an azimuthal body force generated by a rotating magnetic field (RMF). This experimental study is concerned with the secondary meridional flow during the time, if the fluid spins up from rest. Vertical profiles of the axial velocity have been measured by means of the ultrasound Doppler velocimetry. The flow measurements confirm the spin-up concept by Ungarish (J Fluid Mech 347:105–118, 1997) and the continuative study by Nikrityuk et al. (Phys Fluids 17:067101, 2005) who suggested the existence of two stages during the RMF-driven spin-up, in particular the so-called initial adjustment phase followed by an inertial phase which is dominated by inertial oscillations of the secondary flow. Evolving instabilities of the double-vortex structure of the secondary flow have been detected at a Taylor number of 1.24 × 105 verifying the predictions of Grants and Gerbeth (J Fluid Mech 463:229–240, 2002). Perturbations in form of Taylor–Görtler vortices have been observed just above the instability threshold.