We present direct numerical simulations (DNS) of the flow in a finite cylinder d riven by a rotating magnetic field. Employing the rigid-body and low-frequency approximations the mathematical model reduces to the Navier-Stokes equations with a priori known Lorentz force. A sec ond-order finite-element method combined with Adams-Bashforth time integration i s used for discretization. The objective of this study is to identify the dominating structures and mechani sms in the transitional and early turbulent flow regimes. The DNS cover a range up to 10T(ac), where Ta denotes the magnetical Taylor number. The most striking result of the study is the insight in the formation, evolution and finally dissipation of Taylor-Gortler-like vortices that clearly dominate t he turbulence physics and provide an efficient mixing mechanism in this flow. Moreover, large scale flow variations in azimuthal direction could be identified . However, these fluctuations do not contain any harmonic components even in nea r-critical flow.