  The evolution of a Taylor-Green forced magnetohydrodynamic (MHD) system
showing dynamo activity is analyzed via direct numerical simulations. The
statistical properties of the velocity and magnetic field in Eulerian
coordinates and along trajectories of fluid elements (Lagrangian coordinates)
are found to change between the kinematic, non-linear and saturated regime.
Fluid element (tracer) trajectories change from chaotic quasi-isotropic
(kinematic phase) to mean magnetic field aligned (saturated phase). The
probability density functions (PDFs) of the magnetic field change from strongly
non-Gaussian in the kinematic to quasi-Gaussian PDFs in the saturated regime so
that their flatness give a precise handle on the definition of the limiting
points of the three regimes. Also the statistics of the fluctuations of the
kinetic and magnetic energy along fluid trajectories change. All this goes
along with a dramatic increase of the correlation time of velocity and magnetic
field fluctuations experienced by tracers significantly exceeding one turbulent
large-eddy turn-over time. A remarkable consequence is an intermittent scaling
regime of the Lagrangian magnetic field structure functions at unusually long
time scales.
