  We analyse simulations of turbulent, magnetised molecular cloud cores
focussing on the formation of Class 0 stage protostellar discs and the physical
conditions in their surroundings. We show that for a wide range of initial
conditions Keplerian discs are formed in the Class 0 stage already. In
particular, we show that even subsonic turbulent motions reduce the magnetic
braking efficiency sufficiently in order to allow rotationally supported discs
to form. We therefore suggest that already during the Class 0 stage the
fraction of Keplerian discs is significantly higher than 50%, consistent with
recent observational trends but significantly higher than predictions based on
simulations with misaligned magnetic fields, demonstrating the importance of
turbulent motions for the formation of Keplerian discs. We show that the
accretion of mass and angular momentum in the surroundings of protostellar
discs occurs in a highly anisotropic manner, by means of a few narrow accretion
channels. The magnetic field structure in the vicinity of the discs is highly
disordered, revealing field reversals up to distances of 1000 AU. These
findings demonstrate that as soon as even mild turbulent motions are included,
the classical disc formation scenario of a coherently rotating environment and
a well-ordered magnetic field breaks down. Hence, it is highly questionable to
assess the magnetic braking efficiency based on non-turbulent collapse
simulation. We strongly suggest that, in addition to the global magnetic field
properties, the small-scale accretion flow and detailed magnetic field
structure have to be considered in order to assess the likelihood of Keplerian
discs to be present.
