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The influence of turbulence during magnetized core collapse and its consequences on low-mass star formation

[Abridged] Theoretical and numerical studies of star formation have shown that magnetic field (B) has a strong influence on both disk formation and fragmentation; even a relatively low B can prevent these processes. However, very few studies investigated the combined effects of B and turbulence. We study the effects of turbulence in magnetized core collapse, focusing on the magnetic diffusion, the orientation of the angular momentum (J) of the protostellar core, and on its consequences on disk formation, fragmentation and outflows. We perform 3D, AMR, MHD simulations of magnetically supercritical collapsing dense cores of 5 Msun using the MHD code RAMSES. A turbulent velocity field is imposed as initial conditions, characterised by a Kolmogorov power spectrum. Different levels of turbulence and magnetization are investigated, as well as 3 realisations for the turbulent velocity field. Magnetic diffusion, orientation of the rotation axis with respect to B, transport of J, disk formation, fragmentation and outflows formation are studied. The turbulent velocity field imposed as initial conditions contains a non-zero J, responsible for a misalignment of the rotation axis. Turbulence is also responsible for an effective turbulent diffusivity in the vicinity of the core. Both effects are responsible for a significant decrease of the magnetic braking, and facilitate the formation of early massive disks for not too high magnetization. Fragmentation can occur even with mu ~ 5 at late time in contrast with 1 Msun cores for which fragmentation is prevented for such values of mu. Slow asymmetric outflows are launched. They carry a mass which is comparable to the mass within the core. Because of misalignment and turbulent diffusion, massive disk formation is possible though their mass and size are still reduced compared to the hydrodynamical case. We find that for mu >= 5, fragmentation can happen.