Magnetic processes in a collapsing dense core. II Fragmentation. Is there a fragmentation crisis ?

Abridged. A large fraction of stars are found in binary systems. It is therefore important for our understanding of the star formation process, to investigate the fragmentation of dense molecular cores. We study the influence of the magnetic field, ideally coupled to the gas, on the fragmentation in multiple systems of collapsing cores. We present high resolution numerical simulations performed with the RAMSES MHD code starting with a uniform sphere in solid body rotation and a uniform magnetic field parallel to the rotation axis. We pay particular attention to the strength of the magnetic field and interpret the results using the analysis presented in a companion paper. The results depend much on the amplitude, $A$, of the perturbations seeded initially. For a low amplitude, $A=0.1$, we find that for values of the mass-to-flux over critical mass-to-flux ratio, $\mu$, as high as $\mu = 20$, the centrifugally supported disk which fragments in the hydrodynamical case, is stabilized and remains axisymmetric. Detailed investigations reveals that this is due to the rapid growth of the toroidal magnetic field induced by the differential motions within the disk. For values of $\mu$ smaller $\simeq 5$, corresponding to larger magnetic intensities, there is no centrifugally supported disk because of magnetic braking. When the amplitude of the perturbation is equal to $A=0.5$, each initial peak develops independently and the core fragments for a large range of $\mu$. Only for values of $\mu$ close to 1 is the magnetic field able to prevent the fragmentation. Since a large fraction of stars are binaries, the results of low magnetic intensities preventing the fragmentation in case of weak perturbations, is problematic. We discuss three possible mechanisms...