In order for a molecular cloud clump to form stars, some transport of magnetic flux is required from the denser internal regions to the outer regions; otherwise, this can prevent the gravitational collapse. Fast magnetic reconnection, which takes place in the presence of turbulence, can induce a process of reconnection diffusion that has been elaborated on in earlier theoretical works. We have named this process turbulent reconnection diffusion, or simply RD. This paper continues our numerical study of this process and its implications. In particular, we extend our studies of RD in cylindrical clouds and consider more realistic clouds with spherical gravitational potentials (from embedded stars); we also account for the effects of the gas self-gravity. We demonstrate that, within our setup reconnection, diffusion is efficient. We have also identified the conditions under which RD becomes strong enough to make an initially subcritical cloud clump supercritical and induce its collapse. Our results indicate that the formation of a supercritical core is regulated by a complex interplay between gravity, self-gravity, the magnetic field strength, and nearly transonic and trans-Alfvénic turbulence; therefore, it is very sensitive to the initial conditions of the system. In particular, self-gravity helps RD and, as a result, the magnetic field decoupling from the collapsing gas becomes more efficient compared with the case of an external gravitational field. Our simulations confirm that RD can transport magnetic flux from the core of collapsing clumps to the envelope, but only a few of them become nearly critical or supercritical sub-Alfvénic cores, which is consistent with the observations. Furthermore, we have found that the supercritical cores built up in our simulations develop a predominantly helical magnetic field geometry that is also consistent with recent observations. Finally, we have also evaluated the effective values of the turbulent RD coefficient in our simulations and found that they are much larger than the numerical diffusion coefficient, especially for initially trans-Alfvénic clouds, thus ensuring that the detected magnetic flux removal is due to the action of turbulent RD rather than numerical diffusivity.
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