  We model the polarized thermal dust emission from protostellar cores that are
assembled by super-sonic turbulent flows in molecular clouds. Self-gravitating
cores are selected from a three dimensional simulation of super-sonic and
super-Alfvenic magneto-hydrodynamic (MHD) turbulence. The polarization is
computed in two ways. In model A it is assumed that dust properties and grain
alignment efficiency are uniform; in model B it is assumed that grains are not
aligned at visual extinction larger than 3 mag. The main results of this work
are: i) Values of the degree of polarization P between 1 and 10% are typical,
despite the super-Alfvenic nature of the turbulence; ii) A steep decrease of P
with increasing values of the sub-mm dust continuum intensity I is always found
in self--gravitating cores selected from the MHD simulations, if grains are not
aligned above a certain value of visual extinction (model B); iii) The same
behavior is hard to reproduce if grains are aligned independently of visual
extinction (model A); iv) The Chandrasekhar-Fermi formula, corrected by a
factor f=0.4, provides an approximate estimate of the average magnetic field
strength in the cores. Sub-mm dust continuum polarization maps of quiescent
protostellar cores and Bok globules always show a decrease in P with increasing
value of I consistent with the predictions of our model B. We therefore
conclude that sub-mm polarization maps of quiescent cores do not map the
magnetic field inside the cores at visual extinction larger than approximately
3 mag. There is no inconsistency between the results from optical and near-IR
polarized absorption of background stars, and the observed polarization of
sub-mm dust continuum from quiescent cores. In both cases, grains at large
visual extinction appear to be virtually unaligned.
