Planck intermediate results. XX. Comparison of polarized thermal
  emission from Galactic dust with simulations of MHD turbulence
  Polarized emission observed by Planck HFI at 353 GHz towards a sample of
nearby fields is presented, focusing on the statistics of polarization
fractions $p$ and angles $\psi$. The polarization fractions and column
densities in these nearby fields are representative of the range of values
obtained over the whole sky. We find that: (i) the largest polarization
fractions are reached in the most diffuse fields; (ii) the maximum polarization
fraction $p_\mathrm{max}$ decreases with column density $N_\mathrm{H}$ in the
more opaque fields with $N_\mathrm{H} > 10^{21}\,\mathrm{cm}^{-2}$; and (iii)
the polarization fraction along a given line of sight is correlated with the
local spatial coherence of the polarization angle. These observations are
compared to polarized emission maps computed in simulations of anisotropic
magnetohydrodynamical (MHD) turbulence in which we assume a uniform intrinsic
polarization fraction of the dust grains. We find that an estimate of this
parameter may be recovered from the maximum polarization fraction
$p_\mathrm{max}$ in diffuse regions where the magnetic field is ordered on
large scales and perpendicular to the line of sight. This emphasizes the impact
of anisotropies of the magnetic field on the emerging polarization signal. The
decrease of the polarization fraction with column density in nearby molecular
clouds is well reproduced in the simulations, indicating that it is essentially
due to the turbulent structure of the magnetic field: an accumulation of
variously polarized structures along the line of sight leads to such an
anti-correlation. In the simulations, polarization fractions are also found to
anti-correlate with the angle dispersion function $\mathcal{S}$. [abridged]
