  Within ten nearby (d < 450 pc) Gould Belt molecular clouds we evaluate
statistically the relative orientation between the magnetic field projected on
the plane of sky, inferred from the polarized thermal emission of Galactic dust
observed by Planck at 353 GHz, and the gas column density structures,
quantified by the gradient of the column density, $N_H$. The selected regions,
covering several degrees in size, are analyzed at an effective angular
resolution of 10' FWHM, thus sampling physical scales from 0.4 to 40 pc in the
nearest cloud. The column densities in the selected regions range from $N_H
\approx 10^{21}$ to $10^{23}$ cm$^{-2}$, and hence they correspond to the bulk
of the molecular clouds. The relative orientation is evaluated pixel by pixel
and analyzed in bins of column density using the novel statistical tool called
"Histogram of Relative Orientations". Throughout this study, we assume that the
polarized emission observed by Planck at 353 GHz is representative of the
projected morphology of the magnetic field in each region, i.e., we assume a
constant dust grain alignment efficiency, independent of the local environment.
Within most clouds we find that the relative orientation changes progressively
with increasing $N_H$, from preferentially parallel or having no preferred
orientation to preferentially perpendicular. In simulations of
magnetohydrodynamic turbulence in molecular clouds this trend in relative
orientation is a signature of Alfv\'enic or sub-Alfv\'enic turbulence, implying
that the magnetic field is significant for the gas dynamics at the scales
probed by Planck. We compare the deduced magnetic field strength with estimates
we obtain from other methods and discuss the implications of the Planck
observations for the general picture of molecular cloud formation and
evolution.
