  The sizable linear polarization signals produced by the scattering of
anisotropic radiation in the core of the Ca I 4227 $\mathrm{\r{A}}$ line
constitute an important observable for probing the inhomogeneous and dynamic
plasma of the lower solar chromosphere. Here we show the results of a
three-dimensional (3D) radiative transfer complete frequency redistribution
(CRD) investigation of the line's scattering polarization in a
magneto-hydrodynamical 3D model of the solar atmosphere. We take into account
not only the Hanle effect produced by the model's magnetic field, but also the
symmetry breaking caused by the horizontal inhomogeneities and macroscopic
velocity gradients. The spatial gradients of the horizontal components of the
macroscopic velocities produce very significant forward scattering polarization
signals without the need of magnetic fields, while the Hanle effect tends to
depolarize them at the locations where the model's magnetic field is stronger
than about 5 G. The standard 1.5D approximation is found to be unsuitable for
understanding the line's scattering polarization, but we introduce a novel
improvement to this approximation that produces results in qualitative
agreement with the full 3D results. The instrumental degradation of the
calculated polarization signals is also investigated, showing what can we
expect to observe with the Visible Spectro-Polarimeter at the upcoming Daniel
K. Inouye Solar Telescope.
