Planck intermediate results XLIV. Structure of the Galactic magnetic field from dust polarization maps of the southern Galactic cap

Using data from the Planck satellite, we study the statistical properties of interstellar dust polarization at high Galactic latitudes around the south pole (b < -60 degrees). Our aim is to advance the understanding of the magnetized interstellar medium (ISM), and to provide a modelling framework of the polarized dust foreground for use in cosmic microwave background (CMB) component-separation procedures. We examine the Stokes I, Q, and U maps at 353 GHz, and particularly the statistical distribution of the polarization fraction (p) and angle (Psi), in order to characterize the ordered and turbulent components of the Galactic magnetic field (GMF) in the solar neighbourhood. The Q and U maps show patterns at large angular scales, which we relate to the mean orientation of the GMF towards Galactic coordinates (l(0); b(0)) = (70 degrees +/- 5 degrees, 24 degrees +/- 5 degrees). The histogram of the observed p values shows a wide dispersion up to 25%. The histogram Psi of has a standard deviation of 12 degrees about the regular pattern expected from the ordered GMF. We build a phenomenological model that connects the distributions of p and Psi to a statistical description of the turbulent component of the GMF, assuming a uniform effective polarization fraction (p(0)) of dust emission. To compute the Stokes parameters, we approximate the integration along the line of sight (LOS) as a sum over a set of N independent polarization layers, in each of which the turbulent component of the GMF is obtained from Gaussian realizations of a power-law power spectrum. We are able to reproduce the observed p and distributions using a p0 value of 26%, a ratio of 0.9 between the strengths of the turbulent and mean components of the GMF, and a small value of N. The mean value of p (inferred from the fit of the large-scale patterns in the Stokes maps) is 12 +/- 1%. We relate the polarization layers to the density structure and to the correlation length of the GMF along the LOS. We emphasize the simplicity of our model (involving only a few parameters), which can be easily computed on the celestial sphere to produce simulated maps of dust polarization. Our work is an important step towards a model that can be used to assess the accuracy of component-separation methods in present and future CMB experiments designed to search the B mode CMB polarization from primordial gravity waves.