Radio synchrotron polarization maps of the Galaxy can be used to infer the properties of interstellar turbulence in the diffuse warm ionized medium (WIM). In this paper, we investigate the spatial gradient of linearly polarized synchrotron emission (
$| ablatextbf{P}|/|textbf{P}|$) as a tracer of turbulence, the relationship of the gradient to the sonic Mach number of the WIM, and changes in morphology of the gradient as a function of Galactic position in the southern sky. We use data from the S-band Polarization All Sky Survey (S-PASS) to image the spatial gradient of the linearly polarized synchrotron emission ($| abla textbf{P}|/|textbf{P}|$) of the entire southern sky at $2.3$~GHz. The spatial gradient of linear polarization reveals rapid changes of the density and magnetic fluctuations in the WIM due to magnetic turbulence as a function of Galactic position; we make comparisons of these data to ideal MHD numerical simulations. In order to constrain the sonic Mach number ($M_{s}$), we apply a high order moments analysis to the observations and to the simulated diffuse, isothermal ISM with ideal magneto-hydrodynamic turbulence. We find that polarization gradient maps reveal elongated structures, which we associate with turbulence in the ISM. Our analysis corroborates the view of a turbulent WIM in a transonic regime $M_{s}lesssim 2$. Filamentary structures with typical widths down to the angular resolution are seen and the observed morphologies match closely with numerical simulations and in some cases H$alpha$ contours. The $| abla textbf{P}|/|textbf{P}|$ intensity is found to be approximately log-normal distributed. No systematic variations of the sonic Mach number are observed as a function of Galactic coordinates, which is consistent with turbulence in the WIM inferred by the analysis of H$alpha$ data.
The characteristic outer scale of turbulence and the ratio of the random to ordered components of the magnetic field are key parameters to characterise magnetic turbulence in the interstellar gas, which affects the propagation of cosmic rays within t
he Galaxy. We provide new constraints to those two parameters. We use the LOw Frequency ARray (LOFAR) to image the diffuse continuum emission in the Fan region at (l,b) (137.0,+7.0) at 80x70 resolution in the range [146,174] MHz. We detect multi-scale fluctuations in the Galactic synchrotron emission and compute their power spectrum. Applying theoretical estimates and derivations from the literature for the first time, we derive the outer scale of turbulence and the ratio of random to ordered magnetic field from the characteristics of these fluctuations . We obtain the deepest image of the Fan region to date and find diffuse continuum emission within the primary beam. The power spectrum of the foreground synchrotron fluctuations displays a power law behaviour for scales between 100 and 8 arcmin with a slope of (-1.84+/-0.19). We find an upper limit of about 20 pc for the outer scale of the magnetic interstellar turbulence toward the Fan region. We also find a variation of the ratio of random to ordered field as a function of Galactic coordinates, supporting different turbulent regimes. We use power spectra fluctuations from LOFAR as well as earlier GMRT and WSRT observations to constrain the outer scale of turbulence of the Galactic synchrotron foreground, finding a range of plausible values of 10-20 pc. Then, we use this information to deduce lower limits of the ratio of ordered to random magnetic field strength. These are found to be 0.3, 0.3, and 0.5 for the LOFAR, WSRT and GMRT fields considered respectively. Both these constraints are in agreement with previous estimates.
