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We present N-PDFs of 29 Galactic regions obtained from Herschel imaging at high angular resolution, covering diffuse and quiescent clouds, and those showing low-, intermediate-, and high-mass star formation (SF), and characterize the cloud structure using the Delta-variance tool. The N-PDFs are double-log-normal at low column densities, and display one or two power law tails (PLTs) at higher column densities. For diffuse, quiescent, and low-mass SF clouds, we propose that the two log-normals arise from the atomic and molecular phase, respectively. For massive clouds, we suggest that the first log-normal is built up by turbulently mixed H2 and the second one by compressed (via stellar feedback) molecular gas. Nearly all clouds have two PLTs with slopes consistent with self-gravity, where the second one can be flatter or steeper than the first one. A flatter PLT could be caused by stellar feedback or other physical processes that slow down collapse and reduce the flow of mass toward higher densities. The steeper slope could arise if the magnetic field is oriented perpendicular to the LOS column density distribution. The first deviation point (DP), where the N-PDF turns from log-normal into a PLT, shows a clustering around values of a visual extinction of AV (DP1) around 2-5. The second DP, which defines the break between the two PLTs, varies strongly. Using the Delta-variance, we observe that the AV value, where the slope changes between the first and second PLT, increases with the characteristic size scale in the variance spectrum. We conclude that at low column densities, atomic and molecular gas is turbulently mixed, while at high column densities, the gas is fully molecular and dominated by self-gravity. The best fitting model N-PDFs of molecular clouds is thus one with log-normal low column density distributions, followed by one or two PLTs., Comment: Astronomy and Astrophysics in presss

The magnetic field strength in interstellar clouds can be estimated indirectly by using the spread of dust polarization angles ($\delta \theta$). The method developed by Davis 1951 and by Chandrasekhar and Fermi 1953 (DCF) assumes that incompressible magnetohydrodynamic (MHD) fluctuations induce the observed dispersion of polarization angles, deriving $B\propto 1/\delta \theta$ (or, $\delta \theta \propto M_{A}$, in terms of the Alfv\'{e}nic Mach number). However, observations show that the interstellar medium (ISM) is highly compressible. Recently, Skalidis & Tassis 2021 (ST) relaxed the incompressibility assumption and derived instead $B\propto 1/\sqrt{\delta \theta}$ ($\delta \theta \propto M_{A}^2$). We explored what the correct scaling is in compressible and magnetized turbulence with numerical simulations. We used 26 magnetized, ideal-MHD numerical simulations with different types of forcing. The range of $M_{A}$ and sonic Mach numbers $M_{s}$ explored are $0.1 \leq M_{A} \leq 2.0$ and $0.5 \leq M_{s} \leq 20$. We created synthetic polarization maps and tested the assumptions and accuracy of the two methods. The synthetic data have a remarkable consistency with the $\delta \theta \propto M_{A}^{2}$ scaling, which is inferred by ST, while the DCF scaling fails to follow the data. The ST method shows an accuracy better than $50\%$ over the entire range of $M_{A}$ explored; DCF performs adequately only in the range of $M_{A}$ for which it has been optimized through the use of a "fudge factor". For low $M_{A}$, DCF is inaccurate by factors of tens. The assumptions of the ST method reflect better the physical reality in clouds with compressible and magnetized turbulence, and for this reason the method provides a much better estimate of the magnetic field strength over the DCF method., Comment: Accepted for publication in A&A