Change of Magnetic Field$-$Gas Alignment at Gravity-Driven Alfv\'enic Transition in Molecular Clouds: Implications for Dust Polarization Observations

Diffuse striations in molecular clouds are preferentially aligned with local magnetic fields whereas dense filaments tend to be perpendicular to them. When and why this transition occurs remain uncertain. To explore the physics behind this transition, we compute the histogram of relative orientation (HRO) between the density gradient and the magnetic field in 3D MHD simulations of prestellar core formation in shock-compressed regions within GMCs. We find that, in the magnetically-dominated (sub-Alfv\'enic) post-shock region, the gas structure is preferentially aligned with the local magnetic field. For overdense sub-regions with super-Alfv\'enic gas, their elongation becomes preferentially perpendicular to the local magnetic field instead. The transition occurs when self-gravitating gas gains enough kinetic energy from the gravitational acceleration to overcome the magnetic support against the cross-field contraction, which results in a power-law increase of the field strength with density. Similar results can be drawn from HROs in projected 2D maps with integrated column densities and synthetic polarized dust emission. We quantitatively analyze our simulated polarization properties, and interpret the reduced polarization fraction at high column densities as the result of increased distortion of magnetic field directions in trans- or super-Alfv\'enic gas. Furthermore, we introduce measures of the inclination and tangledness of the magnetic field along the line of sight as the controlling factors of the polarization fraction. Observations of the polarization fraction and angle dispersion can therefore be utilized in studying local magnetic field morphology in star-forming regions.