Magnetic fields are known to exhibit different relative orientations with density structures in different density regimes. However, the physical mechanisms behind these relative orientations remain unclear. We investigate the role of the flow features on the relative orientation between the magnetic field and cold neutral medium (CNM) clouds, as well as that of molecular clouds (MCs) as a corollary. We performed three- and two-dimensional (3D+2D) magnetohydrodynamic (MHD) simulations of warm gas streams in the thermally bistable atomic interstellar medium (ISM) colliding with velocities of the order of the velocity dispersion in the ISM to form CNM clouds. In these simulations, we followed the evolution of magnetic field lines to identify and elucidate the physical processes behind their evolution. The collision produces a fast MHD shock, as well as a condensation front roughly one cooling length behind it, on each side of the collision front. A compressive, decelerating velocity field arises between the shock and the condensation fronts, and a cold dense layer forms behind the condensation front. The magnetic field lines, initially oriented parallel to the flow direction, are perturbed by the fast MHD shock, across which the magnetic field fluctuations parallel to the shock front are amplified. The downstream perturbations of the magnetic field lines are further amplified by the compressive downstream velocity gradient between the shock and the condensation front caused by the settlement of the gas onto the dense layer. This process causes the magnetic field to become progressively aligned with the dense layer, leading to the formation of a shear flow around it, due to the field's backreaction on the flow. By extension, we suggest that a tidal stretching velocity gradient, such as that produced in gas infalling into a self-gravitating structure, must straighten the field lines along the accretion flow, orienting them perpendicular to the density structures. We also find that the initially super-Alfv'enic upstream flow becomes trans-Alfv'enic between the shock and the condensation front, and then sub-Alfv'enic inside the condensation. Finally, in 2D simulations with a curved collision front, the presence of the magnetic field inhibits the generation of turbulence by the shear around the dense layer. Our results provide a feasible physical mechanism for orienting the magnetic field parallel to CNM clouds through the action of fast MHD shocks and compressive velocity fields.
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