Relative alignment between gas structures and magnetic field in Orion A at different scales using different molecular gas tracers

Context: Magnetic fields can play crucial roles in high-mass star formation. Nonetheless, the significance of magnetic fields at various scales and their relationship with gas structures is largely overlooked. Aims: Our goal is to examine the relationship between the magnetic field and molecular gas structures within the Orion A giant molecular cloud at different scales and density regimes. Methods: We assess the gas intensity structures and column densities in Orion A by utilizing $^{12}$CO, $^{13}$CO, and C$^{18}$O from Nobeyama observations. Through comparing Nobeyama observations with {\it{Planck}} polarization observations on large scales ($\sim0.6$ pc) and JCMT polarization observations on small scales ($\sim0.04$ pc), we investigate how the role of magnetic fields change with scale and density. Results: We find a similar trend from parallel to perpendicular alignment with increasing column densities in Orion A at both large and small spatial scales. Besides, when changing from low-density to high-density tracers, the relative orientation preference changes from random to perpendicular. The self-similar results at different scales indicate that magnetic fields are dynamically important in both cloud formation and filament formation. However, magnetic fields properties at small scales are relative complicated, and the interplay between magnetic field and star-forming activities needs to be discussed case-by-case.