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tokens: ; Chen et al. 2016 ; Wu et al. 2017b ). For simulations described in the current work, we apply an HRO formulation whereis the magnitude of the angle between N H isocontours (orthogonal to ∇ N H ) and , the simulated polarization pseudovectors. Here, is defined aswhere p = 0.1 is a set constant polarization fraction and χ is the angle in the plane-of-sky derived from the Stokes parameters (see, e.g., Wu et al. 2017b ): The relative orientation angle Φ is then calculated pixel-by-pixel in a column density map. The lowest column density pixels () are ignored, and a gradient threshold for is applied (similar to, e.g., Planck Collaboration et al. 2016 ), to better separate GMC material from diffuse background fields. The remaining pixels are then separated into 25 bins of equal count based on N H . Note that we assume n He = 0.1 n H , yielding a mass of per H. Figure 2 shows column density maps for the noncolliding and colliding models with B = 10, 30, 50 μ G for the density-regulated model. Their respective HROs are also shown for low, medium, and high column density bins. Here, φ represents the smaller angle between the polarization-inferred magnetic field and N H isocontours. Thus, histograms peaking at φ = 0° indicate inferred B -fields preferentially aligned parallel to filamentary structures, while peaks at φ = 90° indicate a preferentially perpendicular alignment. Figure 2. First and third rows: column density maps with the normalized plane-of-sky magnetic polarization field, p , represented with black pseudovectors. The noncolliding cases for B = 10, 30, and 50 μ G are shown in the first row, while the respective colliding cases are shown in the third row. Gas from the density-regulated SF models is shown in each case at t = 5 Myr. Second and fourth rows: histograms of relative orientations are shown directly below their corresponding column-density maps. HROs compare the angle between p and iso- N H contours pixel-by-pixel. The projected map is divided into 25 column-density bins of equal pixel count. HROs for the lowest (1st bin; black), middle (12th bin; blue), and highest (25th bin; red) N H are shown, using angle bins of 15°. The histogram color corresponds to the regions bounded by low (black), intermediate (blue), and high (red) value iso- N H contours as shown in the respective projection map. The thickness of the data points indicates the variance within each bin. Histograms with peaks at 0° correspond to p predominantly aligned with iso- N H contours (i.e., B -fields parallel to filaments), while peaks at 90° correspond to p predominantly perpendicular to iso- N H contours (i.e., B -fields perpendicular to filaments). Gas in the noncolliding, model is slightly preferentially aligned parallel to filaments at high, medium, and low column densities. At , the low- N H gas retains effectively the same orientation behavior, while the mid- and high- N H bins show an increasing trend toward perpendicular alignment. This trend increases