The relation between the true and observed fractal dimensions of turbulent clouds

Observations of interstellar gas clouds are typically limited to two-dimensional (2D) projections of the intrinsically three-dimensional (3D) structure of the clouds. In this study, we present a novel method for relating the 2D projected fractal dimension (Dp) to the 3D fractal dimension (D3D) of turbulent clouds. We do this by computing the fractal dimension of clouds over two orders of magnitude in turbulent Mach number (M = 1–100), corresponding to seven orders of magnitude in spatial scales within the clouds. This provides us with the data to create a new empirical relation between Dp and D3D. The proposed relation is D3D(Dp) = 1 erfc(ξ1 erfc−1 [(Dp − Dp,min)/2] + ξ2) + D3D,min, where the minimum 3D fractal dimension, D3D,min = 2.06 ± 0.35, the minimum projected fractal dimension, Dp,min = 1.55 ± 0.13, 1 = 0.47 ± 0.18, 2 = 0.22 ± 0.07, ξ 1 = 0.80 ± 0.18, and ξ 2 = 0.26 ± 0.19. The minimum 3D fractal dimension, D3D,min = 2.06 ± 0.35, indicates that in the high M limit the 3D clouds are dominated by planar shocks. The relation between Dp and D3D of molecular clouds may be a useful tool for those who are seeking to understand the 3D structures of molecular clouds, purely based upon 2D projected data and shows promise for relating the physics of the turbulent clouds to the fractal dimension.JRB thanks all those who helped with reading and discussing this piece of science, especially Mairead MacKinnon. CF acknowledges funding provided by the Australian Research Council (Discovery Project DP170100603, and Future Fellowship FT180100495),and the Australia–Germany Joint Research Cooperation Scheme (UA-DAAD). RSK acknowledges support from the Deutsche Forschungsgemeinschaft via SFB 881, ‘The Milky Way System’ (subprojects B1, B2, and B8). We further acknowledge high performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo, pr48pi, and GCS Large-scale project 10391), the Partnership for Advanced Computing in Europe (PRACE grant pr89mu), the Australian National Computational Infrastructure (grant ek9), and the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia, in the framework of the National Computational Merit Allocation Scheme and the ANU Allocation Scheme. The simulation software FLASH was in part developed by the DOE-supported Flash Centre for Computational Science at the University of Chicago