2004ApJ...604..196B

The scaling of velocity fluctuation δv as a function of spatial scale L in molecular clouds can be measured from size-line width relations, principal component analysis, or line centroid variation. Differing values of the power-law index of the scaling relation <δv²>^1/2∝L^γ3D_^ in three dimensions are given by these different methods: the first two give γ_3D≃0.5, while line centroid analysis gives γ_3D≃0. This discrepancy has previously not been fully appreciated, as the variation of projected velocity line centroid fluctuations (<δv²_lc>^1/2∝L^γ2D_^) is indeed described, in two dimensions, by γ_2D~0.5. However, if projection smoothing is accounted for, this implies, in three dimensions, that γ_3D~0. We suggest that a resolution of this discrepancy can be achieved by accounting for the effect of density inhomogeneity on the observed γ_2Dobtained from velocity line centroid analysis. Numerical simulations of compressible turbulence are used to show that the effect of density inhomogeneity statistically, but not identically, reverses the effect of projection smoothing in the case of driven turbulence, so that velocity line centroid analysis does indeed predict that γ_2D~γ_3D~0.5. For decaying turbulence, the effect of density inhomogeneity on the velocity line centroids diminishes with time so that at late times, γ_2D~γ_3D+0.5 as a result of projection smoothing alone. Deprojecting the observed line centroid statistics thus requires some knowledge of the state of the flow. This information can be inferred from the spectral slope of the column density power spectrum and a measure of the standard deviation in column density relative to the mean. Using our numerical results, we can restore consistency between line centroid analysis, principal component analysis, and size-line width relations, and we derive γ_3D~0.5, corresponding to shock-dominated (Burgers) turbulence, which also describes the simulations of driven turbulence at scales on which numerical dissipation is negligible. We find that this consistency requires that molecular clouds are continually driven on large scales or are only recently formed.