Modification of projected velocity power spectra by density inhomogeneities in compressible supersonic turbulence

The scaling of velocity fluctuation deltav 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 powerlaw index of the scaling relation [deltav(2)](1/2) proportional to L-gamma3D in three dimensions are given by these different methods: the first two give gamma(3D) similar or equal to 0.5, while line centroid analysis gives gamma(3D) similar or equal to 0. This discrepancy has previously not been fully appreciated, as the variation of projected velocity line centroid fluctuations ([deltav(lc)(2)](1/2) proportional to L-gamma2D) is indeed described, in two dimensions, by gamma(2D) approximate to 0.5. However, if projection smoothing is accounted for, this implies, in three dimensions, that gamma(3D) approximate to 0. We suggest that a resolution of this discrepancy can be achieved by accounting for the effect of density inhomogeneity on the observed gamma(2D) obtained 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 gamma(2D) approximate to gamma(3D) approximate to 0.5. For decaying turbulence, the effect of density inhomogeneity on the velocity line centroids diminishes with time so that at late times, gamma(2D) approximate to gamma(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 gamma(3D) approximate to 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.