The Turbulence Spectrum of Molecular Clouds in the Galactic Ring Survey:
  A Density-Dependent PCA Calibration
  Turbulence plays a major role in the formation and evolution of molecular
clouds. The problem is that turbulent velocities are convolved with the density
of an observed region. To correct for this convolution, we investigate the
relation between the turbulence spectrum of model clouds, and the statistics of
their synthetic observations obtained from Principal Component Analysis (PCA).
We apply PCA to spectral maps generated from simulated density and velocity
fields, obtained from hydrodynamic simulations of supersonic turbulence, and
from fractional Brownian motion fields with varying velocity, density spectra,
and density dispersion. We examine the dependence of the slope of the PCA
structure function, alpha_PCA, on intermittency, on the turbulence velocity
(beta_v) and density (beta_n) spectral indexes, and on density dispersion. We
find that PCA is insensitive to beta_n and to the log-density dispersion
sigma_s, provided sigma_s < 2. For sigma_s>2, alpha_PCA increases with sigma_s
due to the intermittent sampling of the velocity field by the density field.
The PCA calibration also depends on intermittency. We derive a PCA calibration
based on fBms with sigma_s<2 and apply it to 367 CO spectral maps of molecular
clouds in the Galactic Ring Survey. The average slope of the PCA structure
function, <alpha_PCA>=0.62\pm0.2, is consistent with the hydrodynamic
simulations and leads to a turbulence velocity exponent <beta_v>=2.06\pm0.6 for
a non-intermittent, low density dispersion flow. Accounting for intermittency
and density dispersion, the coincidence between the PCA slope of the GRS clouds
and the hydrodynamic simulations suggests beta_v~1.9, consistent with both
Burgers and compressible intermittent turbulence.
