We present a detailed analysis comparing the velocity fields in molecular clouds and the atomic gas that surrounds them in order to address the origin of the gradients. To that end, we present first-moment intensity-weighted velocity maps of the molecular clouds and surrounding atomic gas. The maps are made from high-resolution 13CO observations and 21-cm observations from the Leiden/Argentine/Bonn Galactic HI Survey. We find that (i) the atomic gas associated with each molecular cloud has a substantial velocity gradient---ranging within 0.02 to 0.07 km/s/pc---whether or not the molecular cloud itself has a substantial linear gradient (ii) If the gradients in the molecular and atomic gas were due to rotation, this would imply that the molecular clouds have less specific angular momentum than the surrounding HI by a factor of 1-6. (iii) Most importantly, the velocity gradient position angles in the molecular and atomic gas are generally widely separated---by as much as 130 degrees in the case of the Rosette Molecular Cloud. This result argues against the hypothesis that molecular clouds formed by simple top-down collapse from atomic gas.
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