Examining Turbulence in Galactic Molecular Clouds -- I: A Statistical Analysis of Velocity Structures

We present a systematic analysis of the velocity structure functions (VSFs) of 167 molecular clouds with angular sizes greater than $\sim$176 arcmin$^2$ in three sectors of the Galactic mid-plane. We calculated the 1st- to 3rd-order VSFs and found that 60\% of the VSFs exhibit power-law distributions. The relative power-law exponents are consistent with predictions from intermittent turbulence models. Column density weighting reduces the proportion of power-law VSFs and steepens the VSF slopes, implying a reduction of turbulent energy in high-density regions. All clouds show small-scale intermittency, with slightly stronger intermittency in those molecular clouds showing none power-law VSFs. Negative VSF exponents that may indicate gravitational collapse are not observed in our sample. The scaling exponents of the observed VSFs do not correlate with the virial parameters of the molecular clouds. These two observations suggest that gravity-dominated scales in molecular clouds still need further investigation. Consistent VSF scaling exponents for the molecular clouds with significant power-law VSFs suggest large-scale external driving of turbulence in these molecular clouds. However, the driving mechanisms are likely not universal, as the power-law scaling coefficients in our results show relatively large scatter. The fact that nearly 40\% of the VSFs deviate to some extent from power-law distributions suggests that the influence of local environments on the internal turbulence of molecular clouds may not be negligible.