  Radial velocity fluctuations on the plane of the sky are a powerful tool for
studying the turbulent dynamics of emission line regions. We conduct a
systematic statistical analysis of the H alpha velocity field for a diverse
sample of 9 H II regions, spanning two orders of magnitude in size and
luminosity, located in the Milky Way and other Local Group galaxies. By fitting
a simple model to the second-order spatial structure function of velocity
fluctuations, we extract three fundamental parameters: the velocity dispersion,
the correlation length, and the power law slope. We determine credibility
limits for these parameters in each region, accounting for observational
limitations of noise, atmospheric seeing, and the finite map size. The
plane-of-sky velocity dispersion is found to be a better diagnostic of
turbulent motions than the line width, especially for lower luminosity regions
where the turbulence is subsonic. The correlation length of velocity
fluctuations is found to be always roughly 2% of the H II region diameter,
implying that turbulence is driven on relatively small scales. No evidence is
found for any steepening of the structure function in the transition from
subsonic to supersonic turbulence, possibly due to the countervailing effect of
projection smoothing. Ionized density fluctuations are too large to be
explained by the action of the turbulence in any but the highest luminosity
sources. A variety of behaviors are seen on scales larger than the correlation
length, with only a minority of sources showing evidence for homogeneity on the
largest scales.
