  We present a study of the high rotational bands (J > 2) of H2 toward 4 early
type galactic stars: HD 73882, HD 192639, HD 206267, and HD 207538. In each
case, the velocity dispersion - characterized by the spectrum fitting parameter
b - increases with the level of excitation, a phenomenon that has previously
been detected by the Copernicus and IMAPS observatories. In particular, we show
with 4 sigma confidence that for HD 192639 it is not possible to fit all J
levels with a single b value, and that higher b values are needed for the
higher levels. The amplitude of the line broadening, which can be as high as 10
km s^-1, makes explanations such as inhomogeneous spatial distribution
unlikely. We investigate a mechanism in which the broadening is due to the
molecules that are rotationally excited through the excess energy acquired
after their formation on a grain (H2-formation pumping). We show that different
dispersions would be a natural consequence of this mechanism. We note however
that such process would require a formation rate 10 times higher then what was
inferred from other observations. In view of the difficulty to account for the
velocity dispersion as thermal broadening (T would be around 10,000 K), we
conclude then that we are most certainly observing some highly turbulent warm
layer associated with the cold diffuse cloud. Embedded in a magnetic field, it
could be responsible for the high quantities of CH+ measured in the cold
neutral medium.
