  To interpret H_2 quasar absorption line observations in Damped Ly-alpha
clouds (DLAs), we model the H_2 spatial distribution within a DLA. Based on
numerical simulations of disk structures with parameters similar to those
derived for such absorbers, we calculate the H_2 distribution as a function of
ultraviolet background (UVB) intensity and dust-to-gas ratio. For typical
values of these two quantities we find that the area in which the H_2 fraction
exceeds 10^{-6} (typical observational detection limit) only covers $\la 10$%
of the disk surface, i.e. H_2 has a very inhomogeneous, clumpy distribution
even at these low abundance levels. This explains the relative paucity of H_2
detections in DLAs. We also show the dependence of the covering fraction of H_2
on dust-to-gas ratio and UVB intensity and we comment on the physics governing
the H_2 chemical network at high redshift.
