  We model the $\lya$ forest in a spatially flat, CDM model with $\Omega=0.4$,
with an Eulerian hydrodynamic simulation, and find that the intergalactic,
photoionized gas collapses into sheet-like and filamentary structures which
produce absorption lines like those observed in the $\lya$ forest. A typical
filament is $\sim 1 h^{-1} \mpc$ long with thickness $\sim 50-100 h^{-1} \kpc$
(in proper units), and baryonic mass $\sim 10^{10} h^{-1} \msun$. The gas
temperature increases with time as structures with larger velocities collapse
gravitationally.
  The predicted distributions of column densities and b-parameters of the
absorption lines agree with the observed ones, and their evolution is
consistent with our model if the ionizing background has an approximately
constant intensity between $z=2$ and $z=4$. We also predict the distribution of
transmitted flux and its correlation along a spectrum and on parallel spectra,
and the \heii flux decrement as a function of redshift. We predict a
correlation length of $\sim 80 h^{-1}\kpc$ perpendicular to the line of sight
for features in the Lyman alpha forest.
  In order to reproduce the observed number of lines and average flux
transmission, the baryon content of the clouds may need to be significantly
higher than in previous models because of the low densities we predict. If the
background intensity $\jhi$ is at least that predicted from the observed
quasars, $\Omega_b$ needs to be $\sim 0.025 h^{-2}$, higher than expected by
light element nucleosynthesis; the model also predicts that most of the baryons
at $z>2$ are in $\lya$ clouds, and that the rate at which the baryons move to
more overdense regions is slow.
