  In the nonlinear regime of cosmic clustering, the mass density field of the
cosmic baryon fluid is highly non-Gaussian. It shows different dynamical
behavior from collisionless dark matter. Nevertheless, the evolved field of
baryon fluid is scale-covariant in the range from the Jeans length to a few ten
h^{-1} Mpc, in which the dynamical equations and initial perturbations are
scale free. We show that in the scale-free range, the non-Gaussian features of
the cosmic baryon fluid, governed by the Navier-Stokes equation in an expanding
universe, can be well described by a log-Poisson hierarchical cascade. The
log-Poisson scheme is a random multiplicative process (RMP), which causes
non-Gaussianity and intermittency even when the original field is Gaussian. The
log-Poisson RMP contains two dimensionless parameters: $\beta$ for the
intermittency and $\gamma$ for the most singular structure. All the predictions
given by the log-Poisson RMP model, including the hierarchical relation, the
order dependence of the intermittent exponent, the moments, and the scale-scale
correlation, are in good agreement with the results given by hydrodynamic
simulations of the standard cold dark matter model. The intermittent parameter
$\beta$ decreases slightly at low redshift and indicates that the density field
of baryon fluid contains more singular structures at lower redshifts. The
applicability of the model is addressed.
