  We discuss the generation and statistics of the density fluctuations in
highly compressible polytropic turbulence, based on a simple model and
one-dimensional numerical simulations. Observing that density structures tend
to form in a hierarchical manner, we assume that density fluctuations follow a
random multiplicative process. When the polytropic exponent $\gamma$ is equal
to unity, the local Mach number is independent of the density, and our
assumption leads us to expect that the probability density function (PDF) of
the density field is a lognormal. This isothermal case is found to be singular,
with a dispersion $\sigma_s^2$ which scales like the square turbulent Mach
number $\tilde M^2$, where $s\equiv \ln \rho$ and $\rho$ is the fluid density.
This leads to much higher fluctuations than those due to shock jump relations.
  Extrapolating the model to the case $\gamma \not =1$, we find that, as the
Mach number becomes large, the density PDF is expected to asymptotically
approach a power-law regime, at high densities when $\gamma<1$, and at low
densities when $\gamma>1$. This effect can be traced back to the fact that the
pressure term in the momentum equation varies exponentially with $s$, thus
opposing the growth of fluctuations on one side of the PDF, while being
negligible on the other side. This also causes the dispersion $\sigma_s^2$ to
grow more slowly than $\tilde M^2$ when $\gamma\not=1$. In view of these
results, we suggest that Burgers flow is a singular case not approached by the
high-$\tilde M$ limit, with a PDF that develops power laws on both sides.
