  We performed numerical simulations of supersonic isothermal turbulence driven
by mostly compressive large-scale forcing, using both a static grid and
adaptive mesh refinement with an effective resolution N=768^3. After a
transient phase dominated by shocks, turbulence evolves into a steady state
with an RMS Mach number about 2.5, in which cloud-like structures of over-dense
gas are surrounded by highly rarefied gas. The index of the turbulence energy
spectrum function beta = 2.0 in the shock-dominated phase. As the flow
approaches statistical equilibrium, the spectrum flattens, with beta = 1.9. For
the scaling exponent of the root mean square velocity fluctuation, we obtain
gamma = 0.43 from the velocity structure functions of second order. These
results are well within the range of observed scaling properties for the
velocity dispersion in molecular clouds. Calculating structure functions of
order p=1,...,5, we find for all scaling exponents significant deviations from
the Kolmogorov-Burgers model proposed by Boldyrev. Our results are very well
described by a general log-Poisson model with a higher degree of intermittency,
which implies an influence of the forcing on the scaling properties. Contrary
to previous numerical results for isothermal turbulence, we obtain a skewed
probability density function of the mass density fluctuations that is not
consistent with log-normal statistics and entails a substantially higher
fraction of mass in the density peaks than implied by the Padoan-Nordlund
relation between the variance of the density fluctuations and the Mach number.
In conclusion, it seems necessary to account for the production mechanism of
turbulence in the ISM.
