Context : Supersonic turbulence is ubiquitous in the interstellar medium and plays an important role in contemporary star formation .

Aims : To perform a high-resolution numerical simulation of supersonic isothermal turbulence driven by compressive large-scale forcing and to analyse various statistical properties .

Methods : The compressible Euler equations with an external stochastic force field dominated by rotation-free modes are solved with the piecewise parabolic method .
Both a static grid and adaptive mesh refinement is used with an effective resolution N = 768 ^ { 3 } .

Results : After a transient phase dominated by shocks , turbulence evolves into a steady state with root mean square Mach number \approx 2.2 \ldots 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 \approx 2.0 in the shock-dominated phase .
As the flow approaches statistical equilibrium , the spectrum flattens , with \beta \approx 1.9 .
For the scaling exponent of the root mean square velocity fluctuation , we obtain \gamma \approx 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 , \ldots, 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 .
The spectral index of the quadratic logarithmic density fluctuation \beta _ { \delta } \approx 1.8 .
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 .

Conclusions : Even putting aside further complexity due to magnetic fields , gravity or thermal processes , we question the notion that Larson-type relations are a consequence of universal supersonic turbulence scaling .
For a genuine understanding , it seems necessary to account for the production mechanism of turbulence in the ISM .