Numerical simulations of compressively driven interstellar turbulence I. Isothermal gas

Context. Supersonic turbulence is ubiquitous in the interstellar medium and plays an important role in contemporary star formation. Aims. We performed high-resolution numerical simulations of supersonic isothermal turbulence driven by compressive large-scale forcing and 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 approximate to 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 is beta approximate to 2.0 in the shock-dominated phase. As the flow approaches statistical equilibrium, the spectrum flattens, with beta approximate to 1.9. For the scaling exponent of the root mean square velocity fluctuation, we obtain gamma approximate to 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 significant deviations from the Kolmogorov-Burgers model proposed by Boldyrev for all scaling exponents. Our results are very well described by a general log-Poisson model with a higher degree of intermittency, which implies an influence by the forcing on the scaling properties. The spectral index of the quadratic logarithmic density fluctuation is beta(delta) approximate to 1.8. In contrast 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.