On the structure of the turbulent interstellar atomic hydrogen - I. Physical characteristics. Influence and nature of turbulence in a thermally bistable flow

Aims. We study in some details the statistical properties of the turbulent 2-phase interstellar atomic gas. Methods. We present high resolution bidimensional numerical simulations of the interstellar atomic hydrogen which describe it over 3 to 4 orders of magnitude in spatial scales. Results. The simulations produce naturally small scale structures having either large or small column density. It is tempting to propose that the former are connected to the tiny small scale structures observed in the ISM. We compute the mass spectrum of CNM structures and find that N(M)dM proportional to M(-1.7)dM, which is remarkably similar to the mass spectrum inferred for the CO clumps. We propose a theoretical explanation based on a formalism inspired from the Press & Schecter (1974, ApJ, 187, 425) approach and use the fact that the turbulence within WNM is subsonic. This theory predicts N(M) proportional to M-5/3 in 2D and N(M) proportional to M-16/9 in 3D. We compute the velocity and the density power-spectra and conclude that, although the latter is rather flat, as observed in supersonic isothermal simulations, the former follows the Kolmogorov prediction and is dominated by its solenoidal component. This is due to the bistable nature of the flow which produces large density fluctuations, even when the rms Mach number (of WNM) is not large. We also find that, whereas the energy at large scales is mainly in the WNM, at smaller scales, it is dominated by the kinetic energy of the CNM fragments. Conclusions. We find that turbulence in a thermally bistable flow like the atomic interstellar hydrogen, is somehow different from turbulence in a supersonic isothermal gas. In a companion paper, we compare the numerical results with atomic hydrogen observations and show that the simulations well reproduce various observational features.