Simulating the formation of molecular clouds. II. Rapid formation from turbulent initial conditions

(Abridged). In this paper, we present results from a large set of numerical simulations that demonstrate that H2 formation occurs rapidly in turbulent gas. Starting with purely atomic hydrogen, large quantities of molecular hydrogen can be produced on timescales of 1 -- 2 Myr, given turbulent velocity dispersions and magnetic field strengths consistent with observations. Moreover, as our simulations underestimate the effectiveness of H2 self-shielding and dust absorption, we can be confident that the molecular fractions that we compute are strong lower limits on the true values. The formation of large quantities of H2 on the timescale required by rapid cloud formation models therefore appears to be entirely plausible. We also investigate the density and temperature distributions of gas in our model clouds. We show that the density probability distribution function is approximately log-normal, with a dispersion that agrees well with the prediction of Padoan, Nordlund & Jones (1997). The temperature distribution is similar to that of a polytrope, with an effective polytropic index gamma_eff \simeq 0.8, although at low gas densities, the scatter of the actual gas temperature around this mean value is considerable, and the polytropic approximation does not capture the full range of behaviour of the gas