We discuss the generation and statistics of the density fluctuations in highly compressible polytropic turbulence, based on a simple model and one-dimensional numerical simulations. Observing that density structures tend to form in a hierarchical manner, we assume that density fluctuations follow a random multiplicative process. When the polytropic exponent γ is equal to unity, the local Mach number is independent of the density, and our assumption leads us to expect that the probability density function (PDF) of the density field is a lognormal. This isothermal case is found to be singular, with a dispersion σ 2 s which scales like the square turbulent Mach numberM 2 , where s ≡ ln ρ and ρ is the fluid density. This leads to much higher fluctuations than those due to shock jump relations.Extrapolating the model to the case γ = 1, we find that, as the Mach number becomes large, the density PDF is expected to asymptotically approach a power-law regime, at high densities when γ < 1, and at low densities when γ > 1. This effect can be traced back to the fact that the pressure term in the momentum equation varies exponentially with s, thus opposing the growth of fluctuations on one side of the PDF, while being negligible on the other side. This also causes the dispersion σ 2 s to grow more slowly thanM 2 when γ = 1. In view of these results, we suggest that Burgers flow is a singular case not approached by the high-M limit, with a PDF that develops power laws on both sides. 47.27. Ak, 47.40.Ki, 95.30.Lz

We discuss molecular cloud formation by large-scale supersonic compressions in the diffuse warm neutral medium (WNM). Initially, a shocked layer forms, and within it, a thin cold layer. An analytical model and high-resolution onedimensional simulations predict the thermodynamic conditions in the cold layer. After $1 Myr of evolution, the layer has column density $2:5 ; 10 19 cm À2, thickness $0.03 pc, temperature $25 K, and pressure $6650 K cm À3. These conditions are strongly reminiscent of those recently reported by Heiles and coworkers for cold neutral medium sheets. In the one-dimensional simulations, the inflows into the sheets produce line profiles with a central line of width $0.5 km s À1 and broad wings of width $1 km s À1. Three-dimensional numerical simulations show that the cold layer develops turbulent motions and increases its thickness until it becomes a fully three-dimensional turbulent cloud. Fully developed turbulence arises on times ranging from $7.5 Myr for inflow Mach number M 1; r ¼ 2:4 to >80 Myr for M 1; r ¼ 1:03. These numbers should be considered upper limits. The highest density turbulent gas (HDG, n > 100 cm À3) is always overpressured with respect to the mean WNM pressure by factors of 1.5-4, even though we do not include self-gravity. The intermediate-density gas (IDG, 10 < n/cm À3 < 100) has a significant pressure scatter that increases with M 1, r , so that at M 1; r ¼ 2:4 a significant fraction of the IDG is at a higher pressure than the HDG. Our results suggest that the turbulence and at least part of the excess pressure in molecular clouds can be generated by the compressive process that forms the clouds themselves and that thin CNM sheets may be formed transiently by this mechanism, when the compressions are only weakly supersonic.

We investigate the form of the one-point probability distribution function (pdf) for the density field of the interstellar medium using numerical simulations that successively reduce the number of physical processes included. Two-dimensional simulations of selfgravitating supersonic MHD turbulence, of supersonic self-gravitating hydrodynamic turbulence, and of decaying Burgers turbulence, produce in all cases filamentary density structures and evidence for a power-law density pdf with logarithmic slope around −1.7. This suggests that the functional form of the pdf and the general filamentary morphology are the signature of the nonlinear advection operator.These results do not support previous claims that the pdf is lognormal. A series of 1D simulations of forced supersonic polytropic turbulence is used to resolve the discrepancy. They suggest that the pdf is lognormal only for effective polytropic indices γ = 1 (or nearly lognormal for γ = 1 if the Mach number is sufficiently small), while power laws develop for densities larger than the mean if γ < 1. We evaluate the polytropic index for conditions relevant to the cool interstellar medium using published cooling functions and different heating sources, finding that a lognormal pdf may occur at densities between 10 3 and at least 10 4 cm −3 .Several applications are examined. First, we question a recent derivation of the IMF from the density pdf by Padoan, Nordlund & Jones because a) the pdf does not contain spatial information, and b) their derivation produces the most massive stars in the voids of the density distribution. Second, we illustrate how a distribution of ambient densities can alter the predicted form of the size distribution of expanding shells. Finally, a brief comparison is made with the density pdfs found in cosmological simulations.

Compressible flows with r.m.s. velocities of the order of the speed of sound are studied with direct numerical simulations using a pseudospectral method. We concentrate on turbulent homogeneous flows in the two-dimensional case. The fluid obeys the Navier-Stokes equations for a perfect gas, and viscous terms are included explicitly. No modelling of small scales is used. We show that the behaviour of the flow differs sharply at low compared with high r.m.s. Mach number Ma, with a transition at Ma = 0.3. In the large scales, temporal exchanges between longitudinal and solenoidal modes of energy retain an acoustical character; they lead to a slowing down of the decrease of the Mach number with time, which occurs with interspersed plateaux corresponding to quiescent periods. When the flow is initially supersonic, the small scales are dominated by shocks behind which vortices form. This vortex production is particularly prominent when two strong shocks collide, with the onset of shear turbulence in the region downstream of the collision. However, at the resolutions reached by our code on a 256 × 256 uniform grid, this mechanism proves insufficient to bring vortices into equipartition with shocks in the small-scale tail of the energy spectrum.