The time evolution of the probability density function (PDF) of the mass density is formulated and solved for systems in free-fall using a simple appoximate function for the collapse of a sphere. We demonstrate that a pressure-free collapse results in a power-law tail on the high-density side of the PDF. The slope quickly asymptotes to the functional form P V (ρ) ∝ ρ −1.54 for the (volumeweighted) PDF and P M (ρ) ∝ ρ −0.54 for the corresponding mass-weighted distribution. From the simple approximation of the PDF we derive analytic descriptions for mass accretion, finding that dynamically quiet systems with narrow density PDFs lead to retarded star formation and low star formation rates. Conversely, strong turbulent motions that broaden the PDF accelerate the collapse causing a bursting mode of star formation. Finally, we compare our theoretical work with observations. The measured star formation rates are consistent with our model during the early phases of the collapse. Comparison of observed column density PDFs with those derived from our model suggests that observed star-forming cores are roughly in free-fall.
The probability density function (PDF) of the gas density in subsonic and supersonic, isothermal, driven turbulence is analysed with a systematic set of hydrodynamical grid simulations with resolutions up to 1024 3 cells. We performed a series of numerical experiments with root mean square (r.m.s.) Mach number M ranging from the nearly incompressible, subsonic (M = 0.1) to the highly compressible, supersonic (M = 15) regime. We study the influence of two extreme cases for the driving mechanism by applying a purely solenoidal (divergence-free) and a purely compressive (curl-free) forcing field to drive the turbulence. We find that our measurements fit the linear relation between the r.m.s. Mach number and the standard deviation of the density distribution in a wide range of Mach numbers, where the proportionality constant depends on the type of the forcing. In addition, we propose a new linear relation between the standard deviation of the density distribution σ ρ and the standard deviation of the velocity in compressible modes, i.e. the compressible component of the r.m.s. Mach number M comp . In this relation the influence of the forcing is significantly reduced, suggesting a linear relation between σ ρ and M comp , independent of the forcing, ranging from the subsonic to the supersonic regime.
We model driven, compressible, isothermal, turbulence with Mach numbers ranging from the subsonic (M ≈ 0.65) to the highly supersonic regime (M ≈ 16). The forcing scheme consists both solenoidal (transverse) and compressive (longitudinal) modes in equal parts. We find a relation σ 2 s = b log (1 + b 2 M 2 ) between the Mach number and the standard deviation of the logarithmic density with b = 0.457 ± 0.007. The density spectra follow D(k, M) ∝ k ζ(M) with scaling exponents depending on the Mach number. We find ζ(M) = αM β with a coefficient α that varies slightly with resolution, whereas β changes systematically. We extrapolate to the limit of infinite resolution and find α = −1.91 ± 0.01, β = −0.30 ± 0.03. The dependence of the scaling exponent on the Mach number implies a fractal dimension D = 2 + 0.96M −0.30 . We determine how the scaling parameters depend on the wavenumber and find that the density spectra are slightly curved. This curvature gets more pronounced with increasing Mach number. We propose a physically motivated fitting formula D(k) = D 0 k ζk η by using simple scaling arguments. The fit reproduces the spectral behaviour down to scales k ≈ 80. The density spectrum follows a single power-law η = −0.005 ± 0.01 in the low Mach number regime and the strongest curvature η = −0.04 ± 0.02 for the highest Mach number. These values of η represent a lower limit, as the curvature increases with resolution.
We present a systematic study of the influence of different forcing types on the statistical properties of supersonic, isothermal turbulence in both the Lagrangian and Eulerian frameworks. We analyse a series of high-resolution, hydrodynamical grid simulations with Lagrangian tracer particles and examine the effects of solenoidal (divergence-free) and compressive (curl-free) forcing on structure functions, their scaling exponents, and the probability density functions of the gas density and velocity increments. Compressively driven simulations show significantly larger density contrast, more intermittent behaviour, and larger fractal dimension of the most dissipative structures at the same root mean square Mach number. We show that the absolute values of Lagrangian and Eulerian structure functions of all orders in the integral range are only a function of the root mean square Mach number, but independent of the forcing. With the assumption of a Gaussian distribution for the probability density function of the velocity increments for large scales, we derive a model that describes this behaviour.
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