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Supersonic turbulence occurs in many environments, particularly in astrophysics. In the crucial case of isothermal turbulence, the probability density function (PDF) of the logarithmic density, $s$, is well measured, but a theoretical understanding of the processes leading to this distribution remains elusive. We investigate these processes using Lagrangian tracer particles to track $s$ and $\frac{ds}{dt}$ in direct numerical simulations, and we show that their evolution can be modeled as a stochastic differential process with time-correlated noise. The temporal correlation functions of $s$ and $\frac{ds}{dt}$ decay exponentially, as predicted by the model, and the decay timescale is $\approx$ 1/6 the eddy turnover time. The behavior of the conditional averages of $\frac{ds}{dt}$ and $\frac{d^2s}{dt^2}$ is also well explained by the model, which shows that the density PDF arises from a balance between stochastic compressions/expansions, which tend to broaden the PDF, and the acceleration/deceleration of shocks by density gradients, which tends to narrow it., Comment: 40 pages, 10 figures, to appear in Science Advances

Protoplanetary disks are routinely described as finite mass reservoirs left over by the gravitational collapse of the protostar, an assumption that strongly constrains both disk evolution and planet formation models. We propose a different scenario where protoplanetary disks of pre-main sequence stars are assembled primarily by Bondi-Hoyle accretion from the parent gas cloud. We demonstrate that Bondi-Hoyle accretion can supply not only the mass, but also the angular momentum necessary to explain the observed size of protoplanetary disks, and we predict the dependence of the disk specific angular momentum on the stellar mass. Our results are based on an analytical derivation of the scaling of the angular momentum in a turbulent flow, which we also confirm with a numerical simulation of supersonic turbulence. This new scenario for disk formation and evolution may alleviate a number of observational problems as well as compel major revisions of disk and planet formation models., Comment: Submitted to Nature Astronomy

Stars form through the gravitational collapse of molecular cloud cores. Before collapsing, the cores are supported by thermal pressure and turbulent motions. A question of critical importance for the understanding of star formation is how to observationally discern whether a core has already initiated gravitational collapse or is still in hydrostatic balance. The canonical method to identify gravitational collapse is based on the observed density radial profile, which would change from a Bonnor-Ebert type toward power laws as the core collapses. In practice, due to the projection effect, the resolution limit, and other caveats, it has been difficult to directly reveal the dynamical status of cores, particularly in massive star-forming regions. We here propose a novel, straight-forward diagnostic, namely, the collapsing index (CI), which can be modeled and calculated based on the radial profile of the line width of dense gas. A meaningful measurement of CI requires spatially and spectrally resolved images of optically thin and chemically stable dense gas tracers. ALMA observations are making such data sets increasingly available for massive star-forming regions. Applying our method to one of the deepest dense-gas spectral images ever taken toward such a region, namely, the Orion molecular cloud, we detect the dynamical status of selected cores therein. We observationally distinguished a collapsing core in a massive star-forming region from a hydrostatical one. Our approach would help significantly improve our understanding of the interaction between gravity and turbulence within molecular cloud cores in the process of star formation., Comment: 15 pages, 9 figures, accepted by RAA

While the importance of supernova feedback in galaxies is well established, its role on the scale of molecular clouds is still debated. In this work, we focus on the impact of supernovae on individual clouds, using a high-resolution magneto-hydrodynamic simulation of a region of 250 pc where we resolve the formation of individual massive stars. The supernova feedback is implemented with real supernovae that are the natural evolution of the resolved massive stars, so their position and timing are self-consistent. We select a large sample of molecular clouds from the simulation to investigate the supernova energy injection and the resulting properties of molecular clouds. We find that molecular clouds have a lifetime of a few dynamical times, less then half of them contract to the point of becoming gravitationally bound, and the dispersal time of bound clouds, of order one dynamical time, is a factor of two shorter than that of unbound clouds. We stress the importance of internal supernovae, that is massive stars that explode inside their parent cloud, in setting the cloud dispersal time, and their huge overdensity compared to models where the supernovae are randomly distributed. We also quantify the energy injection efficiency of supernovae as a function of supernova distance to the clouds. We conclude that intermittent driving by supernovae can maintain molecular-cloud turbulence and may be the main process of cloud dispersal. The role of supernovae in the evolution of molecular clouds cannot be fully accounted for without a self-consistent implementation of their feedback., Comment: 33 pages, 23 figures, submitted to ApJ

We address the problem of the origin of massive stars, namely the origin, path and timescale of the mass flows that create them. Based on extensive numerical simulations, we propose a scenario where massive stars are assembled by large-scale, converging, inertial flows that naturally occur in supersonic turbulence. We refer to this scenario of massive-star formation as the "Inertial-Inflow Model". This model stems directly from the idea that the mass distribution of stars is primarily the result of turbulent fragmentation. Under this hypothesis, the statistical properties of the turbulence determine the formation timescale and mass of prestellar cores, posing definite constraints on the formation mechanism of massive stars. We quantify such constraints by the analysis of a simulation of supernova-driven turbulence in a 250-pc region of the interstellar medium, describing the formation of hundreds of massive stars over a time of approximately 30 Myr. Due to the large size of our statistical sample, we can say with full confidence that massive stars in general do not form from the collapse of massive cores, nor from competitive accretion, as both models are incompatible with the numerical results. We also compute synthetic continuum observables in Herschel and ALMA bands. We find that, depending on the distance of the observed regions, estimates of core mass based on commonly-used methods may exceed the actual core masses by up to two orders of magnitude, and that there is essentially no correlation between estimated and real core masses., Comment: 35 pages, 30 figures, ApJ in press

We study density fluctuations in supersonic turbulence using both theoretical methods and numerical simulations. A theoretical formulation is developed for the probability distribution function (PDF) of the density at steady state, connecting it to the conditional statistics of the velocity divergence. Two sets of numerical simulations are carried out, using either a Riemann solver to evolve the Euler equations or a finite-difference method to evolve the Navier-Stokes (N-S) equations. After confirming the validity of our theoretical formulation with the N-S simulations, we examine the effects of dynamical processes on the PDF, showing that the nonlinear term in the divergence equation amplifies the right tail of the PDF and reduces the left one, the pressure term reduces both the right and left tails, and the viscosity term, counter-intuitively, broadens the right tail of the PDF. Despite the inaccuracy of the velocity divergence from the Riemann runs, as found in our previous work, we show that the density PDF from the Riemann runs is consistent with that from the N-S runs. Taking advantage of their much higher effective resolution, we then use the Riemann runs to study the dependence of the PDF on the Mach number, $\mathcal{M}$, up to $\mathcal{M}\sim30$. The PDF width, $\sigma_{s}$, follows the relation $\sigma_{s}^2 = \ln (1+b^2 {\mathcal M}^2)$, with $b\approx0.38$. However, the PDF exhibits a negative skewness that increases with increasing $\mathcal{M}$, so much of the growth of $\sigma_{s}$ is accounted for by the growth of the left PDF tail, while the growth of the right tail tends to saturate. Thus, the usual prescription that combines a lognormal shape with the standard variance-Mach number relation greatly overestimates the right PDF tail at large $\mathcal{M}$, which may have a significant impact on theoretical models of star formation., Comment: Submitted to ApJ

The probabilistic approach to turbulence is applied to investigate density fluctuations in supersonic turbulence. We derive kinetic equations for the probability distribution function (PDF) of the logarithm of the density field, $s$, in compressible turbulence in two forms: a first-order partial differential equation involving the average divergence conditioned on the flow density, $\langle \nabla \cdot {\bs u} | s\rangle$, and a Fokker-Planck equation with the drift and diffusion coefficients equal to $-\langle {\bs u} \cdot \nabla s | s\rangle$ and $\langle {\bs u} \cdot \nabla s | s\rangle$, respectively. Assuming statistical homogeneity only, the detailed balance at steady state leads to two exact results, $\langle \nabla \cdot {\bs u} | s \rangle =0$, and $\langle {\bs u} \cdot \nabla s | s\rangle=0$. The former indicates a balance of the flow divergence over all expanding and contracting regions at each given density. The exact results provide an objective criterion to judge the accuracy of numerical codes with respect to the density statistics in supersonic turbulence. We also present a method to estimate the effective numerical diffusion as a function of the flow density and discuss its effects on the shape of the density PDF., Comment: Accepted for publication in ApJL

We make use of new subgrid model of turbulent mixing to accurately follow the cosmological evolution of the first stars, the mixing of their supernova ejecta, and the impact on the chemical composition of the Galactic Halo. Using the cosmological adaptive mesh refinement code RAMSES, we implement a model for the pollution of pristine gas as described in Pan et al. Tracking the metallicity of Pop III stars with metallicities below a critical value allows us to account for the fraction of Z < Zcrit stars formed even in regions in which the gas' average metallicity is well above Zcrit. We demonstrate that such partially-mixed regions account for 0.5 to 0.7 of all Pop III stars formed up to z = 5. Additionally, we track the creation and transport of "primordial metals" (PM) generated by Pop III supernovae (SNe). These neutron-capture deficient metals are taken up by second-generation stars and likely lead to unique abundance signatures characteristic of carbon-enhanced, metal-poor (CEMP-no) stars. As an illustrative example, we associate primordial metals with abundance ratios used by Keller et al. to explain the source of metals in the star SMSS J031300.36-670839.3, finding good agreement with the observed [Fe/H], [C/H], [O/H], and [Mg/Ca] ratios in CEMP-no Milky Way halo stars. Similar future simulations will aid in further constraining the properties of Pop III stars using CEMP observations, as well as improve predictions of the spatial distribution of Pop III stars, as will be explored by the next generation of ground- and space-based telescopes., Comment: 19 pages, 13 figures, accepted by ApJ Oct 2016

We present a comparison of molecular clouds (MCs) from a simulation of supernova-driven interstellar medium (ISM) turbulence with real MCs from the Outer Galaxy Survey. The radiative transfer calculations to compute synthetic CO spectra are carried out assuming the CO relative abundance depends only on gas density, according to four different models. Synthetic MCs are selected above a threshold brightness temperature value, $T_{\rm B,min}=1.4$ K, of the $J=1-0$ $^{12}$CO line, generating 16 synthetic catalogs (four different spatial resolutions and four CO abundance models), each containing up to several thousands MCs. The comparison with the observations focuses on the mass and size distributions and on the velocity-size and mass-size Larson relations. The mass and size distributions are found to be consistent with the observations, with no significant variations with spatial resolution or chemical model, except in the case of the unrealistic model with constant CO abundance. The velocity-size relation is slightly too steep for some of the models, while the mass-size relation is a bit too shallow for all models only at a spatial resolution $dx\approx 1$ pc. The normalizations of the Larson relations show a clear dependence on spatial resolution, for both the synthetic and the real MCs. The comparison of the velocity-size normalization suggests that the SN rate in the Perseus arm is approximately 70\% or less of the rate adopted in the simulation. Overall, the realistic properties of the synthetic clouds confirm that supernova-driven turbulence can explain the origin and dynamics of MCs., Comment: accepted for publication in ApJ

The compressibility of molecular cloud (MC) turbulence plays a crucial role in star formation models, because it controls the amplitude and distribution of density fluctuations. The relation between the compressive ratio (the ratio of powers in compressive and solenoidal motions) and the statistics of turbulence has been previously studied systematically only in idealized simulations with random external forces. In this work, we analyze a simulation of large-scale turbulence (250 pc) driven by supernova (SN) explosions that has been shown to yield realistic MC properties. We demonstrate that SN driving results in MC turbulence with a broad lognormal distribution of the compressive ratio, with a mean value $\approx 0.3$, lower than the equilibrium value of $\approx 0.5$ found in the inertial range of isothermal simulations with random solenoidal driving. We also find that the compressibility of the turbulence is not noticeably affected by gravity, nor are the mean cloud radial (expansion or contraction) and solid-body rotation velocities. Furthermore, the clouds follow a general relation between the rms density and the rms Mach number similar to that of supersonic isothermal turbulence, though with a large scatter, and their average gas density PDF is described well by a lognormal distribution, with the addition of a high-density power-law tail when self-gravity is included., Comment: Accepted by ApJ

Turbulence is ubiquitous in molecular clouds (MCs), but its origin is still unclear because MCs are usually assumed to live longer than the turbulence dissipation time. Interstellar medium (ISM) turbulence is likely driven by SN explosions, but it has never been demonstrated that SN explosions can establish and maintain a turbulent cascade inside MCs consistent with the observations. In this work, we carry out a simulation of SN-driven turbulence in a volume of (250 pc)$^3$, specifically designed to test if SN driving alone can be responsible for the observed turbulence inside MCs. We find that SN driving establishes a velocity scaling consistent with the usual scaling laws of supersonic turbulence, suggesting that previous idealized simulations of MC turbulence, driven with a random, large-scale volume force, were correctly adopted as appropriate models for MC turbulence, despite the artificial driving. We also find that the same scaling laws extend to the interior of MCs, and that the velocity-size relation of the MCs selected from our simulation is consistent with that of MCs from the Outer-Galaxy Survey, the largest MC sample available. The mass-size relation and the mass and size probability distributions also compare successfully with those of the Outer Galaxy Survey. Finally, we show that MC turbulence is super-Alfv\'{e}nic with respect to both the mean and rms magnetic-field strength. We conclude that MC structure and dynamics are the natural result of SN-driven turbulence., Comment: ApJ, in press

We conduct simulations of turbulent mixing in the presence of a magnetic field, grown by the small-scale dynamo. We show that the scalar gradient field, $\nabla C$, which must be large for diffusion to operate, is strongly biased perpendicular to the magnetic field, ${\mathbf B}$. This is true both early-on, when the magnetic field is negligible, and at late times, when the field is strong enough to back react on the flow. This occurs because $\nabla C$ increases within the plane of a compressive motion, but ${\mathbf B}$ increases perpendicular to it. At late times the magnetic field resists compression, making it harder for scalar gradients to grow and likely slowing mixing., Comment: ApJ Letters (in press)

Turbulent motions are essential to the mixing of entrained fluids and are also capable of amplifying weak initial magnetic fields by small-scale dynamo action. Here we perform a systematic study of turbulent mixing in magnetized media, using three-dimensional magnetohydrodynamic simulations that include a scalar concentration field. We focus on how mixing depends on the magnetic Prandtl number, Pm, from 1 to 4 and the Mach number, M}, from 0.3 to 2.4. For all subsonic flows, we find that the velocity power spectrum has a k^-5/3 slope in the early, kinematic phase, but steepens due to magnetic back reactions as the field saturates. The scalar power spectrum, on the other hand, flattens compared to k^-5/3 at late times, consistent with the Obukohov-Corrsin picture of mixing as a cascade process. At higher Mach numbers, the velocity power spectrum also steepens due to the presence of shocks, and the scalar power spectrum again flattens accordingly. Scalar structures are more intermittent than velocity structures in subsonic turbulence while for supersonic turbulence, velocity structures appear more intermittent than the scalars only in the kinematic phase. Independent of the Mach number of the flow, scalar structures are arranged in sheets in both the kinematic and saturated phases of the magnetic field evolution. For subsonic turbulence, scalar dissipation is hindered in the strong magnetic field regions, probably due to Lorentz forces suppressing the buildup of scalar gradients, while for supersonic turbulence, scalar dissipation increases monotonically with increasing magnetic field strength. At all Mach numbers, mixing is significantly slowed by the presence of dynamically-important small-scale magnetic fields, implying that mixing in the interstellar medium and in galaxy clusters is less efficient than modeled in hydrodynamic simulations., Comment: 16 pages, 8 figures, ApJ, in press

We conduct a comprehensive theoretical and numerical investigation of the pollution of pristine gas in turbulent flows, designed to provide new tools for modeling the evolution of the first generation of stars. The properties of such Population III (Pop III) stars are thought to be very different than later generations, because cooling is dramatically different in gas with a metallicity below a critical value Z_c, which lies between ~10^-6 and 10^-3 solar value. Z_c is much smaller than the typical average metallicity, , and thus the mixing efficiency of the pristine gas in the interstellar medium plays a crucial role in the transition from Pop III to normal star formation. The small critical value, Z_c, corresponds to the far left tail of the probability distribution function (PDF) of the metallicity. Based on closure models for the PDF formulation of turbulent mixing, we derive equations for the fraction of gas, P, lying below Z_c, in compressible turbulence. Our simulation data shows that the evolution of the fraction P can be well approximated by a generalized self-convolution model, which predicts dP/dt = -n/tau_con P (1-P^(1/n)), where n is a measure of the locality of the PDF convolution and the timescale tau_con is determined by the rate at which turbulence stretches the pollutants. Using a suite of simulations with Mach numbers ranging from M = 0.9 to 6.2, we provide accurate fits to n and tau_con as a function of M, Z_c/, and the scale, L_p, at which pollutants are added to the flow. For P>0.9, mixing occurs only in the regions surrounding the pollutants, such that n=1. For smaller P, n is larger as mixing becomes more global. We show how the results can be used to construct one-zone models for the evolution of Pop III stars in a single high-redshift galaxy, as well as subgrid models for tracking the evolution of the first stars in large cosmological simulations., Comment: 37 pages, accepted by ApJ

The interstellar medium in star-forming galaxies is a multiphase gas in which turbulent support is at least as important as thermal pressure. Sustaining this configuration requires continuous radiative cooling, such that the overall average cooling rate matches the decay rate of turbulent energy into the medium. Here we carry out a set of numerical simulations of a stratified, turbulently stirred, radiatively cooled medium, which uncover a fundamental transition at a critical one-dimensional turbulent velocity of ~ 35 km/s. At turbulent velocities below ~35 km/s, corresponding to temperatures below 300,000 K, the medium is stable, as the time for gas to cool is roughly constant as a function of temperature. On the other hand, at turbulent velocities above the critical value, the gas is shocked into an unstable regime in which the cooling time increases strongly with temperature, meaning that a substantial fraction of the interstellar medium is unable to cool on a turbulent dissipation timescale. This naturally leads to runaway heating and ejection of gas from any stratified medium with a one-dimensional turbulent velocity above ~35 km/s, a result that has implications for galaxy evolution at all redshifts., Comment: 16 Pages, 11 figures, ApJ, in press

The first generation of stars had very different properties than later stellar generations, as they formed from a "pristine" gas that was free of heavy elements. Normal star formation took place only after the first stars polluted the surrounding turbulent interstellar gas, increasing its local heavy element concentration, Z, beyond a critical value, Z_c (10^-8 < Z_c <10^-5). Motivated by this astrophysical problem, we investigate the fundamental physics of the pollution of pristine fluid elements in isotropic compressible turbulence. Turbulence stretches the pollutants, produces concentration structures at small scales, and brings the pollutants and the unpolluted flow in closer contact. Our theoretical approach employs the probability distribution function (PDF) method for turbulent mixing. We adopt three PDF closure models and derive evolution equations for the pristine fraction from the models. To test and constrain the theoretical models, we conduct numerical simulations for decaying passive scalars in isothermal turbulent flows with Mach numbers of 0.9 and 6.2, and compute the mass fraction, P(Z_c, t), of the flow with Z < Z_c. In the Mach 0.9 flow, the evolution of P(Z_c, t)$ is well described by a continuous convolution model and dP(Z_c, t)/dt = P(Z_c, t) ln[P(Z_c, t)]/tau_con, if the mass fraction of the polluted flow is larger than ~ 0.1. If the initial pollutant fraction is smaller than ~ 0.1, an early phase exists during which the pristine fraction follows an equation from a nonlinear integral model: dP(Z_c, t)/dt = P(Z_c, t) [P(Z_c, t)-1]/tau_int. The timescales tau_con and tau_int are measured from our simulations. When normalized to the flow dynamical time, the decay of P(Z_ c, t) in the Mach 6.2 flow is slower than at Mach 0.9, and we show that P(Z_c, t) in the Mach 6.2 flow can be well fit using a formula from a generalized version of the self-convolution model., Comment: 30 pages, 8 figure. Accepted by Journal of Fluid Mechanics

We conduct a systematic numerical study of passive scalar structures in supersonic turbulent flows. We find that the degree of intermittency in the scalar structures increases only slightly as the flow changes from transonic to highly supersonic, while the velocity structures become significantly more intermittent. This difference is due to the absence of shock-like discontinuities in the scalar field. The structure functions of the scalar field are well described by the intermittency model of She and L\'{e}v\^{e}que [Phys. Rev. Lett. 72, 336 (1994)], and the most intense scalar structures are found to be sheet-like at all Mach numbers., Comment: 4 pages, 3 figures, to appear in PRE

In many astrophysical environments, mixing of heavy elements occurs in the presence of a supersonic turbulent velocity field. Here we carry out the first systematic numerical study of such passive scalar mixing in isothermal supersonic turbulence. Our simulations show that the ratio of the scalar mixing timescale, $\tau_{\rm c}$, to the flow dynamical time, $\tau_{\rm dyn}$ (defined as the flow driving scale divided by the rms velocity), increases with the Mach number, $M$, for $M \lsim3$, and becomes essentially constant for $M \gsim3.$ This trend suggests that compressible modes are less efficient in enhancing mixing than solenoidal modes. However, since the majority of kinetic energy is contained in solenoidal modes at all Mach numbers, the overall change in $\tau_{\rm c}/\tau_{\rm dyn}$ is less than 20\% over the range $1 \lsim M \lsim 6$. At all Mach numbers, if pollutants are injected at around the flow driving scale, $\tau_{\rm c}$ is close to $\tau_{\rm dyn}.$ This suggests that scalar mixing is driven by a cascade process similar to that of the velocity field. The dependence of $\tau_{\rm c}$ on the length scale at which pollutants are injected into flow is also consistent with this cascade picture. Similar behavior is found for the variance decay timescales for scalars without continuing sources. Extension of the scalar cascade picture to the supersonic regime predicts a relation between the scaling exponents of the velocity and the scalar structure functions, with the scalar structure function becoming flatter as the velocity scaling steepens with Mach number. Our measurements of the volume-weighted velocity and scalar structure functions confirm this relation for $M\lsim 2,$ but show discrepancies at $M \gsim 3$., Comment: Accepted by ApJ