Density Probability Density Function Research Articles

Understanding density fluctuations in supersonic, isothermal turbulence.

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 [Formula: see text] 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 [Formula: see text] decay exponentially, as predicted by the model, and the decay timescale is ≈1/6 the eddy turnover time. The behaviors of the conditional averages of [Formula: see text] and [Formula: see text] are 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.

Ion alfvén velocity fluctuations and implications for the diffusion of streaming cosmic rays

The interstellar medium (ISM) of star-forming galaxies is magnetized and turbulent. Cosmic rays (CRs) propagate through it, and those with energies from ∼ GeV − TeV are likely subject to the streaming instability, whereby the wave damping processes balances excitation of resonant ionic Alfvén waves by the CRs, reaching an equilibrium in which the propagation speed of the CRs is very close to the local ion Alfvén velocity. The transport of streaming CRs is therefore sensitive to ionic Alfvén velocity fluctuations. In this paper we systematically study these fluctuations using a large ensemble of compressible MHD turbulence simulations. We show that for sub-Alfvénic turbulence, as applies for a strongly magnetized ISM, the ionic Alfvén velocity probability density function (PDF) is determined solely by the density fluctuations from shocked gas forming parallel to the magnetic field, and we develop analytical models for the ionic Alfvén velocity PDF up to second moments. For super-Alfvénic turbulence, magnetic and density fluctuations are correlated in complex ways, and these correlations as well as contributions from the magnetic fluctuations sets the ionic Alfvén velocity PDF. We discuss the implications of these findings for underlying “macroscopic” diffusion mechanisms in CRs undergoing the streaming instability, including modeling the macroscopic diffusion coefficient for the parallel transport in sub-Alfvénic plasmas. We also describe how, for highly-magnetized turbulent gas, the gas density PDF, and hence column density PDF, can be used to access information about ionic Alfvén velocity structure from observations of the magnetized ISM.

A synergy of the velocity gradients technique and the probability density functions for identifying gravitational collapse in self-absorbing media

ABSTRACT The velocity gradients technique (VGT) and the probability density functions (PDFs) of mass density are tools to study turbulence, magnetic fields, and self-gravity in molecular clouds. However, self-absorption can significantly make the observed intensity different from the column density structures. In this work, we study the effects of self-absorption on the VGT and the intensity PDFs utilizing three synthetic emission lines of CO isotopologues 12CO (1–0), 13CO (1–0), and C18O (1–0). We confirm that the performance of VGT is insensitive to the radiative transfer effect. We numerically show the possibility of constructing 3D magnetic fields tomography through VGT. We find that the intensity PDFs change their shape from the pure lognormal to a distribution that exhibits a power-law tail depending on the optical depth for supersonic turbulence. We conclude the change of CO isotopologues’ intensity PDFs can be independent of self-gravity, which makes the intensity PDFs less reliable in identifying gravitational collapsing regions. We compute the intensity PDFs for a star-forming region NGC 1333 and find the change of intensity PDFs in observation agrees with our numerical results. The synergy of VGT and the column density PDFs confirms that the self-gravitating gas occupies a large volume in NGC 1333.

Modeling Star Formation as a Markov Process in a Supersonic Gravoturbulent Medium

Abstract Molecular clouds exhibit log-normal probability density functions (PDF) of mass densities, which are thought to arise as a consequence of isothermal, supersonic turbulence. Star formation is then widely assumed to occur in perturbations in which gravitational collapse is faster than the rate of change due to turbulent motions. Here we use direct numerical simulations to measure this rate as a function of density for a range of turbulent Mach numbers, and show that it is faster at high densities than at low densities. Furthermore, we show that both the density PDF and rate of change arise naturally in a simple model of turbulence as a continuous Markov process. The one-dimensional Langevin equation that describes this evolution depends on only two parameters, yet it captures the full evolution seen in direct three-dimensional simulations. If it is modified to include gravity, the Langevin equation also reproduces the rate of material collapsing to high densities seen in turbulent simulations including self-gravity. When generalized to include both temperature and density, similar analyses are likely applicable throughout astrophysics.

Constraining cloud parameters using high density gas tracers in galaxies

Far-infrared molecular emission is an important tool used to understand the excitation mechanisms of the gas in the inter-stellar medium of star-forming galaxies. In the present work, we model the emission from rotational transitions with critical densities n >~ 10^4 cm-3. We include 4-3 < J <= 15-14 transitions of CO and 13CO, in addition to J <= 7-6 transitions of HCN, HNC, and HCO+ on galactic scales. We do this by re-sampling high density gas in a hydrodynamic model of a gas-rich disk galaxy, assuming that the density field of the interstellar medium of the model galaxy follows the probability density function (PDF) inferred from the resolved low density scales. We find that in a narrow gas density PDF, with a mean density of ~10 cm-3 and a dispersion \sigma = 2.1 in the log of the density, most of the emission of molecular lines, emanates from the 10-1000 cm-3 part of the PDF. We construct synthetic emission maps for the central 2 kpc of the galaxy and fit the line ratios of CO and 13CO up to J = 15-14, as well as HCN, HNC, and HCO+ up to J = 7-6, using one photo-dissociation region (PDR) model. We attribute the goodness of the one component fits for our model galaxy to the fact that the distribution of the luminosity, as a function of density, is peaked at gas densities between 10 and 1000 cm-3. We explore the impact of different log-normal density PDFs on the distribution of the line-luminosity as a function of density, and we show that it is necessary to have a broad dispersion, corresponding to Mach numbers >~ 30 in order to obtain significant emission from n > 10^4 cm-3 gas. Such Mach numbers are expected in star-forming galaxies, LIRGS, and ULIRGS. By fitting line ratios of HCN(1-0), HNC(1-0), and HCO+(1-0) for a sample of LIRGS and ULIRGS using mechanically heated PDRs, we constrain the Mach number of these galaxies to 29 < M < 77.

Characterization of ballasted flocs in water treatment using microscopy

Ballasted flocculation is widely used in the water industry for drinking water, municipal wastewater, storm water and industrial water treatment. This gravity-based physicochemical separation process involves the injection of a ballasting agent, typically microsand, to increase the floc density and size. However, the physical characteristics of the final ballasted flocs are still ill-defined. A microscopic method was specifically developed to characterize floc 1) density, 2) size and 3) shape factor. Using this information, probability density functions (PDFs) of the floc settling velocity were calculated. The impacts of the mixing intensity, polymer dosage, microsand size and contact time during the floc maturation phase were assessed. No correlation was identified between the floc diameter, form and density PDFs. The floc equivalent diameter mainly controls the settling velocity (r = 0.94), with the floc density (r = 0.26) and shape factor (r = 0.25) having lower impacts. A velocity gradient of 165 s−1 was optimal to maintain the microsand in suspension while simultaneously maximizing the floc diameter. An anionic high molecular weight polyacrylamide formed 1.5-fold larger aggregates compared with the starch-based polymer tested, but both polymers produced flocs of similar density (relative density = 1.53 ± 0.03). Generally, the floc mean settling velocity is a good predictor of the turbidity removal. An in-depth analysis of the floc characteristics indicates a correlation between the floc size and the largest microsand grain potentially embeddable in the floc structure.

OBSERVATIONAL DIAGNOSTICS OF SELF-GRAVITATING MHD TURBULENCE IN GIANT MOLECULAR CLOUDS

We study the observable signatures of self-gravitating MHD turbulence by applying the probability density functions (PDFs) and the spatial density power spectrum to synthetic column density maps. We find that there exists three characterizable stages of the evolution of the collapsing cloud which we term "early," "intermediate," and "advanced." At early times, i.e. $t<0.15t_{ff}$, the column density has a power spectral slope similar to nongravitating supersonic turbulence and a lognormal distribution. At an intermediate stage, i.e. $0.15t_{ff}< t \leq 0.35t_{ff}$, there exists signatures of the prestellar cores in the shallower PDF and power spectrum power law slopes. The column density PDF power law tails at these times have line of sight averaged slopes ranging from -2.5 to -1.5 with shallower values belonging to simulations with lower magnetic field strength. The density power spectrum slope becomes shallow and can be characterized by $P(k)=A_1k^{\beta_2}e^{-k/k_c}$, where $A_1$ describes the amplitude, $k^{\beta_2}$ describes the classical power law behavior and the scale $k_c$ characterizes the turn over from turbulence dominated to self-gravity dominated. At advanced stages of collapse, i.e. $\approx t>0.35t_{ff}$, the power spectral slope is positive valued, and a dramatic increase is observed in the PDF moments and the Tsallis incremental PDF parameters, which gives rise to deviations between PDF-sonic Mach number relations. Finally, we show that the imprint of gravity on the density power spectrum can be replicated in non-gravitating turbulence by introducing a delta-function with amplitude equivalent to the maximum valued point in a given self-gravitating map. We find that the turbulence power spectrum restored through spatial filtering of the high density material.

Robust functional statistics applied to Probability Density Function shape screening of sEMG data.

Recent studies pointed out possible shape modifications of the Probability Density Function (PDF) of surface electromyographical (sEMG) data according to several contexts like fatigue and muscle force increase. Following this idea, criteria have been proposed to monitor these shape modifications mainly using High Order Statistics (HOS) parameters like skewness and kurtosis. In experimental conditions, these parameters are confronted with small sample size in the estimation process. This small sample size induces errors in the estimated HOS parameters restraining real-time and precise sEMG PDF shape monitoring. Recently, a functional formalism, the Core Shape Model (CSM), has been used to analyse shape modifications of PDF curves. In this work, taking inspiration from CSM method, robust functional statistics are proposed to emulate both skewness and kurtosis behaviors. These functional statistics combine both kernel density estimation and PDF shape distances to evaluate shape modifications even in presence of small sample size. Then, the proposed statistics are tested, using Monte Carlo simulations, on both normal and Log-normal PDFs that mimic observed sEMG PDF shape behavior during muscle contraction. According to the obtained results, the functional statistics seem to be more robust than HOS parameters to small sample size effect and more accurate in sEMG PDF shape screening applications.

ON THE EVOLUTION OF THE DENSITY PROBABILITY DENSITY FUNCTION IN STRONGLY SELF-GRAVITATING SYSTEMS

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 $\mathrm{P}_v(\rho)\propto\rho^{-1.54}$ for the (volume-weighted) PDF and $\mathrm{P}_m(\rho)\propto\rho^{-0.54}$ 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.

Probing the evolution of molecular cloud structure

Aims: We derive the probability density functions (PDFs) of column density for a complete sample of prominent molecular cloud complexes closer than 200 pc. Methods: We derive near-infrared dust extinction maps for 23 molecular cloud complexes, using the "nicest" colour excess mapping technique and data from the 2MASS archive. The extinction maps are then used to examine the column density PDFs in the clouds. Results: The column density PDFs of most molecular clouds are well-fitted by log-normal functions at low column densities (0.5 mag < A_v < 3-5 mag). However, at higher column densities prominent, power-law-like wings are common. In particular, we identify a trend among the PDFs: active star-forming clouds always have prominent non-log-normal wings. In contrast, clouds without active star formation resemble log-normals over the whole observed column density range, or show only low excess of higher column densities. This trend is also reflected in the cumulative PDFs, showing that the fraction of high column density material is significantly larger in star-forming clouds. These observations are in agreement with an evolutionary trend where turbulent motions are the main cloud-shaping mechanism for quiescent clouds, but the density enhancements induced by them quickly become dominated by gravity (and other mechanisms) which is strongly reflected by the shape of the column density PDFs. The dominant role of the turbulence is restricted to the very early stages of molecular cloud evolution, comparable to the onset of active star formation in the clouds.

The nature of the velocity field in molecular clouds - I. The non-magnetic case

We present numerical simulations designed to test some of the hypotheses and predictions of recent models of star formation. We consider a set of three numerical simulations of randomly driven, isothermal, non-magnetic, self-gravitating turbulence with different rms Mach numbers Ms and physical sizes L, but with approximately the same value of the virial parameter, α ≈ 1.2. We obtain the following results: (i) we test the hypothesis that the collapsing centres originate from locally Jeans unstable (‘super-Jeans’), subsonic fragments; we find no such structures in our simulations, suggesting that collapsing centres can arise also from regions that have supersonic velocity dispersions but are nevertheless gravitationally unstable. (ii) We find that the fraction of small-scale super-Jeans structures is larger in the presence of self-gravity. (iii) There exists a trend towards more negative values of the velocity field’s mean divergence in regions with higher densities, implying the presence of organized inflow motions within the structures analysed. (iv) The density probability density function (PDF) deviates from a lognormal in the presence of self-gravity, developing an approximate power-law high-density tail, in agreement with previous results. (v) Turbulence alone in the large-scale simulation (L = 9 pc) does not produce regions with the same size and mean density as those of the small-scale simulation (L = 1 pc). Items (ii)–(v) suggest that self-gravity is not only involved in causing the collapse of Jeans-unstable density fluctuations produced by the turbulence, but also in their formation. We then measure the ‘star formation rate per free-fall time’, SFRff , as a function of Ms for the three runs, and compare with the predictions of recent semi-analytical models. We find marginal agreement to within the uncertainties of the measurements. However, within the L = 9 pc simulation, subregions with similar density and size to those of the L = 1p c simulation differ qualitatively from the latter in that they exhibit a global convergence of the velocity field ∇·v ∼− 0.6 km s −1 pc −1 on average. This suggests that the assumption that turbulence in clouds and clumps is purely random is incomplete. We conclude that (i) part of the observed velocity dispersion in clumps must arise from clump-scale inwards motions, even in driven-turbulence situations, and (ii) analytical models of clump and star formation need to take into account this dynamical connection with the external flow and the fact that, in the presence of self-gravity, the density PDF may deviate from a lognormal.

Variable-density mixing in buoyancy-driven turbulence

The homogenization of a heterogeneous mixture of two pure fluids with different densities by molecular diffusion and stirring induced by buoyancy-generated motions, as occurs in the Rayleigh–Taylor (RT) instability, is studied using direct numerical simulations. The Schmidt number, Sc, is varied by a factor of 20, 0.1 ≤ Sc ≤ 2.0, and the Atwood number, A, by a factor of 10, 0.05 ≤ A ≤ 0.5. Initial-density intensities are as high as 50% of the mean density. As a consequence of differential accelerations experienced by the two fluids, substantial and important differences between the mixing in a variable-density flow, as compared to the Boussinesq approximation, are observed. In short, the pure heavy fluid mixes more slowly than the pure light fluid: an initially symmetric double delta density probability density function (PDF) is rapidly skewed and, only at long times and low density fluctuations, does it relax to a Gaussian-like PDF. The heavy–light fluid mixing process asymmetry is relevant to the nature of molecular mixing on different sides of a high-Atwood-number RT layer. Diverse mix metrics are used to examine the homogenization of the two fluids. The conventional mix parameter, θ, is mathematically related to the variance of the excess reactant of a hypothetical fast chemical reaction. Bounds relating θ and the normalized product, Ξ, are derived. It is shown that θ underpredicts the mixing, as compared to Ξ, in the central regions of an RT layer; in the edge regions, θ is larger than Ξ. The shape of the density PDF cannot be inferred from the usual mix metrics popular in applications. For example, when θ, Ξ ≥ 0.6, characteristic of the interior of a fully developed RT layer, the PDFs can have vastly different shapes. Bounds on the fluid composition using two low-order moments of the density PDF are derived. The bounds can be used as realizability conditions for low-dimensional models. For the measures studied, the tightest bounds are obtained using Ξ and mean density. The structure of the flow is also examined. It is found that, at early times, the buoyancy production term in the spectral kinetic energy equation is important at all wavenumbers and leads to anisotropy at all scales of motion. At later times, the anisotropy is confined to the largest and smallest scales: the intermediate scales are more isotropic than the small scales. In the viscous range, there is a cancellation between the viscous and nonlinear effects, and the buoyancy production leads to a persistent small-scale anisotropy.

Is Thermal Instability Significant in Turbulent Galactic Gas?

We investigate numerically the role of thermal instability (TI) as a generator of density structures in the interstellar medium (ISM), both by itself and in the context of a globally turbulent medium. We consider three sets of numerical simulations: (1) —ows in the presence of the instability only; (2) —ows in the presence of the instability and various types of turbulent energy injection (forcing), and (3) models of the ISM including the magnetic —eld, the Coriolis force, self-gravity and stellar energy injection. Simula- tions in the —rst group show that the condensation process that forms a dense phase (ii clouds ˇˇ) is highly dynamical and that the boundaries of the clouds are accretion shocks, rather than static density discon- tinuities. The density histograms (probability density functions (PDFs)) of these runs exhibit either bimodal shapes or a single peak at low densities plus a slope change at high densities. Final static situ- ations may be established, but the equilibrium is very fragile: small density —uctuations in the warm phase require large variations in that of the cold phase, probably inducing shocks in the clouds. Com- bined with the likely disruption of the clouds by Kelvin-Helmholtz instability, this result suggests that such con—gurations are highly unlikely. Simulations in the second group show that large-scale turbulent forcing is incapable of erasing the signature of TI in the density PDFs, but small-scale, stellar-like forcing causes the PDFs to transit from bimodal to a single-slope power law, erasing the signature of the instability. However, these simulations do not reach stationary regimes, with TI driving an ever- increasing star formation rate. Simulations in the third group show no signi—cant diUerence between the PDFs of stable and unstable cases and reach stationary regimes, suggesting that the combination of the stellar forcing and the extra eUective pressure provided by the magnetic —eld and the Coriolis force over- whelm TI as a density-structure generator in the ISM, with TI becoming a second-order eUect. We emphasize that a multimodal temperature PDF is not necessarily an indication of a multiphase medium, which must contain clearly distinct thermal equilibrium phases, and that this ii multiphase ˇˇ terminology is often inappropriately used. Subject headings: instabilitiesISM: structureturbulence

Density probability distribution in one-dimensional polytropic gas dynamics

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 $\gamma$ 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 $\sigma_s^2$ which scales like the square turbulent Mach number $\tilde M^2$, where $s\equiv \ln \rho$ and $\rho$ is the fluid density. This leads to much higher fluctuations than those due to shock jump relations. Extrapolating the model to the case $\gamma \not =1$, 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 $\gamma<1$, and at low densities when $\gamma>1$. 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 $\sigma_s^2$ to grow more slowly than $\tilde M^2$ when $\gamma\not=1$. In view of these results, we suggest that Burgers flow is a singular case not approached by the high-$\tilde M$ limit, with a PDF that develops power laws on both sides.