Can We Trust the Dust? Evidence of Dust Segregation in Molecular Clouds

We present an extensive numerical study of the time irreversibility of the dynamics of heavy inertial particles in three-dimensional, statistically homogeneous, and isotropic turbulent flows. We show that the probability density function (PDF) of the increment, W(τ), of a particle's energy over a time scale τ is non-Gaussian, and skewed toward negative values. This implies that, on average, particles gain energy over a period of time that is longer than the duration over which they lose energy. We call this slow gain and fast loss. We find that the third moment of W(τ) scales as τ^{3} for small values of τ. We show that the PDF of power-input p is negatively skewed too; we use this skewness Ir as a measure of the time irreversibility and we demonstrate that it increases sharply with the Stokes number St for small St; this increase slows down at St≃1. Furthermore, we obtain the PDFs of t^{+} and t^{-}, the times over which p has, respectively, positive or negative signs, i.e., the particle gains or loses energy. We obtain from these PDFs a direct and natural quantification of the slow gain and fast loss of the energy of the particles, because these PDFs possess exponential tails from which we infer the characteristic loss and gain times t_{loss} and t_{gain}, respectively, and we obtain t_{loss}<t_{gain} for all the cases we have considered. Finally, we show that the fast loss of energy occurs with greater probability in the strain-dominated region than in the vortical one; in contrast, the slow gain in the energy of the particles is equally likely in vortical or strain-dominated regions of the flow.

view Abstract Citations (31) References (23) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The relationship of submillimeter optical depth to 13CO column density in molecular clouds. Righini-Cohen, G. ; Simon, M. Abstract The relationship between dust and molecular gas within a molecular cloud is studied by investigating the correlation of 350-micron and 1-mm optical depths with (C-13)O column density in ten very dense molecular sources that are visibly opaque and probably characterized by an extinction factor well in excess of 20. The cited correlation is examined at a number of points within Sgr B2 and at the peak molecular positions in the other nine sources. A strong correlation is found in each case, and the overall correlations obtained are shown to be real, internally consistent, and not markedly influenced by the assumption that the dust temperature is twice the gas temperature. Contributions to the observed scatter in the correlations are assessed. Publication: The Astrophysical Journal Pub Date: April 1977 DOI: 10.1086/155168 Bibcode: 1977ApJ...213..390R Keywords: Abundance; Infrared Radiation; Interstellar Matter; Microwave Emission; Molecular Gases; Submillimeter Waves; Carbon Monoxide; Cosmic Dust; Gas Density; Interstellar Radiation; Nebulae; Optical Thickness; Astrophysics full text sources ADS |

One issue associated with the use of Large-Eddy Simulation (LES) to investigate the dispersion of small inertial particles in turbulent flows is the accuracy with which particle statistics and concentration can be reproduced. The motion of particles in LES fields may differ significantly from that observed in experiments or direct numerical simulation (DNS) because the force acting on the particles is not accurately estimated, due to the availability of the only filtered fluid velocity, and because errors accumulate in time leading to a progressive divergence of the trajectories. This may lead to different degrees of inaccuracy in the prediction of statistics and concentration. We identify herein an ideal subgrid correction of the a-priori LES fluid velocity seen by the particles in turbulent channel flow. This correction is computed by imposing that the trajectories of individual particles moving in filtered DNS fields exactly coincide with the particle trajectories in a DNS. In this way the errors introduced by filtering into the particle motion equations can be singled out and analyzed separately from those due to the progressive divergence of the trajectories. The subgrid correction term, and therefore the filtering error, is characterized in the present paper in terms of statistical moments. The effects of the particle inertia and of the filter type and width on the properties of the correction term are investigated.

Understanding and predicting the collision rate of small inertial particles in turbulent flows is of importance in many meteorological and industrial processes. For instance, predicting the time of rain initiation of warm clouds, which are known to be turbulent in their core, is still an open problem. Droplets in such clouds, which can be treated as inertial particles, are believed to grow to rain drop by coalescence due to collisions.

Non-spherical particles suspended in fluid flows are subject to hydrodynamic torques generated by fluid velocity gradients. For small axisymmetric particles, the most popular formulation of hydrodynamic torques is that given by Jeffery (Proc R Soc Lond A 102:161–179, 1922), which is valid for uniform shear flow in the viscous Stokes regime. In the lack of simple alternative formulations outside the Stokes regime, the Jeffery formulation has been widely applied to inertial particles in turbulent flows, where it is bound to produce inaccurate results. In this paper we quantify the statistical error incurred when the Jeffery formulation is used to study the motion of elongated axisymmetric particles under nonlinear shear flow conditions. Considering the archetypical case of prolate ellipsoidal particles in turbulent channel flow, we show that error for ellipsoids of the same length, l, as the Kolmogorov scale of the flow, $$\eta _K$$ , is indeed small (order $$1\%$$ ) but increases exponentially up to $$l \simeq 10 \eta _K$$ before becoming almost independent of elongation.

Preferential concentration of inertial particles in turbulent flow is studied by high resolution direct numerical simulations of two-dimensional turbulence. The formation of network-like regions of high particle density, characterized by a length scale which depends on the Stokes number of inertial particles, is observed. At smaller scales, the size of empty regions appears to be distributed according to a universal scaling law.

Understanding the dispersion and the deposition of inertial particles in turbulent flows is a domain of research of utmost practical interest. With advances in computing resources, Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) have become powerful tools for the investigation of particle-laden turbulent flows with the hybrid Eulerian-Lagrangian approach playing a key role in predicting inertial particle dispersion and deposition. Computational intractability that arises due to the need of solving all the scales has restricted DNS to the very low Reynolds number turbulent flows that are not often of practical interest. LES, by solving only the large energy-containing eddies and modeling the small quasi-universal scales, is relaxed from this restriction. Thus, tackling high Reynolds number turbulent flow becomes possible. The use of large-eddy simulation has increased over the years as a promising tool to address these types of problems with the required accuracy at an affordable computing cost. In LES of dispersed turbulent multiphase flows, it has been common that tracking inertial particles in turbulent flows is carried out using only the filtered velocity field. This turned out to be inaccurate for cases dealing with very small, turbulence-responsive particles. For these cases, the timedependent velocity field seen by the inertial particles can be stochastically constructed in a Lagrangian framework. This can be achieved through the use of a stochastic diffusion process such as Langevin models.

We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of this backward dispersion in determining the particle relative velocity is emphasized. We find that our model with a two-phase separation behaviour, an early ballistic phase and a later tracer-like phase, as found by recent simulations for the forward (in time) dispersion of inertial particle pairs, gives good fits to the measured relative speeds from simulations at low Reynolds numbers. In the monodisperse case with identical particles, the generalized acceleration term vanishes and the relative velocity is determined by the generalized shear term. At large Reynolds numbers, our model gives a St1/2-dependence of the relative velocity on the Stokes number St in the inertial range for both the ballistic behaviour and the Richardson separation law. This leads to the same inertial-range scaling for the two-phase separation that well fits the simulation results. Our calculations for the bidisperse case show that, with the friction timescale of one particle fixed, the relative speed as a function of the other particle's friction time has a dip when the two timescales are similar. This indicates that similar-size particles tend to have stronger velocity correlation than different ones. We find that the primary contribution at the dip, i.e. for similar particles, is from the generalized shear term, while the generalized acceleration term is dominant for particles of very different sizes. Future numerical studies are motivated to check the accuracy of the assumptions made in our model and to investigate the backward-in-time dispersion of inertial particle pairs in turbulent flows.

Critical comments to results of investigations of drop collisions in turbulent clouds

We have used a numerical simulation of a turbulent cloud to synthesize maps of the thermal emission from dust at a variety of far-IR and submillimeter wavelengths. The average column density and external radiation field in the simulation is well matched to clouds such as Perseus and Ophiuchus. We use pairs of single-wavelength emission maps to derive the dust color temperature and column density, and we compare the derived column densities with the true column density. We demonstrate that longer wavelength emission maps yield less biased estimates of column density than maps made toward the peak of the dust emission spectrum. We compare the scatter in the derived column density with the observed scatter in Perseus and Ophiuchus. We find that while in Perseus all of the observed scatter in the emission-derived versus the extinction-derived column density can be attributed to the flawed assumption of isothermal dust along each line of sight, in Ophiuchus there is additional scatter above what can be explained by the isothermal assumption. Our results imply that variations in dust emission properties within a molecular cloud are not necessarily a major source of uncertainty in column density measurements.

The formation of stars occurs in the dense molecular cloud phase of the interstellar medium. Observations and numerical simulations of molecular clouds have shown that supersonic magnetised turbulence plays a key role for the formation of stars. Simulations have also shown that a large fraction of the turbulent energy dissipates in shock waves. The three families of MHD shocks --- fast, intermediate and slow --- distinctly compress and heat up the molecular gas, and so provide an important probe of the physical conditions within a turbulent cloud. Here we introduce the publicly available algorithm, SHOCKFIND, to extract and characterise the mixture of shock families in MHD turbulence. The algorithm is applied to a 3-dimensional simulation of a magnetised turbulent molecular cloud, and we find that both fast and slow MHD shocks are present in the simulation. We give the first prediction of the mixture of turbulence-driven MHD shock families in this molecular cloud, and present their distinct distributions of sonic and Alfvenic Mach numbers. Using subgrid one-dimensional models of MHD shocks we estimate that ~0.03 % of the volume of a typical molecular cloud in the Milky Way will be shock heated above 50 K, at any time during the lifetime of the cloud. We discuss the impact of this shock heating on the dynamical evolution of molecular clouds.

We discuss the probability distribution function (PDF) of column density resulting from density fields with lognormal PDFs, applicable to isothermal gas (e.g., probably molecular clouds). For magnetic and nonmagnetic numerical simulations of compressible, isothermal turbulence forced at intermediate scales ( of the box size), we find that the autocorrelation function (ACF) of the density field decays over relatively short distances compared to the simulation size. We suggest that a "decorrelation length" can be defined as the distance over which the density ACF has decayed to, for example, 10% of its zero-lag value, so that the density "events" along a line of sight can be assumed to be independent over distances larger than this, and the central limit theorem should be applicable. However, using random realizations of lognormal fields, we show that the convergence to a Gaussian is extremely slow in the high-density tail. As a consequence, the column density PDF is not expected to exhibit a unique functional shape, but to transit instead from a lognormal to a Gaussian form as the ratio η of the column length to the decorrelation length (i.e., the number of independent events in the cloud) increases. Simultaneously, the variance of the PDF decreases. For intermediate values of η, the column density PDF assumes a nearly exponential decay. For cases with a density contrast of 104, as found in intermediate-resolution simulations, and expected from giant molecular clouds (GMCs) to dense molecular cores, the required value of η for convergence to a Gaussian is at least a few hundred, or, for 106, several thousand. We then discuss the density power spectrum and the expected value of η in actual molecular clouds, concluding that they are uncertain since they may depend on several physical parameters. Observationally, our results suggest that η may be inferred from the shape and width of the column density PDF in optically thin line or extinction studies. Our results should also hold for gas with finite-extent power-law underlying density PDFs, which should be characteristic of the diffuse, nonisothermal neutral medium (with temperatures ranging from a few hundred to a few thousand degrees). Finally, we note that for η ≳ 100, the dynamic range in column density is small (less than a factor of 10), but this is only an averaging effect, with no implication on the dynamic range of the underlying density distribution.

Context. Dust grain dynamics in molecular clouds is regulated by its interplay with supersonic turbulent gas motions. The conditions under which interstellar dust grains decouple from the dynamics of gas in molecular clouds remain poorly constrained. Aims. We first aim to investigate the critical dust grain size for dynamical decoupling, using both analytical predictions and numerical experiments. Second, we aim to set the range of validity of two fundamentally different numerical implementations for the evolution of dust and gas mixtures in turbulent molecular clouds. Methods. We carried out a suite of numerical experiments using two different schemes to integrate the dust grain equation of motion within the same framework. First, we used a monofluid formalism (or often referred to as single fluid) in the terminal velocity approximation. This scheme follows the evolution of the barycentre of mass between the gas and the dust on a Eulerian grid. Second, we used a two-fluid scheme, in which the dust dynamics is handled with Lagrangian super-particles, and the gas dynamics on a Eulerian grid. Results. The monofluid results are in good agreement with the theoretical critical size for decoupling. We report dust dynamics decoupling for Stokes number St &gt; 0.1, that is, dust grains of s &gt; 4 μm in size. We find that the terminal velocity approximation is well suited for grain sizes of 10 μm in molecular clouds, in particular in the densest regions. However, the maximum dust enrichment measured in the low-density material - where St &gt; 1 - is questionable. In the Lagrangian dust experiments, we show that the results are affected by the numerics for all dust grain sizes. At St ≪ 1, the dust dynamics is largely affected by artificial trapping in the high-density regions, leading to spurious variations of the dust concentration. At St &gt; 1 , the maximum dust enrichment is regulated by the grid resolution used for the gas dynamics. Conclusions. Dust enrichment of submicron dust grains is unlikely to occur in the densest parts of molecular clouds. Two fluid implementations using a mixture of Eulerian and Lagrangian descriptions for the dust and gas mixture dynamics lead to spurious dust concentration variations in the strongly and weakly coupled regimes. Conversely, the monofluid implementation using the terminalvelocity approximation does not accurately capture dust dynamics in the low-density regions, that is, where St &gt; 1 . The results of previous similar numerical work should therefore be revisited with respect to the limitations we highlight in this study.

We use the COMPLETE Survey's observations of the Perseus star-forming region to assess and intercompare three methods for measuring column density in molecular clouds: extinction mapping (NIR); thermal emission mapping (FIR); and mapping the intensity of CO isotopologues. The structures shown by all three tracers are morphologically similar, but important differences exist. Dust-based measures give similar, log-normal, distributions for the full Perseus region, once careful calibration corrections are made. We also compare dust- and gas-based column density distributions for physically-meaningful sub-regions of Perseus, and we find significant variations in the distributions for those regions. Even though we have used 12CO data to estimate excitation temperatures, and we have corrected for opacity, the 13CO maps seem unable to give column distributions that consistently resemble those from dust measures. We have edited out the effects of the shell around the B-star HD 278942. In that shell's interior and in the parts where it overlaps the molecular cloud, there appears to be a dearth of 13CO, likely due either to 13CO not yet having had time to form in this young structure, and/or destruction of 13CO in the molecular cloud. We conclude that the use of either dust or gas measures of column density without extreme attention to calibration and artifacts is more perilous than even experts might normally admit. And, the use of 13CO to trace total column density in detail, even after proper calibration, is unavoidably limited in utility due to threshold, depletion, and opacity effects. If one's main aim is to map column density, then dust extinction seems the best probe. Linear fits amongst column density tracers are given, quantifying the inherent uncertainties in using one tracer (when compared with others). [abridged]

Analysing the Galactic plane CO survey with the Nobeyama 45-m telescope, we compared the spectral column density (SCD) of $N_{\rm H_2}$ calculated for the 12CO (J = 1–0) line using the current conversion factor $X_{\rm ^{12}CO}$ to that for the 13CO (J = 1–0) line under the LTE (local thermal equilibrium) assumption in the M16 and W43 regions. Here, SCD is defined by $\mathrm{d}N_{\rm H_2}/\mathrm{d}v$ with $N_{\rm H_2}$ and v being the column density and radial velocity, respectively. It is found that the $X_{\rm ^{12}CO}$ method significantly underestimates the H2 density in a cloud or region, where SCD exceeds a critical value (∼3 × 1021 [H2 cm−2 (km s−1)−1]), but overestimates in lower SCD regions. We point out that the actual CO-to-H2 conversion factor varies with the H2 column density or with the CO line intensity: it increases in the inner and opaque parts of molecular clouds, whereas it decreases in the low-density envelopes. However, in so far as the current $X_{^{12}{\rm CO}}$ is used combined with the integrated 12CO intensity averaged over an entire cloud, it yields a consistent value with that calculated using the 13CO intensity by LTE. Based on the analysis, we propose a new CO-to-H2 conversion relation, $N_{\rm H_2}^* = \int X^*_{\rm CO} (T_{\rm B}) T_{\rm B}\ \mathrm{d}v$, where $X^*_{\rm CO} (T_{\rm B})=(T_{\rm B}/T_{\rm B}^*)^\beta X_{\rm ^{12}CO}$ is the modified spectral conversion factor as a function of the brightness temperature, TB, of the 12CO (J = 1–0) line, and β ∼ 1–2 and $T_{\rm B}^*=12\!-\!16$ K are empirical constants obtained by fitting to the observed data. The formula corrects for the over/underestimation of the column density at low/high CO line intensities, and is applicable to molecular clouds with TB ≥ 1 K (12CO (J = 1–0) line rms noise in the data) from envelope to cores at sub-parsec scales (spatial resolution).