Small-scale clustering of nano-dust grains in supersonic turbulence
MMNRAS , 1–11 (2019) Preprint 17 October 2019 Compiled using MNRAS L A TEX style file v3.0
Small-scale clustering of nano-dust grains in supersonic turbulence
L. Mattsson (cid:63) , J. P. U. Fynbo , , B. Villarroel , Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden The Cosmic Dawn Center (DAWN) Niels Bohr Institute, Lyngbyvej 2, DK-2100 Copenhagen, Denmark Instituto de Astrof´ısica de Canarias (IAC), E-38200 La Laguna, Tenerife, Spain
17 October 2019
ABSTRACT
We investigate the clustering and dynamics of nano-sized particles (nano-dust) in high-resolution (1024 ) simulations of compressible isothermal hydrodynamic turbulence. It iswell-established that large grains will decouple from a turbulent gas flow, while small grainswill tend to trace the motion of the gas. We demonstrate that nano-sized grains may clusterin a turbulent flow (fractal small-scale clustering), which increases the local grain density byat least a factor of a few. In combination with the fact that nano-dust grains may be abundantin general, and the increased interaction rate due to turbulent motions, aggregation involvingnano dust may have a rather high probability. Small-scale clustering will also a ff ect extinctionproperties. As an example we present an extinction model based on silicates, graphite andmetallic iron, assuming strong clustering of grain sizes in the nanometre range, could explainthe extreme and rapidly varying ultraviolet extinction in the host of GRB 140506A. Key words:
ISM: dust, extinction – ISM: clouds – turbulence – hydrodynamics
Nano-sized dust grains, like graphitic particles, polycyclic aromatichydrocarbons (PAHs) or even nano diamonds, are by number themost abundant type of dust in the interstellar medium (ISM). Theirproperties are di ff erent from those of larger grains and bulk ma-terial; they have large proportion of surface atoms and their sizesare actually smaller than (or at least similar to) basic scales suchas the Landau radius, Debye length, De Brooglie wavelength etc.and may therefore show unique optical and kinematic properties(Li & Mann 2012). These grains make up only a tiny fraction ofthe total interstellar dust mass, but are believed to be a significantsource of extinction, in particular if the grains are uniformly andisotropically distributed in the ISM. But if they are concentratedinto small clumps with essentially dust free regions in between, thenet extinction is reduced. Whether nano dust is clustered or not istherefore important. Moreover, nano-sized sp -bonded carbon dusthas for decades been thought to to be the main carrier of the 2175Å extinction feature (see, e.g., Pei 1992; Draine 2003; Jones et al.2013), which indicates that small-scale clustering of such grainsmay be an important phenomenon to consider in order to explainvariations of the ultraviolet (UV) extinction curve in general.Small particles, like nano-dust grains, may form clusteredstructures on scales smaller than the typical length scale of a tur-bulent flow. Such small-scale clustering , a.k.a. preferential con-centration, of particles in turbulent flows is a classical problem in (cid:63) E-mail: [email protected] When we refer to “small scale” in the present paper, we mean scales much fluid mechanics and statistical physics. For incompressible flows,numerous studies have shown that centrifuging of particles awayfrom vortex cores leads to accumulation of particles in conver-gence zones (see, e.g., Maxey 1987; Squires & Eaton 1991; Eaton& Fessler 1994; Bec 2005; Yavuz et al. 2018). Thus, vorticity andinertia of the particles are decisive for the amount of clustering (seeToschi & Bodenschatz 2009, and references therein). The resultantfractal clustering has been studied and simulated quite extensively(see, e.g., Sundaram & Collins 1997; Hogan et al. 1999; Hogan &Cuzzi 2001; Bec 2003; Bec et al. 2007a,b, 2010; Bhatnagar et al.2018a), while it is not well-known to what extent it occurs in com-pressible supersonic turbulence. The latter is of course the regimewhich is interesting from an astrophysical perspective. With den-sity scales typical for interstellar environments, recent results (Hop-kins & Lee 2016; Mattsson et al. 2019) point at small-scale cluster-ing reaching its maximum for grains in the nanometre range (radii a = . . .
100 nm).In astrophysics, studies of clustering of particles in proto-planetary discs are common, but these studies often do not go sig-nificantly below the Kolmogorov scale and may thus not reachthe clustering regime of interest in the ISM (see, e.g., Pan et al.2011; Pan & Padoan 2013, 2014). Downes (2012); Hopkins & Lee(2016); Mattsson et al. (2019) have presented simulations of the tur-bulent dynamics of an interstellar MC including a dust phase fromwhich a consistent picture emerge; small grains (radii (cid:46) . µ m)tend to cluster and follow the gas, while larger grains ( (cid:38) µ m) tend smaller that the characteristic length scale of the flow, which are not smallcompared to, e.g., the particles.c (cid:13) a r X i v : . [ a s t r o - ph . GA ] O c t Mattsson et al. to decouple from the gas flow and not cluster notably. For particlestracing the gas, the density variance due to turbulence will lead toan increased rate of grain growth by accretion of molecules in MCs(often referred to as “dust condensation”) because of the non-lineardependence on molecular-gas density (Mattsson 2019). But if theclustering is strong, the dust will be quite far from uniformly dis-tributed within observable gas structures. Thus, grain-growth pro-cesses which is based on accretion of molecules (e.g., condensationof ices or chemical reactions on grain surfaces) are highly depen-dent on the local environment. Similarly, the collision rate of dustparticles show significant variance due to spatial clustering, leadingto an enhanced growth by coagulation (Zsom & Dullemond 2008;Ormel et al. 2009).To put the above into context, it is important to emphasisethat grain growth is an important dust-formation channel, not theleast as a necessary replenishment mechanism to counteract dustdestruction in the ISM (McKee 1989; Draine 1990). Abundancepatterns in interstellar gas are consistent with dust depletion due tocondensation in MCs (see, e.g., Jenkins 2009; De Cia et al. 2016),and the fact that late-type galaxies seem to have steeper dust-to-gasgradients than metallicity gradients along their discs lend furthersupport to this picture (Mattsson et al. 2012; Mattsson & Andersen2012; Mattsson et al. 2014; V´ılchez et al. 2019).Nano-sized grains are typically not in thermal and radiativeequilibrium with their surroundings, which creates a wide graintemperature distribution (Purcell 1976; Dwek 1986; Draine 2003).Thus, they may a ff ect the infrared flux-to-mass ratio in the sameway as a range of sizes would for large grains in equilibrium(Mattsson et al. 2015). Compared to a homogeneous distribution,small-scale clustering may cause nano grains to emit more long-wavelength radiation relative to the overall extinction they cause.This, in turn, leads to underestimation of nano-dust abundance fromobservations.The present paper aims to explore small-scale clustering ofnano-sized dust grains by direct numerical simulations of hydrody-namic turbulence with Lagrangian inertial particles and discuss afew of its consequences. We model turbulent gas aimang at describing the interior of amolecular cloud (MC). We set up a high-resolution (1024 ) three-dimensional (3D) periodic-boundary box with sides of equal length L = π and solve the standard hydrodynamic equations as de-scribed in, e.g., Mattsson et al. (2019): the momentum equation,the continuity equation and an isothermal condition added as a clo-sure relation, i.e., a constant sound speed c s = = unit velocity).Dust particles are included as inertial particles in 15 size bins with10 − particles in each. We use the P encil C ode , which is anon-conservative, high-order, finite-di ff erence code (sixth order inspace and third order in time). For a more detailed description ofthe code, see Brandenburg & Dobler (2002). For a more detaileddescription of the type of turbulence simulations we employ in thisstudy we refer to Mattsson et al. (2019). To maintain supersonic steady-state turbulence we need some kindof external forcing. We therefore include a white-in-time stochas- tic forcing term with both solenoidal (rotational) and compressiblecomponents. The forcing is applied at low wave-numbers in Fourierspace, where the forcing is a stochastic process integrated usingthe Euler-Maruyama method. In the present study we consider twotypes of forcing; either purely compressive or purely solenoidalforcing (see Table 1) in order to explore if there will be any qualita-tive di ff erences in the dust dynamics. It is known that the resultantgas-density PDFs and the fractal properties of the gas are signifi-cantly di ff erent depending on whether the forcing is compressiveor solenoidal (Federrath et al. 2009, 2010). Inertial particles suspended in a gaseous medium will show delayedresponse to kinetic drag from gas particles. Assuming the interstel-lar dust is accelerated by a turbulent gas flow via an Epstein (1924)drag law, the equation of motion for dust particles isd v d t = u − v τ s , (1)where v and u is the velocities of dust and gas, respectively, and τ s is the stopping time, i.e., the timescale of acceleration (or decel-eration) of the grains. τ s in the Epstein limit depends on the grainradius a and bulk material density ρ gr well as the gas density ρ andthe relative Mach number W s = | u − v | / c s (Schaaf 1963). In thelimit W s (cid:28)
1, we obtain τ s ( W s (cid:28) = (cid:114) π ρ gr ρ ac s ≡ τ s , , (2)where c s has replaced the thermal mean speed of molecules. The W s (cid:28) M s (cid:28)
1. For large M s , we expect W s (cid:29) τ s ( W s (cid:29) = ρ gr ρ a | u − v | . (3)Combining these two limits, we then obtain a convenient formulawhich is su ffi ciently accurate for our purposes (Kwok 1975; Draine& Salpeter 1979), τ s = τ s , (cid:32) + π | u − v | c (cid:33) − / . (4)The second term inside the parentesis can be seen as a correctionfor supersonic flow velocities and compression.Non-inertial particles, a.k.a. tracer particles, will have v = u and be position coupled with the medium in which they reside. Itis often assumed that the approximation v ≈ u is justified when thestopping time is much shorter than the characteristic timescale ofthe flow, which is the case for very small interstellar dust particles.The smallest nano dust (typically a ∼ M s , even a very small amount of inertia can beimportant. A similar approximation, but perhaps better than v ≈ u ,could be assuming that W s (cid:28)
1. Any di ff erence between cluster-ing of tiny particles in incompressible (or nearly incompressible)turbulence and supersonic compressible turbulence is then due tocompressibility and the variance of ρ . If this hypothesis is correct,the clustering of nano dust should be essentially una ff ected by thecorrection for large W s described above.We will use two prescriptions for kinetic Epstein-type dragand in total consider five cases: • Case I: τ s prescription for low W (Eq. 2) and turbulence in-duced by purely compressive forcing. MNRAS , 1–11 (2019) mall-scale clustering of nano-dust grains Table 1.
Basic properties and time-averaged physical parameters of the simulations. All simulation have the mean gas density and isothermal sound speed setto unity, i.e., (cid:104) ρ (cid:105) = c s = f (cid:104) log( ρ min ) (cid:105) (cid:104) log( ρ max ) (cid:105) M rms M max Re / L Drag law Forcing typeCase I 4 . − . ± .
66 1 . ± .
09 3 . ± .
11 9 . ± .
68 206 ± W (cid:28) . − . ± .
03 1 . ± .
08 3 . ± .
11 10 . ± .
05 210 ± W compressiveCase IIb 4 . − . ± .
43 1 . ± .
05 3 . ± .
07 9 . ± .
51 228 ± W solenoidalCase IIIa 4 . − . ± .
60 1 . ± .
07 3 . ± .
15 9 . ± .
86 203 ± . − . ± .
53 1 . ± .
06 3 . ± .
10 9 . ± .
54 228 ± Figure 1.
Projected number density of dust grains. From left to right: smallest (inertial) nano grains in the simulation ( α = . a ≈ α = . a ≈
25 nm) and largest nano grains ( α = . a ≈
100 nm). Upper panels show the run with compressive forcing, while the lowerpanels show the run with solenoidal forcing. • Case IIa: τ s prescription including correction for large W (Eq. 4) and purely compressive forcing. • Case IIb:
Same as Case IIa, but solenoidal instead of com-pressive forcing. • Case IIIa: non-inertial (“tracer”) particles and turbulence in-duced by purely compressive forcing. • Case IIIb:
Same as Case IIIa, but solenoidal instead of com-pressive forcing.The two most realistic cases are IIa and IIb and these two will getsome extra attention in the analysis below.
Nano-sized grains are easily a ff ected by other types of externalforces such as magnetic fields and radiation pressure. If electricallycharged, magnetic fields and ionised gas (protons) will exert forceson the grains, which do not necessarily act in the same directionas the kinetic drag from neutral gas. If exposed to a radiation field, the grains may also experience a systematic directional accelerationdue to momentum transfer from photons to dust grains (“radiationpressure”). Thus, the dynamics and clustering of nano-dust grainscan be very complex.However, as opposed to nano dust in, e.g., the solar wind,where radiation, plasma drag and magnetic fields play crucial roles(see, e.g., Czechowski & Mann 2018), cold MCs in the ISM repre-sent an environment which allows for a significantly simplified de-scription. Before the onset of star formation in an MC, there is littleradiation that can reach the inner parts, where most of the dust islocated. Without a significant radiation field, the gas will maintaina very low degree of ionisation, radiation pressure will be minimal,grains will not carry much electrical charge and thus be mostly un-a ff ected by magnetic fields or Lorentz forces. In the present study,we simulate the conditions in a starless MC, where it is reasonableto assume that kinetic gas drag is the dominant force acting uponthe grains. As soon as star formation sets in the conditions will bevery di ff erent, but we leave that for a future study. MNRAS000
Nano-sized grains are easily a ff ected by other types of externalforces such as magnetic fields and radiation pressure. If electricallycharged, magnetic fields and ionised gas (protons) will exert forceson the grains, which do not necessarily act in the same directionas the kinetic drag from neutral gas. If exposed to a radiation field, the grains may also experience a systematic directional accelerationdue to momentum transfer from photons to dust grains (“radiationpressure”). Thus, the dynamics and clustering of nano-dust grainscan be very complex.However, as opposed to nano dust in, e.g., the solar wind,where radiation, plasma drag and magnetic fields play crucial roles(see, e.g., Czechowski & Mann 2018), cold MCs in the ISM repre-sent an environment which allows for a significantly simplified de-scription. Before the onset of star formation in an MC, there is littleradiation that can reach the inner parts, where most of the dust islocated. Without a significant radiation field, the gas will maintaina very low degree of ionisation, radiation pressure will be minimal,grains will not carry much electrical charge and thus be mostly un-a ff ected by magnetic fields or Lorentz forces. In the present study,we simulate the conditions in a starless MC, where it is reasonableto assume that kinetic gas drag is the dominant force acting uponthe grains. As soon as star formation sets in the conditions will bevery di ff erent, but we leave that for a future study. MNRAS000 , 1–11 (2019)
Mattsson et al.
Figure 2.
Left: distribution of dust particles in a slice of thickness ∆ L / L = /
64 taken through the middle of the simulation box. Right: 3D snapshot fromthe simulation with compressive forcing , showing the dust density distribution of maximally clustered grains ( α = . a ≈
25 nm). Colour coding indicatespecific kinetic energy.
Figure 3.
Left: distribution of dust particles in a slice of thickness ∆ L / L = /
64 taken through the middle of the simulation box. Right: 3D snapshot fromthe simulation with solenoidal forcing , showing the dust density distribution of maximally clustered grains ( α = . a ≈
25 nm). Colour coding indicatespecific kinetic energy.
We employ a “grain-size parameter”, α = ρ gr (cid:104) ρ (cid:105) aL , (5)which is the parameter used by Hopkins & Lee (2016); Mattssonet al. (2019). Because the total gas mass M = (cid:104) ρ (cid:105) L , as well as ρ gr and a are constants, α must also be a constant. It is knownfrom studies of incompressible turbulence that particles of di ff er-ent Stokes number St can be maximally concentrated on di ff erent length scales (Bec 2003; Bec et al. 2007b; Zaichik & Alipchenkov2003). To some extent this may be a reflection of the fact that St,as it is usually defined for incompressible flows, depends on thedissipative scale of the carrier flow (Hogan et al. 1999), which isnot independent of resolution and forcing scale, and particularlyso when the flow is highly compressible and St is not even a uni- In our simulations, the particles are not treated on a mesh, but the meshused for the carrier flow (the gas) still imposes an indirect resolution limit.MNRAS , 1–11 (2019) mall-scale clustering of nano-dust grains versal number on average (Mattsson et al. 2019). The disadvantageof St is that a highly compressible gas with Re (cid:29) (cid:29)
1, but if M rms (cid:29) α may be a better dimensionless measureof grain size than (cid:104) St (cid:105) for supersonic compressible flows. How-ever, α must still be anchored to the physical characteristics of theflow. Following Hopkins & Lee (2016) we can relate L to the soniclength, R s = R MC c | u · u | = R MC M , (6)where R MC is the e ff ective radius of the modelled MC (or partthereof). In the present case we can assume R MC ∼ L .The largest clumps within an MC have characteristic radii ∼ M rms ∼
3, as in the present work,we find that R s ∼ . R s estimatedin Milky Way-like GMCs and the empiric linewidth-size relation M rms ∼ ( R / R s ) / . The resultant scaling relation is L ∼
10 pc (cid:32) M rms (cid:33) (cid:32) R s . (cid:33) . (7)Again following Hopkins & Lee (2016) the physical size of thegrains can be obtained from a = . α (cid:32) L
10 pc (cid:33) (cid:32) (cid:104) n gas (cid:105)
10 cm − (cid:33) (cid:32) ρ gr . − (cid:33) − µ m , (8)where (cid:104) n gas (cid:105) is the average number density of gas particles(molecules). The physical scales introduced in eq. (8) represent thetypical size of large MCs, the characteristic (cid:104) n gas (cid:105) in the cold-phaseISM and the average ρ gr of Galactic interstellar dust. Combiningeqns. (7) and (8) gives , a ∼ α (cid:32) M rms (cid:33) (cid:32) R s . (cid:33) (cid:32) (cid:104) n gas (cid:105)
10 cm − (cid:33) (cid:32) ρ gr . − (cid:33) − µ m . (10)Assuming values of R s , (cid:104) n gas (cid:105) and ρ gr typical for MCs, we have a ∼ α µ m. We use a factor of 0 . µ m when converting from α to a (our standard scaling) but this number of course depends onthe assumed properties. The smallest particles in our simulationshave α = . α = .
6, which corresponds to a ≈ . a ≈ . µ m, respectively. (Nano-dust is usuallydefined as grains with radii a = . . .
100 nm.)
One of the main objectives of the present study is to find the max-imum degree of clustering and at which α this maximum occurs.Centrifuging of particles away from vortex cores leads to accumu-lation of particles in convergence zones which is di ff erent from theincreased number density of dust grains n d due to compression ofthe gas and dust (Maxey 1987; Squires & Eaton 1991; Eaton &Fessler 1994; Bec 2005; Pumir & Wilkinson 2016; Yavuz et al.2018). An alternative definition of the “grain-size parameter” (which takes thecharacteristics of the flow into account) could be˜ α = M ρ gr (cid:104) ρ (cid:105) aR s , (9)but we stick to the definition by Hopkins & Lee (2016) to avoid confusionwhen comparing results. We can quantify clustering of grains using nearest neigh-bour statistics (NNS), obtained with the kd -tree algorithm (Bentley1975) including edge corrections. We can then indirectly determinethe correlation dimension d , a kind of fractal dimension defined as d = d ln N / d ln r as r →
0, where N is the number of particles(grains) surrounding a reference particle within a ball of radius r (Monchaux et al. 2012). Thus, d can be determined from g ( r ), theradial distribution function (RDF), which is related to the two-pointcorrelation function ξ ( r ) = g ( r ) − H ( r ) of the first nearest neighbour dis-tances (1-NNDs) is in turn related to g ( r ). More precisely, H ( r ) = ˜ ng ( r ) exp (cid:34) − (cid:90) r ˜ n g ( s ) 4 π s ds (cid:35) , (11)where ˜ n is the renormalised average number density (Torquato et al.1990; Bhattacharjee 2003). For a homogeneous Poisson process g =
1, so that H ( r ) reduces to the exact result discussed by, e.g.,Chandrasekhar (1943). Clearly, H ( r ) → ˜ ng ( r ) as r →
0. In thislimit g ( r ) ∝ r d − and therefore the h ( r ) = dH ( r ) / dr ∝ r d − . Wecan thus determine d from the NNS by computing h ( r ) or H ( r ),where the latter is obtained by rank-order techniques, which min-imises the error. The agreement between this method and the moredirect metod used by Bhatnagar et al. (2018a) is excellent.A major advantage with the NNS method is that it does notrequire explicit use of a binning radius δ r bin , which in such a caseshould be significantly larger than the distance δ r min between thetwo closest particles. With h ( r ) or H ( r ), we can determine g ( r , α )or ξ ( r , α ) for small r and a given α . To explore g ( r , α ) as a function α only, we must chose an evaluation radius r e , which is sometimes de-fined in terms of the Kolmogorov scale η (see, e.g., Pan et al. 2011).This is reasonable for M rms (cid:46)
1, but since we consider M rms > r e to some fraction of the average 1-NND (cid:104) r (cid:105) . Toensure that we evaluate g ( r , α ) in the power-law regime for small r , we must have r e (cid:46) (cid:104) r (cid:105) and to stay above the resolution limitof the mesh, r e (cid:38) (cid:104) r (cid:105) . (Note that (cid:104) r (cid:105) < δ r min in most cases.) Ifthere is no small-scale clustering, then g ( r e , α ) =
1; if there is, then g ( r e , α ) > Dust extinction depends on how the dust is distributed and thus alsothe degree of clustering. We consider a simplified radiative transfer(RT) problem where we omit absorption and emission from the gasand just calculate the e ff ects on incident light due to dust. The RTequation (RTE) along a column / ray of light then simplifies to, dI λ ds = − ρ ( s ) κ d ,λ ( a , s ) I λ ( s ) , (12)where we can also assume that the dust opacity κ d ,λ is due to pureabsorption, since scattering is typically negligible for a (cid:28) λ/ π .Then, ρ ( s ) κ d ,λ ( a , s ) = π (cid:90) ∞ a n d ( a , s ) Q ext ( a , λ ) da , (13)and the formal solution to (12) can be written I λ ( s ) = I λ (0) exp (cid:34) − π (cid:90) ∞ a n d ( a , s ) Q ext ( a , λ ) da (cid:35) , (14)where Q ext is the ratio between the e ff ective extinction cross-sectionand the geometric cross-section ( π a ) and I λ (0) is the intensity ofthe incident light. The RT for a plane-parallell case (distant radia-tion source) can then be calculated by numerical integration overone geometric dimension of the simulation box. MNRAS , 1–11 (2019)
Mattsson et al.
Figure 4.
Correlation dimension d as function of the grain-size parameter α . The grey diamonds connected by a dashed line in the background showthe results from a simulation of incompressible turbulence by Bhatnagaret al. (2018a). The simulation presented here (see legend) all show a muchwider dip and minimum in d at a lower α . The shaded area marks the nanodust range for a typical physical scaling of the simulations. Figure 5.
Average nearest neighbour distance ratio R ANN as function ofthe grain-size parameter α . As opposed to the case of incompressible turbu-lence, very small particles in compressible supersonic turbulence does notshow R ANN → α →
0, which is merely an e ff ect of the compressibilityof the medium. The resultant d values are presented in Fig. 4. One can clearly seethat there does indeed exist a minimum in d as a function of grainsize also for supersonic compressible turbulence. However, thereis one obvious qualitative di ff erence: the “low- d valley” is signifi-cantly wider and the d minimum occurs at smaller α compared toa simulation of particles in nearly incompressible turbulence (Bhat-nagar et al. 2018a), shown by grey diamonds connected by a dashedline in Fig. 4 . The clustering seems to begin at similar grains sizes Note that in the paper by Bhatnagar et al. (2018a), d is presented asa function of St. Their results were also obtained with the P encil C ode , inwhich particles sizes are defined by α and the unit length and unit density of ( α ∼ ff erent drag-force prescriptions we have tested,we note that the degrees of clustering are very similar. Case I yieldsa d minimum at a somewhat lower α , but since the values plottedin Fig. 4 are based on only a few simulation snapshots [except forthe data taken from Bhatnagar et al. (2018a) which is based on avery long time series] the uncertainties in the d values are compa-rable to the overall di ff erence between the two cases. Furthermore,we note that the hypothesis described in Section 2.2.1, i.e., that W s (cid:28) W s (cid:29)
1, even if justmomentarily. The correction for W s (cid:29) R ANN , is defined as theaverage of the ratio between the measured 1-NND and the corre-sponding quantity for a uniform, isotropic random distribution ofparticles (Poisson process). R ANN is a measure of both clusteringand compression the same time and minima in R ANN and d occurat similar α (compare Figs. 4 and 5). By considering R ANN for non-inertial (“tracer”) particles, the e ff ects of compressional increase of n d can be isolated. In Fig. 5 one can see that R ANN < α and the tracer-particle runs showed that R ANN ∼ . d = Although our main objective here is to study clustering of dustgrains, we have also included two simulations (IIIa and IIIb) withnon-inertial particles. Non-inertial particles have to be handled inseparate runs due to how the P encil C ode is structured.Mass-less particles is the limiting case of very small dustgrains. Without drag forces, the particles will trace the gas flow andthe distance between particles can only be reduced by compressionof the gas in which they reside. An incompressible flow will leavethe initial R ANN unchanged, i.e., R ANN = ff erent result, however.Due to shock compression the gas forms local high-density re-gions, which is where the vast majority of the tracer particles endup. As a result, R ANN ≈ .
8, while the correlation dimension ap-proaches d = not fractallyclustered . The lower values of R ANN is, in fact, a measure of theoverall level of shock compression in the simulations and it is rea-sonable to think that for non-inertial particles, R ANN will scale with M rms , but since we have only considered a narrow range of M rms values (see Table 1), we cannot test this hypothesis here. To study how the kinetics and clustering of grains are connected,we consider the dimensionless specific kinetic energy for dust the simulation. Bhatnagar et al. (2018a) then converted their particle sizesinto St for easier comparison with previous work on clustering of inertialparticles in incompressible turbulence. In Figs. 4 and 5 we have simplyomitted this conversion. MNRAS , 1–11 (2019) mall-scale clustering of nano-dust grains Figure 6.
Distribution of kinetic energyes (in terms of the E kin parameter) and first nearest neighbour distances (1-NND) for dust particles with maximalclustering ( α = . a ≈
25 nm) in the simulations. The left panel shows the simulation of compressive forcing, while the right panel shows the one withsolenoidal forcing. Statistically, there are essentially no di ff erence between the two cases. Figure 7.
Radial distribution functions (RDFs) as functions of α for the cases of compressive (left) and solenoidal (right) stochastic forcing. The RDFs havebeen evaluated at four di ff erent separations r e chosen to be , , , and of the average 1-NND. grains, i.e., E kin = (cid:32) | v | c s (cid:33) . (15)Comparing Figs. 2 (compressive forcing) and 3 (solenoidal forc-ing) it seems that the most clustered grains have the lowest kineticenergies. There is also a qualitative di ff erence between the types offorcing considered. Statistically the cases are actually very similar,which is apparent from Fig. 6. We see that the highest E kin occur inparticles with small separations (short 1-NND) and it is also evidentthat “lonely grains” typically do not have high E kin .The E kin distribution P ( E kin ) is created by a more or lessstochastic process, where grains are ejected from vortices at ar-bitrary directions and positions and may hit other vortices, withdi ff erent rotation, and thus be decelerated or accelerated, as wellejected again in another direction. Snapshots from the simulationsshow particles accumulating in the convergence zones between vor-tices having lost (or not gained very much) kinetic energy. The highest E kin occur in grains that were recently ejected from vor-tices. Isolated dust grains have generally rather average E kin values(see Fig. 6), which is ar result of the stochasticity of the accelera-tion. Other grains with average E kin (which make up the vast ma-jority of grains) are found in, or near, the convergence zones as aresult of the same mechanisms; grains entering vortices will onlystay there for a limited time (Bhatnagar et al. 2016) and this timecan be regarded as a random variable. A majority of the dust grains populate regions which correspondto a total volume smaller than the whole simulation box, i.e., theaverage amount of space in between dust grains is reduced. Thise ff ect can be quantified by the RDF, g ( r , α ), which is peaking atthe α where the R ANN has its minimum (cf. Figs. 7 and 5). Theamplitude of g ( r , α ) depends on the radius r and the implied scaling MNRAS000
Radial distribution functions (RDFs) as functions of α for the cases of compressive (left) and solenoidal (right) stochastic forcing. The RDFs havebeen evaluated at four di ff erent separations r e chosen to be , , , and of the average 1-NND. grains, i.e., E kin = (cid:32) | v | c s (cid:33) . (15)Comparing Figs. 2 (compressive forcing) and 3 (solenoidal forc-ing) it seems that the most clustered grains have the lowest kineticenergies. There is also a qualitative di ff erence between the types offorcing considered. Statistically the cases are actually very similar,which is apparent from Fig. 6. We see that the highest E kin occur inparticles with small separations (short 1-NND) and it is also evidentthat “lonely grains” typically do not have high E kin .The E kin distribution P ( E kin ) is created by a more or lessstochastic process, where grains are ejected from vortices at ar-bitrary directions and positions and may hit other vortices, withdi ff erent rotation, and thus be decelerated or accelerated, as wellejected again in another direction. Snapshots from the simulationsshow particles accumulating in the convergence zones between vor-tices having lost (or not gained very much) kinetic energy. The highest E kin occur in grains that were recently ejected from vor-tices. Isolated dust grains have generally rather average E kin values(see Fig. 6), which is ar result of the stochasticity of the accelera-tion. Other grains with average E kin (which make up the vast ma-jority of grains) are found in, or near, the convergence zones as aresult of the same mechanisms; grains entering vortices will onlystay there for a limited time (Bhatnagar et al. 2016) and this timecan be regarded as a random variable. A majority of the dust grains populate regions which correspondto a total volume smaller than the whole simulation box, i.e., theaverage amount of space in between dust grains is reduced. Thise ff ect can be quantified by the RDF, g ( r , α ), which is peaking atthe α where the R ANN has its minimum (cf. Figs. 7 and 5). Theamplitude of g ( r , α ) depends on the radius r and the implied scaling MNRAS000 , 1–11 (2019)
Mattsson et al.
Figure 8.
Mean dust absorption as a function of grain size ( α ) based on the distributions of maximally clustered grains ( α = . a ≈
25 nm) and assumingmean optical depth (cid:104) τ λ (cid:105) = / Figure 9.
Dust absorption maps based on the distributions of maximally clustered grains ( α = . a ≈
25 nm) and assuming mean optical depth (cid:104) τ λ (cid:105) = / relation is g ∼ r d − as expected for small r . The amplitude at thepeak is also somewhat ( ∼ ff erent particlespecies i and j is ϕ ij ∼ π ( a i + a j ) n i n j ∆ v ij = n i n j C ij , (16)where ∆ v ij is the absolute velocity di ff erence between the collid-ing particles. The average collision kernel (cid:104) C ij (cid:105) is often approxi-mated with (cid:104) C ij (cid:105) ≈ π ( a i + a j ) c s (cid:104)W(cid:105) , where (cid:104)W(cid:105) require somekind of modelling of turbulence and drag (V¨olk et al. 1980). Ob-viously, (cid:104)W(cid:105) does not include information about fractal cluster-ing. By adding the RDF g ( r , a ), we can account for fractal clus-tering. The collision kernel for identical particles i then becomes (cid:104) C i (cid:105) = π a i g ( r e , a i ) c s (cid:104)W(cid:105) (see, e.g., Wang et al. 2000; Pan et al.2011; Pan & Padoan 2014). In Fig. 7 we show g ( r e , α ) for the casesof compressive (left) and solenoidal (right) stochastic forcing, eval- uated at four di ff erent radial separations r e chosen to be , , , and of the average NND (cid:104) r (cid:105) ( r e = (cid:104) r (cid:105) corresponds to the resolutionlimit set by the mesh used for computing the gas flow). The resultsshown in Fig. 7 are qualitatively similar to those by Pan & Padoan(2014). although the scaling is di ff erent. They also indicate that, fora given ρ , there is indeed a special particle size where the interac-tion frequency is higher than for any other size. For a gas density ρ typical of an MC, the nano-dust scale is therefore of special inter-est. In addition to clustering, ϕ ij also increases due to the tur-bulence component of ∆ v ij (see, e.g., Ormel et al. 2009). Smallgrains, which couple well to the flow, will have small ∆ v ij , whiledecoupled large grains are not accelerated very much due to kineticdrag and therefore also have small ∆ v ij . That is, g ( r e , α ) is large forsmall grains, but the collision frequency ϕ coll is limited by ∆ v ij andthe largest increase of the probability of aggregation is obtainedfor accretion of grains with radii a ∼ −
30 Å onto grains with a (cid:38)
100 Å. This follows from the fact that at least one of the collid-
MNRAS , 1–11 (2019) mall-scale clustering of nano-dust grains ing grains must have a significant geometric cross-section, and bothgrains cannot be be tightly coupled to the flow, in order to achievea high ϕ ij (Bhatnagar et al. 2018b). ff ects on dust absorption To study the e ff ects on nano-dust extinction, we consider a simpli-fied RT problem of parallel rays of light incident on the simula-tion box, where we omit absorption and emission from the gas asdescribed in Sect. 2.3. The intensity I λ, i of light surviving the pas-sage through a column of the inhomogeneous distribution of a dustspecies i relative to the mean intensity (cid:104) I λ, i (cid:105) (taken of the wholeprojected area) is a function of the mean optical depth (cid:104) τ λ, i (cid:105) andthe normalised number density of dust.An inhomogeneous distribution of dust leads to a higherthroughput of photons. This e ff ect depends on α , but depends evenmore on (cid:104) τ λ, i (cid:105) (see Fig. 8). The fraction of surviving photons in-creases up to 15% in Case IIa (compressive forcing) for (cid:104) τ λ (cid:105) = I λ, i / (cid:104) I λ, i (cid:105) , whichcoincides with maximal clustering, i.e., the α range where d and R ANN have their minima.It is important to emphasise that the e ff ect described abovemay depend on the geometrical depth of the modelled region. Ifwe consider an elongated rectangular slab, instead of a box, wherethe integration of the RTE is done along the extended axis of theslab, the e ff ective blocking of light will approach the uniform case(provided that the slab is su ffi ciently long). We will end this section by discussing an example of peculiar ex-tinction, which may be due to clustered nano dust. The afterglow ofthe gamma-ray burst GRB 140506A display a very unusual extinc-tion curve that was very steep in the ultraviolet (Fynbo et al. 2014).This peculiar extinction was found to vary between two epochs sep-arated by about 24 h. One possibility for this could be that the thereis very strongly clustered dust in the foreground of the afterglow.The line-of-sight was also peculiar in other ways, indicating a veryhigh density in the intervening material. Fynbo et al. (2014) arguedthat GRB 140506A might be an example of an extreme 2175 Å-feature, much wider and deeper than unusual. However, a follow-up analysis (Heintz et al. 2017) has revealed that the extreme 2175Å-feature scenario seems to be excluded by the data.If extinction in the UV is dominated by nano-sized grains,which are very inhomogeneously distributed in the ISM, thereought to be extinction curves di ff ering between objects separatedby small angles on the sky. This variation should be stronger in theUV than at longer wavelengths as small grains dominate extinc-tion at ultraviolet wavelengths. Concerning the spatial variations ofthe ultraviolet extinction curve the work of Fitzpatrick & Massa(2007) remains the most comprehensive. They report large line-to-sight variations and a lack of a correlation between variations in theultraviolet and infrared parts of the extinction curve.To test whether the extinction in the sightline ofGRB 140506A can be explained with clustered nano dust,we have made a simple extinction model based on silicates (Draine& Li 2001), graphite (Laor & Draine 1993) and metallic iron (Palik1991), assuming all grains have a radius a = Figure 10.
Nano dust model of the extinction towards GRB 140506A.Note that the extinction curve for the GRB tend to coincide with that ofthe Galaxy at long wavelengths. The extreme UV extinction is in the nanodust model mainly due to tiny silicate grains. the middle of the “clustering window” ( a ∼ −
60 nm) impliedby the simulations presented above. We assume that the extinctionis due to a high concentration of nano dust superimposed on anextinction law like the nominal Fitzpatrick & Massa (2007) curve,but scaled down to 25% of the Galactic extinction. Thus, the nanodust is added like a local foreground screen on top of some generichost-galaxy extinction. A good fit to the exceptional extinctioncurve derived by Heintz et al. (2017) is obtained with a nano-dustcomponent consisting of 96% silicate grains, 2% graphite and 2%metallic iron grains (see Fig. 10).The extreme extinction in the sightline of GRB 140506A isunusual and very rare. A plausible, although speculative, explana-tion might be that when an interstellar gas cloud with a high densityof nano dust passes the sightline of the GRB, small-scale cluster-ing may locally cause extreme extinction. The GRB 140506A phe-nomenon should in such a scenario occur only very rarely and, de-pending on the distance between the GRB and the cloud, it mayalso explain fast variability of the extinction.
In the present paper we have investigated the small-scale clusteringof nano-sized dust grains in an interstellar context using 3D highresolution (1024 ) periodic-boundary box simulations of stochasti-cally forced, supersonic steady-state turbulence. The aim has pri-marily been to provide a bench mark for further study by establish-ing how clustering and concentration of particles in compressiblehydrodynamic turbulence is di ff erent from to the incompressiblecase, which is much better understood. A fully realistic model ofclustering of nano-grains in a turbulent MC must include chargingof grains as well as magnetic fields, but this will have to wait untilthe purely kinetic picture is understood.We conclude that kinetic drag on small particles in com-pressible turbulence show maximal clustering at a smaller particlesize α , and display a wider dip in the d – α relation around thatsize ( α ≈ . d < a = . . .
100 nm) assum-
MNRAS000
MNRAS000 , 1–11 (2019) Mattsson et al. ing a scaling of the simulations which represents the conditions inan MC.The RDF g ( r , α ) has a maximum where d has a minimum(at α ≈ . a ∼ −
60 nm. The e ff ective grain-number density increases by atleast a factor 5–6 and the chance the collision kernel increases by asimilar factor. Thus, the theoretical rate of nano-grain coagulationincreases and the chance of building agglomerates starting from apopulation of mostly nano-sized grains without a preceding phaseof grain growth by accretion of molecules goes from being almostimpossible to at least plausible.Simple radiative transfer through columns of the simulationbox, show that the compression of the gas redistributes nano dustin such way that the total throughput of parallel rays of light inci-dent on the box increases compared to the case of homogeneouslydistributed dust and gas. This e ff ect has a clear dependence on theadopted mean optical depth for the dust; for (cid:104) τ λ (cid:105) = / (cid:104) τ λ (cid:105) =
1. There isalso a dependence on α because this e ff ect is only seen when grainscouple well to the gas. In the case of solenoidally forced turbulencewe also see a peak in the radiative throughput at α ≈ .
04, where d has its minimum (corresponding to maximal clustering).Finally, we note that extreme UV extinction, such in the sight-line of GRB 140506A (Fynbo et al. 2014; Heintz et al. 2017), couldbe the result of nano dust. An extinction model based on silicates,graphite and metallic iron, assuming grain sizes in the “clusteringwindow” ( a ∼ −
60 nm), appears to explain the broad and deepUV extinction feature of GRB 140506A, and, possibly, its short-term variations too.
ACKNOWLEDGMENTS
This project is supported by the Swedish Research Council (Veten-skapsrådet), grant no. 2015-04505. The Cosmic Dawn centeris funded by the DNRF. The anonymous reviewer is thankedfor his / her insightful criticism, which helped to improve themanuscript. We also wish to thank our colleagues Akshay Bhat-nagar and Dhrubaditya Mitra for sharing their simulation data andfor all their help and support. REFERENCES
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