Aa33970 18

Partial capture of text on file.

A&A630,A97(2019) Astronomy
https://doi.org/10.1051/0004-6361/201833970 &
©R.-A.Chiraetal. 2019 Astrophysics
Howdovelocitystructure functions trace gas dynamics in
simulated molecular clouds?
1 2,3 4,5,6 1
R.-A. Chira , J. C. Ibáñez-Mejía , M.-M. Mac Low , and Th. Henning
1 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
e-mail: roxana-adela.chira@alumni.uni-heidelberg.de
2 I. Physikalisches Institut, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany
e-mail: ibanez@ph1.uni-koeln.de
3 Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany
4 Department of Astrophysics, American Museum of Natural History, 79th St. at Central Park West, New York, NY 10024, USA
e-mail: mordecai@amnh.org
5 Zentrum für Astronomie, Institut für Theoretische Astrophysik, Universität Heidelberg, Albert-Ueberle-Str. 2,
69120 Heidelberg, Germany
6 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Ave, New York, NY 10010, USA
Received 27 July 2018 / Accepted 9 August 2019
ABSTRACT
Context. Supersonicdisordered ﬂows accompanytheformationandevolutionofmolecularclouds(MCs).Ithasbeenarguedthatthis
is turbulence that can support against gravitational collapse and form hierarchical sub-structures.
Aims. We examine the time evolution of simulated MCs to investigate: What physical process dominates the driving of turbulent
ﬂows?Howcantheseﬂowsbecharacterised?Aretheyconsistentwithuniformturbulence or gravitational collapse? Do the simulated
ﬂowsagreewithobservations?
Methods. WeanalysedthreeMCsthathaveformedself-consistently within kiloparsec-scale numerical simulations of the interstellar
medium (ISM). The simulated ISM evolves under the inﬂuence of physical processes including self-gravity, stratiﬁcation, magnetic
ﬁelds, supernova-driven turbulence, and radiative heating and cooling. We characterise the ﬂows using velocity structure functions
(VSFs) with and without density weighting or a density cutoff, and computed in one or three dimensions. However, we do not include
optical depth effects that can hide motions in the densest gas, limiting comparison of our results with observations.
Results. In regions with sufﬁcient resolution, the density-weighted VSFs initially appear to follow the expectations for uniform
turbulence, with a ﬁrst-order power-law exponent consistent with Larson’s size-velocity relationship. Supernova blast wave impacts
on MCs produce short-lived coherent motions at large scales, increasing the scaling exponents for a crossing time. Gravitational
contraction drives small-scale motions, producing scaling coefﬁcients that drop or even turn negative as small scales become dominant.
Removingthedensity weighting eliminates this effect as it emphasises the diffuse ISM.
Conclusions. We conclude that two different effects coincidentally reproduce Larson’s size velocity relationship. Initially, uniform
turbulencedominates,sotheenergycascadeproducesVSFsthatareconsistentwithLarson’srelationship.Later,contractiondominates
and the density-weighted VSFs become much shallower or even inverted, but the relationship of the global average velocity dispersion
of the MCs to their radius follows Larson’s relationship, reﬂecting virial equilibrium or free-fall collapse. The injection of energy by
shocks is visible in the VSFs, but decays within a crossing time.
Keywords. turbulence – ISM: kinematics and dynamics – ISM: structure – ISM: clouds
1. Introduction remains unclear whether there are particular mechanisms that
dominatethedrivingofturbulencewithinMCs,aseveryprocess
It has long been known that star formation preferentially occurs is supposed to be traced by typical features in the observables.
within molecular clouds (MCs). However, the physics of the Yet, these features are either not seen or are too ambiguous to
star formation process is still not completely understood. It is clearly reﬂect the dominant driving mode. For example, turbu-
obvious that gravity is the key factor for star formation as it lence that is driven by large-scale velocity dispersions during
drives collapse motions and operates on all scales. However, one global collapse (Ballesteros-Paredes et al. 2011a,b; Hartmann
needs additional processes that stabilise the gas or terminate star et al. 2012) produces P-Cygni line proﬁles that have not yet
formation quickly in order to explain the low star formation efﬁ- been observed on scales of entire MCs. Internal feedback, on
ciencies observed in MCs. Although there are many processes the contrary, seems more promising as it drives turbulence from
that act at the different scales of MCs, turbulent support has often small to large scales (Dekel & Krumholz 2013; Krumholz et al.
been argued to be the best candidate for this task. 2014). Observations, though, demonstrate that the required driv-
In the literature, turbulence plays an ambiguous role in the ing sources need to act on scales of entire clouds, which typical
context of star formation. In most of the cases, turbulence is feedback, such as radiation, winds, jets, or supernovae (SNe),
expected to stabilise MCs on large scales (Fleck 1980; McKee & cannot achieve (Heyer & Brunt 2004; Brunt et al. 2009; Brunt &
Zweibel 1992; Mac Low 2003), while feedback processes and Heyer 2013).
shear motions heavily destabilise or even disrupt cloud-like Therehavebeenmanytheoreticalstudiesthathaveexamined
structures (Tan et al. 2013; Miyamoto et al. 2014). However, it the nature and origin of turbulence within the various phases of
A97, page 1 of 21
OpenAccessarticle, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
OpenAccessfundingprovided by Max Planck Society.
A&A630,A97(2019)
the interstellar medium (ISM; Mac Low & Klessen 2004, and 2. Methods
references within). The most established characterisation of tur- 2.1. Cloud models
bulence in general was introduced by Kolmogorov (1941) who
investigated fully developed, incompressible turbulence driven The analysis in this paper is based on a sample of three MCs
on scales larger than the object of interest, and dissipating on identiﬁed within a three-dimensional (3D), magnetohydrody-
scales much smaller than those of interest. In the scope of this namical, adaptive mesh reﬁnement simulation using the FLASH
paper this object is a single MC. MCs are highly compress- code (Fryxell et al. 2000). Paper I and Paper II, as well as
ible, though. Only a few analytical studies have treated this Chira et al. (2018b, Paper III hereafter), describe the simulations
case. She & Lévêque (1994) and Boldyrev (2002), for example, and the clouds in more detail. We summarise the most relevant
generalised and extended the predicted scaling of the decay of properties.
turbulence to supersonic turbulence. Galtier & Banerjee (2011) The numerical simulation models a 1×1×40 kpc3 section
and Banerjee & Galtier (2013) provided an analytic description of the multi-phase, turbulent ISM of a disc galaxy, where dense
of the scaling of mass-weighted structure functions. structures form self-consistently in convergent, turbulent ﬂows
Larson (1981) found a relation between the linewidth σ and PaperI.Themodelincludesgravity–abackgroundgalactic-disc
the effective radius R of MCs. Subsequent investigators have potential accounting for a stellar component and a dark matter
settled on the form of the relation being (Solomon et al. 1987; halo, as well as self-gravity turned on after 250 Myr of simulated
Falgarone et al. 2009; Heyer et al. 2009) time – SN-driven turbulence, photoelectric heating and radiative
1/2 cooling, and magneticﬁelds.Althoughhundredsofdenseclouds
σ∝R . (1) form within the simulated volume, Paper II focussed on three
Goodmanetal. (1998) showed that analysis techniques used clouds, which were re-simulated at a much higher spatial res-
to study this relation could be distinguished by whether they olution. The internal structures of the MCs are resolved using
studied single or multiple clouds using single or multiple tracer adaptive mesh reﬁnement, focussing grid resolution on dense
species. Explanations for this relation have relied on either regions where Jeans unstable structures must be resolved with
turbulent cascades (Larson 1981; Kritsuk et al. 2013a, 2015; a minimum of 4 cells (λJ >4∆xmin). For a maximum resolution
Gnedin 2015; Padoan et al. 2016), or the action of self-gravity of ∆x=0.1 pc, the corresponding maximum resolved density is
3 −3
(Elmegreen 1993, 2007; Vázquez-Semadeni et al. 2006; Heyer 8×10 cm for gas at a temperature of 10 K (e.g. Paper II,
et al. 2009; Ballesteros-Paredes et al. 2011b). Eq. (15)). We deﬁne MCs as regions above a ﬁxed number den-
These can potentially be distinguished by examining the sity threshold with ﬁducial value n =100 cm−3. We chose
cloud
velocity structure function (VSF). Kritsuk et al. (2013a) care- this threshold as it roughly corresponds to the density when CO
fully reviews the argument for Larson’s size-velocity relation becomesdetectable. The MCs have initial masses at the onset of
depending on the turbulent cascade. In short, in an energy cas- self-gravity of 3×103, 4×103, and 8×103 M and are denoted
⊙
cade typical for turbulence, the second-order structure function asM3,M4,andM8,respectively,hereafter.Inthispaper,weusethe
ζ(2) 3
has a lag dependence ℓ with ζ(2)≃1/2. In Ibáñez-Mejía data within (40 pc) subregions centred on the high-resolution
et al. (2016, hereafter Paper I) the authors argue that uniform clouds’ centres of mass, which we map to a uniform grid of 0.1
driven turbulence was unable to explain the observed relation pczonesforanalysis.Forillustrationsofthemorphologiesofthe
in a heterogeneous ISM, but that the relation could be naturally three clouds we refer to Fig. 1 of Paper III.
explained by hierarchical gravitational collapse. It is important to point out that the clouds are embedded
In this paper, we examine the velocity structure functions within a complex turbulent environment, gaining and losing
of three MCs that formed self-consistently from SN-driven tur- mass as they evolve. Paper II described the time evolution of
bulence in the simulations by Paper I and Ibáñez-Mejía et al. the properties of all three clouds in detail, in particular, mass,
(2017, hereafter Paper II). We study how the turbulence within size, velocity dispersion, and accretion rates, in the context
the clouds’ gas evolves. The key questions we address are the of MC formation and evolution within a galactic environment.
following: What dominates the turbulence within the simulated Paper III studied the properties, evolution, and fragmentation
MCs? Does the observed linewidth-size relation arise from the of ﬁlaments that self-consistently condense within these clouds.
turbulent ﬂow? How can structure functions inform us about We paid particular attention to the accuracy of typical stabil-
the evolutionary state of MCs and the relative importance of ity criteria for ﬁlaments, comparing the results to the theoretical
large-scale turbulence, discrete blast waves, and gravitational predictions, showing that simpliﬁed analytic models do not cap-
collapse? ture the complexity of fragmentation due to their simplifying
InSect.2,weintroducethesimulatedcloudsinthecontextof assumptions.
the underlying physics involved in the simulations. Furthermore, 2.2. Velocity structure function
wedescribethetheoreticalbasicsofvelocitystructurefunctions.
Section 3 demonstrates that the velocity structure function is In this paper, we probe the power distribution of turbulence
a useful tool to characterise the dominant driving mechanisms throughout the entire simulated MCs by using the velocity
of turbulence in MCs and can be applied to both simulated structure function (VSF). The VSF is a two-point correlation
and observed data. We examine the inﬂuence of using one- function,
dimensional velocity measurements, different Jeans reﬁnement
levels, density thresholds, density weighting on the applicability Sp(ℓ) = h|∆u|p i (2)
of the velocity structure function, and the results obtained with that measures the mean velocity difference
it in Sect. 4. At the end of that section, we also compare our
results to observational studies. We summarise our ﬁndings and ∆u(ℓ) = u(x + ℓ) − u(x) (3)
conclusions in Sect. 5. The simulation data and the scripts that
this work is based on are published in the Digital Repository of between two points x and x + ℓ, with ℓ being the direction
the American Museum of Natural History (Chira et al. 2018a). vector pointing from the ﬁrst to the second point. The VSF S p is
A97, page 2 of 21
R.-A. Chira et al.:How do velocity structure functions trace gas dynamics in simulated molecular clouds?
usually reported as a function of lag distance, ℓ=|ℓ|, between the so that ζ can be measured from Sp/S3, which typically gives
correlated points. The coherent velocity differences measured by a clearer power-law behaviour. The self-similarity parameter is
the VSF can be produced by both the energy cascade expected deﬁned as
in turbulent ﬂows, and by coherent motions such as collapse, ζ(p)
rotation, or blast waves. Those patterns become more prominent Z(p) = ζ(3). (9)
the higher the value of the power p is (Heyer & Brunt 2004).
For fully developed, homogeneous, isotropic, turbulence the As mentioned before, both Eqs. (5) and (6) return values
VSF is well-described by a power-law relation (Kolmogorov of ζ(3)=1. Therefore, those equations also offer predictions for
1941; She & Lévêque 1994; Boldyrev 2002): Z(p).
S (ℓ) ∝ ℓζ(p). (4) Forthediscussionbelow,wemeasureζ(p)byﬁttingapower-
p law, given by
Kolmogorov (1941) predicts that the third-order exponent, h i ( )
log Sp(ℓ) = log A +ζ(p) log (ℓ), (10)
ζ(3), is equal to unity for an incompressible ﬂow. As a conse- 10 10 10
−5 2π
quence the kinetic energy decays with Ekin(k)∝k 3, with k= ℓ with A being the proportionality factor of the power-law to the
being the wavenumber of the turbulence mode. For a supersonic simulated measurements. We choose the smallest lag of the ﬁt-
ﬂow, however, ζ(3)>1 is expected. Based on Kolmogorov’s ting range to be equal to eight zones, sufﬁciently large to ensure
work, She & Lévêque (1994) and Boldyrev (2002) extended and that our ﬁt does not include the numerical dissipation range. For
generalised the analysis and predicted the following intermit- moredetails of the ﬁtting procedure we refer to Appendix A.
tency corrections to Kolmogorov’s scaling law. For incompress- We follow observational practice and reduce the compu-
ible turbulence with ﬁlamentary dissipative structures She & tational effort of this study by generally focussing on clouds
Lévêque (1994) predict that the VSFs scale with power law deﬁned by a density threshold. However, Paper II shows that
index there is usually no sharp increase in density between the ISM
 !p and the clouds. Instead, the gas becomes continuously denser
p  2 3
 
  towards the centres of mass within the clouds. Consequently,
 
ζ(p) = +2 1− , (5)
 
9  3  our use of a density threshold is a somewhat artiﬁcial bound-
ary between the clouds and the ISM. Observationally, however,
while supersonic ﬂows with sheet-like dissipative structures are introducing a column density (or intensity) threshold is unavoid-
predicted to scale with (Boldyrev 2002) able, be it due to technical limitations (e.g. detector sensitivity)
!p or the nature of the underlying physical processes (for example,
p 1 3 excitation rates, or critical densities). Therefore, we also study
ζ(p) = 9 +1− 3 . (6) howadensitythreshold inﬂuences the VSF and its evolution.
At our ﬁducial density threshold, we actually consider only
It should be noted that both equations return a value of ≤1.5% of the volume in the high resolution cube. To under-
ζ(3)=1, but this is only an exact result for the She & Lévêque stand the inﬂuence of this limitation we set up a test sce-
model, while it is a result of normalisation in the case of nario (see Sect. 3.4) by removing the density threshold (setting
n =0 cm−3) that results in analysing the entire data cube.
Boldyrev. cloud
In the case of compressible turbulence, the energy cascade Details of the method for computing the VSFs in these two cases
can no longer be expressed in terms of a pure velocity differ- are given in Appendix A.
encebecausedensityﬂuctuationsbecomeimportant.Turbulence As in the case without a density threshold it would be too
should then show a cascade in some density-weighted VSF anal- computationally expensive to compute all lags to all zones.
ogous to the incompressible case. Padoan et al. (2016) deﬁned a Thus, we randomly choose a set of 5% of the total number of
density-weighted VSF to attempt to capture this process, which zones as reference points and only compute relative velocities
weuseinoursubsequentanalysis from the entire cube to these zones. By choosing the start-
ing points randomly we ensure that all parts of the cubes are
hρ(x)ρ(x+ℓ)|∆u|pi considered. As a consequence, there is only a small likelihood
Sp(ℓ) = hρ(x)ρ(x+ℓ)i . (7) (5%×1.5%=0.075%)thatanygivenzonechosenwillbewithin
the cloud. Therefore, we emphasise that it is likely that the two
Alternatives have been proposed by Kritsuk et al. (2013b) subsamples (no density threshold and cloud-only) do not have
based on an analysis of the equations of compressible ﬂow that a common subset of starting vectors. Nevertheless, the random
3
should be explored in future work. sample still includes >4×10 zones in the cloud, so the sample
In many cases a three-dimensional computation of the VSF does include information on VSFs of material in the cloud.
cannot be performed because of the observational constraint that
only line-of-sight velocities are available. We therefore compare 3. Results
our three-dimensional (3D) results to one-dimensional (1D),
density-weighted VSFs 3.1. Examples
hρ(x)ρ(x+ℓ)|∆u·e|pi In this section, we present our results on how VSFs reﬂect the
S (ℓ) = i , (8)
p,1D hρ(x)ρ(x+ℓ)i velocity structure within and around MCs.
Figure 1 shows nine examples of density-weighted VSFs
with e representing the unit vector along the i= x-, y-, or z-axis. (Eq. (7)). The ﬁgure shows the VSFs of all three clouds
i
Benzi et al. (1993) introduced the principle of extended (columns) at times of 1.0, 3.0, and 4.2 Myr after the onset of
self-similarity. It proposes that there is a constant relationship gravity. All plots show orders p=1–3. The solid lines show the
betweenthescalingexponentsofdifferentordersatalllagscales ﬁtted power-law relations as given by Eq. (10).
A97, page 3 of 21
A&A630,A97(2019)
M3 M4 M8
t = 1.0 Myr p = 1 t = 1.0 Myr t = 1.0 Myr
p = 2
p = 3
101
p]
1)
s

m
k
(
[ 0
10
)
(
S
10 1
t = 3.0 Myr t = 3.0 Myr t = 3.0 Myr
101
p]
1)
s

m
k
(
[ 0
10
)
(
S
10 1
t = 4.2 Myr t = 4.2 Myr t = 4.2 Myr
101
p]
1)
s

m
k
(
[ 0
10
)
(
S
10 1
100 101 100 101 100 101
[pc] [pc] [pc]
Fig. 1. Examples of VSFs from models (left to right) M3, M4, and M8 as function of lag scale ℓ and order p, based on data with density threshold.
Theexamplesaregivenfortimes(toptobottom)t=1.0Myr,3.0Myr,and4.2Myr.Thedots(connectedbydashedlines)showthevaluescomputed
from the simulations. The solid lines represent the power-law relation ﬁtted to the respective structure functions.
The examples demonstrate that, in general, the measured relation. On larger scales, one observes a local minimum before
VSFs cannot be described by a single power-law relation over the VSFs either increase or stagnate. Additional examples of
the entire range of ℓ. Instead they are composed of roughly three VSFsaregiveninAppendixB.
different regimes: one at small scales at 0.8 pc.ℓ.3 pc, a sec- Theexamples in Fig. 1 and Appendix B illustrate how VSFs
ond one within 3 pc.ℓ.10–15 pc, and the last one at large react to different scenarios that affect the turbulent structure of
scales with 10–15 pc.ℓ.30pc.Weﬁndthatonlythesmalland the entire clouds. All clouds at t=1.0 Myr show the case where
intermediate ranges may be represented by a common power-law turbulence is driven on large scales and naturally decays towards
A97, page 4 of 21

The words contained in this file might help you see if this file matches what you are looking for:

...A astronomy https doi org r chiraetal astrophysics howdovelocitystructure functions trace gas dynamics in simulated molecular clouds chira j c ibanez mejia m mac low and th henning max planck institut fur astronomie konigstuhl heidelberg germany e mail roxana adela alumni uni de i physikalisches universitat zu koln zulpicher stra ph koeln extraterrestrische physik giessenbachstrasse garching department of american museum natural history st at central park west new york ny usa mordecai amnh zentrum theoretische astrophysik albert ueberle str center for computational flatiron institute fifth ave received july accepted august abstract context supersonicdisordered ows accompanytheformationandevolutionofmolecularclouds mcs ithasbeenarguedthatthis is turbulence that can support against gravitational collapse form hierarchical sub structures aims we examine the time evolution to investigate what physical process dominates driving turbulent howcantheseowsbecharacterised aretheyconsistentwithun...

Share

Help

Share

Share to social media

Help

Login Area