# Initial Conditions for Star Formation:
A Physical Description of the Filamentary ISM

Alvaro Hacar Department of Astrophysics, University of Vienna,
Türkenschanzstrasse 17, 1180, Vienna (Austria) Susan E. Clark Department of
Physics, Stanford University, Stanford, California 94305, USA
Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA Fabian Heitsch Department of
Physics & Astronomy, University of North Carolina - Chapel Hill, Chapel Hill,
North Carolina 27599, USA Jouni Kainulainen Chalmers University of Technology,
Department of Space, Earth and Environment, SE-412 93, Gothenburg, Sweden
Georgia V. Panopoulou California Institute of Technology, MC350-17, 1200 East
California Boulevard, Pasadena, CA 91125, USA Daniel Seifried University of
Cologne, I. Physical Institute, Zülpicher Str. 77, 50937 Cologne, Germany
Rowan Smith Jodrell Bank Centre for Astrophysics, Department of Physics and
Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK

###### Abstract

The interstellar medium contains filamentary structure over a wide range of
scales. Understanding the role of this structure, both as a conduit of gas
across the scales and a diagnostic tool of local physics, is a major focus of
star formation studies. We review recent progress in studying filamentary
structure in the ISM, interpreting its properties in terms of physical
processes, and exploring formation and evolution scenarios. We include
structures from galactic-scale filaments to tenth-of-a-parsec scale filaments,
comprising both molecular and atomic structures, from both observational and
theoretical perspectives. In addition to the literature overview, we assemble
a large amount of catalogue data from different surveys and provide the most
comprehensive census of filamentary structures to date. Our census consists of
22 803 filamentary structures, facilitating a holistic perspective and new
insights. We use our census to conduct a meta-analysis, leading to a
description of filament properties over four orders of magnitudes in length
and eight in mass. Our analysis emphasises the hierarchical and dynamical
nature of filamentary structures. Filaments do not live in isolation, nor they
generally resemble static structures close to equilibrium. We propose that
accretion during filament formation and evolution sets some of the key scaling
properties of filaments. This highlights the role of accretion during filament
formation and evolution and also in setting the initial conditions for star
formation. Overall, the study of filamentary structures during the past decade
has been observationally driven. While great progress has been made on
measuring the basic properties of filaments, our understanding of their
formation and evolution is clearly lacking. In this context, we identify a
number of directions and questions we consider most pressing for the field.

##  1 THE FILAMENTARY ISM:
A CENTURY OF DISCOVERIES

“Among the most surprising things in connection with these nebula-filled holes
are the vacant lanes that so frequently run from them for great distances.
These lanes undoubtedly have had something to do with the formation of the
holes and with the nebula in them.” (_Barnard_ , 1907). With these words, E.
E. Barnard first reported the direct connection between filaments (dark
lanes), dense cores (holes), and stars (nebulae). This description precedes
observations of molecular line emission (_Wilson et al._ , 1970), and hints at
a connection between the geometry of interstellar matter and the star
formation process. More than a century and many groundbreaking observations
later, we are still trying to unravel the multiscale physics of star
formation, and the meaning of a ubiquitous ISM geometry: filaments.

A series of theoretical and observational works investigated the basic
physical properties of filaments in molecular clouds in the following decades.
The geometrical simplicity of idealized filaments allowed semi-analytic
derivations of hydrostatic equilibrium solutions (_Stodólkiewicz_ , 1963;
_Ostriker_ , 1964) and quasi-static gravitational fragmentation (_Larson_ ,
1985). These were later extended to include the influence of magnetic fields
(_Nagasawa_ , 1987; _Hanawa et al._ , 1993) and external pressure (_Fiege and
Pudritz_ , 2000). Extinction and infrared (IR) observations illustrated the
filamentary nature of clouds, showing signatures of regular Jeans-like
fragmentation (_Schneider and Elmegreen_ , 1979) and promoting the formation
of cores and stars at high efficiencies (_Hartmann_ , 2002). Polarization
measurements quantified the relative orientation between magnetic fields and
filaments (_Hall_ , 1955; _Vrba et al._ , 1976; _Goodman et al._ , 1990). In
parallel, large-scale molecular maps first explored the internal kinematics of
filaments (_Loren_ , 1989b) typically exhibiting multiple velocity components
(_Duvert et al._ , 1986). Limited in resolution and sensitivity, most of these
early works targeted “prototype” filaments in the solar neighbourhood, such as
the B213-L1495 filament in Taurus (_Mizuno et al._ , 1995), the Ophiuchus
Streamers (_Loren_ , 1989a), or the Integral Shape Filament (ISF) in Orion
(_Bally et al._ , 1987). Later, these studies were extended to other filaments
in our Galaxy, with particular attention paid to filaments with extraordinary
mass (_Schneider et al._ , 2010) or length (_Jackson et al._ , 2010).

The detailed study of filaments was revolutionized by the Herschel Space
Telescope wide-field far-infrared (FIR) continuum maps (_Pilbratt et al._ ,
2010). The unprecedented dynamic range of Herschel highlighted the high degree
of filamentary organization of the gas in molecular clouds (_André et al._ ,
2010; _Molinari et al._ , 2010). Its enhanced sensitivity provided the first
homogeneous measurements of filament masses, radial profiles, and
characteristic radii across entire clouds (_Arzoumanian et al._ , 2011, 2019;
_Palmeirim et al._ , 2013; _Könyves et al._ , 2015). The filaments exhibited a
characteristic density profile with a nearly constant diameter of 0.1 pc
(_Arzoumanian et al._ , 2011). The strong influence of these initial Herschel
results merited a chapter in the previous Protostars & Planets VI (PPVI,
_André et al._ , 2014), becoming one of the most cited papers of this series.

After PPVI, a new generation of studies is transforming our description of
filaments. Continuum and line observations have now identified filaments over
an enormous range of scales and environments, from sub-parsec structures
within clouds (_Hacar et al._ , 2013) up to kpc-sized objects associated with
spiral arms (_Zucker et al._ , 2015), and from the densest high-mass star-
forming regions (_Treviño-Morales et al._ , 2019) to the atomic ISM (_Clark et
al._ , 2014). High-resolution ALMA observations have revealed complex networks
of filaments related to massive star formation (_Peretto et al._ , 2013;
_Hacar et al._ , 2018) and Infrared-Dark Clouds (IRDCs, _Henshaw et al._ ,
2014; _Barnes et al._ , 2021). Novel theoretical works explored different
formation mechanisms for filaments at different scales (_Hennebelle_ , 2013;
_Inoue and Fukui_ , 2013; _Inoue et al._ , 2018; _Chen et al._ , 2016;
_Duarte-Cabral and Dobbs_ , 2017; _Abe et al._ , 2021). In contrast to a
classical static description, modern simulations show how filaments
dynamically evolve and interact with their environments over time (e.g. _Smith
et al._ , 2014b). Zoom-in simulations illustrate filaments with a complex
substructure that greatly depart from idealized cylinders (e.g. _Li and Klein_
, 2019). The widespread detection of filaments in many types of simulations
and observations suggests that while the physical processes that underlie
filament formation must be anisotropic, they are not necessarily unique.

This chapter reviews the state-of-the-art observational and theoretical
research on the filamentary nature of the ISM by the time of the Protostars &
Planets VII conference in 2022. By synthesizing the substantial efforts of the
star formation community over the last 10 years, we aim to provide an updated
description of filaments at different scales and environments. This chapter
introduces a novel meta-analysis of multiple filament surveys presented in the
literature. Our goal is to articulate what is currently known, and to
illuminate the path forward for future studies of ISM filaments. We restrict
our discussion to filamentary structures with linear scales between $\sim$ 500
pc and $\sim$ 0.01 pc, a range of scales that excludes larger-scale structures
like Galactic spiral arms, and structures below the scale of protostellar core
collapse. Our aim is to describe the initial conditions for the formation of
individual stars and clusters in the Milky Way, setting the scene for further
discussions on the origin and evolution of stars and disks (see PPVII reviews
by Pineda et al and Pattle et al).

##  2 DEFINITIONS AND HYDROSTATIC EQUILIBRIUM

The total mass, $M$, and length, $L$, are two properties often derived from
observations of filaments. Both $M$ and $L$ values are typically obtained from
the analysis of maps of the column density, $N$, derived either from continuum
(given a gas-to-dust ratio) or molecular line (assuming a molecular abundance)
observations. Mass and length define a filament’s line mass $m$ (mass per unit
length), typically in units of M⊙ pc-1:

$m=\frac{M}{L}.$ (1)

The gas dynamics inside filaments are inferred from the direct measurement of
the line full-width-half-maximun ($\Delta v$) and line centroids
($v_{\mathrm{LSR}}$) in molecular line spectra. The total gas velocity
dispersion ($\sigma_{\mathrm{tot}}$) along the line-of-sight (LOS) can be
derived from the observed velocity dispersion of a given molecular tracer $i$
($\sigma_{\mathrm{obs,}i}=\Delta v_{i}/\sqrt{8\,\mathrm{ln}\,2}$) via the non-
thermal gas component

$\sigma_{\mathrm{nt}}^{2}=\sigma_{\mathrm{obs,}i}^{2}-\sigma_{\mathrm{th,}i}^{2}\,,$
(2)

with $\sigma_{\mathrm{th,}i}=\sqrt{k_{\mathrm{B}}T/\mu_{i}m_{\mathrm{p}}}$,
where $\mu_{i}$ is the molecular weight of the observed species (e.g. $\mu$
(13CO) = 29), $m_{\mathrm{p}}$ the proton mass, $k_{\mathrm{B}}$ the Boltzmann
constant and $T$ the (independently-determined) gas kinetic temperature. Using
the thermal velocity dispersion of the gas

$c_{\textrm{s}}=\sigma_{\mathrm{th,gas}}=\sqrt{\frac{k_{\mathrm{B}}T}{\mu
m_{\mathrm{p}}}}\,,$ (3)

with $\mu$ = 2.33 or 1.27 for molecular or atomic gas, respectively, and
combining it with $\sigma_{\mathrm{nt}}$ results in the total gas velocity
dispersion

$\sigma_{\mathrm{tot}}^{2}=\sigma_{\mathrm{nt}}^{2}+c_{\mathrm{s}}^{2}\,.$ (4)

Additional parameters describing the global gas motions, such as the velocity
gradient at scale $L$ ($\nabla v_{\mathrm{LSR}}=\Delta v_{\mathrm{LSR}}/\Delta
L$) and centroid dispersions ($\sigma(v_{\mathrm{LSR}})$), can be obtained
from the statistics of the line centroids across filaments. The comparison
between $\sigma_{\mathrm{nt}}$ (or $\sigma v_{\mathrm{LSR}}$) and
$c_{\mathrm{s}}$ indicates whether the non-thermal motions inside filaments
are subsonic ($\sigma_{\mathrm{nt}}<c_{\mathrm{s}}$), transonic
($c_{\mathrm{s}}\leq\sigma_{\mathrm{nt}}\lesssim 2c_{\mathrm{s}}$), or
supersonic ($\sigma_{\mathrm{nt}}>2c_{\mathrm{s}}$).

The critical line mass of a filament is derived from a hydrostatic, isothermal
cylinder model (_Stodólkiewicz_ , 1963; _Ostriker_ , 1964):

$m_{\mathrm{crit}}(T)=\frac{2c_{\mathrm{s}}^{2}}{G}\sim
16.6\left(\frac{T}{10\,\mathrm{K}}\right)\,\mathrm{M}_{\odot}\,\mathrm{pc}^{-1}\,.$
(5)

Filaments can be categorized in terms of gravitational stability, by
considering the ratio of the line mass to a critical line mass (_Ostriker_ ,
1964; _Fischera and Martin_ , 2012a),

$f\equiv\frac{m}{m_{\mathrm{crit}}}\,.$ (6)

Filaments with $f>1$ (supercritical) become radially unstable and must
collapse under their own gravity, while only those with $f<1$ (subcritical)
can remain in hydrostatic equilibrium. This critical line mass plays the same
role as the isothermal Jeans mass in early studies of molecular clouds (e.g.
_Klessen et al._ , 2000): it relies on a series of assumptions (hydrostatic
equilibrium, isolation, isothermality) that may or may not be applicable to
observed filaments. This is readily apparent from the fact that many filaments
have line masses well in excess of the critical value ($f\gg 1$). To include
the contribution from non-thermal motions to the filament’s stability, the
term $c_{\textrm{s}}$ can be replaced by the total velocity dispersion to
obtain the virial line mass

$m_{\mathrm{vir}}(\sigma_{\mathrm{tot}})=\frac{2\sigma_{\mathrm{tot}}^{2}}{G}\sim
465\left(\frac{\sigma_{\mathrm{tot}}}{1\,{\rm
km\,s}^{-1}}\right)^{2}\,\mathrm{M}_{\odot}\,\mathrm{pc}^{-1}\,.$ (7)

This effectively increases the line mass compared to Eq. 5.

Furthermore, the stabilising contribution of the magnetic field can –
approximately – be accounted for by adding the square of the Alfvén speed
$v_{\textrm{A}}=\frac{B}{\sqrt{4\pi\rho}}\,$, i.e. the magnetic pressure
supporting the filament against radial collapse as exerted by a field oriented
parallel to the filament to Eq. 4. If we instead assume that the magnetic
field is oriented perpendicular to the filament, then in the limit of very
strong field $m_{\textrm{vir}}\propto\phi_{\textrm{cl}}$ (_Tomisaka_ , 2014;
_Kashiwagi and Tomisaka_ , 2021), where $\phi_{\textrm{cl}}$ is the magnetic
flux through the filament per unit length. This results in less support
against gravity compared to a parallel field configuration (see also _Seifried
and Walch_ , 2015).

The column density profiles of filaments are often parameterized by Gaussian
functions (see §4.3) or by Plummer-like curves (_Nutter et al._ , 2008;
_Arzoumanian et al._ , 2011),

$N(x)=A_{p}\frac{n_{0}R_{\mathrm{flat}}}{(1+(x/R_{\mathrm{flat}})^{2})^{(p-1)/2}}\,,$
(8)

where $x$ is the projected distance from the filament spine, and $A_{p}$ is a
proportionality constant. The corresponding three-dimensional density
structure

$n(r)=\frac{n_{0}}{(1+(r/R_{\mathrm{flat}})^{2})^{p/2}}$ (9)

can be inferred as function of the central volume density $n_{0}$ and radius
$r$. The filament’s (inner) flat radius is related to its central column
density $N(x=0)=N_{0}$ as $R_{\mathrm{flat}}=N_{0}/(A_{p}n_{0})$. The full-
width-half-maximum FWHM of the column density distribution is

$FWHM=2R_{\mathrm{flat}}\left(2^{2/(p-1)}-1\right)^{1/2}\,.$ (10)

This definition assumes a filament in isolation, i.e. the half-maximum refers
to a background column density of $N=0$.

The isothermal Ostriker filament corresponds to $p=4$, $A_{p}=\pi/2$, and

$R^{2}_{\mathrm{flat}}=\frac{2c_{\mathrm{s}}^{2}}{\pi G\rho_{0}}\,,$ (11)

with $\rho_{0}=\mu m_{\textrm{p}}n_{0}$, which leads to
$N_{0}=\frac{\pi}{2}n_{0}R_{\mathrm{flat}}$ and $FWHM\approx
1.53R_{\mathrm{flat}}$. This solution can be generalized to include the
effects of turbulence, magnetic fields, and rotation (_Nakamura et al._ ,
1993; _Hanawa et al._ , 1993).

According to a linear stability analysis, hydrostatic filaments can fragment
under the influence of gravity if perturbations are larger than their critical
value

$\lambda_{\mathrm{crit}}=3.93\cdot R_{\mathrm{flat}}\,,$ (12)

showing a maximum growth rate for $\lambda_{\mathrm{max}}\sim
2\times\lambda_{\mathrm{crit}}$ (_Stodólkiewicz_ , 1963; _Nagasawa_ , 1987;
_Inutsuka and Miyama_ , 1992).

##  3 A NEW ERA OF GALACTIC SURVEYS

We present an overview of observational studies of filamentary structures on
various scales ($0.01$–$500$ pc) in the ISM. A classification into distinct,
well-separated groups of filaments is difficult or even impossible. For this
reason, we present results sorted by filament families (§3.1–3.8). These
categories are commonly presented in the literature and can sometimes depend
on the technique, resolution, and sensitivity of the observations used.
Consequently, filament families can partly overlap and individual filaments
might belong to several families. We illustrate some characteristic examples
of each of these families in Fig. 1.

Figure 1: Observations exemplifying each of the filament families identified
in section 3. Clockwise from top: (1) Hi filaments (§3.7) shown in a three-
color GALFA-Hi intensity map from _Clark_ (2018b), where red, green, and blue
represent the Hi brightness temperature in sequential velocity channel maps
with $\Delta v\sim 3\mathrm{km}\mathrm{s}^{-1}$. (2) Hub-filament structures
(§3.8) in OMC-1 as observed by ALMA (Credit: ESO/H. Drass/ALMA
(ESO/NAOJ/NRAO)/A. Hacar: http://eso.org/public/news/eso1809/ ; see also
_Hacar et al._ , 2018). (3) IRDC G11.11-0.12 from GLIMPSE (_Kainulainen et
al._ , 2013). (4) “Nessie”, the prototypical giant filament (§3.4) (_Jackson
et al._ , 2010; _Mattern et al._ , 2018b). (5) Herschel 250 $\mu$m maps show
nearby filaments in Taurus, with two levels of inset figures: a zoom-in to a
region containing striations (§3.6) and a region with dense fibers (§3.5).

###  3.1 Filaments in nearby molecular clouds

The filamentary, parsec-scale substructure of molecular clouds closer than
roughly 500 pc has been a subject of intensive study using the _Herschel_
satellite data (_Arzoumanian et al._ , 2011; _Palmeirim et al._ , 2013;
_Benedettini et al._ , 2015; _Arzoumanian et al._ , 2019; _Könyves et al._ ,
2020; _Pezzuto et al._ , 2021), and techniques from dust emission to
extinction and molecular line emission (_Malinen et al._ , 2012; _Hacar et
al._ , 2013; _Seo et al._ , 2015; _Kainulainen et al._ , 2016; _Arzoumanian et
al._ , 2018; _Orkisz et al._ , 2019; _Suri et al._ , 2019). These works employ
a variety of algorithms and techniques to identify filaments from the cloud
maps (see §3.10). Typical methods identify complex networks of structures,
comprising tens or even hundreds of individual, but often connected, (sub-)
parsec-scale filaments. Filaments defined with these methods make up 2-11% of
the area within the cloud maps (_Arzoumanian et al._ , 2019). However, the
filamentary network dominates the mass budget at high column densities
($A_{\mathrm{V}}>7$ mag) and harbors most of the star-forming cores in the
clouds (_Könyves et al._ , 2015, 2020). The filaments are analysed
individually, leading to a census of their properties. Overall, these works
have now built a detailed view of the parsec-scale filamentary networks within
the clouds at a resolution of roughly 10 000 AU (0.05 pc).

_Arzoumanian et al._ (2019) conducted a homogenous, systematic study of the
properties of filaments in dust-emission-derived column density maps of eight
nearby molecular clouds. This census focuses on the basic parameters of the
filaments and their distributions, i.e., line masses, lengths, and radial
profiles. Typically, these filaments span a range of line masses of 5-17 M⊙
pc-1 and lengths of 0.3-0.8 pc. They reach peak column densities of 3-9
$\times$ 1021 cm-2 and dust-based temperatures of 14-16 K. The radial density
profiles of the filaments have received ample attention because of their
connection to the basic physics of hydrostatic cylinders (see §4.3). Studies
based on dust emission maps from Herschel measure the distribution of inner
widths to have the mean value of about 0.1 pc with a typical spread of a
factor of 2 (_Arzoumanian et al._ , 2011, 2019). Similar values have been
measured in C18O filaments (_Orkisz et al._ , 2019; _Suri et al._ , 2019),
while observations of denser gas tracers indicate smaller widths at least for
a subset of structures (_Hacar et al._ , 2018, see also §3.5 and §4.3). The
robustness of these results and whether they indicate a characteristic scale
for filament widths continue to be debated (_Panopoulou et al._ , 2017;
_Ossenkopf-Okada and Stepanov_ , 2019; _Panopoulou et al._ , 2021, see also
§6.5). The radial profiles of the filaments tend to be well-described by
Plummer-like profiles with exponents $p$ around 1.5-2.5. Observations of
polarized dust emission reveal correlations between the orientation of the
magnetic field and the long axis of filaments: filaments at column densities
below $\sim\rm 10^{21}~{}cm^{-2}$ are preferentially parallel to the local
magnetic field, while at $N\gtrsim\rm 10^{22}~{}cm^{-2}$ they are oriented
preferentially perpendicular to the magnetic field (_Planck Collaboration
XXXV_ , 2016; _Jow et al._ , 2018, and §4.5).

###  3.2 Filaments from Galactic plane surveys

The most substantial Galactic plane filament surveys to date have been the
dust-emission based HiGal (_Molinari et al._ , 2010; _Schisano et al._ ,
2020), ATLASGAL surveys (_Schuller et al._ , 2009; _Li et al._ , 2016), and
the 13CO and C18O survey SEDIGISM (_Schuller et al._ , 2017; _Mattern et al._
, 2018a). These surveys covered large, continuous sections of the Galactic
plane, resulting in large samples that enable statistical, Galaxy-wide
studies. For example, about 18,400 individual filaments were characterized
using the HiGal survey (_Schisano et al._ , 2020). However, survey results may
be biased in ways difficult to quantify, due to limited resolution and
sensitivity; for example, typical resolutions of the surveys are roughly
15-30, translating to about 0.25-0.5 pc at 3.5 kpc distance. The mass
sensitivities are tracer- and distance-dependent, but are typically on the
order of a few 100 M⊙. Thus, filaments detected in Galactic plane surveys
likely include _some_ population of filaments similar to those detected within
nearby clouds (§3.1), but also larger-scale filaments, substructure of which
is not resolved at distances of several kpc.

The properties of the filaments detected in the Galactic plane surveys vary
with survey sensitivity. The most sensitive survey to date (HiGal, see
_Schisano et al._ , 2020) finds filaments with typical lengths of about 5-10
pc, masses of 500-1 000 M⊙, line masses of 20-200 M⊙ pc-1, and temperatures of
10-35 K. For comparison, the typical filaments from the ATLASGAL survey are
almost an order of magnitude more massive, reflecting the poorer sensitivity
of ATLASGAL. The majority of ATLASGAL survey emission ($\sim 70\%$, _Li et
al._ , 2016) is in filamentary structures. This is analogous to the result
from nearby clouds that most of the dense gas is in the filamentary network
(§3.1), however, a direct comparison is difficult because of the different
spatial scales the survey probes. So far, only ATLASGAL-identified filaments
have been systematically studied with molecular lines, with measured 13CO
velocity dispersion around 0.6-2 km s-1 (_Mattern et al._ , 2018a). A high
fraction of ATLASGAL filaments are detected in 13CO (2-1), but only a small
minority in C18O (2-1) (_Mattern et al._ , 2018a). Ongoing surveys in CO will
enable substantial improvements in coverage and statistics (_Su et al._ ,
2019).

###  3.3 IRDC filaments

Infrared Dark Clouds (IRDCs) commonly show filamentary morphologies, either
overall, or in parts. Their properties have been investigated mostly in case
studies (_Peretto et al._ , 2014; _Henshaw et al._ , 2014; _Busquet et al._ ,
2016; _Henshaw et al._ , 2016; _Williams et al._ , 2018; _Chen et al._ , 2019;
_Sokolov et al._ , 2019; _Kong et al._ , 2019; _Baug et al._ , 2020; _Li et
al._ , 2021b). The filamentary IRDCs are commonly identified from the
_Spitzer_ /GLIMPSE survey by eye as prominent absorption features. Due to
selection effects, the studied objects are typically located at distances of a
few kiloparsecs (_Rathborne et al._ , 2016; _Rygl et al._ , 2010), reach high
column densities ($N$ $\gtrsim 50\times 10^{21}$ cm-2) and high line masses
($m\gtrsim 100$ M⊙ pc-1), and have lengths between roughly one and a few tens
of parsecs. Detailed studies of the column density structure have uncovered
filamentary networks and hub-filament structures within them (_Busquet et al._
, 2016; _Ohashi et al._ , 2016; _Henshaw et al._ , 2017).

While IRDC filaments have been extensively studied in the continuum, studies
of their kinematics have been more sparse until recently (e.g. _Peretto et
al._ , 2014; _Henshaw et al._ , 2016; _Williams et al._ , 2018; _Chen et al._
, 2019; _Li et al._ , 2021b; _Arzoumanian et al._ , 2022). A homogeneous
analysis of the kinematics of these structures would greatly enhance our
ability to categorize these structures in terms of their
evolutionary/environmental stage. Even less well studied are the magnetic
field properties of IRDCs, with only a few clouds having dedicated
investigations (_Pillai et al._ , 2015; _Santos et al._ , 2016; _Busquet et
al._ , 2016; _Añez-López et al._ , 2020; _Wang et al._ , 2020a; _Tang et al._
, 2019; _Liu et al._ , 2018, see §6 for further details).

###  3.4 Giant Filaments

A recent focus in filament studies has been the search for the longest, most
massive filamentary structures in the Milky Way. The general driver of these
works is to understand the galaxy-scale distribution and organisation of dense
gas that is thought to correspond to kiloparsec-scale continuous structures
commonly seen in external galaxies (_Elmegreen et al._ , 2018). For brevity,
we refer to these structures as ’Giant Filaments’, noting that further sub-
categories have been proposed based on their physical properties (_Zucker et
al._ , 2018) and that their properties may depend on the galactocentric radius
(_Colombo et al._ , 2021). The first systematic works have identified tens of
Giant Filaments (_Ragan et al._ , 2014; _Abreu-Vicente et al._ , 2016; _Zucker
et al._ , 2015; _Wang et al._ , 2015, 2016; _Li et al._ , 2016; _Colombo et
al._ , 2021), varying from tens to hundreds of parsecs in length, the longest
potentially being up to 2 kpc long (_Veena et al._ , 2021). They cover a wide
range of masses up to some 106 M⊙ and their aspect ratios range from about 4
to 100. Typical line masses are around 1000 M⊙ pc-1. The studies use a wide
variety of approaches to identify structures based on dust extinction and
emission signatures and CO line emission. In the abscence of homogeneous
criteria, possible biases arising from different identification mechanisms are
difficult to quantify. Reflecting the variety, the structures are commonly
referred to with various names, e.g., ’Bones’, ’Galactic filaments’, ’Giant
Molecular Filaments’, or ’Large-scale filaments’. Most of these structures are
detected in the galactic plane surveys (at least partially), however, they are
usually broken down into smaller structures by the filament finding algorithms
of those studies. Generally, the Giant Filaments harbor rich sub-structure
consisting of a hierarchy of clumps and cores and smaller-scale filaments
(e.g., _Jackson et al._ , 2010; _Mattern et al._ , 2018b; _Wang et al._ ,
2020b).

_Zucker et al._ (2018) analysed the properties of filaments from different
surveys homogeneously, finding qualitative differences between filament sub-
samples. _Zucker et al._ (2018) identified three distinct categories within
the parameter space of dense gas mass fraction versus aspect ratio: high
aspect ratio giant molecular clouds, networks of dense compact sources, and
highly elongated, high column-density filaments (’the Bone candidates’). These
categories may have differing physical origin and they may be dominated by
different physical processes. Specifically, the high aspect ratio, high
column-density filaments were suggested to be the best candidates to trace the
spiral arms of the Milky Way. For a subset of their objects, _Zucker et al._
(2018) derived steeper radial profiles than what is obtained for nearby
filaments (§3.1).

_Zhang et al._ (2019) also presented a homogeneous analysis of the physical
properties of Giant Filaments, specifically focusing on scaling relations and
star formation activity. Giant filaments follow scaling relations similar to
Larson’s relations and dense gas vs. star formation rate relations similar to
those found in molecular clouds in general. Thus, their study promotes a view
that long filaments are not special in terms of their star formation activity,
but follow the same general scaling behavior as other clouds despite their
elongated morphology.

###  3.5 Dense fibers

The analysis of the gas kinematics in nearby clouds such as B213-L1495 in
Taurus, or the Integral Shape Filament in Orion revealed a new type of dense
molecular filaments at small scales, named dense fibers. First reported by
_Hacar et al._ (2013), fibers are distinguished as velocity-coherent
structures in Position-Position-Velocity space. They have (tran-)sonic
velocity dispersions ($\sigma_{\textrm{tot}}\leq 2\times c_{\textrm{s}}$) and
smooth oscillatory velocity profiles with local velocity variations ($\delta
v_{lsr}\lesssim$ $c_{\textrm{s}}$). Fibers are commonly observed in diffuse
(e.g. C18O) and dense (e.g. N2H+) molecular tracers, and rank among the
lowest-mass molecular filaments ($M$ $\sim$ 5-10 M⊙). They typically show sub-
parsec lengths ($L\lesssim$ 1pc), and have line masses close to critical
($m\sim m_{\textrm{crit}}$; see _Hacar et al._ , 2018, for a statistical
description). Fibers also exhibit high central gas densities $n_{0}>10^{4}$
cm-3, and small characteristic widths (FWHM=0.02-0.1 pc; see _Fernández-López
et al._ , 2014; _Hacar et al._ , 2018; _Monsch et al._ , 2018; _Schmiedeke et
al._ , 2021), making them narrower than most of the Herschel filaments
identified in dust continuum. The first polarization measurements within
fibers indicate random magnetic field orientations with respect to their main
axis (_Doi et al._ , 2020). The magnetic field orientation also changes
between neighbouring fibers and appears different from the magnetic field
detected at scales $>$ 1pc (_Arzoumanian et al._ , 2019; _Doi et al._ , 2020).

Fibers correspond to the fine substructure within larger and more massive
objects identified in the continuum (e.g. _André et al._ , 2014). When
observed at high enough resolution, an increasing number of works report a
rich fibrous substructure in low-mass clouds (_Arzoumanian et al._ , 2013;
_Fehér et al._ , 2016), intermediate mass clusters (_Fernández-López et al._ ,
2014; _Hacar et al._ , 2017a), IRDCs (_Henshaw et al._ , 2014; _Chen et al._ ,
2019; _Sokolov et al._ , 2019), and high-mass star-forming regions (_Hacar et
al._ , 2018; _Treviño-Morales et al._ , 2019; _Shimajiri et al._ , 2019; _Li
et al._ , 2021b; _Cao et al._ , 2021). The presence of an ensemble of of
(sub-)critical fibers ($\Sigma_{\textrm{fibers}}$) in filamentary regions such
as Taurus, Perseus, and Orion might explain the substructure of what are
apparently highly supercritical filaments ($m>m_{\textrm{crit}}$) at larger
scales (i.e. $m\propto\Sigma_{\textrm{fibers}}$; _Hacar et al._ , 2018).

Simulations with high levels of refinement (_Smith et al._ , 2014b; _Moeckel
and Burkert_ , 2015; _Kirk et al._ , 2015; _Li and Klein_ , 2019) show
filament bundles with a rich substructure of fibers. Some of the fibers
extracted in observations may not correspond to real gas structures but rather
to artefacts produced in the line intensity profiles of low-density tracers
(e.g. CO; _Clarke et al._ , 2017; _Zamora-Avilés et al._ , 2017). Nonetheless,
the excellent correspondence between most fibers and the column density
distribution within filaments (_André et al._ , 2014), as well as their
detection in high-density tracers (e.g. N2H+; _Hacar et al._ , 2018), suggests
that artifacts are neglibible in large samples.

Some fibers harbour individual or small groups of cores regularly spaced at
distances consistent with their corresponding Jeans length (_Tafalla and
Hacar_ , 2015). Combined with their (tran-)sonic velocity dispersions up to
parsec-scales, periodic velocity oscillations are found to relate the
positions of these cores with the streaming motions produced by gravitational
fragmentation along their main axis (_Hacar and Tafalla_ , 2011; _Hacar et
al._ , 2017a; _Heigl et al._ , 2018b). Fibers are proposed to be the first
sonic-like structures formed at the end of the turbulent cascade, which then
impart their sonic-like properties to the embedded cores (_Hacar and Tafalla_
, 2011).

###  3.6 Striations

The diffuse parts of several nearby molecular clouds feature elongated
structures termed ‘striations’ (see Herschel images of Taurus, Chamaeleon-
Musca, Polaris Flare, L1642 in _Kirk et al._ , 2013; _Cox et al._ , 2016;
_Miville-Deschênes et al._ , 2010; _Malinen et al._ , 2016, respectively).
Striations were first identified by _Goldsmith et al._ (2008) in a large
dynamic range image derived from CO observations of the Taurus molecular
cloud. Their distinctive characteristics are that, (a) they appear quasi-
periodically spaced and, (b) they are parallel to the magnetic field (_Chapman
et al._ , 2011; _Panopoulou et al._ , 2016; _Malinen et al._ , 2016).
Striations are typically found at column densities of $N(H_{2})\sim
10^{20}-10^{21}$ cm-2. While sometimes observed to be connected with dense,
star-forming filaments (_Palmeirim et al._ , 2013), striations are also found
in areas devoid of denser material (_Goldsmith et al._ , 2008; _Miville-
Deschênes et al._ , 2010).

Striations may be related to flows parallel to the magnetic field that channel
material onto denser filaments (_Palmeirim et al._ , 2013; _Cox et al._ ,
2016). _Heyer et al._ (2016) studied the velocity structure of striations and
found oscillatory behavior in cuts perpendicular to their axis. They proposed
that this might be a signature of Kelvin-Helmholtz instability or of
magnetohydrodynamical (MHD) waves.

In a dedicated numerical study, _Tritsis and Tassis_ (2016) quantified the
observational properties of molecular cloud striations in Taurus. While flows
along/perpendicular to the field or Kelvin-Helmholtz instabilities were unable
to reproduce the observed column density contrast ($\sim$ 25%), the
propagation of MHD waves was able to match the observed properties. The MHD
wave model for the formation of striations has made two predictions that have
so far been confirmed: the existence of normal modes in environments where the
waves are trapped (_Tritsis and Tassis_ , 2018) and the correspondence of the
velocity and column density power spectra which follow the dispersion relation
of MHD waves (found in HI data, _Tritsis et al._ , 2019). _Chen et al._ (2017)
proposed that corrugations of sheets can be caused by the thin shell
instability, yet, this mechanism over-predicts the column density contrast of
observed striations. Striation-like structures are found in numerous
simulations when considering strongly magnetized (trans- to sub-Alfvénic)
media (e.g _Beattie and Federrath_ , 2020, see also §5.1.3).

There is evidence that striations are not a feature of the molecular phase
only, but may also exist in the more diffuse atomic phase of the ISM (e.g.
_Tritsis et al._ , 2019). _Wareing et al._ (2016) performed simulations to
investigate the role of the thermal instability in forming filamentary
structures and found striations in their strongly magnetized models which
reproduced the column density contrast of molecular cloud striations, but
formed in the diffuse atomic medium.

Numerous works identify diffuse filaments parallel to magnetic field lines
(_Planck Collaboration XXXII_ , 2016; _Clark et al._ , 2014; _Inutsuka et al._
, 2015; _Li and Klein_ , 2019), however care must be taken to determine which
of these show the quasi-periodic spacing, low column density contrast and
oscillatory velocity profiles that are characteristic of striations. We note
that the term ‘striations’ has sometimes been used to refer generally to
diffuse filaments parallel to the magnetic field (e.g. _Busquet et al._ ,
2013; _Miettinen_ , 2020), however this choice could lead to confusion. More
work is needed to quantify the properties of striations in a larger sample of
clouds.

###  3.7 Hi Filaments

Sensitive, high-resolution observations of diffuse Hi emission reveal
ubiquitous filamentarity: the sky is patterned with slender, linear Hi
filaments, or Hi “fibers” (_Clark et al._ , 2014). Similarly, fine Hi
filaments are seen in absorption toward the Galactic Center (_McClure-
Griffiths et al._ , 2006). In both cases typical column densities are
$\lesssim 10^{20}\,\mathrm{cm}^{-2}$ (_Kalberla et al._ , 2016, see also). An
even more striking property of the Hi filaments is their excellent parallel
alignment with the local magnetic field orientation, as probed first by
starlight polarization (_McClure-Griffiths et al._ , 2006; _Clark et al._ ,
2014) and later by polarized thermal dust emission (_Clark et al._ , 2015;
_Martin et al._ , 2015; _Kalberla et al._ , 2016). The Hi filaments are well-
aligned with the measured plane-of-sky magnetic field orientation on average,
with some measurable misalignment that may be a useful probe of turbulence in
the nearby ISM (_Huffenberger et al._ , 2020; _Clark et al._ , 2021).

The Hi filaments have very high aspect ratios, particularly as seen in the
$4^{\prime}$-emission measured in the Galactic Arecibo L-Band Feed Array
Survey (GALFA-Hi; _Peek et al._ , 2018, see Fig. 1). However, their magnetic
alignment is still distinctly measurable in lower-angular resolution data like
Hi4PI (_HI4PI Collaboration et al._ , 2016), an all-sky Hi map made by
combining the $9^{\prime}$ EBHIS data (_Winkel et al._ , 2016) with the
$16^{\prime}$ GASS survey (_McClure-Griffiths et al._ , 2009). Polarization
maps “predicted” from Hi geometry via the assumption that Hi filaments are
perfectly aligned with the magnetic field bear a striking resemblance to real
measurements of the polarized dust emission (_Clark_ , 2018a; _Clark and
Hensley_ , 2019).

Because the Hi filaments studied by _Clark et al._ (2014) are particularly
prominent in narrow velocity channels, _Lazarian and Yuen_ (2018) argued that
these structures were “velocity caustics”: an imprint of the turbulent
velocity field uncorrelated with the underlying density field (e.g. _Lazarian
and Pogosyan_ , 2000). This interpretation has been ruled out by a number of
independent analyses. _Clark et al._ (2019) tested the velocity caustics
picture in the diffuse Hi by computing the correlation between velocity
channel emission structure and dust emission traced in broadband measurements
of the FIR, which is insensitive to the gas velocity field. This analysis
finds no measurable velocity caustic effect. Furthermore, _Clark et al._
(2019) find that the ratio of FIR/$N_{\textrm{HI}}$ increases toward
sightlines that contain small-scale structure in the Hi channel map emission.
Indeed the FIR/$N_{\textrm{HI}}$ ratio is positively correlated with the cold
neutral medium (CNM) fraction in the diffuse Hi (_Murray et al._ , 2020), and
with the intensity of small-scale structure (_Kalberla et al._ , 2020).
Moreover, _Peek and Clark_ (2019) showed that the equivalent width of Nai
absorption is more sensitive to the Hi column density in small-scale channel
map structure than to the total Hi column density. Together these results
support the interpretation that the Hi channel map filaments are real density
structures that occupy a colder, denser phase of the Hi gas than the warmer,
more diffuse medium.

Although a prominent feature of the high Galactic latitude sky, Hi filaments
are not confined to diffuse regions of the ISM. The Riegel-Crutcher cloud,
backlit by bright radio emission from the Galactic center, contains many
magnetically aligned Hi filaments that are detected in absorption (_McClure-
Griffiths et al._ , 2006). These filaments are reminiscent of the high
Galactic latitude Hi filaments detected in emission, with similar aspect
ratios and somewhat higher column densities (c.f. _Clark et al._ , 2014).
Surveys of the Galactic plane reveal ubiquitous filamentary structure in Hi,
including extremely massive structures that are apparently coherent both
spatially and spectrally. _Soler et al._ (2020) and _Syed et al._ (2022)
report a particularly extended Hi filament that, assuming the circular
rotation distance estimate of 17 kpc is accurate, would be over 1 kpc long. A
similar giant Hi filament (length 1.1 kpc) was recently reported by _Li et
al._ (2021a). These structures are qualitatively different than the diffuse Hi
filaments for which properties considered in this work (e.g. Table 1) are
derived.

###  3.8 Hubs, ridges, and networks

Filaments can also form complex associations. In the so-called hub-filament
structures (HFS), or simply hubs, multiple filaments extend radially up to
several parsecs away from a central parsec-size clump with column densities of
$>~{}10^{22}$ cm-2 and several hundreds of M⊙ in mass (_Myers_ , 2009a). Line
and continuum observations show converging filaments in a multitude of IRDCs
(e.g. _Peretto and Fuller_ , 2010; _Peretto et al._ , 2013; _Busquet et al._ ,
2013) and Galactic plane clumps (_Kumar et al._ , 2020). Elongated HFS,
sometimes referred to as ridges, have central FWHM up to $\sim$ 0.5 pc
(_Hennemann et al._ , 2012; _Russeil et al._ , 2013) and are among the most
massive filaments in the Galaxy. Observed line masses $m$ $\sim$ 300 M⊙ pc-1
significantly exceed the critical hydrostatic limit $f\gg 1$ (_Schneider et
al._ , 2010; _Hill et al._ , 2011). Nearby examples of these HFS are found in
regions such as NGC 1333 and OMC-1 (_Myers_ , 2009a), DR21 (_Schneider et al._
, 2010), or Mon-R2 (_Treviño-Morales et al._ , 2019).

Hubs and ridges are associated with the earliest phases of high-mass and
cluster formation (see _Motte et al._ , 2018, and references therein). All
nearby young stellar associations exceeding 25 stars pc-2 (_Myers_ , 2009a) as
well as a large fraction of the massive galactic pre- and protostellar clumps
(_Kumar et al._ , 2020) are found in these filamentary systems. Many HFS
appear to be highly dynamic objects. Longitudinal velocity gradients ($\nabla
v_{LSR}>$ 1 km s-1 pc-1) and filamentary accretion flows ($>$ 30 M⊙ Myr-1) are
observed along filaments feeding their central hub (_Kirk et al._ , 2013;
_Peretto et al._ , 2013, 2014; _Hacar et al._ , 2017b; _Baug et al._ , 2018;
_Williams et al._ , 2018; _Chen et al._ , 2019; _Saajasto et al._ , 2019;
_Dewangan et al._ , 2020; _Liu et al._ , 2021; _Ren et al._ , 2021). Evidence
for gas acceleration following free-fall velocity profiles (_Hacar et al._ ,
2017a; _Williams et al._ , 2018), spiral patterns (_Treviño-Morales et al._ ,
2019), and hour-glass shaped and close-to-critical magnetic fields (_Pattle et
al._ , 2017), denote gravity as the main driver of these inflow motions at
parsec-scales (see also §5.3.4).

At sub-parsec scales, densely populated fiber networks are also observed
within hubs such as OMC-1 (_Wiseman and Ho_ , 1998; _Hacar et al._ , 2018),
Serpens (_Fernández-López et al._ , 2014), NGC 6334 (_Arzoumanian et al._ ,
2021; _Li et al._ , 2021b) or DR21 (_Cao et al._ , 2021), among others. Fibers
in these systems show large longitudinal (_Hacar et al._ , 2018) and lateral
(_Dhabal et al._ , 2018; _Chen et al._ , 2020c) velocity gradients, as well as
a variable magnetic field morphology (_Arzoumanian et al._ , 2021; _Pattle et
al._ , 2021), sometimes exhibiting rapid changes with respect to the magnetic
field at cloud scales (_Doi et al._ , 2020). The complexity of these fiber
arrangements appears to increase with total mass of the host clump (_Hacar et
al._ , 2018). The most massive (supra-Jeans) cores in these regions are
typically located at the network nodes (local hubs) (_Hacar et al._ , 2017b;
_Zhang et al._ , 2019) suggesting that junctions, collisions, and
gravitational focusing effects between fibers may become important in dense
systems. Simulations indicate that cores formed in these nodes exhibit larger
masses than those formed during the fragmentation of individual fibers
(_Clarke et al._ , 2020; _Hoemann et al._ , 2021). Additional filamentary
accretion into these nodes may contribute to the rapid assembly of massive
stars and disks (_Banerjee et al._ , 2006; _Smith et al._ , 2011; _Kirk et
al._ , 2015; _Smith et al._ , 2016). High resolution observations of massive
clumps precursors reinforce this interpretation (_Beuther et al._ , 2020).

###  3.9 Other filaments

The study of filaments is necessarily restricted to data with which we can
resolve filaments. Naturally, filaments also exist in other galaxies, and
large-scale filaments have been identified in the Magellanic Clouds (_Fukui et
al._ , 2019; _Tokuda et al._ , 2019) and M100 (_Elmegreen et al._ , 2018).
Another frontier in the study of ISM filaments is the Galactic center, an
extreme environment relative to the rest of the Galactic disk (see PPVII
review by Henshaw et al). The low star formation rate in the Galactic center
relative to its high surface density makes it an intriguing case study for the
theory of star formation. G0.253+0.016 (“The Brick”) is a famous IRDC that has
an unusually high mass, but very little evidence of ongoing star formation
(_Kauffmann et al._ , 2013; _Pillai et al._ , 2015; _Henshaw et al._ , 2019).
For most of this work, we focus on filaments in environments more typical of
the Galactic disk, but we note that the Galactic center is an excellent
testbed for theories of filament formation and evolution (e.g. _Kruijssen et
al._ , 2019).

The ISM is home to a wealth of other “filaments” beyond those considered here.
These include the non-thermal radio filaments at the Galactic center (_Yusef-
Zadeh and Morris_ , 1987; _Heywood et al._ , 2022), radio polarimetric
filaments and depolarization canals (_Haverkorn et al._ , 2000), filaments in
the ionized medium traced in H$\alpha$ (_Planck Collaboration XXV_ , 2016),
polarization gradient filaments _Gaensler et al._ (2011); _Campbell et al._
(2021), and filamentary structures associated with ISM shocks or supernova
remnants (_McCullough and Benjamin_ , 2001; _Bracco et al._ , 2020), including
large angular scale structures associated with supershells.

###  3.10 Filament identification algorithms

New families of filaments have proliferated concomitantly with new filament
identification algorithms. The techniques in use range from specialized,
physically-motivated algorithms to general-purpose feature identification
codes. Each carries particular biases that inform the filament classification
and subsequent analysis. Each algorithm optimizes for the detection of a
certain class of structures: this is the de facto definition of a filament.

DisPerSe (_Sousbie_ , 2011) is a widely-used algorithm for identifying
filaments, especially in continuum data. Originally developed for quantifying
cosmic web structures, it was quickly applied to Herschel observations of
Galactic filaments (_Arzoumanian et al._ , 2011; _Peretto et al._ , 2012;
_Hill et al._ , 2011). DisPerSe’s provenance as a code for identifying
structures in the distribution of large-scale structures informs what it
identifies as “filaments”. The cosmic web is structured by gravitational
forces, and cosmic web filaments connect dark matter halos. This property
motivates DisPerSe’s definition of filaments as ascending 1-manifolds: one-
dimensional structures connecting overdensities in a Morse decomposition of
data. Thus, the application of DisPerSe to filament detection in ISM image
data carries the implicit definition of a filament as a linear structure
connecting image overdensities. For example, DisPerSe identifies filaments on
images consisting entirely of randomly placed cores (_Panopoulou et al._ ,
2014). The output of DisPerSe is one-dimensional: a filament “spine” along
which properties of the structure may be analyzed.

Other algorithms similarly define filaments by their skeletons via some
quantifiable aspect of image morphology. The code getfilaments and its
successor getsf (_Men’shchikov_ , 2013, 2021), which identify filaments via
the application of a series of Gaussian smoothing kernels, define filaments as
structures that are persistent over multiple scales and substantially
anisotropic (elongated along one axis). FilFinder employs a multi-step
preprocessing scheme before defining a filament spine via a medial axis
transform (_Koch and Rosolowsky_ , 2015). These are, in essence, matched
filter algorithms that detect a known signal in the presence of noise by
convolving a measurement (in this case an image of the ISM) with a filter
designed to maximize the signal-to-noise in the presence of the desired
signal. An important caveat is that the definition of a good skeleton has been
entirely subjective. However, recently _Green et al._ (2017) introduced a
measure of skeleton “goodness-of-fit”, the mean structural similarity index,
to remove visual bias in filament identification. _Jaffa et al._ (2018)
introduce the method of $J$-plots, which is based on the moment of inertia, to
characterise the morphology of (already identified) objects in 2D. It
differentiates single filaments, circular or ring-like structures and
condensed cores, at least for simple object geometries.

Identifying filaments in molecular line emission is an inherently three-
dimensional problem, and one that must contend with the ambiguities inherent
in inferring 3D spatial information from position-position-velocity (PPV) data
(e.g., _Beaumont et al._ , 2013). Friends In VElocity (five) approaches this
problem via a friends-of-friends search in PPV space, seeded by high signal-
to-noise spectral components (_Hacar et al._ , 2013, 2018). five then trawls
the PPV cube, defining velocity-coherent structures that meet some additional
criteria. Another approach is to construct 3D filaments from stacks of 2D
filaments identified in velocity channel maps. Fil3D is a recently-developed
algorithm that takes 2D channel map filaments (as delineated by FilFinder or a
similar algorithm) and identifies 3D filaments where such structures are
substantially overlapping and continuous in velocity space (Putman et al. in
prep). Fil3D was motivated by application to filaments in Hi channel maps (Kim
et al. in prep).

Other techniques focus on quantifying filamentarity in image data, rather than
identifying filaments as discrete objects. Many of these algorithms borrow
from edge detection methods in machine vision. As filaments can be understood
as sharp discontinuities in image space, they are highlighted by spatial
gradients (_Koch et al._ , 2013; _Soler et al._ , 2013; _Planck Collaboration
XXXV_ , 2016; _Orkisz et al._ , 2019). The direction of the maximum gradient
is a simple metric for the orientation of a filament, or any other “edge” in
the image plane. One can measure the second derivative of image intensity by
computing the Hessian matrix, which computes the local curvature of structures
(_Molinari et al._ , 2011; _Polychroni et al._ , 2013; _Planck Collaboration
XXXII_ , 2016). A filament can then be defined as an image feature with
negative curvature along one dimension. Each of these methods identifies
filaments at a preferential scale: image gradients and higher-order
derivatives are local operators, and thus high-pass filters. These methods are
most sensitive to filaments with widths that are similar to the scale of the
derivative kernel. Other methods effectively set a minimum spatial scale for
filamentary structure.

Some authors have studied the filamentary ISM by simply applying a high-pass
filter or unsharp mask to image data (_Kalberla et al._ , 2016; _Clark et al._
, 2019). This method does not explicitly parameterize filaments, but removes
smoothly varying emission at large angular scales. To the extent that the
emission studied is inherently filamentary, selecting the small-scale
structure via an unsharp mask has the effect of highlighting filaments. An
unsharp mask is the first step in the Rolling Hough Transform (RHT), which
parameterizes the linearity of image data as a function of orientation (_Clark
et al._ , 2014). The RHT quantifies the probability that any given image pixel
is part of a line in image space at any given orientation. Thus, while not
explicitly a filament finder, it can be used to identify linear features, or
to measure the orientation of filaments identified via other algorithms.

With such a variety of approaches to filament detection, cross-comparison
between various algorithms is a useful way to identify sources of systematic
discrepancy between derived filament properties. _Chira et al._ (2018) compare
DisPerSe, FilFinder, and minimal spanning tree algorithms (e.g., _Wang et al._
, 2016) applied to dendrograms (e.g., _Goodman et al._ , 2009). DisPerSe
returns consistently higher line masses than the other two methods.

One defining characteristic of many of the algorithms described here is that
they are maximally sensitive to filaments at a particular angular scale. Some
algorithms have instead adopted explicit descriptions of multi-scale structure
(e.g. _Ossenkopf-Okada and Stepanov_ , 2019; _Robitaille et al._ , 2019;
_Robitaille et al._ , 2020; _Allys et al._ , 2019). Such treatments describe
structure as a function of angular scale, and are thus capable of quantifying
hierarchical morphology.

##  4 A CENSUS OF MILKY WAY FILAMENTS

Table 1: FILAMENT FAMILIES (I): FUNDAMENTAL PROPERTIES(1)

Family(#) | Targets | $M$ | $L$ | FWHM | $N_{0}$ | $T$ |
---|---|---|---|---|---|---|---
|  |  (M⊙) | (pc) | (pc) |  ( 1021 cm-2) | (K) |
Nearby Filaments | 707 | 2.1 - 13.4 | 0.4 - 1.0 | 0.07 - 0.16 | 3.2 - 9.0 | 13 - 16 |
Galactic Survey | 18854 | 98 - 2.3E3 | 3.6 - 14 | 0.3 - 1.3(?) | 5.9 - 14.4 | 15 - 19 |
IRDCs | 115 | 46 - 239 | 1.2 - 2.8 |  $\sim$ 0.3(?) | 13 - 40 | 16 - 25 |
Giant Filaments | 153 | 7.8E3-1.5E5 | 32 - 83 |  $\sim$ 1(?) | 6.0 - 8.4 | 15 - 19 |
Dense fibers | 127 | 1.5 - 15.0 | 0.2 - 0.5 | 0.03 - 0.10 | 12 - 69 | 10 - 22 |
Striations |  2(∗) | 2.5 - 5.6 | 1.0 | 0.2 - 0.4(?) | 0.5 - 1.0 |  15 - 20(2) |
HI filaments | 2823 | 0.1 - 0.5 | 1.4 - 3.6 | ? |  $<1$(?) |  100(3) |

(1)(1)footnotetext: Values correspond to the interquartile [Q25%,Q75%] range
for each measured quantity. (2) Typical molecular gas temperatures at low
column densities (e.g. _Planck Collaboration et al._ , 2011). (3) Typical
temperature for the CNM. (?) Unknown or poorly constrained. (*) Limited
statistics.

(#)(#)footnotetext: References: Nearby Filaments: _Arzoumanian et al._ (2013,
2019); _Palmeirim et al._ (2013); _Nagahama et al._ (1998); _Kainulainen et
al._ (2016); _Hacar et al._ (2016a); _Chung et al._ (2021). Galactic Plane
Surveys: _Schisano et al._ (2020); _Li et al._ (2016); _Wang et al._ (2016);
_Mattern et al._ (2018a); _Xiong et al._ (2019). IRDCs: _Schneider et al._
(2010); _Hennemann et al._ (2012); _Kainulainen and Tan_ (2013); _Peretto et
al._ (2014); _Lu et al._ (2014); _Beuther et al._ (2015); _Pillai et al._
(2015); _Chen et al._ (2019); _Busquet et al._ (2016); _Santos et al._ (2016);
_Treviño-Morales et al._ (2019); _Leurini et al._ (2019); _Arzoumanian et al._
(2021, 2022). Giant Filaments: _Contreras et al._ (2013); _Ragan et al._
(2014); _Zucker et al._ (2018); _Zhang et al._ (2019); _Colombo et al._
(2021). Dense fibers: _Hacar and Tafalla_ (2011); _Hacar et al._ (2013);
_Hacar et al._ (2016b, 2017b); _Hacar et al._ (2018); _Lee et al._ (2014);
_Tafalla and Hacar_ (2015); _Seo et al._ (2015); _Dhabal et al._ (2019);
_Eswaraiah et al._ (2021); _Schmiedeke et al._ (2021); _Dhabal et al._ (2018);
_Li et al._ (2021b). Striations: _Chapman et al._ (2011); _Panopoulou et al._
(2016). HI Filaments: _Clark et al._ (2014), Putman et al. in prep.

The increasing number of observational results describing the filamentary
properties of the ISM provide a panchromatic view of these gas structures
across our Galaxy. As illustrated in §3, however, the physical characteristics
of these filaments may significantly differ depending on the family,
observational technique, or scales considered in each case. For the first
time, the maturity of this field allows a direct analysis of filament
populations in the ISM beyond studies of individual targets or regions.

We have created a census of filaments across the Milky Way to perform a meta-
analysis of their physical characteristics. We restrict ourselves to the main
physical properties of these filaments typically reported in the literature,
namely (and when available), their total mass ($M$), length ($L$), central
column density ($N_{0}$), width (FWHM), gas kinetic temperature ($T$), total
($\sigma_{\textrm{tot}}$) and non-thermal ($\sigma_{\textrm{nt}}$) velocity
dispersion, internal gradients ($\nabla v_{\textrm{LSR}}$), and magnetic field
strength ($B_{0}$). The assembly of this catalog requires a homogenization of
distinct measurements, including standard conversions into total gas masses
and velocity dispersion from different molecular datasets. Our catalog
includes information of 49 individual observational works, including
continuum, molecular line, and polarization surveys, together with dedicated
works on specific targets. In multiple occasions we combine several
independent but complementary studies describing distinct properties in well
investigated targets (e.g. B213-L1495). Our catalog includes a total of 22803
filaments (entries) across the Galaxy. The mean statistical properties
classified by families, including all references, are listed in Tables 1 &
4.5.

Several caveats should be considered during the analysis of our filament
sample. By construction, our sample is limited by the targets and surveys
included in individual studies in the literature. As a result, the number of
targets per filament family significantly varies across our sample, from the
handful of striations characterized in the literature to the thousands of
filaments in nearby clouds and Galactic plane surveys. Our filament catalog is
also largely inhomogeneous including multiple entries with only partial
measurements. Similarly, some entries may not be unique as the same targets
may be included in multiple catalogs. Due to the availability of these
observations, our sample is dominated in number (83%) by the mass and length
estimates provided by large continuum surveys with a smaller fraction ($<$5%)
of targets including kinematic measurements and magnetic field estimates.

Not all quantities might be equally accurately determined. Continuum and line
observations provide robust measurements of $M$, $L$, and $N_{\textrm{0}}$ of
filaments. In contrast, parameters such as the FWHM are indirectly inferred
from modeling or assuming some filament shape (e.g. uniform cylinder).
Distance uncertainties are not always quoted in the original catalogs, and are
therefore unaccounted for in the majority of the data. Gas kinetic
temperatures are sometimes equated to the dust temperatures as these are
easier to obtain in FIR observations (i.e. $T\approx T_{\textrm{dust}}$).
Other parameters, such as $\sigma_{\textrm{tot}}$ or $\sigma_{\textrm{nt}}$,
might depend on the molecular tracer used. Magnetic fields are inferred via
the Davis-Chandrasekhar-Fermi method (§4.5). While standard practice in the
field, this inhomogeneity in determining parameters should be considered
during the interpretation of our results.

Figure 2: Mass and length measurements for all filaments in our catalog (§4).
Points are colour-coded according to the corresponding filament families
described in §3: Nearby Filaments, Galactic Plane Surveys, IRDCs, Giant
Filaments, Dense Fibers, Striations, and Hi Filaments (see legend). In the
case of Galactic Plane Surveys a subcategory identifies those detected by
ground-based observations (GB). Some prototypical filaments are also included
in this plot (see symbols in legends). Black dotted lines indicate constant
$m$ = 1, 10, 100, 1000 M⊙ pc-1 values, the black dashed line corresponds to
$m_{\textrm{crit}}$(10 K) = 16.4 M⊙ pc-1. The expected maximum filament size
set by the Toomre length for the Galactic midplane at the Sun’s radius is
indicated by a blue dashed line. The red solid line shows $m_{\textrm{vir}}$
(Eq. 21) including turbulent motions. The red dashed line corresponds to the
column density $N_{\textrm{shield}}$ required for self-shielding (Eq. 22).

This meta-analysis provides a panchromatic view of the filamentary properties
at different scales and environments. The compilation reflects not only the
current state-of-the-art but also the large community effort and achievements
describing the filamentary nature of the ISM combining different surveys,
observational techniques, and tracers. We will use this sample to describe the
statistical properties and global trends within and between filament families
(§4.1 \- 4.6). We describe the dynamic properties of filaments (§5) to
interpret the observed correlations (§6) and we identify future avenues for
the study of filaments in the ISM (§7).

###  4.1 Filaments across scales: tracers of ISM physics

The most comprehensive overview of different filaments reported in the
literature can be obtained from the description of their mass and length shown
in Fig. 2. To facilitate their comparisons, different filament families and
prototypes (see §3) are marked in this plot.

Observations report filaments across almost eight orders of magnitude in mass,
from $\sim$ 0.01 to 5$\times$106 M⊙, and four orders of magnitude in length,
from $\sim$ 0.03 to 300 pc. The molecular filaments describe a continuous
distribution in mass and length, including all the nearby filaments and fibers
and extending towards the longest Galactic Plane and Giant Filaments, with an
approximate scaling relation $L\propto M^{0.5}$ (§6.1). Only the HI fibers
clearly depart from this general trend.

The large range in mass and length indicates that filaments possess different
stability and dynamical properties depending on the scale. Most of the
filaments detected at scales $L$ $\gtrsim$ 10 pc show very high $m$ (up to
$\geq 100$ M⊙ pc-1) and thus largely exceed the expected $m_{\textrm{crit}}$
for a hydrostatic filament (Eq. 5). This indicates that these objects are
dynamically evolving. Many of the shorter filaments are closer to, but still
slightly above, $m_{\textrm{crit}}$ with line masses of 15-30 M⊙ pc-1. On the
opposite side, a non negligible fraction of (sub-)parsec scale filaments show
sub-critical $m$ suggesting them to be either evolving or transient
structures. It is thus clear that filaments identified in the literature do
not form a unique or homogeneous population. Instead, different filaments
probe distinct ISM properties depending on the scale, from large cloud
complexes (i.e. Giant Filaments) to small substructures within clouds (e.g.
filaments and fibers in nearby clouds).

Filament prototypes require a special mention. Targets such as B213-L1495,
Musca, Nessie, or DR21 may define the typical (sometimes extreme) properties
of other filaments within the same mass and length ranges. However, it is
unlikely that objects with dissimilar properties are governed by the same
physical processes (see also §5). Giant Filaments such as Nessie are likely
affected by processes at galactic scales (e.g. shear or spiral arms) that have
a small influence in the evolution of small scale filaments within clouds. On
the other hand, it is expected that gravity plays a more dominant role in
massive filaments such as DR21 than in Hi filaments, despite having comparable
sizes (see also the discussion on formation mechanisms in §5.1). The
extrapolation of these prototypes for the interpretation of filaments at
different scales should be avoided or at least treated with extreme caution.

Several observational biases can be identified during the classification and
study of different filaments (see mean values in Tables 1 & 4.5). First,
distance appears as one of the primary discriminators between families. Using
the same observational techniques (e.g. Herschel), Galactic Plane surveys
identify cloud-size filamentary structures much larger in size ($L~{}>$ 10 pc)
and mass ($M~{}>$ 100 M⊙) (e.g. _Schisano et al._ , 2020) than the
corresponding (sub-) parsec long ($L\sim$ 0.5-5 pc) and low-mass ($M\sim$ 1-10
M⊙) filaments reported in nearby studies (e.g. _Arzoumanian et al._ , 2019).
The latter describe the small-scale filaments within molecular clouds likely
unresolved when observed at kpc distances across the Galactic Plane.
Similarly, observational criteria artificially bias the study of smaller
filaments in nearby clouds. For example, limiting the filament length to $L>3$
FWHM in combination with unresolved widths may suppress shorter filaments. The
combination of these effects directly translate into systematic differences in
the corresponding line mass derived in these studies.

Second, sensitivity limits ground-based Galactic surveys to the detection of
filaments with extreme $m$ values ($>$ 500 M⊙ pc-1) (e.g. _Li et al._ , 2016;
_Zucker et al._ , 2019) within the general population of Galactic Plane
targets (see _Schisano et al._ , 2020, for a discussion). Similarly, the use
of molecular tracers might restrict the detection of filaments to the densest
targets at a given scale (e.g. fibers, _Hacar et al._ , 2018).

Third, observational techniques may create artificial distinctions in the
otherwise continuous distribution of filaments across scales. For example,
filamentary IRDCs are observationally identified by their high extinction in
the mid-IR (e.g. _Peretto and Fuller_ , 2010), while their mass and length
cannot be distinguished from an average filament in the Galactic plane.

The observed range of filament lengths is intimately connected to the
fragmentation properties and characteristic timescales of the ISM. The maximum
length of a filament in our Galaxy is limited to $\sim$ 200 pc by the Toomre
instability for a typical midplane gas surface density of 10 M⊙ pc-2 (e.g. _Li
et al._ , 2017). At the opposite end, star formation occurs within parsec-size
filaments at densities $>10^{3}$ cm-3 where gravity dominates their
fragmentation forming quasi-spherical cores (see §5.4). These scale-dependent
properties may change with the Galactocentric distance (with different Toomre
values), elevation with respect to the midplane (e.g. Hi filaments), local
environment (e.g. CMZ), and local gas density (e.g. high-mass clouds).
Functioning as gas compasses, the detailed characterization of the filamentary
properties of the ISM can thus provide unique insights of its most fundamental
physical properties.

###  4.2 Thermal Physics of Filaments

Figure 3: Thermal pressure as function of the filament density. The thermal
equilibrium curve corresponds to a shielding column of $N=10^{21}$ cm-2 and an
interstellar radiation field strength of $G_{0}=1$ (_Wolfire et al._ , 1995).
Symbols are similar to Fig. 2. In the case of HI the temperature and density
are assumed similar to the CNM. The horizontal line indicates the typical
Galactic midplane pressure of $\sim 2\times 10^{4}$ K cm-3 (_Cox_ , 2005).

Filaments cover not only a large parameter space in mass and length (Fig. 2),
but also in thermal pressure (Fig. 3), suggesting that they exist in a wide
variety of environments and evolutionary stages (_Rivera-Ingraham et al._ ,
2017). The thermal pressure of HI filaments is only about $10$% of the
Galactic midplane pressure ($2\times 10^{4}$ K cm-3, _Cox_ , 2005). Magnetic
fields aligned with the HI filaments (_McClure-Griffiths et al._ , 2006;
_Clark et al._ , 2014) could provide the “missing” pressure component. Thermal
pressures substantially above the Galactic midplane pressure may be set by the
filament’s self-gravity, by pressurization due to the “weight” of the ambient
cloud (_Lada et al._ , 2008), or by accretion (§5.2.2). Sustained flow
compressions could also lead to overpressures, with the thermal pressure
balancing the ram pressure of the inflowing gas (_Vázquez-Semadeni et al._ ,
2006). All filaments are close to thermal equilibrium curves (_Wolfire et al._
, 1995; _Koyama and Inutsuka_ , 2002), suggesting that radiative cooling
timescales are short compared to dynamical timescales (_Audit and Hennebelle_
, 2005).

Average filament temperatures vary only slightly between filaments (Tab. 1,
see also Fig. 3). Temperature profiles inferred from dust temperatures
(_Arzoumanian et al._ , 2011; _Palmeirim et al._ , 2013; _Arzoumanian et al._
, 2019) indicate a temperature drop of $\Delta T\sim 2-3$ K toward a
filament’s spine. For the Integral Shape Filament (ISF) in Orion A, _Schuller
et al._ (2021) find temperature changes of $\Delta T\sim 20$ K corresponding
to $100$% in terms of the central temperature when compared to its
environment. Along a filament gas and dust temperatures seem to vary mostly
only within a few Kelvin (_Sokolov et al._ , 2017). Simulations of individual
filaments, including chemistry and radiation attenuation, (_Seifried and
Walch_ , 2016; _Seifried et al._ , 2017a) are consistent with observed
temperature profiles, though some studies show strong temperature increases
toward the center (_Anathpindika and Francesco_ , 2021). Observed and modeled
filament profiles do not support the assumption of isothermality, with
temperatures increasing by at least $\approx 25$% outwards. Such temperature
profiles result in an effective adiabatic exponent of
$\gamma_{\textrm{eff}}<1$, consistent with thermal equilibrium models (_Koyama
and Inutsuka_ , 2002). The effective equation of state
$P\propto\rho^{\gamma_{\textrm{eff}}}$ is relevant for the radial stability of
a filament (_Toci and Galli_ , 2015). Filaments with $\gamma_{\textrm{eff}}>1$
are unconditionally stable. Isothermal (Ostriker) filaments have a critical
point at infinite overdensity, and filaments with $\gamma_{\textrm{eff}}<1$
become critical at finite overdensities. _Toci and Galli_ (2015) state that
observed temperature ranges are smaller than those predicted by polytropic
models and thus motivate non-thermal support, yet this conclusion seems to
depend on the specific value of $\gamma_{\textrm{eff}}$ and on the assumed
filament width.

###  4.3 Radial profiles

The radial density profiles of filaments have received broad attention (e.g.
_Arzoumanian et al._ , 2011) due to the connection to the basic physics in
hydrostatic cylinders directly inferred from three main observables: the
central column density ($N_{0}$), the radial dependence (via the power-law
index $p$), and the full-width half maximum (FWHM) of the column density
profile of filaments (see definitions in §2).

Filaments cover more than three orders of magnitude in column density, from
the atomic and diffuse Hi filaments ($N_{0}<10^{20}$ cm-2) to the densest
filamentary IRDCs and fibers ($N_{0}>5\times 10^{22}$ cm-2) (see Table 1).
Resolved nearby filaments typically show higher $N_{0}$ values ($\sim
6-7\times 10^{21}$ cm-2; _Arzoumanian et al._ , 2011, 2019) than those
detected in Galactic Plane surveys ($\sim 2-3\times 10^{21}$ cm-2; _Schisano
et al._ , 2014). Positive correlations are found between $N_{0}$ and the star-
forming properties of resolved filaments (e.g. star-forming clumps and lines
mass $m$), as well as between $N_{0}$ and the filament column density
background $N_{\textrm{bg}}$ (_Schisano et al._ , 2014; _Arzoumanian et al._ ,
2019).

The radial dependence of the column density profile provides insights about
the stability, equation of state, and pressure confinement of filaments
(§5.2.1). An isothermal hydrostatic cylinder has a radial density profile of
$n\propto r^{-p}$ with $p$ = 4 (_Stodólkiewicz_ , 1963; _Ostriker_ , 1964),
although this dependence is rarely observed in filaments (_Pineda et al._ ,
2010; _Hacar and Tafalla_ , 2011; _Fischera and Martin_ , 2012b; _Monsch et
al._ , 2018; _Zucker et al._ , 2018). Instead, most filaments tend to be well-
described with Plummer-like profiles with exponents around $p\sim$ 1.5-2.5
(_Arzoumanian et al._ , 2011; _Palmeirim et al._ , 2013) which can be
interpreted as a departure from equilibrium (e.g. _Kawachi and Hanawa_ ,
1998).

Shallow ($p<$ 4) profiles are expected for magnetized (_Fiege and Pudritz_ ,
2000; _Tomisaka_ , 2014; _Kashiwagi and Tomisaka_ , 2021; _Kirk et al._ ,
2015) and externally pressurized filaments (_Fischera and Martin_ , 2012a).
Non-isothermal (_Recchi et al._ , 2013; _Smith et al._ , 2014b) or polytropic
(_Gehman et al._ , 1996a) equations of state as well as rotation (_Recchi et
al._ , 2014) will also flatten the profile. The observational distinction
between these scenarios is however hampered by the limited column density
contrast (i.e. $N_{0}$/$N_{\textrm{bg}}$, see above) in most resolved
filaments (see _Arzoumanian et al._ , 2019), and measured values of $p$ may
vary with location in a filament (_Fischera and Martin_ , 2012b).

Figure 4: Mass-length (left panel) and density-length correlations (right
panel) observed at different scales in Orion A (cyan), B213-L1495 (red), and
Musca (green) in comparison to the entire filament population (grey; see also
Fig. 2). Different symbols (see legend) denote the nested structure of these
filamentary regions, from the largest to smallest being filamentary clouds
(squares), parsec-size filaments (crosses), bundles (triangles), sub-filaments
(pentagons), and fibers/sub-filaments (circles). The overall filament
distribution follow the relations $L\propto M^{0.5}$ and $n\propto L^{-1.1}$
(black solid lines). The arrows in the left panel indicate the change in
position under various processes. References & tracers: All: see Tables 1;
Orion A – cloud: FIR continuum (_Lombardi et al._ , 2014; _Großschedl et al._
, 2018), filaments: 13CO (_Nagahama et al._ , 1998), ISF fibers: N2H+ (_Hacar
et al._ , 2018); B213-L1495 – filament: C18O & FIR continuum (_Hacar et al._ ,
2013; _Palmeirim et al._ , 2013), bundles & fibers: C18O (_Hacar et al._ ,
2013), sub-filaments: FIR continuum (_Arzoumanian et al._ , 2019); Musca –
cloud: extinction (_Kainulainen et al._ , 2016), sub-filaments: FIR continuum
(_Arzoumanian et al._ , 2019); DR21 – FIR continuum (_Schneider et al._ ,
2010); NGC1333 – FIR continuum (_Hacar et al._ , 2013); Mon-R2 – FIR continuum
(_Treviño-Morales et al._ , 2019).

The dust-based Herschel studies in local clouds suggest that the distribution
of inner widths has a mean universal value of about FWHM $\simeq$ 0.1 pc with
a typical spread of a factor of 2 (_Arzoumanian et al._ , 2011, 2019;
_Palmeirim et al._ , 2013; _André et al._ , 2014; _Schuller et al._ , 2021).
Similar mean values, although showing larger dispersion, are also found in
molecular observations of resolved filaments using tracers such as C18O
(_Orkisz et al._ , 2019; _Suri et al._ , 2019). The measured filament widths
may, however, depend on the filament scale and density (§6.5). Larger widths
of FWHM $\sim$ 0.26-0.34 pc are reported in more distant ridges (_Hennemann et
al._ , 2012) and Galactic Plane filaments (_Schisano et al._ , 2014) using
Herschel, and in nearby filaments observed in 13CO (_Panopoulou et al._ ,
2014). At the opposite end, narrower widths of FWHM $\lesssim$ 0.05 pc are
measured in denser sub-filaments and fibers using interferometric observations
of dense molecular tracers (_Fernández-López et al._ , 2014; _Hacar et al._ ,
2018; _Dhabal et al._ , 2018; _Monsch et al._ , 2018). The inverse
FWHM-$N_{0}$ relation predicted for pressurized filaments (_Fischera and
Martin_ , 2012a; _Heitsch_ , 2013a) has not been observed yet (_Arzoumanian et
al._ , 2019; _Suri et al._ , 2019). Resolution (_Schisano et al._ , 2014) and
confusion (_Suri et al._ , 2019) may also affect the resulting FWHM
measurements. The interpretation and robustness of a possible characteristic
scale is thus a matter of on-going debate (e.g. _Panopoulou et al._ , 2017;
_Ossenkopf-Okada and Stepanov_ , 2019; _Arzoumanian et al._ , 2019) and
deserves a careful re-evaluation of both physical and observational arguments
(see also § 6.5).

###  4.4 Filaments as hierarchical structures

The self-similar nature of molecular clouds (e.g. _Falgarone et al._ , 1991)
is imprinted in its internal filamentarity. Filaments at different scales are
identified within the same region depending on resolution and density
thresholds (e.g. _Schneider et al._ , 2011). Orion A may serve as an example
(Fig. 4). At large-scales, Orion A appears as an elongated $\sim$ 90 pc-long
cloud (_Großschedl et al._ , 2018) of more than 105 M⊙ (e.g. _Lombardi et al._
, 2014; _Stutz and Kainulainen_ , 2015), comparable to some of the largest
Giant Filaments detected in Galactic Plane surveys. At intermediate scales,
this cloud resolves into dozens of parsec-size filaments with masses between
$\sim$ 102 and 104 M⊙ detected by single-dish observations of diffuse-gas
tracers (e.g. _Nagahama et al._ , 1998), including the famous Integral Shape
Filament (_Bally et al._ , 1987; _Johnstone and Bally_ , 1999). Denser
molecular tracers show most of these filaments are heavily structured into
sub-parsec-like sub-filaments with a few M⊙ in mass at interferometric
resolutions (_Wiseman and Ho_ , 1998; _Hacar et al._ , 2018; _Monsch et al._ ,
2018; _Suri et al._ , 2019).

As a result, the choice of scale may result in diverging interpretations of
the same region. In Fig. 4 (left panel) we display the position in the $M-L$
plane of different filamentary structures reported in Orion A (cyan),
B213-L1495 (red), and Musca (green) using a variety of observational
techniques and resolutions (see references in the figure caption). When
considered at parsec-scales, filamentary clouds (squares) and large filaments
(plus symbols) systematically exhibit line masses $m\sim 50-300$ M⊙ pc-1
suggesting them to be super-critical ($f\gg 1$) and therefore gravitationally
unstable. Smaller bundles of fibers (triangles), sub-filaments (pentagons),
and individual fibers (circles) within the same region show line masses closer
to mcrit ($f\sim 1$) and appear to remain radially stable. This observed
hierarchy allows us to connect the near and far observational results (see
§4.1). According to the left panel of Fig. 4, the massive filaments reported
in Galactic Plane surveys are likely resolved out into smaller (trans-)
critical filaments similar to those detected in nearby clouds once observed at
higher resolutions (see also _Schisano et al._ , 2014), questioning the
interpretation of their stability in terms of their global line mass. Examples
of this behaviour are already present in the literature (_Henshaw et al._ ,
2014; _Chen et al._ , 2019; _Shimajiri et al._ , 2019, among others).

Filaments at different scales effectively sample distinct gas densities in the
ISM. Fig. 4 (right panel) compares the filament lengths with their effective
gas densities, where the latter is calculated as $\langle n\rangle$ =
$m/(\pi(\frac{FWHM}{2})^{2})$ in those cases where estimates of the filament
FWHM were available. Overall, the mean gas density in molecular filaments
increases with decreasing length. Filaments identified at parsec scales have
average densities of $\sim 10^{3}-10^{4.5}$ cm-3 compared to the $\gtrsim
10^{5}$ cm-3 seen in sub-parsec filaments and fibers. The relation between
filament density and length for molecular filaments (black solid line, right
panel in Fig. 4) is given by a least-square fit

$\langle n\rangle\simeq(1.2\pm 0.3)\times 10^{4}\left(\frac{L}{{\rm
pc}}\right)^{-1.1\pm 0.1}\,{\rm cm}^{-3}$ (13)

despite a relatively large ($\sim 1$ dex) dispersion due to the choice of
different tracers and mass thresholds.

The observed filament $M-L$ distribution shows a sub-linear relation

$L\propto M^{0.5\pm 0.2}$ (14)

(black, solid line, left panel). Qualitatively speaking this dependence is
expected if large-scale filaments are actually networks of smaller structures
(see §6.1 for details). Indeed, nearby filaments are characterized by some
authors as multi-fractal structures embedded within the mono-fractal cloud gas
(_Ossenkopf-Okada and Stepanov_ , 2019; _Robitaille et al._ , 2019;
_Robitaille et al._ , 2020; _Yahia et al._ , 2021). Beyond their morphological
identification, this nested structure of filaments within filaments should be
considered for the interpretation of both observations and simulations.

In general, this hierarchical structure could arise from both top-down
fragmentation, as in _Hacar et al._ (2013), and from a bottom-up process
whereby gravitational collapse assembles larger filaments from gas containing
pre-existing filamentary structure, as in _Smith et al._ (2014b). In reality
both large and small filaments may be formed simultaneously. Hierarchical
structure formation would move the distribution diagonally downwards to the
left of Fig. 4 (left panel; see arrows) as opposed to other mechanisms such as
longitudinal collapse (moving down), ablation (moving left) or
stretching/shear (moving to the top left).

While individual clouds tend to follow a relatively tight $M-L$ relation (see
color-coded symbols for each cloud), the normalization of the individual
relations varies between clouds. The normalization is set by the line mass of
the largest hierarchical structure, and it may be determined by the specific
cloud environment (see §6.1-6.3).

Ridges and hubs (crosses in Fig. 4) appear to depart from the mass-length and
density-length correlations observed in most other filamentary clouds, showing
approximately an order of magnitude larger masses and effective densities than
filaments with comparable lengths. The large contraction timescales (§6.2) and
small velocity gradients observed at parsec scales (§4.6) rule out the
longitudinal collapse of larger filaments as the main origin for these
structures (i.e. a vertical displacement in the left panel of Fig. 4).
Instead, local accretion and/or merging of multiple filaments (see _Myers_ ,
2009a; _Kumar et al._ , 2020) or cloud-cloud collisions (_Inoue and Fukui_ ,
2013; _Inoue et al._ , 2018; _Fukui et al._ , 2021) will all move filaments
rightward in the plot. In combination with homologous collapse (see _Peretto
et al._ , 2013), this appears to be the more plausible mechanism for the
origin of these massive filamentary clouds.

The origin of the mass-size relation $L\propto M^{\alpha}$ has been
classically attributed to the hierarchical nature of molecular clouds
imprinted by turbulence and fragmentation (e.g. _Larson_ , 1981; _Myers_ ,
1983; _Stutzki et al._ , 1998; _Kauffmann et al._ , 2010b; _Roman-Duval et
al._ , 2010). Clouds are expected to follow a mean mass-size relation such as
$L\propto M^{0.52}$ (or $M\propto L^{1.9}$), equivalent to the third Larson’s
relation ($n\propto L^{-1.1}$; _Larson_ , 1981), also found in simulations
(e.g. _Beattie et al._ , 2019). Shallower distributions are found in different
cloud surveys ($\alpha=0.27-0.4$; _Elmegreen and Falgarone_ , 1996) while
steeper correlations and different normalizations are found in resolved clouds
and clumps (see _Kauffmann et al._ , 2010a, b; _Barnes et al._ , 2021,
$\alpha\sim 0.5-0.7$). These variations can be at least partly attributed to
observational biases such as sensitivity and superposition (e.g. _Ballesteros-
Paredes et al._ , 2019). Our filament survey shows a similar global mass-size
relation with approximately $L\propto M^{0.5}$ while individual clouds may
present slightly steeper values.

###  4.5 Magnetic fields

The properties of the magnetic field within and around filamentary structures
are most commonly measured via polarized dust emission or stellar
polarization, which both trace the magnetic field orientation as projected on
the plane of the sky (POS). Our catalog contains estimates of the POS magnetic
field strength, $B_{\textrm{POS}}$, derived from observations of polarized
dust emission via the Davis-Chandrasekhar-Fermi (DCF) method (_Davis_ , 1951;
_Chandrasekhar and Fermi_ , 1953). This method relies on measuring the
dispersion of polarization angles, as well as the density and the velocity
dispersion of the gas, and has a number of limitations (e.g., _Ostriker et
al._ , 2001; _Heitsch et al._ , 2001b; _Cho and Yoo_ , 2016; _Liu et al._ ,
2021; _Skalidis and Tassis_ , 2021). Furthermore, for several objects, the DCF
results are obtained on a different scale (both larger and smaller) than the
filament width. In addition, although magnetic fields have been mapped in many
star-forming regions, the magnetic field strength is typically not measured
for individual filaments.

Fig. 5 shows the magnetic field strength against the column density of
filaments in our catalog. Most of the measurements in the catalog refer to a
single cloud from (_Arzoumanian et al._ , 2021, yellow diamonds). For
comparison, we also show the magnetic field measurements from _Crutcher_
(2012) (grey dots), which have been obtained using observations of the Zeeman
effect (probing the line-of-sight magnetic field). The black dashed line
indicates where the ratio of the mass to the magnetic flux is equal to the
critical value (_Spitzer_ , 1968; _Mouschovias and Spitzer_ , 1976), which
roughly matches the data points at high $N$ ($\gtrsim 10^{22}$ cm-2). While
the data in the filament catalog seem consistent with the Zeeman observations,
more statistics are needed to quantify the agreement.

In addition to the field strength, the orientation of the magnetic field with
respect to the filament axis can also inform us about the effect of the field
on filaments (e.g. _Nagai et al._ , 1998; _Li et al._ , 2013; _Palmeirim et
al._ , 2013; _Malinen et al._ , 2016; _Panopoulou et al._ , 2016; _Planck
Collaboration XXXV_ , 2016; _Soler and Hennebelle_ , 2017; _Jow et al._ ,
2018; _Fissel et al._ , 2019; _Soler_ , 2019). Low-column density Hi filaments
(§3.7) tend to be aligned parallel to the local magnetic field orientation
(_McClure-Griffiths et al._ , 2006; _Clark et al._ , 2014, 2015). Statistical
evalulations of the relative orientation between the magnetic field and
filamentary structures (_Soler et al._ , 2013; _Jow et al._ , 2018) reveal a
strong dependence with column density: magnetic fields are preferentially
parallel to column density structures at low $N$ but can become preferentially
perpendicular to filaments at high $N$. The transition in relative orientation
appears at $N\simeq 10^{21-22}$ cm-2, which approximately coincides with the
column density at which the magnetic field strength starts to increase (Fig. 5
and _Crutcher_ , 2012). For an interpretation of these results we refer to
§5.1.3 (see also the chapter by Pattle et al. in this book).

The number of magnetic field measurements in the literature and thus in our
catalog is rather limited compared to other properties of filaments (e.g.
mass, length, velocity dispersion). Given the potential wealth of information
stored in the magnetic field structure and strength on the evolution of
filaments (see §5.1.3), a more detailed investigation of the magnetic field
properties in individual filamentary objects is warranted.

Figure 5: Magnetic field strength versus filament column density. The field
strength in our catalog (colored, large symbols) is obtained via the DCF
method (except the data point for HI). The dashed black line corresponds to a
mass-to-flux ratio equal to the critical one. The data from our catalog seem
to be broadly consistent with data from _Crutcher_ (2012) (gray dots and error
bars).

l—c—c—c—c—c—c—c 65em FILAMENT FAMILIES (II): DYNAMIC PROPERTIES(1) Family(#) &
$\sigma_{\textrm{tot}}$ $\sigma_{\textrm{NT}}$ $m$ $f$ $\nabla
v_{\textrm{LSR,}\parallel}$ $B_{0}$
(km s-1) (km s-1) (M⊙ pc-1) $(=m/m_{\textrm{crit}})$ (km s-1 pc-1) ($\mu$G)
Nearby Filaments & 0.28 - 1.32 0.21 - 1.30 4.4 - 18.6 0.2 - 0.7 1.02 - 1.60 70
- 70
Galactic Survey Fils. 1.02 - 1.55 1.00 - 1.54 23 - 191 0.8 - 6 0.22 - 0.55 ?
IRDCs 0.40 - 0.62 0.35 - 0.62 31 - 115 0.9 - 2.5 0.20 - 0.90 45 - 500
Giant Filaments 1.1 - 3.57 2.43 - 3.93 250 - 2513 15 - 35 0.04 - 0.08 ?
Dense fibers 0.26 - 0.36 0.15 - 0.24 9.3 - 33.5 0.3 - 1.4 0.35 - 3.23 40 -
1000(2)
Striations 0.71 - 0.90 ? 2.5 - 5.6 0.1 - 0.2 ? 26 - 30
HI fibers 0.88(?) 0.65(?) 0.05 - 0.15 ? ? 5(3)
(1)Values correspond to the interquartile [Q25%,Q75%] range for each measured
quantity. (2) Inferred from the surrounding medium. (3) Standard $B_{0}$ value
for the diffuse medium ($n<10^{3}$ cm-3, _Crutcher et al._ , 2010). (?)
Unknown or poorly constrained. (#) See Table 1 for references.

###  4.6 Gas kinematics

Figure 6: Velocity dispersion in units of the sound speed
($\sigma_{\textrm{tot}}/c_{\textrm{s}}$) as function of the filament length
$L$ (left panel) and line mass $m$ (right panel). The observed correlations of
$\frac{\sigma_{\textrm{tot}}}{c_{\textrm{s}}}=\left(1+\frac{L}{0.5\mathrm{pc}}\right)^{0.5}$
(Eq. 15) and $\frac{\sigma_{\textrm{tot}}}{c_{\textrm{s}}}\propto m^{0.5}$ are
indicated by dashed lines. For comparison, the previously reported
$\sigma_{\textrm{tot}}/c_{\textrm{s}}\propto m^{0.36}$ by _Arzoumanian et al._
(2013) is also indicated in the right panel. In both plots, the
$\sigma_{\textrm{tot}}/c_{\textrm{s}}=1$ is indicated by a horizontal line.

The gas velocity field at different scales within filaments can be
characterized using both line-of-sight ($\sigma_{\textrm{tot}}$ and
$\sigma_{\textrm{nt}}$) and POS ($\nabla v_{\textrm{LSR}}$) measurements (see
definitions in §2). We report the change of
$\sigma_{\textrm{tot}}/c_{\textrm{s}}$ with the total filament length $L$ in
the left panel of Fig. 6. We use $c_{\textrm{s}}$ for each filament
individually for normalisation. However, due to the small temperature
variations found between individual filaments (see §4.2), using a constant
$c_{\textrm{s}}$ for all filaments would not change our overall findings.

Resolved nearby filaments and fibers show total velocity dispersions close to
the sonic speed ($\sigma_{\textrm{tot}}/c_{\textrm{s}}\simeq 1-2$) (e.g.
_Hacar and Tafalla_ , 2011; _Hacar et al._ , 2013; _Arzoumanian et al._ ,
2013). On the other hand, larger Galactic Plane and Giant Filaments exhibit
highly supersonic velocity dispersions
($\sigma_{\textrm{tot}}/c_{\textrm{s}}\gg 1$) (e.g. _Wang et al._ , 2016;
_Mattern et al._ , 2018a). The large dynamic range covered by our sample
allows an empirical parametrization of the observed
$\sigma_{\textrm{tot}}$-$L$ scaling in filaments as

$\frac{\sigma_{\textrm{tot}}}{c_{\textrm{s}}}=\left(1+\frac{L}{0.5\,\mathrm{pc}}\right)^{0.5}$
(15)

(dashed line in the left panel of Fig. 6) with uncertainties of $\pm 0.2$ pc
for the length normalization and $\pm 0.2$ for the exponent. The relative
increase of $\sigma_{\textrm{tot}}$ with respect to $c_{\textrm{s}}$ can be
attributed to an increasing contribution of non-thermal motions at larger
scales as $\sigma_{\textrm{nt}}/c_{\textrm{s}}\propto L^{0.5}$, reminiscent of
the velocity dispersion-size relationship observed in the ISM (_Solomon et
al._ , 1987, see also §5.3.2).

As shown in the right panel of Fig. 6, a similar correlation is observed in
terms of the line mass $m$. Filaments with $m<$ 100 M⊙ pc-1 exhibit sonic-like
$\sigma_{\textrm{tot}}$ values while increasing values are seen in targets
with $m~{}>$ 100 M⊙ pc-1. The change of $\sigma_{\textrm{tot}}$ with $m$ was
first investigated by _Arzoumanian et al._ (2013), finding that
$\sigma_{\textrm{tot}}/c_{\textrm{s}}\propto m^{0.36}$ (dotted line in the
right panel of Fig. 6). Our new results indicate a somewhat steeper
correlation with

$\sigma_{\textrm{tot}}/c_{\textrm{s}}\propto m^{0.5}$ (16)

(dashed line). The interpretation of this $\sigma_{\textrm{tot}}-m$
correlation is, however, non-trivial (see §5.2.2) and subject to strong
observational biases. While velocity dispersions in local filaments are
obtained from resolved, single-velocity components of density-selective
tracers, measurements in large filaments in Galactic Plane surveys typically
make use of more diffuse tracers (e.g. 13CO; _Mattern et al._ , 2018a), and
average large-scale motions (see below) as well as multiple velocity
components blended within a beam (e.g. _Sokolov et al._ , 2019).

Velocity gradients $\nabla v_{\textrm{LSR}}$ can be employed to quantify the
bulk gas motions inside filaments. We find systematic variations of $\nabla
v_{\textrm{LSR}}$ with $L$ as shown in Fig. 7. Nearby parsec-scale filaments
show typical longitudinal gradients along their main axis of $\nabla
v_{\textrm{LSR,}\parallel}\sim$ 1-2 km s-1 pc-1 (_Loren_ , 1989b; _Bally et
al._ , 1987; _Schneider et al._ , 2010; _Kirk et al._ , 2013; _Lu et al._ ,
2018; _Hu et al._ , 2021, for additional references see also §3.8). While
cloud-size IRDCs (e.g. _Tackenberg et al._ , 2014; _Sokolov et al._ , 2019)
and large-scale filaments (_Ragan et al._ , 2014) show similar or even smaller
$\nabla v_{\textrm{LSR},\parallel}$ values, shorter fibers (at scales $<$ 1
pc) show larger gradients up to $\nabla v_{\textrm{LSR},\parallel}\gtrsim$ 10
km s-1 pc-1 (_Lee et al._ , 2013; _Hacar et al._ , 2018; _Dhabal et al._ ,
2018; _Chen et al._ , 2020c, b) following an approximate $\nabla
v_{\textrm{LSR},\parallel}\propto L^{-1}$ relationship (dashed line). For an
interpretation of the observed correlation we refer to §5.3.1.

We find no clear correlation between $\nabla v_{\textrm{LSR},\parallel}$ and
$m$ (not shown) suggesting that most filaments do not show a global homologous
collapse (see also §6.2) with only few potential cases reported in the
literature (e.g. _Zernickel et al._ , 2013). Similarly, gradients associated
with edge effects (_Burkert and Hartmann_ , 2004) appear to be an exception
(_Dewangan et al._ , 2019). Local collapse can nonetheless become dominant, as
converging velocity gradients at scales of $\sim$ 1 pc are seen in hub-like
structures (e.g. _Peretto et al._ , 2013; _Kirk et al._ , 2013; _Hacar et al._
, 2017b, see also §3.8)

Figure 7: Parallel velocity gradients $\nabla v_{\textrm{LSR,}\parallel}$ as
function of the filament length $L$. The velocity gradient follows roughly a
$L^{-1}$ relation (black dashed line).

##  5 FILAMENTS AS DYNAMIC STRUCTURES

Measured line masses (§4.1) and internal non-thermal motions (§4.6) indicate
that filaments are highly dynamical structures, significantly different from
the hydrostatic case outlined in §2. Furthermore, their hierarchical nature
(§4.4) suggests that filaments and their interstellar environments both
exhibit a complex 3D structure that greatly departs from idealized cylinders,
which complicates the interpretation of POS-projected observations. As
illustrated in Fig. 8, simulations provide unique insights into the gas
velocity field, internal mass distribution, and magnetic field structure of
filaments in space and time. This section reviews the status of numerical
simulations in shaping our current interpretation of the filamentary structure
of the ISM.

Figure 8: Left panel: Filamentary density structure (depicted with various
isocontours between $\sim$100 and 2000 cm-3) of a simulated molecular cloud.
The red arrows show the complex velocity field in the vicinity of the
filamentary structures suggesting features like accretion along filaments (A),
accretion onto filaments (B1, B2) and accretion onto a hub-like system (C).
Right panel: Volume rendering of the density structure in a simulated
molecular cloud showing cold and dense filaments (blue-green) as well as warm
and diffuse gas in its surrounding (pink). Both the density structure as well
as representative fields lines associated with filaments show the internal
complexity of these objects, e.g. a kinked/bow-like field structure towards
the spine of filaments. Simulations are taken from _Seifried et al._ (2017b);
_Seifried et al._ (2020)

###  5.1 Filament formation mechanisms

Filaments are definitionally anisotropic, which suggests that their formation
is at least affected – if not controlled – by anisotropic agents, such as
large-scale gas motions sweeping up material (§5.1.1), shock interactions
induced by turbulence (§5.1.2), magnetic fields (§5.1.3), feedback (§5.1.4),
or global environments such as Galactic spiral arms (§5.1.5). The abundance of
anisotropic physics across a wide range of physical scales may explain why
filamentarity is ubiquitous in the ISM. The relative unimportance of thermal
pressure at least on large scales in molecular clouds gives credence to
gravitational contraction as a key ingredient in filament evolution (_Vázquez-
Semadeni et al._ , 2019). Moreover, many mechanisms create networks of
filaments rather than single objects, which contributes to the hierarchy of
filaments throughout the ISM.

####  5.1.1 Classical Formation via sheets

In the classical picture, filaments can form in infinite sheets due to
gravitational instabilities (equivalent to the Jeans instability in 3D;
_Tomisaka and Ikeuchi_ , 1983; _Larson_ , 1985; _Miyama et al._ , 1987a, b;
_Nagai et al._ , 1998; _Van Loo et al._ , 2014), once the wavelength of a
perturbation is above a few times the thickness of the sheet. For a sheet in
pressure equilibrium with a thickness of approximately the Jeans length, this
results in filaments with near-critical line masses (_Miyama et al._ , 1987a).

However, under realistic conditions, the planar shock fronts that form in the
ISM (e.g. due to cloud-cloud collisions, stellar feedback-driven bubbles, or
cascading turbulent motions) will not be infinite in their lateral extension.
The behaviour of a finite sheet is markedly different (_Li_ , 2001; _Burkert
and Hartmann_ , 2004; _Hartmann and Burkert_ , 2007). In general, the sheet
will undergo global collapse, and gravitational focusing will lead to a
concentration of mass at the edge of the sheet rather than fragmentation
inside the sheet. Elongated sheets will collapse into filamentary structures
with density enhancements at either end.

####  5.1.2 Turbulence-induced formation

In the absence of gravity, interactions of sheets can lead to the formation of
filaments. This has been seen for more than two decades in simulations of
turbulent boxes intended to represent parts of molecular clouds (e.g. _Porter
et al._ , 1994; _Padoan and Nordlund_ , 1999; _Heitsch et al._ , 2001a;
_Padoan et al._ , 2001; _Pudritz and Kevlahan_ , 2013; _Hennebelle_ , 2013;
_Li and Klein_ , 2019; _Federrath_ , 2016; _Federrath et al._ , 2021, and many
more). The turbulent motions must be both supersonic and super-Alfvénic in the
magnetised case. They create shock planar fronts (sheets), which can collide
obliquely to form a filament at the intersection with a turbulent wake behind
it (_Pudritz and Kevlahan_ , 2013).

Turbulence is a natural mechanism for forming networks of filaments that trace
the underlying gas velocity field. The turbulent energy cascade generates an
internal hierarchy of filaments from the driving scale downwards, as seen in
observations (see §4.4). A turbulent origin has also been proposed to explain
the intertwined fibers inside filaments, although their exact origin remains
under debate: In a top-down scenario, the large-scale coherence and parallel
organization of fibers with respect to the main filament reported in regions
such as B213-L1495 suggest that these structures might form out of their
parental filament (fray and fragment scenario; _Tafalla and Hacar_ , 2015;
_Clarke et al._ , 2017). Alternatively, some simulations favour a bottom-up
approach, with small subsonic filaments forming in a turbulent medium first,
and then being gathered by collapse and shear flows (fray and gather scenario;
_Smith et al._ , 2016).

####  5.1.3 Magnetic field-assisted formation

Since magnetic fields exert anisotropic forces, they can assist the formation
of filaments in various ways depending on the relative strength of the
magnetic field with respect to the gas motions.

Shock compression of a magnetised clump: If a magnetised clump is hit by a
shock wave, the clump is swept up and becomes deformed in a kink-like fashion
together with the magnetic field threading the clump. The resultant kinked
magnetic field structure focuses the gas flow in the post-shock gas towards
the convex point of the deformed shock, leading to the formation of an
elongated structure (_Inoue and Fukui_ , 2013; _Vaidya et al._ , 2013; _Inoue
et al._ , 2018; _Fukui et al._ , 2021). _Abe et al._ (2021) argue that this
mechanism is dominant when the unshocked gas is supersonic and super-Alfvénic,
but becomes sub-Alfvénic behind the shock.

Stretching of overdensities: Magnetic fields may affect filament formation
most strongly in the diffuse atomic gas, and in the lower-density molecular
gas as traced by e.g. 12CO, where gas motions are sub- to trans-Alfvénic. In
such regions, magnetic fields introduce anisotropy because motions
perpendicular to the magnetic field are impeded by the Lorentz force, while
gas can move freely along the field lines. Thus overdensities will be
stretched out by (turbulent) shear flows preferentially along field lines
(_Hennebelle_ , 2013; _Inoue and Inutsuka_ , 2016; _Xu et al._ , 2019). This
mechanism may explain the observed predominance of filaments that are parallel
to the local magnetic field at column densities below $\sim 10^{21-22}$ cm-2
(§4.5). This is particularly pronounced for the Hi filaments, which are well
aligned with the Galactic magnetic field (§3.7). These possibly originate from
overdensities which are created by thermal instability and are subsequently
stretched out (_Inoue and Inutsuka_ , 2016; _Wareing et al._ , 2021).

Converging flow-assisted formation: Around $N\simeq 10^{21-22}$ cm-2 the
preferred relative orientation of filaments and magnetic fields switches from
parallel to perpendicular (see §4.5), a trend also seen in a number of
simulations including turbulence and strong magnetic fields (_Heitsch et al._
, 2001a; _Ostriker et al._ , 2001; _Li and Nakamura_ , 2004; _Nakamura and Li_
, 2008; _Collins et al._ , 2011; _Hennebelle_ , 2013; _Soler et al._ , 2013;
_Chen and Ostriker_ , 2015; _Li et al._ , 2015; _Chen et al._ , 2016; _Zamora-
Avilés et al._ , 2017; _Mocz and Burkhart_ , 2018; _Li and Klein_ , 2019;
_Chen et al._ , 2020a; _Seifried et al._ , 2020; _Dobbs and Wurster_ , 2021,
but see also _Hennebelle and Inutsuka_ 2019 for a recent review). The
observation that high-column density filaments are preferentially
perpendicular to the ambient magnetic field admits multiple interpretations
for how magnetic fields, gas flows, and gravity might interact in the filament
formation process.

_Soler and Hennebelle_ (2017) argue that the transition from a parallel to a
perpendicular field orientation indicates an anisotropic and converging
(accretion) flow. The anisotropy is introduced by a dynamically important
magnetic field on larger scales (causing aligned magnetic fields and
filaments); the converging flow can be caused by either shocks (_Körtgen and
Soler_ , 2020) or gravitational collapse (_Seifried et al._ , 2020). For the
latter case, the relative orientation of the magnetic field is related to its
relative strength with respect to gravity and in the end to the formation of
filamentary structures: for high (column) densities ($n$ $\simeq$ 102 \- 103
cm-3, $N$ $\simeq$ 1021-22 cm-2) gravity starts to dominate over the magnetic
force (_Seifried et al._ , 2020; _Ibáñez-Mejía et al._ , 2021). For the
converging flow mechanism to work the motions are globally sub-Alfvénic but
become super-Alfvénic on smaller scales (if they were globally super-Alfvénic,
we would end up with the turbulence-induced formation mechanism described in
§5.1.2). _Chen et al._ (2016) argue that locally super-Alfvénic motions alone
are sufficient to create filaments perpendicular to the magnetic field. More
recent simulations find that super-Alfvénic motions are necessary but not
sufficient (_Soler and Hennebelle_ , 2017; _Seifried et al._ , 2020).

A common conclusion of these analyses is that the geometrical configuration of
the magnetic field with respect to the filament can provide insights into its
formation mechanism. An example is the bow-like structure of magnetic fields
around filamentary structures, which is caused when a dynamically weak field
is dragged along with the filament (see right panel of Fig. 8). This
configuration is found in both simulations (e.g. _Hartmann_ , 2002; _Inoue et
al._ , 2018; _Li and Klein_ , 2019; _Reissl et al._ , 2021), and observations
(_Heiles_ , 1997; _Heiles and Robishaw_ , 2009; _Tahani et al._ , 2018, 2019,
2022). Helical magnetic fields, suggested to explain the radial stability of
filaments (_Fiege and Pudritz_ , 2000), have – to the authors’ knowledge – to
date not been found in any simulation and would also be difficult to
distinguish from a bow-like structure (_Reissl et al._ , 2021, see also
_Tahani et al._ , 2019 for a comparison between different field geometries
around Orion).

####  5.1.4 Feedback-induced filaments

Filaments can also be shaped (and enhanced) by the pressure effects of
mechanical and radiation feedback on pre-existing density inhomogeneities on
the cloud surfaces and edges. Two types of filaments occur in the context of
feedback. Filaments orthogonal to the wind direction will show asymmetric
radial profiles (_Peretto et al._ , 2012) that can be enhanced by the effects
of radiation (_Suri et al._ , 2019). On the other hand, pillar-like filaments
elongated along the wind direction exhibit signatures of head compression
(e.g. _Bally et al._ , 2018) and Kelvin-Helmholtz instabilities (_Berné et
al._ , 2010). In both cases magnetic fields can play a crucial role in shaping
the filaments (_Ntormousi et al._ , 2017). While initially constructive, the
continuous action of these feedback mechanisms eventually destroys these
filamentary structures via photo-dissociation, erosion, and ablation of their
molecular material over time (e.g. _Goicoechea et al._ , 2016).

These wind-shaped filaments are typically low-mass, are found at parsec scales
within clouds, and are located in the vicinity of the high-mass stars that
supply winds and radiation (e.g. _Suri et al._ , 2019). Examples of filaments
sculpted by externally-applied forces have been classically reported in
connection to HI expanding shells (e.g. L204; _Heiles_ , 1988) and stellar
winds (e.g. Oph Streamers; _Loren_ , 1989b). High-sensitivity atomic ([Cii];
_Pabst et al._ , 2020) and molecular observations (CO; _Suri et al._ , 2019)
have identified these feedback-induced filaments at the walls of Hii regions.

####  5.1.5 Shear, Differential rotation and Galactic Dynamics

A common feature of the above models is the presence of shear due to
turbulence, which enhances the filamentarity of gas structures. However, the
mechanisms discussed in §5.1.2 and §5.1.3 usually only consider turbulent
motions within molecular clouds. Simulations have shown that on larger scales
($\sim$100 pc or more) Galactic rotation can play an important role in forming
the Giant Filaments discussed in §3.4 (e.g. _Wada et al._ , 2002; _Smith et
al._ , 2014a). For example, _Duarte-Cabral and Dobbs_ (2017) zoomed into the
Giant Filament population of an isolated galaxy simulation to show that Giant
Filaments form predominantly through galactic shear.

Giant Filament formation is aided by instabilities such as the Parker
instability (e.g. _Kim et al._ , 2002; _Machida et al._ , 2013; _Körtgen et
al._ , 2018), and the thermal instability which rapidly drives gas from the
warm to the cold phase (_Hennebelle and Audit_ , 2007; _Kim et al._ , 2010).
These filaments are then further elongated by differential rotation as they
move between arms as, due to the their length ($\gtrsim$ 100 pc), the ends of
the filaments are at different galactic radii (_Smith et al._ , 2020). Within
spiral arms, dense filamentary clouds known as ‘bones’ such as the Nessie
filament are proposed to form from gas compressed as it falls into the Milky
Way spiral potential (_Goodman et al._ , 2014; _Zucker et al._ , 2015, 2019).

###  5.2 Environment

Filaments do not exist in isolation. Fig. 3 (see also §4.2) suggests that –
depending on the line mass – the pressure of the ambient gas (§5.2.1) or
accretion (§5.2.2) driven by the filament’s gravitational potential might play
a role in filament structure and evolution.

####  5.2.1 External Pressure

Whether pressure confinement is required to prevent the filament from
expanding depends on the effective equation of state (_Toci and Galli_ ,
2015), on the line mass (_Fischera and Martin_ , 2012a), the global
gravitational potential (_Ballesteros-Paredes et al._ , 2009) and on the
magnetic field geometry (_Fiege and Pudritz_ , 2000). The Galactic midplane
pressure of $\sim 2\times 10^{4}$ K cm-3 (_Cox_ , 2005; _Blitz and Rosolowsky_
, 2006) provides a lower limit for the external pressure. Magnetic pressure
(_Hennebelle_ , 2013, see also _Kalberla et al._ , 2016 for HI filaments) and
ram pressure (_Audit and Hennebelle_ , 2005; _Duarte-Cabral and Dobbs_ , 2017;
_Chira et al._ , 2018) of the ambient gas can provide additional pressure
confinement.

Ram pressure confinement suggests dynamical or even transient filaments, since
it entails a mass inflow leading to accretion (_Heitsch_ , 2013b, §5.1). Ram
pressure exerted by turbulent flows (_Audit and Hennebelle_ , 2005; _Duarte-
Cabral and Dobbs_ , 2017; _Chira et al._ , 2018) in combination with magnetic
stresses (_Hennebelle_ , 2013) can provide a confining pressure.

Assuming a truncated Ostriker filament, the role of pressure confinement can
be quantified by the steepness of the column density profile (_Fischera and
Martin_ , 2012a) as a function of the criticality parameter $f$ (Eq. 6).
Beyond the inner radius $R_{\textrm{flat}}$ (Eq. 11), filaments show column
density profiles consistent with power-law slopes of $1.5<p<2.5$, though
steeper slopes approaching $p=4$ have been measured (see §4.3 and reference
therein). It remains to be seen – for example by correlating the power-law
slope $p$ with the estimated central pressure – whether flatter profiles
indeed indicate stronger confinement as predicted by _Fischera and Martin_
(2012a). The effective equation of state (_Toci and Galli_ , 2015), turbulence
(_Gehman et al._ , 1996a) or magnetic fields (_Fiege and Pudritz_ , 2000) may
also play a role for shaping the filament profiles.

Pressure confinement can also be interpreted as radial compression of a
filament at constant line mass. How a filament responds to an external
pressure increase depends on the effective equation of state and
magnetization. Isothermal filaments cannot be squeezed into collapse, since
the internal and gravitational energy increase with the same dependence on the
radius (_Fiege and Pudritz_ , 2000), yet, pressurization affects the critical
and maximum wave number for longitudinal fragmentation (Eq. 12; _Nagasawa_ ,
1987). For equations of state softer than isothermal
($\gamma_{\textrm{eff}}<1$), increasing the external pressure leads to
collapse (_Toci and Galli_ , 2015) and affects the longitudinal fragmentation
and resulting shape of cores (_Heigl et al._ , 2018b; _Hosseinirad et al._ ,
2018; _Anathpindika and Francesco_ , 2021; _Motiei et al._ , 2021).

####  5.2.2 Accretion onto filaments

Line observations of the medium surrounding filaments show velocity gradients
often roughly perpendicular to the main axis of the filament, suggestive of
gas accretion onto the filament (_Schneider et al._ , 2010; _Kirk et al._ ,
2013; _Palmeirim et al._ , 2013; _Beuther et al._ , 2015; _Dhabal et al._ ,
2018; _Williams et al._ , 2018; _Shimajiri et al._ , 2019; _Bonne et al._ ,
2020; _Chen et al._ , 2020c; _Gong et al._ , 2021). Such ongoing accretion
onto filaments is also evident in 3D simulations of filamentary molecular
clouds (see positions labelled B1 and B2 in the left panel of Fig. 8).
Estimates of the associated accretion rates, $\dot{m}$, range from a few $10$
to a few $100$ M⊙ Myr-1 pc-1 (_Kirk et al._ , 2013; _Palmeirim et al._ , 2013;
_Schisano et al._ , 2014; _Bonne et al._ , 2020; _Gong et al._ , 2021). These
estimates are somewhat higher but still in rough agreement with simulations of
filaments located inside molecular clouds giving accretion rates of a few to a
few 10 M⊙ Myr-1 pc-1(_Gómez and Vázquez-Semadeni_ , 2014; _Chira et al._ ,
2018). Variations in the observed accretion rates can in part be attributed to
environmental effects, as $\dot{m}$ is found to correlate with the column
density (_Heitsch_ , 2013a; _Gómez and Vázquez-Semadeni_ , 2014). This is
supported by the observation that higher-mass filaments are preferentially
embedded in environments with a higher background column density (_Rivera-
Ingraham et al._ , 2016).

The above accretion rates lead to accretion timescales

$\tau_{\textrm{acc}}\equiv\frac{m}{\dot{m}},$ (17)

on which a filament would double its mass, as short as 0.1 Myr. The
implications of such short accretion timescales for the evolution of filaments
will be discussed further in §5.4.3 and §6.2.

The geometry of accretion flows and their relation to the ambient magnetic
field are still under discussion. _Shimajiri et al._ (2019) argue that the
accretion in B211/B213 in Taurus occurs within a sheet-like structure (see
_Chen et al._ , 2020b, for a simulation counterpart), consistent with a
formation scenario via sheets (see §5.1.1). _Palmeirim et al._ (2013) suggest
that accretion occurs along the striations which are oriented perpendicular to
the main filaments B211/213 but parallel to the local magnetic field (see also
e.g. _Hennemann et al._ , 2012). This relative orientation of magnetic fields
and filaments was first proposed by _Nagai et al._ (1998) and subsequently
confirmed in numerous observational and theoretical works (see §5.1.3 and the
review article of Pattle et al. in this book). This geometry suggests a
magnetic field strong enough to channel the gas flow.

Mass accretion can drive turbulence in molecular clouds (_Klessen and
Hennebelle_ , 2010; _Goldbaum et al._ , 2011). While 3D simulations show that
mass accretion leads to turbulence within a filament (_Seifried and Walch_ ,
2015; _Clarke et al._ , 2017; _Heigl et al._ , 2018a), _Heigl et al._ (2020)
point out that the resulting turbulent pressure does not contribute to the
stability of the filament since the turbulent pressure profile is flat. If
accretion is driven by gravity, invoking accretion-driven turbulence as a
stabilization mechanism against gravitational collapse conflicts with energy
conservation. Turbulence driven by (gravitational) accretion implies that
kinetic energy flows from the outside in, and is dissipated on a crossing
timescale (_Mac Low et al._ , 1998; _Klessen and Hennebelle_ , 2010). If
accretion-driven turbulence were to support the filament against collapse, the
turbulent kinetic energy ($m\sigma_{\textrm{nt}}^{2}$) would have to grow
faster than the gravitational energy ($|m^{2}G|$), a proposition difficult to
reconcile with energy conservation, since it is the gravitational potential
that drives the accretion in the first place. Moreover, the conversion
efficiency between the kinetic energy of the inflow and the resulting
turbulent energy is only a few percent (_Klessen and Hennebelle_ , 2010),
resulting in trans- to mildly supersonic turbulence (_Seifried and Walch_ ,
2015; _Clarke et al._ , 2017; _Heigl et al._ , 2018a; _Heigl et al._ , 2020)
in agreement with observations (_Hacar and Tafalla_ , 2011; _Hacar et al._ ,
2013; _Arzoumanian et al._ , 2013, see §4.6). Therefore, accretion cannot
result in filament stabilization, but it may induce fragmentation (§5.4).

###  5.3 Filament Dynamics

####  5.3.1 Coherence

One of the characteristic features of filaments is their coherence. This is
illustrated in two ways 1) in the POS velocity gradients along their length,
and 2) in the LOS velocity dispersion where there is a transition to sonic
motions at small scales (see §5.3.2).

We first consider the large-scale longitudinal coherence shown in Fig. 7 in
the observed velocity gradients of our filament families. Two distinct classes
emerge. On scales larger than $\sim 1$ pc, average POS velocity gradients of
only 1 km s-1 pc-1 or less are observed, independent of the filament’s length.
This is particularly apparent for the Giant Filaments on 100 pc scales. This
large-scale longitudinal coherence is likely due to two factors. First, it is
often an explicit condition in the algorithm used to identify filaments,
particularly in the case of Giant Filaments (e.g. _Ragan et al._ , 2014; _Wang
et al._ , 2015; _Zucker et al._ , 2015; _Wang et al._ , 2016). Second,
coherency may simply be a necessary feature of survival, as any larger
gradient may lead to a rapid destruction of the filament. Simulations of
galactic-scale filaments (e.g. _Duarte-Cabral and Dobbs_ , 2016; _Smith et
al._ , 2020) exhibit similar coherency in large-scale structures.

However, velocity gradients rise on smaller scales. Fig. 7 shows that below
scales of 1 pc gradients can exceed values of $10$ km s-1 pc-1 and there is a
tentative indication that the dispersion scales inversely with the filament
length, $\nabla v_{\textrm{LSR},\parallel}\propto L^{-1}$. Such an effect
could be due to a number of reasons. First, larger gradients along the
filament could be a sign of global collapse, possibly also connected to
fragmentation. Second, feedback effects such as outflows may disrupt filaments
locally on small scales. _Smith et al._ (2020) investigated the velocity
gradients along filamentary cloud networks formed as a result of galactic-
scale dynamics (their Figure 12) and found a similar distribution to that seen
in Fig. 7. In this case the gradients at small scales were high in both i) a
massive filament system undergoing significant collapse and fragmentation and
ii) a highly turbulent filament system undergoing disruption by supernovae -
suggesting that both mechanisms play a role in the velocity dispersion scaling
behaviour.

An additional contribution to the increasing velocity gradient at small scales
could arise from projection effects. Considering a hypothetical filament with
length $L$ and constant velocity gradient $g$, one obtains
$L_{\textrm{obs}}=L\cos\alpha$ and $g_{\textrm{obs}}=g\tan\alpha$, where
$\alpha$ is the angle between the filament and the POS. It follows that
$\alpha$ = arccos($L_{\textrm{obs}}/L$) = arccos($x$) and thus
$g_{\textrm{obs}}$ = $g$ tan(arccos($x$)) = $g$ $\frac{\sqrt{1-x^{2}}}{x}$
which, for $x\ll 1$, gives 1/$x$. Hence, the observed gradient would scale as
1/$x$, i.e. $\nabla v_{\textrm{LSR},\parallel}\propto$ $L^{-1}$.

####  5.3.2 The $\sigma_{\textrm{tot}}-L$ relation and the sonic regime

Coherence is also observed in the velocity dispersion of filaments, but only
on small scales ($\leq$ 1 pc) were
$c_{\textrm{s}}<\sigma_{\textrm{tot}}<2c_{\textrm{s}}$ (Fig. 6). Such
behaviour is reminiscent of the transition to sonic coherence seen in star
forming cores at $\sim 0.1$ pc (e.g. _Goodman et al._ , 1998). Despite the
complications in measuring filament widths (§4.3, §6.5), it is clear that the
FWHM of filaments with $L<1$ pc is typically less than $0.2$ pc (see §6.5),
and hence in the sonic regime. Subsonic filaments are a natural consequence of
supersonic turbulent shocks (see §5.1.2). This would imply that the transition
to coherence in molecular clouds occurs before star-forming cores are formed,
i.e. at the filament-formation stage (_Hacar and Tafalla_ , 2011). Hence, the
cores simply inherit their subsonic nature when they are formed by filament
fragmentation (§5.4).

However, $\sigma_{\textrm{tot}}$ becomes increasingly supersonic for larger
filament lengths (Fig. 6, Eq. 15). The scaling $\sigma\propto L^{0.5}$ is
reminiscent of the second Larson relation (_Larson_ , 1981). For a constant
column density $\Sigma\propto R_{\textrm{flat}}$, such an exponent would be
consistent with an energy equipartition of
$E_{\textrm{k}}/|E_{\textrm{p}}|=\sigma^{2}R_{\textrm{flat}}/2GM\sim 1/2$ as
expected from the Virial theorem.

Non-thermal motions are commonly interpreted as evidence for internal
turbulence, potentially driven by large-scale accretion flows. Interpretation
of such non-thermal motions is further complicated as different tracers are
used (e.g. N2H+ for small, nearby filaments, and 13CO for large, distant
filaments) resulting in a larger column of gas and more diffuse material being
included in the measurement of larger structures. Given the hierarchical
nature of the filamentary ISM, this can lead to superposition effects. Better-
resolved nearby clouds break into networks of filaments. If a network of small
trans-sonic fibers with different centroid velocities were averaged into a
single beam, the measured linewidth would be supersonic (_Hacar et al._ ,
2016a).

####  5.3.3 Filament rotation and angular momentum

In addition to their longitudinal gradients $\nabla
v_{\textrm{LSR},\parallel}$ (§4.6 and §5.3.1), orthogonal velocity gradients
inside filaments are also reported for both fiber-like (_Fernández-López et
al._ , 2014; _Dhabal et al._ , 2018) and nearby filaments (_Palmeirim et al._
, 2013) with values of $\nabla v_{\textrm{LSR},\perp}\lesssim$ 10 km s-1 pc-1.
These gradients could represent a continuation of the accretion onto the
filaments from larger scales (§5.2.2). Organized, large-scale velocity
gradients are observed in some cloud-size filaments ($\nabla
v_{\textrm{LSR},\perp}\sim 2~{}\mathrm{km}~{}\mathrm{s}^{-1}\mathrm{pc}^{-1}$,
_Álvarez-Gutiérrez et al._ , 2021) and massive IRDCs ($\nabla
v_{\textrm{LSR},\perp}>20$ km s-1 pc-1; _Beuther et al._ , 2015).

Turbulence simulations also show filament velocity gradients (_Smith et al._ ,
2016; _Chen et al._ , 2020b). Orthogonal gradients may be related to filament
formation mechanisms such as self-gravity or turbulent shocks, which can be
distinguished by comparing the transverse kinetic and gravitational energies
at different radii (_Chen et al._ , 2020b). Velocity gradients across
filaments could also be interpreted as rotation along the long axis of the
filament (_Levshakov et al._ , 2016; _Álvarez-Gutiérrez et al._ , 2021).
Rotation can provide additional support against collapse (_Recchi et al._ ,
2014) and may play an important role in determining the angular momentum of
cores formed within them (_Hsieh et al._ , 2021). Alternative explanations for
perpendicular velocity gradients include magnetic reconnection events from
clump collisions with misaligned magnetic fields (_Kong et al._ , 2021).

####  5.3.4 Accretion flows and velocity oscillations along filaments

The observed longitudinal velocity gradients discussed in §4.6 and §5.3.1
often extend coherently along large parts of the filament, with values from
$\sim$0.5 km s-1 pc-1 to a few 10 km s-1 pc-1. In particular in hub systems,
accretion flows appear directed towards the intersection point, supporting
their gravitational origin (see §3.8 and the location labelled C in the left
panel of Fig. 8). Together with the radial extent and the local density of the
filament, velocity gradients enable estimation of the accretion rates at which
cores accumulate mass. Accretion rates estimated from the aforementioned
observations cover a wide range from 101 – 104 M⊙ Myr-1.

The mass growth of cores via accretion along filaments was first proposed by
_Gehman et al._ (1996b) and studied subsequently in a number of numerical and
theoretical works (_Balsara et al._ , 2001; _Banerjee et al._ , 2006; _Myers_
, 2009b; _Smith et al._ , 2011, 2016; _Gómez and Vázquez-Semadeni_ , 2014;
_Gómez et al._ , 2018; _Li and Klein_ , 2019). Such an accretion flow along a
filament is also visible in the left panel of Fig. 8 at position A. _Gómez and
Vázquez-Semadeni_ (2014) suggest that the accretion flow along the filament is
fed by the accretion onto the filament (such an accretion flow conversion
might occur between location B1 and C in the left panel of Fig. 8). Due to
mass conservation, the accretion rate increases along the filament towards the
central core reaching values up to 10-4 M⊙ yr-1. Evidence for similar
accretion flow patterns has been seen in observed filaments (_Schneider et
al._ , 2010; _Kirk et al._ , 2013; _Chen et al._ , 2020c).

Besides coherent velocity gradients over the entire length of a filament,
velocity gradients may take the form of oscillations along the filament spine
(_Heiles_ , 1988; _Loren_ , 1989b; _Hacar and Tafalla_ , 2011; _Henshaw et
al._ , 2014; _Hacar et al._ , 2017a; _Barnes et al._ , 2018; _Liu et al._ ,
2019; _Sokolov et al._ , 2019). According to _Henshaw et al._ (2020), velocity
oscillations are present at different scales in the ISM, from sub-parsec to
kpc scales. Indeed, theoretical studies by _Gritschneder et al._ (2017) have
shown that long-lived, stable oscillations can be driven in hydrostatic
filaments due to the action of gravity upon small bends within the filament.
Large-scale velocity oscillations have been proposed to also produce stellar
ejections via a slingshot mechanism under the presence of helical magnetic
fields (_Stutz and Gould_ , 2016).

In a few tantalising cases velocity gradient and density oscillations appear
to have the same wavelength $\lambda$ but be systematically shifted by
$\lambda/4$ (_Hacar and Tafalla_ , 2011; _Henshaw et al._ , 2016). This could
be evidence for accretion flows within filaments onto cores, which in turn
suggests gravitational instabilities as the origin for these motions (_Gehman
et al._ , 1996b; _Heigl et al._ , 2016). However, such a “dephased” behaviour
between the velocity and density perturbations is not observed in all cases
(_Tafalla and Hacar_ , 2015). As stated by _Gehman et al._ (1996b), this could
be due to turbulent motions which superimpose a different velocity pattern on
the gravitational flow thus breaking the correspondence of velocity and
density oscillations.

###  5.4 Fragmentation

Accretion onto and along filamentary structures both increase the gas density,
either locally (e.g. at the intersection points of filaments in hub systems)
or globally along the entire length of the filament. This directly influences
the instantaneous fragmentation properties by decreasing the Jeans length of
the structures. Furthermore, as discussed in §5.3, filaments are turbulent,
and supersonic compression creates local over-densities and thus seeds
fragmentation before the turbulent energy decays (_Mac Low et al._ , 1998;
_Mac Low_ , 1999). Hence, fragmentation most likely does not occur in the
perfectly smooth hydrostatic filaments as described in §2, but instead within
a hierarchy of structures which have complex motions. The end point of this
fragmentation process is the formation of dense cores that may go on to form
stars (see also §6.6). Does filamentary fragmentation determine the properties
of these cores?

####  5.4.1 Edge fragmentation

Theoretical studies on the evolution of quiescent filaments of finite length
have shown that gravitational focusing causes an enhanced collapse at either
end of the filament (_Bastien_ , 1983; _Burkert and Hartmann_ , 2004; _Pon et
al._ , 2011, 2012; _Toalá et al._ , 2012; _Clarke and Whitworth_ , 2015;
_Hoemann et al._ , 2021). The timescale for the filament to fully collapse
along its long axis is proportional to the free-fall time,
$\tau_{\textrm{ff}}=\sqrt{3\pi/(32G\rho_{0})}$, modified by the aspect ratio
$A~{}\equiv L/\textrm{FWHM}$ of the filament (_Pon et al._ , 2012; _Toalá et
al._ , 2012),

$\tau_{\textrm{long}}=\frac{\sqrt{32A}}{\pi}\tau_{\textrm{ff}}=1.9A^{1/2}\left(\frac{n_{0}}{10^{3}\,\mathrm{cm}^{-3}}\right)^{-1/2}\,\mathrm{Myr}.$
(18)

The actual onset of edge collapse, however, should be observable much earlier
(_Seifried and Walch_ , 2015).

However, observationally there are only a few filaments observed with clear