# The impact of shear on the rotation of Galactic plane molecular clouds

Raffaele Rani,1 Jia-Lun Li,2 Toby J. T. Moore,3 David J. Eden,4 Andrew J.
Rigby,5 Geumsook Park,6,7,8 Yueh-Ning Lee 1
1Center of Astronomy and Gravitation, Department of Earth Sciences, National
Taiwan Normal University, 88, Sec. 4, Ting-Chou Rd., Wenshan District,
Taipei 116, Taiwan R.O.C.
2 Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan
R.O.C.
3Astrophysics Research Institute, Liverpool John Moores University, IC2,
Liverpool Science Park, 146 Brownlow Hill, Liverpool, L3 5RF, UK
4Armagh Observatory and Planetarium, College Hill, Armagh, BT61 9DB, UK
5School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
6Telepix Co., Ltd., 17, Techno 4-ro, Yuseong-gu, Daejeon 34013, Republic of
Korea
7Research Institute of Natural Sciences, Chungnam National University, 99
Daehak-ro, Yuseong-gu, Daejeon 34134, Republic of Korea
8Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu,
Daejeon 34055, Republic of Korea E-mail<EMAIL_ADDRESS>(RR)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Stars form in the densest regions of molecular clouds, however, there is no
universal understanding of the factors that regulate cloud dynamics and their
influence on the gas-to-stars conversion. This study considers the impact of
Galactic shear on the rotation of giant molecular clouds (GMCs) and its
relation to the solenoidal modes of turbulence. We estimate the direction of
rotation for a large sample of clouds in the ^13CO/C^18O (3-2) Heterodyne
Inner Milky Way Plane Survey (CHIMPS) and their corresponding sources in a new
segmentation of the ^12CO(3-2) High-Resolution Survey (COHRS). To quantify the
strength of shear, we introduce a parameter that describes the shear’s ability
to disrupt growing density perturbations within the cloud. Although we find no
correlation between the direction of cloud rotation, the shear parameter, and
the magnitude of the velocity gradient, the solenoidal fraction of the
turbulence in the CHIMPS sample is positively correlated with the shear
parameter and behaves similarly when plotted over Galactocentric distance.
GMCs may thus not be large or long-lived enough to be affected by shear to the
point of showing rotational alignment. In theory, Galactic shear can
facilitate the rise of solenoidal turbulence and thus contribute to
suppressing star formation. These results also suggest that the rotation of
clouds is not strictly related to the overall rotation of the disc, but is
more likely to be the imprint of Kelvin-Helmholtz instabilities in the
colliding flows that formed the clouds.

###### keywords:

molecular data – Physical Data and Processes – ISM: kinematics and dynamics –
Interstellar Medium (ISM) – Nebulae– ISM: clouds –

††pubyear: 2022††pagerange: The impact of shear on the rotation of Galactic
plane molecular clouds–A

##  1 Introduction

The efficiency and rate at which molecular gas is converted into stars
determine the evolution and the observable properties of galaxies. Compressive
motions associated with large-scale instabilities (gravitational instabilities
upon entering regions with differing densities, shear originating from the
differential galactic rotation, expanding superbubbles created by supernova
explosions Elmegreen, 1995; McKee & Ostriker, 2007) induce the transition from
atomic to molecular gas in the diffuse interstellar medium ISM and the
formation of giant molecular clouds.

In particular, the Kelvin–Helmholtz Instability (KHI) is a well-known
instability arising at the interface of shearing fluids (Drazin P., 1981). In
the ISM, the presence of shearing flows can lead to the formation of KHI and
may cause the formation of clouds. KHIs may induce rotation (Fleck & Jr, 1989)
and contribute to turbulence and/or change the properties and structures of
the molecular cloud through turbulence and mixing (Berné et al., 2010; Röllig
et al., 2011; Sahai et al., 2012; Meidt et al., 2018, 2019). These mechanisms
lead to the formation of shock-bounded layers between convergent flows, a
process that induces fragmentation through non-linear instabilities (Vishniac,
1994). Numerical simulations of this scenario show that fully developed
turbulence arises in the shock-driven layers (Hunter et al., 1986; Klein &
Woods, 1998; Heitsch et al., 2009; Inoue & Inutsuka, 2012; Pudritz & Kevlahan,
2013; Inutsuka et al., 2015). This turbulent state is maintained throughout
the duration of a stream collision and its fragmentation into molecular
clouds. Cloud structure can also be changed by cloud-cloud collisions, which
yield larger and more massive clouds (Tan, 2000).

The internal turbulence of molecular clouds originates from a dissipative
energy cascade in compressible turbulent flows. At every scale, the fraction
of the energy that is not dissipated through shocks is transmitted to smaller-
scale structures (Kornreich & Scalo, 2000). In this framework, the relatively
high star-formation efficiency (SFE) observed in disc clouds is linked to the
prevalence of compressive (curl-free) turbulent modes. In contrast, the low
SFE that characterises clouds in the Central Molecular Zone (CMZ) is related
to the shear-driven solenoidal (divergence-free) component (Longmore et al.,
2013). A similar analysis of the Orion B molecular cloud (Orkisz et al., 2017)
found that the turbulence is mostly solenoidal, consistent with the low star-
formation rate associated with the cloud. These solenoidal modes are, however,
position-dependent and vary with scale within the Orion B cloud, with motions
around the main star-forming regions being strongly compressive. Thus,
significant inter-cloud variability of the compressive/solenoidal mode
fractions may be a decisive agent of variations in the SFE.

This framework suggests that the magnitude of shear in galaxies or the shear-
induced cloud-cloud collisions contributes to the SFE and star formation rate
(SFR Silk, 1997; Tan, 2000). Models (Tan, 2000) and simulations (Weidner et
al., 2010; Colling et al., 2018) yield contrasting evidence on the relation
between enhanced/reduced SFR and high/low shear. Although, shear-induced
cloud-cloud collisions promote higher SFRs, accounting for magnetic fields and
stellar feedback in hydrodynamical simulations seems to complicate the
relationship between star formation and shear (Colling et al., 2018). If a
weaker strength of shear should enable higher rates of star formation,
feedback from the first generation of stars will dissipate the gas in
molecular clouds over shorter timescales, resulting in lower SFEs. A study
(Watson et al., 2012) of 20 bulgeless galaxies showed no correlation between
SFE and the galaxies’ circular velocities. Irregular galaxies also show poor
correlations between the shear magnitude and the SFRs (Hunter et al., 1998).

The analysis of shear-quantifying parameters in a sample of clouds in the
Galactic Ring Survey (Jackson et al., 2006a) found no evidence that shear
plays a significant role in opposing gravitational collapse. In addition, the
shear parameter of the clouds (see sub-section 4.2) does not depend on the
Galactic environment and no correlations between this measure (estimated on a
Galactic rotation model) and several indicators of their star formation
activity was found (Dib et al., 2012).

If shear enhances the solenoidal modes of turbulence in clouds by inducing a
velocity gradient in molecular clouds, it could be a factor responsible for
the observed decline in the relative fraction of power in turbulent solenoidal
modes (the solenoidal fraction) with Galactocentric distances (Rani et al.,
2022, shear being assumed to be stronger closer to the Galactic centre) in
the^13CO/C^18O ($J=3-2$) Heterodyne Inner Milky Way Plane Survey, CHIMPS Rigby
et al. (2016). In this framework, shear could also be a major contributor to
the negative correlation between the solenoidal fraction and the SFE in CHIMPS
Rani et al. (2022). If this were the case and shear really induced a full-
scale velocity gradient in molecular clouds, we ought to be able to quantify
this effect by considering the overall rotation of molecular clouds. Shear-
induced rotation would then also introduce full-scale solenoidal modes.
However, although a cloud’s overall rotation indeed contributes to the
increase of the solenoidal modes over the entire structure (see below), local
vorticity and the gas motions in larger, looser, more rarefied envelopes also
result in higher values of the overall solenoidal modes (Orkisz et al., 2017).
The solenoidal fraction, used in our studies to quantify the solenoidal
turbulence in fact, provides an estimate of the relative power in the
solenoidal component of turbulence within an individual cloud without
distinguishing how these motions originate.

Any cloud rotation induced by shear related to Galactic rotation would
manifest as an alignment of the clouds’ axes of rotation with the axis of the
Galaxy. This alignment would then be lost with the diminishing impact of shear
at larger Galactocentric distances.

In this work, we consider the sample of CHIMPS clouds produced by (Rani et
al., 2022) together with a novel sample of ^12CO(3-2) sources in the CO High
Resolution Survey, COHRS (Dempsey et al., 2013; Park et al., 2022) to quantify
the strength of shear through induced cloud rotation. Furthermore, making use
of shear parameters defined by Dib et al. (2012), we attempt to find a
connection between shear, SFE and the solenoidal fraction in CHIMPS clouds.

Section 2 provides a brief description of the surveys chosen and the
construction of a new COHRS catalogue, including emission extraction and
distance assignments. Section 4.1 introduces the methods used to identify
velocity gradients and the directions of the rotation axes in first-moment
maps. It also provides the definition of a parameter used to quantify the
effect of shear. Finally, the results of the analysis are presented and
discussed in Sections 5 and 6.

##  2 Data

###  2.1 Surveys

The ^13CO/C^18O ($J=3-2$) Heterodyne Inner Milky Way Plane Survey (CHIMPS) is
a spectral survey of the $J=3-2$ rotational transitions of ^13CO at 330.587
GHz and C^18O at 329.331 GHz. The survey encompasses approximately 19 square
degrees of the Galactic plane, ranging in longitudes from
$27\aas@@fstack{\circ}5$ to $46\aas@@fstack{\circ}4$ and with latitudes
$|b|<0\aas@@fstack{\circ}5$. It was conducted with an angular resolution of 15
arcseconds using observations carried out at the James Clerk Maxwell Telescope
(JCMT) in Hawaii (Rigby et al., 2016). Both isotopologues were observed
concurrently (Buckle et al., 2009) using the Heterodyne Array Receiver
Programme (HARP) together with the Auto-Correlation Spectral Imaging System
(ACSIS). The data collected are structured into position-position-velocity
(PPV) cubes, each with velocity channels of 0.5 km s-1 and a bandwidth of 200
km s-1. The velocity range varies, covering from $-50<v_{\rm LSR}<150$ km s-1
at $28^{\circ}$ to $-75<v_{\rm LSR}<125$ km s-1 at $46^{\circ}$, in order to
accommodate the Galactic velocity gradient associated with the spiral arms in
the kinematic local standard of rest (e.g., Dame et al., 2001). The mean root
mean square (rms) sensitivities of the ^13CO survey are $\sigma(T_{\rm
A}^{*})\approx 0.6$ K per velocity channel, while for C^18O we have
$\sigma(T_{A}^{*})\approx 0.7$ K, where $T_{A}^{*}$ represents the antenna
temperature corrected for ohmic losses inside the instrument, spillover,
rearward scattering, and atmospheric attenuation (Rigby et al., 2016).

The CO Hi-Resolution Survey (COHRS) mapped the ^12CO ($3-2$) emission in the
Inner Milky Way plane, covering latitudes
$10\aas@@fstack{\circ}25<l<17\aas@@fstack{\circ}5$ with longitudes
$|\,b\,|\leq 0\aas@@fstack{\circ}25$ and
$17\aas@@fstack{\circ}5<l<50\aas@@fstack{\circ}25$ with $|\,b\,|\leq
0\aas@@fstack{\circ}25$ (Park et al., 2022). This particular region was
selected to match a set of important surveys including CHIMPS, the Galactic
Ring Survey (GRS; Jackson et al., 2006b), the FOREST Unbiased Galactic plane
Imaging survey with the Nobeyama 45-m telescope (FUGIN; Umemoto et al., 2017),
the Galactic Legacy Infrared Mid Plane Survey Extraordinaire (GLIMPSE;
Churchwell et al., 2009), the Bolocam Galactic Plane Survey (BGPS; Aguerre et
al., 2011), and the Herschel Infrared Galactic Plane Survey (Hi-GAL; Molinari
et al., 2016). COHRS observations were also performed at JCMT with HARP at
$345.786$ GHz and ACSIS set to a 1-GHz bandwidth, yielding a frequency
resolution of $0.488$ MHz ($0.42$ km s-1). The survey covers a velocity range
between $-30$ and $155$ km s-1, with a spectral resolution of $0.635$ km s-1
and angular resolution of $16.6$ arcsec (FWHM), producing a mean rms of
$\approx 0.7$ K in $T_{A}^{*}$.

The COHRS data (second release) are publicly
available111https://doi.org/10.11570/22.0078. In our analysis, we construct a
sub-sample of the full set of COHRS sources by only considering those COHRS
sources that contain the emission peaks of CHIMPS clouds. This set comprises
452 clouds. Hence this catalogue will be simply referred to as the COHRS
catalogue/survey.

##  3 Constructing a new COHRS catalogue

###  3.1 Cloud extraction

To identify molecular clouds in the COHRS data, we employ the Spectral
Clustering for Interstellar Molecular Emission Segmentation (SCIMES) algorithm
(Colombo et al., 2015). This image-segmentation method encodes the global
hierarchical structure of emission within a molecular-line datacube into a
dendrogram. This emission dendrogram is produced through the Python package
for astronomical dendrograms (Astrodendro, Astropy Collaboration et al., 2013,
2018). SCIMES then uses similarity criteria to analyse the dendrogram by
recasting it as a weighted complete graph (with vertices corresponding to the
leaves of the dendrogram and weighted edges representing the chosen affinity
relations between the leaves). The SCIMES algorithm then uses spectral
clustering on the Laplacian of the affinity matrix representing the graph to
partition the dendrogram into separate components. These clusters define a
segmentation of the emission into individual clouds.

Because of the variable weather conditions and the varying number of active
receptors during the 4 semesters of COHRS observations, the original cubes do
not present an entirely uniform sensitivity across the whole survey (Park et
al., 2022). To avoid high-noise regions being incorrectly identified as clouds
(i.e. false positives) and to prevent the loss of real signal-to-noise sources
in regions of low background (false negatives), the SCIMES extraction is
performed on the signal-to-noise ratio (SNR) cubes instead of brightness-
temperature data (Moore et al., 2015; Eden et al., 2017). This novel COHRS
catalogue thus differs from the one constructed by Colombo et al. (2019) for
the first COHRS data release for which the actual emission maps (masked for
given SNR thresholds) were used for the SCIMES segmentation.

Since the area covered by both CHIMPS and COHRS is too large to be analysed as
a single datacube and since our analysis of shear focuses on the largest ^12CO
structures in the emission as they are more likely to be impacted by the
differential rotation of the Galaxy, we organise the original datacubes into
18 regions of $4^{\circ}$ longitude each, with resolution degraded by a factor
2 in all 3 axes. Each pair of adjacent regions overlaps by $2^{\circ}$. This
wide overlap allows for any source in contact with the edges of any region to
be completely included and accounted for in the adjacent one (see below).
After the extraction, only clouds that contained at least one voxel with SNR
$\geq 10$ are retained. The parameters defining the emission dendrogram are
chosen as multiples of the background $\sigma_{\mathrm{rms}}$ (with
$\sigma_{\mathrm{rms}}=1$ in SNR cubes). We set each branch of the dendrogram
to be defined by a SNR change (min_delta) of $3$ and to contain at least 5
voxels ($\mathtt{min\\_npix}=5$). Any value of the SNR below
$3\sigma_{\mathrm{rms}}$ ($\mathtt{min\\_val}=3$) is not considered.

Figure 1: Prescription for cloud removal in the overlapping area of adjacent
regions (between edge A and B in the figure). In each region, we remove all
clouds in contact with the edges of the field of observation. Clouds that
touch the B edge are assigned to the right region (light green area), while
those t between edges B and C are assigned to the next region (grey and shaded
green areas). The process is repeated on all regions with the exception of the
last region on the right, for which all the clouds between A and B are kept.
Green ticks and red crosses indicate the clouds that are removed and those
that are left, respectively.

The catalogue is rid of spurious sources and noise artefacts that are left
after extraction by applying an additional filter. This mask removes those
sources that either cover fewer than 4 pixels in each spatial direction or
four velocity channels. This requirement ensures that each cloud is fully
resolved in each direction and that the selection does not include sources
with too small a field size.

Clouds touching the edges of the field of observation are removed from the
catalogues. To avoid double counting of sources in the overlapping areas
between regions, the sources to remove in each region are selected according
to the recipe shown in Figure 1. The two-degree overlap was chosen after
visual inspection to ensure that the recipe would yield a unique set of
sources for all the 18 regions COHRS is divided into. The final COHRS
catalogue comprises 3271 sources.

###  3.2 Distances

The distances and physical properties of the sources identified through the
segmentation of CHIMPS are taken from the SCIMES catalogue introduced in Rani
et al. (2023). Helio- and Galactocentric distances to COHRS sources are
assigned through direct comparison with the aforementioned CHIMPS catalogue.
COHRS and CHIMPS sources are matched by considering the position of the COHRS
peak emission in the CHIMPS segmentation assignment cubes. A source whose peak
lies within a CHIMPS cloud is ascribed the distance of that cloud. Unassigned
sources are not required in the present study, however they can be
givendistances by calculating their position with a Galactic rotation curve
model (Brand & Blitz, 1993; Reid et al., 2016).

###  3.3 Size and mass

We estimate the size of the CHIMPS and COHRS sources by considering an
‘equivalent’ radius that approximates the spatial extension of a cloud.
Following Rigby et al. (2019), we define the equivalent radius
$R_{\mathrm{eq}}$ as the radius of the circle whose area ($A_{c}$) is
equivalent to the projected area of the source scaled to its distance,

$R_{\mathrm{eq}}=\sqrt{A_{c}/\pi}.$ (1)

The values of the equivalent radii associated with the SCIMES and COHRS
sources were calculated directly from the values of the exact areas produced
by the Astrodendro dendrogram statistics tools. Figure 2 displays the
distribution of the equivalent radii of sources in the two surveys. Although
12CO (3–2) emission detected some slightly larger structures than 13CO (3–2),
the mean size of the sources in the two surveys takes on similar values. We
notice that the lower optical depths of ^13CO imply that, although these
clouds are more centrally concentrated, $R_{\mathrm{eq}}$ does not account for
the intensity distribution, and so it returns an upper limit for the cloud
size. Due to the high opacity of ^12CO and a resulting lower effective
critical density, the ^12CO transition probes significantly lower H_2 volume
densities in the clouds than ^13CO (Tang et al., 2013; Roueff et al., 2020;
Mazumdar et al., 2021). Therefore, the COHRS map delineates broader scales,
whereas the emission of ^13CO is responsive to gas with higher column density,
thus delineating the denser, more compact clumps. As larger structures are
expected to be more susceptible to the impact of shear, the COHRS sample is
instrumental in revealing any preferential rotation induced by shear.

The mass of CHIMPS clouds can be then estimated through the column-density
cubes (Rigby et al., 2019). The H_2 mass of the cloud is estimated by
considering the mean mass per H_2 molecule, taken to be 2.72 times the mass of
the proton, accounting for a helium fraction of 0.25 (Allen, 1973), and an
abundance of $10^{6}$ H_2 molecules per ^13CO molecule (Draine, 2011).

COHRS masses are expressed in terms of the molecular gas luminosity and
calculated through a ^12CO luminosity-mass conversion factor
$M=\alpha_{\mathrm{CO}}L_{\mathrm{CO}}$, with
$\alpha_{\mathrm{{}^{12}CO(1-0)}}=4.35\,\mathrm{M}_{\odot}\,\mathrm{pc}^{-2}\,\mathrm{km}^{-1}\mathrm{s}$,
assuming a mean molecular weight of $2.72m_{\mathrm{H}}$ per hydrogen molecule
(Colombo et al., 2019).

Figure 2: Distribution of equivalent radii across CHIMPS (red) and COHRS
(blue) matching samples. The vertical lines indicate the mean values.  Figure
3: Distribution of masses across CHIMPS (red) and COHRS (blue). The vertical
lines indicate the mean values.

The distribution of masses in COHRS (Fig. 3) peaks at higher values and
exhibits an overall greater number of sources at higher mass values than
CHIMPS. This behaviour may be enhanced by the different methods used in the
mass estimation.

##  4 Cloud rotation, shear and turbulence

###  4.1 Velocity gradients and rotation angles

If shear is the dominant force behind cloud rotation, at least at short
Galactocentric distances where shear is stronger, we should expect clouds to
rotate with axes parallel to the axis of rotation of the Galaxy (but with
opposite direction, see Fig. 4). Cloud rotation gives rise to a velocity
gradient across a molecular cloud. Although several processes (such as
turbulence Burkert & Bodenheimer, 2000) may induce velocity gradients on
different scales in molecular clouds, we assume a full-scale gradient be an
indicator of cloud rotation as it may be induced by Galactic shear. Measuring
the overall velocity gradient is the most reliable and systematic approach to
the detection of rotation in large samples of clouds (Braine et al., 2018).
The evaluation of the velocity gradient and the subsequent determination of
the direction of the cloud’s axis of rotation is performed in two steps.
First, for each cloud, we calculate the first moment of the velocity of the
emission map in the frame of reference of the centre of mass of the cloud
(thus by shifting the velocity axis by the velocity of the cloud’s centroid)

$v(l,b)=\frac{\sum_{i}I_{i}v_{i}dv}{\sum_{i}I_{i}dv},$ (2)

where all non-zero emission voxels $i$ in the cloud contribute to the
summations with their velocity $v_{i}$ and emission $I_{i}$. All velocity
channels have the same size $dv$. This yields an emission-weighted velocity
value for each position ($l,b$) in the cloud, Considering a frame of reference
centred at the average values of the map, we now fit a plane (Imara & Blitz,
2011):

$v(l,b)=C_{1}l+C_{2}b+C_{3}$ (3)

with $C_{1}=\partial v/\partial l$, $C_{2}=\partial v/\partial b$, being the
velocity gradients in $l$ and $b$, and $C_{3}$ is a constant ($C_{3}=0$ in the
frame of reference of the centre of mass of the cloud).

The magnitude of the linear velocity gradient is then

$\Omega=\frac{(C_{1}^{2}+C_{2}^{2})^{\frac{1}{2}}}{d},$ (4)

where $d$ is the distance of the source (see also Fig. 8 and 9).

The direction of the velocity gradient (direction of increasing velocities)
with respect to the positive direction of Galactic longitude axis (in the
$l-b$ plane) can be estimated as the angle

$\theta=\arctan\left(\frac{C_{2}}{C_{1}}\right).$ (5)

The angle $\theta$, measured in degrees also represents the angle between the
axes of cloud rotation, perpendicular to the direction of the velocity
gradient, and the rotation of the Galaxy as shown in Fig. 4.

Since we lack information on the inclination $i$ of the individual clouds to
our line of sight, the measurement of the velocity gradient underestimates the
actual value by a factor $\sin(i)^{-1}$ (Phillips, 1999; Imara & Blitz, 2011).
Thus, the values of $\theta$ calculated through equation 5 represent a lower
bound for the sample, with the alignment with the Galactic rotation axis
worsening when the actual inclination of the cloud rotation is considered. The
lack of information on cloud inclination on the other hand leads to
underestimating the measured magnitude of the velocity gradient ($\Omega$)
and, consequently, the angular velocity of a cloud’s rotation (Imara & Blitz,
2011), making these measures much less reliable than the rotation angle for
the proposed analysis of shear . The method presented in this section focuses
on determining the direction of rotation of each cloud in the sample. As the
direction of the axis of rotation is ultimately determined by the overall
direction of the velocity gradient, this, in turn, accounts for the strengths
of the internal motion of the substructure within the cloud.

Figure 4: Definition of the direction of rotation of a cloud (see text). The
angle $\theta$ is defined with reference to the direction of rotation induced
by Galactic shear.

In our study, we consider the angle $\theta$. defined in Fig. 4 as
$\theta=90^{\circ}-\tilde{\theta}$, with $\tilde{\theta}$ derived from fitting
the first-moment map with a plane, as in equations 3 and 5.

The angle $\theta$ quantifies the difference in orientation between the
direction of cloud rotation and the direction of shear-induced rotation.

###  4.2 Quantifying the impact of shear

Assuming that overdensities in the molecular ISM accrete mass through
streaming motions along magnetic field lines, their growth then depends solely
on the interplay between gravity and the local level of galactic shear
(Elmegreen, 1993; Hunter et al., 1998). In this framework, since magnetic
fields can transfer the angular momentum away from the cloud, self-gravity and
shear play a more relevant role in core formation than the competition between
self-gravity, pressure, and the Coriolis force (which is represented by the
Toomre-Q parameter (Toomre, 1964).

In a scenario in which mass accretion i is determined by the competition
between their self-gravity and the local strength of Galactic shear, the
growth rate of the density perturbations can be expressed as

$P_{r}=\frac{\pi G\Sigma}{\sigma_{v}}\ {\rm s}^{-1},$ (6)

where $\Sigma$ is the local gas surface density, $\sigma_{v}$ the velocity
dispersion, and $G$ the gravitational constant. Perturbations grow most
efficiently when $P_{r}<|1/A|$, where $A$ is the Oort constant

$A=0.5\left(\frac{V}{R_{\mathrm{gc}}}-\frac{dV}{dR_{\mathrm{gc}}}\right),$ (7)

where $R_{\mathrm{gc}}$ is the Galactocentric distance of the source and $V$
the rotation velocity of the gas at a given Galactocentric radius. In
particular, for a source of size $L$ with centroid located at
$R_{\mathrm{gc}}$.

$A=0.5\left(\frac{V(R_{\mathrm{gc}})}{R_{\mathrm{gc}}}-\frac{|V(R_{\mathrm{gc}}+L/2)-V(R_{\mathrm{gc}}-L/2)|}{L}\right).$
(8)

From the Oort constant, we can define a critical surface density; this is the
threshold beyond which a density perturbation in the ISM becomes so
significant that it cannot be erased by shear:

$\Sigma_{\mathrm{sh}}=\frac{\alpha_{A}A\sigma_{v}}{\pi G},$ (9)

with $\alpha_{A}=\ln(Q)/2$. The factor $Q$ represents the growth factor that
perturbations in the diffuse ISM must have to overcome the disruptive impact
of shear. Hunter et al. (1998) quantified $Q$ to equal 100. This value yields
the density contrast between the diffuse ISM with densities $\sim
0.1-1\,\mathrm{cm}^{-3}$ and the molecular phase $\gtrsim
100\,\mathrm{cm}^{-3}$.

The critical surface density can thus be used to define a shear parameter for
gravitational instabilities:

$S=\frac{\Sigma_{\mathrm{sh}}}{\Sigma}=\frac{\alpha_{A}A\sigma_{v}}{\pi
G\Sigma},$ (10)

Shear will disrupt density perturbations at values of $S>1$ while being
ineffective at $S<1$.

The quantities that enter the definition of $S$ above are taken from the
catalogues produced by SCIMES (in particular the velocity dispersion and the
exact area of each source, adjusted for the distance). The masses in CHIMPS
are derived from the column density data cubes by scaling each voxel in the
source with the assigned distance. The surface density is then taken to be the
ratio between the mass and the exact projected area of the source (Rani et
al., 2023). COHRS surface densities are calculated using the luminosity masses
reported in the COHRS catalogue (Park et al., 2022).

###  4.3 Solenoidal fraction

Our turbulence analysis is based on the the statistical method developed by
Brunt et al. (2010) and Brunt & Federrath (2014), which allows us to quantify
the relative fraction of power in the solenoidal modes of turbulence present
in a molecular cloud from emission and column density observations. The main
idea behind the method is to reconstruct the properties of a three-dimensional
source from the information contained in its observed two-dimensional line-of-
sight projection.

According to the Helmholtz theorem, a three-dimensional vector field can be
decomposed into a divergence-free (solenoidal or transverse). component and a
curl-free (compressive, parallel) component. It can be shown through symmetry
arguments and the local orthogonality of the Helmholtz components that, under
the assumption of statistical isotropy, the projected line-of-sight component
of the three-dimensional field is proportional to the solenoidal component of
the full field in $k_{z}=0$ cut in Fourier space. The solenoidal fraction or
the amount of power in the solenoidal modes of the full field is then defined
as the ratio of the variances of the projected component and the full field,
which can be written in terms of the power spectra of these fields (2D and 3D)
through the Parceval theorem. Therefore, the power spectrum of the projected
component is a direct measure of the power spectrum of the solenoidal
component of the full field. In the case of optically thin gas, this relation
translates naturally into a function of the power spectra of the zeroth
($W_{0}$) and first ($W_{1}$) moments of velocity in which the three-
dimensional field in question is the momentum density’,$\rho\mathbf{v}$. In
emission data cubes, the solenoidal fraction, $R$ of each molecular cloud can
be practically estimated as

$R=\Bigg{[}\frac{\langle W_{1}^{2}\rangle}{\langle
W_{0}^{2}\rangle}\Bigg{]}\Bigg{[}\frac{\langle W_{0}^{2}/\langle
W_{0}\rangle^{2}\rangle}{1+B_{1}(\langle W_{0}^{2}\rangle/\langle
W_{0}\rangle^{2}-1)}\Bigg{]}\Bigg{[}g_{21}\frac{\langle W_{2}\rangle}{\langle
W_{0}\rangle}\Bigg{]}^{-1}B_{2}$ (11)

where

$B_{1}=\frac{(\sum_{k_{x}}\sum_{k_{y}}\sum_{k_{z}}f(k))-f(0)}{\sum_{k_{x}}\sum_{k_{y}}f(k))-f(0)},$
(12)

and

$B_{2}=\frac{\sum_{k_{x}}\sum_{k_{y}}\sum_{k_{z}}f_{\perp}(k)\frac{k_{x}^{2}+k_{y}^{2}}{k^{2}}}{\sum_{k_{x}}\sum_{k_{y}}f_{\perp}(k)}.$
(13)

Figure 5: Distribution of the shear parameter $S$ with Galactocentric distance
in the CHIMPS (red) and COHRS (blue) samples. The vertical and horizontal
solid black lines mark the $5.5$-kpc and $S=1$ values respectively. These
reference values are used to define a $R_{\mathrm{gc}}$-limited and an
$S$-limited subsample of CHIMPS sources (see text). The right panel shows the
binned means and the standard errors of the mean (S.E.M.) of the distributions
of $S$ over the Galactocentric distance.

with $f(k)$ and $f_{\perp}(k)$ being the angular (azimuthal) averages of the
power spectra of the zeroth and first moments (notation after Orkisz et al.,
2017). The angle brackets denote the spatial averages of the velocity moments
$W_{0}$, $W_{1}$, and $W_{2}$, calculated in the reference frame of the centre
of mass of the cloud. The statistical correction factor $g_{21}$ accounts for
the correlations between the variations of $\rho$ and $\mathbf{v}$. If $\rho$
and $\mathbf{v}$ are uncorrelated, $g_{21}=1$. The constant $g_{21}$ can be
expressed In terms of density, velocity and the spatial average of the density
$\rho_{0}$, by the variance of the three-dimensional volume density
$\langle(\rho/\rho_{0})^{2}\rangle$

$g_{21}=\frac{\langle\rho^{2}v^{2}\rangle/\langle\rho^{2}\rangle}{\langle\rho
v^{2}\rangle/\langle\rho\rangle}=\Bigg{\langle}\frac{\rho^{2}}{\rho_{0}^{2}}\Bigg{\rangle}^{\epsilon},$
(14)

where $\epsilon$ is a is a small positive constant which is the exponent of
the power law expressing the relation between the variance of the velocity
$\sigma_{v}^{2}$ and the density $\rho$ (Brunt et al., 2010; Brunt &
Federrath, 2014). In our study of the CHIMPS sample, we use the values of the
solenoidal fraction published by Rani et al. (2022). As the COHRS ^12CO
emission violates the requirement of being optically thin, the COHRS source
will not be considered when discussing the solenoidal fraction.

##  5 Results

Fig. 5 shows how the shear parameter $S$ changes with Galactocentric distance
in the CHIMPS and COHRS samples. The plot suggests that in both samples shear
has a stronger impact on clouds at shorter distances from the Galactic centre,
declining outwards. Fig. 5 also emphasises source-crowding within the main
spiral arms, with large peaks in source concentration at $\sim 4.5$ and $\sim
6.5$ kpc. These are the locations of the Scutum and Sagittarius arms seen from
the Galactic Centre. The smaller peak at $\sim 7.5$ kpc corresponds to the
Perseus arm. These areas encompass the broadest ranges of $S$. The spikes in
$S$ observed at spiral arms location result from the larger variety of clouds
collected in these regions.

Figure 6: Distributions of the inclinations of the axes of rotation of the
sources in the CHIMPS (left panel) and COHRS (right panel) samples with
respect to the axis of rotation of the Galaxy.

The greatest change in $S$ occurs between $3.5$ (the shortest Galactocentric
distance probed by CHIMPS) and $5.5$ kpc. COHRS sources exhibit larger values
of $S$ overall, but exhibit the same trend with distance as their CHIMPS
counterparts. In both samples, $S$ peaks at the inner edge of the Inner Galaxy
($\sim 3$ kpc) and falls off with increasing distance from the Galactic
centre. This trend seem to reflect the behaviour of the solenoidal fraction
described by Rani et al. (2022), where the relative fraction of power in the
solenoidal modes of turbulence peaks at $3.5-4$ kpc, the edge of the region
swept by the rotation of the Galactic bar, and declines outwards with a
shallow gradient (see Appendix A). The similarity in the behaviour of $R$ and
$S$ with Galactocentric distance suggests a connection between them.

Figure 7: Distributions of rotation angles with Galactocentric distance.
Figure 8: Top panel: Top panel: Cloud rotation angles (as defined in Fig. 4 as
a function of the solenoidal fraction $R$ for the full CHIMPS survey (red
dots) and $R_{\mathrm{gc}}$-limited subsample (lime triangles). Middle panel:
The positive correlation between the shear parameter ($S$) and the solenoidal
fraction ($R$) in CHIMPS clouds. The picture highlights the higher values of
both $S$ and $R$ in the $R_{\mathrm{gc}}$-limited sample. Bottom panel:
Distribution of the magnitude of the velocity gradients as a function of the
solenoidal fraction $R$ for the full CHIMPS survey (red dots), the
$R_{\mathrm{gc}}$-limited subsample (green triangles), and the S-limited
samples (cyan triangles). Figure 9: Left panel: Distribution of the magnitude
of the velocity gradients as a function of the equivalent radii of the sources
for the full CHIMPS survey (red dots), the $R_{\mathrm{gc}}$-limited subsample
(green triangles), and the S-limited samples (cyan triangles). The
distribution in the COHRS sample is shown in the right panel.

Where $S\geq 1$, shear is strong enough to disrupt the
perturbations/overdensities that seed star formation. It is in sources with
$S$ above this threshold that shear is most likely to induce an overall cloud
rotation. In the analysis that follows, we will consider a subset of CHIMPS
sources defined by $S\geq 1$ (hence referred to as the ‘$S$-limited’ set).
This set contains the sources for which shear can potentially disrupt the
gravitational collapse of clouds. We also investigate the set of sources with
Galactocentric distances between 3.5 and 5.5 kpc. This
’$R_{\mathrm{gc}}$-limited’ subset comprise those sources with the greatest
variation in $S$. The former subset includes 550 sources and the latter 926.

The distribution of the angles of the clouds’ rotation axes with respect to
the anti-parallel direction of rotation of the Galaxy ($\theta$, see Fig. 4)
is shown in Fig. 6. No preferred direction of rotation is apparent in any of
the CHIMPS samples considered. The ^12CO sources show accumulations of sources
around $100^{\circ}$ and $300^{\circ}$. These directions however are not in
agreement with shear-induced rotation. Fig. 7 show the distribution of
rotation angles with the Galactocentric distance. Even in this case, no
correlation is found between distance and $\theta$ (Spearman test with p-value
$>>0.001$) suggesting that even at shorter distances from the Galactic center
no obvious alignment is achieved.

The solenoidal fraction, $R$, is positively correlated with the shear factor
(Fig. 8) in both the full CHIMPS sample (Spearman correlation coefficient
$r=0.34$, $p$-value $\ll 10^{-3}$) and the two subsamples
($R_{\mathrm{gc}}$-limited: $r=0.36$; $S$-limited: $r=0.15$, both also with
p-values $\ll 10^{-3}$). This result is in agreement with the idea that
stronger shear promotes the rotational modes in the gas by enhancing both the
overall or partial rotation of a cloud. Both $R$ and $S$ are positively
correlated with the size of the CHIMPS clouds measured by the equivalent
radius (Spearman $r=0.31$ and $0.17$, respectively, with $p$-value $\ll 0.001$
for both). The shear-size correlation is much more marked in the COHRS sample,
for which we find $r=0.72$. These correlations suggest that the outer, less
dense layers of a molecular cloud are more sensitive to shear. Although, no
correlation is found between the clouds’ overall direction of rotation and the
solenoidal fraction $R$ (Spearman test with p-value $>>0.001$, Fig. 8) in any
of the CHIMPS samples, $R$ is positively correlated with $S$ (Spearman
$r=0.34$ with $p$-value $\ll 0.001$). This correlation is also reflected in
the distribution of $S$ with Galactocentric distance (Fig. 5) which reflects
the declining values of $R$ past the edge of the Inner Galaxy shown in Rani et
al. (2022). Removing the location dependence of both $S$ and $R$ through as
partial Spearman correlation test reveals positive correlation ($r=0.3$ with
p-value $\ll 0.001$) between these the two quantities.

Fig. 8 highlights that, within the complex structure of CHIMPS molecular
clouds, the solenoidal modes of turbulence are more likely to arise from local
perturbations rather than from shear-driven global contributions. The impact
of shear on $R$ in the form of a contribution to vorticity is not detected by
the magnitude of directed gradient in the distribution of velocity in first-
moment maps. We also find a slightly negative correlation between the gradient
magnitude and $R$ in our three samples (full CHIMPS: $r=-0.13$,
$R_{\mathrm{gc}}$-limited: $r=-0.11$, $S$-limited $r=-0.17$, with p-values
$\ll 10^{-3}$) and between the gradient magnitude and the equivalent radius of
the sources (Fig. 9, Spearman $r=-0.15$, p-values $\ll 0.001$). The latter
result indicates the higher probability of finding more well-defined gradients
in smaller clouds.

##  6 Discussion and Conclusions

Molecular clouds form as a result of the condensation of lower-density atomic
ISM gas, thereby acquiring its turbulent and shear-driving motions (Meidt et
al., 2018, 2019). Galactic dynamics can thus either stabilise clouds (Meidt et
al., 2013) or compress them, facilitating star formation (Jeffreson &
Kruijssen, 2018). The solenoidal modes of turbulence, in particular, have been
shown to be linked to reduced star formation efficiency in molecular clouds
(Rani et al., 2022). The relative fraction of power in the solenoidal modes of
turbulence (the solenoidal fraction, $R$) peaks at the edge of the Inner
Galaxy (the region swept by the rotation of the Galactic bar) and declines
with a shallow gradient with Galactocentric distance. The strength of shear
originating from the differential rotation of the Galaxy follows a similar
trend (see section 5). To quantify the strength of shear, we introduced the
shear factor $S$, which defines the ability of density perturbations to grow
within the cloud. The higher the value of $S$, the more likely shear is to
disrupt the growth of over-density and gravitational collapse (Dib et al.,
2012). Both CHIMPS and COHRS clouds display the same trend with $S$ peaking at
the inner edge of the Inner Galaxy and decaying with Galactocentric distance.
The similarities between the distributions of $R$ and $S$ over Galactocentric
distances suggest that, theoretically, shear may act as a facilitator of
solenoidal turbulence. This connection is strengthened by the positive
correlation between the shear parameter and the solenoidal fraction in CHIMPS
clouds for which the solenoidal fraction has been calculated. The declining
values of $R$ and $S$ with increased Galactocentric distances also agree with
the studies of the dense gas fraction within Galactic-plane molecular clouds
(Longmore et al., 2013; Urquhart et al., 2013) pointing to the heightened
shear of the CMZ as the cause of higher turbulent gas pressure, which raises
the density threshold for star formation. A clear example of this phenomenon
is the low SFE in the cloud G0.253+0.016 which appears to be caused by a
prevalence of shear-driven solenoidal turbulence (Federrath et al., 2016).

The parameter $S$ exhibits overall higher values in COHRS, a fact that is
compatible with the larger size of the structures traced by ^12CO emission
($S$ is positively correlated with the equivalent radius $R_{\mathrm{eq}}$) in
both the CHIMPS and COHRS samples). Despite the correlation between the shear
parameter and Galactocentric distance, our study of the clouds’ rotation
(Braine et al., 2018) in both the CHIMPS and COHRS samples found no preferred
direction of rotation induced by Galactic shear (anti-parallel to the axis of
rotation of the Galaxy) at all Galactocentric distances (Fig. 7). The absence
of correlation between rotation angles and the Galactocentric distance
indicates that cloud rotation is not sensitive to the strength of shear even
at distances where $S>1$ (Fig. 5). At this value of $S$, shear becomes strong
enough to prevent the gravitational collapse of dense cores. Although this
condition is satisfied for a large portion of CHIMPS sources and the majority
of COHRS clouds at Galactocentric distances shorter than 7 kpc, the magnitude
of Galactic shear does not suffice to induce an overall cloud rotation. This
finding hints at shear within the Galactic disc not being strong enough to
induce rotation at scales at the scale of the largest clouds in the CHIMPS and
COHRS samples. The rotation of these clouds thus must be imprinted at the time
of their formation or subsequently. An alternative hypothesis would see most
clouds at distances $<7$ kpc being too short-lived (Jeffreson et al., 2018b,
a) for shear to induce a preferred rotation in a statistically significant
number of them.

Our study also found that the magnitude of the velocity gradient in the
sources is negatively correlated to both the $R$ and the equivalent radius. As
the shear due to the Galactic rotation is not producing any preferred
direction in the clouds it can not be the mechanism responsible for these
correlation, and thus ’local perturbations’ are the most likely candidates to
explain these trends. These must be greater at lower $R_{\mathrm{gc}}$,
however, so cloud or/and flow collisions are more influential at smaller
Galactic radii.

To calculate the angle of rotation with respect to the direction of the
Galactic rotation we introduce a rigid-body approximation by fitting the
velocity gradient with a plane. The negative correlation between the magnitude
of the velocity gradient and the size ($R_{\mathrm{eq}}$) of the source
emphasises that larger sources may exhibit more complex structures, such as
velocity flows that change significantly in different regions within the
cloud. In this framework, the velocity distribution appears to be dominated by
internal gas motions originating upon cloud formation (Vishniac, 1994), cloud-
cloud collisions (Tanvir & Dale, 2020) or produced by stellar feedback (Fall
et al., 2020), even in the subset of clouds with shear parameters greater than
unity. The slightly negative correlation between $R$ and the magnitude of the
velocity gradient may imply that turbulence in smaller sources with more
structured gradients is, in general, less solenoidally dominated than sources
that possess more extended envelopes (both $R$ and $S$ are positively
correlated with size). In the case of sources that extend over tens of
parsecs, shear may increase the relative fraction of solenoidal modes in the
turbulence by inducing rotation in the most extended envelopes.

Together, these results, along with the overall lack of a preferred direction
of rotation in all samples we have examined would suggest that the
differential rotation of the Galactic disc has little impact on molecular
clouds both during their formation and after they formed. The rotation of the
clouds would thus be inherited as an imprint of the instabilities (principally
Kelvin-Helmoltz instabilities) that lead to the formation of the clouds or
result from their history of mergers rather than arising from instabilities
induced by Galactic shear.

## Acknowledgements

G.P. was supported by the National Research Foundation of Korea through grants
NRF-2020R1A6A3A01100208 & RS-2023-00242652. This work was also partly
supported by the Korea Astronomy and Space Science Institute grant funded by
the Korea government(MSIT) (Project No. 2022-1-840-05).

## Data availability

The CHIMPS catalogue used for this paper are available from the archives of
the CHIMPS (Rigby et al., 2019). The COHRS catalogue constructed for the
analysis in this article is available to download from the CANFAR
archive222https://www.canfar.net/citation/landing?doi=23.0019.

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##  Appendix A The solenoidal fraction across CHIMPS sources

To facilitate the comparison between the trend shown in Fig. 5 and the
behaviour of the solenoidal fraction, $R$ across the sample considered in this
study, we reproduce the main result obtained by Rani et al. (2022). As Fig. 10
shows, $R$ peaks at the 3–4-kpc bin. Although the confirmation of this trend
is required by analysing sources at lower longitudes, this result is
consistent with the disc becoming stable against gravitational collapse at
these radii.

The 3–4-kpc distance bin marks the boundary between the inner Galaxy and the
region of influence of the Galactic bar, which in extragalactic systems has
been observed to quench star formation (James & Percival, 2016).

Figure 10:  Distributions of the solenoidal fraction with Galactocentric
distance in CHIMPS. The size of the bins is adjusted to the number of source
(Rani et al., 2022). The horizontal lines represent the mean value within each
bin, while the vertical bars indicate the standard error of the mean.