Observational evidence for local vertical constraining of H i by molecular cloud complexes

A massive molecular cloud complex represents local gravitational potential that can constrain the vertical distribution of surrounding stars and gas. This pinching effect results in the local corrugation of the scale height of stars and gas which is in addition to the global corrugation of the mid-plane of the disc. For the first time, we report observational evidence for this pinching on the H i vertical structures in the Galactic region (20 ◦ < 𝑙 < 40 ◦ ), also called W41–W44 region. The H i vertical distribution is modelled by a double Gaussian profile that physically represents a narrow dense gas distribution confined to the mid-plane embedded in a wider diffuse H i . We find that the estimate of the H i scale height distribution of wider components, shows corrugated structures at the locations of molecular complexes, as theoretically predicted in literature. While the narrow component is less affected by the pinching, we found a hint of the disc being disrupted by the active dynamics in the local environment of the complex, e.g., supernova explosions. Molecular complexes of mass of several 10 6 M ⊙ , associated with the mini-starburst region W43 and the supernova remnant W41 show the strongest evidence for the pinching; here a broad trough, with an average width of ∼ 400 pc and height ∼ 300 pc, in the disc thickness of the wider component is prominently visible. Searching for similar effect on the stars as well as in the location of other complexes in the Milky Way and other galaxies will be useful to establish this phenomenon more firmly.


INTRODUCTION
The vertical distribution of stars and gas in disc galaxies is a wellexplored topic due to its significance in our understanding of the galactic potential and dynamical effects.Evidence from an enormous amount of Galactic and extra-galactic observations established that the vertical distribution of stars and gas rather being smooth and uniform, in fact, is generally uneven in a complex manner, which is caused by multiple phenomena like disc flaring, warping, disc bending and others (Burke 1957;Hunter & Toomre 1969;Quiroga 1974;Dickey & Lockman 1990;Narayan et al. 2020;Nandakumar et al. 2022).Though most of these are identified as global phenomena that lead to perturbations of the entire disc structure, a much smaller number of studies has focused on the dynamical effects that perturb the stellar and gas vertical distribution locally, meaning over a certain region of the disc.Jog & Narayan (2001) investigate one such local dynamical effect that massive molecular cloud complexes are creating on the vertical distribution of stars and gas, which they refer to as the constraining or "pinching" of the disc.Much of the Milky Way's interstellar molecular gas is contained in Giant Molecular Clouds (GMCs) that have masses of 10 4 to 10 6 M ⊙ and sizes from tens to hundred parsecs (Dame et al. 1986;Scoville & Sanders 1987).
★ Email: meeran@iisc.ac.in † Email:nroy@iisc.ac.in ‡ Email:cjjog@iisc.ac.in § Email:kmenten@mpifr-bonn.mpg.de These are further segregated into large molecular cloud complexes or clusters (Rivolo et al. 1986).Jog & Narayan (2001) theoretically modelled the self-consistent response of the stellar and gas disc to the gravitational potential of such a complex and showed that the scale height of both the stars and H i gas becomes reduced by a factor of ∼ 3 compared to the case in which the complex was absent.They show that the constraining effect in the gas can extend over a distance of ∼ 500 pc from the complex centre along the mid-plane.Given that the magnitude of the resulting broad trough-like features in the scale height distribution is significant, it should make these features accessible to observations.
In this work, we investigate this constraining effect on the vertical distribution of H i gas surrounding molecular clouds in various complexes in the Milky Way.For this we first briefly introduce the vertical distribution in a typical galactic disc.Knowing the average vertical distribution of various disc components is crucial in understanding galactic dynamics, however, obtaining the three-dimensional distribution of the interstellar medium (ISM) through observations has its challenges in both the Milky Way and in external galaxies.Highly inclined edge-on galaxies are best for probing the vertical structures of the disc by photometry.Modelling the vertical distribution by an exponential function, De Geyter et al. (2014) reported an average scale height of the stellar disc of ∼ 510 pc for 12 edge-on galaxies.Using another sample of 34 edge-on spiral galaxies from Kregel et al. (2002), they obtained similar estimates of the stellar scale height of 570 pc.Inspecting the surface brightness profile determined in dif-ferent optical bands reveals growing evidence for an increase in the stellar scale height along their major axes (de Grĳs & Peletier 1997;Teague 2019;Kasparova et al. 2020).A flaring stellar disc is shown to be a generic feature from theoretical modelling of a multi-component galactic disc in hydrostatic balance (Narayan & Jog 2002a,b;Sarkar & Jog 2018, 2019).De Geyter et al. (2014) estimate an average scale height of 460 pc for dust emission in the same sample of 12 edge-on galaxies, similar to the values found in previous studies (Xilouris et al. 1999;Bianchi 2007).Zheng et al. (2022) studied the vertical distribution of the H i gas disc of 15 highly inclined nearby galaxies from the Continuum Halos in Nearby Galaxies -an EVLA Survey (CHANG-ES; Irwin et al. 2012).The photometric H i scale height they measured for this sample ranges from ∼ 700 pc to 2 kpc.Assuming the disc to be in hydrostatic equilibrium, the H i scale height of low inclination galaxies can also be estimated from observables like gas velocity dispersion and surface density (Olling 1995(Olling , 1996;;Bacchini et al. 2019).The H i scale height of several spiral and dwarf galaxies has been estimated by Patra (2020a) and Patra (2020b), who found values around a few 100 parsecs near the centre, which extended to a few kiloparsecs in outer regions.A recent study by Randriamampandry et al. (2021) on H i rich galaxies selected from the BLUEDISK, THINGS, and VIVA surveys found that the average scale height of H i within the extent of the stellar disk is ∼ 400 pc.A similar analysis has been carried out to estimate the molecular scale height by a few authors (Wilson et al. 2019;Patra 2019;Bacchini et al. 2019).Molecular gas is found to be much more constrained in thin discs around the mid plane, with the H 2 distributions having scale heights ranging from ∼ 20 pc to 100 pc in the inner parts that flare up to ∼ 200 pc in the outer regions.Physically, the scale height of a disc component is a measure of the equilibrium between the local vertical gravitational force and the pressure gradient, as given by the equation of hydrostatic equilibrium (e.g., Rohlfs 1977).A realistic disc consisting of stars and gas that are gravitationally coupled within the dark matter halo has been modelled theoretically.This model gives a self-consistent vertical distribution including the scale height, for each of the disc components, in good agreement with the observed data (Narayan & Jog 2002b;Kalberla 2003;Sarkar & Jog 2018).
In the Milky Way, the scale height of stellar and gas components is estimated directly from their number density distribution.Using the photometric data from various surveys like SDSS, XSTPS-GAC etc., numerous studies have derived the Galactic structure parameters like scale height and scale length (Bland-Hawthorn & Gerhard 2016;Chen et al. 2017).These results suggest the presence of a thin disc and a thick disc component for the stellar distribution with exponential scale heights of ∼ 250 pc and ∼ 700 pc, respectively.With the advent of extensive observational data of the Galactic plane by numerous surveys at multiple wavelengths including the infrared, optical and radio ranges, multiple attempts have been made to map the three-dimensional distribution of dust and gas in the Galactic disc (Drimmel & Spergel 2001;Nakanishi & Sofue 2003;Misiriotis et al. 2006;Levine et al. 2006;Marshall et al. 2006;Jones et al. 2011;Marasco et al. 2017;Li et al. 2018;Sofue 2022).The vertical structure of the Galactic H i disc, studied by various authors, is found to have a thickness that remains smooth in an inner radius of < 5 kpc and increases radially towards the outer Galaxy (Oort 1962;Lockman et al. 1986;Boulares & Cox 1990;Dickey & Lockman 1990;Malhotra 1995;Narayan & Jog 2002b;Nakanishi & Sofue 2003;Kalberla & Dedes 2008;Kalberla & Kerp 2009).Additionally, the vertical distribution of gas is observed to have complex components like an extended halo.Dickey & Lockman (1990) approximate the H i vertical distribution within Galactocentric radii, , of 0.4 ⊙ ≲  ≲  ⊙ , by a combination of a double Gaussian with best fitting Full Width at Half Maximum (FWHM) of 212 pc, 530 pc respectively and an exponential function with scale height of 403 pc. ⊙ is the distance of the Sun from the Galactic center.Like in external galaxies, the Galactic molecular hydrogen has a much thinner distribution with average FWHM values starting from ∼ 60 pc near the Galactic centre that increase radially up to 140 pc beyond a radius of 2 kpc (Sanders et al. 1984;Dame et al. 1987;Clemens et al. 1988;Bronfman et al. 1988;Malhotra 1994;Dame et al. 2001;Jackson et al. 2006;Heyer & Dame 2015).From the three-dimensional analysis of Milky Way dust, Li et al. (2018) find the global scale height of the dust emission to be 103 pc, which agrees with previous results (Drimmel & Spergel 2001;Marshall et al. 2006;Jones et al. 2011).In addition to the scale height of the neutral medium, using multiple interstellar tracers, the vertical distributions of the warm ionized medium and that of the hot ionized medium in the Galactic disc were found to have exponential scale heights of ∼ 300 − 1000 pc and ∼ 2 kpc -5 kpc, respectively (Reynolds 1984;Sembach & Savage 1992;Shelton & Cox 1994;Savage et al. 1997Savage et al. , 2000;;Ferrière 2001;Langer et al. 2014).
Most of the above-mentioned papers estimate a global value or an azimuthally averaged radial profile of the scale height.However, to detect a localized phenomenon like pinching, an estimation of the disc thickness at a specific location or region of the Galaxy is essential.Despite the availability of numerous high-sensitivity Galactic gas surveys and having the advantage of resolving the structures much better than in external galaxies, three-dimensional mapping of the gas density distribution in the Milky Way is challenging due to the location of the Sun.The geometrical transformations from the observed brightness temperature, which is a function of Galactic latitude, longitude and radial velocity to a coordinate system that describes the Milky Way disc (e.g., (, , )), strongly depends on the shape of the Milky Way rotation curve (Nakanishi & Sofue 2003;Kalberla et al. 2007;Kalberla & Dedes 2008).However, this transformation requires an accurate determination of the Milky Way's rotation curve (Kalberla et al. 2007).Even with a better estimate of the rotation curve (Reid et al. 2019), due to geometrical reasons, the distance along the line of sight direction frequently cannot be uniquely determined in the inner Galaxy due to the near-far distance ambiguity (e.g., Nakanishi & Sofue 2003).However, at a tangent point, where the line of sight at a particular longitude is tangential to the circle of Galactocentric radius R, the distance to that point can be determined without any ambiguity.
In this work, we explore the vertical distribution of neutral gas in the region of Galactic longitude, , from 20 • to 40 • (hereafter the W41-W44 region), which is the pilot region of the GLOSTAR (A Global View of Star Formation in the Milky Way, Brunthaler et al. 2021) survey and the MeGaPlug (Metrewave Galactic Plane with the uGMRT, Dokara et al. 2023) survey.This region hosts several well-known extended H ii regions and supernova remnants like W41, W42, W43 and W44, first identified by their radio emission (Westerhout 1958).These sources are serendipitously situated around the tangent point in the Galactic disc and hence the W41-W44 region enables us to estimate the vertical structure parameters from observations with less uncertainty.The study by Medina et al. (2019) of the "pilot region" of the GLOSTAR-VLA survey identifies numerous extended and compact radio sources in the 28 • <  < 36 • range that include the star-forming complex W43 and the supernova remnant (SNR) W44.A complementary large-scale Galactic plane survey, MeGaPluG, covering bands in the low-frequency radio regime (< 1 GHz) is planned for the uGMRT and in a precursor study, Dokara et al. (2023) conducted uGMRT observation of the W43/W44 region in the 300 − 750 MHz frequency range.A plethora of multi-wavelength Galactic surveys with high sensitivity have been conducted or are underway, addressing many aspects of Galactic astronomy and in particular star formation on a global scale, while detailed studies addressing local dynamical effects the newly formed stars and their nascent GMCs exert on the surrounding interstellar medium (ISM) are rare.The rest of the paper is divided as follows.A brief review on the theoretical predictions of the pinching effect made by Jog & Narayan (2001) is given in Section 2. Section 3 presents details about the W41-W44 region and describes the observational data used in our analysis.The method of estimating the scale height of H i distribution in this region is explained in Section 4. Section 5 and Section 6 discuss the findings of our analysis.

VERTICAL CONSTRAINING EFFECT OF MOLECULAR CLOUD COMPLEX ON STARS AND H i GAS
A typical molecular cloud complex (also known as GMC complex) has a mass of ∼ 10 7 M ⊙ and a size of a few 100 pc, and an oblate spheroidal shape, and a central total scale height of about 120 pc (Rivolo et al. 1986).Jog & Narayan (2001) noted that such a massive, extended complex dominates its local gravitational field.The average mass density of molecular gas within a complex is ∼ 1M ⊙ pc −3 , which is six times larger than the dynamical or total mid-plane density, or the Oort limit of 0.15 M ⊙ pc −3 (Oort 1960).Thus the stars and gas in the surrounding galactic disc would respond to the additional, dominant gravitational field of the complex and would tend to get clustered towards the mid-plane, with a resulting decrease in their vertical scale heights.Jog & Narayan (2001) obtained the redistributed self-consistent stellar vertical distribution by solving together the joint Poisson equation and the hydrostatic balance equation for an isothermal stellar disc taking account of the gravitational force due to stars and the complex, while keeping the surface density of the stellar disc at a given galactocentric radius constant.The redistributed stellar distribution is shown to be constrained closer to the mid-plane, such that at the centre of the complex, the stellar mid-plane density increases by a factor of 2.6 and the corresponding vertical scale height (Half Width Half Maximum (HWHM) of the vertical stellar distribution) decreases by a factor of 3.4 compared to the undisturbed disc.Similarly the interstellar H i gas distribution also shows a similar constraining effect, here the H i gas gravity is also included in determining its vertical structure (see Fig 6 in Jog & Narayan 2001).
A surprising result is that the vertical constraining or pinching effect is felt over a large radial distance ∼ 500 pc from the complex centre.This is due to the extended mass distribution in a complex.The gravitational force due to a spheroidal complex of constant density is maximum at its centre.Hence the constraining effect of the complex normal to the mid-plane is maximum at the centre of the complex and tapers off gradually with radial distance from the centre.The H i scale height shows a significant decrease of a factor ∼ 3 at the centre of the complex, while the decrease is smaller but still ∼ 10%, even at 500 pc.The vertical scale height of stars and H i would therefore appear locally corrugated over a scale of a few 100 pc around a complex.The H i and stellar scale heights would have a broad troughlike appearance with respect to the centre of the complex, unlike a uniform scale height distribution in an undisturbed disc.
A similar vertical constraining effect has been shown to apply to the general disc distribution at any point in the Milky Way (Narayan & Jog 2002a;Sarkar & Jog 2018) with a focus on the inner and the outer Galaxy, respectively.In these papers, a self-consistent vertical distribution has been obtained theoretically for a realistic multi-component disc consisting of gravitationally coupled stars, interstellar H i and H 2 gas in the gravitational field of the dark matter halo.Due to the additional gravitational force of the other disc components and the halo, each disc component is shown to be constrained closer to the mid-plane, and the stellar density profile is shown to be steeper than sech 2 .Thus the resulting scale height values at any radius are smaller than in a single-component disc.We stress that in contrast to the case of the complex where the additional force due to the complex on the disc is localized around a complex in the disc, here the gravitational force of the disc components and the halo is not localized to a given region.Rather the net vertical distribution of all disc components in the coupled disc is constrained (w.r.t their one-component values) at each radius.Results from this model are shown to agree well with observations.In particular, the resulting H i gas scale height values as a function of radius for the inner Galaxy agree well with observations as shown by Narayan & Jog (2002a).However, the local corrugation of H i scale heights due to a molecular cloud complex as predicted by Jog & Narayan (2001) has not been observationally confirmed so far.

SAMPLE AND OBSERVATION
3.1 The W41-W44 region Dame et al. (1986) identified 26 GMCs with masses greater than 5 × 10 5 M ⊙ in the inner Galaxy (i.e., within  = −12 • and 60 • ).In the Galactic longitude range encompassing W41, W42, W43 and W44 (20 • <  < 40 • ) they found around 17 bright CO sources in their survey.A total of 11 of these GMCs have masses greater than 10 5 M ⊙ and sizes ranging from ∼ 60 pc to 120 pc (see their Table 2).In the following, we discuss the properties of these star-forming regions and SNRs in the W41-W44 regions and their associated molecular cloud complexes.W43 is a so-called mini-starburst region that is associated with one of the most massive GMC complexes in the Galaxy, which has a total H 2 gas mass of ∼ 7.1 × 10 6 M ⊙ (Nguyen Luong et al. 2011;Park et al. 2023).Nguyen Luong et al. (2011) conducted an extensive multi-wavelength study of the W43 region and identified a large coherent complex of molecular clouds within an approximate diameter of ∼ 140 pc.The overall mass contained in the high-density molecular clumps, possible sites of star formation, traced by 870 m dust emission in W43 is around ∼ 8.4 × 10 5 M ⊙ .Among the more than 20 clouds in this complex (Solomon et al. 1987;Rathborne et al. 2009), W43-Main ( = 30.8• ) and W43-South ( = 29.9• ) are the most massive and host most of the denser clumps (Park et al. 2023).Nguyen Luong et al. (2011) found that the main LSR velocity range of gas that is tightly associated with W43 ranges from 80 to 110 km s −1 .They did a Gaussian fitting in this velocity range to the 13 CO spectrum integrated over 1.8 • × 0.8 • area that covers the entire W43, giving a peak at ∼ 95.9 km s −1 and FWHM of ∼ 22.3 km s −1 .The line of sight distance to the complex has been estimated to be 6 kpc, which is placed near the intersection of the Scutum Arm and the Galactic bar (Nguyen Luong et al. 2011;Zhang et al. 2014).
The SNR W44, located at ,  = 34.7 • , −0.4 • at distance of ∼ 3.2 kpc, is surrounded by a few GMCs whose CO emission peaks at ∼ 45 km s −1 (Seta et al. 1998(Seta et al. , 2004;;Cardillo et al. 2014).A GMC complex at this location has been found to have a mass of 1.8 × 10 6 M ⊙ , a size 52 pc and V LSR = 44 km s −1 (Dame et al. 1986).An H ii region G034.8 − 0.7 is also identified in the vicinity of the SNR W44.Various millimetre and infra-red observations of this environment have shown that parts of the associated molecular clouds are shocked by the propagation of supernovae blast wave (Reach et al. 2005;Paron et al. 2009;Sashida et al. 2013).
W42 is an obscured giant H ii region, characterized as a corehalo structure with an approximate size of ∼ 4.3 pc located at  = 25.38 • and  = −0.18• (Woodward et al. 1985;Garay et al. 1993;Blum et al. 2000;Deharveng et al. 2010;Park et al. 2023). 13CO emission at LSR velocities of 58 − 69 km s −1 is ascribed to a cloud physically associated to the W42 H ii region (Anderson et al. 2009;Ai et al. 2013).The distance to W42 has been estimated as ∼ 3.8 kpc (Anderson et al. 2009).Various infrared images that trace the warm dust emission towards W42 reveal a bipolar nebular morphology of size ∼ 10 pc (Deharveng et al. 2010).

Observations and data preparation
To observationally probe the constraining effect, one needs to investigate the variation in the vertical distribution of H i gas surrounding the molecular complexes in the W41-W44 region.To trace the atomic hydrogen content, we use H i 21cm line emission, which has been extensively observed with high sensitivity in different parts of the Milky Way in multiple H i surveys (e.g., Kalberla et al. 2005;Stil et al. 2006;HI4PI Collaboration et al. 2016).Among the data for various CO rotational transition lines that trace the molecular hydrogen content in the clouds at different physical conditions, we use those of the 12 CO  = 1 − 0 line in our analysis whose, observed sky brightness commonly taken as a proxy to the molecular hydrogen content in the molecular ISM (Bolatto et al. 2013).Being the lowest rotational transition, the 12 CO line also traces the diffuse cloud structures contained in the complexes.We obtained the publicly available data from two Galactic surveys that trace the H i 21cm and 12 CO emission in the W41-W44 region, which are described in the following subsections.

H i data
To study the neutral hydrogen distribution in the W41-W44 region, we use the archival data from the all-sky database of the Galactic H i 4 survey (HI4PI, HI4PI Collaboration et al. 2016).This comprises H i 21cm observations from two surveys, the Effelsberg-Bonn H i Survey (EBHIS, Kerp et al. 2011;Winkel et al. 2016) and the Galactic All-Sky Survey (GASS, McClure-Griffiths et al. 2009;Kalberla et al. 2010;Kalberla & Haud 2015), both of which share similar sensitivity and angular resolution.The combined HI4PI data covers a radial velocity range from −600 km s −1 to 600 km s −1 with a spectral resolution of 1.49 km s −1 .The angular resolution of 16.2 ′ and a noise level of ∼ 43 mK, make HI4PI the most sensitive H i survey that covers the full sky.

CO data
Dame et al. ( 2001) released a complete survey in the 12 CO  = 1 − 0 of the entire Milky Way by combining their previous large-scale CO surveys.We use the 12 CO data taken from the First quadrant as part of this survey, which was observed with the CfA 1.2 m telescope.
The data has a velocity coverage ranging from −0.5 to 271 km s −1 with a spectral resolution of 0.65 km s −1 and an angular resolution of 7.5 ′ .The data have a sensitivity of 0.18 K covering the sky in Galactic latitudes || < 32 • which is adequate for probing the vertical structures in our analysis.

Velocity integrated intensity maps
Figure 1 shows the H i intensity distribution of the W41-W44 region integrated over the velocity range 30 − 120 km s −1 .Overlaid black contours represent the integrated 12 CO intensity map in the same velocity range.White dotted circles marked the area from which the representative H i and 12 CO spectra are extracted, which are given in figure 2. Figure 1 shows the integrated H i and CO integrated emission, which comprises emission from gas clouds located at multiple positions along the line of sight.Additionally, the individual GMC complexes are emitting CO emission at different LSR velocities, indicating that they are located at different line of sight distances.Hence to investigate the constraining effect imposed by the individual molecular cloud complexes on their surrounding gas and also to minimize the uncertainty introduced in the distance estimation, we did the following.The position-position-velocity cubes obtained from the HI4PI and CO surveys have been integrated along the velocity axis from 30 to 120 km s −1 in multiple bins, each having a width of 15 km s −1 .The dot-dashed vertical lines in figure 2 demarcate each velocity bin.Hence each intensity map essentially represents a two-dimensional "slice", normal to the line of sight direction towards the W41-W44 region.Table 1 shows the velocity range for each slice and the corresponding line of sight distance.Appendix A describes the estimation of distance from observed LSR velocity.The bottom panel of figure 2 shows the estimated line of sight distance as a function of LSR velocity at the position of W43 ( = 30.8• ).
Due to the kinematic distance ambiguity, it is difficult to separate the emission from gas located at the near and the far distances, which both contribute to emission in the same velocity range.Because the angular scale of the structures representing far emission is smaller than that coming from a closer distance, the observed thickness of the disc will be better determined by emission from a distance nearer to us.Hence in this analysis, we consider the near distance as the line of sight distance to each slice.We also note that this uncertainty is minimal at V LSR ≳ 90 km s −1 for the W41-W44 region.Unlike in other Galactic regions where the kinematic distance varies drastically with the Galactic longitude at the same LSR velocity, in the W41-W44 region the near kinematic distance remains the same (See figure A1).In other words, for a particular slice, the variation of the line of sight distance along the Galactic longitude is small enough so that we can assume it to be a constant for the entire W41-W44 region.Hence the near line of sight distance is given in table 1. Due to the geometry, the near kinematic distances increase in a linear fashion with V LSR towards the W41-W44 direction and each slice is on an average ∼ 870 pc distance apart.We estimate the distribution of H i disc thickness in different slices and the following section discusses the estimation in detail.

SCALE HEIGHT ESTIMATION
The vertical density distribution of a single component isothermal disc in hydrostatic equilibrium follows a sech 2 profile (Spitzer 1942).
In addition to the sech 2 profile, exponential and Gaussian profiles are also used in much of the literature to model the vertical distribution of the stars and gas.The scale height is defined as the HWHM of the vertical profile.The density or mass-weighted rms heights (the second moment of the density distribution) are also considered as vertical height estimation in the literature (Koyama & Ostriker 2009;Hill et al. 2012).In the following, we give the details of the approach that we choose to estimate the thickness of the disc for a particular slice.
Each slice represents a density distribution of gas, (, ), at the Galactic coordinates  and  which is located at a distance D away from us.At each Galactic longitude , we take the density distribution along the latitude  and use it to determine the disc thickness.We model the distribution by a Gaussian profile defined as follows, where the FWHM is given by √ 8 ln 2 times the standard deviation,  Gauss .Here  = D tan(b) is the height above the midplane in units of parsec. 0 is the position at which the distribution is at its peak density,  0 .The FWHM calculated from the best fitting  Gauss is considered as the estimate of H i disc thickness in this analysis.Note that the estimated FWHM is double the H i scale height of the disc.Therefore, any undulations in the H i scale height, if present, will be readily detectable in the distribution of FWHM values.We also estimate the density-weighted rms of the height,  ℎ , as follows, where z is the density-weighted mean position of the distribution.Though this is not a reliable quantity to directly compare with the scale height estimation from observations, it characterises the thickness of the disc with less bias than modelling it with a function.Thus estimated  ℎ from eq 2 multiplied by a factor of √ 8 ln 2 can be considered as a proxy to the disc thickness.We fit the vertical distribution () by a single Gaussian as given in eq. 1 and obtained the best fitting parameters  0 ,  0 and  Gauss for each value of .Hence the estimated FWHM from  Gauss (blue line with pentagon marker) with error bars as a function of Galactic longitude  is given in figure 4 for "Slice 5".The error bar represents the uncertainty of the fitting parameter estimates.The top panel shows the H i intensity map integrated over the velocity range in Slice 5 and the black contours are the respective integrated CO intensities.Assuming  = 30 • as the origin, we converted the Galactic longitudes to physical distance in units of parsec.These values are mentioned on the top of the x-axis.The green line with square markers shows the density-weighted rms height for the H i disc at each .
In the 90−105 km s −1 velocity range, the W43 complex dominates in 12 CO emission, while the GMC associated with W41 also contribute a significant amount of emission.In addition, a high velocity component towards the direction W42 is significantly contributing in this slice (See right panel of figure 2).However, the molecular clouds contributing to this velocity component are not physically connected with the H ii regions of W41 and they are located behind W42, around a distance of 3 kpc.We call this molecular cloud, which has significant 12 CO emission in slices 5 and 6, "W42-Back".From the top panel of figure 4, we see that W43 has as well compact high-density H 2 structures between  = 29 • and 31 • .However, from the best fitting FWHM of the single Gaussian profile, we do not see any hint of a decrease in the best fitting FWHM, which manifests the constraining effect.Interestingly, an apparent increase in the disc thickness is visible around the complex centre (i.e., at  = 30.6• ).Similarly, around the GMCs W41 and W42-Back, we do not see any hint of constraining effect from the best fitting FWHM.However, H i rms height distribution (green squares) shows interesting features around GMCs.The rms height variation around W43 shows a ∼ 700 pc wide trough-like feature ( = 27.5 • − 35 • ).Similarly, around the location of W41 and W42-Back, we see another trough which is of similar size between  = 21 • and 27 • .This is clearly manifesting the gravitational pull that these complexes exert on the surrounding H i gas, as discussed in Jog & Narayan (2001).The distribution of rms height in the entire W41-W44 complex eventually shows that the overall thickness of the H i disc ranges from ∼ 400 pc to 700 pc.In contrast, the FWHM from the single Gaussian fitting varies from ∼ 150 pc to 250 pc in this slice.These findings indicate the presence of an extended diffuse H i disc with lower density compared to the gas residing near the mid-plane.
The concept of vertical stratification within the atomic hydrogen disc has been known for long time and many attempts have led to the identification of multiple components (usually two) with discrete H i scale heights (Radhakrishnan et al. 1972;Baker & Burton 1975).By fitting the Galactic H i vertical distribution within Galactocentric radii of 0.4 ⊙ ≲  ≲  ⊙ with a combination of two Gaussians and an exponential tail, Dickey & Lockman (1990) derived the thickness of H i components.The Gaussians have FWHM values of 212 pc and 530 pc, where the narrow component has a peak density ∼ 4 times that of the wider component.The exponential tail has a thickness of 403 pc with a peak density equal to one-sixth of that of the narrow component.Since the GMC complexes in the W41-W44 regions are also located within the same range of Galactocentric radii, these results strongly suggest the necessity of modelling the vertical distribution by at least a double Gaussian profile.Hence we repeat the exercise by fitting a double Gaussian defined as follows, where the FWHM of each component can be obtained by multiplying √ 8 ln 2 to the respective s.Various numerical simulations demonstrated that the vertical distribution of ISM is heavily influenced by The 12 CO contours represent 0.6, 2, 5, 7, 9 × 10.5 K km s −1 .The bottom panel shows the H i disc thickness estimated using different methods.FWHM values from the single Gaussian fitting and rms height are plotted as blue pentagons and green squares, respectively.The rms height plotted here is  ℎ multiplied by √ 8 ln 2, so that it act as proxy to the overall disc thickness.FWHM of the narrow and wide components from double Gaussian fitting are given as black stars and red down triangles, respectively.All the FWHM values are plotted with error bars, which represent the fitting uncertainties.Two trough-like features in the FWHM wide distribution, annotated by an arrow, are found at the location of W43 and an extended structure contributed by W41 and a molecular cloud located at the back of the W42, (hence 'W42-Back').The total H 2 mass obtained from the total 12 CO emission in the region of the trough is mentioned in parentheses in units of 10 6 M ⊙ .dynamical processes such as turbulent mixing, stellar feedback etc (Kritsuk et al. 2017;Chen et al. 2023).MHD simulations of a vertical column of magnetised ISM by Hill et al. (2012) shows that the supernovae are the primary dynamical drive that leads to vertical stratification of three-phase ISM, where gas components at different temperature regime have distinct scale heights.However, separating the three phases through observations remains challenging due to several factors reasons including difficulties in temperature measurements, velocity crowding and components overlapping in velocity etc (Bhattacharjee et al. 2023).Hence, in this analysis, we only focus to distinguish between H i close to the disc and H i distributed over a thicker region, by modelling the vertical distribution by the double Gaussian profile.The two components of the disc obtained from the fitting procedure may not necessarily represent any distinct gas phases.
We use the parameters from the single Gaussian fitting and the rms height estimates to feed the initial guesses while modelling double Gaussian profile.It is possible that for some region, two Gaussian are not required and a single Gaussian model is adequate to fit the observational data.In such cases, the wider Gaussian profile will be considering the values lying within the noise of the data for modelling and the resultant peak density,   0 , will be only a few times the noise floor.Additionally, in such cases, the narrow Gaussian component is expected to have fitting parameters similar to those of a single Gaussian.Based on these criteria, we performed fitting with both single and double Gaussian models for the vertical distribution at different longitudes.Wherever the wider component's fitting parameters (especially   0 ) are insignificant with respect to the noise floor and the fitted values of narrow components (  0 and  narrow ) match those of a single Gaussian within 85%, we assume that the disc can be modelled by a single Gaussian profile and have a single scale height at that longitude.The thus estimated FWHM of H i narrow and wider Gaussian components at different longitudes are represented as red and black points, respectively, with error bars in figure 4. Here, the error bars correspond to the uncertainty in the fitting estimates.Note that, at values of  for which a single Gaussian represents the vertical distribution better than the double Gaussian profile, the FWHM of the wider component is replaced with that of the fitted single Gaussian.Such  values can be easily identified from figure 4; the FWHM of the single Gaussian and wider components match at those locations.Interestingly, in this figure 4, such positions are highly correlated with the locations of the W41 and W43 GMC complexes.This indicates the lack of a significant thick diffuse component of atomic hydrogen gas around these complexes.In other words, gas located nearer to the complex is getting pulled towards the mid-plane by its gravitational potential.As we move farther away from the complex, the effect of the potential on diffuse gas will diminish,leaving it as an extended halo.This eventually indicates the pinching or constraining effect that Jog & Narayan (2001) have found through theoretical modelling.
We determine the different estimates of H i disc thickness as a function of  with a pixel size of 0.08 • .To reduce noise and hence to make prominently visible the trough-like features, all the H i disc thickness estimates have been smoothed by averaging in uniform bins.For the averaging, we have chosen a bin size of ∼ 0.3 • , which is similar to the angular resolution of the H i data (∼ 0.26 • ).The broad troughs between  = 21 • and 29 • and a narrow trough between  = 30 • and 33 • in the FWHM of H i wide component provide clear evidence for vertical constraining that the GMC complexes of W41, W42-Back and W43 respectively are imposing on the H i disc.We annotated those locations where pinching is visible in figure 4 by an arrow and the name of the respective GMC complex.The H 2 mass in units of 10 6 M ⊙ obtained from measuring the integrated intensity of 12 CO(1 − 0) emission in the region where the trough is visible is mentioned in figure 4 underneath the name of the respective complex.We use the conversion factor 2.75 × 10 20 (cm 2 K km s −1 ) −1 given by Bloemen et al. (1986) to calculate the H 2 column density and further H 2 mass.We repeat this exercise for other slices mentioned in Table 1 and plots similar to Fig 4 for other slices are given in Appendix B. In the following section, we discuss about our findings on the pinching effect by various complex.Jog & Narayan (2001), the gravitational effect of molecular complexes results in local corrugations in the thickness of stellar and gas disc.These are the additional variations on top of the global corrugation of the disc mid-plane.Our analysis shows strong evidence for the pinching effect in the W41-W44 region.The broad trough in the FWHM of the wide component (red points in figure 4), between  = 21 • and 29 • is a strong manifestation of the gravitational constraint imposed by the molecular gas in W41 and W42-Back.In this velocity range (90 − 105 km s −1 ), both W41 and W42-Back have significant 12 CO emission in the form of an elongated structure along the mid-plane with a few high-density clumps at the cloud centre (i.e., at  = 23.03• and  = 25.38 • ).The location of a broad minimum (from 23 • to 26 • ) in the trough aligns with the position of these cloud centres.With respect to the thickness of the disc far away from the clouds, the FWHM of the wider component is reduced by a factor of 2.5 around the complex centre.The molecular hydrogen mass from total 12 CO emission comprising the entire region where the trough is visible is 6.4 × 10 6 M ⊙ .The width of the trough is around ∼ 850 pc.A similar trough that however is less wide (∼ 470 pc) is also visible around W41 in the next lower velocity range, i.e., Slice 4 (75−90 km s −1 ) (See figure B4).Interestingly, in this velocity range the 12 CO spectrum of W42 doesn't show any significant emission while W41 has the strongest emission.Clearly, we can say that the pinching in thickness of the wider component seen in this slice is solely caused by the molecular cloud complex in W41.The similarity in the length of the dense elongated 12 CO structure in W41 (∼ 500 pc) and the width of the trough is also strongly favouring the evidence for pinching.It is to be noted that similar trough-like features are also visible in the rms-height distribution around W41 in both slices.As we go to the other slices where the LSR velocity decreases or the slices become nearer to us, the trough-like feature is gradually vanishing.This is clearly because the W41 does not have significant molecular hydrogen content at those distances.At high velocity ranges 105 < V LSR < 120 km s −1 (i.e., Slice 6), W42-Back and W41 have moderate amount of emission which causes another trough in FWHM wide and rms height in similar longitude range.The emission from molecular clouds associated with W42 is significant in Slice 2, however, a pronounced deviation in either FWHM wide or rms height is not detected.

As shown in
W43 is the most massive complex in the entire region under consideration; it has its peak CO emission in Slice 5. From figure 4, we see a narrow dip near W43 from  = 30 • to 33 • .The GMC complex in W43 is positively attracting the surrounding gas, which results in the existence of a single denser H i structures confined around the mid-plane.The width of the trough is ∼ 320 pc and the disc thickness is reduced by a factor of ∼ 2 at the minima.At Slice 3 and Slice 4, W43 have concentrated dense CO structures in a similar longitude range and a decrease in the FWHM wide is visible from  = 29 • to 33 • in figure B3 and B4 respectively.The average width of the trough is around 350 pc in both cases and the denser structures are located near the centre of the trough.A similar narrow trough can be seen in Slice 6, though not as prominent as in other slices.Again, wherever the pinching effect by the W43 complex is detected from the FWHM wide distribution, their respective rms height distribution also shows a coinciding trough-like feature, but lesser in magnitude.The 12 CO spectra towards the direction of W43 show a low velocity component from 30 to 55 km s −1 in figure 2 and Nguyen Luong et al. (2011) identified this to be originated from diffuse clouds located in the front along the line of sight, unrelated to W43 complex.In Slice 4, we see dense structures in 12 CO emission around  = 30 • , which are originating from these front clouds.The narrow dip in FWHM wide from  = 29 • to 32 • is caused by the constraining effect of these clouds, which we name "W43-Front".The total  2 mass from this W43-Front is 2 × 10 5 M ⊙ and the FWHM is decreased by a factor of 1.5.This finding strongly suggests the possibility of detecting the constraining effect of less massive molecular clouds or complexes, where the trough in disc thickness is generally lower in magnitude.
Unlike the other cases, for W44, the effect of pinching is only detected in the velocity range 45 − 60 km s −1 .The width of the trough detected in Slice 2 around W44 is about 165 pc and the thickness of the diffuse H i component decreases by a factor of three.The main velocity component of W44 is 30 − 60 km s −1 , with the denser CO structures emitting around 45 km s −1 .Although a trough feature is visible around W44 in the other velocity range 30 − 45 km s −1 , a strong increase in the FWHM at the exact location of the W44 complex, prevents us from a confirmed detection for pinching in Slice 1.In contrast to the other complexes, which are confined to the midplane of the Galactic disc, W44 has a filament-like structure that extends along the vertical axis.The molecular gas associated with the SNRs W44 has been found to be strongly shocked (Seta et al. 2004;Sashida et al. 2013).Indeed > 25 km s −1 wide high-velocity CO line wings suggest a violent interaction between the molecular cloud and the SNR.Additionally, ultra-high velocity emissions in CO J= 3−2, traced by line widths ∼ 100 km s −1 are also detected toward W44 (Sashida et al. 2013).Possibly such a violent environment has disrupted the entire gas distribution surrounding it.Eventually this results in making the surrounding vertical disc structure more complex and hence detecting constraining effect become more difficult.Table 2 summarises the details about the detected troughs in our analysis.In addition to the GMC complexes of our interest, there are enormous amounts of 12 CO emission at other Galactic longitudes.For instance, in Slice 1, the longitude range from 25 • to 31 • comprises an elongated dense molecular gas structure having a total mass of ∼ 2 × 10 5 M ⊙ .This cloud structure creates a trough in a similar longitude range, with a similar fractional decrease in thickness.Similarly, the narrow dips centred at 28 • and 37.5 • in Slice 4, are also possibly caused by the constraining effect of the dense gas clouds present in the respective location.Hence these local corrugations that are prominently visible in the thickness of H i diffuse disc, are in very good agreement with the predictions by Jog & Narayan (2001).
To quantify the correlation between the H i scale height variation and GMCs, we estimate the Spearman correlation coefficient between the disc thickness parameters and CO intensity.For every value of , we estimate the integrated CO intensity summed over the entire vertical axis.For any slice, this integrated   is proportional to the total molecular gas density summed over the entire vertical axis at a particular longitude over the respective velocity range.Note that the angular resolution of the H i and CO data are different and hence to have a position-wise comparison between the H i disc thickness and CO intensity, we did a uniform binning along the longitude axis with a bin width of 0.3 • .Hence the bin averaged values of different quantities are considered for the correlation study.We separate the entire W41-W44 region into three regions so that the effect of the individual complex can be quantified.For example, we consider the Galactic longitudes from  = 20 • to  = 27 • , where the W41 and W42-Back clouds are located.Figures 4 and 5 show the scatter plot between the integrated   and FWHM wide (left column), FWHM narrow (middle column) and the H i rms height (right column) for W41-W42 region.Each row corresponds to different slices.Points in the scatter plots are colour-coded with distance from the centre of the complex (here it is  = 23.5 • ).The Spearman correlation coefficient between the respective quantities is mentioned in each plot.The coefficient ranks the correlation between two quantities, where a value of 1 corresponds to a strong correlation and 0 corresponds to zero association between them.A Spearman correlation coefficient of −1 represents a strong anti-correlation between two quantities.From figure 4 and figure 5, we can see that FWHM of the wide component is anti-correlated with the integrated CO emission in the velocity range 60 − 105 km s −1 .Respective values of Spearman coefficient −0.63, −0.80 and −0.81 at different slices in this velocity range depict the following picture.In regions where the thickness of the H i wide component is smaller, the molecular gas has a higher mass.In other words, high-density CO regions are reducing the thickness of H i wide component, which is nothing but the effect of pinching.In slices where anti-correlation is strongest (Slice 4 and Slice 5), we can easily see from the scatter plot of H i FWHM wide and integrated   , the points that are lying in the high-density tail is essentially located around the complex centre.The decreasing trend in the anti-correlation as it goes towards the lower velocity components suggests that the pinching effect is less due to lower molecular mass.Similarly, the H i rms height shows a very strong anti-correlation with the integrated 12 CO intensity in similar velocity ranges, with Spearman coefficient −0.82, −0.86 and −0.91 in each slice.The negative correlation of FWHM wide and rms height with the H 2 density in these velocity ranges quantitatively confirms the detection of vertically constrained H i disc by molecular complexes in W41 and W42-Back.The Spearman coefficient of −0.68 between the H i rms height and the integrated CO intensity in Slice 6, also favours the pinching effect by W41, W42-Back together.We did a similar analysis in the other two complexes and their estimates of the Spearman correlation coefficients are given in table 3.
The middle column of figure 4 and figure 5 is quantitatively representing any association between the narrow disc thickness and molecular gas density, if it exit.Though any noticeable trend of neither correlation nor anti-correlation is visible, the following are some exceptions.The correlation between FWHM narrow and integrated  CO in W41-W42 region at velocity range 60 − 75 km s −1 is 0.9.Leahy & Tian (2008) identify a large molecular CO cloud at 77 km s −1 interacting with the supernova W41, at a kinematic distance 3.9 − 4.5 kpc.The strong correlation of 0.9 between the molecular gas and narrow disc is probably the resultant effect of this interaction.Similarly, the surrounding gas around W41 at a velocity range of 45 − 60 km s −1 is found to be violently disrupted by supernovae shock, which could possibly result in a small hump-like feature seen in the thickness of narrow disc between 20 • to 25 • (Black stars in figure B3).The fractional increase in the thickness is around a factor of two.Similarly, at the location of W43 in Slice 5, where the complex has its brightest emission in 12 CO, an apparent increase in the scale height of the narrow disc by two times, is visible from 30 • to 32 • .A positive rank correlation with   = 0.6 at the location of W43 in Slice 5 is a result of this variation in FWHM narrow .The dense H i gas that is seen confined to the mid-plane within ≲ 200 pc can easily get disrupted by the local dynamics happening within the high star-forming regions like W41 and W43 and thus could possibly enhance its thickness.A positive Spearman coefficient of   = 0.75 between narrow disc thickness and CO intensity around the W44 region in velocity slice 2, where the related SNR is located, is also indicating that the narrow disc is disturbed by supernova interaction.
Each of these slices is separated by an average distance of ∼ 870 pc.Therefore, the identified troughs within a particular slice are attributed to the cumulative effect of all the sub-complex structures contained within the complex which are distributed over this 870 pc along the line of sight.A close look at the 12 CO spectrum in Figure 2 reveals that the peak 12 CO emissions from complexes W41, W42, and W44 are falling at the edges of the velocity bins.Hence the possible constraining effect in these complexes may not be prominently visible in these slices.To resolve this, a two-dimensional distribution of the H i disc thickness must be estimated, as a function of Galactic   longitude,  and line-of-sight distance, D. Subsequently, this estimated distribution of H i disc thickness should be compared to the CO content within each respective complex.We have attempted to estimate the two-dimensional distribution of the H i disc thickness in the W41-W44 region using the following procedure.We integrate the H i and 12 CO emissions along the same velocity range of 30 to 120 km s −1 in multiple bins, now each has a smaller width of 3 km s −1 and hence generated numerous slices.Further, a similar exercise was repeated to estimate the H i disc thickness for each slice.
With the known model of rotation curve as discussed in appendix A, is an equivalent representation of the face-on view of the W41-W44 region.The close alignment between the high-density CO contours and the locations where FWHM wide are small around W41, W42 and W43 provides further confirmation of our findings.The prominent decrease in the H i rms height around W41, W42 and W43 supports the anti-correlation trend that was seen between the high-density H 2 and H i disc thickness previously.Interestingly, despite the presence of high-density molecular content in the W44 region, no hint of a pinching effect was detected.This suggests that strong local stellar feedback (Sashida et al. 2013) suppresses the constraining effect in this region.

DISCUSSION AND CONCLUSION
We report for the first time, the observational evidence for the local vertical constraining in H i by the Giant Molecular Cloud complexes in the W41-W44 region.The local corrugations in the thickness of the diffuse H i disc detected in various velocity slices are verifying the theoretical predictions by Jog & Narayan (2001).The narrow component represents a distribution of compact dense H i gas along the mid-plane of the disc and no prominent correlation between its thickness and CO emission is detected.However, we see a few indications, where the FWHM of the narrow component is enhancing around W41 and W43, possibly caused by feedback from stellar ac-tivity.Though such local environmental activities could dominate the effect of potential of molecular complex at smaller heights from the mid-plane, at larger heights where the diffuse H i gas is present, gravitational attraction by the complex dominates.The strong negative correlation (i.e., −0.94 <   < −0.6) between the rms height and CO intensities in the entire W41-W44 region is quantitatively emphasizing this.
Theoretically, a constant density complex shows the strongest pinching effect at its centre of the complex along the mid-plane (Jog & Narayan 2001).However, for a real disc, the molecular cloud complex consists of several sub-complexes, each located at a different distance from the Sun and each with a different extent and density.Being located inside the Galaxy, from observations it is difficult to disentangle the constraining effect by individual complexes.Various trough-like distributions detected in each slice (as listed in table 2), are the convoluted contribution by such sub-complexes which are spread across ∼ 800 − 1000 pc distance along the line of sight.The W43 GMC complex contributes to the 12 CO emission in velocity range, 60 < V LSR < 120 km s −1 , representing distinct molecular sub-complexes located at distances ranging from 4 kpc to 7 kpc from us.The trough detected around  ∼ 30 • − 32.5 • at different slices in these velocities gives the cleanest evidence for the constraining effect.Similarly, the GMC complex associated with W41 shows strong confirmation for the pinching effect in the same velocity ranges.The width of the various troughs given in table 2 depends on the molec-ular mass distribution in the respective longitudes and the massive distribution by W41 and W42-Back creates the most wide trough.
The uniform binning along the LSR velocity ranges reduces the complexity and uncertainties that can be introduced in the transformation of Galactic coordinates to physical length scales.However, since different GMC complexes are located at slightly different distances from the Sun, it is possible that peak CO emission of individual clouds aligns with the edges of the adjacent velocity bins.For instance, in figure 2 , the W44 complex has its dominant 12 CO content around 45 km s −1 , which is located at the margin of Slice 1 and 2. As discussed in section 5, unlike other GMC complexes, evidence for pinching is not prominent for W44.However, the twodimensional distribution of FWHM of H i wide component and H i rms height estimated as in figure 6, rules out this limitation of the velocity binning and we see that irrespective of the presence of very high density 12 CO content in this complex, any hint of constraining effect in W44 is not detected even in the wide component of H i disc.This indicates that the local stellar feedback acting in this region is dominant over the gravitational potential that the complex imposes.
• The H i vertical density distribution in W41-W44 region (20 • <  < 40 • ) which spans over around 5 kpc along the line of sight is modelled; high-density atomic gas located around the mid-plane is found to be embedded in a diffuse low-density gas distribution present at larger height from the Galactic plane.
• By inspecting the multiple regions located at different distances from us, we infer the 2-D distribution of vertical thickness of H i disc in W41-W44.The typical variation of the thickness of the H i wide component ranges from 200 pc to 700 pc, whereas that of the narrow disc is between the 50 pc and 200 pc.
• We see that the diffuse H i gas distributed around the Galactic plane is highly sensitive to the gravitational potential imposed by the various GMC complexes and results in the trough-like features in the FWHM of the H i wider component.This eventually lead to local corrugations in the H i scale height distribution.
• The W43 and W41 are the dominant GMC complexes that strongly show the evidence for pinching effect in the form of a troughlike feature in scale height.The constraining effect by W43 is visible from 29 • to 32 • in Galactic longitudes and spans over a 3 kpc distance along the line of sight of direction.The GMC complex associated with W41 shows a pinching effect in similar line-of-sight distances and in the longitude range from 21 • to 25 • .
• The typical width of the trough varies from ∼ 150 to ∼ 850 pc and it highly depends on the molecular hydrogen content in the respective velocity range.On average, the disc thickness decreases by a factor of two to three near the minima of the trough, with respect to the thickness of the H i disc in the unaffected regions (i.e., ∼ 700−800 pc).
• The biggest trough that was detected in our analysis is created by the 6.4 × 10 6 M ⊙ massive elongated molecular gas distribution mainly contributed by the complex W41 and a molecular cloud W42-Back.It has an approximate width of 850 pc and a height 350 pc.
• The smallest GMC that caused the pinching effect is a physically not connected molecular cloud/complex located in front of the W43.The 2×10 5 M ⊙ massive gas clouds create one of the smallest troughs detected with a width of 165 pc and a height of 200 pc.This enhances the possibility of detection of effect in low mass molecular clouds or complexes.
• The narrow component of the H i disc is found to be not affected by the pinching, in fact, there are a few indications it can get disturbed by the stellar feedback in the star-forming complexes.
• The H i rms height which characterizes the overall extent of the disc is found to be anti-correlated (i.e.,   ≲ −0.6) with molecular gas content, in the entire W41-W44 region, for distance 3 kpc onwards.This depicts the global constraining effect caused by the molecular gas distribution in the mid-plane.
A similar constraining effect would also be expected to be seen in stars, as shown by (Jog & Narayan 2001).Thus, such massive molecular cloud complexes and the associated constraining of stars and H i gas around each of these would constitute a substantial local perturbation to the gravitational field in the disc.This can further affect galactic disc dynamics, for example, it could lead to the heating of stellar velocity dispersion (Jog & Narayan 2001).Similar molecular cloud complexes are also seen in other galaxies, for example, in M83 (Lord & Kenney 1991), M100 (Rand 1995), and M81 (Brouillet et al. 1998).Hence the locally non-uniform or corrugated distribution of H i and stars predicted by Jog & Narayan (2001) is a generic phenomenon.Our findings reveal a possibility of probing these effects in other cloud complexes within the Galaxy, as well as in external galaxies.However, the complexity in modelling the 3-D distribution of gas within the Milky Way, makes the probing attempt challenging.Nevertheless, being the only system where such a local dynamical effect can be investigated in detail, similar investigations need to be carried out in another part of the Galaxy.

Figure 1 .Figure 2 .
Figure1.The H i 21cm line intensity distribution of the W41-W44 region integrated over the LSR velocity range 30 − 120 km s −1 is shown in colours.Overlaid black contours represent the 12 CO 1 − 0 intensity integrated over the same velocity range.White dotted circles represent the positions and areas from which the H i and CO spectra are extracted for each cloud.Values of CO contours are 2, 3, 5, 6, 8, 9, 10 × 30 K km s −1 .

Figure 3 .
Figure 3.The top panel shows the H i intensity map integrated over the velocity range in Slice 5 and the respective CO intensity map overlaid as black contours.The 12 CO contours represent 0.6, 2, 5, 7, 9 × 10.5 K km s −1 .The bottom panel shows the H i disc thickness estimated using different methods.FWHM values from the single Gaussian fitting and rms height are plotted as blue pentagons and green squares, respectively.The rms height plotted here is  ℎ multiplied by √ 8 ln 2, so that it act as proxy to the overall disc thickness.FWHM of the narrow and wide components from double Gaussian fitting are given as black stars and red down triangles, respectively.All the FWHM values are plotted with error bars, which represent the fitting uncertainties.Two trough-like features in the FWHM wide distribution, annotated by an arrow, are found at the location of W43 and an extended structure contributed by W41 and a molecular cloud located at the back of the W42, (hence 'W42-Back').The total H 2 mass obtained from the total 12 CO emission in the region of the trough is mentioned in parentheses in units of 10 6 M ⊙ .

Figure 4 .Figure 5 .
Figure 4. Correlation study in W41-W42 region ( = 20 • − 27 • ) at different slices.The scatter plots between the integrated   and FWHM wide (left column), FWHM narrow (middle column) and H i rms height (right column).Each row corresponds to different slices.Points are colour-coded with distance from the centre of the cloud.

Figure 6 .
Figure 6.Density plot showing the 2D distribution of FWHM wide (left) and rms height (right) in the entire W41-W44 region in polar coordinates where the radial axis is the line of sight distance from the Sun in units of kpc and angular coordinate is the Galactic longitude in degrees (Note that for the display purpose, the angular coordinate is scaled by a factor of two).The integrated  CO levels are plotted as contours and the values of the respective contour are mentioned in the color bar.The locations of the four complexes are also annotated in the figure.

Figure B1 .
Figure B1.The distribution of H i disc thickness in Slice 1 (Velocity range 30 − 45 km s −1 ).The plotting convention is similar to that of Figure 4.The values of 12 CO contours are 0.6, 2, 5, 7, 9 × 8.2 K km s −1 .The rms height plotted here is  ℎ multiplied by √ 8 ln 2, so that it act as proxy to the overall disc thickness.

Figure B2 .
Figure B2.The distribution of H i disc thickness in Slice 2. The values of 12 CO contours are 0.6, 2, 5, 7, 9 × 6.7 K km s −1 .The locations where the pinching effect is visible are annotated by an arrow with the name of the cloud that causing it.The total H 2 mass obtained from the total 12 CO emission in the region of the trough is mentioned in the bracket in units of 10 6 M ⊙ .Two troughs in the FWHM wide is found at GMC associated with W44 and another molecular cloud located in the front of the W43, (hence 'W43-Front').

Figure B3 .
Figure B3.The distribution of H i disc thickness in Slice 3. The values of 12 CO contours are 0.6, 2, 5, 7, 9 × 7.2 K km s −1 .Two troughs are visible in the FWHM wide at the location of W41 and W43.

Figure B4 .
Figure B4.H i scale height distribution in Slice 4. The values of 12 CO contours are 0.6, 2, 5, 7, 9 × 8.4 K km s −1 .Two troughs are visible in the FWHM wide , one around W41, W42-Back and the other around W43.

Figure B5 .
Figure B5.The distribution of H i disc thickness in Slice 6.The values of 12 CO contours are 0.6, 2, 5, 7, 9 × 9.5 K km s −1 .Two troughs are visible in the FWHM wide , one around W41, W42-Back and the other around W43.

Table 1 .
The velocity range of each slice and respective line of sight distance (near) estimated are tabulated here.We also mentioned the respective complexes which is emitting 12 CO significantly in each velocity range.

Table 2 .
The table summarises the details about the detected troughs in H i FWHM wide distribution.

Table 3 .
Spearman correlation coefficient between the various quantities for individual complex regions.Each row corresponds to the different slices.