Relative Velocities between 13CO Structures within 12CO Molecular Clouds

Velocity fields of molecular clouds (MCs) can provide crucial information on the merger and split between clouds, as well as their internal kinematics and maintenance, energy injection and redistribution, and even star formation within clouds. Using the CO spectral lines data from the Milky Way Imaging Scroll Painting survey, we measure the relative velocities along the line of sight (ΔV LOS) between 13CO structures within 12CO MCs. Emphasizing MCs with double and triple 13CO structures, we find that approximately 70% of ΔV LOS values are less than ∼1 km s−1, and roughly 10% of values exceed 2 km s−1, with a maximum of ∼5 km s−1. Additionally, we compare ΔV LOS with the internal velocity dispersion of 13CO structures ( σ13CO,in ) and find that about 40% of samples in either double or triple regime display distinct velocity discontinuities, i.e., the relative velocities between 13CO structures are larger than the internal line widths of 13CO structures. Among these 40% samples in the triple regime, 33% exhibit signatures of combinations through the two-body motion, whereas the remaining 7% show features of configurations through the multiple-body motion. The ΔV LOS distributions for MCs with double and triple 13CO structures are similar, as well as their ΔV LOS/ σ13CO,in distributions. This suggests that relative motions of 13CO structures within MCs are random and independent of cloud complexities and scales.

The properties of MCs have been extensively researched and analyzed in the literature.This includes their masses, sizes, velocity dispersion, etc., and the scaling relations between cloud sizes, velocity dispersion, surface densities, etc. (e.g.Larson 1981; Dame et al. 1986;Solomon et al. 1987;Heyer et al. 2009;Roman-Duval et al. 2010;Rice et al. 2016;Miville-Deschênes et al. 2017;Riener et al. 2020;Rani et al. 2023).Moreover, the column density probability distribution (e.g.Vazquez-Semadeni 1994;Ma et al. 2021Ma et al. , 2022) ) and the velocity structure function (e.g.Heyer & Brunt 2004;Heyer et al. 2006) are also used to investigate the distributions of column densities and velocity fields in MCs.These studies provide a comprehensive understanding of the kinematic, dynamic, structural, and evolutionary characteristics of MCs.
Our recent series of studies based on a large sample of MCs from the Milky Way Imaging Scroll Painting (MWSIP) CO survey (Su et al. 2019), find that as MCs grow in scale, they tend to exhibit more complex filamentary networks (Yuan et al. 2021, hereafter Paper I) and host greater numbers of 13 CO structures (Yuan et al. 2022, hereafter Paper II).These 13 CO structures have areas that generally do not exceed 70% of the MC's 12 CO emission areas (Paper II).In addition, we revealed a preferred spatial separation among individual 13 CO structures, which is independent of the MC's scale (Yuan et al. 2023a, hereafter Paper III).Furthermore, we found that the relative motions between 13 CO structures are the primary contributor to the total velocity dispersions of MCs (Yuan et al. 2023b, hereafter Paper IV).Based on these observed results, an alternative picture for the assembly and destruction of MCs has been proposed.It suggests that the regularly spaced 13 CO structures serve as the building blocks for MCs, and the transient processes of MCs occur through slow mergers or splits among these fundamental blocks.Meanwhile, these processes do not significantly alter MCs' density structures but do affect their global velocity fields.Numerical models indicate that mergers between clouds are gentle and do not heavily impact the density structures of clouds, but they can result in higher velocity dispersions of MCs (Dobbs et al. 2011(Dobbs et al. , 2015;;Jeffreson et al. 2021;Skarbinski et al. 2023), which are consistent with our pictures.
The picture above provides crucial insights into the build-up process of MC.It is mainly based on the spatial distribution and motions of internal 13 CO structures.The motions of gas displays distinct characteristics, such as noticeable variations in velocity, which likely result from the process of merging or splitting. 13CO struc-tures are considered as the fundamental building blocks of material transfer between clouds as described in Paper III and Paper IV.By analyzing velocity fields of 13 CO structures within clouds, we can gain key clues to resolve the gas motions within clouds and better understand the material transfer processes between clouds.It also sheds light on the energy injection sources driving the kinematics of MCs, and the processes of energy redistribution and material gathering for star formation.
In this research, we focus on the relative velocities of the 13 CO structures.This paper is organized as follows: Section 2 presents the data from the MWISP CO survey, along with the identification of 12 CO molecular clouds and their harbored 13 CO structures.Section 3 mainly describes the results, including the distributions of line-of-sight relative velocities between 13 CO structures within MCs, as well as ratios between relative velocities and velocity dispersions of 13 CO structures within MCs, the fractions of MCs displaying distinct velocity discontinuities.In Section 4, we discuss the observational bias in our results and compare our observed results with previously simulated works.Section 5 summarizes our findings.
2. DATA 2.1.The 12 CO(J=1-0) and 13 CO(J=1-0) spectral lines data from the MWISP survey The Milky Way Imaging Scroll Painting (MWISP) survey is an ongoing northern Galactic plane CO survey, which is performed by the 13.7m telescope at Delingha, China, and observes the 12 CO, 13 CO, and C 18 O lines at the transition J=1-0, simultaneously.A detailed description of the performance of the telescope and its 3×3 multibeam sideband-separating Superconducting Spec-troScopic Array Receiver (SSAR) system is given in Su et al. (2019); Shan et al. (2012).The observational strategy and raw data processing are also introduced in Su et al. (2019).The half-power beamwidth (HPBW) of the antenna at the frequencies of 115 GHz is ∼ 50 ′′ .The typical system temperature is ∼ 250 K at a line frequency of the 12 CO line (115.271GHz) in the upper sideband and ∼ 140 K at 13 CO (110.201GHz) and C 18 O (109.782GHz) lines in the lower sideband, respectively.The total bandwidth of 1 GHz with 16,384 channels provides a spectral resolution of 61 kHz per channel, resulting in a velocity resolution of about 0.16 km s −1 for 12 CO lines and 0.17 km s −1 for 13 CO and C 18 O lines.The typical RMS achieved in 12 CO and 13 CO lines are ∼ 0.5 K and ∼ 0.3 K, respectively.
In this work, the 12 CO and 13 CO lines data are from the MWISP survey and cover about 450 deg 2 region with the Galactical longitude l from 104 • .75 to 150 • .25, the Galactical latitude |b| < 5 • .25, and the line-of-sight velocity of −95 km s −1 < V LSR < 25 km s −1 .These 12 CO and 13 CO lines emission data also have been analyzed in our previous series of works in Yuan et al. (2021Yuan et al. ( , 2022Yuan et al. ( , 2023a,b),b).
2.2.The 12 CO molecular clouds and their internal 13 CO structures In our analysis, the 12 CO molecular cloud is defined as a set of adjacent voxels in the position-position-velocity (PPV) space with observed 12 CO(1-0) line intensities exceeding a certain threshold.The Density-based Spatial Clustering of Applications with Noise (DBSCAN) algorithm, designed to discover clusters with arbitrary shapes in large spatial databases (Ester et al. 1996), was employed to identify MCs in the 12 CO data cube by Yan et al. (2020).This algorithm combines both intensity levels and continuity of signals, which is appropriate for the extended and irregular shapes of MCs.Three parameters are used in the DBSCAN to extract the 12 CO MCs, one parameter of cutoff determines the line intensity threshold, while the other two parameters of ϵ and MinPts define the connectivity of the extracted structures.A core point within extracted structures satisfies that its adjacent points within a certain radius (ϵ) have to exceed a threshold number (MinPts).The border point is inside the ϵ-radius of a core point, but its adjacent points within the radius of ϵ do not exceed the number of MinPts (Yan et al. 2020).The parameters of cutoff =2σ (σ is the rms noise, whose value is ∼ 0.5 K for the 12 CO line emission), MinPts=4, and ϵ=1 are used for the identification of 12 CO clouds, as suggested in (Yan et al. 2020).Moreover, post-selection criteria are utilized to avoid noise contamination, which includes: (1) the total number of voxels in each extracted structure is greater than 16; (2) the peak intensity of extracted voxels must be higher than the cutoff value adding 3σ; (3) the angular area of the extracted structure must be larger than one beam size (2×2 pixels ∼ 1 ′ ); and (4) the number of velocity channels must be greater than 3. Using aforementioned parameters and criteria, a catalog of 18,190 12 CO molecular clouds was identified in the above region by the DBSCAN algorithm (Yan et al. 2021).We have visually inspected and classified these 18,190 MCs into filaments and nonfilaments in Paper I. Additionally, the dependence of extracted MC samples on the finite angular resolution, sensitivity of observed spectral lines, and different algorithms have been systematically investigated in Yan et al. (2022).
Individual 13 CO structures are defined as connected voxels in the PPV space, meanwhile the 13 CO line intensities on these voxels must exceed a certain thresh-  1.
In addition, according to the spiral structure model of the Milky Way (Reid et al. 2016(Reid et al. , 2019;;Xu et al. 2023), the MC samples are further divided into two groups, i.e., near and far.The MC samples in the near group have central velocities ranging from -30 to 25 km s −1 , which are mainly distributed in the local arm and have kinematical distances of ∼ 0.5 kpc.The MC samples in the far group have central velocities in a range of (-95 -30) km s −1 , most of which are located in the Perseus arm and their kinematical distances concentrate on ∼ 2 kpc (Reid et al. 2016).Considering these kinematical distances, the physical scale for the MC with an angular size of 1 ′ in the local arm is about 0.15 pc, and this value is ∼ 0.6 pc for that in the perseus arm.The number of MC samples distributed in the near and far groups has also been listed in Table 1.

RELATIVE VELOCITIES BETWEEN 13 CO STRUCTURES WITHIN MOLECULAR CLOUDS
In our previous studies (Paper III and IV), we have proposed that regularly spaced 13 CO structures serve as the fundamental units of the gas transfer between clouds.Additionally, mergers or splits between clouds Yuan et al.
have an impact on their velocity fields, resulting in two distinct features: (1) relatively discontinuous velocity fields with blue and redshifted velocities, and (2) velocity structures with two or more velocity components, leading to higher velocity dispersions.Based on these features, we aim to investigate the relative velocities between individual 13 CO structures within MCs and the imprints on the build-up processes of MCs they provide.
To achieve this, we first determine the relative velocities between each pair of 13 CO structures within the MCs.The relative velocities between 13 CO structures are defined as the absolute differences between their centroid velocities.Thus the relative velocity (∆V LOS ) between jth and (j − 1)th 13 CO structures along the line of sight can be calculated as: where V cen, 13 CO,j represents the centroid velocity of the jth 13 CO structure, the sum Σ jth runs over all voxels within the jth 13 CO structure, the T13 CO,ji and V13 CO,ji are the brightness temperature and line-of-sight velocity of 13 CO emission at the ith voxel in the jth 13 CO structure.
Secondly, we compare the relative velocities with the internal velocity dispersion of 13 CO structures.The internal velocity dispersion of 13 CO structures (σ13 CO,in ) within a cloud are defined as: where the sum Σ cloud j runs over the whole individual 13 CO structures within a 12 CO cloud, the σ13 CO,j is the velocity dispersion within the jth 13 CO structure, is the integrated flux of 13 CO line emission for the jth 13 CO structure.
Furthermore, if the relative velocities between 13 CO structures within a cloud satisfy: i.e. the relative velocities between 13 CO structures are greater than the internal linewidth of 13 CO structures, they are characterized as the distinct velocity discontinuities (∆V dis ).

Molecular clouds with double 13 CO structures
Each MC with double 13 CO structures has a single ∆V LOS , as illustrated in Figure 1, making it easier to analyse the relative movement between 13 CO structures within a cloud.Here we focus on the 443 MC samples with double 13 CO structures, whose physical properties are shown in Figures A1 and A2.Based on the kinematical distances of these clouds, the interquartile ranges of their physical scales are 1 -2 pc for those in the near group and 3.5 -5.6 pc for the far group.
Figure 2 presents the distributions of ∆V LOS and ∆V LOS /σ13 CO,in ratios for these MCs.The quantiles at 0.05, 0.25, 0.5, 0.75, and 0.95 and the mean values of the ∆V LOS and ∆V LOS /σ13 CO,in are listed in Tables 2 and  3, respectively.Approximately 70% of ∆V LOS in the 'Double' samples are less than 1 km s −1 , with less than 10% having values greater than 2 km s −1 and reaching a maximal value of approximately 5 km s −1 .The distributions of ∆V LOS in the near and far groups have similar patterns, but the greater ∆V LOS (3 -5 km s −1 ) have slightly higher probabilities to be observed in the far group.The quantiles of ∆V LOS in the far group are ∼ 1.3 times greater than those in the near group, probably due to the beam dilution effects on clouds with different distances, as previously discussed in Section 4.1 in Paper III, which also cause the linear separations among 13 CO structures in the far group to be ∼ 3 times those in the near group.
For the distribution of ∆V LOS /σ13 CO,in , approximately 50% of values are less than 2, and around 70% are less than 3.About 10% of samples exhibit ∆V LOS /σ13 CO,in greater than 5, with several samples reaching ∼ 12.The ratio values (∆V LOS /σ13 CO,in ) and their quantiles for the samples in the near and far groups display consistent distributions.Among these clouds, about 42% of ∆V LOS are identified as distinct velocity discontinuities (∆V dis ), where ∆V dis > √ 8ln2σ13 CO,in , as listed in Table 4.We also visually examine the velocity fields for these clouds with ∆V dis , whose velocity fields are derived from the first moment of the 12 CO and 13 CO line emission, and present their averaged spectral lines of 12 CO and 13 CO line emission.The velocity fields of these clouds exhibit the following features: (1) relatively discontinuities on velocity fields, with one 13 CO structure displaying blueshifted velocity and the other redshifted; (2) the averaged 13 CO spectral lines resolving into two velocity components, as shown in several cases in Figure B1.Based on these observations, about 42% of MCs in the 'double' regime show the signs of ongoing mergers or splits between clouds.As mentioned above, approximately 20% of the whole MC samples contain more than two 13 CO structures.The analysis of relative velocities between multiple 13 CO structures becomes increasingly complex as the number of 13 CO structures increases.Here we focus on the 185 MC samples with triple 13 CO structures, whose physical properties are presented in Figures A1 and A2.Based on the kinematical distances of these clouds, the interquartile ranges of their physical scales are 1.5 -2.5 pc for those in the near group and 4.7 -7.0 pc for the far group.

Relative velocities between each two 13 CO structures
An individual MC with triple 13 CO structures contains three ∆V LOS values between each two 13 CO structures, as illustrated in Figure 1.The distributions of the ∆V LOS values within these MC samples in both near and far groups are presented in Figure 3.The distributions of their maximum ∆V LOS (∆V max ) within each sample are also presented, along with the quantiles at 0.05, 0.25, 0.5, 0.75, and 0.95, and the mean values of the ∆V LOS and ∆V max , which are tabulated in Table 2.We find that the distributions of ∆V LOS in the MCs in the 'triple' and 'double' regimes are similar, and their quantiles and mean values are also close for both near and far groups.However, the ∆V max , whose median value is about 0.92 km s −1 in the near group and 1.3 km s −1 in the far group, is significantly larger than those values for MCs in the 'double' regime.It should be noted that ∆V max is the maximum value for the ∆V LOS between each pair of the 13 CO structures.Therefore, the larger ∆V max is likely due to statistical probabilities for the larger number of the 13 CO structure pairs in the 'triple' regime.
Figure 3 displays distribution of the ratios between the ∆V LOS with σ13 CO,in and the ratios between ∆V max and σ13 CO,in for MCs with triple 13 CO structures.In Table 3, the quantiles at 0.05, 0.25, 0.5, 0.75, and 0.95, as well as the mean values of these ∆V LOS /σ13 CO,in and ∆V max /σ13 CO,in values, are also listed.The distributions of ∆V LOS /σ13 CO,in for MCs with triple 13 CO structures are remarkably similar to those for MCs with double 13 CO structures, with a nearly consistent interquartile range of ∼ 0.9 -3.4 and a median value of ∼ 2, in either the near or far group.However, the ∆V max /σ13 CO,in values in the near group (median value of 3.26) and far group (median value of 2.77) are significantly greater than those values in MCs with double 13 CO structures, which can be attributed to the greater ∆V max .Overall, the distributions of the ∆V LOS for both 'double' and 'triple' regime samples are similar, as well as their ∆V LOS /σ13 CO,in distributions.Such similarities suggest that the relative motions of 13 CO structures within clouds are regulated by the fundamental processes that are independent of structure complexities and cloud scales.

Relative velocities between 13 CO structures connected by MST
For a cloud containing three 13 CO structures, the relative velocities between each pair of them do not take the spatial distribution of 13 CO structures into account.In order to examine whether the relative motion between 13 CO structures is primarily random motion or systematic motion, we connect the centroid coordinates of 13 CO structures together using the minimal spanning tree (MST), which minimizes the sum of the angular separation between 13 CO structures.Furthermore, we analyzed the relative velocities (∆V MST ) between the connected 13 CO structures.There are two ∆V MST values in each MC with triple 13 CO structures, as illustrated in Figure 1.
The distributions of ∆V LOS and ∆V MST for the 185 MCs in the triple regime, are showed in Figure 4. We find that their distributions are nearly accordant, espe-cially for their normalized probability densities.Such similarity means the relative motions between 13 CO structures are random and regulated by fundamental processes.Turbulent flows are a promising candidate for this process, as they are driven by dynamical processes in different scales, including galactic differential rotation and shear (Kim et al. 2006;Bonnell et al. 2006;Dobbs & Bonnell 2008), large-scale instabilities (Wada et al. 2002;Tasker & Tan 2009), and stellar feedback from the supernova explosions (Brunt et al. 2009;Skarbinski et al. 2023;Watkins et al. 2023) and HII regions (Silk 1985;Krumholz et al. 2006).

Distinct velocity discontinuities within MCs
Among 185 MCs with triple 13 CO structures, 21.1% of these MCs have one ∆V dis , 33.5% of samples exhibit two ∆V dis , and 7% of them display three ∆V dis , as listed in Table 4.We further conducted visual inspections of the velocity fields (first moment maps of CO emission) of these clouds and the averaged spectral lines of their 12 CO and 13 CO emissions.In Figure B2, we present the velocity fields for clouds with two ∆V dis , which have two 13 CO structures with redshifted (blueshifted) velocities, and the other 13 CO structure exhibits blueshifted (redshifted) velocities.Furthermore, the averaged spectra of 13 CO emission are resolved into two velocity components.Figure B3 shows the velocity fields for clouds with three ∆V dis , featuring three different velocity fields, each harboring one 13 CO structure.The averaged spectra of 13 CO emission are also decomposed into three velocity components.
We suggest that the build-up processes for MCs containing triple 13 CO structures could be a result of two clouds combination, one with double 13 CO structures and the other with single 13 CO structure (two-body mode), or the assembly of three clouds, each with a single 13 CO structure (multiple-body mode).Figure 5 illustrates the velocity fields of cloud interaction through two-body and multiple-body modes, respectively.In the two-body mode, one 13 CO structure exhibits redshifted (blueshifted) velocity, while the other two 13 CO structures display the blueshifted (redshifted) velocity, with two ∆V dis between one 13 CO structure and the other two 13 CO structures within clouds.In the multiplebody mode, each 13 CO structure has a distinct velocity, resulting in three discontiguous velocity fields in the cloud's velocity fields.Thus three ∆V dis between each pair of 13 CO structures are expected within the cloud.That is, the velocity fields of clouds are constructed by velocities from these 13 CO structures and exhibit three discontiguous velocities.  .Upper panel: number distributions of relative velocities (∆VLOS) between each two 13 CO structures in the MC samples having triple 13 CO structures and their ratios with internal velocity dispersions of 13 CO structures (∆VLOS/σ13 CO,in ).Lower panel: number distributions of the maximum relative velocity (∆Vmax) within each MC and their ratios with internal velocity dispersions (∆Vmax/σ13 CO,in ).The vertical-dashed lines show their median values.In the upper-right corner of each panel, the corresponding normalized probability densities are presented.The normalized probability densities are presented as log scales, and the values in x-axis are binned as linear scales.
Thus, in the 'triple' regime, about 40.5% of MCs show distinct velocity discontinuities, with 33.5% of clouds displaying the signatures of the two-body mode and only 7% of clouds presenting the multiple-body mode.This suggests that cloud mergers or splits tend to occur between two MCs.Horie et al. (2023) simulated the fraction of mass from each progenitor within a colliding GMC and found the sum of the two most significant fractions is mostly close to 1, also suggesting a two-body mode for the majority of MC collisions.We should note that 21.1% of MCs have one ∆V dis between the arbitrary two 13 CO structures, but do not exhibit the distinct discontinuity in velocity field caused by the bulk motions of 13 CO structures.These clouds are not included into the MCs in triple regime showing distinct velocity discontinuities.
In summary, the similarity of the ∆V LOS distribution in the double and triple regimes indicates the relative motion between 13 CO structures is random and independent of cloud scales.Additionally, a considerable portion of distinct velocity discontinuities between 13 CO structures (∆V dis ) is observed within 12 CO MCs.These results are coincident with our previous findings on regularly spaced 13 CO structures within MCs (Paper III) and distinctly demonstrate on how the relative motion of 13 CO structures provides a dominant contribution to the kinetic energy of MCs (Paper IV).Thus these results further support our previously proposed picture that 13 CO structures act as the building blocks of MCs, and the transient processes of MCs proceed by slow mergers or splits among these fundamental blocks.

Observational bias
Our observed MC samples are identified as the contiguous structures in the position-position-velocity (PPV) space, which have 12 CO(1-0) emission intensities above a certain threshold.However, it is important to note that the structure identification in the PPV intensity structures, compared with the real structures in the position-position-position (PPP), may have some bias due to the effects of projection.This bias is attributed to two main factors.Firstly, distinct structures in the  Note-The quantiles at 0.05, 0.25, 0.5, 0.75, and 0.95 for the ∆VLOS/σ13 CO,in and ∆Vmax/σ13 CO,in in their sequential data.The MC samples are reported in Table 2. PPP space that move at similar velocities along the line of sight can be projected as a contiguous structure in the PPV space.Secondly, a single structure in the PPP space that has distinct velocity gradients can result in multiple PPV structures.Unfortunately, these ambiguities that are caused by projection cannot be directly resolved through a simple method in observations.To mitigate this issue, we combine the observed and simulated results.
In observations, the number of velocity components per spectrum along the line of sight can give us a global insight into the level of velocity crowding in the observed region of sky.Our MC samples are located in the second Galactic quadrant with l = 104  PPV space of 12 CO emission within this region.Figure 6 shows the number distribution of 18,190 12 CO clouds per line of sight.We find that on ∼ 40% of this sky region, there is no cloud on the line of sight.In addition, on another ∼ 40% of the region, there is only one cloud along the line of sight.In approximately 15% of this area, two clouds overlap on the line of sight, and ∼ 5% of positions intersect three or four clouds.Furthermore, we count the number of molecular clouds along the line of sight, which are in the velocity range of (-95 -30) km s −1 (Far) and (-30 25) km s −1 (Near), respectively.As shown in Figure 6, approximately 50% of positions have no cloud and ∼ 40% have only one cloud on the line of sight in the near range.However, on the line of sight in the far range, ∼ 85% of areas have no cloud and ∼ 15% have only one cloud.Additionally, Riener et al. (2020); Miville-Deschênes et al. ( 2017) also have revealed that in most of the sky, apart from the vicinity of the Galactic center, there are only single or double velocity components along the line of sight.Considering that ∼ 10% of positions are covered by more than one cloud in the near velocities range of (-30 25) km s −1 , the likelihood for our clouds, 95% of which have 12 CO velocity span less than 10 km s −1 , being projected together by multiple clouds on the line of sight is roughly 2%.
In numerical simulations, the relationship between the PPP density structures and PPV intensity structures has been explored in different aspects.In a simulated barred-spiral galaxy, Pan et al. (2015) found that ∼ 70% of clouds had single counterparts in both PPP and PPV data sets.At the MC's scale, Beaumont et al. (2013) suggested that structures traced in 12 CO lines were more affected by overlap than those traced in 13 CO lines due to the opacity of 12 CO lines.The scaling relations, such as mass-size and size-linewidth, were found to be quite robust to projection by Ballesteros-Paredes & Mac Low (2002); Beaumont et al. (2013).In terms of velocity and density structures, Pichardo et al. (2000) reveal that the PPV features are more representative of the line of sight velocity than the density field.Burkhart et al. (2013) further found that the dominant structures in the PPP and PPV are strictly linked to supersonic gas.For our MC samples, which are extracted in the contiguous 12 CO emission in the PPV space of the Milky Way, we primarily focus on their internal velocity structures traced by the 13 CO emission.Their velocity dispersions are greater than ∼ 0.5 km s −1 , as listed in Table 2 of Yuan et al. (2023b).Comparing the sound speed c s = (kT kin /µm H ) 1/2 ∼ 0.2 km s −1 for gas kinetic temperature T kin =10 K and mean molecular weight µ=2.33, thus these MC samples are thought to be supersonic.Therefore, based on these simulated results, the line of sight velocities of clouds are most likely represented by their velocity structures traced by the 13 CO lines.

Comparison with simulated works
The observed relative velocity between a pair of 13 CO structures is the absolute difference between the observed centroid velocity of each 13 CO structure, which is the three-dimensional (3D) velocity of each 13 CO structure ( ⃗ V 3D ) projected along the line of sight.We determine an angle of θ j between the direction of ⃗ V 3D,j at the jth 13 CO structure and the line of sight, thus the ob-
Based on our previous works, it appears that the distinct velocity discontinuities between 13 CO structures are most likely a result of cloud merging or splitting.In order to gain further insight into cloud motion and its relation to the internal 13 CO structure motion, it is necessary to systematically in further.Numerical simulations of the cloud's motion can provide useful insight at this point.For instance, the distribution of relative velocities for the cloud-cloud collisions or mergers (CCCs) has been analyzed in a simulated galaxy (Horie et al. 2023).The distribution of the cloud collision speeds exhibits a similar trend to that of the observed relative velocities between internal 13 CO structures (Horie et al. 2023).However, the simulated values of cloud collision speeds are distributed at a peak of ∼ 7 km s −1 , with about 65% of them being less than 10 km s −1 (Horie et al. 2023). Nonetheless, Skarbinski et al. (2023) have found that about 80% of mergers occur at a relative velocity of less than 5 km s −1 , implying that cloud mergers have a greater impact on aggregating mass into larger molecular complexes than generating shocks and further altering density structures of the merged cloud.This finding was also discussed in Dobbs et al. (2015); Jeffreson et al. (2021).In observation, Fukui et al. (2021) summarized the recent observational results on about 65 high-mass star-forming regions triggered by CCCs and found that their median collision velocity is ∼ 5 km s −1 .The distribution of internal relative velocities between 13 CO structures is fairly consistent with that for the merged speeds of clouds in these simulated works and observed results.
Furthermore, it is worth noting that the observed velocity discontinuities in this research are interpreted as the signatures of mergers or splits undergoing within MCs.However, we cannot exclude the possibilities of rotation motion, which likely arises from the turbulent flows from the shearing action of differential Galactic rotation (Fleck & Clark 1981;Belloche 2013;Arroyo-Chávez & Vázquez-Semadeni 2022).At least, our observed results also provide a constraint for the velocity gradients on the rotation motion of clouds.Additionally, further analysis on the distribution of the angular momentum from cloud scales down to internal substructures are helpful in resolving the process of the angular momentum redistribution within MCs.

CONCLUSIONS
Our study analyzes a sample of 443 12 CO MCs with double 13 CO structures, and 185 12 CO MCs with triple 13 CO structures from the MWISP CO survey.Our objective is to investigate the relative velocities on the line of sight between 13 CO structures (∆V LOS ) within 12 CO clouds and identify the proportion of MCs exhibiting instinct velocity discontinuities.Our findings can be summarized as follows: 1. Approximately 70% of ∆V LOS values are less than ∼ 1 km s −1 , and roughly 10% of values exceed 2 km s −1 with the maximum value reaching ∼ 5 km s −1 .For the ratios between ∆V LOS values and internal velocity dispersions of 13 CO structures (∆V LOS /σ13 CO,in ), approximately 70% of them are less than 3 and about 10% are greater than 5, with several samples reaching ∼ 12.
2. The distributions of ∆V LOS for MCs with double and triple 13 CO structures are similar, as well as their ∆V LOS /σ13 CO,in .Additionally, in the triple regime, distributions of ∆V LOS between arbitrary two 13 CO structures and pairs of 13 CO structrues connected by the minimal spanning tree are also similar.Such similarities suggest that the relative motions of 13 CO struc-tures within clouds are random and regulated by the fundamental processes that are independent of structure complexities and cloud scales.Turbulent flows are a promising candidate, as they can be driven by dynamical processes in different scales.
3.About 40% of MCs in either double or triple regimes exhibit distinct velocity discontinuities, with relative velocities between 13 CO structures greater than the internal linewidths of 13 CO structures.The velocity fields for this portion of MC samples present redshifted and blueshifted velocities that come from the bulk motions of 13 CO structures.
4. Among the 40% of samples in the triple regime showing distinct velocity discontinuities, ∼ 33% exhibit the signatures of bulk motions through the two-body motion, whereas the remaining ∼ 7% show the feasures of combinations through the multiple-body motion.This suggests that cloud mergers or splits tend to occur between two MCs.
5. The indication of random motion among 13 CO structures and a considerable portion of distinct velocity discontinuities between 13 CO structures with 12 CO clouds provide further evidence for our previously proposed picture that 13 CO structures act as the building blocks of MCs, and the transient processes of MCs proceed by slow mergers or splits among these fundamental blocks.
This research made use of the data from the Milky Way Imaging Scroll Painting (MWISP) project, which is a multi-line survey in 12 CO/ 13 CO/C 18 O along the northern galactic plane with PMO-13.7mtelescope.We are grateful to all of the members of the MWISP working group, particulaly the staff members at the PMO-13.7mtelescope, for their long-term support.LY acknowledges Haoran Feng for his support on some data analysis scripts.

Figure 1 .Figure 2 .
Figure1.Relative velocities along the line of sight between 13 CO structures (∆VLOS) in MCs with double and triple 13 CO structures.A single MC with double 13 CO structures has one ∆VLOS.A single MC with triple 13 CO structures have three ∆VLOS between each two 13 CO structures or two ∆VMST between each pair of 13 CO structures connected by the minimal spanning tree (MST) algorithm.

Figure 3
Figure3.Upper panel: number distributions of relative velocities (∆VLOS) between each two 13 CO structures in the MC samples having triple 13 CO structures and their ratios with internal velocity dispersions of 13 CO structures (∆VLOS/σ13 CO,in ).Lower panel: number distributions of the maximum relative velocity (∆Vmax) within each MC and their ratios with internal velocity dispersions (∆Vmax/σ13 CO,in ).The vertical-dashed lines show their median values.In the upper-right corner of each panel, the corresponding normalized probability densities are presented.The normalized probability densities are presented as log scales, and the values in x-axis are binned as linear scales.

Figure 4 .
Figure4.The distribution of relative velocities between 13 CO structures for 185 MCs in the 'triple' regime.The gray histograms represent the whole 555 ∆VLOS from the arbitrary two of 13 CO structures within 185 samples.The green histograms mean the 370 ∆VMST between the 13 CO structures connected through the minimal spanning three (MST) algorithm, there are two ∆VMST within a cloud.The corresponding vertical-dashed lines show their median values.In the upper-right corner of each panel, the corresponding normalized probability densities are presented.The normalized probability densities are presented as log scales, and the values in x-axis are binned as linear scales.

Figure 5 .Figure 6 .
Figure 5. Schematic illustrations of the two kinds of combination modes for MCs with triple 13 CO structures.

Figure A2 .
Figure A2.Number distribution of angular areas, velocity spans, integrated fluxes, and peak intensities of 12 CO line emission for MC samples in the far group, most of which are distributed in the Perseus arm with kinematical distances of ∼ 2 kpc.The gray histograms represent 280 12 CO MCs with double 13 CO structures, and the green histograms represent 105 12 CO MCs with triple 13 CO structures.

Figure B1 .
Figure B1.Velocity fields of MCs with double 13 CO structures, whose relative velocities are characterized as distinct velocity discontinuities.Left panel: the color map represents the first moment map (velocity field) of 12 CO emission of MCs, and the cyan contours show the moment zero map (velocity-integrated intensity) of 13 CO line emission, ranging from 10% to 90% in increments of 20% of its maximum value.Middle panel: the color map represents the first moment map (velocity field) of 13 CO emission of MCs, and the cyan contours show the moment zero map (velocity-integrated intensity) of 12 CO line emission, ranging from 10% to 90% in increments of 20% of its maximum value.Right panel: the averaged spectral lines for the extracted 12 CO line emission (black) and 13 CO line emission (cyan) within MCs.The number noted in the upper-left corner is the number ID for MC samples.

Figure B2 .
Figure B2.Same as Figure B1, but for the MC with triple 13 CO structures, where two distinct velocity discontinuities between each two of 13 CO structures are determined.

Table 1 .
Number of MC sources in different groups.

Table 2 .
Radial relative velcotities (∆VLOS) between 13 CO structures within 12 CO MC samples.Note-The quantiles at 0.05, 0.25, 0.5, 0.75, and 0.95 for the ∆VLOS (km s −1 ) in their sequential data.The 'Double' represents the 443 MCs with double 13 CO structures and the 'Triple' corresponds to the 185 MCs having three 13 CO structures.The 'Near' and 'Far' represent the MC samples in the near and far groups, respectively.The ∆VLOS is the relative velocity on the line of sight between 13 CO structures.There is one ∆VLOS in each 'Double' sample and three ∆VLOS in each 'Triple' sample.The '∆Vmax' means the maximum value in the three ∆VLOS of each 'Triple' sample.

Table 3 .
The ratios between relative velcotities and internal velocity dispersions (∆VLOS/σ13 CO,in ) of 13 CO structures within 12 CO MC samples.

Table 4 .
Fractions of MC samples with the distinct velocity discontinuities.∆V dis ) and having one ∆V dis in MC samples with double 13 CO structures.Also, the fractions of MCs not having ∆V dis (zero) and having one, two, and three ∆V dis in MC samples with triple 13 CO structures.
Yuan et al.
This research was supported by the National Natural Science Foundation of China through grant 12303034 & 12041305 and the Natural Science Foundation of Jiangsu Province through grant BK20231104.MWISP was sponsored by the National Key R&D Program of China with grant 2023YFA1608000 & 2017YFA0402701 and the CAS Key Research Program of Frontier Sciences with grant QYZDJ-SSW-SLH047.Data Availability.The extracted 12 CO line data for the 18,190 12 CO clouds and the extracted 13 CO line data within the 2851 12 CO clouds are publicly available at DOI:10.57760/sciencedb.j00001.00427.