Understanding the Kinetic Energy Deposition within Molecular Clouds

According to the structures traced by 13CO spectral lines within 12CO molecular clouds (MCs), we investigate the contributions of their internal gas motions and relative motions to the total velocity dispersions of 12CO MCs. Our samples of 2851 12CO MCs harbor a total of 9556 individual 13CO structures, among which 1848 MCs (∼65%) have one individual 13CO structure and the other 1003 MCs (∼35%) have multiple 13CO structures. We find that the contribution of the relative motion between 13CO structures ( σ13CO,re ) is larger than that from their internal gas motion ( σ13CO,in ) in ∼62% of 1003 MCs in the “Multiple” regime. In addition, we find that σ13CO,re tends to increase with the total velocity dispersion ( σ12CO,tot ) in our samples, especially for MCs with multiple 13CO structures. This result provides a manifestation of macroturbulence within MCs, which gradually becomes the dominant way of storing kinetic energy along with the development of MC scales.


INTRODUCTION
Molecular clouds (MCs) are the fundamental components within galaxies and also the sites of star formation. Thus understanding the dynamic evolution of MCs is crucial for understanding how the large-scale gas gradually gathers into the stellar system. Several scenarios of the formation and evolution of MCs have been proposed, mainly including the top-down, bottom-up, and transient pictures. The top-down picture depicts that the large-scale clouds monotonously fragment into a hierarchy of successively small-scale clouds as the density rises and the Jeans mass decreases, due to the gravitational instabilities (Oort 1954;Lin & Shu 1964;Goldreich & Lynden-Bell 1965). The bottom-up models are the agglomeration or coagulation between smaller clouds to construct larger clouds monotonously (Field & Saslaw 1965;Kwan & Valdes 1983Tomisaka 1984Tomisaka , 1986. In the transient picture, MCs formed in the diffuse ISM through local compressions induced by the different scales of converging flows, appear to evolve in either direction Passot et al. 1995;Ballesteros-Paredes et al. 1999;Vázquez-Semadeni et al. 2006;Heitsch et al. 2006;Beuther et al. 2020). Other interesting theories on the formation and evolution of MCs also have been discussed (e.g. Dobbs & Baba 2014;Ballesteros-Paredes et al. 2020;Chevance et al. 2023). Note that these mechanisms are not necessarily mutually exclusive and may have a role at various stages of the dynamical evolution of MCs (Dobbs 2008;Tasker & Tan 2009;Dobbs & Pringle 2013;Dobbs et al. 2015;Ballesteros-Paredes et al. 2020;Jeffreson et al. 2021). However, the observational supports for these scenarios are still insufficient. Revealing the gas kinematics within MCs on scales ranging from MCs to their internal substructures are able to provide more observational constrains to the mechanisms acting on the MCs evolution.
One of the most fundamental and influential descriptions on the dynamical states of MCs is parameterized by the Larson relations in Larson (1981), which is the power-law relationship between the global velocity dispersions (δV/km s −1 ) and spatial sizes (L/pc) of MCs. The δV-L relation was investigated in different scales from the cloud-to-cloud (Larson 1981;Solomon et al. 1987;Heyer et al. 2009), cloud-to-subregion (Myers 1983), to the velocity structure functions in individual clouds (Heyer & Brunt 2004;Heyer et al. 2006;Brunt et al. 2009). However, the velocity dispersions of MCs in these relations are their total velocity dispersions, which are induced by the thermal and nonthermal gas motion. Taking the complex environment of the Galaxy into account, the molecular gas motion can be tied to the largescale galactic-dynamical processes, e.g., the differential rotation and shear (Bonnell et al. 2006), and the stellar feedback, e.g., the expansion of HII regions or supernova explosions (Skarbinski et al. 2023). The nonthermal motions are composed of systematic and turbulent gas motion, and the turbulent motion contains both microturbulent and macroturbulent (Zuckerman & Evans 1974;Silk 1985;Hacar et al. 2016). We still lack information on the contributions of these different motions to the total velocity dispersion.
Our previous work in Yuan et al. (2022) (Paper II) has identified the 13 CO structures within 2851 12 CO MCs using the CO lines data from the Milky Way Imaging Scroll Painting (MWISP) survey (Su et al. 2019). We further investigated the spatial distribution of these 13 CO structures, and found that there is a preferred spatial separation between 13 CO structures in these 12 CO clouds (Yuan et al. 2023) (Paper III). In addition, a scaling relation between the 12 CO cloud area and its harbored 13 CO structure count also has been revealed. According to these observational results, we propose an alternative picture for the dynamical evolution of MCs: the regularly-spaced 13 CO structures as the fundamental units build up the MCs, and the assembly and destruction of MCs proceeds under this fundamental unit. This picture implies the contribution of the relative motion between 13 CO structures in the total velocity structures should not be ignorable. Meanwhile, the decomposition of the gas motion within MCs is essential to verify this picture.
In this paper, we aim to decompose the total velocity dispersion into different components, which are mainly from the internal and relative movements of the 13 CO structures within MCs. Section 2 mainly describes the 12 CO and 13 CO lines data from the MWISP survey, the extracted 12 CO MC samples and their internal 13 CO structures. In section 3, we present the results, including the systematic velocities derived by the 12 CO and 13 CO line emission for each MC, the distributions of the total velocity dispersion and its components for each MC, and the correlations between the total velocity dispersion and its components. In section 4, we discuss the effects of the derived systematic velocities and the optical depths of 12 CO lines on our results. Meanwhile, we spec-ulate about the micro-turbulent and macro-turbulent motions within MCs and also provide the implication on the dynamic evolution of MCs.
2. DATA 2.1. 12 CO(J=1-0) and 13 CO(J=1-0) spectral lines data from the MWISP survey The 12 CO and 13 CO at J=1-0 transition lines are from the Milky Way Imaging Scroll Painting (MWISP) survey, which is an ongoing northern Galactic plane CO survey. This CO survey is carried out on the 13.7m telescope at Delingha, China, and observes 12 CO, 13 CO, and C 18 O(J=1-0) lines simultaneously. A detailed description on the performance of the telescope and its multibeam receiver system is given in Su et al. (2019); Shan et al. (2012). The observational strategy and the raw data reduction are also introduced in Su et al. (2019). The half-power beamwidth (HPBW) is ∼ 50 ′′ at 115 GHz. The typical noise temperature is ∼ 250 K at 12 CO lines and ∼ 140 K at 13 CO and C 18 O lines. A velocity resolution is about 0.16 km s −1 for 12 CO line and 0.17 km s −1 for 13 CO and C 18 O lines, with a typical rms level of ∼ 0.5 K for 12 CO lines and ∼ 0.3 K for the 13 CO lines, respectively.
In this work, we utilize the 12 CO and 13 CO line emission in the second Galactic quadrant with the Galactical longitude from 104 • .75 to 150 • .25, the Galactical latitude |b| < 5 • .25, and the 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 Yuan et al. (2021); Yuan et al. (2022); Yuan et al. (2023).
2.2. The 12 CO molecular clouds and 13 CO structures The 12 CO molecular cloud in this work is defined as a set of contiguous voxels in the position-position-velocity (PPV) space with 12 CO(1-0) line intensities above a certain threshold. A catalog of 18,190 12 CO molecular clouds has been identified from the 12 CO line emission in the above region, using the Density-based Spatial Clustering of Applications with Noise (DBSCAN) algorithm (Ester et al. 1996;Yan et al. 2021). The DBSCAN algorithm was designed to discover clusters in arbitrary shapes (Ester et al. 1996), and further developed to identify the 12 CO MCs by Yan et al. (2020). This method combines both intensity levels and connectivity of signals to extract structures, which is fit for the extended and irregular shapes of MCs. These 18,190 MCs were visually inspected and also took a morphological classification, which was mainly classified into filaments and nonfilaments in Yuan et al. (2021) (Paper I).
An individual 13 CO structure in this work is characterized by a set of connected voxels in the PPV space having 13 CO line intensities above a certain threshold, which is extracted using the DBSCAN algorithm within the boundaries of 12 CO clouds. Among the total 18,190 12 CO clouds, 2851 12 CO clouds are identified to have 13 CO structures (Yuan et al. 2022). The properties of these extracted 13 CO structures have been systematically analyzed in Papers II and III. The performance of the DBSCAN algorithm and the parameters used for the extraction of 12 CO clouds and 13 CO structures are described in detail in Appendix A. The extracted 12 CO line data for 18,190 12 CO clouds and the extracted 13 CO line data within the 2851 12 CO clouds are available at DOI:10.57760/sciencedb.j00001.00427. In this work, we focus on these 2851 12 CO molecular clouds and their internal 13 CO structures.

RESULTS
The goal of this work is to decompose the contributions of the internal and relative gas motion to the total velocity dispersion of a MC, according to the gas movements of its internal 13 CO structures. Our samples of 2851 MCs harbor a total of 9566 individual 13 CO structures, among which 1848 MCs (65%) have one 13 CO structure and the other 1003 MCs (35%) have more than one 13 CO structure. According to this, the whole clouds are separated into two regimes, i.e. single and multiple. In a cloud, we define its internal region having both 12 CO and 13 CO emission as the 13 CO-bright region, which contains the whole individual 13 CO structures within a MC, and the region with the 12 CO line emission but not the 13 CO emission as the 13 CO-dark region.

Systematical velocities of 12 CO molecular clouds
To calculate the total velocity dispersion of a MC, its systematic velocity due to the differential rotation of the Galaxy needs to be determined first. According to the 12 CO and 13 CO line emission of MCs, we can calculate their centroid velocities to represent the systematic velocities of MCs. Using the 12 CO and 13 CO emission in a position-position-velocity (PPV) space for each cloud, we derive the centroid velocity for a MC as follows (Rosolowsky & Leroy 2006): where the V12 CO,i and T12 CO,i are the line-of-sight velocity and brightness temperature of 12 CO emission at the ith voxel in the PPV space of the whole 12 CO cloud, the V13 CO,i and T13 CO,i are those values of the 13 CO emission in the 13 CO-bright region of a MC, the sum Σ cloud i runs over all voxels within the PPV space of a 12 CO cloud, the sum Σ 13 CO−bri i runs over all voxels within the 13 CO-bright region of a 12 CO cloud. Figure 1 shows the differences between the centroid velocities calculated by the 12 CO and 13 CO emissions for each cloud. The quantiles of the differences between V cen, 12 CO and V cen, 13 CO for the MCs in the 'all', 'single', and 'multiple' regimes are listed in Table 1, respectively. We find that the differences between V cen, 12 CO and V cen, 13 CO are less than ∼ 0.65 km s −1 for 90% of all the samples and less than ∼ 0.15 km s −1 for 50% of the samples. The differences between V cen, 12 CO and V cen, 13 CO for the MCs in the 'multiple' regime are a bit more scattered than those in the MCs in the 'single' regime. The differences for 50% of MCs in the 'multiple' regime are within ∼ 0.2 km s −1 , while this value is ∼ 0.15 km s −1 for the MCs in the 'single'. The median values are close to zero for the MCs with either single or multiple 13 CO structures. Overall, the differences between V cen, 12 CO and V cen, 13 CO for most MCs concentrate in a narrow range, especially compared with the 12 CO velocity span of MCs with a median value of 4.0 km s −1 (Yuan et al. 2022). The centroid velocity represents the systematical motion of the total gas in a single MC, although 12 CO emission is easy to be optically thick, while it can trace most gas in a MC. Thus we first take the V cen, 12 CO as the systematic velocity (V sys ) of MCs in the following calculations. The effects of the V cen, 12 CO or V cen, 13 CO as the systematic velocities (V sys ) of MCs on the results also have been analyzed in the discussion.

Decomposition of the total velocity dispersion in each MC
Furthermore, we calculate the total velocity dispersion and its compositions from the gas motions in each MC. The total velocity dispersion of a MC (σ12 CO,tot ) is calculated using its 12 CO line emission as: where the sum Σ cloud i runs over all voxels within the PPV space of a 12 CO cloud.
The velocity dispersion for the gas in the 13 CO-bright region (σ13 CO,tot ) is calculated as: where the sum Σ 13 CO−bri i runs over all voxels within the 13 CO-bright region of a cloud. Multiple Figure 1. The distributions of the differences between the centroid velocities from 12 CO emission (V cen, 12 CO ) and 13 CO emission (V cen, 13 CO ) in MCs. Among that, the 'All' represents the whole 2851 MCs, and the 'Single' and 'Multiple' correspond to the MCs having single and multiple (more than one) 13 CO structures, respectively. In the middle and right panels, each dot represents a 12 CO MC. The colors on these dots represent the distribution of the probability density function (2D-PDF) of their 12 CO MCs, which are calculated utilizing the kernel-density estimation through Gaussian kernels in the PYTHON package of scipy.stats.gaussian kde. The cyan-dashed lines indicate the lines with V cen, 12 CO = V cen, 13 CO . Note-The quantiles at 0.05, 0.25, 0.5, 0.75, and 0.95 for the differences between V cen, 12 CO (km s −1 ) and V cen, 13 CO (km s −1 ) in their sequential data. The 'All' represent the whole 2851 MCs, and the 'Single' and 'Multiple' correspond to the MCs having single and multiple 13 CO structures, respectively.
For a cloud, it is made up of the 13 CO-bright and the 13 CO-dark regions. The 13 CO-bright region includes a single or multiple (more than one) individual 13 CO structures, thus σ13 CO,tot can be decomposed into the internal gas motion within 13 CO structures and the relative motion between 13 CO structures. Thus, for a cloud having 13 CO structures with the number of j, the σ13 CO,tot can be further decomposed as: where the σ13 CO,in represents the internal gas motion within the 13 CO structures in a 12 CO cloud, the σ13 CO,re represents the relative motion between the 13 CO structures in a 12 CO cloud. The σ13 CO,j is the velocity dispersion within the jth 13 CO structure, the V cen, 13 CO,j is the centroid velocity of 13 CO emission in 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. Among that, is the integrated flux of 13 CO line emission for the jth 13 CO structure. Figure 2 shows the distributions of the calculated σ12 CO,tot , σ13 CO,tot , σ13 CO,in , and σ13 CO,re for each cloud. For comparison, these values for the MCs in the single and multiple regimes are also presented in Figure 2. Moreover, their quantiles at 0.05, 0.25, 0.5, 0.75, 0.95 and the mean values are listed in Table 2. We find that the values of σ12 CO,tot and σ13 CO,tot in the 'multiple' samples are systematically larger than those in the 'single' regime. The σ13 CO,tot is composed of the σ13 CO,re and σ13 CO,in . We find that the differences between the σ13 CO,re for the clouds in the 'multiple' and the 'single' are more obvious than those differences between the σ13 CO,in in the 'single' and 'multiple' regimes, as shown in Figure 2. From the values listed in Table  2, the σ13 CO,re in the 'multiple' are larger than those in the 'single' by a factor of ∼ 3, this factor is about 1.5 for the σ13 CO,in . Figure 3 shows the increasing trends of these velocity dispersions at different quantiles as listed in Table 2. For the whole 2851 MCs, ∼ 65% of them are in the 'single' regime and the rest ∼ 35% belong to the 'multiple' regime. We find the increasing trend of σ13 CO,tot is similar to the σ12 CO,tot , either in the multiple or single regime. Nevertheless, the increasing trends of σ13 CO,re and σ13 CO,in are different, the σ13 CO,re increases with a steeper slope than that from the σ13 CO,in . In the whole sample, we find ∼ 40% of them having the σ13 CO,re larger than the σ13 CO,in , the fraction is ∼ 62% in the 'multiple' samples and ∼ 20% in the 'single' samples, respectively. From above results, those indicate the increase of σ13 CO,tot is mainly attributed to the increase of σ13 CO,re instead of the σ13 CO,in , especially for the MC in the 'multiple' regime.

Correlations between kinetic energy compositions
For a MC, the total velocity dispersion (σ 2 12 CO,tot ) reflects the whole 12 CO gas motion, the σ 2 13 CO,tot is the velocity dispersion for the gas motion within its 13 CObright region. Meanwhile, the σ 2 13 CO,tot is further decomposed into σ 2 13 CO,re and σ 2 13 CO,in , corresponding to the internal and relative motions of 13 CO structures, respectively. An interesting question concerns how these different components change with the increase of total velocity dispersion.
In Figure 4, we present the correlations between the velocity dispersions of σ 2 13 CO,tot , σ 2 13 CO,re and σ 2 13 CO,in with σ 2 12 CO,tot in different MC samples, respectively. The Spearman's rank correlation coefficients (R-value) for these relations are also noted. We find the σ 2 13 CO,tot positively correlate with the σ 2 12 CO,tot for all the samples with a R-value of 0.85. Meanwhile, this R-value of 0.9 for the MCs in the 'multiple' regime is larger than the R-value of 0.77 in the 'single' regime. The σ 2 13 CO,tot is further decomposed into the σ 2 13 CO,re and σ 2 13 CO,in . The R-value is 0.68, 0.77, and 0.49 for the relations between σ 2 13 CO,re and σ 2 12 CO,tot for the MC samples in the 'all', 'multiple', and 'single', respectively. The corresponding R-value is 0.60, 0.59, and 0.49 for these relations between σ 2 13 CO,in and σ 2 12 CO,tot . We find that the σ 2 13 CO,re are more positively correlated with σ 2 12 CO,tot in the 'multiple' regime, while the σ 2 13 CO,in don't significantly increase with σ 2 12 CO,tot , either in the 'single' or 'multiple' regime. That further indicates the increase of σ 2 12 CO,tot is mainly attributed to the increase of σ 2 13 CO,re , i.e. the relative motion between 13 CO gas structures, especially for the MCs in the 'multiple' regime.
We should note that the statistical error of the σ13 CO,re is influenced by the number of 13 CO structures within the MC. In our 2851 MC samples, ∼ 65% of clouds harbor a single 13 CO structure, about 15% of them have double 13 CO structures, and the rest ∼ 20% have at least three 13 CO structures, as presented in our Paper II. Thus the calculated σ 2 13 CO,re of MCs are scattered in statistics and it should be reliable for MCs having at least ten 13 CO structures. In Figure 5, we highlight the 125 clouds having at least ten 13 CO structures. For these 125 MCs, the correlation between their σ 2 13 CO,re and σ 2 12 CO,tot has a higher R-value of 0.88, however, the R-value is 0.43 for the relation between their σ 2 13 CO,in and σ 2 12 CO,tot . That further demonstrates the relative motion between 13 CO structures within clouds, instead of their interior gas motion, is more positively correlated with their global velocity dispersions.
Since the two types of gas motions from the 13 CO structures, i.e. their relative motions (σ 2 13 CO,re ) and internal motions (σ 2 13 CO,in ), exhibit the different correlations with σ 2 12 CO,tot , especially for the MC samples in the 'Multiple' regime. We further look into the fractions of the σ 2 13 CO,re and σ 2 13 CO,in within σ 2 tot, 13 CO , and reveal how the fractions change with the increases of σ 2 13 CO,tot . Figure 6 shows the relations between σ 2 13 CO,re /σ 2 13 CO,tot and σ 2 13 CO,tot for the MCs in the 'multiple' regime, as well as the relations between σ 2 13 CO,in /σ 2 13 CO,tot and σ 2 13 CO,tot . We find that the fractions of σ 2 13 CO,re within σ 2 13 CO,tot tend to be ∼ 80 percent, which also has a slight increase as the increase of σ 2 13 CO,tot , the left ∼ 20 percent contributions are from the σ 2 13 CO,in . This means the relative motions of 13 CO gas structures tend to be the dominant form to store the kinetic energy for MCs in the 'multiple' regime.  In order to ensure the effects of the V cen, 12 CO or V cen, 13 CO as systematic velocity on our results, here, we take the V cen, 13 CO as the systematic velocities of MCs and further calculate the velocity dispersions following Eq.5. For the MCs in the 'single' regime, the systematic velocities(V cen, 13 CO ) of MCs are consistent with the centroid velocities of their single 13 CO structures. Thus the relative velocity between the 13 CO structure and systematic velocity for a MC in the 'single' regime is equal to zero. The MCs in the 'All' regime can be divided into MCs in the 'Multiple' and 'Single' regimes. In Figure B1, we show the relations between σ 2 12 CO,tot and σ 2 13 CO,tot for the MCs, which are in the 'All', 'Multiple', and 'Single', respectively. The R-values are 0.81 for the MCs in the 'Multiple' and 0.41 for those in the 'Single'. That implies that the σ 2 13 CO,tot tend to increase with σ 2 12 CO,tot for MCs in the 'Multiple' regime. However, the σ 2 13 CO,tot for the MC in the 'Single' regime, which is equal to its σ 2 13 CO,in , doesn't significantly increase with σ 2 12 CO,tot . In addition, the σ 2 13 CO,tot is decomposed into σ 2 13 CO,re and σ 2 13 CO,in . Figure B2 presents the correlations between the σ 2 13 CO,tot , σ 2 13 CO,re , σ 2 13 CO,in with the σ 2 12 CO,tot for all the samples, respectively. We find that the σ 2 13 CO,re tends to increase with σ 2 12 CO,tot having a Rvalue of 0.71, while the σ 2 13 CO,in doesn't have a clear trend with σ 2 12 CO,tot , whose R-value is 0.54. Furthermore, we decompose the σ 2 13 CO,tot for the MCs in the 'Multiple' regime. Figure B3 presents the correlations between the σ 2 13 CO,tot , σ 2 13 CO,re , σ 2 13 CO,in with the σ 2 12 CO,tot for the samples in the 'Multiple', respectively. For the MCs in the 'Multiple' regime, the σ 2 13 CO,tot also tends to increase with σ 2 12 CO,tot , whose R-value is 0.81. Meanwhile, the R-value is 0.71 for the relation between σ 2 13 CO,re and σ 2 12 CO,tot and 0.54 for the relation between σ 2 13 CO,in and σ 2 12 CO,tot . The relations between σ 2 13 CO,re and σ 2 12 CO,tot are consistent either in the 'all' or 'multiple' regime, due to that the σ 2 13 CO,re is zero for in MCs in the 'single' regime. In addition, the relations between σ 2 13 CO,in and σ 2 12 CO,tot have the same R-value either in the 'all' or 'multiple'. The σ 2 13 CO,re is always more positively correlated with the σ 2 12 CO,tot than σ 2 13 CO,in for the MCs in the 'multiple' regime.
Thus, either V cen, 12 CO or V cen, 13 CO is defined as the systematic velocity for a MC, we find the relative motion between 13 CO structures gradually provides the primary contributions to the σ 2 12 CO,tot with the development of MC scales.

Effects of optical depths
Since the opacity can severely affect the observed linewidths in the optically thick CO line emission, e.g. 12 CO(1-0) lines (Phillips et al. 1979;Hacar et al. 2016;Pineda et al. 2008). It is necessary to explore and constrain the influence of the opacity broadening on the derived σ12 CO,tot . The spectral profiles of 12 CO and 13 CO lines emission are described by the radiative transfer equation (Rohlfs & Wilson 1996): where J ν (T ) = ( hν/k exp(hν/kT )−1 ), T ex is the excitation temperature of the lines, T bg = 2.7 K is the cosmic microwave background temperature, τ12 CO and τ13 CO are the line opacities at the 12 CO and 13 CO emission, respectively. In the 13 CO-bright region, we assume that the excitation temperature of 13 CO(1-0) line is the same as that for the 12 CO(1-0) line, then combine the Eq. 6 and Eq. 7 as: . Correlations between σ 2 13 CO,tot , σ 2 13 CO,re , and σ 2 13 CO,in with σ 2 12 CO,tot for the MC samples in the 'all', 'multiple', and 'single' regimes, respectively. Each dot in the panels represents a 12 CO MC. The colors on these dots represent the distribution of the probability density function (2D-PDF) of their 12 CO MCs. The corresponding Spearman's rank correlation coefficient (R-value) is noted in each panel.
To visualize the distribution of the 12 CO and 13 CO line emission in a MC, In Figure C4 and C5, we present the distributions of the velocity-integrated intensities of 12 CO (I12 CO ) and 13 CO (I13 CO ) emission for two MC samples and also the distributions of the pixel numbers in the intervals of the values (I12 CO , I13 CO , τ13 CO ). We find that the 12 CO line intensities in the pixels within the 13 CO-dark region vary smoothly, especially comparing with those values within the 13 CO-bright region. Thus it is reasonable to determine the excitation temperature in the periphery of 13 CO-bright region (T 0 ) to be consistent with that in the 13 CO-dark region. Further the optical opacity of 12 CO emission in the 13 CO-dark region can be estimated as: where T mb, 12 CO,0 and τ12 CO,0 represent the brightness temperature and the optical depth of 12 CO line emission lie at the periphery of the 13 CO-bright region. We adopt the minimum value of τ12 CO within the 13 CO-bright region as the τ12 CO,0 , and its corresponding T mb, 12 CO is defined as T mb, 12 CO,0 . Thus using Eq.8 and Eq.9, we drived the opacities of 12 CO line emission in the 13 CO-bright and 13 COdark region of each MC, respectively. Figure 7 shows the distributions of the minimal (τ12 CO,min ), maximal (τ12 CO,max ), and mean opacities (τ12 CO ) of the 12 CO emission in the whole region in each MC, as well as in its 13 CO-dark and 13 CO-bright regions, respectively. The medians for these values are also noted in each Interior Figure 5. Correlations between σ 2 13 CO,re with σ 2 12 CO,tot (left panel) and σ 2 13 CO,in with σ 2 12 CO,tot (right panel) for the MC samples, respectively. The green dots in the panels represent 12 CO MCs harboring at least ten 13 CO structures, the corresponding Spearman's rank correlation coefficient (R-value) is noted in each panel. The gray crosses are for the left 12 CO MCs, whose number of 13 CO structures is less than 10. Relation between the ratio of σ 2 13 CO,re over σ 2 13 CO,tot with the σ 2 13 CO,tot . Right panel: Relation between the ratio of σ 2 13 CO,in over σ 2 13 CO,tot with the σ 2 13 CO,tot . Each dot in both panels represents a MC in the 'multiple' regime. The colors on the dots represent the distribution of the probability density function (2D-PDF) of their 12 CO MCs. The black-dashed lines show the values of σ 2 13 CO,re /σ 2 13 CO,tot = 0.8 in the left panel and σ 2 13 CO,in /σ 2 13 CO,tot = 0.2 in the right panel.
panel. Theτ12 CO in the whole regions of MCs have a median value of 4.9, this value is 21 in the 13 CObright regions of MCs and 1.3 in the 13 CO-dark regions. That indicates the 12 CO emission in the 13 CO-bright region is optically thick, while that in the 13 CO-dark region is slightly optically-thick. From the τ12 CO,max distribution, we find that ∼ 90% of our samples with τ12 CO,max less than 80. That indicates the most of 13 CO emission(X( 13 CO):X( 12 CO)=1:76.7) is optically thin in these MCs. In addition, the median of τ12 CO,min in the 13 CO-bright regions of our MC samples is 10.8, which may be the critical value between the 13 CO-bright and 13 CO-dark regions, under our observational sensitivities of 0.25 K at a velocity resolution of ∼ 0.2 km s −1 for the 13 CO line emission. Since the 12 CO line emission in the 13 CO-bright region is optically thick, the line broadening produced by the line opacity can be defined as (Phillips et al. 1979): where ∆V and ∆V int are referred to as the observed and the intrinsic velocity width (FWHM), respectively. We use the τ12 CO map to estimate the β map for each cloud according to Eq.10. Figure 7 shows the distributions of the minimal (β min ), maximal (β max ), and mean β (β) values in the whole region of each MC, as well as in the 13 CO-dark and 13 CO-bright regions, respectively. Thē β in the whole regions of MCs range from ∼ 1 to 2.1 with a median of 1.33, the median value is 2.2 and 1.15 for theβ in the 13 CO-bright regions and the 13 CO-dark regions of MCs, respectively. We use the value ofβ in the whole region of each cloud to mitigate the effect of 12 CO line opacity on the velocity dispersion of 12 CO emission, following σ12 CO,int = σ12 CO,tot /β. Figure C6 shows the relations between the σ 2 13 CO,tot , σ 2 13 CO,re , σ 2 13 CO,in with the σ 2 12 CO,int . We find that the Spearman's rank correlation coefficients (R-value) in the relations between σ 2 13 CO,re and σ 2 12 CO,int are higher than those for the relations between σ 2 13 CO,re and σ 2 12 CO,tot . However, the R-value for the relations between σ 2 13 CO,in and σ 2 12 CO,int are lower than those from the relations between σ 2 13 CO,in and σ 2 12 CO,tot . That means our conclusion that the relative motions between 13 CO structures mainly account for the increases of σ 2 12 CO,int in the MCs with 'multiple' 13 CO structures is still established, after taking the opacity effects into account.

Kinetic energy in the molecular cloud: microscopic versus macroscopic
Larson's relation between MCs, a power-law relation between their total velocity dispersions and spatial sizes, is similar to the Kolmogorov law for incompressible turbulence (Larson 1981). This suggests the interstellar turbulent energy cascade, i.e. the energy range of random motion produced on a large spatial scale, then cascades to smaller scales in the inertial range and finally dissipates in the damping range (Baker 1976;Fleck 1980). However, the MCs are quite inhomogeneous and highly structured, consisting of numerous denser clumps and cores, Larson's relations are thought to be violated in these dense regions within MCs Ballesteros-Paredes et al. 2011;Traficante et al. 2018).
Our main observational result is that the relative motions between distinct 13 CO structures gradually dominate the total velocity dispersions of MCs with the development of MC scales. The random motion between discrete internal structures is a manifestation of macroscopic turbulence (macro-turbulence), which is thought to primarily determine the line widths of spectral profiles (Zuckerman & Evans 1974;Silk 1985;Kwan & Sanders 1986;Hacar et al. 2016). The existence of dense structures and bulk kinetic energy on various scales provide evidence for the compressibility of MCs. Such a model also would allow a fairly long time scale for damping the motions of internal structures and further tend to prevent the entire cloud from collapsing rapidly (Zuckerman & Evans 1974). The macro-turbulent could be generated by either large-scale or small-scale converging flows (Klessen et al. 2000). Taking the complex Galactic environment into consideration, the gas dynamics within MCs manifested as the macro-turbulence could be driven by the hierarchical gravitational collapse (Zuckerman & Evans 1974;Baker 1976;Ballesteros-Paredes et al. 2011), stellar feedbacks, such as stellar wind and outflows (Silk 1985), supernova explosions (Watkins et al. 2023;Skarbinski et al. 2023), and spiral shocks (Bonnell et al. 2006;Falceta-Gonçalves et al. 2015).

Implications on the dynamic evolution of molecular clouds
In our series of works using a large sample of 12 CO MCs, we have investigated these MCs in different aspects, including their morphologies (Paper I), dense gas fractions traced by 13 CO lines (Paper II), spatial distributions of their internal 13 CO structures (Paper III), and internal kinetic energy depositions in this work. The morphologies of 12 CO MCs are developing from nonfilaments (e.g. clumpy and extended structures) to filaments with the increases of their scales. After extracting the relatively dense gas structures traced by 13 CO lines within each 12 CO cloud, we find that the 13 CO gas contents within 12 CO MCs are confined by the scales of 12 CO MCs (Paper II). Furthermore, we also reveal that there is a preferred spatial separation between 13 CO structures within these 12 CO MCs, independent of their spatial scales. In this work, we further find that the relative motions of 13 CO structures gradually provide the primary contributions to the total velocity dispersions of MCs, as the development of MC scales. Combining these observational results, we find that MCs tend to exhibit complex and filamentary morphology, local density enhancements ( 13 CO gas structures), structural stabilization (regularly-spaced 13 CO gas structures), and relative motions between 13 CO structures, along with the development of MC scales.
Taking the dynamic environment of the Galaxy into account, e.g. the converging gas flows driven by galactic differential rotation and shear, large-scale instabilities, and stellar feedback from the supernova explosions and HII regions. Some numerical simulations predict that converging flows could induce shock compressions and produce local density enhancements in the diffuse ISM (Scalo et al. 1998;Vazquez-Semadeni et al. 1995;Ballesteros-Paredes et al. 1999;Mac Low & Klessen 2004;McKee & Ostriker 2007;Ballesteros-Paredes et al. 2020), and also are the primary drivers of cloud mergers and splits (Tasker & Tan 2009;Dobbs et al. 2012;Dobbs & Pringle 2013;Dobbs et al. 2015;Skarbinski et al. 2023;Watkins et al. 2023). We propose an alternative picture for the assembly and destruction of MCs: the regularlyspaced 13 CO structures are thought to be the building bricks of MCs, the dynamic build-up and destruction of MCs proceeds under this fundamental brick.
We further compare this picture with the simulated results in terms of the development of the velocity structures, density structures, and morphology of MCs. Firstly, we compare the relative velocities in the simulated mergers and the relative velocities between 13 CO structures within MCs. In our MC samples, we find ∼ 90% of the radially relative velocity between internal 13 CO structures is less than 5 km s −1 , as presented in Paper III. Also, 95% of MCs have velocity dispersions of σ12 CO,tot less than 1.66 km s −1 , as listed in Table 2. These observational results are consistent with that mergers are typically slow, occurring at relative speeds of ≲ 5 km/s (∼ 3 times the internal cloud velocity dispersion) predicted in the simulation of Dobbs et al. (2015); Skarbinski et al. (2023), which is thought to be unlikely to cause the shocks (Balfour et al. 2015(Balfour et al. , 2017Liow & Dobbs 2020). In addition, the relative motion between 13 CO structures provides the main contribution to the increases in the global velocity dispersion for MCs, this is also consistent with the cloud mergers leading to the higher velocity dispersions of MCs, as simulated in Dobbs et al. (2011Dobbs et al. ( , 2015; Jeffreson et al. (2021); Skarbinski et al. (2023).
Secondly, the slow merger also is consistent with the existence of a preferred separation between 13 CO structures within MCs, independent of the MC scales. The mergers are typically slow and gentle so that it is unable to induce the shocks and to further change the distributions of their density structures (Balfour et al. 2015(Balfour et al. , 2017Liow & Dobbs 2020). That is also consistent with the numerical results of slow mergers that do not have a strong impact on the density structures of clouds in the works of Dobbs et al. (2015); Jeffreson et al. (2021).
Lastly, Dobbs et al. (2015) discussed that the merger of clouds tends to result in an even further elongated cloud, i.e. the smaller clouds gently merge onto the ends of the larger clouds, which tend to align with spiral arms and interact along their minor axis. This process is also coincident with that clouds tend to exhibit from non-filaments to filaments as increasing with cloud scales (Yuan et al. 2021).
According to the comparisons between our observational facts with the simulated results, we suggest the assembly and destruction of MCs: the regularly-spaced 13 CO structures are thought to be the building bricks of MCs, and the dynamic build-up of MCs proceed by slow mergers among these fundamental bricks, this process does not suffer a significant change on their density structures, but have an impact on their global velocity structures.

CONCLUSIONS
Using a sample of 2851 12 CO molecular clouds, inside which a total of 9566 13 CO gas structures are identified, we investigate the relations between the internal and relative gas motions of 13 CO structures with the total velocity dispersions of each 12 CO cloud, respectively. Our main conclusions are as follows: 1. The centroid velocities calculated by 12 CO and 13 CO lines emission are nearly consistent, their differences are less than ∼ 0.65 km s −1 in 90% of the whole MC samples and less than ∼ 0.15 km s −1 in 50% of the samples.
2. The increasing trend of σ13 CO,tot is similar to that of σ12 CO,tot . For its components of σ13 CO,re and σ13 CO,in , the σ13 CO,re increases with a slope, which is steeper than that from σ13 CO,in .
3. The relation between σ 2 13 CO,re and σ 2 12 CO,tot is more positively correlated than that between σ 2 13 CO,in and σ 2 12 CO,tot . This provides a clear trend of macroturbulence becoming the dominant component of kinetic energy with the development of MC scales.
4. Comparing our observational results with the simulations, we propose a picture on the assembly and destruction of MCs: the regularly-spaced 13 CO structures are thought to be the building bricks of MCs, and the transient processes of MCs proceed by slow mergers among these fundamental bricks, during which the density structures of MCs do not vary significantly, but their global velocity structures are influenced.
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-13.7m telescope. We are grateful to all of the members of the MWISP working group, particulaly the staff members at the PMO-13.7m telescope, for their long-term support. The DBSCAN algorithm identifies a set of consecutive voxels (points) in the PPV cube as a molecular cloud. The extracted voxels need to have 12 CO line intensities above a certain threshold and connect with each other. There are three input parameters: cutoff, ϵ, and MinPts. The parameter of 'cutoff' determines the line intensity threshold. The ϵ and MinPts are for confining the connectivity of extracted structures. A core point is a point within the extracted contiguous structure, which have the adjacent points exceeding a number threshold in a certain radius. The 'MinPts' determines the number threshold of adjacent points and the ϵ determines the radius of the adjacence. The border points of extracted structures are inside the ϵ-neighborhood of core points, but do not include the 'MinPts' neighbors in its ϵ-neighborhood (Ester et al. 1996). We adopt the cutoff = 2σ (σ is the rms noise, whose value is ∼ 0.5 K for 12 CO line emission), MinPts=4, and ϵ=1 in the DBSCAN algorithm for the identification of 12 CO clouds, as suggested in Yan et al. (2020). In addition, the post-selection criteria are also utilized to avoid noise contamination. That includes:(1) the total number of voxels in each extracted structure is larger than 16; (2) the peak intensity of extracted voxels is higher than the 'cutoff' adding 3σ; (3) the angular area of the extracted structure is larger than one beam size (2×2 pixels ∼ 1 arcmin 2 ); (4) the number of velocity channels needs to be larger than 3. The performance of different DBSCAN parameters on the extracted structures is presented in Yan et al. (2020Yan et al. ( , 2022. The observational effects, including the finite angular resolution and sensitivity of the observed spectral lines data, on the extracted 12 CO clouds also have been systematically investigated in Yan et al. (2022). In addition, the MC samples extracted by the DBSCAN algorithm also have been compared with those identified by other clustering algorithms, e.g., HDBSCAN and SCIMES, in Yan et al. (2022).
The DBSACN parameters used for the extraction of 13 CO structures are identical to the above parameters for 12 CO clouds, except for the post-selection criteria of the peak intensities higher than the 'cutoff' adding 2σ. In addition, the σ is ∼ 0.25 K for 13 CO line emission. We also compare the performance of three methods, including clipping, moment mask, and DBSCAN, on the extraction of 13 CO structures in paper II.

B. THE CENTROID VELOCITIES OF 13 CO LINE EMISSION AS SYSTEMATIC VELOCITIES OF MCS
The centroid velocities of 13 CO line emission (V cen, 13 CO ) are defined as the systematic velocities of MCs, which are used in Eq.5 to calculate the velocity dispersions. We present the relations between the σ 2 13 CO,tot , σ 2 13 CO,re , and σ 2 13 CO,in with the σ 2 12 CO,tot , for the MC samples. The MCs in the 'All' regime can be divided into MCs in the 'single' and 'multiple' regimes. Figure B1 shows that relations between σ 2 13 CO,tot and σ 2 12 CO,tot for the MC samples in the 'All', 'Multiple', and 'Single' regimes, respectively. Meanwhile, the σ 2 13 CO,tot is decomposed into σ 2 13 CO,re and σ 2 13 CO,in . Figure B2 presents the relations between the σ 2 13 CO,tot , σ 2 13 CO,re , σ 2 13 CO,in with the σ 2 12 CO,tot for all the MC samples, respectively. Also, Figure B3 demonstrates the same relations, but for the MCs in the multiples.  Figure B1. The correlations between the σ 2 13 CO,tot with the σ 2 12 CO,tot for the MC samples in the 'All', 'Multiple', and 'Single' regimes, respectively. The Vsys used in Eq.5 is the V cen, 13 CO of each MC. Each dot in the panels represents a 12 CO MC. The colors on these dots represent the distribution of the probability density function(2D-PDF) of their 12 CO MCs. The corresponding Spearman's rank correlation coefficient (R-value) is noted in each panel.  Figure B2. The correlations between the σ 2 13 CO,tot , σ 2 13 CO,re , and σ 2 13 CO,in with σ 2 12 CO,tot , respectively, for all the samples. The Vsys used in Eq.5 is the V cen, 13 CO of each MC. Each dot in the panels represents a 12 CO MC. The colors on these dots represent the distribution of the probability density function(2D-PDF) of their 12 CO MCs. The corresponding Spearman's rank correlation coefficient (R-value) is noted in each panel.  Figure B3. The correlations between the σ 2 13 CO,tot , σ 2 13 CO,re , and σ 2 13 CO,in with σ 2 12 CO,tot , respectively, for the samples in the 'multiple' regime. The Vsys used in Eq.5 is the V cen, 13 CO of each MC. Each dot in the panels represents a 12 CO MC. The colors on these dots represent the distribution of the probability density function(2D-PDF) of their 12 CO MCs. The corresponding Spearman's rank correlation coefficients (R-value) is noted in each panel.

C. EFFECTS OF OPTICAL DEPTHS ON THE RELATIONS
In Figure C4 and C5, we present the distributions of the velocity-integrated intensities of 12 CO (I12 CO ) and 13 CO (I13 CO ) emission for two MC samples in our catalog, and also the distributions of the pixel numbers in the intervals of the values (I12 CO , I13 CO , and τ13 CO ). The values of τ13 CO in the 13 CO-bright regions of two MC samples are calculated as Eq. 8. Figure C6 shows the relations between the σ 2 13 CO,tot , σ 2 13 CO,re , and σ 2 13 CO,in with σ 2 12 CO,int , where σ12 CO,int = σ12 CO,tot /β τ to mitigate the effect of 12 CO line opacity on the velocity dispersions of 12 CO line emission. The averaged spectra for the extracted 12 CO and 13 CO lines emission within this MC. The vertical black-dashed line delineates the centroid velocity of 12 CO line emission (V cen, 12 CO , km s −1 ) calculated as Eq. 1. The length of the horizontal black-dashed line shows the value of √ 8ln2 × σ12 CO,tot , which corresponds to the FWHM velocity-width of 12 CO spectral line, and the σ12 CO,tot is calculated as Eq. 5. The vertical green-dashed line delineates the centroid velocity of 13 CO line emission (V cen, 13 CO , km s −1 ) calculated as Eq. 2. The length of the horizontal green-dashed line shows the value of √ 8ln2 × σ13 CO,tot , which corresponds to the FWHM velocity-width of 13 CO spectral line, and the σ13 CO,tot is calculated as Eq. 5.  Figure C5. Same as Figure C4, but for another MC named G123.291-0.539-44.15 in our catalog. The green contours in the left panel range from 10% to 90% stepped by 20% of the maximum value (11.9 K km s −1 ). . Relations between the velocity dispersion from different components(σ 2 13 CO,tot , σ 2 13 CO,re , and σ 2 13 CO,in ) with the σ 2 12 CO,int , whose values are revised the effect of optical depths as σ12 CO,int = σ12 CO,tot /βτ . Their Spearman's rank correlation coefficients (R-value) are noted in each panel.