<table><caption>**Table 3.** Number of cells and fractions of cells on each refinement level for the stratified box simulations.</caption><thead><tr><th>Name</th><th>N<sub>tot</sub></th><th>f(4 pc)</th><th>f(2 pc)</th><th>f(1 pc)</th><th>f(0.5 pc)</th><th>f(0.25 pc)</th></tr></thead><tbody><tr><td>SILCC-1pc-3μG</td><td>2.9 × 10<sup>7</sup></td><td>0.04</td><td>0.16</td><td>0.80</td><td>—</td><td>—</td></tr><tr><td>SILCC-1pc-6μG</td><td>3.2 × 10<sup>7</sup></td><td>0.03</td><td>0.18</td><td>0.79</td><td>—</td><td>—</td></tr><tr><td>SILCC-0.5pc-3μG</td><td>2.0 × 10<sup>8</sup></td><td>0.004</td><td>0.02</td><td>0.09</td><td>0.89</td><td>—</td></tr><tr><td>SILCC-0.25pc-3μG</td><td>3.4 × 10<sup>8</sup></td><td>0.003</td><td>0.01</td><td>0.05</td><td>0.46</td><td>0.47</td></tr><tr><td>LS-no-driving</td><td>6.9 × 10<sup>7</sup></td><td>0.22</td><td>0.13</td><td>0.19</td><td>0.25</td><td>0.21</td></tr><tr><td>LS-weak-driving</td><td>8.0 × 10<sup>7</sup></td><td>0.19</td><td>0.11</td><td>0.20</td><td>0.27</td><td>0.22</td></tr><tr><td>LS-medium-driving</td><td>8.1 × 10<sup>7</sup></td><td>0.19</td><td>0.12</td><td>0.20</td><td>0.27</td><td>0.22</td></tr><tr><td>LS-strong-driving</td><td>4.6 × 10<sup>7</sup></td><td>0.34</td><td>0.20</td><td>0.17</td><td>0.16</td><td>0.13</td></tr></tbody></table>

<figure><img src="image_16.png" /><figcaption>**Fig. 15.** Comparison of the cloud mass spectrum in two RAMSES test simulations with a similar setup to simulation LS using different refinement strategies.</figcaption></figure>

resolve gravitationally collapsing structures well, they capture less of the turbulent cascade to small spatial scales that occurs in the more diffuse ISM. This results in them missing a population of small structures created by turbulence.

To verify this, we compare the results of two RAMSES test simulations of the stratified box type which apply different refinement criteria. The first simulation uses a targeted refinement strategy, similar to the one used in the LS simulations listed in Table 1, which focusses on following the gravitational collapse of dense objects. This is done by requiring the local Jeans length and mass, which vary with scale, to be resolved by a certain number of cells. The second simulation has more abundant refinement, meaning that refinement is triggered more frequently in lower density gas. This is achieved by setting a uniform limit on the mass that can be contained in a cell. This results in many more cells and thus significantly longer computation time. Figure 15 compares the mass spectra of the clouds extracted in these two runs. We see that the abundant strategy is able to resolve many more low-mass structures and has a mass spectrum with a shape comparable to what we found for the SILCC runs. The run with the selective refinement strategy has a peak at larger values and a low-mass tail which becomes more prominent

over time as gravitational collapsing structures develop. This test indeed confirms that the difference in refinement strategy results in a different mass spectrum for SILCC and LS.

## 5.2. Cloud extraction parameters and other algorithms

There are many ways to extract clouds from simulations and the results depend on the implementation and chosen parameters of the extraction algorithm. In Appendix A.4 we test the dependence of the mass spectrum on the chosen parameters of the HOP algorithm. Notably, the saddle merging criterion is found to have a significant effect on the slope. If we do not use the saddle merging criterion, the mass spectrum is closer to a log-normal rather than having a power-law high-mass tail. Without saddle merging, the mass spectrum will depend strongly on resolution, since resolved substructures will be counted as individual objects. When enabling saddle merging, the slope of the power-law decreases with decreasing merging density value. When merging everything that touches, using the most aggressive merging strategy possible with $\rho_{saddle} = \rho_{threshold}$, the mass spectrum is significantly shallower than the value of $\rho_{saddle} = 10\rho_{threshold}$ we adopted in this work. The majority of small objects are absorbed into large cloud complexes.

The mechanics of HOP with the most aggressive saddle merging is similar to how the friends-of-friends (FOF) algorithm operates. For particle simulations, FOF dictates that two particles with a density above the required threshold belong to the same group if they are within a specified distance from one another, typically taken to be some fixed fraction of the mean inter-particle separation. For grid simulations, one typically requires the cells to be neighbours. Iffrig & Hennebelle (2017) used FOF to identify clouds in kpc-sized boxes similar to the LS simulations used in this work. They applied a slightly higher density threshold of $50 \text{ cm}^{-3}$ and find that the slope of the mass spectrum depends on the strength of the magnetic field. Without magnetic field the fitted slope is around -1 or slightly steeper. For an initial field of $3-6 \text{ µG}$, comparable to the field strengths used in the stratified box simulations analysed in this work, they obtain shallower slopes, about -0.7 to -0.8. These values are shallower than what we recovered with HOP, in line with the difference observed when changing the saddle merge criterion.

The mass-size relation is also affected. Iffrig & Hennebelle (2017) report that their results are in line with the observed slope of 2.36, somewhat shallower than the general trend of -3 we found in Sect. 3. In Fig. 16, we see that we indeed find a shallower mass-size relation when extracting the clouds with a FOF-like algorithm. Small objects are merged into larger ones, tilting the slope.
<table><caption>**Table 3.** Number of cells and fractions of cells on each refinement level for the stratified box simulations.</caption><thead><tr><th>Name</th><th>N<sub>tot</sub></th><th>f(4 pc)</th><th>f(2 pc)</th><th>f(1 pc)</th><th>f(0.5 pc)</th><th>f(0.25 pc)</th></tr></thead><tbody><tr><td>SILCC-1pc-3μG</td><td>2.9 × 10<sup>7</sup></td><td>0.04</td><td>0.16</td><td>0.80</td><td>—</td><td>—</td></tr><tr><td>SILCC-1pc-6μG</td><td>3.2 × 10<sup>7</sup></td><td>0.03</td><td>0.18</td><td>0.79</td><td>—</td><td>—</td></tr><tr><td>SILCC-0.5pc-3μG</td><td>2.0 × 10<sup>8</sup></td><td>0.004</td><td>0.02</td><td>0.09</td><td>0.89</td><td>—</td></tr><tr><td>SILCC-0.25pc-3μG</td><td>3.4 × 10<sup>8</sup></td><td>0.003</td><td>0.01</td><td>0.05</td><td>0.46</td><td>0.47</td></tr><tr><td>LS-no-driving</td><td>6.9 × 10<sup>7</sup></td><td>0.22</td><td>0.13</td><td>0.19</td><td>0.25</td><td>0.21</td></tr><tr><td>LS-weak-driving</td><td>8.0 × 10<sup>7</sup></td><td>0.19</td><td>0.11</td><td>0.20</td><td>0.27</td><td>0.22</td></tr><tr><td>LS-medium-driving</td><td>8.1 × 10<sup>7</sup></td><td>0.19</td><td>0.12</td><td>0.20</td><td>0.27</td><td>0.22</td></tr><tr><td>LS-strong-driving</td><td>4.6 × 10<sup>7</sup></td><td>0.34</td><td>0.20</td><td>0.17</td><td>0.16</td><td>0.13</td></tr></tbody></table>

<figure><img src="image_16.png" /><figcaption>**Fig. 15.** Comparison of the cloud mass spectrum in two RAMSES test simulations with a similar setup to simulation LS using different refinement strategies.</figcaption></figure>

resolve gravitationally collapsing structures well, they capture less of the turbulent cascade to small spatial scales that occurs in the more diffuse ISM. This results in them missing a population of small structures created by turbulence.

To verify this, we compare the results of two RAMSES test simulations of the stratified box type which apply different refinement criteria. The first simulation uses a targeted refinement strategy, similar to the one used in the LS simulations listed in Table 1, which focusses on following the gravitational collapse of dense objects. This is done by requiring the local Jeans length and mass, which vary with scale, to be resolved by a certain number of cells. The second simulation has more abundant refinement, meaning that refinement is triggered more frequently in lower density gas. This is achieved by setting a uniform limit on the mass that can be contained in a cell. This results in many more cells and thus significantly longer computation time. Figure 15 compares the mass spectra of the clouds extracted in these two runs. We see that the abundant strategy is able to resolve many more low-mass structures and has a mass spectrum with a shape comparable to what we found for the SILCC runs. The run with the selective refinement strategy has a peak at larger values and a low-mass tail which becomes more prominent

over time as gravitational collapsing structures develop. This test indeed confirms that the difference in refinement strategy results in a different mass spectrum for SILCC and LS.

## 5.2. Cloud extraction parameters and other algorithms

There are many ways to extract clouds from simulations and the results depend on the implementation and chosen parameters of the extraction algorithm. In Appendix A.4 we test the dependence of the mass spectrum on the chosen parameters of the HOP algorithm. Notably, the saddle merging criterion is found to have a significant effect on the slope. If we do not use the saddle merging criterion, the mass spectrum is closer to a log-normal rather than having a power-law high-mass tail. Without saddle merging, the mass spectrum will depend strongly on resolution, since resolved substructures will be counted as individual objects. When enabling saddle merging, the slope of the power-law decreases with decreasing merging density value. When merging everything that touches, using the most aggressive merging strategy possible with $\rho_{saddle} = \rho_{threshold}$, the mass spectrum is significantly shallower than the value of $\rho_{saddle} = 10\rho_{threshold}$ we adopted in this work. The majority of small objects are absorbed into large cloud complexes.

The mechanics of HOP with the most aggressive saddle merging is similar to how the friends-of-friends (FOF) algorithm operates. For particle simulations, FOF dictates that two particles with a density above the required threshold belong to the same group if they are within a specified distance from one another, typically taken to be some fixed fraction of the mean inter-particle separation. For grid simulations, one typically requires the cells to be neighbours. Iffrig & Hennebelle (2017) used FOF to identify clouds in kpc-sized boxes similar to the LS simulations used in this work. They applied a slightly higher density threshold of $50 \text{ cm}^{-3}$ and find that the slope of the mass spectrum depends on the strength of the magnetic field. Without magnetic field the fitted slope is around -1 or slightly steeper. For an initial field of $3-6 \text{ µG}$, comparable to the field strengths used in the stratified box simulations analysed in this work, they obtain shallower slopes, about -0.7 to -0.8. These values are shallower than what we recovered with HOP, in line with the difference observed when changing the saddle merge criterion.

The mass-size relation is also affected. Iffrig & Hennebelle (2017) report that their results are in line with the observed slope of 2.36, somewhat shallower than the general trend of -3 we found in Sect. 3. In Fig. 16, we see that we indeed find a shallower mass-size relation when extracting the clouds with a FOF-like algorithm. Small objects are merged into larger ones, tilting the slope.
<figure><img src="image_17.png" /><figcaption>**Fig. 16.** Mass-size relation for the LS stratified boxes and M51 galaxy when using our default HOP parameters versus FOF-like merge criteria. Coloured contour lines are the same as in Fig. 6a.</figcaption></figure>

Another approach is to use dendograms, which describe the hierarchy of structures in the form of a tree. Tress et al. (2021) extracted clouds from their M51 galaxy, that we also use in the comparison in this work, using SCIMES (Colombo et al. 2015). This stands for Spectral Clustering for Interstellar Molecular Emission Segmentation and is a technique mainly used in observations. It is based on the cluster analysis of dendrograms of 3D (in the PPV or PPP space) data cubes that performs particularly well for the identification of large structures such as the molecular clouds in spectroscopical observations of large fields in the Milky Way (e.g. Colombo et al. 2015; Schuller et al. 2017; Benedettini et al. 2020). SCIMES considers the dendrogram tree of the 3D structures in the data cube (using the implementation of Rosolowsky et al. 2008 for astronomical data-sets) and groups different leaves together into ‘clusters’ of leaves, based on one or more criteria. We note that the parameters of the extraction in Tress et al. (2021) are different than ours, which means that we should not over-interpret details in the numbers. They use the extraction based on molecular hydrogen with a threshold number density of $n_{H_2,min} = 1 \text{ cm}^{-3}$. This does not simply translate into a total number density as shown in detail in e.g. Seifried et al. (2017) for the same chemical network. Furthermore, the dendrogram methods works on a grid-based data set, which requires the Arepo cells to be mapped onto a (locally) uniform grid. Whereas, this is not expected to yield systematically different results if the resolution is high enough, it might easily shape the outliers in the distributions. Nonetheless, many of the extracted cloud properties are well in line with our values. The bulk of the cloud masses extracted in (Tress et al. 2021) broadly range from $10^3–10^5 M_\odot$, slightly higher but well within the scatter given the differences for the parameters. The sizes of the clouds peak at ~10 pc, which is close to our sizes. Both the mass-size relation and the size-velocity dispersion relation are in line with the trends found in this work. Furthermore, the virial parameters are very similar ($\alpha_{vir} \sim 10$) as well as their flat scaling with mass. This illustrates that different cloud extraction methods do not always have to disagree.

## 6. Summary and conclusions

In this paper, we extracted clouds from multi-physics ISM simulations using a set of four different types of (M)HD simulations (SILCC, Girichidis 2021; LS, Colman et al. 2022; M51, Tress et al. 2021; Ramses-F20, Brucy 2022). The set covers a large range in spatial scales from 0.25 pc for the smallest cells in patches of the ISM up to a full galactic disks with a diameter of 30 kpc. Furthermore, the simulations were performed using different codes (FLASH, RAMSES and AREPO) with distinctive numerical techniques and physics ingredients of varying complexity. For all data sets, we used a standardised 3D density-based extraction method built around the HOP algorithm.

We find that many properties of the extracted clouds follow robust trends across scales and very smooth transitions between the different simulations despite the vast differences in resolution, physics included, size of the box and setup geometry. In particular the relation between mass, size, shape and velocity dispersion (see Sects. 3 and 4.1 and Figs. 4, 6, 7 and 9) show remarkably consistent scaling properties. This suggests some universality of these relations, and that gravity and turbulence, which are equally present in all simulations, are the key drivers of (molecular) cloud properties.

More specifically, our results can be summarised as follows:

- The geometry of the extracted clouds ranges from roundish object to complex filamentary networks. Consequently, the precise shape is only poorly represented by simple quantities such as the sphericity or the triaxiality. However these quantities, computed from an ellipsoidal approximation of the shape of the clouds, are useful to characterise their 3D extents. The sphericity and triaxiality are very consistent across all simulations and across the number of cells per cloud within each simulation with a sphericity of 0.25–0.4. This is not an artefact of the extraction algorithm, which is able to detect all complex shapes from the simulations.

- The mass spectra of the clouds follows a remarkably consistent and universal distribution: the normalised number of clouds per unit area of interstellar gas results in a common shape of the high-mass end of the distributions with a slope of $dN/d\ln M \sim -1$ across 6 orders of magnitude. No scaling of individual simulations was required to match this universal high-mass distribution across the simulations.

- The slope of the mass distribution in our simulations agrees with theoretical predictions but differs from other numerical work and observations. We show that this discrepancy can easily originate from the extraction method used.

- The low-mass end of the distributions is sensitive to the applied refinement criterion. An aggressive refinement, such as used in SILCC that does not only act on a small volume around collapsing regions, allows for both higher resolution at intermediate densities and a better representation of the turbulent cascade. Consequently, in this case many more structures form in the presence of and driven by turbulent motions. The resulting mass distributions show the universal scaling down to a small multiple of the resolution limit of the simulation. This naturally comes at a much higher numerical cost. The original LS simulations use a more localised refinement, which results in a peak of the mass spectrum at significantly higher masses. However, tests with similarly aggressive refinement show a consistent evolution with SILCC.

- The relation between cloud mass and size follows a smooth power law of the form $M \propto R^3$ across simulations. The
<figure><img src="image_17.png" /><figcaption>**Fig. 16.** Mass-size relation for the LS stratified boxes and M51 galaxy when using our default HOP parameters versus FOF-like merge criteria. Coloured contour lines are the same as in Fig. 6a.</figcaption></figure>

Another approach is to use dendograms, which describe the hierarchy of structures in the form of a tree. Tress et al. (2021) extracted clouds from their M51 galaxy, that we also use in the comparison in this work, using SCIMES (Colombo et al. 2015). This stands for Spectral Clustering for Interstellar Molecular Emission Segmentation and is a technique mainly used in observations. It is based on the cluster analysis of dendrograms of 3D (in the PPV or PPP space) data cubes that performs particularly well for the identification of large structures such as the molecular clouds in spectroscopical observations of large fields in the Milky Way (e.g. Colombo et al. 2015; Schuller et al. 2017; Benedettini et al. 2020). SCIMES considers the dendrogram tree of the 3D structures in the data cube (using the implementation of Rosolowsky et al. 2008 for astronomical data-sets) and groups different leaves together into ‘clusters’ of leaves, based on one or more criteria. We note that the parameters of the extraction in Tress et al. (2021) are different than ours, which means that we should not over-interpret details in the numbers. They use the extraction based on molecular hydrogen with a threshold number density of $n_{H_2,min} = 1 \text{ cm}^{-3}$. This does not simply translate into a total number density as shown in detail in e.g. Seifried et al. (2017) for the same chemical network. Furthermore, the dendrogram methods works on a grid-based data set, which requires the Arepo cells to be mapped onto a (locally) uniform grid. Whereas, this is not expected to yield systematically different results if the resolution is high enough, it might easily shape the outliers in the distributions. Nonetheless, many of the extracted cloud properties are well in line with our values. The bulk of the cloud masses extracted in (Tress et al. 2021) broadly range from $10^3-10^5 M_\odot$, slightly higher but well within the scatter given the differences for the parameters. The sizes of the clouds peak at ~10 pc, which is close to our sizes. Both the mass-size relation and the size-velocity dispersion relation are in line with the trends found in this work. Furthermore, the virial parameters are very similar ($\alpha_{vir} \sim 10$) as well as their flat scaling with mass. This illustrates that different cloud extraction methods do not always have to disagree.

## 6. Summary and conclusions

In this paper, we extracted clouds from multi-physics ISM simulations using a set of four different types of (M)HD simulations (SILCC, Girichidis 2021; LS, Colman et al. 2022; M51, Tress et al. 2021; Ramses-F20, Brucy 2022). The set covers a large range in spatial scales from 0.25 pc for the smallest cells in patches of the ISM up to a full galactic disks with a diameter of 30 kpc. Furthermore, the simulations were performed using different codes (FLASH, RAMSES and AREPO) with distinctive numerical techniques and physics ingredients of varying complexity. For all data sets, we used a standardised 3D density-based extraction method built around the HOP algorithm.

We find that many properties of the extracted clouds follow robust trends across scales and very smooth transitions between the different simulations despite the vast differences in resolution, physics included, size of the box and setup geometry. In particular the relation between mass, size, shape and velocity dispersion (see Sects. 3 and 4.1 and Figs. 4, 6, 7 and 9) show remarkably consistent scaling properties. This suggests some universality of these relations, and that gravity and turbulence, which are equally present in all simulations, are the key drivers of (molecular) cloud properties.

More specifically, our results can be summarised as follows:

- The geometry of the extracted clouds ranges from roundish object to complex filamentary networks. Consequently, the precise shape is only poorly represented by simple quantities such as the sphericity or the triaxiality. However these quantities, computed from an ellipsoidal approximation of the shape of the clouds, are useful to characterise their 3D extents. The sphericity and triaxiality are very consistent across all simulations and across the number of cells per cloud within each simulation with a sphericity of 0.25–0.4. This is not an artefact of the extraction algorithm, which is able to detect all complex shapes from the simulations.

- The mass spectra of the clouds follows a remarkably consistent and universal distribution: the normalised number of clouds per unit area of interstellar gas results in a common shape of the high-mass end of the distributions with a slope of $dN/d\ln M \sim -1$ across 6 orders of magnitude. No scaling of individual simulations was required to match this universal high-mass distribution across the simulations.

- The slope of the mass distribution in our simulations agrees with theoretical predictions but differs from other numerical work and observations. We show that this discrepancy can easily originate from the extraction method used.

- The low-mass end of the distributions is sensitive to the applied refinement criterion. An aggressive refinement, such as used in SILCC that does not only act on a small volume around collapsing regions, allows for both higher resolution at intermediate densities and a better representation of the turbulent cascade. Consequently, in this case many more structures form in the presence of and driven by turbulent motions. The resulting mass distributions show the universal scaling down to a small multiple of the resolution limit of the simulation. This naturally comes at a much higher numerical cost. The original LS simulations use a more localised refinement, which results in a peak of the mass spectrum at significantly higher masses. However, tests with similarly aggressive refinement show a consistent evolution with SILCC.

- The relation between cloud mass and size follows a smooth power law of the form $M \propto R^3$ across simulations. The
normalisation reflects the density threshold used by the
extraction algorithm.

- The internal velocity dispersion also scales very smoothly across the setups. For the large clouds, the trend is a power law $\sigma \propto R^{0.75}$. For clouds smaller than ~0.5 pc we find a deviation from the power-law scaling towards higher velocity dispersions. The median value for $\sigma$ seem to asymptotically approach ~0.5 km s$^{-1}$. A detailed analysis shows that this base level of internal velocity dispersion results from supernova-driven turbulence rather than gravitationally driven collapse. The small (low-mass) clouds with higher $\sigma$ show a higher mass-weighted temperature and reduced fraction of molecular gas, which is in line with turbulence driving from the hot and warm phases into the clouds.

- A comparison with observational catalogues shows a similar trend of a power-law scaling of $\sigma$ with size and a flat distribution for small clouds with a base level of internal velocity dispersion of approximately 1 km s$^{-1}$. The scaling for the simulated clouds of $\sigma \propto R^{0.75}$ is steeper compared to the observed scaling of $\sigma \propto R^{0.6}$, which does not necessarily mean an inconsistency in the dynamics of clouds, but could also be related to the differently inferred sizes of the clouds between simulations and observations.

- The scaling of the virial parameter $\alpha_{vir}$ with mass agrees well with the $\alpha_{vir} \propto M^{-0.5}$ scaling found in observations of both molecular clouds and dense clumps. However, it is difficult to say whether this is a robust physical result or merely a consequence of the selection effects affecting both observational and numerical measurements of $\alpha_{vir}$.

- The mean cloud temperatures do not agree well between simulations, primarily due to differences in the treatment of photoelectric heating and stellar feedback in the different models. However, the good agreement we find in other cloud properties despite this difference in temperatures suggests that the thermal pressure of the gas has only a small influence on these properties.

- Care must be taken when comparing mean cloud properties between catalogues, as different simulations (or observations) may trace different parts of a universal cloud hierarchy. Catalogue averages reflect the resolution and volume under study.

The catalogues we produced along with the analysis tools are publicly available in the Galactica database<fnref n="4" />.

*Acknowledgements.* We thank the anonymous referee for their useful comments. This project was funded by the European Research Council under ERC Synergy Grant ECOGAL (grant 855130), led by Patrick Hennebelle, Ralf Klessen, Sergio Molinari and Leonardo Testi. The authors acknowledge Interstellar Institute's program "I16" and the Paris-Saclay University's Institut Pascal for hosting discussions that nourished the development of the ideas behind this work. The authors gratefully acknowledge computing resources provided by the Ministry of Science, Research and the Arts (MWK) of the State of Baden-Württemberg through bwHPC and the German Science Foundation (DFG) through grants INST 35/1134-1 FUGG and 35/1597-1 FUGG, and they thank for data storage at SDS@hd funded through grants INST 35/1314-1 FUGG and INST 35/1503-1 FUGG. RSK and SCOG furthermore acknowledge financial support from the Heidelberg Cluster of Excellence STRUCTURES in the framework of Germany's Excellence Strategy (EXC-2181/1-390900948), and from the German Ministry for Economic Affairs and Climate Action in project MAINN (funding ID 50002206). CRediT author statement (more information about the roles defined by the CRediT system can be found here: https://www.elsevier.com/researcher/author/policies-and-guidelines/credit-author-statement): Tine Colman: Conceptualisation, Software, Formal Analysis, Investigation, Writing – Original Draft. Noé Brucy: Conceptualisation, Formal Analysis, Investigation,

Writing – Original Draft, Visualisation, Data Curation. Philipp Girichidis: Conceptualisation, Formal Analysis, Investigation, Writing – Original Draft. Simon C. O. Glover: Conceptualisation, Writing – Original Draft. Milena Benedettini: Resources, Writing – Review & Editing. Juan D. Soler: Writing – Review & Editing. Robin G. Tress: Resources, Writing – Review & Editing. Alessio Traficante: Writing – Review & Editing. Patrick Hennebelle: Conceptualisation, Resources, Writing – Review & Editing, Funding Acquisition. Ralf S. Klessen: Conceptualisation, Writing – Review & Editing, Funding Acquisition. Sergio Molinari: Conceptualisation, Funding Acquisition. Marc-Antoine Miville-Deschênes: Resources, Writing – Review & Editing

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Dubey, A., Fisher, R., Graziani, C., et al. 2008, in ASP Conf. Ser., 385, Numerical Modeling of Space Plasma Flows, eds. N. V. Pogorelov, E. Audit, & G. P. Zank, 145

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<footnote n="4">⁴ http://www.galactica-simulations.eu/db/ISM/CLOUD_COMP</footnote>
normalisation reflects the density threshold used by the
extraction algorithm.

- The internal velocity dispersion also scales very smoothly across the setups. For the large clouds, the trend is a power law $\sigma \propto R^{0.75}$. For clouds smaller than ~0.5 pc we find a deviation from the power-law scaling towards higher velocity dispersions. The median value for $\sigma$ seem to asymptotically approach ~0.5 km s$^{-1}$. A detailed analysis shows that this base level of internal velocity dispersion results from supernova-driven turbulence rather than gravitationally driven collapse. The small (low-mass) clouds with higher $\sigma$ show a higher mass-weighted temperature and reduced fraction of molecular gas, which is in line with turbulence driving from the hot and warm phases into the clouds.

- A comparison with observational catalogues shows a similar trend of a power-law scaling of $\sigma$ with size and a flat distribution for small clouds with a base level of internal velocity dispersion of approximately 1 km s$^{-1}$. The scaling for the simulated clouds of $\sigma \propto R^{0.75}$ is steeper compared to the observed scaling of $\sigma \propto R^{0.6}$, which does not necessarily mean an inconsistency in the dynamics of clouds, but could also be related to the differently inferred sizes of the clouds between simulations and observations.

- The scaling of the virial parameter $\alpha_{vir}$ with mass agrees well with the $\alpha_{vir} \propto M^{-0.5}$ scaling found in observations of both molecular clouds and dense clumps. However, it is difficult to say whether this is a robust physical result or merely a consequence of the selection effects affecting both observational and numerical measurements of $\alpha_{vir}$.

- The mean cloud temperatures do not agree well between simulations, primarily due to differences in the treatment of photoelectric heating and stellar feedback in the different models. However, the good agreement we find in other cloud properties despite this difference in temperatures suggests that the thermal pressure of the gas has only a small influence on these properties.

- Care must be taken when comparing mean cloud properties between catalogues, as different simulations (or observations) may trace different parts of a universal cloud hierarchy. Catalogue averages reflect the resolution and volume under study.

The catalogues we produced along with the analysis tools are publicly available in the Galactica database<fnref n="4" />.

**Acknowledgements.** We thank the anonymous referee for their useful comments. This project was funded by the European Research Council under ERC Synergy Grant ECOGAL (grant 855130), led by Patrick Hennebelle, Ralf Klessen, Sergio Molinari and Leonardo Testi. The authors acknowledge Interstellar Institute's program "I16" and the Paris-Saclay University's Institut Pascal for hosting discussions that nourished the development of the ideas behind this work. The authors gratefully acknowledge computing resources provided by the Ministry of Science, Research and the Arts (MWK) of the State of Baden-Württemberg through bwHPC and the German Science Foundation (DFG) through grants INST 35/1134-1 FUGG and 35/1597-1 FUGG, and they thank for data storage at SDS@hd funded through grants INST 35/1314-1 FUGG and INST 35/1503-1 FUGG. RSK and SCOG furthermore acknowledge financial support from the Heidelberg Cluster of Excellence STRUCTURES in the framework of Germany's Excellence Strategy (EXC-2181/1-390900948), and from the German Ministry for Economic Affairs and Climate Action in project MAINN (funding ID 50002206). CRediT author statement (more information about the roles defined by the CRediT system can be found here: https://www.elsevier.com/researcher/author/policies-and-guidelines/credit-author-statement): Tine Colman: Conceptualisation, Software, Formal Analysis, Investigation, Writing – Original Draft. Noé Brucy: Conceptualisation, Formal Analysis, Investigation,

Writing – Original Draft, Visualisation, Data Curation. Philipp Girichidis: Conceptualisation, Formal Analysis, Investigation, Writing – Original Draft. Simon C. O. Glover: Conceptualisation, Writing – Original Draft. Milena Benedettini: Resources, Writing – Review & Editing. Juan D. Soler: Writing – Review & Editing. Robin G. Tress: Resources, Writing – Review & Editing. Alessio Traficante: Writing – Review & Editing. Patrick Hennebelle: Conceptualisation, Resources, Writing – Review & Editing, Funding Acquisition. Ralf S. Klessen: Conceptualisation, Writing – Review & Editing, Funding Acquisition. Sergio Molinari: Conceptualisation, Funding Acquisition. Marc-Antoine Miville-Deschênes: Resources, Writing – Review & Editing

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# Appendix A: Structure finding algorithm

The full algorithm, including test setups, can be found in the *ecogal_tools* repository on gitlab. It is built around the HOP clump finder, as described and implemented by Eisenstein & Hut (1998). Here we give a summary of their implementation, discuss our adaptation and parameter choices for application to hydrodynamics simulations and present some test results.

## Appendix A.1: Finding groups around local peaks

First, the list of particles is read from the input file. For this step, we need to provide HOP with the coordinates x, y, and z of each particle or cell<fnref n="5" />, as well as their mass and density. If we do not have density information, we can ask HOP to calculate it for us by applying a smoothing kernel. However due to grid codes already providing densities and SPH codes possibly using a variety of kernels, we think it more straightforward to leave the density calculation to the user in the case of hydrodynamics simulations. While we typically perform the structure finding on the density field, it can in theory be applied on any field.

A search tree is constructed based on the particle positions to allow for efficient neighbour searching. For each particle, the highest density neighbour among the Nhop closest neighbours (including the particle itself) is identified. We keep the default value of Nhop=16. Eisenstein & Hut (1998) verified that the result is insensitive to this parameter for particle simulations. For grid simulations, a minimum of 7 is required to probe the direct neighbours of a fully refined region. Adopting large value (e.g. 27 which would probe also the diagonal neighbours in a fully refined region) could lead to artificial merging of structures that are separated by cells that are not at the deepest refinement level. The larger Nhop, the larger the computation time.

Next, for each particle we follow the chain of densest neighbours. For the highest density neighbour, we find its own highest density neighbour, then find the highest density neighbour of that particle, etc. We proceed in this way along the chain until we eventually reach a particle that is its own densest neighbour. The particle we started with is then assigned to the group of the final particle. For example, we start at particle A which has a densest neighbour B. We then go to particle B and see that the densest neighbour of B is C. Arriving at C, we see that there is no particle around C that is denser than C, meaning that C is its own densest neighbour. Particle A is then assigned to the group of C. The name 'hop' actually refers to this process of hopping between neighbours. The result is a collection of peak patches around local density maxima. The resulting segmentation is outputted to file as an intermediate result, which can be reused later.

To prepare the field for the next step, HOP also investigates the boundaries of the identified structures. For each particle, we check whether any of the closest Nmerge neighbours belongs to a different structure. If this is indeed the case, we found a boundary and its density, called the saddle density, is defined as the average between the particle and the neighbour that is in the other group. For each pair of structures, the highest saddle density is recorded and outputted as an intermediate result. We set Nmerge=8, larger than the default value of 4, to account for the regularity of the

neighbour locations in grid codes. When provided with information on the size of the computational domain, HOP accounts for periodic boundaries when determining neighbours.

## Appendix A.2: Altering and merging groups

Once HOP has identified all the peak patches, we move to the regrouping phase executed by the REGROUP algorithm (Eisenstein & Hut 1998). This serves to exclude low density particles, to reduce noise and to merge structures according to user specified criteria. For example, a cloud could consist of multiple peaks which are blended together. Or we could have a cell with an unusually high density fluctuation with nothing else around it which should be considered as noise. Four parameters control this phase.

- A density threshold rho_thresh is set to allow only cells with a density above this value to be included in structures.

- A required peak density rho_peak is set to identify an individual structure, as opposed to noise or low amplitude fluctuations inside another structure.

- A maximum allowed saddle density rho_saddle is set to identify an individual structure as opposed to the patch being part of a larger entity.

- The structure has to have a minimum number of cells min min_cells.

First, all particles which have a density below rho_thresh are removed from peak patches. Usually the value of this parameter is determined by the type of structure we want to obtain. We typically adopt the thresholds 1 cm⁻³ for cloud complexes, 30 cm⁻³ for clouds and 1000 cm⁻³ for clumps. It is important to note that generally these thresholds are arbitrary and that varying them can give interesting additional information. From an observational point of view, we can set the threshold to approximate the point where we can detect a certain tracer, e.g. CO. This, in turn is determined by the chemical species under study and the sensitivity of the detector.

Next, the algorithm merges structures based on the remaining criteria. If the peak density is not sufficient, observations would not pick it up as an individual structure. There are now several possibilities depending on whether it has neighbours or not. If the structure has a neighbour that does qualify as an individual entity and the saddle is above the saddle threshold, the structure is merged with this neighbour. If there are multiple neighbours, the one with the largest saddle density is chosen. When there are no suitable neighbours, the structure does not stand out enough from the background and is discarded.

The peak density criterion can be thought of as a required signal-to-noise ratio. For example, we could require the peak of the structure to be at least twice as strong as the background density rho_thresh to be distinguishable. The saddle threshold criterion is responsible for merging substructures back together. If the saddle density is higher than rho_saddle, we decide the structure is part of the neighbour with which it shares this saddle, rather than being an individual structure. When looking at clouds for example, we indeed expect a cloud to have substructure with multiple local density peaks inside one cloud. This criterion can also help to reduce differences when comparing simulations with different resolutions. When the resolution is higher, naturally more local density peaks are found and so we need to group them to recover the full cloud which envelopes multiple clumps. This also implies that the simulations must be able to resolve densities of at least this chosen saddle density.

<footnote n="5">⁵ For regular grids we take the cell centre. For moving mesh codes, one can either pick the centre of mass in a cell or the location of the mesh generating point. In this work we opted for the latter. We note that in AREPO a mesh generating point is steered towards the centre of mass of the cell, so both are equivalent to first order.</footnote>
# Appendix A: Structure finding algorithm

The full algorithm, including test setups, can be found in the *ecogal_tools* repository on gitlab. It is built around the HOP clump finder, as described and implemented by Eisenstein & Hut (1998). Here we give a summary of their implementation, discuss our adaptation and parameter choices for application to hydrodynamics simulations and present some test results.

## Appendix A.1: Finding groups around local peaks

First, the list of particles is read from the input file. For this step, we need to provide HOP with the coordinates x, y, and z of each particle or cell<fnref n="5" />, as well as their mass and density. If we do not have density information, we can ask HOP to calculate it for us by applying a smoothing kernel. However due to grid codes already providing densities and SPH codes possibly using a variety of kernels, we think it more straightforward to leave the density calculation to the user in the case of hydrodynamics simulations. While we typically perform the structure finding on the density field, it can in theory be applied on any field.

A search tree is constructed based on the particle positions to allow for efficient neighbour searching. For each particle, the highest density neighbour among the Nhop closest neighbours (including the particle itself) is identified. We keep the default value of Nhop=16. Eisenstein & Hut (1998) verified that the result is insensitive to this parameter for particle simulations. For grid simulations, a minimum of 7 is required to probe the direct neighbours of a fully refined region. Adopting large value (e.g. 27 which would probe also the diagonal neighbours in a fully refined region) could lead to artificial merging of structures that are separated by cells that are not at the deepest refinement level. The larger Nhop, the larger the computation time.

Next, for each particle we follow the chain of densest neighbours. For the highest density neighbour, we find its own highest density neighbour, then find the highest density neighbour of that particle, etc. We proceed in this way along the chain until we eventually reach a particle that is its own densest neighbour. The particle we started with is then assigned to the group of the final particle. For example, we start at particle A which has a densest neighbour B. We then go to particle B and see that the densest neighbour of B is C. Arriving at C, we see that there is no particle around C that is denser than C, meaning that C is its own densest neighbour. Particle A is then assigned to the group of C. The name 'hop' actually refers to this process of hopping between neighbours. The result is a collection of peak patches around local density maxima. The resulting segmentation is outputted to file as an intermediate result, which can be reused later.

To prepare the field for the next step, HOP also investigates the boundaries of the identified structures. For each particle, we check whether any of the closest Nmerge neighbours belongs to a different structure. If this is indeed the case, we found a boundary and its density, called the saddle density, is defined as the average between the particle and the neighbour that is in the other group. For each pair of structures, the highest saddle density is recorded and outputted as an intermediate result. We set Nmerge=8, larger than the default value of 4, to account for the regularity of the

neighbour locations in grid codes. When provided with information on the size of the computational domain, HOP accounts for periodic boundaries when determining neighbours.

## Appendix A.2: Altering and merging groups

Once HOP has identified all the peak patches, we move to the regrouping phase executed by the REGROUP algorithm (Eisenstein & Hut 1998). This serves to exclude low density particles, to reduce noise and to merge structures according to user specified criteria. For example, a cloud could consist of multiple peaks which are blended together. Or we could have a cell with an unusually high density fluctuation with nothing else around it which should be considered as noise. Four parameters control this phase.

- A density threshold rho_thresh is set to allow only cells with a density above this value to be included in structures.

- A required peak density rho_peak is set to identify an individual structure, as opposed to noise or low amplitude fluctuations inside another structure.

- A maximum allowed saddle density rho_saddle is set to identify an individual structure as opposed to the patch being part of a larger entity.

- The structure has to have a minimum number of cells min min_cells.

First, all particles which have a density below rho_thresh are removed from peak patches. Usually the value of this parameter is determined by the type of structure we want to obtain. We typically adopt the thresholds 1 cm⁻³ for cloud complexes, 30 cm⁻³ for clouds and 1000 cm⁻³ for clumps. It is important to note that generally these thresholds are arbitrary and that varying them can give interesting additional information. From an observational point of view, we can set the threshold to approximate the point where we can detect a certain tracer, e.g. CO. This, in turn is determined by the chemical species under study and the sensitivity of the detector.

Next, the algorithm merges structures based on the remaining criteria. If the peak density is not sufficient, observations would not pick it up as an individual structure. There are now several possibilities depending on whether it has neighbours or not. If the structure has a neighbour that does qualify as an individual entity and the saddle is above the saddle threshold, the structure is merged with this neighbour. If there are multiple neighbours, the one with the largest saddle density is chosen. When there are no suitable neighbours, the structure does not stand out enough from the background and is discarded.

The peak density criterion can be thought of as a required signal-to-noise ratio. For example, we could require the peak of the structure to be at least twice as strong as the background density rho_thresh to be distinguishable. The saddle threshold criterion is responsible for merging substructures back together. If the saddle density is higher than rho_saddle, we decide the structure is part of the neighbour with which it shares this saddle, rather than being an individual structure. When looking at clouds for example, we indeed expect a cloud to have substructure with multiple local density peaks inside one cloud. This criterion can also help to reduce differences when comparing simulations with different resolutions. When the resolution is higher, naturally more local density peaks are found and so we need to group them to recover the full cloud which envelopes multiple clumps. This also implies that the simulations must be able to resolve densities of at least this chosen saddle density.

<footnote n="5">⁵ For regular grids we take the cell centre. For moving mesh codes, one can either pick the centre of mass in a cell or the location of the mesh generating point. In this work we opted for the latter. We note that in AREPO a mesh generating point is steered towards the centre of mass of the cell, so both are equivalent to first order.</footnote>
<figure><img src="image_18.png" /><figcaption>Fig. A.1: Illustration of structure identification and merging by HOP on a hypothetical density field. First a collection of peak patches around local maxima is identified. Then, the green structure is rejected because its peak density is not high enough. The cyan structure is merged into the blue one for the same reason. Next, the yellow structure is merged into the red one because the saddle density between the structures is larger than the saddle threshold, indicating that they are connected structures.</figcaption></figure>

Finally, we discard structures which consist of fewer than the minimum required number of cells. By using this criterion we can select structures that are sufficiently resolved.

The full process is illustrated in Figure A.1.

## Appendix A.3: Calculating structure properties

Once we have identified which cells/particles belong to which structure, we calculate the structure properties. Users of our wrapper can easily select what type of properties they want to calculate. One can also easily extend the code to add additional properties of interest. Here we list some elemental ones that are studied in this work.

### Appendix A.3.1: Position, mass, density and magnetic field

The total mass $M$ is given by the sum of the masses of each particle or grid cell:

$$M = \sum_{i} m_{i} = \sum_{i} \rho_{i} V_{i}, \qquad (A.1)$$

where $\rho_i$ is the density and $V_i$ is the volume of the cell $i$. The average density is defined as

$$\rho = \frac{\sum_{i} m_{i}}{\sum_{i} V_{i}}. \qquad (A.2)$$

We also determine the maximum density inside the structure and the corresponding peak coordinates. The centre-of-mass coordinates are given by

$$\mathbf{x} = \sum_{i} m_{i} \mathbf{x}_{o,i} / M, \qquad (A.3)$$

where we take into account periodic boundaries by first re-centring around the peak location (and assuming the structure is significantly smaller than the box size of the simulation).

### Appendix A.3.2: Size and shape

Once we know the centre of mass, we construct the inertia matrix using the centre-of-mass coordinates

$$I = \begin{bmatrix} \sum_{i} m_{i}(y_{i}^{2} + z_{i}^{2}) & -\sum_{i} m_{i}x_{i}y_{i} & -\sum_{i} m_{i}x_{i}z_{i} \\ -\sum_{i} m_{i}x_{i}y_{i} & \sum_{i} m_{i}(x_{i}^{2} + z_{i}^{2}) & -\sum_{i} m_{i}y_{i}z_{i} \\ -\sum_{i} m_{i}x_{i}z_{i} & -\sum_{i} m_{i}y_{i}z_{i} & \sum_{i} m_{i}(x_{i}^{2} + y_{i}^{2}) \end{bmatrix}, \quad (A.4)$$

and find its eigenvalues $\lambda_1 \le \lambda_2 \le \lambda_3$ through diagonalisation. We approximate the shape of the clouds by considering the ellipsoid with an uniform density that has the same inertia matrix

eigenvalues. The half-axes $a \le b \le c$ of this ellipsoid are given by

$$a = \sqrt{\frac{5}{2M}(\lambda_1 + \lambda_2 - \lambda_3)} \qquad (A.5)$$

$$b = \sqrt{\frac{5}{2M}(\lambda_1 + \lambda_3 - \lambda_2)}$$

$$c = \sqrt{\frac{5}{2M}(\lambda_2 + \lambda_3 - \lambda_1)}.$$

We tested visually that the corresponding ellipsoid matches well with the extent of the clouds. While this gives good results, it is worth recalling that clouds have a much more complex shape compared to a simple ellipsoid. We define the average size as the geometric mean of the three half-axes multiplied by 2,

$$L = 2 (abc)^{1/3}. \qquad (A.6)$$

It is important to keep in mind that this definition corresponds to a diameter, rather than a radius.

### Appendix A.3.3: Kinematic quantities

The centre-of-mass velocity (a.k.a. the bulk velocity) is simply the mass-weighted average velocity of the cells in the structure:

$$\bar{v} = \frac{\sum_{i} m_{i} \mathbf{v}_{i}}{M}. \qquad (A.7)$$

The velocity dispersion is obtained using

$$\sigma = \sqrt{\frac{\sum_{i} m_{i} |\mathbf{v}_{i} - \bar{\mathbf{v}}|^{2}}{M}}. \qquad (A.8)$$

## Appendix A.4: Choosing merge parameters

To determine suitable merge parameters, we first investigate the effect of each individual criterion on the extraction results. More specifically, we look at their influence on the cloud mass spectrum. Figure A.2 shows the results of our tests on LS-weak-driving, a stratified box simulation run with RAMSES. Requiring a minimum number of cells for a structure means we only select the larger objects, as seen in the top panel. As long as we are not too demanding, we recover the same high-mass tail. The middle panel shows that peak merging on its own has a minor effect. It merges small structures with low peak-to-background ratios into larger structures, producing a spectrum which is somewhat more top heavy. Finally, the saddle merging
<figure><img src="image_18.png" /><figcaption>Fig. A.1: Illustration of structure identification and merging by HOP on a hypothetical density field. First a collection of peak patches around local maxima is identified. Then, the green structure is rejected because its peak density is not high enough. The cyan structure is merged into the blue one for the same reason. Next, the yellow structure is merged into the red one because the saddle density between the structures is larger than the saddle threshold, indicating that they are connected structures.</figcaption></figure>

Finally, we discard structures which consist of fewer than the minimum required number of cells. By using this criterion we can select structures that are sufficiently resolved.

The full process is illustrated in Figure A.1.

## Appendix A.3: Calculating structure properties

Once we have identified which cells/particles belong to which structure, we calculate the structure properties. Users of our wrapper can easily select what type of properties they want to calculate. One can also easily extend the code to add additional properties of interest. Here we list some elemental ones that are studied in this work.

### Appendix A.3.1: Position, mass, density and magnetic field

The total mass $M$ is given by the sum of the masses of each particle or grid cell:

$$M = \sum_{i} m_{i} = \sum_{i} \rho_{i} V_{i}, \qquad (A.1)$$

where $\rho_i$ is the density and $V_i$ is the volume of the cell $i$. The average density is defined as

$$\rho = \frac{\sum_{i} m_{i}}{\sum_{i} V_{i}}. \qquad (A.2)$$

We also determine the maximum density inside the structure and the corresponding peak coordinates. The centre-of-mass coordinates are given by

$$\mathbf{x} = \sum_{i} m_{i} \mathbf{x}_{o,i} / M, \qquad (A.3)$$

where we take into account periodic boundaries by first re-centring around the peak location (and assuming the structure is significantly smaller than the box size of the simulation).

### Appendix A.3.2: Size and shape

Once we know the centre of mass, we construct the inertia matrix using the centre-of-mass coordinates

$$I = \begin{bmatrix} \sum_{i} m_{i}(y_{i}^{2} + z_{i}^{2}) & -\sum_{i} m_{i}x_{i}y_{i} & -\sum_{i} m_{i}x_{i}z_{i} \\ -\sum_{i} m_{i}x_{i}y_{i} & \sum_{i} m_{i}(x_{i}^{2} + z_{i}^{2}) & -\sum_{i} m_{i}y_{i}z_{i} \\ -\sum_{i} m_{i}x_{i}z_{i} & -\sum_{i} m_{i}y_{i}z_{i} & \sum_{i} m_{i}(x_{i}^{2} + y_{i}^{2}) \end{bmatrix}, \qquad (A.4)$$

and find its eigenvalues $\lambda_1 \le \lambda_2 \le \lambda_3$ through diagonalisation. We approximate the shape of the clouds by considering the ellipsoid with an uniform density that has the same inertia matrix

eigenvalues. The half-axes $a \le b \le c$ of this ellipsoid are given by

$$a = \sqrt{\frac{5}{2M}(\lambda_{1} + \lambda_{2} - \lambda_{3})} \qquad (A.5)$$

$$b = \sqrt{\frac{5}{2M}(\lambda_{1} + \lambda_{3} - \lambda_{2})}$$

$$c = \sqrt{\frac{5}{2M}(\lambda_{2} + \lambda_{3} - \lambda_{1})}.$$

We tested visually that the corresponding ellipsoid matches well with the extent of the clouds. While this gives good results, it is worth recalling that clouds have a much more complex shape compared to a simple ellipsoid. We define the average size as the geometric mean of the three half-axes multiplied by 2,

$$L = 2 (abc)^{1/3}. \qquad (A.6)$$

It is important to keep in mind that this definition corresponds to a diameter, rather than a radius.

### Appendix A.3.3: Kinematic quantities

The centre-of-mass velocity (a.k.a. the bulk velocity) is simply the mass-weighted average velocity of the cells in the structure:

$$\bar{v} = \frac{\sum_{i} m_{i} v_{i}}{M}. \qquad (A.7)$$

The velocity dispersion is obtained using

$$\sigma = \sqrt{\frac{\sum_{i} m_{i} |v_{i} - \bar{v}|^{2}}{M}}. \qquad (A.8)$$

## Appendix A.4: Choosing merge parameters

To determine suitable merge parameters, we first investigate the effect of each individual criterion on the extraction results. More specifically, we look at their influence on the cloud mass spectrum. Figure A.2 shows the results of our tests on LS-weak-driving, a stratified box simulation run with RAM-SES. Requiring a minimum number of cells for a structure means we only select the larger objects, as seen in the top panel. As long as we are not too demanding, we recover the same high-mass tail. The middle panel shows that peak merging on its own has a minor effect. It merges small structures with low peak-to-background ratios into larger structures, producing a spectrum which is somewhat more top heavy. Finally, the saddle merging
<figure><img src="image_19.png" /></figure>

Fig. A.4: Study of the influence of resolution on the structure mass spectrum for LS-weak-driving. We limit the loading of the AMR tree to a certain refinement level, as indicated in the legend. Increasing the refinement by one level means the smallest spatial scale resolved becomes a factor two smaller and the volume of a cell is a factor 8 smaller.

Now that we understand the impact of the merge parameters, we can make an educated choice. We want to obtain a catalogue of structures that are resolved, to avoid having to calculate the properties of unresolved structures close to the resolution limit. We therefore choose minCells = 100, which accurately recovers the high-mass slope while neglecting a large amount of unresolved objects. Since the peak density criterion plays only a minor role, we can apply any desired signal-to-noise ratio. As a default value, we set rho_peak to be twice rho_thresh. The saddle threshold is the most difficult parameter to choose, and one might want to vary it depending on the application. Our tests show that a value of ten times rho_thresh gives good results. This selects structures in a specified density range and allows for the comparison of simulations with different resolutions, provided the saddle density is resolved in the lowest resolution run. Figure A.3 verifies the mass distribution does not change dramatically when slightly varying the chosen merge parameters.

## Appendix A.5: Testing resolution

In Figure A.4 we demonstrate that the high-mass part of the cloud mass spectrum does not depend on the resolution. The cut-off at low masses does depend on resolution, especially since the minimum requested number of cells is kept the same.

## Appendix A.6: Testing low density cutoff

In Figure A.5 we demonstrate that it is possible to recover the same cloud mass spectrum when neglecting low density parts of the simulation in the group finding phase of HOP. This is particularly interesting for simulations which contain many particles or cells and are thus memory heavy.

<figure><img src="image_20.png" /></figure>

Fig. A.5: Study where we limited the input for HOP to cells above a certain density threshold relative to the extraction threshold. We provide the resulting mass spectrum for LS-weak-driving.

<figure><img src="image_21.png" /></figure>

Fig. A.6: HOP detection of simple structures in a synthetic volume. Various spherical or filamentary shapes were correctly detected by HOP.

## Appendix A.7: Tests on idealised structures

To test how good HOP is at identifying structures and recovering shape information, we run the extraction algorithm on a synthetic volume populated by simple structures (spheres and filaments). HOP was able to correctly identify all structures and recover the shape parameters (see Section 3.3). This is illustrated by Fig. A.6.
<figure><img src="image_19.png" /></figure>

Fig. A.4: Study of the influence of resolution on the structure mass spectrum for LS-weak-driving. We limit the loading of the AMR tree to a certain refinement level, as indicated in the legend. Increasing the refinement by one level means the smallest spatial scale resolved becomes a factor two smaller and the volume of a cell is a factor 8 smaller.

Now that we understand the impact of the merge parameters, we can make an educated choice. We want to obtain a catalogue of structures that are resolved, to avoid having to calculate the properties of unresolved structures close to the resolution limit. We therefore choose minCells = 100, which accurately recovers the high-mass slope while neglecting a large amount of unresolved objects. Since the peak density criterion plays only a minor role, we can apply any desired signal-to-noise ratio. As a default value, we set rho_peak to be twice rho_thresh. The saddle threshold is the most difficult parameter to choose, and one might want to vary it depending on the application. Our tests show that a value of ten times rho_thresh gives good results. This selects structures in a specified density range and allows for the comparison of simulations with different resolutions, provided the saddle density is resolved in the lowest resolution run. Figure A.3 verifies the mass distribution does not change dramatically when slightly varying the chosen merge parameters.

## Appendix A.5: Testing resolution

In Figure A.4 we demonstrate that the high-mass part of the cloud mass spectrum does not depend on the resolution. The cut-off at low masses does depend on resolution, especially since the minimum requested number of cells is kept the same.

## Appendix A.6: Testing low density cutoff

In Figure A.5 we demonstrate that it is possible to recover the same cloud mass spectrum when neglecting low density parts of the simulation in the group finding phase of HOP. This is particularly interesting for simulations which contain many particles or cells and are thus memory heavy.

<figure><img src="image_20.png" /></figure>

Fig. A.5: Study where we limited the input for HOP to cells above a certain density threshold relative to the extraction threshold. We provide the resulting mass spectrum for LS-weak-driving.

<figure><img src="image_21.png" /></figure>

Fig. A.6: HOP detection of simple structures in a synthetic volume. Various spherical or filamentary shapes were correctly detected by HOP.

## Appendix A.7: Tests on idealised structures

To test how good HOP is at identifying structures and recovering shape information, we run the extraction algorithm on a synthetic volume populated by simple structures (spheres and filaments). HOP was able to correctly identify all structures and recover the shape parameters (see Section 3.3). This is illustrated by Fig. A.6.
## Appendix B: Parameters of the galactic simulations

Table B.1 summarises the initial conditions used for the Ramses-F20 and the M51 simulations.

<table><caption>Table B.1: Initial conditions for the full galaxy simulations, with $N_{\text{part}}$ being the resolution in number of particles, $H_i$ the scale height, $\sigma_i$ the initial velocity dispersion at a scale of $0.1H_i$ (when relevant), $T_i$ the initial temperature, and $Q_{\text{min}}$ the minimal value for the Toomre parameter. The gas particles are then used to generate the initial gas density, and the other are used directly.</caption><thead><tr><th colspan="2">Name</th><th>Ramses-F20</th><th>M51</th></tr></thead><tbody><tr><td colspan="2">Total mass</td><td>112 × 10<sup>10</sup> M<sub>☉</sub></td><td>71 × 10<sup>10</sup> M<sub>☉</sub></td></tr><tr><td rowspan="3">Dark matter halo</td><td>Profile</td><td>NFW sphere (Navarro et al. 1996)</td><td>Spheroidal (Hernquist 1990)</td></tr><tr><td>Mass</td><td>107 × 10<sup>10</sup> M<sub>☉</sub></td><td>60.4 × 10<sup>10</sup> M<sub>☉</sub></td></tr><tr><td>$N_{\text{part,DM}}$</td><td>10<sup>6</sup></td><td>10<sup>6</sup></td></tr><tr><td rowspan="4">Thin stellar disk</td><td>Mass</td><td>3.4 × 10<sup>10</sup> M<sub>☉</sub></td><td>4.77 × 10<sup>10</sup> M<sub>☉</sub></td></tr><tr><td>$N_{\text{part,★}}$</td><td>10<sup>6</sup></td><td>10<sup>6</sup></td></tr><tr><td>$H_{i,★}$</td><td>3.4 kpc</td><td>2.26 kpc</td></tr><tr><td>$Q_{\text{min}}$</td><td>1.5</td><td>0.7</td></tr><tr><td rowspan="6">Gaseous disk</td><td>Mass</td><td>8.5 × 10<sup>9</sup> M<sub>☉</sub></td><td>5.3 × 10<sup>9</sup> M<sub>☉</sub></td></tr><tr><td>$N_{\text{part,g}}$</td><td>10<sup>6</sup></td><td>1.8 × 10<sup>8</sup></td></tr><tr><td>$T_i$</td><td>10<sup>4</sup> K</td><td>10<sup>4</sup> K</td></tr><tr><td>$H_{i,g}$</td><td>3.4 kpc</td><td>2.26 kpc</td></tr><tr><td>$\sigma_i$</td><td>20 km s<sup>-1</sup></td><td>—</td></tr><tr><td rowspan="3">Stellar bulge</td><td>Mass</td><td>4.2 × 10<sup>9</sup> M<sub>☉</sub></td><td>5.3 × 10<sup>9</sup> M<sub>☉</sub></td></tr><tr><td>$N_{\text{part,b★}}$</td><td>125000</td><td>10<sup>5</sup></td></tr><tr><td>$H_{i,b★}$</td><td>0.34 kpc</td><td>0.09 kpc</td></tr></tbody></table>

## Appendix C: Fit of the mass spectrum

We fitted the mass spectrum in the appropriate range for each individual simulation.

<figure><img src="image_22.png" /><figcaption>Fig. C.1: Fits of the individual cloud mass spectra</figcaption></figure>
## Appendix B: Parameters of the galactic simulations

Table B.1 summarises the initial conditions used for the Ramses-F20 and the M51 simulations.

<table><caption>Table B.1: Initial conditions for the full galaxy simulations, with $N_{\text{part}}$ being the resolution in number of particles, $H_i$ the scale height, $\sigma_i$ the initial velocity dispersion at a scale of $0.1H_i$ (when relevant), $T_i$ the initial temperature, and $Q_{\text{min}}$ the minimal value for the Toomre parameter. The gas particles are then used to generate the initial gas density, and the other are used directly.</caption><thead><tr><th colspan="2">Name</th><th>Ramses-F20</th><th>M51</th></tr></thead><tbody><tr><td colspan="2">Total mass</td><td>112 × 10<sup>10</sup> M<sub>☉</sub></td><td>71 × 10<sup>10</sup> M<sub>☉</sub></td></tr><tr><td rowspan="3">Dark matter halo</td><td>Profile</td><td>NFW sphere (Navarro et al. 1996)</td><td>Spheroidal (Hernquist 1990)</td></tr><tr><td>Mass</td><td>107 × 10<sup>10</sup> M<sub>☉</sub></td><td>60.4 × 10<sup>10</sup> M<sub>☉</sub></td></tr><tr><td>$N_{\text{part,DM}}$</td><td>10<sup>6</sup></td><td>10<sup>6</sup></td></tr><tr><td rowspan="4">Thin stellar disk</td><td>Mass</td><td>3.4 × 10<sup>10</sup> M<sub>☉</sub></td><td>4.77 × 10<sup>10</sup> M<sub>☉</sub></td></tr><tr><td>$N_{\text{part,★}}$</td><td>10<sup>6</sup></td><td>10<sup>6</sup></td></tr><tr><td>$H_{i,★}$</td><td>3.4 kpc</td><td>2.26 kpc</td></tr><tr><td>$Q_{\text{min}}$</td><td>1.5</td><td>0.7</td></tr><tr><td rowspan="6">Gaseous disk</td><td>Mass</td><td>8.5 × 10<sup>9</sup> M<sub>☉</sub></td><td>5.3 × 10<sup>9</sup> M<sub>☉</sub></td></tr><tr><td>$N_{\text{part,g}}$</td><td>10<sup>6</sup></td><td>1.8 × 10<sup>8</sup></td></tr><tr><td>$T_i$</td><td>10<sup>4</sup> K</td><td>10<sup>4</sup> K</td></tr><tr><td>$H_{i,g}$</td><td>3.4 kpc</td><td>2.26 kpc</td></tr><tr><td>$\sigma_i$</td><td>20 km s<sup>-1</sup></td><td>—</td></tr><tr><td rowspan="3">Stellar bulge</td><td>Mass</td><td>4.2 × 10<sup>9</sup> M<sub>☉</sub></td><td>5.3 × 10<sup>9</sup> M<sub>☉</sub></td></tr><tr><td>$N_{\text{part,b★}}$</td><td>125000</td><td>10<sup>5</sup></td></tr><tr><td>$H_{i,b★}$</td><td>0.34 kpc</td><td>0.09 kpc</td></tr></tbody></table>

## Appendix C: Fit of the mass spectrum

We fitted the mass spectrum in the appropriate range for each individual simulation.

<figure><img src="image_22.png" /><figcaption>Fig. C.1: Fits of the individual cloud mass spectra</figcaption></figure>
# Appendix D: Variations with galactic radius

As mentioned in Section 3.1.4 and shown in Figure 5, we did not find a difference in the mass spectrum for the inner versus outer galaxy in our two galaxy simulations, with a boundary between inner and outer zone set at 7 kpc. However, Tress et al. (2021) did find that the masses of clouds in the very centre of the simulated M51 galaxy are typically larger. In Figure D.1 we compare the cumulative distribution for clouds in the inner 1 kpc of our galaxies versus that of clouds located in the remainder of the galaxy. For the cloud mass (top panel) in M51, we recover the qualitative result of Tress et al. (2021). Clouds also have higher velocity dispersions, as can be seen in the bottom panel. The Ramses-F20 galaxy on the other hand, has slightly less massive clouds in the centre and the distribution of velocity dispersions is very similar in both sub-regions. We caution that the statistics in the central region of both galaxies are rather poor.

The weight of the stellar and dark matter component is higher in the centre than in the outskirts. We would expect this to influence the mean statistical properties of the clouds. A more efficient channelling of the gas along the deeper galactic potential in the centre could lead to more massive structures. However this effect is modulated by other aspects, such as higher turbulence and shear which can easily disrupt large clouds. The gas fraction $M_{gas}/(M_{gas} + M_{stars})$ of the M51 simulation is overall lower than that of Ramses-F20 (see Table B.1). As a consequence, the impact of the weight of the non-gaseous components is higher in M51. This could provide a possible explanation for the different behaviour of the mass spectrum in both galaxies.

Another key difference between the two galaxies is their morphology. In particular, M51 has a central bar while Ramses-F20 does not (see Figure 1). Observations have shown that gas in the centre of barred galaxies has an increased velocity dispersion (Sun et al. 2020). We indeed recover this behaviour in the bottom left panel of Figure D.1.

<figure><img src="image_23.png" /><figcaption>Fig. D.1: Comparison between key cloud properties in the centre of the galaxies versus those in the main body. The threshold between the two regions is taken at a galactic radius of 1 kpc. We show the cumulative distribution of cloud mass (top) and cloud velocity dispersion (bottom).</figcaption></figure>
**Appendix D: Variations with galactic radius**

As mentioned in Section 3.1.4 and shown in Figure 5, we did not find a difference in the mass spectrum for the inner versus outer galaxy in our two galaxy simulations, with a boundary between inner and outer zone set at 7 kpc. However, Tress et al. (2021) did find that the masses of clouds in the very centre of the simulated M51 galaxy are typically larger. In Figure D.1 we compare the cumulative distribution for clouds in the inner 1 kpc of our galaxies versus that of clouds located in the remainder of the galaxy. For the cloud mass (top panel) in M51, we recover the qualitative result of Tress et al. (2021). Clouds also have higher velocity dispersions, as can be seen in the bottom panel. The Ramses-F20 galaxy on the other hand, has slightly less massive clouds in the centre and the distribution of velocity dispersions is very similar in both sub-regions. We caution that the statistics in the central region of both galaxies are rather poor.

The weight of the stellar and dark matter component is higher in the centre than in the outskirts. We would expect this to influence the mean statistical properties of the clouds. A more efficient channelling of the gas along the deeper galactic potential in the centre could lead to more massive structures. However this effect is modulated by other aspects, such as higher turbulence and shear which can easily disrupt large clouds. The gas fraction $M_{gas}/(M_{gas} + M_{stars})$ of the M51 simulation is overall lower than that of Ramses-F20 (see Table B.1). As a consequence, the impact of the weight of the non-gaseous components is higher in M51. This could provide a possible explanation for the different behaviour of the mass spectrum in both galaxies.

Another key difference between the two galaxies is their morphology. In particular, M51 has a central bar while Ramses-F20 does not (see Figure 1). Observations have shown that gas in the centre of barred galaxies has an increased velocity dispersion (Sun et al. 2020). We indeed recover this behaviour in the bottom left panel of Figure D.1.

<figure><img src="image_23.png" /><figcaption>Fig. D.1: Comparison between key cloud properties in the centre of the galaxies versus those in the main body. The threshold between the two regions is taken at a galactic radius of 1 kpc. We show the cumulative distribution of cloud mass (top) and cloud velocity dispersion (bottom).</figcaption></figure>