Dynamical cooling of galactic discs by molecular cloud collisions -- Origin of giant clumps in gas-rich galaxy discs

TL;DR

This paper proposes that molecular cloud collisions cause dynamical cooling in gas-rich galactic discs, facilitating gravitational instability and clump formation, and introduces a parameter to quantify this cooling process.

Contribution

It introduces a new model linking molecular cloud collisions to disc cooling and instability, providing a quantitative dissipation parameter for galaxy simulations.

Findings

Cloud collision cooling is effective when disc surface density exceeds one-third of cloud surface density.

Dynamical cooling via cloud collisions can trigger clump formation in gas-rich discs.

The model can be incorporated into numerical simulations as a subgrid process.

Abstract

Different from Milky-Way-like galaxies, discs of gas-rich galaxies are clumpy. It is believed that the clumps form because of gravitational instability. However, a necessary condition for gravitational instability to develop is that the disc must dissipate its kinetic energy effectively, this energy dissipation (also called cooling) is not well-understood. We propose that collisions (coagulation) between molecular clouds dissipate the kinetic energy of the discs, which leads to a dynamical cooling. The effectiveness of this dynamical cooling is quantified by the dissipation parameter $D$, which is the ratio between the free-fall time $t_{\rm ff}\approx 1/ \sqrt{G \rho_{\rm disc}}$ and the cooling time determined by the cloud collision process $t_{\rm cool}$. This ratio is related to the ratio between the mean surface density of the disc $\Sigma_{\rm disc}$ and the mean surface density of molecular clouds in the disc $\Sigma_{\rm cloud}$. When $D <1/3$ (which roughly corresponds to $\Sigma_{\rm disc} < 1/3 \Sigma_{\rm cloud}$), cloud collision cooling is inefficient, and fragmentation is suppressed. When $D > 1/3$ (which roughly corresponds to $\Sigma_{\rm disc} > 1/3 \Sigma_{\rm cloud}$), cloud-cloud collisions lead to a rapid cooling through which clumps form. On smaller scales, cloud-cloud collisions can drive molecular cloud turbulence. This dynamical cooling process can be taken into account in numerical simulations as a subgrid model to simulate the global evolution of disc galaxies.

Tables

Table 1. Table 1: List of variable definitions. The microscopic variables are the variables defined on the cloud scale, and the macroscopic variables are the variables defined on the disc scale.

Variable Name	Definition
Microscopic variables
$r_{\mathrm{cloud}}$	Typical size of a molecular cloud
$m_{\mathrm{cloud}}$	Typical mass of a molecular cloud
${\Sigma }_{\mathrm{cloud}}$	Typical surface density of a molecular cloud. ${\Sigma }_{\mathrm{cloud}}\approx {\Sigma }_{\mathrm{cloud},\mathrm{crit}}$
${\Sigma }_{\mathrm{cloud},\mathrm{crit}}$	Critical surface density of a molecular cloud to survive against photodissociation (Eq. 11)
${\sigma }_{v,\mathrm{cloud}}$	Velocity dispersion of a molecular cloud
Macroscopic variables
$n_{\mathrm{cloud}}$	Mean number density of molecular clouds
${\sigma }_{v,\mathrm{disc}}$	Velocity dispersion of the galactic disc
${\rho }_{\mathrm{disc}}$	Mean density of the galactic disc
${\rho }_{\mathrm{disc},\mathrm{crit}}$	Critical density for effective cloud collision cooling
${\Sigma }_{\mathrm{disc}}$	Surface density of the galactic disc
$t_{\mathrm{cool}}\approx t_{\mathrm{collide}}$	Cloud collision cooling time $\approx$ cooling time
${\lambda }_{\mathrm{cloud}}$	Mean free path of molecular clouds
$G$	Gravitational constant
$Z$	Metallicity
$G_0$	Interstellar radiation field
${\Omega }_{\mathrm{kep}}$	Angular frequency of galactic (which is approximately the epicyclic frequency)
$t_{\mathrm{kep}}$	Disc dynamical time $t_{\mathrm{kep}}\approx 3{\Omega }_{\mathrm{kep}}^{-1}$
$t_{\mathrm{ff},\mathrm{disc}}=\sqrt{1/G{\rho }_{\mathrm{disc}}}$	Local mean free-fall time of the disc
$H_{\mathrm{disc}}$	Disc scaleheight
$Q_{\mathrm{disc}}$	Toomre $Q$ parameter of the disc
$Q_{\mathrm{gas}}$	Toomre $Q$ parameter of the gas component of the disc
$f_H$	Disc scaleheight correction due to self-gravity and gravitational force from the stars, $f_H\approx 1$
$D$	Dissipation parameter (proposed in this paper), dissipative systems have $D>1$, clump formation requires $D>1/3$

Table 2. Table 2: A list of the estimated parameters for different galactic environments.

Region	${\Sigma }_{\mathrm{disc}}$	${\Sigma }_{\mathrm{cloud}}$	$D={\Sigma }_{\mathrm{disc}}/{\Sigma }_{\mathrm{cloud}}$
Milky Way interarm region	$1M_{\odot }{\mathrm{pc}}^{-2}$	$12M_{\odot }{\mathrm{pc}}^{-2}$	0.1
Milky Way molecular ring	$10M_{\odot }{\mathrm{pc}}^{-2}$	$12M_{\odot }{\mathrm{pc}}^{-2}$	1
M51 Interarm region	$4M_{\odot }{\mathrm{pc}}^{-2}$	$12M_{\odot }{\mathrm{pc}}^{-2}$	0.3
M51 outer spiral arm	$15M_{\odot }{\mathrm{pc}}^{-2}$	$12M_{\odot }{\mathrm{pc}}^{-2}$	1
Giant clumps in gas-rich galaxies	$100M_{\odot }{\mathrm{pc}}^{-2}$	$30M_{\odot }{\mathrm{pc}}^{-2}$	3

Equations

$${\lambda }_{\mathrm{cloud}}=\frac{n_{\mathrm{cloud}}^{-1}}{s_{\mathrm{cloud}}}={\rho }_{\mathrm{disc}}^{-1}{m}_{\mathrm{cloud}}{r}_{\mathrm{cloud}}^{-2},$$

$$t_{\mathrm{cool}}\approx t_{\mathrm{collide}}\approx {\lambda }_{\mathrm{cloud}}/{\sigma }_{\mathrm{disc}}={\rho }_{\mathrm{disc}}^{-1}{m}_{\mathrm{cloud}}{r}_{\mathrm{cloud}}^{-2}{\sigma }_{\mathrm{v},\mathrm{disc}}^{-1}=\frac{{\Sigma }_{\mathrm{cloud}}}{{\rho }_{\mathrm{disc}}{\sigma }_{\mathrm{v},\mathrm{disc}}},$$

$$t_{\mathrm{ff}}\approx \sqrt{\frac{1}{G{\rho }_{\mathrm{disc}}}}.$$

$$D=\frac{t_{\mathrm{cool}}^{-1}}{t_{\mathrm{ff}}^{-1}}=\frac{{\rho }_{\mathrm{disc}}^{1/2}{\sigma }_{\mathrm{v},\mathrm{disc}}}{G^{1/2}{\Sigma }_{\mathrm{cloud}}},$$

$${\rho }_{\mathrm{disc}}>{\rho }_{\mathrm{disc},\mathrm{crit}}=\frac{G{\Sigma }_{\mathrm{cloud}}^2}{{\sigma }_{\mathrm{v},\mathrm{disc}}^2},$$

$${\dot{e}}_{\mathrm{kin}}\approx e_{\mathrm{kin}}/t_{\mathrm{cool}}.$$

$$t_{\mathrm{cool}}\approx \frac{{\Sigma }_{\mathrm{cloud}}}{{\rho }_{\mathrm{disc}}{\sigma }_{\mathrm{v},\mathrm{disc}}},$$

$$H_{\mathrm{disc}}=\frac{f_{\mathrm{H}}{\sigma }_{\mathrm{v},\mathrm{disc}}}{{\Omega }_{\mathrm{kep}}},$$

$$t_{\mathrm{cool}}=f_{\mathrm{H}}\frac{{\Sigma }_{\mathrm{cloud}}}{{\Sigma }_{\mathrm{disc}}}\times \frac{1}{{\Omega }_{\mathrm{kep}}}\approx \frac{{\Sigma }_{\mathrm{cloud}}}{{\Sigma }_{\mathrm{disc}}}\times \frac{1}{{\Omega }_{\mathrm{kep}}}.$$

$$D=\frac{t_{\mathrm{cool}}^{-1}}{t_{\mathrm{ff}}^{-1}}=\frac{{\Sigma }_{\mathrm{disc}}}{f_{\mathrm{H}}{\Sigma }_{\mathrm{cloud}}},$$

$${\Sigma }_{\mathrm{cloud},\mathrm{crit}}\approx \frac{12}{Z}{M}_{\odot }{\mathrm{pc}}^{-2},$$

$$D=\frac{t_{\mathrm{cool}}^{-1}}{t_{\mathrm{ff}}^{-1}}=\frac{{\rho }_{\mathrm{disc}}^{1/2}{\sigma }_{\mathrm{v},\mathrm{disc}}}{G^{1/2}{\Sigma }_{\mathrm{cloud}}}\approx \frac{{\Sigma }_{\mathrm{disc}}}{f_{\mathrm{H}}{\Sigma }_{\mathrm{cloud}}}\approx \frac{Z}{12f_{\mathrm{H}}}\frac{{\Sigma }_{\mathrm{disc}}}{M_{\odot }\mathrm{p}{\mathrm{c}}^{-2}},$$

$${\dot{e}}_{\mathrm{disc}}=\frac{{\sigma }_{\mathrm{v},\mathrm{disc}}^2}{t_{\mathrm{cool}}}\approx \frac{{\rho }_{\mathrm{disc}}{\sigma }_{\mathrm{v},\mathrm{disc}}^3}{{\Sigma }_{\mathrm{cloud}}}.$$

$$\sigma_{\rm v,cloud}\approx\sigma_{\rm v,disc}\big{(}\frac{\rho_{\rm disc}}{\Sigma_{\rm cloud}}\big{)}^{1/3}r_{\rm cloud}^{1/3}\approx\sigma_{\rm v,disc}\big{(}\frac{\Sigma_{\rm disc}}{\Sigma_{\rm cloud}}\big{)}^{1/3}\big{(}\frac{r_{\rm cloud}}{H_{\rm disc}}\big{)}^{1/3}\;,$$

$$D=\frac{t_{\mathrm{cool}}^{-1}}{t_{\mathrm{ff}}^{-1}}=\frac{{\rho }_{\mathrm{disc}}^{1/2}{\sigma }_{\mathrm{v},\mathrm{disc}}}{G^{1/2}{\Sigma }_{\mathrm{cloud}}}\approx \frac{{\Sigma }_{\mathrm{disc}}}{{\Sigma }_{\mathrm{cloud}}}\approx \frac{1}{12Z}\frac{{\Sigma }_{\mathrm{disc}}}{M_{\odot }\mathrm{p}{\mathrm{c}}^{-2}},$$

$$H_{\mathrm{disc}}=\frac{f_{\mathrm{H}}{\sigma }_{\mathrm{v},\mathrm{disc}}}{{\Omega }_{\mathrm{kep}}}.$$

$$p_{\mathrm{disc}}\approx {\rho }_{\mathrm{disc}}{\sigma }_{\mathrm{v},\mathrm{disc}}^2\approx ({\Sigma }_{\mathrm{disc}}/H_{\mathrm{disc}})\times {\sigma }_{\mathrm{v},\mathrm{disc}}^2,$$

$$p_{\mathrm{gravity}}\approx a_{\mathrm{z}}{\Sigma }_{\mathrm{disc}}\approx {\Omega }_{\mathrm{kep}}^{2}r(H_{\mathrm{disc}}/r){\Sigma }_{\mathrm{disc}},$$

$$H_{\mathrm{disc}}=\frac{{\sigma }_{\mathrm{v},\mathrm{disc}}}{{\Omega }_{\mathrm{kep}}},$$

$$\frac{p_{\mathrm{self}-\mathrm{gravity}}}{p_{\mathrm{gravity}}}\approx \frac{1}{Q_{\mathrm{gas}}},$$

$$Q_{\mathrm{gas}}=\frac{{\sigma }_{\mathrm{v},\mathrm{gas}}{\Omega }_{\mathrm{kep}}}{\text{uppi}G{\Sigma }_{\mathrm{gas}}}.$$

Full text

††pubyear: 2015††pagerange: Dynamical cooling of galactic discs by molecular cloud collisions – Origin of giant clumps in gas-rich galaxy discs–Dynamical cooling of galactic discs by molecular cloud collisions – Origin of giant clumps in gas-rich galaxy discs

1 Introduction

Star formation occurs in the cold phase of the galactic interstellar medium (ISM), which exhibits diverse morphologies in different galaxies. The Milky Way is an example where molecular gas is mostly distributed in molecular clouds whose sizes are much smaller than the disc scaleheight. In contrast to this, in high-redshift, gas-rich galaxies, molecular gas seems to concentrate in giant (kpc-size), massive clumps, and majority of the star formation occurs in the clumps (Abraham et al., 1996; Noguchi, 1998; Elmegreen & Elmegreen, 2005; Genzel et al., 2008; Guo et al., 2015; Elmegreen et al., 2009; Förster Schreiber et al., 2009) 111Sometimes the giant clumps appear in chains, and these galaxies are called “chain galaxies” (Cowie et al., 1995; van den Bergh et al., 1996; Elmegreen et al., 2004b, a; Conselice et al., 2005), and Dalcanton & Shectman (1996) suggested that the chain-like morphology is due the projected view of an edge-on galaxy.. Unlike molecular clouds, the clumps have sizes that are often comparable to the disc scaleheight. How do they form?

One promising clump formation path is the gravitational instability, where the growth of perturbations on the disc scale leads to structure formation. In fact, the sizes of the observed clumps are often comparable to the Toomre length (Fisher et al., 2017a), suggesting that gravitational instability is at work. The short-time evolution of the gravitational instability is determined by the Toomre $Q$ parameter, where $Q_{\rm disc}=\sigma_{\rm v,disc}\Omega_{\rm disc}/\uppi G\Sigma_{\rm disc}$, $\sigma_{\rm v,disc}$ is the velocity dispersion, $\Sigma_{\rm disc}$ is the disc surface density, and $\Omega_{\rm disc}$ is disc angular frequency. In Toomre’s formalism, fragmentation is driven by gravity (which is characterized by $\Sigma_{\rm disc}$), which is balanced by a combination of supported caused by local motions $\sigma_{\rm v,disc}$ and shear ($\Omega_{\rm disc}$, which is roughly the epicyclic frequency). Discs with $Q_{\rm disc}<1$ are unstable against non-axisymmetric instabilities and should fragment.

It has been believed that the clump formation is largely controlled by the Toomre parameter (Kereš et al., 2005; Dekel et al., 2009a, b; Inoue et al., 2016) or a Toomre-like formalism, e.g. Romeo et al. (2010). However, the Toomre formalism seems to provide a still incomplete description to the clump formation process: observationally, the connection between a smaller Toomre $Q$ parameter and a clumpy disc has not been established. On the contrary, observations indicate that galactic discs tend to have $Q_{\rm disc}\approx 1$ with an observed uncertainty of a factor of two, and this is almost independent on whether a disc is clumpy or not 222There are some variations of the Toomre $Q$ parameter in different galaxies, with clumpy galaxies having smaller $Q$ (e.g. Fisher et al., 2014). But it remains a theoretical issue how to define the Toomre $Q$ in a disc that consists of both gas and stars accurately (Romeo, 1992; Wang & Silk, 1994; Elmegreen, 2011; Romeo & Wiegert, 2011; Inoue et al., 2016). Therefore, some of the observed variations in Toomre $Q$ might arise from this theoretical uncertainty. Besides, the variation is much smaller than the difference in the $D$ parameter that what we discuss in this paper, and we believe that it is the difference in the $D$ parameter that leads to different evolution. (Hitschfeld et al., 2009; Puech, 2010; Cacciato et al., 2012; Fisher et al., 2014; Romeo & Falstad, 2013; Romeo & Mogotsi, 2017) 333Where, to properly evaluate the $Q$ parameter, one should properly combine the dynamical contribution of the stars and the gas, see e.g. Romeo & Falstad (2013); Romeo & Mogotsi (2017).. This cannot be explained within the Toomre formalism, but is believed to the the result of a self-regulated disc evolution. Since the discs are already self-regulating systems, a better theory of clump formation should take this into account.

Apart from this, more recent theoretical studies also suggested that the efficiency of disc cooling plays a more fundamental role in clump formation. Gammie (2001); Johnson & Gammie (2003) carried out a set of local, shearing box simulations of disc evolution, and found that (a) the balance between heating and cooling processes tends to regulate the disc to a state where $Q_{\rm disc}\approx 1$ . They also found that (b) it is the cooling time that determines the long-time evolution of such discs: if cooling is inefficient (cooling time is long compared to the dynamical time), the disc should evolve into a gravo-turbulent state where turbulence maintained by accretion can effectively suppress the overly-rapid fragmentation, and the disc is dominated by fluffy, filamentary structures that are constantly smeared apart by rotation; when cooling is efficient (cooling time is short compared to the dynamical time), gas in the disc rapidly dissipates kinetic energy and fragments into a few giant clumps.

The theory of Gammie (2001) seems to provide a good description the evolution of disc galaxies. First, both simulations (e.g. Hopkins et al., 2012; Lehnert et al., 2013; Goldbaum et al., 2016) and observations (Hitschfeld et al., 2009; Puech, 2010; Cacciato et al., 2012; Fisher et al., 2014) seem to indicate that discs tend to have $Q_{\rm disc}\approx 1$ , consistent with self-regulation. Secondly, we do observe a similar dichotomy in the morphology of the observed galaxies: in Gammie (2001), a disc can be either filamentary or clumpy, and this seems to correspond to the fact that in galaxies gas either organises into smaller clouds that can be aligned on a larger scale (e.g. the Milky Way where they form larger filaments) or into giant clumps (e.g. the gas-rich clumpy galaxies). This resemblance poses the question: can we explain the morphological difference of galaxies using the theory of Gammie (2001)? What is the major cooling mechanism in these discs?

The cooling of a galactic disc remains an open issue. We emphasize that the disc is dynamically supported by some motions characterized by a velocity dispersion $\sigma_{\rm v,disc}$, and the disc-scale cooling should be able to reduce this velocity dispersion. When the disc is supported by thermal pressure, $\sigma_{\rm v,disc}$ is related to the temperature of the gas; when the disc is supported by some random motions, $\sigma_{\rm v,disc}$ is the velocity dispersion of these random motions. Previously, the dominate cooling mechanism of these discs is not well-identified: Dekel et al. (2009b); Mayer et al. (2016) identified the radiative cooling time of the atomic ISM as the cooling time relevant for disc fragmentation, and Wada et al. (2002); Elmegreen & Burkert (2010); Klessen & Hennebelle (2010); Elmegreen (2011); Forbes et al. (2014) 444They are based on earlier models, such as Lynden-Bell & Pringle (1974); Lin & Pringle (1987a, b). assumed galactic disc turbulence model, and identified the turbulence dissipation time as the disc cooling time. We argue that neither are adequate. This is because in a real galactic disc (we will be focusing on gas-rich discs where more than half of the gas is molecular, see e.g. Tacconi et al. (2010)), the local dynamical support comes from the quasi-random motions of the molecular gas. To cool down a disc one must reduce this random motion. Dekel et al. (2009b) mistakenly believed that relevant cooling time is the radiative cooling time of the atomic gas. However, they overlooked the critical fact that even if one has successfully converted the atomic gas into the molecular gas through the radiative cooling, the disc can still be supported by motions of the molecular ISM. The kinetic energy contained in these motions cannot be reduced with the radiative cooling, and additional dynamical cooling channels are needed. Another set of models are based on the picture of galactic disc turbulence (Wada et al., 2002; Elmegreen & Burkert, 2010; Klessen & Hennebelle, 2010; Elmegreen, 2011; Forbes et al., 2014). These models correctly identified that to cool down the disc, one must reduce the velocity dispersion of the ISM. However, they made the assumption that the dynamics of the multi-phase ISM can be described by the turbulence model – an assumption that is widely accepted yet not justified: the standard theory of turbulence has only been shown to be valid for non-self-gravitating gas with well-defined equation of states. In a real galactic disc, gas is separated into different phases where molecular gas has a density that is two orders of magnitudes larger than the density of the ambient ISM (See also Sec. 2.1). This density contrast makes many assumptions of the galactic disc turbulence model invalid. For example, in the standard picture of galactic disc turbulence, ideally, one requires energy to cascade from the larger scales continuously down to the Kolmogorov microscales where it dissipates into heat. But in a real galactic disc, due to the density contrast between the molecular clouds and ambient gas as well as the fact that molecular gas forms centrally-condensed structures, it is practically very difficult for the kinetic energy in the warm ISM to cascade into molecular clouds 555E.g. Ibáñez-Mejía et al. (2016) found that the due to the density contrast between the molecular ISM and the warm neutral medium, it is difficult for supernova feedback to drive molecular cloud turbulence; in the analytical study of Li (2017), the author found that when a region which has $\rho\sim r^{-2}$ where $\rho$ is the density and $r$ is the radius, it is difficult for turbulence to cascade from the outside to the inside. . Another consequence of such a density contrast is that if a dense molecular cloud travels in a galactic disc with a diffuse ambient medium, in most cases the cloud would neglect the drag from the ambient medium and travel on its own (as has been estimated by Marochnik et al., 1983). One must take this dynamical detachment induced by phase separation into account in realistic galactic disc models. Previously, some models simply stepped over the cooling issue by assuming isothermal equations of states (Behrendt et al., 2015, 2016). They simulated gravitationally unstable isothermal discs, and found that the disc first fragments into rings and then into clouds, and in the final step the clouds merge to form clumps. The possibility of merging clouds found in their simulations is an interesting one. However, since they have assumed an isothermal equation of state, how cooling leads to disc fragmentation remains unclear.

We consider a picture where a galactic disc consists of an ensemble of colliding molecular clouds. We assume the existence of these clouds, and study the consequence of cloud-cloud collisions on the evolution of the galactic discs. We argue that a dynamical cooling occurs because of the collisions between the clouds, and on the macroscopic level this cooling can lead to the formation of the clumps. The previous studies on cloud-cloud collisions in galaxies (Kwan & Valdes, 1983; Marochnik et al., 1983; Tomisaka, 1986; Tan, 2000; Vollmer & Beckert, 2002; Dobbs, 2008) mainly considered the evolution of clouds in fixed galactic potentials and focused on properties of the clouds. In their restricted settings, clouds are usually not allowed to back-react on the disc dynamics. These models are valid for Milky Way like galaxies where disc surface densities are low, but are not valid for gas-rich discs. In this paper, we carry their analyses into the non-linear regime, where the cloud-cloud collisions that occur on smaller scales are able to back-react on the disc dynamics on the large scale. When the collision rate is high, clumps form as a result of the collective motion of the molecular clouds included by cloud collision cooling.

2 The model

We describe the properties of a system consists of coagulating clouds, and derive the time-scale for cloud-cloud collisions to significantly remove kinetic energy from the system. We evaluate the stability of galactic discs that consist of coagulating clouds, and show that when the mean surface density of a disc is comparable or larger than the mean surface density of molecular clouds in the disc, cloud collision cooling is effective. A list of variables can be found in table 1.

2.1 General picture

In our colliding clouds picture, molecular clouds are surrounded by a hot ambient medium. The clouds are mainly composed of H2 gas, and the ambient medium includes gas in both the warm neutral phase and the warm ironized phase.

The structure formation in such a medium takes several steps. First, one needs to create molecular gas, and second, one needs to assemble the molecular gas into structures on different scales. We will be focusing on the structure formation, and therefore we assume that there is a continuous of supply of small clouds which eventually coagulate to form larger clouds. In reality, there are several ways to form these small clouds: If the cold gas from out of the dynamically-triggered thermal instability, the structure formation is likely to occur in a bottom-up way, with smaller structures being created first and larger structures being built through the coagulation of smaller structures. This is largely caused by the shape of ISM cooling curve, and details concerning this can be found in McCourt et al. (2016). Another mechanism to produce small clouds is stellar feedback, which disrupts larger clouds into smaller ones. Given these possibilities, we are able to assume the existence of a population of clouds, and focus on the growth of structures in such a clumpy medium.

Our next step is to study the structure formation. In this paper, the term cloud refers to single, centrally condensed structures whose sizes are smaller than the disc scaleheight, and more complex structures whose sizes are comparable to the disc scaleheight are called complexes (in Milky Way-like galaxies) and clumps (in gas-rich discs), respectively. Observationally, clouds appear as close-to-isolated objects, and complexes and clumps contain a significant amount of sub-structures (e.g. see Fisher et al., 2017b, for the case of the clumps) and Nguyen Luong et al. (2011, for the case of a complex).666We must acknowledge that it is still difficult to come us with more rigorous definitions to these objects. For example, many of the complexes in the Milky Way are also called clouds in some papers. Here, we make distinction since observations do indicate that clouds can organise themselves on larger scales. An example of this can be found in the M51 galaxy (Dobbs, 2008), see also our Fig. 4. Since the clumps and complexes are more structured, we propose that they are built by quickly assembling the clouds on larger scales, and in this paper we study this assembly process.

To understand how structures form, one needs to describe the multi-phased structure of the ISM. We assume that the ISM of gas-rich discs can be described as an ensemble of molecular clouds, and the properties of the ISM on the large scale can be derived by considering the interactions between the molecular clouds. This is justified because we are interested in gas-rich discs, where more than half of the gas is molecular (Tacconi et al., 2010). In these discs, the majority of the mass and hence the kinetic energy is contained in molecular gas. Because the molecular gas has densities that are much higher 777E.g. the molecular gas a density of $>10^{3}\rm cm^{-3}$, which is already $\sim$ 100 times larger than the density of the surrounding warm neutral medium. Between these two phases, there exists the so-called cold neutral medium which contains mostly cold neutral hydrogen of a temperature of $\sim 100$ K, but this phase contains very little mass, and is usually not dynamically important. than the density of the ambient medium, we can neglect the drag force between the molecular clouds and the ambient medium (Marochnik et al., 1983), and threat the molecular ISM as an ensemble of clouds that interact through cloud-cloud collisions.

Based on this, we propose that to describe the dynamical properties of such a multi-phased medium, it is sufficient to construct a model that consists of an ensemble of molecular clouds, where the properties of the medium on the large scale can be determined by considering the collisions between the clouds. In our view, the collisions of clouds have two major consequences on the disc evolution. The first and perhaps most obvious consequence of cloud-cloud collisions is that they lead to the formation of larger structures. This has been previously discussed (Kwan & Valdes, 1983; Marochnik et al., 1983; Tomisaka, 1986; Tan, 2000; Vollmer & Beckert, 2002; Dobbs, 2008). Note that these collisions can produce clouds characterized by a power-law mass spectrum, similar to what is observed. But we argue that there is another consequence of cloud-cloud collisions, namely that these collisions should lead to a dissipation of kinetic energy in the disc, where the disc kinetic energy is converted into molecular cloud turbulence through these collisions. Following Elmegreen (1989), we name this process cloud collision cooling. We propose that there are two regimes within which a galactic disc would evolve: there is almost-dissipationless regime, where larger clouds are built through the gradual coagulation of smaller clouds. Since the clouds are well-separated, the rate of cloud-cloud collisions is low. This regime is relevant for galaxies like the Milky Way where the disc is less massive. On top of this, we propose that there is a new regime, where the rate of cloud-cloud collisions is so high that the energy dissipation due to clouds-cloud collisions has a non-negligible effect on the disc dynamics. In this dissipative regime, cooling can effectively remove kinetic energy from the disc, and this loss of kinetic energy will enable disc to contract form a larger scale where the clouds can coagulate into clumps. As a result, each clump should contain a significant number of overlapping structures reminiscent of the colliding clouds. The difference between these two regimes is illustrated in Fig. 1. The paper is an investigation of the effect of cloud collision cooling on the evolution of galactic discs in different regimes.

We do not explicitly consider the effect of stellar feedback. Together with accretion, stellar feedback is a source of heating to the disc, which maintains its velocity dispersion. Here, we simply assume that both accretion and stellar feedback would drive the velocity dispersion of the disc which ensures $Q_{\rm disc}\approx 1$, and we study the effect of cloud collision cooling on the subsequent evolution of such a $Q_{\rm disc}\approx 1$ disc 888For spiral galaxies, $Q_{\rm disc}$ should be computed by combining the contributions from the gas and the stars (Romeo, 1992; Wang & Silk, 1994; Elmegreen, 2011; Romeo & Wiegert, 2011; Romeo & Falstad, 2013; Romeo & Mogotsi, 2017). . Since cloud collision cooling mechanism that we are discussing is independent on how the clouds are created or disrupted, the link we establish between cloud collision cooling and disc fragmentation is independent on the details of feedback and cloud destruction. However, we should note that stellar feedback is still important for our model as it disrupts bigger clouds into smaller clouds, which helps to sustain a population of clouds of different masses among which collisions should occur.

2.2 Disc cooling through cloud-cloud

collisions

To obtain a quantitative understanding of the underlying physics, we assume that a typical cloud has a mass $m_{\rm cloud}$ and a size $r_{\rm cloud}$. In fact our results do not depend on the assumed the masses and sizes, but only on the mass-weighted mean surface density of the cloud. We further assume that when averaged over a larger volume (whose size is much larger than the size of a molecular cloud, but is comparable to the disc scaleheight), the molecular galactic disc as a whole has a local mean density of $\rho_{\rm disc}$ and velocity dispersion $\sigma_{\rm v,disc}$.

The number density of clouds is thus $n_{\rm cloud}=\rho_{\rm disc}/m_{\rm cloud}$. The mean free path of clouds in the disc is

[TABLE]

where $s_{\rm cloud}\approx r_{\rm cloud}^{2}$ is the surface area of a cloud .

Since cloud-cloud collisions are inelastic (such that each collision removes a significant portion of the kinetic energy), the typical time for the system to lose kinetic energy can be estimated from the typical time for a cloud to collide with another cloud, which is

[TABLE]

where $\Sigma_{\rm cloud}$ is the mean surface density of the clouds 999In the Milky Way, the velocity dispersion of the spiral arm at $\sim 5\;\rm kpc$ is around 30 km/s, and mean density of the molecular disc is $0.02\rm\;M_{\odot}\;\rm pc^{-3}$. Assuming that the clouds have $10\;\rm M_{\rm\odot}\;\rm pc^{-2}$ (Sec. 2.4), we derive a cloud mean free path of $\sim 500\;\rm pc$, and a cloud collision time of $\sim 20\;\rm Myr$. These values are taken from Nakanishi & Sofue (2006), and the estimate cloud collision time agrees qualitatively with the value derived by Dobbs et al. (2015), which is $\sim 10\;\rm Myr$.. For comparison, the free-fall time of the gas is

[TABLE]

The collision time depends on both the bulk properties of the gas such as the mean density $\rho_{\rm disc}$, the mean velocity dispersion $\sigma_{\rm disc}$, as well as the properties of the individual clouds $m_{\rm cloud}$ , $r_{\rm cloud}$. When the mean surface densities of the molecular clouds $\Sigma_{\rm cloud}=m_{\rm cloud}/r_{\rm cloud}^{2}$ and the velocity dispersion of the systems $\sigma_{\rm v,disc}$ are fixed, the cloud collision cooling time is proportional to $\rho_{\rm disc}^{-1}$, and the free-fall time is proportional to $\rho_{\rm disc}^{-1/2}$, it is clear that cloud collision cooling will be important at regions where the mean density of the disc $\rho_{\rm disc}$ is sufficiently high. Whether the disc is dissipative or not is quantified by the dissipation parameter

[TABLE]

which is the ratio between the rate of cloud collision cooling $t_{\rm cool}^{-1}$ and the rate of gravitational collapse $t_{\rm ff}^{-1}$. Systems with large $D$ are dissipative. Note that $D$ is only dependent on the mean surface density of the cloud. Therefore, our equations are accurate when the clouds in a galaxies share a common surface density. When the clouds have different surface densities, since we are interested in the dynamical properties of molecular gas on the large scale, $\Sigma_{\rm cloud}$ should be mass-weighted mean surface density the molecular clouds. We explain in Sec. 2.4 that a constant $\Sigma_{\rm cloud}$ for a galaxy is likely a good assumption.

Fig. 2 plots the cloud collision time and the free-fall time as a function of mean density $\rho_{\rm disc}$. One can identify these two regimes: when the density is low, the free-fall time is shorter than the characteristic time for two clouds to collide, $D<1$, and the kinetic energy contained in the system is roughly conserved in a free-fall time. When the density is large enough e.g.

[TABLE]

$D>1$, collisions between the clouds can effectively remove the kinetic energy from the system within a dynamical time, and the system is dissipative. If such a system has a kinetic energy density of $e_{\rm kin}$, the cooling rate can be estimated via

[TABLE]

Therefore, collisions between molecular clouds that occur on the microscopic scale can reduce the velocity dispersion of the system, and constitute as a macroscopic, dynamical cooling channel. This cooling channel is different from radiative cooling, which merely removes the thermal energy but not the kinetic energy. Following Elmegreen (1989)101010Where he also derived the dispersion relation of such a clumpy medium., we name this dynamical cooling process “cloud collision cooling”.

One might wonder the effect of cloud destruction due to stellar feedback: what would happen if the cloud is destroyed by stellar feedback before it can collide with another cloud? Although the finite lifetime of molecular clouds is an importance factor to consider when one wishes to build an intuitive picture of disc evolution, whether the cloud lifetime is longer or shorter than the cloud collision time does not affect our cooling rate estimate. This is because our cloud collision cooling mechanism works as long as the collisions lead to energy dissipation, and this is independent on if the clouds can survive between two successive encounters or not.

2.3 Impact of cooling on disc fragmentation

The ratio between cooling time and dynamical time determines disc evolution: Gammie (2001); Johnson & Gammie (2003) simulated the fragmentation of self-gravitating gas discs, and demonstrated that the evolution of the disc in the nonlinear regime is determined by the ratio between the cooling time $t_{\rm cool}$ and $3\Omega_{\rm kep}^{-1}$ (which is roughly the disc dynamical time $t_{\rm kep}$). The cooling time plays a determining role in the dynamics of the disc: when $t_{\rm cool}<3\Omega_{\rm kep}^{-1}\approx t_{\rm kep}$, cooling is not significant, the disc evolves into a gravo-turbulent state where gas forms long, filamentary structures that are constantly sheared apart by disc rotation; when $t_{\rm cool}<3\Omega_{\rm kep}^{-1}\approx t_{\rm kep}$, cooling is important, and the disc quickly fragments into clumpy gas condensations.

The cooling time of the disc on the macroscopic scale is comparable to the cloud collision time on the microscopic scale, which is (see Eq. 2)

[TABLE]

We can further simply the expression of $t_{\rm cool}$ using some additional ansatz: When the disc reaches hydrostatic equilibrium along the vertical direction, we can estimate its scaleheight (see Appendix. A)

[TABLE]

where $f_{\rm H}\approx 1$ is a numerical factor taking into account the additional compression contributed from self-gravity and the gravitational force from the stars. For order-of-magnitude analysis, we assume $f_{\rm H}\approx 1$. Combining Eq. 7 and Eq. 8:

[TABLE]

According to Gammie (2001), to decide if a disc would fragment or not, one needs to compare the cooling time $t_{\rm cool}$ with the disc dynamical time $t_{\rm kep}\approx 3\Omega_{\rm kep}^{-1}$. However, for a disc that is self-regulating, $Q_{\rm disc}\approx 1$, the local free-fall time of the disc gas is always comparable to the disc dynamical time ($t_{\rm ff}\approx 1/3t_{\rm kep}\approx\Omega_{\rm kep}^{-1}$) 111111The free-fall time is $t_{\rm ff}\approx 1/\sqrt{G\rho_{\rm disc}}$, and the disc dynamical time is $t_{\rm kep}\approx 1/\Omega_{\rm kep}$. Using Eq. 8, $\rho_{\rm disc}\approx\Sigma_{\rm disc}/H_{\rm disc}\approx\Omega_{\rm kep}\Sigma_{\rm disc}/\sigma_{\rm v,disc}\approx G^{-1}\Omega_{\rm kep}^{2}/Q_{\rm disc}$. Since $Q_{\rm disc}\approx 1$, $t_{\rm ff}\approx 1/\sqrt{G\rho_{\rm disc}}\approx Q_{\rm disc}^{1/3}\Omega_{\rm kep}^{-1}\approx 1/3t_{\rm kep}$ .. Thus, instead of evaluating the ratio between $t_{\rm cool}$ and $t_{\rm kep}\approx 3\Omega_{\rm kep}^{-1}$, one can also compare the $t_{\rm cool}$ with the local free-fall time $t_{\rm ff}$ to decide the disc fragmentation. In fact, the ratio between cooling time $t_{\rm cool}$ and the local free-fall time $t_{\rm ff}$ is related to the $D$ parameter that we have derived before ($t_{\rm cool}/3\Omega_{\rm kep}\approx t_{\rm cool}/3t_{\rm ff}=1/3D$)! In other words, $D$ can also be used to decide if a disc will fragment into clumps, and clump-forming discs have $D>1/3$. Using Eq. 9,

[TABLE]

where it is simply determined by the ratio between the mean surface density of the galactic disc $\Sigma_{\rm disc}$ and the mean surface density of molecular clouds in the disc $\Sigma_{\rm cloud}$. Fragmenting discs should have $D>1/3$ (therefore $\Sigma_{\rm disc}>1/3\Sigma_{\rm cloud}$).

2.4 Surface densities of clouds

Our next goal is to determine the “typical” surface density of the clouds. In a galaxy, the clouds can have different masses and sizes. The typical surface density is determined from the mass-weighted mean of the surface densities of the individual clouds. We argue that this mean value is mainly determined by the metallicity of a galaxy.

Our argument is based on the analysis by Krumholz et al. (2008, 2009) where they considered the atomic-to-molecular transition and they self-consistently modelled H2 formation, star formation, photodissociation and shielding. They found that the limiting surface density of atomic-to-molecular transition is determined mainly by the metallicity, and is only weakly dependent on the interstellar radiation field. Besides, the fact that a cold phase must exist also provides addition constraints to the strength of the interstellar radiation field. As a result, the critical surface density for H i/H2 transition is merely a function of the metallicity. Following them, Sternberg et al. (2014) found that the following formula can provide a reasonably good description to the observed H i/H2 transition at Galactic and extragalactic observations (e.g. Lee et al., 2012; Bialy et al., 2017) 121212Some observations indicate higher values of the typical surface density of the clouds. For the Milky Way, the reported surface density of the clouds ranges from $40M_{\odot}\;\rm pc^{-2}$ (Heyer et al., 2009) to $200M_{\odot}\;\rm pc^{-2}$ (Roman-Duval et al., 2010). As Roman-Duval et al. (2010); Heyer & Dame (2015) noted, the major uncertainty comes from the different ways to define the cloud boundary, and these values should be considered as upper limits. These surveys also miss a significant fraction of the total mass due to source extraction (Solomon & Rivolo, 1989). Because of these, the values derived from the dedicated studies such as Lee et al. (2012); Bialy et al. (2017) are more reliable.:

[TABLE]

where $Z$ is measured in terms of solar abundance.

Eq. 11 predicts the typical surface density a cloud should have in a galaxy. If a cloud has a surface density that is lower than the value predicted by Eq. 11, it is vulnerable to photodissociation. On the other hand, if a cloud has a much higher surface density, star formation in such a cloud would destroy the cloud within a relatively short time, making these high surface density clouds rare. In fact, as Fall et al. (2010) pointed out, most of the clouds in a galaxy should have surface densities that are not much higher than the value predicted by Eq. 11.

One can look for evidence of this almost-constant surface density from observations: in the Milky Way, it is relatively easy to measure the surface density of the well-resolved clouds, and these observations do indicate that the HI/H2 occurs at the typical surface density predicted by Eq. 11. Recently, Bialy et al. (2017) carried out a similar study on the W43 molecular complex. This is an extremely complicated region, where the gas is already evolved. They found that the total measured surface density of can be separated into contributions from a few velocity components, and on average, each component has a surface density that is close to the $12M_{\odot}\;\rm pc^{-2}$, agreeing well with Eq. 11. The still limited observational evidences seem to support the idea that the clouds do stay close to the threshold predicted by Eq. 11.

For gas-rich discs, since the individual clouds are not resolved, it remains an assumption that Eq. 11 predicts the characteristic cloud surface density. We believe in this assumption since the physical arguments are still valid in these environment. In practise, even if the cloud surface density deviate from the prediction of Eq. 11, it is still acceptable since we are mainly interested in comparing Milky-Way-like galaxies with gas-rich clumpy galaxies whose surface densities are two orders of magnitudes higher. Compared to this variation, the deviation of the surface density of real clouds from the one predicted by Eq. 11 is probably negligible.

We should also emphasize that since clumps are objects that are formed from the disc instability enabled by cloud collision cooling where the clouds are allowed to organized on a larger scales, we do not expect Eq. 11 to hold for the clumps. Instead, since each clump contains a significant number of overlapping clouds, we expect $\Sigma_{\rm cloud}>>\Sigma_{\rm cloud}$.

2.5 Condition of clump formation

Based on the previous discussions, we rewrite $D$ parameter as

[TABLE]

where in the last step we have used Eq. 11. In practise, one can express the $D$ parameter with a few different expressions: $D=\rho_{\rm disc}^{1/2}\sigma_{\rm v,disc}/G^{1/2}\Sigma_{\rm cloud}$ is a rigours definition, and $D\approx\Sigma_{\rm disc}/\Sigma_{\rm cloud}$ is an approximate expression assuming that the disc reaches hydrostatic balance along the vertical direction. In converting $D$ into the approximate expression, we have assumed that the disc reaches hydrostatic equilibrium in the vertical dimension and assumed that the gravitational force from the stars are not significant. In this paper, we will be mainly using the approximate expression more frequently as it is more convent, and a comment on the errors can be found in Sec. 2.3 and Appendix A.

The evolution of a disc is therefore determined by the $D$ parameter, which is related to the ratio between the surface density of a galactic disc $\Sigma_{\rm disc}$ and the typical surface density of molecular clouds in the disc $\Sigma_{\rm cloud}$. There are these distinct regimes of disc evolution: When $D<1/3$, cloud collision cooling is inefficient, and gas is stretched into filamentary structures due to shear. When $D>>1/3$, cloud collision cooling is efficient, and the disc fragments into clumps.

What occurs in these different regimes? When $D<1/3$, the clouds do collide, but the rate of the collision is relatively low, and energy dissipation from the cloud-cloud collisions is negligible. Thus the impact of cloud collision cooling on the disc dynamics is relatively insignificant. When $D>>1/3$, $\Sigma_{\rm disc}>>\Sigma_{\rm cloud}$, one should see clouds overlap along the individual line of sights 131313To help with the imagination, one is invited to consider a picture where $\rho_{\rm cloud}>\rho_{\rm disc}$. However, since disc is relatively thick, the clouds are overlapping along the individual line of sight, such that $\Sigma_{\rm disc}>\Sigma_{\rm cloud}$. . In this regime, clouds should collide with each other at a high frequency, and cloud-cloud collisions can cool down the disc within a dynamical time. The dissipation of kinetic energy leads the disc to contract, and this drives the collective motion of the clouds that leads to the formation of the giant clumps. Therefore, what distinguishes a clumpy galaxies with normal spiral galaxies is the value of the $D$ parameter.

In this paper, we are interested in the effect of cloud collision cooling on the formation of structure, and to fulfil our purpose $\Sigma_{\rm disc}$ should be the mean surface density of the disc before fragmentation occurs. For gas-rich discs, to evaluate the $D$ parameter, one needs to make an average across the whole disc or over different radii. We are also interested in the formation of regular-spaced molecular complexes in spiral galaxies. In this case, one should evaluate the $D$ parameter using the mean surface density of gas measured along the the spiral arms.

Note that the dissipation parameter $D$ is proportional to the metallicity (Eq. 12). In other words, galaxies with higher metallicities will dissipate their kinetic energy faster compared to galaxies with lower metallicities. This can be understood as follows: when one increases the metallicity, the molecular clouds require a lower surface density to shield against the background radiation field, the clouds are thus less condensed. They have lower surface densities, and are more like to collide with each other. This leads to a higher rate of cloud-cloud collisions and a more efficient disc dynamical cooling.

2.6 Kinetic energy budged and driving of molecular cloud

turbulence

In previous models of galactic discs (e.g. Romeo et al., 2010; Elmegreen & Burkert, 2010; Klessen & Hennebelle, 2010; Elmegreen, 2011; Forbes et al., 2014), it has been assumed that the kinetic energy in the disc can be efficiently dissipated into heat due to turbulence dissipation. In our model, the kinetic energy of the disc is contained in the random motions of the clouds, and it is converted into molecular cloud turbulence due to cloud-cloud collisions. In the final step, molecular cloud turbulence is dissipated into heat. The cloud-cloud collisions become a first step in the whole energy conversion process, followed by turbulence dissipation. This is illustrated in Fig. 3. Can we explain the observed level of molecular cloud turbulence with cloud-cloud collisions? As an estimate, the energy dissipation rate (of unit $\rm erg\;g^{-1}\;s^{-1}$) of a galactic disc due to cloud-cloud collisions is (Eq. 6)

[TABLE]

Molecular clouds in the Milky Way and presumably other galaxies are characterized by the Larson relation (Larson, 1981; Mac Low & Klessen, 2004) where, roughly speaking $\sigma_{\rm v,cloud}\sim l^{1/3}$ where $\sigma_{\rm v}$ is the velocity dispersion of a molecular cloud and $l$ is the size. Assuming the kinetic energy of the clouds are dissipated due to turbulence, one can determine the normalisation of the Larson relation: $\sigma_{\rm v}\sim\dot{e}_{\rm cloud}^{1/3}l^{1/3}$ where $\dot{e}_{\rm cloud}$ is the mean energy dissipation rate of molecular cloud turbulence 141414Here we are interested in the normalisation of the Larson relation, and do not consider the effect of fractal-like underlying density structures on the slope of the Larson relation. Details concerning this can be found in Kritsuk et al. (2007).. Assuming that the amount of energy a molecular cloud receives due to cloud-cloud collisions can be effectively converted to turbulence, $\dot{e}_{\rm cloud}\approx\dot{e}_{\rm disc}$, thus

[TABLE]

where $H_{\rm disc}$ is the disc scaleheight. If cloud-cloud collisions constitute as the major driver of molecular cloud turbulence, the level of turbulence seen in molecular clouds is linked to the properties of the galactic discs on the large scale! For discs of gas-rich galaxies where the surface densities are high, we expect higher normalisations of the Larson’s relation due to the increased cloud-cloud collision rate. Indeed, we do observe such an increase of turbulence in gas-rich disc (e.g. Swinbank et al., 2011). Although the authors have attributed the cause of this increased level of turbulence to the high ambient pressure, according to our model, it can also be explained by an increase rate of cloud-cloud collisions. We note that a similar concept has been previously proposed by Tan et al. (2013) (followed by Li et al. (2017a)) where they emphasized that collisions inject momentum into the clouds, and here we have made a different assumption where collisions are mainly injecting the kinetic energy which then dissipates due to turbulence. We plan to further investigate this issue in a forthcoming paper.

3 Connection to observations

According to our analysis, whether a disc will fragment into clumps or not is determined by the $D$ parameter. To estimate $D$, we collect data from the literature, and the collected values are listed in Table 2. In our analysis, the values of $\Sigma_{\rm cloud}$ are estimated using Eq. 11, and arguments for this can be found in Sec. 2.4. How $\Sigma_{\rm disc}$ is estimated is dependent on the question we are interested in: for clumpy galaxies, we are interested in the fragmentation of the disc, and the value of $\Sigma_{\rm disc}$ is estimated by averaging over the whole disc. For spiral galaxies, since we are interested in the fragmentation of different parts, the values of $\Sigma_{\rm disc}$ are evaluated by averaging over the spiral arms and the interarm regions, respectively.

For gas in interarm region of the Milky Way and M51, $D<1/3$ (which roughly corresponds to $\Sigma_{\rm disc}<1/3\;\Sigma_{\rm cloud}$), cloud collision cooling is inefficient (where, by definition, the time for a cloud to collide with another cloud is longer than the disc dynamical time). The gas clouds coagulate slowly, and at the same time they are sheared apart by the disc differential rotation. This slow coagulation process can produce clouds characterized by power-law mass distributions where small clouds dominate in number Kwan & Valdes (1983) 151515We note, however, that this can also be produced by theory assuming turbulence fragmentation (e.g. Hopkins, 2013). However they assumed an barotropic equation of state, and did not treat the multi-phase ISM.. In this regime, one expects to see only a small fraction of clouds undergoing cloud-cloud collisions. In fact, such collision candidates have been reported in the literate (Duarte-Cabral et al., 2011; Nakamura et al., 2012; Torii et al., 2015; Fukui et al., 2017; Gong et al., 2017). Such collisions can also produce rotating clouds, like those already observed in Li et al. (2017b); Liu (2017). But we should note that in the majority of the cases one should expect to observe a smaller cloud nudging a larger cloud (Dobbs et al., 2015). Because of shear, one should also see long gas filaments that are parallel to the disc mid-plane. These filaments are also seen (Li et al., 2013; Goodman et al., 2014; Ragan et al., 2014; Wang et al., 2015; Zucker et al., 2015; Wang et al., 2016; Li et al., 2016; Abreu-Vicente et al., 2016), and are predicted by theories (Pringle et al., 2001; Smith et al., 2014; Dobbs, 2015).

At some high-density regions such as the spiral arms, $\Sigma_{\rm disc}$ starts to approach $1/3\Sigma_{\rm cloud}$ and the velocity dispersion is also locally enhanced due to the spiral density wave. As a result, cloud-cloud collisions become frequent. Molecular clouds start to collide frequently within a relatively short time. The cloud collision cooling can reduce the overall velocity dispersion of molecular gas in the discs which leads to dynamical cooling, and this dynamical cooling produces the complexes. However, in this particular case, for cloud-cloud collisions to be frequent enough, one needs processes such as the spiral density wave to enhance the density beforehand. As a result, these structures are formed along the spiral arms. It is believed that wiggle-like regular-spaced structures seem on the arms of the grand-design spiral galaxy M51 are produced in this way (Dobbs, 2008) 161616We note that there are alternative explanation to these wiggles, where they come from the wiggle (Kelvin-Helmholtz) instability (Wada & Koda, 2004; Dobbs & Bonnell, 2006; Shetty & Ostriker, 2006; Kim & Ostriker, 2006). It seems unclear how Kelvin-Helmholtz instability can develop in such a multi-phased ISM where the equation of states are poorly defined. Therefore, we prefer the explanation provided Dobbs (2008) where these spurs are produces by cloud collisions. . Molecular complexes in the Milky Way such as W43 and W51 might also belong to this category. In this regime, cloud collision cooling has some influences on the disc dynamics. But these influences are restricted to the close vicinity of the spiral arms where the densities are enhanced.

For a massive, gas-rich disc, the cloud collision time is much shorter than the disc dynamical time ($D>>1/3$, $\Sigma_{\rm disc}>>1/3\Sigma_{\rm cloud}$ 171717Where, if one could resolve these clouds in a galaxy, one should see the clouds overlap along the line of sights., typically the cloud collision time is shorter than the disc dynamical time by a factor of 3–10) at almost everywhere in the disc. Cloud collision cooling rapidly removes kinetic energy from the disc and allows the high-density regions (typically of $\sim$ kpc size) to contract from the large scale. The contraction retrospectively enhances the cloud collision rate. This positive feedback enables the masses of high-density regions to grow rapidly, and it leads to the formation of clumps. The clump formation is thus an non-linear process, where cloud collision cooling starts to back-react on the disc dynamics on the large scale and enables these large perturbations to grow. In this cooling-efficient regime, one should be able to observe clumps whose sizes are comparable to the Toomre length (Toomre, 1964, $l_{\rm Toomre}\approx 2\uppi G\Sigma_{\rm disc}/\Omega_{\rm disc}^{2}$, which characterises the size of the large-scale perturbations in the discs. Perturbations larger than the Toomre length are suppressed due to shear), as has been reported by a recent observation (Fisher et al., 2017a). We note that since the clumps are already the end results of the cloud collision cooling process, their surface densities should be much larger than $\Sigma_{\rm cloud}$, where we also expect the clumps to contain a significant amount of overlapping sub-structures, as has been seen by Fisher et al. (2017b).

4 Conclusions

We study the evolution of galactic discs, and aim to understand the formation mechanism of the kpc-sized giant clumps commonly observed in gas-rich discs. We argue that for these discs, self-regulation leads to a Toomre $Q$ parameter that is close to unity, and on top of this, efficient dynamical cooling is a necessary condition for clump formation.

We approximate a galactic disc ISM as a medium that contains an ensemble of colliding clouds, and study its evolution. We propose that the major dynamical cooling channel of such a galactic disc is the coagulation (collision) between molecular clouds. When the cloud-cloud collision rate is high enough, this dynamical cooling (called cloud collision cooling, see also Elmegreen (1989)) can back-react on the disc, which enables it to contract on the large scale. In spiral galaxies this produces molecular complexes seen on spiral arms, and in gas-rich discs this produces kpc-sized molecular clumps.

The effectiveness of cloud collision cooling is characterized by the $D$ (dissipation) parameter:

[TABLE]

where $\Sigma_{\rm disc}$ is the mean surface density of the galactic disc, and $\Sigma_{\rm cloud}$ is the typical surface density of a molecular cloud in the disc, and $Z$ is the metallicity. The $D$ parameter is a fundamental parameter that determines the disc evolution: When $D<1/3$ ($t_{\rm cool}>3\Omega_{\rm kep}^{-1}$, and roughly $\Sigma_{\rm disc}<1/3\Sigma_{\rm cloud}$), the disc should enter a state where shear stretches gas into long, filamentary structures. When $D>1/3$ ($t_{\rm cool}<3\Omega_{\rm kep}^{-1}$, roughly $\Sigma_{\rm disc}>1/3\Sigma_{\rm cloud}$), molecular clouds in a disc form a system that is highly dissipative where energy is dissipated due to cloud collision cooling (cloud-cloud collisions), and the disc should fragment into clumps/complexes. Since cloud-cloud collisions have played a vital role, the clumps/complexes formed in this way should contain a significant amount of overlapping sub-structures reminiscent of the colliding clouds. In our picture, the collisions have played different roles on different scales: on large scales, cloud collision cooling can back-react on the disc and enables the formation of giant clumps commonly observed in gas-rich discs, and on smaller scales, cloud-cloud collisions drive molecular cloud turbulence.

In a more general sense, we have made an effort to understand how the multi-phase ISM dissipates its kinetic energy. The current analysis is carried out in a simplified model, and the dissipation rate estimates are accurate only in the order of magnitude sense. Nevertheless, when the gas is separated into different phases, we expect system to be almost-dissipationless when $D<<1/3$ and dissipative when $D>>1/3$, and we expect the transition to occur when the surface density of the galactic disc starts to exceed the mean surface density of the molecular clouds by much. Understanding this transition is of crucial importance for understanding the interplay between the ISM and the disc dynamics, and this issue deserves further studies.

Theoretically, one can derive the cloud collision cooling rate using local, shearingbox-like simulations. This would be a crucial test to our model. Once this is verified to a better accuracy, the cooling time estimates can be incorporated into simulations of lower resolutions as subgrid models, and into analytical models to study the fragmentation in different regimes. Such approaches have been previously made Booth et al. (2007), and one should make parameter studies and extend it to simulate formation of the clumps.

Observationally, one can test our scenario by constraining the properties of the molecular clouds and study their interactions in different galactic environments. The properties of the clouds can be constrained by high-resolution CO observations. The global energy budged can be verified by comparing the cloud velocity dispersions with the predictions made in Sec. 2.6. The actual cloud collision process can be traced using transitions of shock-tracing molecules such as SiO and methanol (see also the prediction from a recent chemical study by Bisbas et al. (2017)). Systematic studies of cloud evolution along these directions will help to constrain the dynamical cooling and understand disc evolution.

Acknowledgements Guang-Xing Li thanks Xun Shi for discussions and for comments on the paper, and thanks Manuel Behrendt and Andi Burket for introducing him the problem and for sharing their views. Alexei Kritsuk is acknowledged for a discussion at MIAPP (Munich) where he introduced the Marochnik et al. (1983) paper, and Masataka Fukugita is acknowledged for a discussion on HI/H2 transition. Jens Kauffman and Pavel Kroupa are acknowledge for critical comments. Bruce Elmegreen is acknowledged for an in-depth discussion on the multi-phase ISM. Finally, the referee must be thanked the a careful reading of the paper, and for the pertinent comments. This paper is supported by the Deutsche Forschungsgemeinschaft (DFG) priority program 1573 ISM-SPP.

Appendix A Disc scaleheight corrections

To convert $D=\rho_{\rm disc}^{1/2}\sigma_{\rm v,disc}/G^{1/2}\Sigma_{\rm cloud}$ to the approximate expression $D\approx\Sigma_{\rm disc}/\Sigma_{\rm cloud}$, one needs to estimate the thickness of the molecular disc (Eq. 8)

[TABLE]

The derivation and error analysis of Eq. 15 is present below.

We have assumed that the scaleheight of the disc is determined by the balance between pressure in the disc, which is of order

[TABLE]

and the vertical compression due to the gravity from the dark matter

[TABLE]

where $a_{z}$ is the gravitational acceleration along the vertical direction. Here, the have used the galaxy rotation velocity to estimate the acceleration (where $\rm r$ is the radial direction and $z$ is the height). $a_{\rm r}=\Omega_{\rm kep}^{2}r$, and $a_{\rm z}\approx H_{\rm disc}/r\ \times a_{\rm r}$.

One can estimate the scaleheight of the disc using using $p_{\rm disc}\approx p_{\rm gravity}$, which gives

[TABLE]

where one has neglected the self-gravity of the molecular gas and the gravitational force from the stars.

We can prove that these effects are not important: the pressure due to self-gravity is $p_{\rm self-gravity}\approx G\Sigma_{\rm disc}^{2}$, and one can prove that

[TABLE]

where $Q_{\rm gas}$ is the defined as

[TABLE]

It is easy to see that self-gravity is not important when $Q_{\rm gas}>1$, and is only comparable to the gravitational force from the dark matter halo when $Q_{\rm gas}=1$. When $Q_{\rm gas}>>1$, Eq. 15 provides an extremely good estimate to the disc thickness, and when $Q_{\rm gas}\approx 1$, Eq. 15 overestimates the disc scaleheight by $\sqrt{2}-1\approx 0.4$. In either cases the error is not significant. The galaxies that we are studying have $Q_{\rm gas}\gtrsim 1$ (Romeo & Falstad, 2013; Romeo & Mogotsi, 2017), and as a result Eq. 15 is always a good estimate to the disc thickness. In additional to these, the gravitational force from the stars can further reduce the disc thickness, but after going through a similar analysis, one can prove that the effect is also negligible for our purpose.