The dynamical evolution of molecular clouds near the Galactic Centre – II. Spatial structure and kinematics of simulated clouds

We carry out simulations of massive gas clouds in the CMZ environment (Dale et al., 2019) using the state-of-the-art smoothed particle hydrodynamics (SPH) code gandalf (Hubber et al., 2018). The simulations considered here include self-gravity using an octal tree, as well as hydrodynamics and artificial viscosity (Monaghan, 1997), with the gas following a barytropic equation of state with a critical density of $2.5\times 10^{7}~\mbox{${\rm cm}^{-3}$}$ (assuming a mean molecular weight per particle of $\mu_{\rm p}=2.35$). Gas above a density of $2.5\times 10^{6}~\mbox{${\rm cm}^{-3}$}$ is converted into sink particles, implying that the effective equation of state is isothermal, with an adopted temperature of $T=65~{\rm K}$ to represent the high temperature of CMZ gas (e.g. Ao et al., 2013; Ginsburg et al., 2016; Krieger et al., 2017). At the adopted numerical resolution (see below), these sink particles do not represent individual stars, but stellar subclusters. We assign no further physical meaning to these particles, because we are interested in following the large-scale morphology and kinematics of the clouds. The sink particles are only included to prevent the simulations from reaching excessively high (and computationally costly) densities.

The presented simulations do not include stellar feedback or magnetic fields, because we aim to study the pure gravo-hydrodynamics of gas clouds on an eccentric orbit around the Galactic Centre. These mechanisms would act to slow down or truncate star formation. As a result, the simulated clouds will always collapse to form stars at an efficiency much higher than observed or obtained by simulations that do include feedback and/or magnetic fields. While the present paper does not report on any of the star formation characteristics of the simulations, we quantify the relative influence of orbital dynamics on the star formation efficiency in Dale et al. (2019).

The full suite of simulations presented in Dale et al. (2019) contains 36 models of $10^{6}$ SPH particles each, consisting of 12 different sets of initial conditions followed in three different tidal environments. Here we consider a total of 5 different simulations that represent the subset of ‘tidally-virialised’ clouds from Dale et al. (2019, also see below) and are moving on the Kruijssen et al. (2015) orbit in the gravitational potential generated by the mass distribution from Launhardt et al. (2002), with a vertical flattening parameter $q_{\Phi}=0.63$ (see Appendix A of Kruijssen et al. 2015). For comparison, we also consider five ‘control runs’, in which the clouds are moving on a circular orbit in the same background potential. These simulations follow the evolution of spherical clouds with a Gaussian initial density profile, which we truncate at the radius $R_{\rm t}$, where the local volume density has dropped to 5 per cent of the central value (corresponding to 2.45 standard deviations or 1.70 times the half-mass radius $R_{\rm h}$ of the truncated Gaussian). The clouds have a divergence-free initial turbulent velocity field, i.e. without any net compression or expansion. We have carried out several random realisations of the velocity field and select the one with a net angular velocity opposite to the orbital motion of $1~\mbox{${\rm Myr}$}^{-1}$. This way, the spin angular momentum and angular velocity are roughly consistent with what a cloud would have if it were given an infinitesimal perturbation to induce its contraction towards its centre of mass within the shearing interstellar medium in the CMZ potential (cf. its shear-driven angular velocity of $r{\rm d}\Omega/{\rm d}r\approx-0.7~\mbox{${\rm Myr}$}^{-1}$, caused by the fact that the circular velocity increases sub-linearly with galactocentric radius). This is also consistent with the regular spacing of clouds observed on the 100-pc stream (Henshaw et al., 2016b).

The global properties of the clouds are uniquely set by the mean volume density $\rho$, virial parameter $\mbox{$\alpha_{\rm vir}$}\equiv 2T/|V|$ (with $T$ the kinetic energy and $V$ the gravitational potential energy),²²2For this definition, virial equilibrium is achieved at $\mbox{$\alpha_{\rm vir}$}=1$. and velocity dispersion $\sigma$, implying that the masses and radii vary. We first generate clouds with $\rho=\{0.4,1.3,3.8\}\times 10^{3}~\mbox{${\rm cm}^{-3}$}$, $\mbox{$\alpha_{\rm vir}$}=\{0.87,2.6,7.8\}$, and $\sigma=\{6.3,12.7,19.0\}~\mbox{${\rm km}~{\rm s}^{-1}$}$, with the intermediate values representing the fiducial model (these choices are justified below). Relative to the fiducial model, we obtain six non-fiducial models by changing one parameter at a time. All non-fiducial models are derived from the fiducial model by scaling the particle masses, as well as their position and velocity vectors, such that the resulting set of initial conditions is homologous. We then isotropically adjust the velocity dispersions such that the internal turbulent energy balances the compressive tidal energy due to the background potential. Without such a ‘tidal virialisation’ (as in the ‘self-virialised’ models of Dale et al. 2019) by elevating the turbulent velocity dispersion and the corresponding virial ratio, the clouds undergo rapid gravitational collapse. This tidal virialisation fixes the virial parameter, because it becomes inversely related to the cloud density, thus leaving us with five models in total. The resulting initial velocity dispersions and virial parameters of the clouds span $\sigma=12.1$–$41.2~\mbox{${\rm km}~{\rm s}^{-1}$}$ and $\mbox{$\alpha_{\rm vir}$}=3.1$–$27.4$. All parameters describing the initial conditions are summarised in Table 1.

The initial column and volume densities are chosen to be similar to those observed in the CMZ upstream of the dust ridge (at $-0\aas@@fstack{\circ}7<l<0\aas@@fstack{\circ}0$ and $-0\aas@@fstack{\circ}1<b<0\aas@@fstack{\circ}1$), which are thought to have recently condensed out of the diffuse medium (Henshaw et al., 2016b), spanning $\Sigma=0.5$–$1.5\times 10^{3}~\mbox{M${}_{\odot}$}~\mbox{${\rm pc}$}^{-2}$ and $\rho=2$–$7\times 10^{3}~\mbox{${\rm cm}^{-3}$}$, respectively (Henshaw et al., 2017). While these densities are already similar to those used in our initial conditions, it is important to note that these observations probe projected radii smaller by a factor of $\sim 7$. Using the observed size-linewidth relation $\sigma\propto R^{0.7}$ (e.g. Shetty et al., 2012; Kauffmann et al., 2017a) and assuming virial equilibrium ($M\propto R\sigma^{2}$), we expect that $\Sigma\propto R^{0.4}$ and $\rho\propto R^{-0.6}$. Extrapolating the densities of all clouds from Henshaw et al. (2017) to the spatial scales of the simulated clouds, we obtain $\Sigma=0.8$–$3.3\times 10^{3}~\mbox{M${}_{\odot}$}~\mbox{${\rm pc}$}^{-2}$ and $\rho=0.6$–$3.2\times 10^{3}~\mbox{${\rm cm}^{-3}$}$, in excellent agreement with Table 1. Similarly, the velocity dispersions are consistent with the observed size-linewidth relation (e.g. Shetty et al., 2012; Kruijssen & Longmore, 2013; Kauffmann et al., 2017a), which at the radii listed in Table 1 require $\sigma=5$–$40~\mbox{${\rm km}~{\rm s}^{-1}$}$. Again, this matches the range spanned by our set of initial conditions, indicating that the simulations capture the full range of physical conditions observed upstream of the dust ridge.

We deliberately model clouds with masses higher than those of most of the observed CMZ clouds, so that we can model the condensation of such clouds out of a more extended gas reservoir (cf. Henshaw et al., 2016b). We initialise the clouds shortly after the apocentre passage preceding the dust ridge in the Kruijssen et al. (2015) model, at coordinates $\{l,b\}=\{-0\aas@@fstack{\circ}65,-0\aas@@fstack{\circ}07\}$ and a radius of $r_{0}=90~\mbox{${\rm pc}$}$ (corresponding to $t_{0}=-2.46~\mbox{${\rm Myr}$}$ in Appendix C of Kruijssen et al., 2015). Each simulation then follows a single cloud through pericentre, across the dust ridge, past the position of Sgr B2. This way, the simulations capture the evolution of the clouds across a complete radial orbital oscillation, from their diffuse state near apocentre through the dynamical perturbations induced by their orbit in the gravitational potential. Throughout the paper, we use a non-rotating coordinate system centred on the origin of the Galactic coordinate system, i.e. on $\{l,b\}=\{0\aas@@fstack{\circ}0,0\aas@@fstack{\circ}0\}$, with the $x$-axis pointing towards negative Galactic longitudes, the $y$-axis pointing from the observer to the Galactic centre (with Sgr A^∗ at $y=0$), and the $z$-axis pointing towards positive Galactic latitudes.

In order to perform a first, visual comparison of the simulated clouds to the observed CMZ clouds on the dust ridge, we combine a number of snapshots from the different simulations, taken at different times to match the positions of the observed clouds, and generate column density maps of the gas with densities $\rho\geq 10^{4}~\mbox{${\rm cm}^{-3}$}$. This density threshold is chosen to suppress the (initially) extended cloud envelopes in the simulations and represent the high-density gas found in the CMZ, as traced by the bulk of the dust emission (e.g. Longmore et al., 2013a) and molecules such as HCN and NH₃ (e.g. Mills & Morris, 2013; Rathborne et al., 2014a; Krieger et al., 2017). We consider three projections of the simulation snapshots, i.e. top-down $\{l,y\}$, plane-of-sky $\{l,b\}$, and position-velocity $\{l,\mbox{$v_{\rm los}$}\}$.³³3Throughout this paper, we adopt a distance to the Galactic Centre of $d=8.3~\mbox{${\rm kpc}$}$ (Reid et al., 2014) when converting between angular and physical sizes, for which $\{1\degr,1\arcmin,1\arcsec\}\approx\{145,2.41,0.040\}~\mbox{${\rm pc}$}$.

The snapshots are selected to visually best represent the real CMZ clouds in the $\{l,b\}$ plane in terms of their spatial extents, column densities, and orientations. This is a necessary step, because there are no constraints on the initial conditions of the clouds. The relevant question to ask is thus whether any of them reproduce the $\{l,b,\mbox{$v_{\rm los}$}\}$ structure of the observed dust ridge clouds. Even though this optimises the agreement between simulations and observations, it is an interesting comparison to make, because the initial conditions of the simulations span the full range of physical conditions seen in the progenitors of the dust ridge clouds. As we will see below, the simulated evolution of this variety of progenitors reproduces the observed variety of dust ridge clouds, which is an important test of the model’s self-consistency. This enables the interpretation of several key observables in terms of the underlying physical processes. In doing so, we caution that this interpretation should be carried out in terms of global characteristics or systematic trends only. Direct comparisons between any observed and simulated cloud should be made with caution, as the detailed structure (e.g. individual filamentary structures and positions of protostellar cores) is determined entirely by the specific realisation of the initial conditions.

The adopted snapshots are summarised in Table 2 and their relative positioning in the $\{l,b\}$ plane is shown in Figure 1. These clouds represent the main ingredients of the dust ridge (the Brick or G0.253+0.016, Clouds c/d/e/f, and Sgr B2), and also include a low-density cloud upstream of the Brick (which we dub the ‘Pebble’, at $\{l,b\}=\{0\aas@@fstack{\circ}11,0\aas@@fstack{\circ}00\}$) and the downstream continuation of the Sgr B2 complex (‘Sgr B2+’, at $\{l,b\}=\{0\aas@@fstack{\circ}72,-0\aas@@fstack{\circ}08\}$). Three different projections of the combined maps are shown in Figure 2. An animated version of this figure is available as online Supporting Information. For comparison, Figure 2 also includes contours showing the observed gas column densities derived from the cold dust (HiGAL, Battersby et al. in prep.) and contours showing the observed position velocity structure in ¹³CO(2-1) (Ginsburg et al., 2016).⁴⁴4Throughout this work, we transform velocities from the local standard of rest to the rest frame of Sgr A* by adding $U_{\odot}=14~\mbox{${\rm km}~{\rm s}^{-1}$}$ (Schönrich, 2012), consistently with Kruijssen et al. (2015). These have considerably lower spatial resolutions ($25\arcsec$ and $30\arcsec$, respectively, corresponding to $1.0~\mbox{${\rm pc}$}$ and $1.2~\mbox{${\rm pc}$}$) than the simulations ($\sim 0.1~\mbox{${\rm pc}$}$), which leads to trivial differences in spatial structure between the simulations and observations. None the less, these maps provide a good point of reference, because they can be converted into physical units with relatively few assumptions.⁵⁵5 We adopt a mean molecular weight per ${\rm H}_{2}$ molecule of $\mu_{{\rm H}_{2}}=2.8$ to convert dust column densities to mass densities. To translate the ¹³CO flux to a mass density, we assume a ¹³CO-to-H₂ conversion factor of $\alpha({\rm{}^{13}CO})=22.8~\mbox{M${}_{\odot}$}~\mbox{${\rm pc}$}^{-2}~({\rm K}~\mbox{${\rm km}~{\rm s}^{-1}$})^{-1}$ (Schuller et al., 2017; Cormier et al., 2018), which is particularly appropriate in high-column density environments like the CMZ. A number of qualitative properties of the simulated clouds immediately stand out. We list these here and quantify several of them further in Section 3 below.

1. 1.

   Fragmentation and line-of-sight extension: The top panel of Figure 2 shows that the clouds collapse and fragment as the turbulent energy dissipates. At the same time, the external gravitational potential causes the fragments within each cloud to roughly maintain their mutual separation, thus inhibiting global collapse across the full $2R_{\rm t}=15$–$50~\mbox{${\rm pc}$}$ size scales modelled here. The most massive of these fragments match the masses of young massive cluster progenitors observed on the dust ridge (see e.g. Walker et al., 2015) and proceed to collapse. The extensions towards the observer at high longitudes result from the shear-driven dispersal of the outer layers of the clouds, which may explain observational hints of expanding outer layers in the Brick (Rathborne et al., 2014a; Bally et al., 2014). These extensions are a common feature across most of our simulations after a few cloud dynamical times, which strongly suggests that the real CMZ clouds at positive Galactic longitudes may have significant depth along the line of sight, spanning a typical length scale comparable to their major axis in the plane of the sky. Similar, radially extended ‘feathers’ are prevalent in dust extinction maps of gas clouds in extragalactic centres (e.g. Peeples & Martini, 2006). Our simulations suggest that such feathers arise due to shearing motions in the galactic potential. When projected along the line of sight, these feathers could plausibly be (mis)interpreted as e.g. the base of a spiral arm or a cloud-cloud collision due to line-of-sight confusion or multiple velocity components.

2. 2.

   Vertical compression, large-scale filamentary structure, and inclination: The middle panel of Figure 2 shows that the clouds attain a vertically compressed morphology just after they have passed through pericentre (which occurs at $l=-0\aas@@fstack{\circ}16$). This is mainly caused by the tidal field, which is considerably more compressive in the vertical direction than azimuthally (see Section 3.1). The compression generates layers and leaves the clouds highly substructured, with flocculent filamentary features running in the longitudinal direction, much akin to the structure of the Brick as observed with ALMA (Rathborne et al., 2015) and of clouds b, d, and e as observed with the Submillimeter Array (SMA, Walker et al., 2018). The tilt of the clouds towards positive longitudes and latitudes arises due to a torque experienced by the clouds as they move through pericentre above the Galactic plane. On its approach to pericentre passage, the leading end of the cloud arrives first and is pulled upwards. This combination of inclination and flattening is a common feature across all of our simulations and matches that of the observed contours at the positions of the Brick, clouds e/f, Sgr B2, and Sgr B2+.

3. 3.

   Velocity gradient, shear, and kinematic complexity: The bottom panel shows that the clouds develop a velocity gradient with a direction opposite to the orbital motion. This counter-gradient is driven by shear, which causes the side of the cloud facing the Galactic centre to move faster than the side facing away from the Galactic centre (see Section 3.2). Again, this is a common feature of all models and qualitatively matches the velocity gradient of the Brick, clouds e/f, and the Sgr B2 complex. However, it is also quite sensitive to the realisation of the initial velocity field, which was chosen to be consistent with the onset of contraction from a shearing gas reservoir (see Section 2.1). Clouds with zero initial spin angular momentum exhibit significantly weaker velocity gradients. Finally, this panel also reveals significant kinematic complexity in the Sgr B2 region, which is caused by the superposition of clouds along the line of sight as the orbit curves off, away from the observer. This matches the observed, complex kinematic structure of the cloud complex spanned by Sgr B2 and Sgr B2+ (e.g. Henshaw et al., 2016a) and provides a plausible alternative to the currently canonical interpretation of this complexity as a key piece of evidence for a cloud-cloud collision (e.g. Mehringer et al., 1993; Hasegawa et al., 1994; Sato et al., 2000; Jones et al., 2008). We find that no such collision is necessary to generate the large degree of kinematic complexity in the Sgr B2 region. Instead, this is naturally reproduced by the combination of fragmentation and the orbital geometry.

In summary, the clouds’ evolution in a background potential and the pericentre passage initially turns them into inclined, sub-structured, spinning pancakes, before the combination of fragmentation and orbital curvature generates highly complex $\{l,b,\mbox{$v_{\rm los}$}\}$ structures. We find that a wide range of properties of the observed dust ridge clouds (contours in Figure 2) are reproduced by making a suitable selection from the set of simulations. As stated before, this is an interesting result, because the initial conditions of the simulations were chosen to be representative of the observed range of physical conditions (i.e. column densities, volume densities, and velocity dispersions) in the clouds upstream from pericentre (see Section 2.1). This shows that the morphology and kinematics of the dust ridge clouds are consistent with their deformation due to the background potential and the preceding pericentre passage as modelled here, under the plausible condition that their initial properties were similar to those currently observed upstream.

More broadly, Figure 2 illustrates how the evolution of molecular clouds in the CMZ is closely coupled to their orbital dynamics. As will be quantified further in the next sections, these results can be generalised to any (extra)galactic centre where the rotation curve turns over from being flat to $v\propto r^{\beta}$ with $1/2<\beta<1$, which is far from unusual (see e.g. Miyamoto & Nagai, 1975; Rubin et al., 1980; Persic & Salucci, 1995; Erroz-Ferrer et al., 2016). For any rotation curve with such a turnover, the inflow of gas from larger galactocentric radii is predicted to stall due to a decrease of the shear, which leads to the accumulation of the gas in a ring-like structure like the 100-pc stream (Krumholz & Kruijssen, 2015). During this accumulation, the orbital motion of the gas is synchronised by hydrodynamic forces, after which the stream fragments into clouds undergoing synchronised, semi-ballistic motion on an eccentric orbit that retains some memory of the non-zero radial inflow velocity (this behaviour has also been found in three-dimensional simulations, see e.g. Emsellem et al., 2015).⁶⁶6The eccentric stream thus exists within the ring-like region of minimal (but non-zero) shear, which is $45<r/\mbox{${\rm pc}$}<115$ in the gravitational potential of Launhardt et al. (2002, see Section 3.2 below and ) and closely matches the pericentre and apocentre radii of the stream ($r_{\rm p}=60~\mbox{${\rm pc}$}$ and $r_{\rm a}=120~\mbox{${\rm pc}$}$, respectively).

Our simulations show the clouds on the stream are torqued, compressed, and stretched, each of which has the potential to fundamentally change their subsequent evolution. Given the strength of these interactions, the dynamical coupling with the host galaxy may induce collapse and star formation in certain ‘hotspots’, which appear as evolutionary streamlines of progressively advanced states of star formation and feedback. Such evolutionary sequences may be ubiquitous in extragalactic centres too, but they are unlikely to be strictly monotonic – Table 2 and Figure 2 show that some diversity of initial conditions is needed to reproduce the observed CMZ clouds, indicating that the responses of these clouds to the background potential and pericentre passage may differ. In the next section, we quantify how the morphology and kinematics of the clouds are affected by their orbital dynamics.

The morphological evolution of the clouds is quantified in Figure 3 for all five simulations. For reference, a colour composite of the inner CMZ with the orbital model highlighted is included as the top panel. The figure shows several key (observable) properties of the simulated clouds as a function of Galactic longitude, i.e. the dimensions of the clouds in the plane of the sky, their aspect ratios, and their plane-of-sky column densities. All quantities are calculated using the gas with densities $\rho\geq 10^{4}~\mbox{${\rm cm}^{-3}$}$ as in Figure 2, which represents the centrally concentrated components of the clouds and is traced by most of the dust emission (Longmore et al., 2013a), as well as molecular line tracers (such as HCN and NH₃, see e.g. Mills & Morris, 2013; Rathborne et al., 2014a). The dimensions of the clouds in the Galactic longitudinal ($\delta x$) and latitudinal ($\delta z$) directions are obtained by calculating the centre of mass in each coordinate ($x_{\rm cm}$ or $z_{\rm cm}$) and finding the corresponding intervals that enclose half of the mass (i.e. $x_{\rm cm}\pm\delta x$ or $z_{\rm cm}\pm\delta z$). The aspect ratios follow as $\delta z/\delta x$.⁷⁷7Note that we use Cartesian coordinates $\{x,z\}$ to denote positions within a given simulation snapshot, whereas $\{l,b\}$ refer to position along the orbit and thus include the motion and time evolution of the clouds. Finally, the average column densities of the clouds ($\Sigma$) are calculated in a mass-weighted fashion over the rectangle spanned by a cloud’s half-mass intervals (thus together enclosing less than half of its mass) as

with $\Sigma_{\rm local}(x,z)$ the local column density at the coordinate $\{x,z\}$. The weighting by mass emphasises the column density at which most of the mass resides and makes it insensitive to the total spatial extent of the clouds. This is desirable, because the simulations follow the condensation of clouds from a gas reservoir that is initially larger than the observed clouds (see Table 1 and Section 2.1). In addition, we prevent dilution by focussing a window on the cloud centre of mass. By taking a mass-weighted average over this region of interest rather than an area-weighted one, we predict the typical column density expected to be observed in CMZ clouds.

The kinematic evolution of the clouds is quantified in Figure 4 for all five simulations. For reference, a colour composite of the inner CMZ with the orbital model highlighted is included as the top panel. The figure shows several key properties of the simulated clouds as a function of Galactic longitude, i.e. their line-of-sight velocity dispersions, their line-of-sight velocity gradients in Galactic longitude, and their spin angular momenta. As before, the quantities are calculated for the gas with densities $\rho\geq 10^{4}~\mbox{${\rm cm}^{-3}$}$ (cf. Figure 2), except for the spin angular momentum (see below). The line-of-sight velocity dispersions ($\sigma_{\rm los}$) are obtained by calculating the standard deviation of the line-of-sight velocities of the particles within the rectangle spanned by the one-dimensional half-mass extents, i.e. $\{x_{\rm cm}\pm\delta x,z_{\rm cm}\pm\delta z\}$. This is equivalent to taking the mass-weighted average of the line-of-sight velocity dispersion at each coordinate $\mbox{$\sigma_{\rm los,local}$}(x,z)$ as

Likewise, we calculate the line-of-sight velocity gradient (${\rm d}\mbox{$v_{\rm los}$}/{\rm d}l$) by carrying out an orthogonal linear regression to the $\{l,\mbox{$v_{\rm los}$}\}$ distribution of the gas map within $\{x_{\rm cm}\pm\delta x,z_{\rm cm}\pm\delta z\}$ (Boggs & Rogers, 1990). Again, this is equivalent to a mass-weighted fit to the position-velocity image. Finally, the spin angular momenta ($L_{z}$) are calculated without making any cuts in density or position, because this is a physical quantity used to describe the entire cloud and interpret the predicted velocity gradients, rather than an observable quantity. Given a population of $N_{\rm part}$ particles, we obtain

where $m_{i}$ is the particle mass, $\textbf{{r}}_{xy,i}$ is the position vector in the Galactic plane relative to the cloud’s centre of mass, and $\textbf{{v}}_{xy,i}$ is the velocity vector in the Galactic plane relative to the cloud’s centre of motion.

The presented simulations capture the orbital and internal dynamics of the clouds, but omit several physical mechanisms that are potentially important, such as magnetic fields, cosmic rays, detailed chemistry, and stellar feedback. While this may obstruct drawing any conclusions concerning the absolute star formation rates and efficiencies of the clouds, it does allow us to isolate and characterise the morphological and dynamical evolution of the clouds, which is the main focus of this paper. Above all, this enables us to perform controlled experiments and obtain a systematic understanding of the interplay between cloud evolution and galactic dynamics in the CMZ, with implications for galactic centres in general.

Stellar feedback is not expected to strongly affect the structure and dynamics of the clouds prior to advanced gravitational collapse and widespread star formation, which is not achieved until the position of Sgr B2. Therefore, it is a reasonable omission in the context of this work. Chemistry is an important ingredient when generating synthetic line emission maps for direct comparison to observations (in the context of the CMZ, see e.g. Bertram et al., 2016). Clearly, a detailed chemical network and appropriate radiative transfer modelling enables the generation of considerably more realistic maps than the volume density limit of $\rho\geq 10^{4}~\mbox{${\rm cm}^{-3}$}$ that we adopted to create Figure 2. However, we note that the CMZ clouds host notoriously complex chemistry, excitation conditions, and optical depth effects, to the extent that clouds have a different appearance in each molecular line (e.g. Rathborne et al., 2015). Accurately modelling the molecular chemistry necessary for following the high-density gas tracers often used to observed CMZ clouds remains out of reach for the foreseeable future. Importantly, a considerable part of our analysis focuses on kinematics. Observational velocity measurements are less affected by molecular abundances or chemistry than integrated intensity measurements. Finally, magnetic fields have been suggested to affect the dynamics of CMZ clouds on sub-pc scales (e.g. Pillai et al., 2015). However, this is based on the total magnetic field strength. In the absence of a coherent magnetic field, the unordered (turbulent) magnetic field mainly acts as a source of pressure (for recent discussions, see e.g. Li & Henning, 2011; Pillai et al., 2015; Federrath et al., 2016). Estimates of the turbulent magnetic field strength yield Alfvénic Mach numbers ${\cal M}_{\rm A}>1$ (Federrath et al., 2016), indicating that supersonic turbulence dominates the cloud structure. Our results substantiate this finding – the fact that our simulations reproduce the observed spatial structure and kinematics of CMZ clouds, suggests that magnetic fields may not be dynamically important, but instead trace the turbulent flow in the CMZ clouds.

Another assumption worth discussing is the initial density profile of the clouds. The adopted Gaussian density profile does not represent a hydrostatic equilibrium solution, meaning that some part of the clouds’ evolution may be caused by their initial progression towards equilibrium. Unfortunately, the presence of an external gravitational potential implies that even density profiles satisfying hydrostatic equilibrium in isolation, such as cored power laws (e.g. McKee & Ostriker, 2007; Keto & Caselli, 2010), would still require an initial equilibration phase. To answer whether a significant part of the observed cloud evolution is caused by the choice of initial density profile, we therefore compare the simulations discussed here to isolated control runs in Dale et al. (2019),¹³¹³13Note that these control runs do not include any background potential and are distinct from the circular-orbit control runs discussed in Appendix A. finding that the evolutionary trends presented in this work are unique to the clouds evolving in the background potential of the CMZ. Reassuringly, this shows that any influence of the initial density profile on the results is subdominant relative to the role of the external potential in governing cloud evolution.

A final caveat is the choice of orbital model. Other parameterisations or dynamical models for the 100-pc stream in the CMZ exist (e.g. Sofue, 1995; Molinari et al., 2011; Ridley et al., 2017). However, out of all these models, the Kruijssen et al. (2015) model provides the closest match to the position-position-velocity structure of the dust ridge (and the 100-pc stream in general, also see Henshaw et al., 2016a), which motivates its use in this paper. Perhaps most importantly, many of the identified ways in which the orbital dynamics affect cloud evolution are not sensitive to the details of the orbital solution, but are set by the global properties of the gravitational potential. For instance, the sizes, aspect ratios, velocity dispersions, velocity gradients, and spin angular momenta are all most strongly affected by the instantaneous tidal field and shear, while carrying second-order imprints of the pericentre passage on the adopted orbital model. Therefore, we predict that many of our findings should also persist in alternative orbital geometries (e.g. Sormani et al., 2018).

We now briefly compare the properties of the simulated clouds to those of the observed dust ridge clouds. The first of these comparisons is quantitative, as it considers the column densities and velocity dispersions as a function of position along the dust ridge as in Figure 3 and Figure 4, mirroring the evolution with orbital time in our simulations. The second comparison follows in Section 4.3 and focuses on a single snapshot of our high-density simulations at the position of the Brick.

For the first of the above two comparisons, we derive observed column densities from the HiGAL cold dust map (Battersby et al. in prep.; see the middle panel of Figure 2) and velocity dispersions from the HNCO $4(0,4)-3(0,3)$ data obtained with the Mopra CMZ survey (Jones et al., 2012). Henshaw et al. (2016a) selected this molecular line as their primary tracer of the gas kinematics, because HNCO is widespread in the CMZ, suffers minimally from self-absorption in high column density regions, and does not exhibit hyperfine structure.¹⁴¹⁴14Note that the latter two points minimise any spurious broadening of the spectral line profiles when fitting them with a Gaussian. Due to its widespread nature, it likely traces the bulk of the gas in clouds (Jones et al., 2012), necessitating that we include all gas present in the simulations, instead of imposing a density cut as before. To ensure an appropriate comparison between the column densities and velocity dispersions of observed and simulated clouds, we subject them to a highly analogous analysis. The observations and simulations are first convolved to a common spatial resolution of $1\arcmin$. The column densities are then evaluated at a single $\{l,b\}$ coordinate per cloud. For the simulations, we select the coordinate with the highest column density within a $5\times 5~\mbox{${\rm pc}$}$ area centred on the cloud centre of mass. For the observations, this coordinate is taken to correspond to the cloud centre, which is identified by fitting a two-dimensional Gaussian to each cloud in the HiGAL column density map. The column densities are then obtained directly from the simulated and observed maps, at the pixel closest to the the selected coordinate. The uncertainty on the observed column densities is taken to be a factor of 2.

The velocity dispersions of the simulated clouds are measured by calculating the mass-weighted line-of-sight velocity dispersion within a square of $7.2~\mbox{${\rm pc}$}$ (i.e. $0\aas@@fstack{\circ}05$) in width, again centred on the cloud centre of mass. This averaging scale matches that over which the observed velocity dispersions are extracted. When measuring the observed velocity dispersions, it is crucial to avoid contamination by the complex large-scale kinematic structure of the CMZ, with multiple streams intersecting along the line of sight (e.g. Kruijssen et al., 2015; Henshaw et al., 2016a). Therefore, we fit the HNCO spectra across the entire CMZ obtained using the spectral line fitting code SCOUSE (Henshaw et al., 2016a). For these fits, we adopt a spectral averaging area with a width of $7.2~\mbox{${\rm pc}$}$ (i.e. $0\aas@@fstack{\circ}05$). We centre the spectral averaging areas on the cloud coordinates obtained above, reject velocity components unassociated with the clouds,¹⁵¹⁵15Specifically, we remove components with negative line-of-sight velocities relative to the local standard of rest, or velocities offset from the Kruijssen et al. (2015) orbit by more than $40~\mbox{${\rm km}~{\rm s}^{-1}$}$. We also exclude components containing less than 25 per cent of the flux of the brightest component. and calculate the intensity weighted average velocity dispersion of these components.

The middle panel of Figure 5 compares the observed column densities in the Brick, clouds b, c, d, e, and f, Sgr B2, and Sgr B2+ to the total range spanned by the simulations (grey-shaded area) at each Galactic longitude or time. Neither the observations nor the simulations show any significant trends of increasing or decreasing column density with longitude (with the exception of the single data point marking Sgr B2). For the simulations, we see that the total range spanned by the five different sets of initial conditions from Table 1 is wider than the amplitude of any trend across the full longitude range (as in Figure 3). The absolute column densities found in the simulations reproduce the observed range, with the exception of Sgr B2. There are two possible reasons for this. Firstly, it may indicate the superposition of several clouds along the line of sight (as found in Section 2.2). Secondly, our high-density simulation achieves an extreme star formation efficiency of nearly 90 per cent near the position of Sgr B2. This is unphysical and results from the absence of mechanisms that may slow down collapse and star formation, such as stellar feedback and magnetic fields. Including the mass locked in sink particles increases the upper envelope of the grey-shaded area by a factor of 5–10, making it consistent with Sgr B2. Therefore, we conclude that the high column density of Sgr B2 should be addressed in future simulations employing a more complete physical model.

In the bottom panel of Figure 5, we compare the observed velocity dispersions of the same set of clouds to the range covered in our simulations, again as a function of Galactic longitude. The observed velocity dispersions fall within the total range of simulated velocity dispersions. In addition, both the simulations and the observed clouds suggest a weak increase of $\sigma_{\rm los}$ at $l>0\aas@@fstack{\circ}2$, with Pearson and Spearman correlation coefficients for the observed data points of $r\approx 0.5$. In the simulations, this is caused by a combination of (tidally-induced) gravitational collapse and shear (see Section 3.2.1). The generally good agreement at these longitudes contrasts with a slight ($2$–$3~\mbox{${\rm km}~{\rm s}^{-1}$}$) underprediction of the observed velocity dispersion of cloud c. It is plausible that the difference results from differences in the initial conditions. If true, this would imply that cloud c had initial conditions similar to the high-density and high-velocity dispersion simulations in Table 1, which exhibit the highest velocity dispersions and define the upper bound on the grey-shaded area in the bottom panel of Figure 5. However, two important caveats are in order. Firstly, the observed velocity dispersions of CMZ clouds may vary by up to a factor of $\sim 2$ depending on the spectral line used, even if they have comparable critical excitation densities (e.g. Rathborne et al., 2015). With HNCO, we use a tracer showing widespread emission in the CMZ that likely traces most mass in clouds. It thus provides a good match to the simulations presented here, but it remains possible that other spectral lines are more appropriate. Secondly, we reiterate that the presented simulations capture a limited number of physical processes. The inclusion of additional physics, such as stellar feedback or chemistry, may add further scatter and widen the predicted ranges represented by the grey-shaded areas in Figure 5. We plan to address the influence of these processes in future work.

We now turn to a qualitative comparison of the observed 3 mm (89 GHz) dust continuum ALMA observations of the Brick (Rathborne et al., 2015) to the high-density simulation snapshot at the position listed in Table 2, which provides the best match to the Brick among the simulations and snapshots presented in this work. To carry out this comparison, we generate a ‘simulated’ ALMA observation using the CASA software package (McMullin et al., 2007), adopting the same setup with which the Brick was observed by Rathborne et al. (2015). This is done using the simobserve and simanalyze tasks in casa. These tasks allow us to perform synthetic ALMA observations of an input image by first simulating the observation based on user-defined input parameters and subsequently generating a set of corresponding visibilities. These visibility data are then imaged using the clean task.

To prepare the simulated map, we first convert it from units of $\mbox{M${}_{\odot}$}~\mbox{${\rm pc}$}^{-2}$ to Jy. To do this, we assume a distance of $d=8.3~\mbox{${\rm kpc}$}$ as in the rest of this paper, a dust temperature of $T=20~{\rm K}$, an observing frequency of $\nu=89~{\rm GHz}$, a gas-to-dust ratio of 100, and an emissivity index of $\beta=1.75$. The input parameters for this simulated observation are based on those presented in Rathborne et al. (2015, ALMA project ID 2011.0.00217.S). We use the buildConfigurationFile task to generate antenna configuration files from the measurement sets of the real data. The data were taken across six execution blocks, each with varying antenna configurations. To replicate the observations, we simulate six separate observations, each corresponding to a different configuration. A mosaic of $72\arcsec\times 162\arcsec$ with a central position of $\{l,b\}=\{0.275,0.030\}$ is ‘observed’,¹⁶¹⁶16Because the simulated cloud is lopsided (see Figure 2), the central coordinates are offset from the centre of the Brick by $0\aas@@fstack{\circ}026$. That way, the field of view is focused on where most of the mass resides in the simulation. with four spectral bands set at 87.2, 89.1, 99.1, and 101.1 GHz, each with 1875 MHz bandwidth, thus providing a combined bandwidth of 7.5 GHz. Each execution is run for 40 minutes on-source and assumes a precipitable water vapour of ${\rm PWV}=1.5~{\rm mm}$.

Figure 6 shows the original column density map of the high-density simulation at the position of the Brick, its simulated observation that has been generated with CASA according to the procedure above,¹⁷¹⁷17For consistency with most of the analysis presented in this work, we have used the same density threshold of $\rho\geq 10^{4}~\mbox{${\rm cm}^{-3}$}$ to produce the images of the simulation. Material at lower densities is unlikely to contribute strongly to the ALMA maps, because emission on large angular scales is filtered out by the interferometer. and the dust continuum ALMA image of the real Brick from Rathborne et al. (2015), all on the same (linear) intensity scale indicated by the colour bar. This comparison provides several relevant insights. Firstly, the setup used to observe the Brick with ALMA only recovers part of the flux and structure from the simulation. Relative to the left-hand panel of Figure 6, the middle panel especially misses emission near the edges of the field of view, where the sensitivity drops. However, the brightest cores and filamentary structures are recovered well. A qualitative¹⁸¹⁸18We reiterate that a one-to-one, absolute comparison between the simulated and observed maps is not meaningful, because the precise structure of the simulated map depends entirely on the specific realisation of the simulation’s initial conditions. Even between identical realisations of the initial conditions, differences will develop during the simulation due to micro-scale chaos and stochasticity (e.g. Keller et al., 2019). comparison of these to the true Brick dust column density map in the right-hand panel shows remarkable agreement. Other than the observed core at $\{l,b\}=\{0\aas@@fstack{\circ}261,0\aas@@fstack{\circ}016\}$, which is known to host a water maser marking the onset of star formation and is saturated on the colour scale of Figure 6 at $2.4~{\rm mJy}~\mbox{${\rm arcsec}$}^{-2}$, the brightest cores in both the simulated and real Brick are of the order $0.5~{\rm mJy}~\mbox{${\rm arcsec}$}^{-2}$. In addition, both maps show core sizes of $0.1$–$0.2~\mbox{${\rm pc}$}$, connected by pc-scale, flocculent filamentary structures. Globally, they also follow a similar morphology, which manifests itself in the form of comparable inclinations, aspect ratios, and curvature. In both maps, the filamentary structures run along the major axis near the middle of the cloud (which we attribute to the vertical tidal compression in Section 2.2) and fan out to cover the full width of the cloud towards its extremities in the top left and bottom right.

The above, cursory comparison of the simulated and real Brick clouds will be followed up with a more detailed analysis in future work. For instance, the size-linewidth relation, spatial and velocity power spectra, fractal dimension, and column density PDF are ideally suited observables for a thorough, quantitative comparison of these simulations to the observed CMZ clouds. In particular, an ALMA Large Programme covering the entire CMZ with a spatial resolution, sensitivity, and spectral setup as in Figure 6 is both observationally feasible and would enable systematic comparisons of these quantities as a function of (orbital) position to the simulations.

Throughout this paper, we have drawn a direct comparison between the simulated clouds and observations of real CMZ clouds. This comparison shows that the presented simulations quantitatively reproduce a surprising variety of properties of the observed dust ridge clouds, from their column densities and velocity dispersions (Section 4.2) to the velocity gradient (Section 3.2.2) and spatially-resolved structure (Section 4.3) of the Brick. These results add to the qualitative discussion of a much wider variety of observables in Section 2.2 and justify the use of these simulations as interpretative tools for constraining which physical mechanisms govern the baryon cycle in the CMZ. The quantitative character of these simulations opens up a variety of future observational tests of the physical predictions and hypotheses put forward in this work. Here, we focus on predictions of our models that would benefit from additional observational follow-up work. In addition, we discuss how broadly the predictions of our models should apply.

In principle, the physical processes modelled in this work should apply generally. We have adopted initial conditions representative of the clouds some $0.75~\mbox{${\rm Myr}$}$ upstream of the dust ridge, situated on the other side of the preceding pericentre passage, at negative Galactic longitudes and positive Galactic latitudes (cf. Henshaw et al., 2016b). Based on their relatively low densities relative to other CMZ clouds, these clouds are assumed to have recently condensed out of the diffuse medium, possibly due to their recent arrival on the 100-pc stream. If the progenitors to other dense, dust ridge-like gas structures have similar properties, we would expect that these sequences of dense gas clouds also follow the behaviour found in our simulations. In this context, specific targets of interest could be the $20$ and $50~\mbox{${\rm km}~{\rm s}^{-1}$}$ clouds. These clouds are passing through pericentre at a radius of $r=60$–$70~\mbox{${\rm pc}$}$ (Kruijssen et al., 2015), represent the high-density end of a part of the 100-pc stream that starts more diffusely near Sgr C, and exhibit signs of star formation (e.g. Ho et al., 1985; Mills et al., 2011; Lu et al., 2015, 2017). It is quite plausible that the morphological and kinematic trends with longitude identified in this work are found not only in the dust ridge, but at least qualitatively also in the $20$ and $50~\mbox{${\rm km}~{\rm s}^{-1}$}$ clouds.

We emphasise that the trends with Galactic longitude (or equivalently with time relative to accretion or pericentre passage) predicted by Figure 3 and Figure 4 are not necessarily monotonic. The range of initial conditions was chosen to represent a reasonable range based on the plausible precursor clouds to the dust ridge. As such, the spread between the model curves in Figure 3 and Figure 4 provides an uncertainty range. For some quantities (e.g. the column density and velocity dispersion, see Section 4.2), this range exceeds the total magnitude of the trend, implying that even opposite trends may be observed if the initial conditions of the clouds vary systematically with longitude. Notwithstanding this caveat, we do predict a weak, but measurable trend of increasing velocity dispersions towards the highest longitudes, which is tentatively confirmed by the observations shown in Figure 5 (also see Krieger et al. 2017). Other quantities (e.g. the aspect ratio and spin angular momentum) exhibit well-defined trends that represent firm predictions of this work. A systematic study of the presented observables as a function of position along the 100-pc stream should allow recently-condensed sequences of clouds to be identified. These sequences may cover any range of evolutionary phases between the initial condensation out of the diffuse medium and their eventual disruption by stellar feedback. However, the finite lengths of such correlated segments mean they are unlikely to span the complete evolutionary timeline at a single moment in time. Correlating the occurrence of these sequences with pericentre passages will show whether the eccentricity of the orbit plays a defining role in regulating cloud morphology in kinematics or, as expected based on these simulations, plays a relevant yet sub-dominant role next to the presence of the external gravitational potential.

Finally, the advent of ALMA now enables extragalactic CMZs to be observed at a sensitivity and spatial resolution similar to those obtained of the Galactic CMZ in the pre-ALMA era. Provided that the gravitational potential, as parameterised through the power law slope of the enclosed mass profile, and the orbital eccentricity of the gas streams in such CMZs are similar to those adopted here, our predictions should apply directly to these systems too. Discussed observables that may feasibly be obtained in face-on CMZs are the clouds’ longitudinal extents ($\delta x$) and velocity dispersions ($\sigma_{\rm los}$, under the assumption that the trends with longitude are stronger than the deviations from velocity isotropy). In addition, galaxies with low inclinations provide a unique opportunity to quantify the extents of clouds in the galactic plane, which we predict to be substantial. The predictions for the cloud aspect ratios, column densities, and velocity gradients are specific to observations through the orbital plane. Highly-inclined, edge-on galaxies such as NGC 253 (e.g. Sakamoto et al., 2011) would enable a comparison to these observables, in addition to the clouds’ longitudinal extents and velocity dispersions.

Above all, the simulations presented in this paper constitute a rich data set that can be used to shed light on a wide range of open questions. While we have considered several key quantities and observables here, future work will extend our analysis to e.g. the star formation efficiencies, virial parameters, and size-linewidth relations. Together with its companion paper (Dale et al., 2019), the present work sets the first step towards an in-depth physical understanding of the close interaction between galactic dynamics, cloud evolution, and star formation, both in the CMZ of the Milky Way and in extragalactic centres.