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|  Le Wu is currently an associate professor and Ph.D. supervisor at the Hefei
University of Technology (HFUT), China. She received the Ph.D. degree from the
University of Science and Technology of China (USTC). Her general area of
research interests is data mining, recommender systems and social network
analysis. She has published more than 50 papers in referred journals and
conferences. Dr. Le Wu is the recipient of the Best of SDM 2015 Award, the
Distinguished Dissertation Award from China Association for Artificial
Intelligence (CAAI) 2017, and the Youth Talent Promotion Project from China
Association for Science and Technology.
---|---
|  Xiangnan He is a professor at the University of Science and Technology of
China (USTC). He received his Ph.D. in Computer Science from the National
University of Singapore (NUS). His research interests span information
retrieval, data mining, and multi-media analytics. He has over 80 publications
that appeared in several top conferences such as SIGIR, WWW, and MM, and
journals including TKDE, TOIS, and TMM. His work has received the Best Paper
Award Honorable Mention in WWW 2018 and ACM SIGIR 2016. He is in the editorial
board of journals including Frontiers in Big Data, AI Open etc. Moreover, he
has served as the PC chair of CCIS 2019 and SPC/PC member for several top
conferences including SIGIR, WWW, KDD, MM, WSDM, ICML etc., and the regular
reviewer for journals including TKDE, TOIS, TMM, etc.
---|---
|  Xiang Wang is now a research fellow at National University of Singapore. He
received his Ph.D. degree from National University of Singapore in 2019. His
research interests include recommender systems, graph learning, and
explainable deep learning techniques. He has published some academic papers on
international conferences such as KDD, WWW, SIGIR, and AAAI. He serves as a
program committee member for several top conferences such as KDD, SIGIR, WWW,
and IJCAI, and invited reviewer for prestigious journals such as TKDE, TOIS,
TNNLS, and TMM.
---|---
|  Kun Zhang received the PhD degree in computer science and technology from
University of Science and Technology of China, Hefei, China, in 2019. He is is
currently a faculty member with the Hefei University of Technology (HFUT),
China. His research interests include Natural Language Understanding,
Recommendation System. He has published several papers in refereed journals
and conferences, such as the IEEE Transactions on Systems, Man, and
Cybernetics: Systems, the ACM Transactions on Knowledge Discovery from Data,
AAAI, KDD, ACL, SIGIR, WWW, ICDM. He received the KDD 2018 Best Student Paper
Award.
---|---
|  Meng Wang received the BE and PhD degrees from USTC, in 2003 and 2008,
respectively. He is a professor with HFUT. His current research interests
include multimedia content analysis, computer vision, and pattern recognition.
He has authored more than 200 book chapters, journal, and conference papers in
these areas. He is the recipient of the ACM SIGMM Rising Star Award 2014. He
is an associate editor of the IEEE Transactions on Knowledge and Data
Engineering, the IEEE Transactions on Circuits and Systems for Video
Technology, and the IEEE Transactions on Neural Networks and Learning Systems.
He is an IEEE Fellow and IAPR Fellow.
---|---


# Dynamically Driven Inflow onto the Galactic Center and its Effect upon
Molecular Clouds

H Perry Hatchfield  University of Connecticut, Department of Physics, 196A
Auditorium Drive, Unit 3046, Storrs, CT 06269 USA  Mattia C. Sormani
Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische
Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany  Robin G.
Tress  Universität Heidelberg, Zentrum für Astronomie, Institut für
Theoretische Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany
Cara Battersby  University of Connecticut, Department of Physics, 196A
Auditorium Drive, Unit 3046, Storrs, CT 06269 USA  Rowan J. Smith  Jodrell
Bank Centre for Astrophysics, Department of Physics and Astronomy, University
of Manchester, Oxford Road, Manchester M13 9PL, UK  Simon C.O. Glover
Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische
Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany  Ralf S.
Klessen  Universität Heidelberg, Zentrum für Astronomie, Institut für
Theoretische Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany
Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches
Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany

###### Abstract

The Galactic bar plays a critical role in the evolution of the Milky Way’s
Central Molecular Zone (CMZ), driving mass inward toward the Galactic Center
via gas flows known as dust lanes. To explore the interaction between the CMZ
and the dust lanes, we run hydrodynamic simulations in arepo, modeling the
potential of the Milky Way’s bar in the absence of gas self-gravity and star
formation physics, and we study the flows of mass using Monte Carlo tracer
particles. We estimate the efficiency of the inflow via the dust lanes,
finding that only about a third (30$\pm$12%) of the dust lanes’ mass initially
accretes onto the CMZ, while the rest overshoots and accretes later. Given
observational estimates of the amount of gas within the Milky Way’s dust
lanes, this suggests that the true total inflow rate onto the CMZ is 0.8$\pm
0.6$ M⊙ yr-1. Clouds in this simulated CMZ have sudden peaks in their average
density near the apocenter, where they undergo violent collisions with
inflowing material. While these clouds tend to counter-rotate due to shear,
co-rotating clouds occasionally occur due to the injection of momentum from
collisions with inflowing material ($\sim 52\%$ are strongly counter-rotating,
and $\sim 7\%$ are strongly co-rotating of the 44 cloud sample). We
investigate the formation and evolution of these clouds, finding that they are
fed by many discrete inflow events, providing a consistent source of gas to
CMZ clouds even as they collapse and form stars.

##  1 Introduction

The process of star formation is governed by a complex interweaving of
physical mechanisms including self-gravity, galactic scale dynamical effects,
turbulence, stellar feedback and magnetic fields (McKee & Ostriker, 2007;
Federrath, 2015; Klessen & Glover, 2016). Studying the individual effects and
interaction of these mechanisms and how they vary as a function of environment
is necessary to understand star formation as it occurs across the cosmos.

A vast range of physical scales are relevant to the processes that govern star
formation, ranging from the kiloparsec scales typical of galactic features
such as spiral arms and bars, down to the $<0.01$ parsec scales of individual
protostellar systems. Hydrodynamic simulations have proven to be an excellent
tool for studying the interplay of gas self-gravity, turbulence (e.g.
Federrath & Klessen 2012), galactic dynamics (e.g. Kim et al. 2012; Armillotta
et al. 2019; Kruijssen et al. 2019; Tress et al. 2020a; Smith et al. 2020; Li
et al. 2021), star formation and feedback processes such as supernovae and
stellar winds across this range of scales (e.g. Federrath et al. 2014; Walch
et al. 2015; Girichidis et al. 2016; Gatto et al. 2017; Rahner et al. 2019;
Rosen & Krumholz 2020; Tress et al. 2020b; Sormani et al. 2020).
Observationally, however, the smallest of these scales can only be studied
within the Milky Way. Since there is a limited variety of conditions within
our Galaxy, this provides only a small window into the environmental variation
of star formation as it occurs across the cosmos.

The Milky Way’s center provides a unique opportunity to study the physics of
star formation on sub-parsec scales within a more extreme environment than
those nearby within the Milky Way’s disk. The Central Molecular Zone (CMZ),
which is defined as the region within Galactocentric radius $R\approx 250$ pc,
hosts a reservoir of $\sim$ 3-5$\times$107 M⊙ in molecular gas (Morris &
Serabyn, 1996; Dahmen et al., 1998), with its clouds exhibiting high average
densities and gas temperatures relative to the Galactic Disk (e.g. Mills &
Morris 2013; Ginsburg et al. 2016; Krieger et al. 2017; Mills et al. 2018),
large velocity dispersions (e.g. Henshaw et al. 2016; Federrath et al. 2016;
Kauffmann et al. 2017; Henshaw et al. 2019), and intense magnetic fields (e.g.
Crutcher et al. 1996; Pillai et al. 2015). These conditions bear similarities
to some high redshift ($z\sim 2$) galaxies in which such detailed observations
are impossible (Kruijssen & Longmore, 2013).

The dynamics of the galactic environment are understood to be vital for
modeling the mass flows, star formation rates and efficiencies, and molecular
cloud life cycles within galaxy centers (e.g. Sormani et al., 2019; Tress et
al., 2020b; Sormani et al., 2020; Orr et al., 2021). In particular, tidal
compression effects during pericenter passage have been suggested as a
mechanism for triggering star formation in molecular clouds on CMZ orbits
(Kruijssen et al., 2015; Jeffreson et al., 2018; Kruijssen et al., 2019; Dale
et al., 2019).

The Milky Way is known to host a stellar bar (e.g. Blitz & Spergel 1991;
Binney et al. 1997; Wegg & Gerhard 2013). This bar generates a highly non-
axisymmetric gravitational field within the Galaxy’s inner few kpc, which
induces strongly non-circular gas motions (Fux, 1999; Kim et al., 2011, 2012;
Li et al., 2015; Sormani et al., 2015b; Ridley et al., 2017; Sormani et al.,
2018a). The bar potential admits two families of closed orbits that are of
particular relevance for the gas flows: the x1 orbits, which occur in the
outer parts, have the semimajor axis aligned with the bar and become more
eccentric as the Galactic Center is approached; and the mildly elliptical x2
orbits, which occur closer to the Center and have semimajor axis perpendicular
to the bar (Binney et al., 1991; Sormani et al., 2015a). The self-interaction
of gas on the innermost x1 orbit induces shocks that efficiently drive
material from the x1 to the x2 orbits, where the majority of CMZ gas settles
(Sormani et al., 2015a). The shocked streams of gas transitioning between
these orbital families are known as the bar "dust lanes" by analogy with
similar structures observed in many barred spiral galaxies (such as NGC 1097,
NGC 1300, and many others, see for example Martini et al. 2003a, b). Position-
velocity features interpreted as the dust lanes have been identified in the CO
maps of the Galactic Center (e.g. Fux, 1999; Marshall et al., 2008), and are
believed to be the primary avenue by which the Milky Way’s CMZ gains mass
(e.g. Kim et al. 2012; Tress et al. 2020b).

To measure this bar-driven inflow rate, Sormani & Barnes (2019) estimate the
mass and inwards velocity of material in the dust lanes by interpreting 12CO
observations of longitude-velocity (l-v) structures near the Galactic Center
using a simple geometrical model that is motivated by the hydrodynamical
simulations from Sormani et al. (2018b). The 12CO intensity from the
structures that they identify as dust lane candidates is translated to a mass
using an XCO conversion factor, and then the gas radial velocity is calculated
using the geometrical model and adopting recent constraints on the bar’s
position-angle relative to the Sun (Wegg & Gerhard, 2013; Bland-Hawthorn &
Gerhard, 2016). With this model, they find a mass inflow rate of
$2.7^{+1.5}_{-1.7}$ M⊙yr-1. However, this estimate assumes that all inflowing
mass in the dust lanes will accrete onto the CMZ on the timescale suggested by
its inward velocity (i.e., its distance from the CMZ divided by its radial
speed). They acknowledge (see their Section 4.1.4) that a significant fraction
of the dust lane material likely overshoots the CMZ and will be added to the
CMZ’s mass on a longer timescale, and that as a consequence their simple model
consequently tends to overestimate the inflow rate.

In this work, we run a hydrodynamic simulation of the flows of gas in the
presence of the Milky Way’s barred gravitational potential in order to measure
the inflow efficiency of the dust lanes. We also use these simulations to
study how the properties of molecular clouds are affected by the dynamics of
the Galactic Center. The key novelty of this simulation is that we include
tracer particles that are weighted by mass throughout the entire simulation.
This allows us to study in detail the mass flows, and to keep track of the
properties of molecular clouds as a function of time and reconstruct their
history. We do not include gas self-gravity, star formation or stellar
feedback to better isolate the effect of the external gravitational field and
orbital properties on cloud properties.

The structure of this paper is as follows. In Section 2 we summarize the
numerical setup for our simulated CMZ. Section 3 describes our tracer particle
method for following material falling into the CMZ, and presents the results
from our measurement of the inflow efficiency. In Section 4, we use tracer
particles to follow the evolution, angular momenta, and build-up of individual
clouds in the CMZ, and we report the results of this cloud-tracing analysis.
We discuss the possible implications of these results in Section 5. Finally,
we conclude and sum up in Section 6.

##  2 Simulation Design

We use the moving-mesh code arepo (Springel, 2010; Weinberger et al., 2020) to
re-simulate the model of Sormani et al. (2019) with the addition of Monte
Carlo tracer particles (MCTP, or just tracer particles). These allow us to
follow the flow of mass between cells and track the properties of clouds self-
consistently between simulation snapshots. Here we briefly summarize the
numerical design of these simulations. A more detailed description of the
simulation design can be found in Sormani et al. (2018b) and Sormani et al.
(2019), while a description of the implementation of the MCTP can be found in
Genel et al. (2013).

arepo solves the equations of hydrodynamics in three-dimensions in the absence
of magnetic fields111arepo can also be used to model magnetohydrodynamical
flows, but we do not make use of these capabilities in this study. and gas
self-gravity:

$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0$ (1)
$\frac{\partial\rho\mathbf{v}}{\partial
t}+\nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v}+P\mathbf{I})\mathbf{v}=-\rho\nabla\Phi$
(2) $\frac{\partial(\rho e)}{\partial t}+\nabla\cdot[(\rho
e+P)\mathbf{v}]=\dot{Q}+\rho\frac{\partial\Phi}{\partial t}$ (3)

where $\rho$ is the density of the gas parcel, $\mathbf{v}$ is the fluid
velocity, $P$ is the thermal pressure field, $\mathbf{I}$ is the identity
matrix, $\boldsymbol{\Phi}$ is the external gravitational potential tuned to
fit the properties of the Milky Way, $e=e_{\rm
thermal}+\boldsymbol{\Phi}+\mathbf{v}^{2}/2$ is the total energy per unit mass
and $e_{\rm thermal}$, the thermal energy per unit mass, and $\dot{Q}$
describes the rate of change in the internal energy of the cell due to
radiative and chemical cooling and heating. We assume an ideal gas equation of
state, obeying

$P=(\gamma-1)\rho e_{\rm thermal}$ (4)

with an adiabatic index $\gamma=5/3$, a good approximation for both the atomic
and cold molecular hydrogen composing the vast majority of the gas content in
our regions of interest (Sormani et al., 2018b). We use the NL97 chemical
network from Glover & Clark (2012) to follow the interaction and evolution of
chemical species in our simulated ISM. This network combines a model for
hydrogen chemistry from Glover & Mac Low (2007) with a simple model for CO
formation and destruction developed by Nelson & Langer (1997). It tracks the
non-equilibrium abundances of the species H, H2, H+, C+, O, and CO between
each timestep. The distributions of several species is shown for a sample
timestep in Figure 1.

Figure 1: Face-on density slices through the plane of $z=0$ for the
distribution of four chemical species traced by our chemical network. From top
to bottom CO, H2, H and H+ are shown. The gas in the Galactic Center is highly
molecular, with high densities mainly concentrated in the dust lanes and in
the central x2 orbits.

We initialize the simulation with an initial axisymmetric density profile
described by

$\rho_{\rm
init}(R,z)=\frac{\Sigma_{0}}{4z_{d}}\text{exp}\bigg{(}-\frac{R_{m}}{R}-\frac{R}{R_{d}}\bigg{)}\text{sech}^{2}\bigg{(}\frac{z}{2z_{d}}\bigg{)},$
(5)

for Galactocentric radius $R$ and height $z$, $\Sigma_{0}=50$ M⊙pc-2,
$z_{d}=85$ pc, $R_{d}=7$ kpc, and $R_{m}=1.5$ kpc. These parameters have been
chosen to match the observed radial distribution of gas in the Milky Way
(Kalberla & Dedes, 2008; Heyer & Dame, 2015). In order to cut down on
computational expense, the outer reaches of the Galactic Disk (all material at
$R\geq 10$ kpc) are removed from the simulation’s initial conditions. Gas
beyond this radius has a negligible effect on the gas dynamics at smaller
Galactic radii over the relevant timescale. No noise is added to the initial
distribution of gas, and all asymmetries and turbulent flows observed in the
gas flows are caused by the imposed barred Galactic potential.

In this work we use the same rigidly rotating external barred potential as
used by Sormani et al. (2019). This potential is composed of components for
the halo, disk, bulge, and a bar rigidly rotating with pattern speed
$\Omega_{p}=40$ km s-1kpc-1 whose properties are consistent with observational
constraints (Launhardt et al., 2002; Sormani et al., 2015b; McMillan, 2017;
Portail et al., 2017; Sanders et al., 2019; Sormani et al., 2020; Bland-
Hawthorn & Gerhard, 2016). For more details on the explicit form and
construction of the gravitational potential, we refer the reader to appendix A
of Sormani et al. (2019). The non-axisymmetric potential term, corresponding
to the stellar bar, is introduced slowly from the beginning of the simulated
time in order to avoid transient dynamics in the early gas distribution. The
barred potential is fully developed at $t\approx 150$ Myr, and all of the
analysis is performed on simulation snapshots later than this time. The
simulation is run for $\sim$30 Myr after the barred potential is fully active.

The spatial resolution of our simulation varies depending on the local
density. Cells are refined down to a target cell mass of $25$ M⊙. To achieve
this, arepo divides any cells with mass a factor of two larger than the target
mass, and merges any adjacent cells that are a factor of two less massive so
that cells are kept on average close to the target mass. We also implement a
minimum effective cell radius ($r_{\rm eff}=({3V_{\rm cell}}/{4\pi})^{1/3}$
for cell volume $V_{\rm cell}$) of $0.1$ pc to prevent excessive refinement
and computational slow-down in areas of extremely high density. Cells with a
volume below this cell volume are not permitted to divide into smaller cells.
This refinement scheme allows for a higher spatial resolution in regions of
high density while lowering the computational cost of including large portions
of lower density material outside of the Galactic center region. The
distribution of cell spatial resolution is shown in Figure 2, which shows that
the densest gas is resolved with cells of radius $\sim 0.1$ pc.

To trace the mass flows in this simulation, we use the Monte Carlo tracer
particle implementation presented in Genel et al. (2013). The tracers are
massless particles associated, at any given timestep, with a given arepo cell,
known as that tracer’s “parent cell”. These tracers do not have individual
positions or velocities of their own, instead adopting those of their parent
cell. Using the instantaneous finite volume fluxes, which are already
calculated for each Voronoi cell face between every active timestep, we
determine a mass-flux weighted number of tracers to be transferred across each
cell interface. This allows the tracers to effectively follow the fluid flow.
The Monte Carlo draw occurs in order to probabilistically distribute the
finite number of tracers in each cell according to the mass fluxes. Without
the tracer particles, the mass flow cannot be easily followed between multiple
timesteps because arepo is not a purely Lagrangian code. More details about
the implementation of this procedure and its advantages over other tracer
particle schemes can be found in Genel et al. (2013). Tracers are injected
into each cell in the simulation in a mass-weighted fashion at a simulation
time of 100 Myr, with roughly 1 tracer per 10 M⊙, rounded up. Given our
adopted target mass, this means that we have roughly 2-3 tracers per
simulation cell, with more massive cells having more tracers. These tracers
then move according to the mass fluxes for the remaining simulated time.

Figure 2: Visualizations of the distribution of cell spatial resolution across
the simulation. The left-hand panel shows cell radius from top down slice
through $z=0$, showing how the spatial resolution is higher for cells in
regions of interest such as the dust lanes and x2 orbits in the CMZ. The
snapshot has been rotated such that the sun’s position is directly to the
left. The right-hand panel shows the distribution of cell radius as a function
of density, highlighting the volume cutoff corresponding to an effective
radius of 0.1 pc to prevent runaway cell division and computational slowdown.
Cells with volumes smaller than half this cutoff are derefined in the next
timestep, and are not permitted to sub-divide further. The right-hand panel
indicates that resolution scale is largely a proxy for cell density. The bar
has been rotated 20∘ relative to other figures to include more of the dust
lane region in the frame. Figure 3: A schematic illustration of the simulated
gas flow features at simulation time $t=149$ Myr, where the bar major axis has
been oriented horizontally. The dotted circle indicates the CMZ, with phase
angles labeled according to the notation used in this work. The red arrows
denote the flow of gas along the near-side and far-side dust lanes. An example
of the blurring of the dust lane components is highlighted in the upper left-
hand side, while a more distinct configuration is highlighted on the lower
right-hand side. The direction to the observer (the Sun) is indicated.

##  3 The Inflow Efficiency of Dust Lane Gas Accreting into the CMZ

While the Galactic bar is an efficient mechanism for driving gas to smaller
Galactocentric radii, not all of the gas in the dust lanes accretes directly
into the innermost 250 pc of the Galaxy on the timescale naively implied by
its inwards velocity. Some of the high-velocity gas transported by the dust
lane shocks will overshoot, and be accreted at a later time. This inflow
inefficiency has been noted in the past. For example, the results of the
hydrodynamical model of the barred galaxy NGC 1530 presented in Regan et al.
(1997) show that not all of the material traveling along that galaxy’s dust
lanes is efficiently accreted into the CMZ, and some instead overshoots and
interacts with the opposite dust lane. However, the precise fraction of
overshooting gas calculated by Regan et al. (1997) was inaccurate due to an
inherited mistake in the treatment of the bar using the CMHOG code (Kim et
al., 2012). This overshooting phenomenon is also visible by eye in previous
simulations of the Milky Way’s Galactic Center as well (e.g. Sormani et al.
2018b), but could not be quantified because individual mass elements cannot be
followed within arepo’s quasi-Lagrangian framework. In this section, we
measure the inflow efficiency of inflowing gas by following samples of tracer
particles selected from the dust lanes.

###  3.1 The Morphology of the Dust Lanes

Due to the complexity and variability of the dust lanes, it is not immediately
obvious how we should define the inflow efficiency. To do so effectively, we
must consider the morphology of the dust lanes and how matter accretes onto
the CMZ. In the simulations presented in this work, the dust lanes appear to
generally have multiple components that span a range in velocity and impact
parameter with respect to the Galactic Center. Multiple, spatially distinct
dust lanes are not a general feature of simulations of gas flow in barred
galaxies, but similar velocity features that resemble these components are
visible in position-velocity maps of CO within the Milky Way’s Galactic Center
(see the features labeled L1 and L3 in figure 1 and their simulated
counterparts D1 and D2 in figure 3 of Sormani et al. 2019; see also Liszt
2008). The specific morphology of the simulated dust lanes depends on the
details of the gravitational potential used in the simulation and the sub-grid
feedback physics, and additionally varies as a function of simulation time. A
simplified schematic of a sample dust lane morphology and a general
description of the dust lane region and CMZ that emerges from these
simulations is shown in Figure 3.

The simulations of Tress et al. (2020b), which are identical to the
simulations presented here except for the inclusion of gas self-gravity and
supernova feedback (and in particular utilize the same external gravitational
potential and initial gas distribution) show that supernova feedback can
spatially blur together the two dust lanes. They also show that the total mass
inflow rate toward the CMZ from the dust lanes is not dependent on the
inclusion of gas self-gravity or star formation feedback. This suggests that
the blurring of the dust lanes does not affect the average inflow rate, which
is ultimately controlled by the torques of the bar potential as they remove
angular momentum from the gas. Therefore, it should not affect the average
correction factor that we aim to derive in this work.

Given this understanding of the dust lane morphology and of the inflow
mechanism, we measure the inflow efficiency of our simulated Galactic Center
using three different methods. The first method compares the rate derived from
an approach resembling that of Sormani & Barnes (2019) to the actual CMZ
growth rate. In the second method, we measure the probability that a randomly
selected parcel of mass in the dust lane will be accreted the first time it
passes the CMZ. In the final method, we constrain the number of times
infalling tracers pass within 250 pc of the Galactic Center without remaining
within the CMZ indefinitely (or whether they never get accreted). As we will
see below, the values of the inflow efficiency obtained by these three
different methods agree with each other within their uncertainties.

###  3.2 Measuring the Bulk Inflow Efficiency of the Dust Lanes

Firstly, we measure the true inflow rate in the simulations by measuring the
mass growth of the CMZ as a function of time. The total gas mass of the CMZ is
obtained by summing up the mass of all cells with Galactocentric radius $<250$
pc as a function of time, and the inflow rate is calculated as the rate of
change of this total mass. We find that while the instantaneous inflow rate
can vary within a factor of $\sim 2$, the average value over a 30 Myr window
is $\sim$0.7 M⊙ yr-1.

We then calculate a “naive estimate” of the inflow rate by assuming that all
of the inflowing mass in the dust lanes will accrete onto the CMZ within the
time suggested by its radially inward velocity. In other words, for each cell
in the dust lane region that has an inward radial velocity greater than 50 km
s-1, we consider its projected velocity toward the Galactic Center and its
mass, and treat each of these parcels of mass as if they will accrete within a
time $t=R_{\text{GC}}/v_{\text{in}}$ where $R_{\text{GC}}$ is the
Galactocentric Radius of the cell and $v_{\text{in}}$ is the radial component
of the velocity toward the Galactic Center. However, this method is likely to
overestimate the accretion rate because it assumes that all the gas accretes
the first time that it passes close to the CMZ, while in reality it will
sometime overshoot as noted above. Considering the three independent dust lane
samples selected throughout the simulation, we obtain estimates of the inflow
rate ranging between 1.5-1.8 M⊙ yr-1 with a mean of 1.7 M⊙ yr-1.

Therefore, taking the ratio of the true net mass inflow rate of $\sim$0.7 M⊙
yr-1 to the naive estimate of $\sim$1.7 M⊙ yr-1, we obtain an inflow
efficiency of $\epsilon\approx 0.41$. This value fluctuates on smaller
($\sim$1 Myr) timescales, but these variations are averaged out over the
timescale used in this calculation (30 Myr). The other methods that are
described below will give an indication of the small timescale variations.

###  3.3 Measuring the Instantaneous Inflow Efficiency of the Dust Lanes

A second way to measure the inflow efficiency is to consider the probability
that any given parcel of mass in the dust lanes will be accreted onto the CMZ
on its next pass within 250 pc of the Galactic Center. This approach also
allows us to better estimate the variability of the inflow efficiency.

Figure 4: An example of the selection of inflowing material used to measure
the inflow efficiency in Section 3.3. The tracer particles are selected from
all cells within 100 pc long segments of the near and far-side dust lanes
(shown as blue and red rectangles, respectively) that meet a velocity criteria
($v<50$ km s-1) to exclude cells that are not associated with the dust lanes.
The cells hosting selected tracers are colored blue and red corresponding to
the near and far-side dust lanes, and the corresponding colors in Figure 6.
Modest changes to the shape, orientation, and length of the selection regions
do not substantially change the inflow efficiency measured. To calculate
$\langle\epsilon_{\rm DL}\rangle$ in Section 3.3, 25 pairs of tracer regions
(for a total of 50 samples) are selected from different timesteps throughout
the simulation. The blue selection of tracers is used as an example in Section
3.4.

We select 25 pairs of tracer particle samples (one from the near and one from
from the far-side dust lane for each used timestep, for a total of 50
independent samples each containing $\sim 10^{5}$ tracer particles), spaced
$\sim$2.5 Myr apart to avoid excessive correlations. For each timestep used,
the tracer particles are selected from a 100 pc long segment of the dust lanes
starting at a radial distance of about 300 pc from the Galactic Center. An
example of this selection for a single timestep is shown in Figure 4, and a
gallery of snapshots following this same sample of tracers (the blue
selection) as they overshoot and accrete is shown in Figure 5. Because the
number of tracer particles injected into each cell is proportional to the cell
mass, the sample effectively tracks the mass flows of the gas initially
contained in the selected region. We ensure that the sample only contains
material associated with the dust lanes by rejecting any tracers associated
with cells having small ($v<50$ km s-1) Galactocentric radial velocities. This
velocity minimum also ensures that the 20 Myr window will be more than
sufficient to observe the overshooting of the tracers because it would take a
particle moving at $v>50$ km s-1 less than 10 Myr to cross the 250 pc radius
ring used to define our CMZ, and the vast majority of tracers in the dust lane
are moving significantly faster than that. Using these tracer particle
samples, we calculate the average instantaneous inflow efficiency
$\langle\epsilon_{\rm DL}\rangle$ as

$\langle\epsilon_{\rm DL}\rangle=\bigg{\langle}\frac{\min(N_{\rm
250pc})_{20\rm Myr}}{N_{\rm tot}}\bigg{\rangle},$ (6)

where $\min(N_{\rm 250pc})_{20\rm Myr}$ is the local minimum number of tracer
particles contained within 250 pc of the Galactic Center ($N_{\rm 250pc}$)
within a 20 Myr window after the tracers’ first approach toward the CMZ,
$N_{\rm tot}$ is the total number of tracers in the sample, and the angle
brackets denote an average over the 50 samples of tracers. The time span of 20
Myr is only used to limit the time that we follow each tracer sample to cut
down on unnecessary computational expense. The 20 Myr window that we used to
select the minimum of $N_{\rm 250pc}$ begins only after the selected tracers
undergo their initial approach to the CMZ. This measurement of the local
minimum is visualized in Figure 6, which shows the percentage of tracer
particles at a given Galactocentric radius. We see that the tracers initially
approach the CMZ (at which time the 20 Myr window begins, labeled 1) before
partially overshooting (labeled 2) and eventually re-accreting over time
(labeled 3). The local minimum at which $\min(N_{\rm 250pc})_{20\rm Myr}$ is
calculated approximately corresponds to the point labeled 2.

Figure 5: Six snapshots depicting the tracer inflow over time for one of the
50 tracer selections, plotted in blue over that timestep’s molecular hydrogen
number density. The snapshot is rotated such that the Sun’s position is
directly to the left, as in Figure 2. The blue tracers correspond to the same
selection used to generate the left-hand panel of Figure 6. In this case the
tracers are selected from the near-side dust lane, in which the two velocity
components are blurred together. The top middle panel shows the material on
its initial infalling trajectory, labeled 1. In the top right-hand panel, the
overshooting material is labeled 2, and the rest of the material is undergoing
a violent collision as it accretes onto the CMZ.

We find, averaging the near and far-side dust lane, that $\langle\epsilon_{\rm
DL}\rangle=0.30\pm 0.12$. Most of the gas that overshoots the CMZ collides
with the opposite dust lane, and after $\sim$25 Myr, the majority of the
overshooting mass of the initial parcel of gas has been accreted to within 250
pc of the Galactic Center. The uncertainty in the measurement of
$\langle\epsilon_{\rm DL}\rangle$ reflects the standard deviation of the 50
tracer samples considered.

Figure 6: The Galactocentric radial distribution of gas selected from
infalling material in the dust lanes. The left-hand panel is the near-side
(closer to the Sun) dust lane, and the right-hand panel shows the far-side
dust lane. A dashed line shows the mean tracer particle radius at each
timestep, and the solid black line is the percentage of tracer particles
located within 250 pc of the Galactic Center. The blue and red contours show
the radius containing the radius range that contains 33%, 66% and 100% of the
tracer particles both above and below the mean tracer value. In both panels,
three points of interest are labeled, and correspond to similar labels in
other figures (such as Fig. 5). Feature 1 shows the initial accretion event in
which most of the gas passes within 250pc of the Galactic Center. Feature 2
highlights the dip in the percentage due to overshooting gas, and the inflow
efficiency is measured as the local minimum around this point. Feature 3
designates the re-accretion as gas begins to fall in toward the CMZ along the
opposite dust lane.

###  3.4 Constraining the Inflow Efficiency using Individual Tracer Histories

Lastly, we can constrain the inflow efficiency by considering how many times a
typical parcel of inflowing mass will overshoot before finally accreting onto
the CMZ. We can provide an upper limit on the inflow efficiency by measuring
the cumulative Galactocentric phase angle traveled by each tracer particle
within the sample of inflowing tracer particles before they each finally
accrete onto the CMZ. In this case, by “finally accretes”, we mean the time at
which each tracer arrives within 250 pc of the Galactic Center and stays
within that radius for the remaining simulated time.

To perform this experiment, we consider a sample of tracers from the near-side
dust lane. The sample used is identical to the blue sample in Figure 4. We
follow the evolution of these tracers and identify the time at which they are
“finally accreted”. The evolution of this tracer particle sample is shown in
Figure 7, in which the tracer particles are binned according to the simulation
time and their Galactocentric radius. To quantify the “final accretion” time
for each particle, we determine the simulation time at which the particle
enters a Galactocentric radius of 250 pc and does not leave that radius for
longer than 4 Myr. The window is chosen to be 4 Myr because >95% of the mass
in the dust lanes is moving with an inward velocity greater than 150 km s-1.
Moving constantly at that speed, it would take the tracers only $\sim 3$Myr to
cross the 500 pc diameter of the Galactic center. Employing this simple
method, we can make the generalizing assumption that overshooting tracers will
not be counted as accreted in this way.

Figure 7: The evolution of infalling tracer particles (the blue selection from
Figure 4) as a function of simulation time and Galactocentric radius. Many of
the tracer particles accrete onto the CMZ upon their first approach, while
others overshoot, traveling to larger Galactocentric radii before either
accreting or overshooting again. While the majority of the tracers are
accreted within the 30 Myr time span simulated, about $\sim$23% of the tracers
do not accrete within the time span.

With the final accretion times of each tracer in hand, we can measure the
number of “passes” by the the CMZ that each tracer takes to finally accrete.
We measure this by the cumulative Galactocentric phase angle transited by each
tracer particle. A histogram of the cumulative phase angle traveled before
accretion is shown in Figure 8. Of the initial sample of $2\times 10^{4}$
tracer particles, about 40% accrete on the first pass (meaning their accretion
phase angle is close to 0), then about 25% of the remaining tracers accrete on
the next pass. The number of tracers accreting on subsequent passes seems to
diminish, as might be expected.

However, $\sim$23% of the tracer particles do not accrete onto the CMZ within
the span of time simulated. Of the tracers that are not yet accreted,
virtually all remain in the bar region, and will be accreted at later times.
While this makes it difficult to measure the ”average“ inflow efficiency using
this method, it does allow us to place a lower limit on the number of passes,
constraining the inflow efficiency. It appears from Figure 8 that between two
and three overshooting events are typically required for the CMZ to accrete
the majority of the tracer particles sampled. Because fewer than half of the
tracer particles accrete on the first pass, the inflow efficiency cannot be
greater than 0.5.

If the tracer samples used in this experiment are characteristic of the dust
lanes more generally, then we would expect an inflow efficiency of
$\epsilon_{\text{DL}}\simeq 1/(\text{\\# passes})$. Since more than two passes
are required to accrete more than 75% of the tracer particles (including those
that do not accrete within the measured time), we know that the inflow
efficiency is less than $\epsilon_{\text{DL}}\simeq 0.5$. While this is not a
precise measurement of the efficiency, it does provides an intuitive upper
bound and an interpretation of the more quantitatively rigorous estimates of
$\epsilon_{\text{DL}}$ in Sections 3.2 and 3.3.

Figure 8: The distribution of final accretion phase angles for the sample of
2$\times 10^{4}$ tracer particles selected from the near-side dust lane. The
accretion phase angle is measured cumulatively from the the time at which the
tracers are selected until the time at which the tracer enters a
Galactocentric radius of 250 pc and stays within that radius for at least 4
Myr. About 40% of the tracers are accreted upon their first pass
(corresponding to a final accretion phase angle of $\sim 0$). Another
$\sim$25% of the total tracer sample is accreted upon the next pass, having
transited $\pi$ rad, and $\sim$13% more accrete again after the next pass,
having transited 2$\pi$ rad. Of the initial sample, $\sim$23% do not accrete
within the amount of time simulated, either escaping out further into the disk
or accreting at times past the limit of what was simulated.

###  3.5 Complexities of Measuring the Inflow Efficiency

The values for the inflow efficiency generated in the last three sections each
have their own advantages and disadvantages. The bulk inflow efficiency
derived in Section 3.2 is the simplest and most intuitive way of measuring
what fraction of mass in the dust lanes actually accretes as a function of
time, but averages away variations in the efficiency. The accretion phase
angles derived in Section 3.4 permit us to understand more clearly how the
overshooting gas re-accretes over time but is limited by the time window
simulated, which would have to be very long to capture the accretion of the
entire tracer sample. The instantaneous inflow efficiency derived in Section
3.3 provides a middle ground, where we can measure the inflow efficiency for a
large number of independent samples of tracer particles while observing their
evolution for the requisite span of time to derive $\epsilon_{\text{DL}}$. In
this section we address a couple of further complexities of the instantaneous
inflow efficiency.

For instance, there is a possibility that $\epsilon_{\rm DL}$ may vary if we
select smaller subsections of the dust lane. To address this, we determined
the value of $\epsilon_{\rm DL}$ as a function of the length of the tracer
selection region used. We found that the variations in $\epsilon_{\rm DL}$
resulting from changes to the size of the tracer selection region are small
compared to the overall difference between the two dust lanes and the overall
morphological variations of the dust lanes. In general, subdivided regions of
tracers behave close to the mean behavior of the larger region that contains
them. This may in part be due to the lack of self-gravity and supernova
feedback which act to disrupt and fragment the infalling material. Feedback
processes which result in clumpier streams of intermittent inflow might lead
to a more variable $\epsilon_{\rm DL}$ for individual parcels of gas within
the dust lanes, though the “smooth” dust lanes simulated in this work are able
to reproduce a total inflow rate into the CMZ consistent with simulations that
include star formation feedback and gas self-gravity (see Tress et al. 2020b).

Furthermore, the initial conditions of the gas in the disk and further out in
the dust lanes may have a significant effect on the inflow rate because they
determine the amount of gas available to flow along the dust lanes. Factors
such as spiral arm structure, merger history, and any other changes to the
gravitational potential may lead to different initial conditions for infalling
gas, and therefore greater variations in the inflow rate. Quantifying these
variations would require a dedicated experiment with variations to initial
conditions of the simulation, which is an effort that is beyond the scope of
this work.

As simulation time progresses, we observe changes in the effective impact
parameter of the gas flowing along the dust lanes relative the x2 orbits which
have a time-varying effect on the inflow efficiency measured. The dust lanes
appear to alternate between a state mostly intersecting the CMZ (with a
higher-than-average inflow efficiency) and a state mostly overshooting (with a
lower-than-average inflow efficiency). When the bar potential first matures
($t\approx 150$ Myr), most of the material in the dust lanes flows closer to
the CMZ, whereas after about 5-10 Myr the impact parameter of most of the gas
is larger and the majority of material overshoots. After another 10-15 Myr,
the dust lanes again mostly intersect the CMZ, with a smaller impact
parameter.

These observed variations are taken into account in the average value and
remain in agreement with the reported uncertainty of $\epsilon_{\rm DL}$. We
do not observe signatures of oscillations in the inflow efficiency on
timescales greater than those resolved within the timescale of this
simulation. While the instantaneous efficiency changes throughout the
simulation due to the varying morphology of the dust lane region, the reported
value for $\epsilon_{\rm DL}$ can be considered to be an average value for
correcting the bulk inflow rate of the entire dust lane region.

##  4 Following Gas on CMZ Orbits

###  4.1 Cloud Identification and Tracing

In the previous sections, we characterized how gas gets accreted onto the CMZ
within the simulated Galactic Center. We will now turn our attention to the
behavior of clouds on orbits within the CMZ. Giant molecular clouds are
hierarchical structures, and one’s method of determining meaningful boundaries
for such structures necessarily depends on the nature of the intended
analysis. Within the simulated CMZ presented in this work, high density
structures resembling molecular clouds form and evolve along their orbits of
the Galactic Center. Overdensities that form in these clouds do not have the
capacity to collapse past our imposed cell volume floor and they cannot form
stars. With no feedback processes to disrupt the ISM, the resulting clouds
have absolute masses and densities higher than would be expected of the
astrophysical ISM. However, the lack of gas self-gravity and feedback allows
us to directly investigate the isolated effects of the external potential on
the dynamical evolution of the clouds. To this end, we must define a scheme to
identify clouds and track their evolution in time.

We adopt a simple criterion for cloud identification: a volume density
boundary. Each corresponds to an isolated structure with a boundary satisfying
a density requirement. In this context, “isolated” only means that the cloud
is not subdivided into multiple separate components by the volume density
boundary within the spatial extent initially chosen. We identify five such
structures in the CMZ at a simulation time of 153 Myr using an initial volume
density threshold of $10^{5}$ cm-3. These clouds are lettered alphabetically
(A-E), and are visualized in Figure 9.

Figure 9: A total gas density slice through $z=0$ of the simulated CMZ. The
cloud structures discussed in Section 4 are highlighted in red as they are
initially selected. The clouds are lettered for comparison with Figures 10 and
14. The extent of the region illustrated here corresponds roughly to the
dotted circle in Figure 3.

The spatial extent of each of these clouds is somewhat sensitive to these
requirements. However, the mass-weighted mean cloud properties analyzed in
this work are dominated by their high density components which are largely
resilient to changes in this identification procedure (see Section 4.2).

At each timestep after tracer injection, we follow a simple procedure to
determine which cells should be associated with the cloud in the next time
step:

  1. 1.

Identify the unique IDs of tracer particles belonging to cells in the cloud in
the present timestep.

  2. 2.

In the next timestep, locate all of the cells who are parents of these
tracers. A tracer’s parent cell is the cell with which its position is
associated at a given timestep.

  3. 3.

Check which of these parent cells satisfy the minimum volume density condition
and a locality condition to prevent over-expansion of the cloud due to
outliers.

  4. 4.

Label the cells that satisfy these conditions as associated with the cloud and
record their properties.

  5. 5.

Repeat this process for the next timestep, and so on.

This procedure allows the cloud to grow, shrink and change at each timestep,
permitting new tracers to be advected into cloud-associated cells while other
tracers may migrate far enough from the cloud to be excluded from the
structure in future timesteps. The locality condition applied in the third
step serves to exclude outlier tracers that happen to be advected rapidly away
from the cloud association. These outliers are rare ($<0.1\%$ of the cloud’s
population) but if such a condition is neglected, or is too relaxed, then
these outlier tracers cause the cloud to “grow” rapidly and in a way that is
not representative of the bulk cloud motion. If the condition is too strict,
the cells on the exterior of the cloud may be excluded at each timestep and
the cloud may be forced to shrink at each timestep, which will lead to much
smaller structures than the initially identified clouds. We impose a condition
excluding all tracers hosted by cells with a radial distance (relative to the
center-of-mass of the cloud) more than 50% larger than the 2$\sigma$ radius
that contains 95% of the tracer particles. This allows the cloud to evolve,
shrink and grow, while discarding statistically insignificant outlier tracers
that would cause rapid over-expansion.

###  4.2 Properties of Clouds on x2 Orbits

By considering the properties of our identified clouds as they evolve along
their x2 orbits in the absence of self-gravity, star formation and stellar
feedback, we can glean some understanding of how CMZ clouds are affected by
the external potential. While we emphasize again that these simulated
structures may not consistently match the properties of observed Galactic
Center clouds due to the limited physics included in the simulation, the
observed dynamical effects ought to have a similar impact on real clouds
subject to a similar external potential.

In Figure 10 we present the evolution of several key properties for the five
simulated clouds as they progress along x2 orbits. The clouds’ densities seem
to have sharply peaked maxima, varying significantly on a $\sim$1-2 Myr
timescale. The Galactocentric radius of the clouds is plotted on the second
row of panels to highlight the apocenter and pericenter passage of the clouds,
which is discussed in more detail in Section 5.3.

The evolution of the total cloud masses is plotted in the third row, which
shows that most of the clouds do not undergo rapid or substantial changes to
their total mass as they orbit. Two of the clouds (D and E) seem to gradually
lose mass, as their outermost material is stripped away by the powerful shear
affecting clouds in the CMZ. This shear affects cloud E so severely that it
eventually falls below the density criteria and effectively dissolves. The
velocity dispersion of the clouds is calculated as the average standard
deviation of the cloud velocity, $\langle\sigma\rangle=\text{std}(v_{\rm
COM})$ where $v_{\rm COM}$ is the center of mass subtracted velocity of each
cell and the angle brackets denote a mass-weighted average over each cell in
the cloud. The evolution of this dispersion is plotted as a function of time
and orbital phase in the bottom row of Figure 10. While the velocity
dispersion of the clouds does vary, it does not seem to correlate with any of
the other orbital properties. However, one should bear in mind that the
dispersion would surely be strongly affected by the inclusion of gas self-
gravity, star formation and feedback physics, all of which are absent in our
simulations.

Figure 10: The evolution of cloud properties for the five traced clouds
highlighted in Figure 9. For each row, the left-hand panel shows the
quantity’s evolution as a function of time and the right-hand panel shows the
same quantity as a function of orbital position-angle. The four rows display
(from top to bottom) the cloud average number density, the Galactocentric
radius of the cloud, the total cloud mass and finally the average velocity
dispersion of the cloud. The apparent coincidence of density maxima with
apocenter and pericenter passage (apocenter near an orbital phase of $-3\pi/4$
and $\pi/4$, and pericenter near $3\pi/4$ and $-\pi/4$) is quantified and
analyzed in Section 5

###  4.3 Cloud Angular Momenta

The clouds identified in Section 4.1 appear to have varying degrees of spin
angular momentum, many showing signatures of counter-rotation with respect to
the orbital angular momentum, while some appear to have short-lived co-
rotation. To quantify this, we compare the significance of the cloud
rotational motion to the velocity dispersion of the cloud using the following
dimensionless ratio:

$\displaystyle\frac{\Omega_{\rm eff}R}{\sigma}=\frac{1}{M_{\rm
tot}R\sigma}\sum_{i}[m_{i}\vec{v}_{i}\times\vec{r}_{\rm i}]_{z}\,,$ (7)

where $\Omega_{\rm eff}$ is an effective rotational velocity, $R$ is the
cloud’s effective radius, calculated as $R=\sqrt{\frac{1}{M_{\rm
tot}}\sum_{i}m_{i}r_{i}^{2}}$ with $m_{i}$ and $r_{i}$ being the mass and
center-of-mass radius of each cell composing the cloud respectively, $\sigma$
is the cloud’s mean velocity dispersion, $\vec{v}_{i}$ is the velocity of the
$i^{\rm th}$ particle in the cloud relative to the cloud center-of-mass rest
frame, $\vec{r}_{i}$ is the $i^{\rm th}$ cell’s position relative to the cloud
center-of-mass, $z$ indicates the out-of-Galactic-plane axis, and $M_{\rm
tot}$ is the summed mass of all cells composing the cloud. The velocity
dispersion of each cloud, $\sigma$ is calculated as the Pythagorean sum of the
standard deviations for the $x$, $y$, and $z$ components of the velocities in
the center-of-mass rest frame,
$\sigma=\sqrt{\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}}$. A cloud that has
$\text{abs}(\Omega_{\rm eff}R/\sigma)>1$ displays ordered rotation which is
more significant than the random motions within the cloud. Given the clockwise
orbit for these clouds, a negative value for this ratio designates co-rotation
and a positive value corresponds to counter-rotation.

We study the distribution of cloud angular momenta for a sample of 44 clouds
isolated independently at different timesteps throughout the simulation. The
results are presented in Figure 11. These clouds are selected using the same
initial selection criteria used to identify the clouds in Section 4.1, but are
not followed in time because the material at larger center-of-mass radii in
each cloud contributes significantly to the overall angular momentum, and thus
the resulting angular momenta are sensitive to the radial cutoff that is used
to follow the clouds in time. Instead, we re-select the clouds at later times
but do not follow the angular momentum evolution of each cloud individually.

The majority of these clouds display significant counter-rotation, although a
minority display co-rotation. Only 3 out of 44 clouds ($\sim 7\%$) show
significant co-rotation, whereas 23 out of 44 ($\sim 52\%$) are significantly
counter-rotating. The remainder have insignificant spin angular momentum.
There is evidence for examples of counter-rotating clouds in the Milky Way’s
CMZ (e.g. Federrath et al., 2016), but the spin angular momentum properties of
such clouds are difficult to disentangle given their complex kinematic
environment. Counter-rotation is a common feature observed in other recent
simulations of the Galactic Center clouds (Dale et al., 2019; Kruijssen et
al., 2019).

Figure 11: A histogram of the significance of rotation, $\Omega_{\rm
eff}R/\sigma$, for 44 clouds identified at different times throughout the
simulation. The significance is calculated as the ratio of the effective
rotational velocity of the cloud to its velocity dispersion, described in
Section 5.4. There are a greater number of counter-rotating clouds ($\sim
52\%$ of the sample), which is in agreement with previous studies of cloud
rotation in the CMZ. However, significantly co-rotating clouds do occur,
albeit much less frequently ($\sim 7\%$ of the sample).

###  4.4 Tracing the Origin of CMZ Clouds

We also consider the origin of the mass composing a particular cloud, which is
isolated using the same methods described in Section 4.1 but followed
backwards in time. By flagging all of the tracers associated with this cloud
and identifying the parent cells of those tracers at earlier time steps, we
can track the origin and evolution of the distribution of mass which
eventually composes the cloud.

Figure 12 shows the evolution of the tracers that will eventually constitute
this cloud, binned as a function of time and Galactocentric radius (RGC). The
cloud is selected from the snapshot corresponding to the right-most column of
pixels at $t\approx 178$ Myr. At this time, the cloud is tightly localized at
R${}_{\rm GC}\approx 100$ pc, occupying a typical x2 orbit. There is a
concentration of tracer particles that primarily constitutes the cloud,
remaining on its orbit for the duration of the simulation, while occasionally
accreting the material streaming in from the dust lane. These accretion events
are visible as the long arcs of tracer-density arriving from large values of
RGC, labeled as feature 1. About a tenth of the cloud’s future mass originates
outside the CMZ just $10$ Myr before the cloud is selected, meaning a minimum
$\sim$10% of its tracers originate at R${}_{\rm GC}>250$ pc. Because the
clouds produced in this simulation are more massive than their astrophysical
counterparts, this fraction of mass amounts to $\sim 2\times 10^{6}$M⊙ for
this example cloud and should be thought of as a lower limit percentage for
more realistic molecular clouds in the CMZ.

Figure 12: The origin and evolution of mass tracers for a cloud identified
late in the simulation, at $t\approx 178$ Myr. The intensity corresponds to
the number of tracer particles that end up in the selected cloud as a function
of time and Galactocentric radius. This distribution shows how a concentrated
seed of the cloud persists within the CMZ (the high number of tracers
oscillating around 100 pc in Galactocentric radius), while gradually accreting
gas as it falls in along the dust lanes. A minimum of $\sim$10% of the mass of
the cloud originated outside the CMZ within the relevant timescale. Several
features are numbered for comparison with their counterparts in other figures.
Feature 1 indicates one of many instances of inflowing material that is
destined to become part of the cloud’s mass. In total, these inflows
contribute $\sim 2\times 10^{6}$M⊙ to the cloud seed’s mass over 10 Myr.
Feature 2 highlights a portion of this inflowing material that overshoots,
“bouncing” back to higher Galactocentric radii before merging with the cloud.
Feature 3 indicates the re-accretion of the overshooting material as it
combines with the cloud seed.

##  5 Results and Discussion

###  5.1 The Efficiency of Bar Driven Inflow

A natural application for our estimate of the inflow efficiency is to
calculate the mass accretion rate of the Milky Way’s CMZ using observational
estimates of the amount of mass within the Milky Way’s dust lanes. Of the
three methods that we have used to quantify the inflow efficiency, the
instantaneous inflow efficiency calculated in Section 3.3 best captures its
variability as a function of time and dust lane morphology. Since all three
methods agree within one standard deviation, we quote this instantaneous
inflow efficiency for the purposes of correcting the observationally derived
inflow rate from Sormani & Barnes (2019). Using this value of
$\langle\epsilon_{\rm DL}\rangle$ we calculate the adjusted CMZ mass inflow
rate as

$\dot{M}_{\rm CMZ}=\langle\epsilon_{\rm DL}\rangle\dot{M}_{\rm DL}$ (8)

where $\dot{M}_{\rm DL}$ is the total mass inflow rate under the assumption
that the CMZ accretes all inflowing mass from the dust lanes, and
$\langle\epsilon_{\rm DL}\rangle$ is the average dust lane accretion
efficiency calculated in Section 3.3. We apply the observationally determined
mass inflow rate of $\dot{M}_{\rm DL}=2.7^{+1.5}_{-1.7}$ calculated by Sormani
& Barnes (2019), which we use to find an adjusted CMZ mass inflow rate of
$\dot{\text{M}}_{\rm CMZ}=0.8\pm 0.6$ M⊙yr-1. The errors in this value include
uncertainties on the position-angle of the bar relative to the sun’s position,
the XCO factor used to determine the mass content of the dust lanes, and the
variability measured in this work for the dust lane inflow efficiency. The
instantaneous value in the inflow efficiency for any given parcel of dust lane
gas will be influenced by the initial conditions of inflowing gas, but the
resulting average inflow rate is not strongly dependent on the inclusion of
supernovae feedback or gas self-gravity (Tress et al., 2020b).

This CMZ mass inflow measurement solely accounts for the inflow rate from the
dust lanes, and does not account for the loss of mass from supernova-driven
outflows, which may drive winds out of the disk in the absence of AGN feedback
(e.g. Crocker, 2012; Crocker et al., 2015; Armillotta et al., 2019). We also
do not account for the effects of the supermassive black hole Sgr A* (e.g. Su
et al., 2010; Wardle & Yusef-Zadeh, 2014), which may have, in the past, been a
key mechanism for driving outflows from the Galactic Center.

Many models of the CMZ and its star formation rate depend upon the inflow rate
(Longmore et al., 2013b; Kruijssen & Longmore, 2014), and in turn these models
inform our understanding of how the Milky Way’s stellar bulge evolves
(Kormendy & Kennicutt, 2004). The value calculated in this work is consistent
with, but lower than previous measurements of the inflow rate, which was
reported as $\dot{M}=1-3$ M⊙yr-1 by Armillotta et al. (2019) and
$\dot{M}\simeq 1$ M⊙yr-1 by Tress et al. (2020b). Both of these previous
measurements were performed using simulations including gas self-gravity, star
formation, and feedback, suggesting that these processes do not affect the
inflow efficiency considerably. The simulation used in this work to derive the
inflow efficiency was tuned to replicate a Milky Way-like CMZ, but the
prevalence of dust lanes observable in other barred spiral galaxies suggests
that a similar efficiency might be applicable to other galaxies with similar
morphology. A generalized understanding of how the inflow efficiency changes
as a function of the parameters of the gravitational potential could be
explored in future work, but lies beyond the scope of this paper.

###  5.2 Accreted Gas is Well Distributed Across the CMZ

One of the consequences of the extreme shearing that occurs in the dust lanes
and the partial overshooting of inflowing material is that a single parcel of
gas in the dust lane will not be neatly deposited into an x2 orbit as it
accretes, nor will it be neatly integrated into a single existing CMZ cloud
structure. Instead, the infalling material is well distributed across the CMZ,
both due to shearing in the dust lane and the overshooting phenomenon. These
effects can be seen in Figure 5, where the tracer distribution becomes
increasingly elongated and eventually fragments during its accretion.

Figure 13: The binned distribution of infalling material as a function of
Galactocentric phase angle and time. The initial parcel of tracers is tightly
packed in phase angle when it is selected at a simulation time of 150 Myr
(labeled as feature 1), but is rapidly dispersed over the next 5 Myr. The
portion of overshooting gas, labeled as feature 2, appears as the portion of
tracers which remain at a constant position-angle of about 0 for the $\sim$15
Myr before re-accretion on the far-side of the CMZ. The re-accretion is
labeled as feature 3.

We quantify the distributed accretion of gas by measuring the phase angle of
tracers from a single parcel of gas in the dust lane. We select samples of
tracers identically to criteria described in Section 3.3, and follow their
motion. In Figure 13, we show the time-evolving histogram of the tracer
distribution as a function of Galactocentric phase angle as a proxy for how
well the tracer particles have spread throughout the CMZ. The tracers
initially occupy a narrow range of phase space, as the dust lane itself covers
only a small position-angle. Upon approach, the distribution of tracers
spreads out as the gas is sheared apart, entering the CMZ over an extended
time. Star-forming clouds in an environment fed by such a mechanism may be
continually fueled by this sheared out, accreting gas on timescales similar to
the timescales on which they collapse and form stars ($\sim$ a few Myr).
Additionally, the material that overshoots is largely re-accreted within the
next 10-15 Myr. This infalling material is once again sheared apart upon its
approach along the opposite dust lane. By the time it has been fully accreted,
it is well distributed in position-angle. This re-accretion event is labeled
with a (3) in Figures 6, 12, and 13.

This dynamically driven mechanism for mixing dust lane gas into the CMZ may
influence the formation and evolution of molecular clouds, particularly the
metallicity distribution of the Galactic Center. If individual structures in
the dust lanes are sheared apart and accreted gradually over a large position-
angle, then we would expect to see a CMZ that is continually fueled by
material with a disk-like metallicity. This may influence the interpretation
of the chemical composition and evolution of CMZ clouds (e.g. Tanaka et al.,
2011).

###  5.3 Apocenter Passage Drives Cloud Density Maxima

Using the procedure discussed in Section 4.1, we are able to track the
evolution of the long-lived clouds on CMZ x2 orbits and investigate how their
properties are affected by the dynamics of their environment. The density
evolution of these clouds in Figure 10 displays sharply peaked oscillations in
the average cloud density on a timescale similar to the orbital period. The
density increases do not consistently occur at the same position in orbital
phase, but occur more frequently at the apocenter of the cloud’s orbit. Figure
14 shows the concentration of density maxima for each of the five clouds’
evolution binned according to their orbital phase angle. A higher
concentration of these significant peaks occur while the cloud is near
apocenter, and a smaller number occur after pericenter. Many of these density
fluctuations are relatively small. If we consider only the maxima that peak
50% higher than the time-averaged cloud density (the blue histogram in Figure
14), then this pattern becomes more tightly focused on the orbit apocenter.
The correlation of density maxima with orbital position is resilient to
changes in the parameters of our cloud-following algorithm (e.g., as the cloud
density criteria and the radial cutoff), as well as changes in the time
resolution used between tracked snapshots. While the importance of pericenter
passage and the density fluctuations that may be driven by that component of
the orbit have been previously studied in the context of inducing star
formation (Longmore et al., 2013a; Kruijssen et al., 2015, 2019; Dale et al.,
2019; Jeffreson et al., 2018), we find that apocenter passage plays a more
significant role in pushing gas to higher densities and can potentially
trigger gravitational collapse.

Other properties of these clouds do not oscillate similarly. The total mass
for each of the five clouds does not exhibit sharply peaked local maxima,
instead they either remain relatively constant (e.g. cloud A in Figure 10) or
they gradually change (e.g. clouds D and E in the same figure). The gradual
decrease in mass observed in clouds D and E is due to the powerful shearing
out of the clouds over time, while the few more sudden changes are due to
cloud-cloud interactions and collisions with inflowing material. These effects
are not sufficient to destroy the most of the cloud structures, with the
exception of cloud E which dissipates due to shear over about 20 Myr. For more
massive clouds, it appears that dynamics alone do not dominate cloud lifetimes
in the CMZ. This result is in good agreement with recent analytical studies of
cloud life cycles in this environment by Jeffreson & Kruijssen (2018) and
Jeffreson et al. (2018), who find that gravitational collapse and star
formation should be the dominant processes controlling cloud lifetimes in the
CMZ.

These density maxima are likely to be the result of multiple mechanisms.
Firstly, as gas slows down near the apocenter, the clouds are compressed as in
a traffic jam. However, if this were a uniform effect that occurs at each
apocenter passage, then one should expect fluctuations in the density to occur
consistently and have a similar magnitude at each apocenter passage. Because
these average density peaks do not occur uniformly at each apocenter passage,
this compression mechanism is insufficient to fully explain the observed
behavior. A second mechanism driving these density fluctuations is that
collisions with inflowing material tend to occur near the orbit apocenter. Gas
accreting onto the CMZ displays inwards radial velocities ranging between 150
and 300 km s-1. Generally, cloud-cloud collisions are capable of driving
molecular clouds to high densities, and trigger gravitational collapse and
efficient star formation, particularly in tumultuous environments such as the
Galactic Center (e.g Tsuboi et al., 2015; Uehara et al., 2019). The inflowing
material is clumpy, and appears more clumpy in simulations that include self-
gravity (Sormani et al., 2020; Tress et al., 2020b), which means that not
every apocenter passage will guarantee a significant collision. This may help
to explain the non-uniformity of the density fluctuations seen in Figure 10.

Figure 14: Upper panel: the normalized histogram of local maxima in average
cloud density as a function of orbital position-angle. Bottom panel: The phase
angle evolution of Galactocentric radius for the five sample clouds identified
in Section 4. The distribution of the entire sample of density maxima (the red
histogram) peak at an orbital position-angle corresponding to apocenter
passage. This pattern is more pronounced if we only consider the local density
maxima that exceed 50% more than the mean cloud density (the blue histogram).
The vertical dashed red lines indicate the position of typical apocenter
passage in both panels.

###  5.4 Counter-rotation and co-rotation of CMZ clouds

In Section 4.3, we find that a large portion of clouds in our simulation have
significant angular momenta, most of them counter-rotating relative to the
clockwise orbital angular momentum of the CMZ’s x2 orbits ($\sim 52\%$
counter-rotating, $\sim 7\%$ co-rotating, the rest with insignificant
rotation). The rotation of molecular clouds (e.g. Phillips, 1999; Imara &
Blitz, 2011), and more specifically the counter-rotation of clouds on CMZ
orbits, is an expected result. This rotation is understood to be largely
generated by the strong shear acting on clouds in this gravitational
potential. However, this shear exclusively drives counter-rotation. Since the
simulation does not include the effects of star formation and feedback that
may play a role in disrupting cloud angular momenta in the astrophysical ISM,
some other mechanism must be responsible for altering the angular momenta of
these clouds to induce the infrequent co-rotation that we observe.

Each of the three most significant examples of co-rotating clouds (those with
$|\Sigma_{\rm eff}|R/\sigma$>1) occur in the quarter orbit past apocenter
passage, after a collision with inflowing material from the dust lanes. As
inflowing gas from the dust lanes collides violently with gas at the apocenter
of x2 orbits, large quantities of angular momentum are transferred into CMZ
clouds, generating short-lived co-rotation. The shear induced by the
gravitational potential rapidly counteracts this short-lived co-rotation, and
within a half-orbit the cloud displays either insignificant spin angular
momentum or counter-rotation222Angular momentum is not a conserved quantity of
a particle moving in a non-axisymmetric barred potential. Ultimately, all of
the excess angular momentum of the gas is absorbed by the Galactic bar..
Furthermore, the velocity gradients of these clouds might be significantly
disrupted by physics not included in this work, such as the shocks from nearby
supernovae or magnetic fields.

Some observed clouds in the CMZ show signs of counter-rotation, such as the
rotation indicated by the velocity gradient of “The Brick” (Rathborne et al.,
2014; Federrath et al., 2016; Kruijssen et al., 2019). The spin angular
momenta for most CMZ clouds are not well constrained. It is possible that co-
rotating clouds, resulting from inflow collisions, occur in the Milky Way’s
Galactic Center.

###  5.5 The Bar Driven Feeding of CMZ Clouds

While the shocks from supernovae and other stellar feedback effects are likely
to influence the mixing of material within the CMZ, there is also a
significant amount of mixing driven by dynamical forces. The strong shearing
that is present for material on CMZ orbits and the interaction with the
inflowing material on the dust lane shocks that are described in Section 4.4
lead to an environment where molecular clouds are constantly mixing with the
surrounding gas, even in the absence of mixing effects from supernovae, due to
the inflow from the dust lanes, which is in turn driven by the Galactic bar.

The time evolution of the tracers in Figure 12 suggests a continually
accreting cloud seed formed from the interaction of multiple CMZ clouds, as
well as multiple distinct inflow events over a period of time of $\sim 3-4$
Myr, which is slightly longer than the expected collapse and star formation
timescale suggested by Jeffreson et al. 2018. If star formation in CMZ clouds
are expected to collapse and form stars on timescales of $\sim$1-4 Myr, then
the inflow driven by the bar will supply $\sim 0.8-5\times 10^{5}$M⊙ of non-
CMZ material to the clouds in the CMZ within that timescale. These accretion
events occur near the apocenter of the cloud’s orbit and may contribute fresh
gas to sites of ongoing star formation, where winds from young stars may
already have blown away the natal gas cloud.

##  6 Summary

We have performed hydrodynamic simulations of the Milky Way’s CMZ with the aim
of isolating the effects of the Milky Way’s potential on gas inflow and
properties of molecular clouds in this environment. We re-simulate the models
of Sormani et al. (2019), using a barred Milky Way’s external potential with
the addition of MCTP which advect between arepo’s Voronoi cells proportional
to the flow of mass, permitting us to quantitatively follow mass flows. Our
main findings are:

  * •

We determine the inflow efficiency for gas accreting onto the CMZ from the bar
dust lanes. This efficiency is found to be 30$\pm$12%, and varies with
fluctuations in the density and morphology of the dust lanes. We calculate the
efficiency using three methods, all of which agree within the quoted
variability. Using this, we calculate an adjusted value for the CMZ’s observed
inflow rate, originally calculated by Sormani & Barnes (2019), and find that
$\dot{M}_{\rm cmz}=0.8\pm 0.6$ M⊙ yr-1.

  * •

By following the evolution of clouds in the absence of self-gravity and star
formation physics, we find that the barred potential drives sharp peaks in the
average cloud density for CMZ clouds on x2 orbits. These sudden changes to
cloud density occur most frequently as the clouds pass their orbit’s
apocenter. This may be due to a combination of the orbital properties, as well
as collisions with inflowing gas from the dust lanes, which intersect the x2
orbits at their apocenter.

  * •

In the absence of self-gravity and star formation physics, clouds in the
Galactic Center have significant spin angular momentum oriented such that they
counter-rotate. This effect has been noted previously and is attributed to the
strong shear acting on CMZ clouds. However, we note the presence of short-
lived co-rotation in a small subset of clouds (only three co-rotating clouds
in a 44 cloud sample, or $\sim 7\%$). This co-rotation seem to be driven by
high-velocity collisions with inflowing material because these rare co-
rotating clouds occur within the quarter-orbit after apocenter passage.

  * •

We find that, due to both the partial overshooting and the powerful shear
within the dust lane, the inflowing material is rapidly distributed across the
entire CMZ within $\sim$4 Myr. This same effect also informs the origin of the
material constituting clouds in the Galactic Center, which are constantly
exchanging mass with their surroundings and have a steady diet of fresh gas
from the dust lanes. Using our determined inflow rate, we find that CMZ clouds
are supplied $\sim 0.8-5.5\times 10^{6}$M⊙ of gas from the dust lanes within a
4 Myr timescale. This could help to inform the interpretations of metallicity
distribution evolution of gas and stars in the CMZ, and suggests that stellar
nurseries could efficiently gain fuel for further star formation during and
well after the initial onset of gravitational collapse.

These results are observed in a simulated ISM that isolates the effects of
dynamical sources from the effects of self-gravity, star formation, and star
formation feedback. The observed effects are largely driven by the external
barred gravitational potential and by violent cloud-cloud collisions that
occur as high-velocity gas from the dust lanes crashes into dense clouds in
the CMZ near the apocenter of their orbits. These collisions may greatly
impact the life cycle and evolution of molecular clouds in the CMZ.

We thank the anonymous referee for their helpful and constructive feedback
that has improved this work. H.P.H. gratefully acknowledges support from the
SOFIA Archival Research Program (program ID 09_0540). H P.H. also thanks the
LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF
Cybertraining grant #1829740, the Brinson Foundation, and the Moore
Foundation; his participation in the program has benefited this work. C.B. and
H.P.H. gratefully acknowledge support from the National Science Foundation
under Award No. 1816715. M.C. S., R.G.T., S.C.O.G., and R.S.K. acknowledge
support from the German Research Foundation (DFG) via the collaborative
research center (SFB 881, Project-ID 138713538) the Milky Way System
(subprojects A1, B1, B2, and B8), from the Heidelberg Cluster of Excellence
“STRUCTURES” in the framework of Germany’s Excellence Strategy (grant
EXC2181/1, Project-ID 390900948), and from the ERC via the ERC Synergy Grant
“ECOGAL” (grant 855130). The project made use of computing resources provided
by the state of Baden-Württemberg through bwHPC and the German Research
Foundation (DFG) through grant INST 35/1134-1 FUGG. Data are stored at SDS@hd
supported by the Ministry of Science, Research and the Arts Baden-Württemberg
(MWK) and the German Research Foundation (DFG) through grant INST 35/1314-1
FUGG. R.J.S. gratefully acknowledges an STFC Ernest Rutherford fellowship
(grant ST/N00485X/1) and H.P.C. from the Durham DiRAC supercomputing facility
(grants ST/P002293/1, ST/R002371/1, ST/S002502/1, and ST/R000832/1). This
research made use of astropy, a community-developed core Python package for
astronomy (Astropy Collaboration et al., 2013, 2018) and NumPy (Van Der Walt
et al., 2011).

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