# Rapid dust growth during hydrodynamic clumping due to streaming instability

Ryosuke T. Tominaga RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako,
Saitama 351-0198, Japan  Hidekazu Tanaka Astronomical Institute, Graduate
School of Science, Tohoku University, 6-3, Aramaki, Aoba-ku, Sendai 980-8578,
Japan

(Received –; Accepted –)

###### Abstract

Streaming instability is considered to be one of the dominant processes to
promote planetesimal formation by gravitational collapse of dust clumps. The
development of streaming instability is expected to form dust clumps in which
the local dust density is strongly enhanced and even greater than the Roche
density. The resulting clumps can collapse to form planetesimals. Recent
simulations conducted long-term simulations and showed that such strong
clumping occurs in a wider parameter space than previously expected. However,
the indicated timescale for strong clumping can be on the order of tens to
hundreds Keplerian periods. In this paper, we estimate the growth time of dust
grains during the pre-clumping phase. We find that the dust growth
considerably proceeds before the strong clumping because even the moderate
clumping due to streaming instability increases the local dust-to-gas ratio
$\gtrsim 10$. Depending on the gas sound speed, the dust collision velocity
can be kept below $\sim 1\;\mathrm{m/s}$ once sufficiently strong dust
clumping occurs. Thus, even silicate grains might have the potential to grow
safely toward the size whose Stokes number is unity during the clumping. Our
results demonstrate the importance of local dust coagulation during the dust
clumping due to streaming instability.

hydrodynamics — instabilities — protoplanetary disks

##  1 Introduction

There are two promising processes for planetesimal formation: direct
collisional growth of dust grains (e.g., Ormel et al., 2007; Okuzumi et al.,
2012; Kataoka et al., 2013; Krijt et al., 2016; Arakawa & Nakamoto, 2016;
Garcia & Gonzalez, 2020; Kobayashi & Tanaka, 2021) and gravitational collapse
of a dust layer (e.g., Safronov, 1972; Goldreich & Ward, 1973; Sekiya, 1983;
Youdin & Shu, 2002). The latter process is expected to occur once a sufficient
amount of dust is concentrated against turbulent diffusion in a disk (e.g.,
Cuzzi et al., 1993; Sekiya, 1998). Some dust-driven instabilities have been
proposed as the dust concentration mechanism (e.g., Youdin & Goodman, 2005;
Johansen & Youdin, 2007; Youdin, 2011; Takahashi & Inutsuka, 2014; Tominaga et
al., 2021, 2022a, 2022b). Among the dust-driven instabilities, streaming
instability has been extensively studied analytically and numerically (e.g.,
Youdin & Goodman, 2005; Youdin & Johansen, 2007; Johansen & Youdin, 2007;
Johansen et al., 2007; Krapp et al., 2019; Chen & Lin, 2020; Umurhan et al.,
2020; Paardekooper et al., 2020, 2021; McNally et al., 2021; Zhu & Yang, 2021;
Yang & Zhu, 2021; Carrera et al., 2021, 2022). Streaming instability occurs on
a spatial scale much smaller than the gas scale height. The nonlinear
evolution causes turbulent motion and transient/intermittent formation of
azimuthally elongated filamentary structures (e.g., Johansen & Youdin, 2007).
The resulting dust-dense regions collapse self-gravitationally once the local
dust density exceeds the Roche density $\rho_{\mathrm{R}}$ (e.g., Johansen et
al., 2007; Simon et al., 2016) and the self-gravity overcomes turbulent
diffusion (Gerbig et al., 2020; Klahr & Schreiber, 2020, 2021). This combined
process of streaming instability and the subsequent self-gravitational
collapse is expected to explain planetesimal formation (e.g., Dra̧żkowska et
al., 2016; Nesvorný et al., 2019; Gole et al., 2020; Gerbig et al., 2020).

Previous studies investigated the condition for streaming instability to cause
strong dust clumping where the maximum dust density
$\rho_{\mathrm{d},\mathrm{max}}$ exceeds the Roche density (e.g., $\sim$
several hundred times gas density for low-mass disks)111 The Roche density is
$\rho_{\mathrm{R}}\equiv 9\Omega^{2}/4\pi G\simeq 3\times
10^{2}\rho_{\mathrm{g}}\times(Q/55)$, where $\Omega$ is the Keplerian angular
velocity, $G$ is the gravitational constant, $\rho_{\mathrm{g}}$ is the
midplane gas density, and $Q$ is the Toomre’s $Q$ value of the gas disk. ,
which is necessary for the self-gravitational collapse of the dust clump
(Carrera et al., 2015; Yang et al., 2017; Li & Youdin, 2021). Carrera et al.
(2015) and Yang et al. (2017) found that the dust-to-gas surface density ratio
should be a few times higher than 0.01 for the strong clumping, which also
depends on the Stokes number of dust, $\mathrm{St}$. Gerbig et al. (2020)
discussed that the critical value for the gravitational collapse to proceed
also depends on the gas pressure gradient and the Toomre’s $Q$ value of the
gas disk. Recently, Li & Youdin (2021) revisited the condition by performing
high-resolution long-term simulations. They found that the strong clumping
occurs even for lower dust-to-gas ratios during the time evolution over ten to
hundreds Keplerian periods (see their Tables 1 and 2). The critical dust-to-
gas ratio at the midplane is about $\simeq 0.35-2.5$ in their simulations for
$10^{-3}\lesssim\mathrm{St}\lesssim 1$ (see their Figure 4). This may indicate
that streaming instability and subsequent clumping operate more easily than
previously thought, leading to planetesimal formation.

In such a long-term evolution, dust coagulation may be more important. The
coagulation timescale is only a few tens of Keplerian periods for the dust-to-
gas surface density ratio of 0.01 (e.g., Brauer et al., 2008), which can be
shorter than the time required for the strong clumping to occur. Besides, the
coagulation may be effective even in moderately high density regions around
much denser clumps. However, a possible combined process of coagulation and
streaming instability is not well studied. We note that there are studies that
investigated how disk evolution with coagulation produces a region where
streaming instability operates (e.g., Dra̧żkowska et al., 2016; Carrera et
al., 2017). What they considered is that dust coagulation and streaming
instability occur one after the other: the dust sizes are determined by dust
coagulation through the global disk evolution and are independent from the
onset of streaming instability and the resulting local turbulence/clumping. In
contrast to these studies, we investigate the possible impact of dust growth
during the local process of streaming instability, which could change the
planetesimal formation efficiency assumed in the previous models. In this
paper, we demonstrate that dust coagulation considerably proceeds in
moderately dense regions or clumps before the strong clumping.

We consider dust growth toward the size of $\mathrm{St}=1$. The collision
velocity provides the insight into the growth efficiency. When dust grains are
so large that the Brownian motion is insignificant, the collision velocity is
determined by the drift motion and the turbulent motion. The drift-induced
collision velocity of similar-sized dust is on the order of $\mathrm{St}\eta
v_{\mathrm{K}}$, where $v_{\mathrm{K}}$ is the Keplerian velocity,
$\eta\sim(c_{\mathrm{s}}/v_{\mathrm{K}})^{2}$ is a measure of the radial
pressure gradient, and $c_{\mathrm{s}}$ is the gas sound speed (e.g.,
Weidenschilling, 1977). For $\mathrm{St}\sim 1$, this velocity is $\sim
54\;\mathrm{m/s}$ in the minimum mass solar nebula (Hayashi, 1981) and lower
in a disk that is optically thick to the stellar irradiation (e.g., Chiang &
Goldreich, 1997). The turbulence-induced collision velocity depends on
$\mathrm{St}$ and the turbulence strength $\alpha$ (Shakura & Sunyaev, 1973).
When dust grains are large enough, the velocity is on the order of
$\sqrt{\alpha\mathrm{St}}c_{\mathrm{s}}$ for similar-sized grains (Ormel &
Cuzzi, 2007). The collision velocity is thus $\sim 10-30\;\mathrm{m/s}$ for
$\alpha=10^{-3}$, $\mathrm{St}=0.1-1$, and $c_{\mathrm{s}}=1\;\mathrm{km/s}$.
It has been suggested that such high velocity collisions can lead to
fragmentation that limits the dust growth (e.g., Weidenschilling & Cuzzi,
1993; Blum & Wurm, 2008). Even when the fragmentation is less efficient, the
radial drift can limit the dust growth: the drift time is so short that dust
grains reach the central star before they grow in size significantly. Before
streaming instability operates, we naively expect that dust grains grow to the
fragmentation- or drift-limited sizes. We thus address whether or not the
clumping due to streaming instability can assist further dust growth.

An advantage of dust coagulation in the clumping regions is lower collision
velocity (Johansen et al., 2009; Bai & Stone, 2010; Schreiber & Klahr, 2018).
Schreiber & Klahr (2018) showed that the local velocity dispersion of the same
sized dust grains can be on the order of $10^{-3}c_{\mathrm{s}}$ or much lower
in the clumping regions for $\mathrm{St}=0.1$ (e.g., see Figure 11 therein).
Bai & Stone (2010) also showed low collision velocities in the simulations
where dust clumping is efficient (their R10Z3 run, see Figures 6 and 10
therein). It is also indicated that large-scale turbulence is ineffective on
clump scales once dust grains are locally concentrated (Klahr & Schreiber,
2020). This means that, even if the dust growth is limited by fragmentation
before the clumping, dense regions shielded from the large-scale turbulence
enable further dust growth owing to the lower-speed collisions. Besides, the
clumping leads to the reduced drift velocity (see Figures 7 and 8 in Bai &
Stone, 2010)222 Bai & Stone (2010) also found that the multiple-species effect
reduces the drift velocity of each dust species (see solid and dashed (dash-
dotted) lines in their Figure 8 for 2D (3D) simulations). We focus on the
effect of high dust densities due to clumping (see the panels of R10Z1 and
R10Z3 runs as well as Figure 5 therein). , which is due to the aerodynamical
backreaction from dust to gas (Nakagawa et al., 1986). Thus, such dense
regions are also preferable for dust grains to overcome the drift barrier.

This paper is organized as follows. We describe our model in Sections 2 and 3.
In Section 2, we describe the duration time before the strong clumping via
streaming instability (a pre-clumping period) and the coagulation timescale.
In Section 3, we describe models for dust collision velocities in clumping
regions, which is based on the previous simulations of streaming instability.
We then compare the coagulation timescale and the pre-clumping period in
Section 4. We show that coagulation proceeds faster than the clumping. In
Section 5, we give conclusions and brief discussion.

##  2 Timescale Model

Figure 1: Duration time of the pre-clumping phase reported in Li & Youdin
(2021). They define the pre-clumping phase using (1) the time at which the
first sedimentation phase ends ($t_{\mathrm{sedi-tr}}$) and (2) the time at
which the dust density exceeds $2\rho_{\mathrm{R}}/3$ ($t_{\mathrm{pre-cl}}$).
These values are collected from their Table 2. The difference $t_{\mathrm{pre-
cl}}-t_{\mathrm{sedi-tr}}$ that we plot is the pre-clumping period. We refer
to the simulations that adopt $\mathrm{St}\geq 10^{-2}$ and show
$t_{\mathrm{pre-cl}}>t_{\mathrm{sedi-tr}}$. Color of filled circles represents
$\mathrm{St}$ adopted in their simulations (see Table 1 in Li & Youdin, 2021).
In this figure, the pre-clumping periods are given by
$t_{\mathrm{sim}}-t_{\mathrm{sedi-tr}}$ for weak/moderate clumping cases (see
text for details, and see also Table 1 in Li & Youdin (2021)). Li & Youdin
(2021) also conducted simulations with $\mathrm{St}=10^{-3}$ (their Z4t0.1,
Z3t0.1, and Z2t0.1 runs) and show the pre-clumping periods $\gtrsim
475\Omega^{-1}$ (see their Tables 1 and 2), which is longer than
$\tau_{\mathrm{SI,sat}}\Omega^{-1}$ in our fiducial model.

###  2.1 Pre-clumping periods

We consider a situation where streaming instability has developed into the
nonlinear phase and the dust clumping gradually and/or intermittently
increases the local dust density as observed in numerical simulations (e.g.,
Johansen & Youdin, 2007; Johansen et al., 2007; Bai & Stone, 2010; Yang &
Johansen, 2014; Yang et al., 2018; Schaffer et al., 2018; Xu & Bai, 2022). As
mentioned in the previous section, such evolution is expected when the dust-
to-gas mass ratio at the midplane is $\gtrsim 0.35-2.5$ (Li & Youdin, 2021).
Li & Youdin (2021) found the critical dust-to-gas mass ratio of $\simeq
0.35-1$ for $\mathrm{St}\gtrsim 0.02$ (see their Figure 4 and the sudden
change of the critical value at $0.01<\mathrm{St}<0.02$). For smaller
$\mathrm{St}$, larger dust-to-gas ratio is required for the strong clumping,
and it takes longer time than for $\mathrm{St}=0.1-1$ according to Li & Youdin
(2021) (see their Tables 1 and 2). Such situations will be more preferable for
coagulation to operate before the strong clumping. We thus focus on the dust
evolution for $\mathrm{St}\gtrsim 0.01$.

To investigate the possible effect of dust coagulation in clumps before the
strong clumping, we estimate the coagulation timescale and compare it to the
duration time of the pre-clumping phase (pre-clumping periods). We represent
the pre-clumping period as $\tau_{\mathrm{SI,sat}}\Omega^{-1}$, where
$\tau_{\mathrm{SI,sat}}$ is a numerical factor, and $\Omega$ is the Keplerian
angular velocity. In this period, we assume that turbulence driven by
streaming instability is nearly saturated and that azimuthally elongated
filaments have developed.

Li & Youdin (2021) defined the pre-clumping phase as a phase between the first
sedimentation 333 Their simulations show an increase of the dust scale height
after the first settling. They include this phase in the first phase that they
call the transient sedimentation phase. ($t<t_{\mathrm{sedi-tr}}$, where $t$
here denotes their simulation time) and the strong clumping phase
($t>t_{\mathrm{pre-cl}}$)444 Li & Youdin (2021) define the starting time of
the strong clumping phase as the time when the maximum dust density first
exceeds $2\rho_{\mathrm{R}}/3$.. Table 2 in Li & Youdin (2021) shows
$t_{\mathrm{sedi-tr}}$ and $t_{\mathrm{pre-cl}}$ for each run. In Figure 1, we
plot the pre-clumping period, $t_{\mathrm{pre-cl}}-t_{\mathrm{sedi-tr}}$, for
$\mathrm{St}\geq 0.01$ using the reported values in their table. We refer to
the runs that show strong clumping and $t_{\mathrm{pre-cl}}>t_{\mathrm{sedi-
tr}}$ (below the gray line). We also refer to their simulations that only show
weak/moderate clumping (i.e.,
$\rho_{\mathrm{d},\mathrm{max}}<\rho_{\mathrm{R}}$; above the gray dashed
line). For such cases, we plot $t_{\mathrm{sim}}-t_{\mathrm{sedi-tr}}$, where
$t_{\mathrm{sim}}$ is the simulation time reported in Table 1 of Li & Youdin
(2021) (e.g., Z0.3t30 run). Although Li & Youdin (2021) call them non-clumping
cases (or weak clumping), the local dust-to-gas ratio increases to $\sim 10$
in some of them (e.g., their Z0.3t30 and Z0.4t100 runs; see their interactive
figure of the online version of Li & Youdin (2021)). We thus call them as the
moderate clumping in the present paper. Such moderate clumping is enough for
dust growth to proceed efficiently, which is shown in Section 4.

According to Li & Youdin (2021) and Figure 1, the pre-clumping period ranges
between $\sim 10\Omega^{-1}-1000\Omega^{-1}$. For $\mathrm{St}<0.3$, the pre-
clumping period is greater than $100\Omega^{-1}$. Although
$\tau_{\mathrm{SI,sat}}$ should depend on the background
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ and $\mathrm{St}$, we just assume it to
be a constant for simplicity. In this paper, we adopt
$\tau_{\mathrm{SI,sat}}\Omega^{-1}=100\Omega^{-1}$ as the fiducial pre-
clumping period.

###  2.2 Dust growth time

If we assume the perfect sticking, the coagulation timescale is given by

$\displaystyle t_{\mathrm{coag}}=a\left(\frac{da}{dt}\right)^{-1}$
$\displaystyle=3m\left(\frac{dm}{dt}\right)^{-1},$ (1)
$\displaystyle=3\frac{m}{\rho_{\mathrm{d}}4\pi a^{2}\Delta v},$ (2)

where $a$ and $m$ denote the size and the mass of a spherical dust grain,
$\rho_{\mathrm{d}}$ is the dust density, $\Delta v$ is the dust-dust collision
velocity. Here, we assume equal-mass collisions for simplicity (see also
Section 5). We focus on the mass-dominating dust sizes. If the number density
distribution of dust $dn(a)/da$ is proportional to $a^{-3.5}$ or shallower,
the mass-dominating dust size roughly corresponds to the largest dust size.
Adopting the size distribution of $d\ln n/d\ln a=-3.5$, Yang & Zhu (2021)
conducted numerical simulations of polydisperse streaming instability and
found dust segregation that concentrates large dust grains in dense regions.
Since we focus on the dust growth in dense regions resulting from streaming
instability, the assumption of the equal-mass collisions of the mass-
dominating dust grains may be valid.

In the Epstein regime, the Stokes number, $\mathrm{St}\equiv
t_{\mathrm{stop}}\Omega$, is given as follows:

$\mathrm{St}=\sqrt{\frac{\pi}{8}}\frac{\rho_{\mathrm{int}}a}{\rho_{\mathrm{g}}c_{\mathrm{s}}}\Omega,$
(3)

where $\rho_{\mathrm{g}}$ is the gas density. We then rewrite the coagulation
timescale in terms of $\mathrm{St}$:

$\tau_{\mathrm{coag}}\equiv
t_{\mathrm{coag}}\Omega=\sqrt{\frac{8}{\pi}}\frac{\rho_{\mathrm{g}}}{\rho_{\mathrm{d}}}\frac{\mathrm{St}c_{\mathrm{s}}}{\Delta
v}.$ (4)

The coagulation timescale depends on the dust-to-gas ratio
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$. An increase in
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ reduces $\tau_{\mathrm{coag}}$, i.e.
accelerates dust coagulation. If we assume no clumping and that large scale
turbulence determines $\Delta v$, one obtains
$\tau_{\mathrm{coag}}\sim\Sigma_{\mathrm{g}}/\Sigma_{\mathrm{d}}$,555 We also
assumed the following: (1) the vertical density distributions of gas and dust
are the Gaussian profile, (2) the dust scale height $H_{\mathrm{d}}$ is
determined by the large scale turbulence, $H_{\mathrm{d}}\sim
H\sqrt{\alpha/\mathrm{St}}$, where $H\equiv c_{\mathrm{s}}/\Omega$ is the gas
scale height (e.g., Dubrulle et al., 1995), and (3)
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ in Equation (4) is the midplane dust-to-
gas ratio. where $\Sigma_{\mathrm{g}}$ and $\Sigma_{\mathrm{d}}$ are the
surface densities of gas and dust (e.g., Brauer et al., 2008).

To take the effect of fragmentation into account, we also utilize the sticking
efficiency $p_{\mathrm{eff}}$ introduced in Okuzumi & Hirose (2012) and
Okuzumi et al. (2016):

$p_{\mathrm{eff}}\equiv\mathrm{min}\left(1,-\frac{\ln\left(\Delta
v/v_{\mathrm{frag}}\right)}{\ln 5}\right),$ (5)

where $v_{\mathrm{frag}}$ is the critical fragmentation velocity. We then
scale the coagulation timescale with the sticking efficiency as follows:

$\tau_{\mathrm{coag,eff}}=\tau_{\mathrm{coag}}/p_{\mathrm{eff}}.$ (6)

The critical fragmentation velocity has been investigated by both laboratory
experiments and numerical simulations (e.g., Blum & Wurm, 2000, 2008; Wada et
al., 2009, 2013; Hasegawa et al., 2021). However, the value of
$v_{\mathrm{frag}}$ still seems uncertain. We thus adopt
$v_{\mathrm{frag}}=10\;\mathrm{m/s}$ as a fiducial case (see also Section 5).
This value has been used in the model treating coagulation and planetesimal
formation due to streaming instability (e.g., Dra̧żkowska et al., 2016).

##  3 Collision velocity model

Once we model $\Delta v$ as a function of dust properties, we can estimate how
efficiently coagulation proceeds using Equation (6). As a simple model, we
consider the following form of the collision velocity:

$\Delta v\propto
c_{\mathrm{s}}\mathrm{St}^{A}(\rho_{\mathrm{d}}/\rho_{\mathrm{g}})^{B},$ (7)

where $A$ and $B$ are constant. We assume the velocity dispersion of dust
grains to be $\Delta v$ in this work. The velocity dispersion has been
measured in the previous numerical simulations (e.g., Johansen et al., 2009;
Bai & Stone, 2010; Schaffer et al., 2018; Schreiber & Klahr, 2018; Yang & Zhu,
2021). We follow Schreiber & Klahr (2018) and Bai & Stone (2010) in this work
since (1) Schreiber & Klahr (2018) showed the dependence of the velocity
dispersion on the dust-to-gas ratio in small-domain simulations (e.g., see
Figure 5 therein), and (2) Bai & Stone (2010) showed the velocity dispersion
for larger $\mathrm{St}$ with the
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$-dependence (e.g., see Figure 13
therein).

###  3.1 Model 1: Schreiber & Klahr (2018)

Schreiber & Klahr (2018) conducted 2D simulations with $\mathrm{St}=0.01$ and
0.1 and with the background dust-to-gas ratio ranging from 0.1 to $10^{3}$.
Some of their simulations adopted very small domain size of $10^{-3}H$, where
$H$ is the gas scale height. Velocity dispersion measured in such a small
domain is suitable to discuss dust collisions at small spatial scales. They
derived velocity dispersion with respect to the domain-averaged velocity and
one with respect to the cell-averaged velocity. The former could include
relative velocities between individual dust clumps. We refer to the latter,
which seems to be more important for dust collisions. We note that their
simulations treat a single dust species, and thus the measured velocity
dispersion is for the same dust sizes. This is sufficient for the purpose of
this work since the previous coagulation simulations showed that the dust
growth is dominated by similar-sized dust collisions (e.g., Okuzumi et al.,
2012).

They found that the velocity dispersion is almost constant for
$1\lesssim\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\lesssim 10$, and $\sim
10^{-3}c_{\mathrm{s}}$ for $\mathrm{St}=0.1$. On the other hand, the velocity
dispersion is proportional to $(\rho_{\mathrm{d}}/\rho_{\mathrm{g}})^{-1}$ for
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\gtrsim 10$ (Figures 5, 11, 16 and 21
therein). As for the $\mathrm{St}$-dependence, their simulations show a
relatively weak dependence with $0<A<1$ (e.g., see their Figures 11 and 21 for
the runs with the radial and vertical dimensions). Based on these results, we
model the velocity dispersion ($\Delta v=\Delta v_{1}$) as follows:

$\Delta v_{1}=\begin{cases}\displaystyle
10^{-3}c_{\mathrm{s}}\left(\frac{\mathrm{St}}{10^{-1}}\right)^{0.5}&\displaystyle\left(\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{g}}}<10\right)\\\
\displaystyle
10^{-3}c_{\mathrm{s}}\left(\frac{\mathrm{St}}{10^{-1}}\right)^{0.5}\left(\frac{\rho_{\mathrm{d}}/\rho_{\mathrm{g}}}{10}\right)^{-1}.&\displaystyle\left(\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{g}}}\geq
10\right)\end{cases}$ (8)

This collision velocity $\Delta v_{1}$ is lower than the maximum drift speed
$\eta v_{\mathrm{K}}$:

$\eta\equiv-\frac{1}{2}\left(\frac{c_{\mathrm{s}}}{v_{\mathrm{K}}}\right)^{2}\frac{d\ln
P}{d\ln r},$ (9)

where $P$ is the gas pressure and $r$ is the radial distance from the central
star (e.g., Adachi et al., 1976; Weidenschilling, 1977). The maximum drift
velocity is thus $\eta v_{\mathrm{K}}\sim(c_{\mathrm{s}}/v_{\mathrm{K}})\times
c_{\mathrm{s}}$. The disk aspect ratio $c_{\mathrm{s}}/v_{\mathrm{K}}$ is
$\sim 10^{-2}-10^{-1}$ (e.g., D’Alessio et al., 1999). Thus, $\Delta v_{1}$ is
about ten times lower than the maximum drift speed. This collision velocity
becomes even lower in the clumping regions of
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}>10$. This low collision velocity is
favorable for dust growth without significant fragmentation.

Figure 2: Comparison of our collision velocity model $\Delta v_{2}$ and the
simulation results of Bai & Stone (2010)
($c_{\mathrm{s}}=0.99\;\mathrm{km/s}$). Solid and dash-dotted lines show the
collision velocities measured in the R10Z3 run of Bai & Stone (2010) with 2D
and 3D domains, respectively (see their Figure 13). Dashed lines show our
model (Equation (10)). We show the collision velocities of each $\mathrm{St}$
with different color.

Figure 3: Ratio of the pre-clumping period $\tau_{\mathrm{SI,sat}}$ and the
coagulation timescale $\tau_{\mathrm{coag,eff}}$. We assume the gas
temperature of $50\;\mathrm{K}$ ($c_{\mathrm{s}}\simeq 0.4\;\mathrm{km/s}$) as
an example. We note that we take into account the effect of the possible
imperfect sticking (Equation (6)). The blue (red) region indicates that the
growth time is shorter (longer) than the pre-clumping period before the strong
clumping. We naively expect that the dust evolves along the white region
($\tau_{\mathrm{SI,sat}}\sim\tau_{\mathrm{coag,eff}}$) on this
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}-\mathrm{St}$ plane (see also Figure 4).

###  3.2 Model 2: Bai & Stone (2010)

The simulations in Schreiber & Klahr (2018) are limited to the cases of
$\mathrm{St}=0.01$ and 0.1. As shown in the subsequent section, the clumping
due to streaming instability can promote further dust growth: dust grains
accumulated locally in a clump collide with each other, and their Stokes
number will increase further. Thus, in addition to Model 1, we also refer to
Bai & Stone (2010) who performed multiple-dust-size simulations with larger
dust grains. The simulations of Bai & Stone (2010) also treats the vertical
stratification in 2D and 3D domains while the simulations of Schreiber & Klahr
(2018) considered unstratified disks with smaller domains.

They assumed the uniform dust mass distribution over their assumed size ranges
(see Section 2.3 therein). Among their simulations with different size ranges
and total metallicities, we refer to the R10Z3 run, where the size range is
$0.1\leq\mathrm{St}\leq 1$ and the metallicity is 0.03. The reasons for
referring to this run are as follows. First, the previous studies on dust
coagulation in a disk show that the dust mass distribution tends to be top-
heavy (e.g., Brauer et al., 2008; Birnstiel et al., 2012; Okuzumi et al.,
2012). The uniform mass distribution in the numerical simulation of streaming
instability would be valid if the size range is narrow and covers the mass-
dominating sizes, as noted in Bai & Stone (2010). The R10Z3 run is one of the
runs with the narrowest size range in the logarithmic space. Second, the focus
of our study is the dust coagulation in the pre-clumping phase, and thus we
need the collision velocity in dust dense regions. The R10Z3 run shows
efficient dust concentration in both 2D and 3D simulations while the other
runs labeled R21Z3 and R30Z3 show very small volume fraction of the dust dense
region in 3D (see Figure 6 therein). The high volume fraction of the dust
dense region in the R10Z3 run will ensure better statistics of the collision
velocity.

The dust collision velocity in the R10Z3 run is shown in their Figure 13. We
focus on the equal mass collision for clear comparison with Model 1. According
to their results, the equal mass collision velocity is larger for larger dust
(Figure 2). As in Schreiber & Klahr (2018), Bai & Stone (2010) also show the
decrease of the collision velocity in the dust-dense region. Based on their
data, we construct the following model as Model 2 ($\Delta v=\Delta v_{2}$):

$\Delta v_{2}=\begin{cases}\displaystyle 1.7\times
10^{-2}c_{\mathrm{s}}\left(\frac{\mathrm{St}}{1}\right)&\displaystyle\left(\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{g}}}<20\right)\\\
\displaystyle 1.7\times
10^{-2}c_{\mathrm{s}}\left(\frac{\mathrm{St}}{1}\right)\left(\frac{\rho_{\mathrm{d}}/\rho_{\mathrm{g}}}{20}\right)^{-0.4}.&\displaystyle\left(\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{g}}}\geq
20\right)\end{cases}$ (10)

We derive the prefactor of $1.7\times 10^{-2}$ and the
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$-dependence by fitting $\Delta
v_{2}/\mathrm{St}$ to the collision velocities of $\mathrm{St}=10^{-0.5}$ and
$1$ in the high-density regions of $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}>20$.
We note that their simulations show higher abundance of particles of
$\mathrm{St}=10^{-0.5}$ and $1$ than particles of $\mathrm{St}=10^{-1}$ (see
Figure 6 of Bai & Stone, 2010), which may indicate better statistics for
larger particles. We thus use the larger particle data for the fitting. In
Figure 2, we compare $\Delta v_{2}$ and the simulation data of Bai & Stone
(2010) for $\mathrm{St}=10^{-1},\;10^{-0.5}$, and $1$. Our model
underestimates the collision velocity for $\mathrm{St}=10^{-1}$ and
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}>10^{2}$. Thus, the collisional growth of
dust particles of $\mathrm{St}=10^{-1}$ is more efficient than our model
predicts unless the critical fragmentation velocity is low. Our model
overestimates the collision velocity of $\mathrm{St}=1$ for
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\lesssim 10$, and thus the coagulation
efficiency is overestimated by a factor of a few. The collision velocity of
$\mathrm{St}=10^{-0.5}$ is better represented by our model.

The collision velocity $\Delta v_{2}$ is greater than $\Delta v_{1}$, which
may be due to the vertical motion in the stratified disk. The collision
velocity can be comparable to the maximum drift velocity for
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}<20$ or only slightly lower. Nevertheless,
the dust enhancement of $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}>20$ reduces the
collision velocity and makes fragmentation less efficient. This is thus
favorable for dust growth (see also Johansen et al., 2009).

Figure 4: Schematic figure to show dust evolution on the
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}-\mathrm{St}$ plane. We assume Model 1 for
the collision velocity as an example. The dust moves upward if the coagulation
timescale is shorter than $t_{\mathrm{SI,sat}}$ (the blue region). For
$\tau_{\mathrm{SI,sat}}<\tau_{\mathrm{coag,eff}}$, the clumping increases the
dust density and thus the dust moves rightward on the
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}-\mathrm{St}$ plane (the red region).
Therefore, we naively expect that the dust first moves to the boundary of the
blue and red regions and then moves along it, which is independent from
initial values of $\mathrm{St}$ and $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$. The
gray region indicates the maximum density that can be expected via the strong
clumping due to streaming instability.

##  4 Results

###  4.1 Dust evolution on $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}-\mathrm{St}$
plane

Figure 3 shows $\tau_{\mathrm{SI,sat}}/\tau_{\mathrm{coag,eff}}$ as a function
of $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ and $\mathrm{St}$. The collision
velocity used to plot the left panel is the Model 1 while that for the right
panel is the Model 2. We assume the gas temperature of $50\;\mathrm{K}$ and
thus $c_{\mathrm{s}}\simeq 0.4\;\mathrm{km/s}$ as an example. We note that the
gas temperature affects the coagulation timescale only through
$p_{\mathrm{eff}}$ (see also Equation (4)). We find that, for the high dust-
to-gas ratio, coagulation proceeds before the strong clumping
($\tau_{\mathrm{SI,sat}}>\tau_{\mathrm{coag,eff}}$). The strong clumping
dominates over coagulation when $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ is
relatively small or $\mathrm{St}$ is sufficiently large. We note that,
regardless of low collision velocities, the dust growth timescale in clumps is
comparable to or shorter than the growth timescale in the non-clumping case
where $t_{\mathrm{coag}}\sim 100\Omega^{-1}$ for
$\Sigma_{\mathrm{d}}/\Sigma_{\mathrm{g}}=0.01$ (Brauer et al., 2008).

One can see how the dust properties evolve with time on the
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}-\mathrm{St}$ plane (see also Figure 4).
In the blue region ($\tau_{\mathrm{SI,sat}}>\tau_{\mathrm{coag,eff}}$), the
coagulation is more efficient than the dust clumping. This means that
$\mathrm{St}$ increases faster than $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$, and
thus the dust evolution is in the vertical direction and upward on the
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}-\mathrm{St}$ plane. On the other hand, in
the red region, the dust clumping is faster than the coagulation, which means
the efficient increase of $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$. The dust
evolution is then in the horizontal direction and rightward. In this way, we
naively expect that the dust moves first to the critical line of
$\tau_{\mathrm{SI,sat}}\sim\tau_{\mathrm{coag,eff}}$ and then roughly along
it, which is independent from initial values of $\mathrm{St}$ and
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$.

In the case of the Model 1 (the left panel of Figure 3), the critical line is
inclined for $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}<10$, and thus both
coagulation and clumping appear to proceed. Once the dust-to-gas ratio becomes
larger than 10, the coagulation dominates over the clumping for
$10^{-2}\leq\mathrm{St}\leq 1$. The coagulation timescale in this regime is
given by

$\tau_{\mathrm{coag,eff}}=p_{\mathrm{eff}}^{-1}\sqrt{\frac{8}{\pi}}10^{1.5}\sqrt{\mathrm{St}},$
(11)

which is independent from $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$. Thus, the
coagulation timescale is comparable to $\tau_{\mathrm{SI,sat}}$ for the
following stopping time:

$\mathrm{St}=p_{\mathrm{eff}}^{2}\frac{\pi}{8}10^{-3}\tau_{\mathrm{SI,sat}}^{2}\simeq
4p_{\mathrm{eff}}^{2}\left(\frac{\tau_{\mathrm{SI,sat}}}{100}\right)^{2}.$
(12)

For the perfect sticking case ($p_{\mathrm{eff}}=1$), the clumping dominates
over the coagulation for $\mathrm{St}>0.36$ and $\mathrm{St}>4$ when
$\tau_{\mathrm{SI,sat}}$ is 30 and $100$, respectively. In this region, the
dust density increases efficiently (see also Figure 4).

In the case of the Model 2, the coagulation timescale is comparable to
$\tau_{\mathrm{SI,sat}}=100$ for $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\sim 1$
when dust grains are relatively small ($\mathrm{St}\lesssim 0.3$). Thus,
coagulation dominates the dust evolution, and the dust moves along the
vertical white region. As the dust becomes larger ($\mathrm{St}\gtrsim 0.3$),
the collision velocity becomes high, and collisions lead to the imperfect
sticking. As a result, the clumping dominates for
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\sim 1$ and $\mathrm{St}\gtrsim 0.3$,
leading to an increase in the dust density. For high dust-to-gas ratios
($\rho_{\mathrm{d}}/\rho_{\mathrm{g}}>10$), the coagulation again becomes
faster than the clumping. Therefore, we expect coagulation to be important for
Model 2 as well. Bai & Stone (2010) also discussed the combined effect of
coagulation and the clumping, indicating the positive feedback between them
(see Section 6 therein). Our results are consistent with their discussion.

Figure 5: Dependence of the critical line
($\tau_{\mathrm{SI,sat}}=\tau_{\mathrm{coag,eff}}$) on
$\tau_{\mathrm{SI,sat}}$. The solid and dotted lines represent cases where we
adopt Model 1 and Model 2, respectively. As in Figure 3, we assume the gas
temperature of 50 K to plot this figure. The critical $\mathrm{St}$ for which
the solid line becomes horizontal (see the case of
$\tau_{\mathrm{SI,sat}}=30,\;50$) is given by Equation (12).

Figure 6: Collision velocities as a function of $\mathrm{St}$ and
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ for $T=100\;\mathrm{K}$ and
$200\;\mathrm{K}$, for which $c_{\mathrm{s}}$ is $\simeq 0.6\;\mathrm{km/s}$
and $\simeq 0.8\;\mathrm{km/s}$, respectively. The adopted critical
fragmentation velocity in each case is $v_{\mathrm{frag}}=10\;\mathrm{m/s}$
and $v_{\mathrm{frag}}=3\;\mathrm{m/s}$, which is based on the fact that
silicate grains are often assumed to be more fragile than water ice (but see
also Kimura et al., 2015; Steinpilz et al., 2019). The white line marks the
critical line ($\tau_{\mathrm{SI,sat}}=\tau_{\mathrm{coag,eff}}$). The arrows
show the direction of the dust evolution on
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}-\mathrm{St}$ plane. Dust grains suffer
fragmentation in a wider parameter space than in Figure 3 because of higher
temperature and smaller $v_{\mathrm{frag}}$. Nevertheless, dust growth time is
shorter than the pre-clumpting period space once the local
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ increases beyond 15 in the Model 1 for
$T=200\;\mathrm{K}$ (20 in the Model 2 for $T=100\;\mathrm{K}$). In the Model
2 with $T=200\;\mathrm{K}$ (the bottom right panel), the critical line is
close to the fragmentation-limited line ($\Delta v=v_{\mathrm{frag}}$) for
$\mathrm{St}\gtrsim 0.2$.

Figure 7:  Isolines of the collision velocity of the Model 2 $\Delta v_{2}$
with $\mathrm{St}=1$ on $r-\rho_{\mathrm{d}}$ plane. We adopt
$120\;\mathrm{K}(r/1\;\mathrm{au})^{-3/7}$ (Chiang & Youdin, 2010) and plot
lines of $3\;\mathrm{m/s},\;6\;\mathrm{m/s}$ and $10\;\mathrm{m/s}$. Note that
the collision velocity decreases as $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$
increases for $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\geq 20$ (Equation (10)).

In Figure 5, we show the $\tau_{\mathrm{SI,sat}}$-dependence of the critical
line ($\tau_{\mathrm{SI,sat}}=\tau_{\mathrm{coag,eff}}$) for both Model 1
(solid lines) and Model 2 (dotted lines). As shown by Equation (12), the
critical line for the Model 1 becomes horizontal at $\mathrm{St}\simeq 0.36$
and $\mathrm{St}=1$ for $\tau_{\mathrm{SI,sat}}=30$ and $50$, respectively.
Thus, the clumping takes over the coagulation earlier in the
smaller-$\tau_{\mathrm{SI,sat}}$ cases. In the Model 2, larger
$\tau_{\mathrm{SI,sat}}$ leads to wider ranges of $\mathrm{St}$ where
coagulation dominates the clumping (see the regions where the dotted lines are
inclined). Thus, even in the presence of the imperfect sticking, coagulation
can be the mechanism to govern the dust evolution.

According to Schreiber & Klahr (2018), the dust-to-gas ratio increases by a
factor of $\sim$10-100 from the background value via the dust clumping in the
nonlinear phase of turbulence due to streaming instability. Other studies also
show a similar increase (e.g., Yang & Johansen, 2014; Yang et al., 2017).
Although Li & Youdin (2021) show that the maximum dust density
$\rho_{\mathrm{d},\mathrm{max}}$ can be $\sim 10^{3}\rho_{\mathrm{g}}$ in
their simulations, the dust-to-gas ratio has a time variation and oscillates
in $\sim 10^{2}\rho_{\mathrm{g}}-10^{3}\rho_{\mathrm{g}}$ (e.g., see Figures 3
and 12 therein). Thus, it would be reasonable to expect
$\rho_{\mathrm{d},\mathrm{max}}/\rho_{\mathrm{g}}$ to be $\sim 10^{2}$ on
average if the clumping starts at $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\sim 1$.
This means that, even if the dust moves to the clumping regime (e.g., the red
region in Figure 4), the dust density stops increasing around
$\rho_{\mathrm{d},\mathrm{max}}$ (the gray region in Figure 4) unless many
resulting filaments merge so that the density increases further (see also Li
et al., 2018, for the dependence of the maximum density on the mass
reservoir).

The fate of the dust clumps depends on whether or not
$\rho_{\mathrm{d},\mathrm{max}}$ is greater than the Roche density
$\rho_{\mathrm{R}}\equiv 9\Omega^{2}/4\pi G$, where $G$ is the gravitational
constant. On the one hand, the dust clumps have the potential to collapse
self-gravitationally for $\rho_{\mathrm{d},\mathrm{max}}>\rho_{\mathrm{R}}$.
For the self-gravitational collapse, it is also required that the self-gravity
of the dust clump dominates over the turbulent diffusion (e.g., Gerbig et al.,
2020; Klahr & Schreiber, 2020, 2021). Klahr & Schreiber (2020) found that dust
clumps larger than $60-120\;\mathrm{km}$ can collapse, which is based on the
solar nebula model of Lenz et al. (2020). On the other hand, for
$\rho_{\mathrm{d},\mathrm{max}}<\rho_{\mathrm{R}}$, the dust evolves only
through coagulation. Although the coagulation in such a regime takes a longer
time than $\tau_{\mathrm{SI,sat}}\Omega^{-1}$, the large dust-to-gas ratio
leads to lower collision velocity, and dust grains will avoid significant
fragmentation (see also Tominaga et al., 2022b, for a similar process).
Besides, the dust drift becomes slower for $\mathrm{St}>1$, which also helps
dust grains to grow beyond the drift barrier.

In the above estimate, we assume $T=50\;\mathrm{K}$, and thus the results are
applicable in the context of icy planetesimal formation. To investigate the
temperature dependence, we also estimate
$\tau_{\mathrm{SI,sat}}/\tau_{\mathrm{coag,eff}}$ and the collision velocity
for $(T,v_{\mathrm{frag}})=(100\;\mathrm{K},\;10\;\mathrm{m/s})$ and
$(200\;\mathrm{K},\;3\;\mathrm{m/s})$. We adopt smaller $v_{\mathrm{frag}}$ in
the case of $T=200\;\mathrm{K}$ since silicate grains are often assumed to be
more fragile than water ice (but see also Kimura et al., 2015; Steinpilz et
al., 2019). Figure 6 shows the critical lines and the collision velocities. In
the case of the Model 1 with $T=100\;\mathrm{K}$ (the top left panel), the
critical line is the same as in Figure 3, since $p_{\mathrm{eff}}$ is unity in
the plotted region and $\tau_{\mathrm{coag}}$ is independent of
$c_{\mathrm{s}}$ (see Equations (6) and (8)). In the case of the Model 2 (the
top right panel), the region where the imperfect sticking occurs is larger
than in Figure 3. Nevertheless, once $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$
increases beyond $\simeq 20$, the coagulation dominates over the clumping even
for large dust of $\mathrm{St}\sim 1$.

In the cases of $(T,v_{\mathrm{frag}})=(200\;\mathrm{K},\;3\;\mathrm{m/s})$,
dust grains suffer fragmentation even with the Model 1 since $\Delta v_{1}$
becomes high (e.g., $\simeq 1\;\mathrm{m/s}$ for $\mathrm{St}>0.1$ and
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}<10$; see the bottom left panel). The dust
growth is thus delayed compared to the perfect sticking case. Nevertheless,
the collision velocity is still lower than 3 m/s by a factor of $\lesssim 0.6$
(e.g., $\Delta v_{1}/v_{\mathrm{frag}}\simeq 0.6$ for
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}=10$ and $\mathrm{St}=0.5$; see Equation
(8)). For the Model 2, the collision velocity is very close to
$v_{\mathrm{frag}}$ for $\mathrm{St}\gtrsim 0.2$ along the critical line (the
bottom right panel). Fragmentation significantly slows the dust growth down,
and thus the increase in $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ via streaming
instability dominates over coagulation in a wider parameter space than in the
top panels and in Figure 3. If the strong clumping or the merger of filaments
leads to $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\sim 10^{3}$, silicate dust
grains can grow toward $\mathrm{St}=1$ for the Model 2 (see Section 4.2).

We also estimate the drift timescale during the dust evolution along the white
line in Figure 6. The drift timescale $t_{\mathrm{drift}}$ is given by

$t_{\mathrm{drift}}\equiv\frac{r}{|v_{\mathrm{drift}}|}=\frac{\left(1+\epsilon\right)^{2}+\mathrm{St}^{2}}{2\mathrm{St}\eta\Omega},$
(13)

where we use the drift velocity $v_{\mathrm{drift}}$ that depends on both
$\mathrm{St}$ and $\epsilon\equiv\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$
(Nakagawa et al., 1986). The timescale of the dust evolution along the white
line is on the order of $\tau_{\mathrm{SI,sat}}\Omega^{-1}$ since the
coagulation timescale is comparable to the clumping timescale. Streaming
instability and the coagulation in clumps will be ineffective if
$t_{\mathrm{drift}}$ is smaller than $\tau_{\mathrm{SI,sat}}\Omega^{-1}$ from
the beginning. We thus consider
$t_{\mathrm{drift}}>\tau_{\mathrm{SI,sat}}\Omega^{-1}$ in the initial state
$(\epsilon,\;\mathrm{St})=(\epsilon_{0},\;\mathrm{St}_{0})$. Assuming the
change in $\eta$ to be insignificant compared to $\mathrm{St}$ and
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$, we obtain an increasing factor of the
drift timescale with respect to the initial value:

$\displaystyle\frac{\tau_{\mathrm{drift}}(\epsilon,\;\mathrm{St})}{\tau_{\mathrm{drift}}(\epsilon_{0},\;\mathrm{St}_{0})}$
$\displaystyle=\frac{\mathrm{St}_{0}}{\mathrm{St}}\frac{\left(1+\epsilon\right)^{2}+\mathrm{St}^{2}}{\left(1+\epsilon_{0}\right)^{2}+\mathrm{St}_{0}^{2}},$
$\displaystyle\sim\frac{\mathrm{St}_{0}}{\mathrm{St}}\left(\frac{1+\epsilon}{1+\epsilon_{0}}\right)^{2}.$
(14)

where $\tau_{\mathrm{drift}}\equiv t_{\mathrm{drift}}\Omega$ and we assumed
$(1+\epsilon)^{2}\gg\mathrm{St}^{2}$ and
$(1+\epsilon_{0})^{2}\gg\mathrm{St}_{0}^{2}$ considering the dust evolution
along the white lines in Figure 6. In the case of the top left panel of Figure
6, the white line shows $\mathrm{St}\propto\epsilon^{2}$, and the drift
timescale is almost constant along the white line. The slopes of the white
lines on the other panels are shallower for $\Delta v/v_{\mathrm{frag}}>0.2$,
meaning
$\tau_{\mathrm{drift}}(\epsilon,\;\mathrm{St})>\tau_{\mathrm{drift}}(\epsilon_{0},\;\mathrm{St}_{0})$
for $\epsilon>\epsilon_{0}$. Therefore, the drift speed of dust grains does
not increase during their growth in clumps and the clumping. The clumping can
help dust grains to overcome the drift barrier (see also Bai & Stone,
2010).666 Because of the $r$-dependence of $\Omega$,
$t_{\mathrm{drift}}(\epsilon,\;\mathrm{St})$ can be shorter than
$t_{\mathrm{drift}}(\epsilon_{0},\;\mathrm{St}_{0})$. However, the coagulation
timescale and the clumping timescale also scale with $\Omega^{-1}$ (see Brauer
et al., 2008, for the coagulation timescale). We thus ignore the factor of
$\Omega$.

###  4.2  On the possible fragmentation limit during streaming instability

For the Model 1, we find that fragmentation affects the dust growth, but it
just moderately slows the dust growth down. The collision velocity of the
Model 2 can be, however, very close to $v_{\mathrm{frag}}=3\;\mathrm{m/s}$
even for $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\gg 10$. This indicates that, in
the case of the Model 2, efficient fragmentation regulates or limits the dust
growth unless the clumping increases $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$
sufficiently.

To see where and when the fragmentation prevents the dust growth beyond
$\mathrm{St}=1$, we adopt a simple temperature profile,
$T=120\;\mathrm{K}(r/1\;\mathrm{au})^{-3/7}$ (Chiang & Youdin, 2010), and see
the $r-$dependence of the collision velocity of the Model 2. Since the
velocity also depends on $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$, we assume
$\mathrm{St}=1$ and plot isolines of the collision velocity. From Equation
(10) with $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\geq 20$ and $\mathrm{St}=1$,
one gets a formula of an isoline, $\Delta v_{2}(\mathrm{St}=1)=\Delta v$,

$\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{g}}}\simeq
75\left(\frac{c_{\mathrm{s}}}{1\;\mathrm{km/s}}\right)^{2.5}\left(\frac{\Delta
v}{10\;\mathrm{m/s}}\right)^{-2.5}.$ (15)

In Figure 7, we plot three isolines of the collision velocity of the Model 2
$\Delta v_{2}(\mathrm{St}=1)$ on the $r-\rho_{\mathrm{d}}$ plane. We note that
the collision velocity decreases as $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$
increases, and also note that the vertical axis of Figure 7 represents the
dust-to-gas ratio in dust clumps. For the adopted temperature model, the
collision velocity can be lower than 3 m/s at $r\lesssim 0.5\;\mathrm{au}$
($T\gtrsim 160\;\mathrm{K}$) if the dust-to-gas ratio in a clump exceeds $\sim
800$. If $v_{\mathrm{frag}}$ is $\simeq 6\;\mathrm{m/s}$ for silicate dust
with 0.1 $\mu\mathrm{m}$-sized monomers (Wada et al., 2009), the required
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ is about five times smaller (see also
Equation (15)). Growth of silicate dust grains is even easier if their surface
energy is ten times higher than previously assumed, which is indicated in
Kimura et al. (2015) and Steinpilz et al. (2019).

For the temperature profile we adopted, the collision velocity of the Model 2
for $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\geq 20$ is lower than 10 m/s beyond
$\simeq 1.6\;\mathrm{au}$. Thus, regardless of higher collision velocity than
the Model 1, the growth of icy dust grains will not be inhibited in the
turbulent state due to streaming instability.

##  5 Conclusions and Discussion

Based on the previous numerical studies of streaming instability, we model the
collision velocity and compare the coagulation timescale and the duration time
of the pre-clumping phase (the pre-clumping period). We show that even
moderately increased dust density due to streaming instability promotes dust
coagulation (Figure 3). It is expected that dust evolves roughly along the
line of $\tau_{\mathrm{coag,eff}}\sim\tau_{\mathrm{SI,sat}}$ on the
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}-\mathrm{St}$ plane (Figure 4). The
combination of dust growth and the clumping might allow dust growth toward and
beyond $\mathrm{St}=1$ if the clumping leads to sufficient deceleration of
dust drift. Our results highlight the importance of numerical simulations that
consider both coagulation and streaming instability.

Once dust grains grow beyond $\mathrm{St}=1$ via the rapid coagulation,
streaming instability becomes inefficient to sustain the clumping state
because of weak dust-gas coupling. Such large dust grains will then settle
toward the midplane, gradually increasing the midplane dust density. If a
sufficient amount of large dust grains form and settle, planetesimal formation
via gravitational instability will occur (e.g., Michikoshi et al., 2010;
Michikoshi & Kokubo, 2017). If the increase of the midplane dust density is
too slow, the evolution of the large dust grains might be regulated by erosion
or mass transfer through collisions with remaining smaller dust grains (Krijt
et al., 2015; Hasegawa et al., 2021).

One important parameter in this study is the pre-clumping period
$\tau_{\mathrm{SI,sat}}$. Although we assume $\tau_{\mathrm{SI,sat}}$ to be
constant, their simulations show shorter periods for larger dust sizes (see
their Tables 1 and 2, and Figure 1 of the present paper). Therefore, the dust
growth during the pre-clumping phase might accelerate the clumping. This
positive feedback process was also indicated in Bai & Stone (2010) and will
operate even if $\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ increases only up to a
few or $\sim 10$ by the moderate clumping (e.g., Z0.6t2 and Z0.3t30 runs in Li
& Youdin (2021)) since dust grains should grow gradually. The efficiency of
the feedback will depend on a production rate of larger dust grains due to the
coagulation in clumps, which we will address in our future work.

In the present model calculations, we assume equal-mass collisions, where the
collision velocity is primarily due to turbulent motion. If we consider
unequal-mass collisions, the coagulation timescale may become shorter since
dust grains also collide through relative drift motion. We note that
collisions in dense clumps are still necessary to avoid fragmentation since
collision velocity of unequal-mass dust grains is higher outside the clumps
(see Johansen et al., 2009; Bai & Stone, 2010). Dense clumps will also help
dust grains to overcome the drift barrier since the drift velocity is reduced
in dense regions (Nakagawa et al., 1986; Bai & Stone, 2010). On the other
hand, recent studies show that streaming instability is less efficient when
one includes a dust size distribution (Krapp et al., 2019; Paardekooper et
al., 2020; McNally et al., 2021; Zhu & Yang, 2021; Yang & Zhu, 2021). This may
indicate that it takes longer time to achieve the strong clumping. Therefore,
coagulation will be even more effective in this case than the clumping, which
further motivates us to consider coagulation during the pre-clumping phase.
Further quantitative studies are necessary since the impact of the size
distribution depends on its slope and shape (Zhu & Yang, 2021; McNally et al.,
2021).

In our fiducial case, we assume $T=50\;\mathrm{K}$ and consider icy dust
grains (see also the top panels of Figure 6 for $T=100\;\mathrm{K}$).
Numerical simulations of dust aggregate collisions showed
$v_{\mathrm{frag}}\simeq 30-100\;\mathrm{m/s}$ for water ice with a monomer
size of $0.1\;\mu\mathrm{m}$ (e.g., Wada et al., 2009; Hasegawa et al., 2021).
The fragmentation velocity can be about two times smaller than these values
for a monomer size of $\simeq 0.2\;\mu\mathrm{m}$, which is in the possible
range of monomer sizes observationally indicated by Tazaki & Dominik (2022).
The assumed $v_{\mathrm{frag}}$ in this work is lower than these values,
meaning that the efficiency of coagulation will be higher than we estimated
(see also Figure 7). Therefore, coagulation should be taken into account when
one investigates icy planetesimal formation via streaming instability.

In the case of silicate grains, the coagulation efficiency will be lower (the
bottom panels of Figure 6) since usually assumed $v_{\mathrm{frag}}$ for
silicate is lower. Regardless of small $v_{\mathrm{frag}}$, we show that the
collision velocity of the Model 1 is kept below 3 m/s for
$c_{\mathrm{s}}\simeq 0.8\;\mathrm{km/s}$ in moderately clumping regions with
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}>10$. This suggests that even silicate
dust grains can grow toward the size of $\mathrm{St}=1$. In the case of the
Model 2 (the bottom right panel of Figure 6), silicate dust grains can grow
toward $\mathrm{St}=1$ at the strong clumping phase with
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}\sim 10^{3}$. This required value of
$\rho_{\mathrm{d}}/\rho_{\mathrm{g}}$ strongly depends on $v_{\mathrm{frag}}$
(see Figure 7 and Equation (15)). The silicate dust will grow more easily
during the clumping if $v_{\mathrm{frag}}$ is $\simeq 6\;\mathrm{m/s}$ for
silicate dust with 0.1 $\mu\mathrm{m}$-sized monomers (Wada et al., 2009) or
higher (Kimura et al., 2015; Steinpilz et al., 2019). In our future work, we
will address time evolution of dust sizes and collision velocities during the
clumping in more detail.

We thank Xuening Bai for providing us with their simulation data that we used
to verify our collision velocity model and for helpful comments. We also thank
the anonymous referee for constructive comments that helped us to improve the
manuscript. This work was supported by JSPS KAKENHI Grant Nos. 21K20385
(R.T.T.), 19K03941 (H.T.). R.T.T. is also supported by RIKEN Special
Postdoctoral Researchers Program.

## References

  * Adachi et al. (1976) Adachi, I., Hayashi, C., & Nakazawa, K. 1976, Progress of Theoretical Physics, 56, 1756, doi: 10.1143/PTP.56.1756
  * Arakawa & Nakamoto (2016) Arakawa, S., & Nakamoto, T. 2016, ApJ, 832, L19, doi: 10.3847/2041-8205/832/2/L19
  * Bai & Stone (2010) Bai, X.-N., & Stone, J. M. 2010, ApJ, 722, 1437, doi: 10.1088/0004-637X/722/2/1437
  * Birnstiel et al. (2012) Birnstiel, T., Klahr, H., & Ercolano, B. 2012, A&A, 539, A148, doi: 10.1051/0004-6361/201118136
  * Blum & Wurm (2000) Blum, J., & Wurm, G. 2000, Icarus, 143, 138, doi: 10.1006/icar.1999.6234
  * Blum & Wurm (2008) —. 2008, ARA&A, 46, 21, doi: 10.1146/annurev.astro.46.060407.145152
  * Brauer et al. (2008) Brauer, F., Dullemond, C. P., & Henning, T. 2008, A&A, 480, 859, doi: 10.1051/0004-6361:20077759
  * Carrera et al. (2017) Carrera, D., Gorti, U., Johansen, A., & Davies, M. B. 2017, The Astrophysical Journal, 839, 16, doi: 10.3847/1538-4357/aa6932
  * Carrera et al. (2015) Carrera, D., Johansen, A., & Davies, M. B. 2015, A&A, 579, A43, doi: 10.1051/0004-6361/201425120
  * Carrera et al. (2021) Carrera, D., Simon, J. B., Li, R., Kretke, K. A., & Klahr, H. 2021, AJ, 161, 96, doi: 10.3847/1538-3881/abd4d9
  * Carrera et al. (2022) Carrera, D., Thomas, A. J., Simon, J. B., et al. 2022, ApJ, 927, 52, doi: 10.3847/1538-4357/ac4d28
  * Chen & Lin (2020) Chen, K., & Lin, M.-K. 2020, ApJ, 891, 132, doi: 10.3847/1538-4357/ab76ca
  * Chiang & Youdin (2010) Chiang, E., & Youdin, A. N. 2010, Annual Review of Earth and Planetary Sciences, 38, 493, doi: 10.1146/annurev-earth-040809-152513
  * Chiang & Goldreich (1997) Chiang, E. I., & Goldreich, P. 1997, ApJ, 490, 368, doi: 10.1086/304869
  * Cuzzi et al. (1993) Cuzzi, J. N., Dobrovolskis, A. R., & Champney, J. M. 1993, Icarus, 106, 102, doi: 10.1006/icar.1993.1161
  * D’Alessio et al. (1999) D’Alessio, P., Calvet, N., Hartmann, L., Lizano, S., & Cantó, J. 1999, ApJ, 527, 893, doi: 10.1086/308103
  * Dra̧żkowska et al. (2016) Dra̧żkowska, J., Alibert, Y., & Moore, B. 2016, A&A, 594, A105, doi: 10.1051/0004-6361/201628983
  * Dubrulle et al. (1995) Dubrulle, B., Morfill, G., & Sterzik, M. 1995, Icarus, 114, 237, doi: 10.1006/icar.1995.1058
  * Garcia & Gonzalez (2020) Garcia, A. J. L., & Gonzalez, J.-F. 2020, MNRAS, 493, 1788, doi: 10.1093/mnras/staa382
  * Gerbig et al. (2020) Gerbig, K., Murray-Clay, R. A., Klahr, H., & Baehr, H. 2020, ApJ, 895, 91, doi: 10.3847/1538-4357/ab8d37
  * Goldreich & Ward (1973) Goldreich, P., & Ward, W. R. 1973, ApJ, 183, 1051, doi: 10.1086/152291
  * Gole et al. (2020) Gole, D. A., Simon, J. B., Li, R., Youdin, A. N., & Armitage, P. J. 2020, ApJ, 904, 132, doi: 10.3847/1538-4357/abc334
  * Hasegawa et al. (2021) Hasegawa, Y., Suzuki, T. K., Tanaka, H., Kobayashi, H., & Wada, K. 2021, ApJ, 915, 22, doi: 10.3847/1538-4357/abf6cf
  * Hayashi (1981) Hayashi, C. 1981, Progress of Theoretical Physics Supplement, 70, 35, doi: 10.1143/PTPS.70.35
  * Johansen et al. (2007) Johansen, A., Oishi, J. S., Mac Low, M.-M., et al. 2007, Nature, 448, 1022, doi: 10.1038/nature06086
  * Johansen & Youdin (2007) Johansen, A., & Youdin, A. 2007, ApJ, 662, 627, doi: 10.1086/516730
  * Johansen et al. (2009) Johansen, A., Youdin, A., & Mac Low, M.-M. 2009, ApJ, 704, L75, doi: 10.1088/0004-637X/704/2/L75
  * Kataoka et al. (2013) Kataoka, A., Tanaka, H., Okuzumi, S., & Wada, K. 2013, A&A, 554, A4, doi: 10.1051/0004-6361/201321325
  * Kimura et al. (2015) Kimura, H., Wada, K., Senshu, H., & Kobayashi, H. 2015, ApJ, 812, 67, doi: 10.1088/0004-637X/812/1/67
  * Klahr & Schreiber (2020) Klahr, H., & Schreiber, A. 2020, ApJ, 901, 54, doi: 10.3847/1538-4357/abac58
  * Klahr & Schreiber (2021) —. 2021, ApJ, 911, 9, doi: 10.3847/1538-4357/abca9b
  * Kobayashi & Tanaka (2021) Kobayashi, H., & Tanaka, H. 2021, ApJ, 922, 16, doi: 10.3847/1538-4357/ac289c
  * Krapp et al. (2019) Krapp, L., Benítez-Llambay, P., Gressel, O., & Pessah, M. E. 2019, ApJ, 878, L30, doi: 10.3847/2041-8213/ab2596
  * Krijt et al. (2015) Krijt, S., Ormel, C. W., Dominik, C., & Tielens, A. G. G. M. 2015, A&A, 574, A83, doi: 10.1051/0004-6361/201425222
  * Krijt et al. (2016) —. 2016, A&A, 586, A20, doi: 10.1051/0004-6361/201527533
  * Lenz et al. (2020) Lenz, C. T., Klahr, H., Birnstiel, T., Kretke, K., & Stammler, S. 2020, A&A, 640, A61, doi: 10.1051/0004-6361/202037878
  * Li & Youdin (2021) Li, R., & Youdin, A. N. 2021, ApJ, 919, 107, doi: 10.3847/1538-4357/ac0e9f
  * Li et al. (2018) Li, R., Youdin, A. N., & Simon, J. B. 2018, ApJ, 862, 14, doi: 10.3847/1538-4357/aaca99
  * McNally et al. (2021) McNally, C. P., Lovascio, F., & Paardekooper, S.-J. 2021, MNRAS, 502, 1469, doi: 10.1093/mnras/stab112
  * Michikoshi & Kokubo (2017) Michikoshi, S., & Kokubo, E. 2017, ApJ, 842, 61, doi: 10.3847/1538-4357/aa7388
  * Michikoshi et al. (2010) Michikoshi, S., Kokubo, E., & ichiro Inutsuka, S. 2010, The Astrophysical Journal, 719, 1021, doi: 10.1088/0004-637X/719/2/1021
  * Nakagawa et al. (1986) Nakagawa, Y., Sekiya, M., & Hayashi, C. 1986, Icarus, 67, 375, doi: 10.1016/0019-1035(86)90121-1
  * Nesvorný et al. (2019) Nesvorný, D., Li, R., Youdin, A. N., Simon, J. B., & Grundy, W. M. 2019, Nature Astronomy, 3, 808, doi: 10.1038/s41550-019-0806-z
  * Okuzumi & Hirose (2012) Okuzumi, S., & Hirose, S. 2012, ApJ, 753, L8, doi: 10.1088/2041-8205/753/1/L8
  * Okuzumi et al. (2016) Okuzumi, S., Momose, M., Sirono, S.-i., Kobayashi, H., & Tanaka, H. 2016, ApJ, 821, 82, doi: 10.3847/0004-637X/821/2/82
  * Okuzumi et al. (2012) Okuzumi, S., Tanaka, H., Kobayashi, H., & Wada, K. 2012, ApJ, 752, 106, doi: 10.1088/0004-637X/752/2/106
  * Ormel & Cuzzi (2007) Ormel, C. W., & Cuzzi, J. N. 2007, A&A, 466, 413, doi: 10.1051/0004-6361:20066899
  * Ormel et al. (2007) Ormel, C. W., Spaans, M., & Tielens, A. G. G. M. 2007, A&A, 461, 215, doi: 10.1051/0004-6361:20065949
  * Paardekooper et al. (2020) Paardekooper, S.-J., McNally, C. P., & Lovascio, F. 2020, MNRAS, 499, 4223, doi: 10.1093/mnras/staa3162
  * Paardekooper et al. (2021) —. 2021, MNRAS, 502, 1579, doi: 10.1093/mnras/stab111
  * Safronov (1972) Safronov, V. S. 1972, Evolution of the protoplanetary cloud and formation of the earth and planets.
  * Schaffer et al. (2018) Schaffer, N., Yang, C.-C., & Johansen, A. 2018, A&A, 618, A75, doi: 10.1051/0004-6361/201832783
  * Schreiber & Klahr (2018) Schreiber, A., & Klahr, H. 2018, ApJ, 861, 47, doi: 10.3847/1538-4357/aac3d4
  * Sekiya (1983) Sekiya, M. 1983, Progress of Theoretical Physics, 69, 1116, doi: 10.1143/PTP.69.1116
  * Sekiya (1998) —. 1998, Icarus, 133, 298, doi: 10.1006/icar.1998.5933
  * Shakura & Sunyaev (1973) Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
  * Simon et al. (2016) Simon, J. B., Armitage, P. J., Li, R., & Youdin, A. N. 2016, ApJ, 822, 55, doi: 10.3847/0004-637X/822/1/55
  * Steinpilz et al. (2019) Steinpilz, T., Teiser, J., & Wurm, G. 2019, ApJ, 874, 60, doi: 10.3847/1538-4357/ab07bb
  * Takahashi & Inutsuka (2014) Takahashi, S. Z., & Inutsuka, S.-i. 2014, ApJ, 794, 55, doi: 10.1088/0004-637X/794/1/55
  * Tazaki & Dominik (2022) Tazaki, R., & Dominik, C. 2022, A&A, 663, A57, doi: 10.1051/0004-6361/202243485
  * Tominaga et al. (2021) Tominaga, R. T., Inutsuka, S.-i., & Kobayashi, H. H. 2021, The Astrophysical Journal, 923, 34, doi: 10.3847/1538-4357/ac173a
  * Tominaga et al. (2022a) Tominaga, R. T., Kobayashi, H., & Inutsuka, S.-i. 2022a, ApJ, 937, 21, doi: 10.3847/1538-4357/ac82b4
  * Tominaga et al. (2022b) Tominaga, R. T., Tanaka, H., Kobayashi, H., & Inutsuka, S.-i. 2022b, ApJ, 940, 152, doi: 10.3847/1538-4357/ac97e8
  * Umurhan et al. (2020) Umurhan, O. M., Estrada, P. R., & Cuzzi, J. N. 2020, ApJ, 895, 4, doi: 10.3847/1538-4357/ab899d
  * Wada et al. (2013) Wada, K., Tanaka, H., Okuzumi, S., et al. 2013, A&A, 559, A62, doi: 10.1051/0004-6361/201322259
  * Wada et al. (2009) Wada, K., Tanaka, H., Suyama, T., Kimura, H., & Yamamoto, T. 2009, ApJ, 702, 1490, doi: 10.1088/0004-637X/702/2/1490
  * Weidenschilling (1977) Weidenschilling, S. J. 1977, MNRAS, 180, 57, doi: 10.1093/mnras/180.1.57
  * Weidenschilling & Cuzzi (1993) Weidenschilling, S. J., & Cuzzi, J. N. 1993, in Protostars and Planets III, ed. E. H. Levy & J. I. Lunine (University of Arizona Press, Tucson, Arizona), 1031
  * Xu & Bai (2022) Xu, Z., & Bai, X.-N. 2022, ApJ, 924, 3, doi: 10.3847/1538-4357/ac31a7
  * Yang & Johansen (2014) Yang, C.-C., & Johansen, A. 2014, ApJ, 792, 86, doi: 10.1088/0004-637X/792/2/86
  * Yang et al. (2017) Yang, C. C., Johansen, A., & Carrera, D. 2017, A&A, 606, A80, doi: 10.1051/0004-6361/201630106
  * Yang et al. (2018) Yang, C.-C., Mac Low, M.-M., & Johansen, A. 2018, ApJ, 868, 27, doi: 10.3847/1538-4357/aae7d4
  * Yang & Zhu (2021) Yang, C.-C., & Zhu, Z. 2021, MNRAS, 508, 5538, doi: 10.1093/mnras/stab2959
  * Youdin & Johansen (2007) Youdin, A., & Johansen, A. 2007, ApJ, 662, 613, doi: 10.1086/516729
  * Youdin (2011) Youdin, A. N. 2011, ApJ, 731, 99, doi: 10.1088/0004-637X/731/2/99
  * Youdin & Goodman (2005) Youdin, A. N., & Goodman, J. 2005, ApJ, 620, 459, doi: 10.1086/426895
  * Youdin & Shu (2002) Youdin, A. N., & Shu, F. H. 2002, ApJ, 580, 494, doi: 10.1086/343109
  * Zhu & Yang (2021) Zhu, Z., & Yang, C.-C. 2021, MNRAS, 501, 467, doi: 10.1093/mnras/staa3628