Clusters of heavy particles in two-dimensional Keplerian turbulence Fabiola A. Gerosaand H elo se M eheut Universit e C ote dAzur Observatoire de la C ote dAzur CNRS Laboratoire Lagrange Nice France

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Clusters of heavy particles in two-dimensional Keplerian turbulence
Fabiola A. Gerosa∗and H´elo¨ıse M´eheut
Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange, Nice, France
J´er´emie Bec
Universit´e Cˆote d’Azur, Inria, CNRS, Sophia-Antipolis, France
Mines Paris, PSL University, CNRS, Cemef, Sophia-Antipolis, France
Protoplanetary disks are gaseous systems in Keplerian rotation around young stars, known to
be turbulent. They include a small fraction of dust from which planets form. In the incremental
scenario for planet growth, the formation of kilometer-size objects (planetesimals) from pebbles is a
major open question. Clustering of particles is necessary for solids to undergo a local gravitational
collapse.
To address this question, the dynamic of inertial particles in turbulent flows with Keplerian rotation
and shear is studied. Two-dimensional direct numerical simulations are performed to explore sys-
tematically two physical parameters: the rotation rate, which depends on the distance to the star,
and the particle response time, which relates to their size.
Shear is found to drastically affect the characteristics of the turbulent flow destroying cyclones and
favoring the survival of anticyclones. Faster Keplerian rotation enhances clustering of particles.
For intermediate sizes, particles concentrate in anticyclones. These clusters form in a hierarchical
manner and merge together with time. For other parameter values, solids concentrate on fractal
sets that get more singular with rotation. The mass distribution of particles is then found to be
multifractal with small dimensions at large orders, intriguing for triggering their gravitational col-
lapse. Such results are promising for a precise description and better understanding of planetesimal
formation.
I. INTRODUCTION
A stellar system appears after the gravitational col-
lapse of a molecular cloud. While most matter accumu-
lates in the young star, a tiny fraction (less than a few
percents) sediments in a protoplanetary disk, which ro-
tates around the star and serves as a stage for the forma-
tion of planets, moons, asteroids, and other small bodies.
Such disks are mainly composed of gas, but also contain
some sub-micrometer dust particles (≈1% in mass) that
are the building blocks of larger solid bodies. One of the
major open questions in the theory of planet formation is
the emergence of planetesimals (objects with sizes of the
order of kilometers) that are subsequently able to further
accrete solids and even capture the surrounding gas. In-
deed classical scenarios, in which dust particles grow in
a hierarchical, sequential manner [1], are facing several
difficulties.
A first issue relates to the outcomes of collisions be-
tween solids [2]. At sizes larger than centimeters,1colli-
sions between pebbles are expected to involve too violent
impact velocities. They can hardly lead to coagulation,
but rather result in bouncing or fragmentation, making
hierarchical growth by far too slow and inefficient at such
sizes. A second issue comes from the radial drift that is
experienced by solid particles [3]. Protoplanetary disks
are indeed expected to have a large-scale radial pressure
∗fabiola.gerosa@oca.eu
1The sizes given in this paragraph vary with distance to the star.
Examples given here correspond to 1 Astronomical Unit.
gradient, so that the gas rotates with a slightly sub-
Keplerian speed. Solid particles are insensitive to this
pressure gradient and rotate with the exact Keplerian
velocity. They are thus dragged by the gas that slows
them down, making them drift radially toward the cen-
tral star. This process occurs on too short time scales to
form planetesimals by sequential growth.
Drift and fragmentation make the growth of planetes-
imals difficult. Nevertheless, it is generally assumed that
such effects can be overcome by a direct self-gravitational
collapse of dust particles that can bypass hierarchical co-
alescence. This requires both gravitational interactions
to be dominant over the dispersion induced by the Kep-
lerian shear, and velocity differences between dust grains
to be sufficiently feeble. This could occur during the
settling of solids in the disk’s equatorial plane [4]. We
focus here on another scenario involving spatial inho-
mogeneities, whose presence provides favorable condi-
tions. Self-gravity can indeed become dominant in re-
gions where the density of solids is large and their veloc-
ity dispersion narrow [5]. In such a scenario a major role
is played by the turbulence of the carrier gas [6, 7]. It is
even largely admitted that the back reaction of the par-
ticles on the gas could itself trigger turbulence. Under-
lying mechanisms involve the streaming instability [8] or
other resonant drag instabilities, in which the decoupling
between gas and particles (through settling, drift, buoy-
ancy, or magnetic waves) strongly amplifies tiny fluctua-
tions in the concentration of solids [9]. Recent evidences
have however been obtained that, on the one hand, an
already developed turbulent state impedes the growth
of such instabilities [10, 11] and that, on the other hand,
arXiv:2210.13147v2 [astro-ph.EP] 3 Feb 2023
2
they prohibitively require starting from already large par-
ticle sizes [12].
Besides planet formation, the origin and nature of tur-
bulence in protoplanetary disks, and how it couples to
particle dynamics, is key to several other astrophysical
questions. For instance, disks are known to have a rather
short lifetime (of the order of a few millions years). The
dissipation of angular momentum by the gas molecular
viscosity is orders of magnitude too low to explain this
fast accretion onto the central star. Yet turbulence is
an excellent candidate to transport matter on shorter
timescales, hence providing estimates on the expected
turbulent intensity of disks. Transport mechanisms in
protoplanetary disks, possible instabilities and the de-
velopment of a turbulent state are still the topic of in-
tense work (see, e.g., [13, 14] for a review). Whether it
originates from magneto-rotational effects or purely com-
pressible or incompressible hydrodynamical instabilities,
disk turbulence is confined, in rotation, stratified, and
thus generally displays fairly strong two-dimensional fea-
tures, as for instance long-lived vortical structures [15].
The presence of such vortices might clearly impact dust
accretion and consequently the formation of planetesi-
mals. While heavy inertial particles are usually expelled
from rotating regions by Maxey’s centrifugal effect [16],
the presence of shear and rotation in protoplanetary disks
actually has an opposite influence. Numerical [17, 18],
analytical [19, 20] and experimental [21] studies indeed
give evidence that dust particles cluster in some of the
structures of the flow. More specifically, solid particles
experience a Coriolis force that can either accelerate their
ejection from cyclones (vortices with the same sign as
global rotation) or, when strong enough, overcome cen-
trifugal forces in anticyclones. This pushes the particles
toward the core of anticyclonic vortices, possibly creating
extremely dense point clusters that are excellent candi-
dates for gravitational instabilities. These mechanisms
have been known for years, but their study remained es-
sentially qualitative or limited to model flows. Under-
standing further what is the impact of vortex clustering
on planetesimal formation requires more quantitative in-
sights, such as estimates on involved timescales in the
presence of turbulent fluctuations, dependence on phys-
ical parameters, nature of the associated mass distribu-
tion, etc. The present work aims providing insights on
such questions.
We present in this paper 2D direct numerical simula-
tions with the idea of shedding some lights on the com-
plex problem of dust dynamics in turbulent protoplan-
etary disks. We investigate the clustering properties of
solid particles in a forced, developed turbulent incom-
pressible flow subject to Keplerian rotation and shear.
We follow the “shearing box” approach, which consists
in solving locally the Navier–Stokes equation with a mean
constant shear and with periodic boundary conditions on
a domain that is distorted in the direction of the mean
velocity. A number of simulations are performed vary-
ing systematically the two main physical parameters of
the problem: the particle response time τp, which mea-
sures their inertia and their lag on the gas flow, and the
rotation rate Ω, which prescribes the mean shear. We
then analyse quantitatively various dynamical and sta-
tistical features of solid particles, and in particular their
clustering properties when flow structures, typical of a
turbulent flow in rotation, are present. Tools borrowed
from the study of dynamical systems, such as Lyapunov
exponents and fractal dimensions, are used to character-
ize and quantify particle clusters.
Before focusing on particles, we find that the mean
shear has noticeable impacts on the properties of the tur-
bulent flow. Both energy and enstrophy budgets, as well
as energy and enstrophy spectra, change substantially
with increasing rotation. Strong rotation also leads to
pronounced anisotropies in the flow and thus to increased
skewness and kurtosis of the vorticity distribution. Shear
is found to cause the preferential formation and survival
of anticyclonic vortices. Concerning particles, we find
that at specific values of the physical parameters, they
can form point clusters located inside anticyclones. This
strong clustering requires large-enough values of the ro-
tation rate Ω and intermediate values of the particle re-
sponse time τp. These clusters are found to form in a hi-
erarchical manner and to merge one with the other when
time increases. We propose a simple kinetic model that
catches most aspects of their evolution and distribution.
Outside this extreme regime, solid particles can never-
theless concentrate on dynamically evolving fractal sets
whose dimensions non-trivially depend on the rotation
rate. Evidence is obtained that the mass distribution of
particles is then multifractal, with a dimension that de-
creases and saturates at large orders. Such a behavior is
key to quantify the probability to get a large local den-
sity of solids and to trigger self-gravitating instabilities.
In the strong clustering regime, such instabilities would
occur during transients.
This paper is organised as follows. We first introduce
and characterise in section II the flow used for the nu-
merical simulations and we explain in section III the con-
sidered dynamics of particles. In Section IV we show
the results of the simulations, highlighting the different
behaviors observed for the distribution of inertial parti-
cles. In section V and section VI the mass distribution of
particles is inspected in the regime of fractal and strong
clustering, respectively. Finally, in section VII we con-
clude the paper summarizing the results of this study and
connecting it back to the astrophysical context.
II. THE TWO-DIMENSIONAL SHEARING BOX
Dust and gas dynamics in protoplanetary disks, as in
most astrophysical situations, involve gravity and rota-
tion. The balance between gravity and centrifugal forces
defines the Keplerian angular velocity, which is given by
Ω(r) = pGM?/r3(1)
3
where M?is the mass of the central star, Gis the gravita-
tional constant, and ris the distance to the central star.
This velocity profile corresponds to a rotating flow with
a strong shear (see Fig. 1).
FIG. 1. Schematic view of a protoplanetary disk and of the
rotating shearing box that we here consider.
A. Model and equations of motion
We focus on a small box whose edge is located at a
distance r0from the star and which is rotating with the
reference angular velocity Ω(r0) (see Fig. 1). We then
write the dynamics in the co-moving coordinates x=r−
r0and y=r0(θ−Ω(r0)t), where θdenotes the azimuthal
angle. When the reference distance r0is much larger that
the box size, curvature terms can be neglected. Local
variations of the Keplerian angular velocity are shearing
our small box. We indeed have Ω(r)−Ω(r0)≈(r−
r0)d
drΩ(r0) = −3
2Ω(r0)x/r0. In the co-rotating frame,
gravity and centrifugal terms balance at r=r0. Away
from r0, a tidal force and a Coriolis term must be added.
This model, introduced by Hill [22] in the late 70’s, leads
to write the momentum conservation for the gas as:
∂tv+v·∇v=−1
ρg∇p+ν∇2v−2Ω×v+ 3Ω2xex,(2)
where vis the total gas velocity in the co-moving frame.
ρgand νare its mass density and kinematic viscosity,
respectively. We have here denoted Ω = Ω(r0) = const
and Ω= Ω ez. For an incompressible flow, the equation
of continuity reads:
∇ · v= 0,(3)
which hence prescribes the pressure pin (2). The use of
incompressibility is justified by the fact that, at the small
scales that we consider, density waves are less important.
Therefore, density can be assumed constant over time.
Note that averaging (2) leads to the mean shear h∂xvyi=
−3
2Ω.
We focus here on two-dimensional flows. Several con-
siderations partly legitimize this choice. A first idea
comes from the limit of small Rossby numbers Ro =
U/(LΩ) 1, where Uand Lare characteristic veloc-
ity and length scales of the flow. The Taylor–Proudman
theorem [23, 24] ensures that any motion that occurs
on timescales longer than Ω−1becomes independent of
the zcoordinate. The flow is then approximately two-
dimensional and characterized by the well-known Taylor
columns [25], that are long columnar eddies oriented par-
allel to the rotation axis. A second idea comes from the
fact that protoplanetary disks are expected to be highly
stratified in the vertical direction z. This could be an-
other mechanism by which the gas dynamics becomes
almost two-dimensional. In this case, rather than being
long and columnar, eddies are now flat with the shape
of pancakes. Their formation originates then from in-
ternal gravity waves with group velocity perpendicular
to the gravitational force [26]. Both kinds of structures,
columns and pancakes, typically coexist in the presence
of both rotation and stratification [27]. However, the
global dynamics of protoplanetary disks actually involve
baroclinic amplifications, thermal transfers, radial strat-
ification, elliptical instabilities, etc., that are all affecting
the formation and survival of coherent eddies [28]. Study-
ing such instabilities is beyond the scope of our work.
We rather focus on two-dimensional solutions of (2) with
turbulent fluctuations and vortex dynamics maintained
in a developed, statistically steady state thanks to the
addition of an external random forcing.
In our settings, the fluid velocity solves the two-
dimensional incompressible Navier–Stokes equation. The
divergence-free turbulent fluctuations u=v+3
2Ωxey
are such that the associated vorticity ω=∇T·u≡
∂xuy−∂yuxsolves
∂tω+v· ∇ω=ν∇2ω−α ω +fω.(4)
Note that, because of incompressibility, the Coriolis force
does not contribute to the vorticity dynamics. We are
here applying a stochastic forcing fωin order to main-
tain a developed turbulent state. This forcing is assumed
Gaussian, with zero mean, homogeneous, isotropic, white
in time, and with spatial correlations concentrated at
large scales. As common in two-dimensional turbulence,
we suppose that vorticity experiences a linear friction
with coefficient α. Such a term prevents kinetic energy
from piling up in the flow. It typically originates, either
from the friction between different layers in the strati-
fied case, or from the effect of boundaries when the two-
dimensional effective dynamics corresponds to a vertical
average.
B. Direct enstrophy cascade
Figures 2a, b, and c show snapshots of the gas vor-
ticity field (normalized by τ−1
ω) for various values of the
rotation frequency Ω. One clearly observes, at a quali-
tative level, that shear sustains anticyclonic vortices (ro-
tating in the opposite direction of Ω,i.e. with a negative
sign, in blue). Because the space average of the fluctu-
ating vorticity ωis conserved and equal to 0, positive
values (in red) distribute in the background as denser
4
and denser filaments. Another observation is that shear
tends to stretch and align anticyclones with the direction
of the mean flow. At large values of the rotation, struc-
tures typically look like ellipses. We will turn back later
to estimate how their aspect ratio depends on the shear
rate.
0246
x
0
5
10
15
20
25
y
(a)
0246
x
(b)
0246
x
(c)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
−4
−3
−2
−1
0
1
2
3
4
Vorticity τωω
FIG. 2. Three different snapshots of the gas vorticity field
obtained in the absence of rotation and shear (a), for an in-
termediate value of Ω (b), and for a large value of Ω (c).
The developed turbulent state that we observed in
Fig. 2 is characterized by two inviscid quadratic invari-
ants, namely the kinetic energy E=1
2hkuk2iand the
enstrophy Z=1
2hω2i, whose budgets satisfy
dE
dt=0=−2νZ − 2αE − 3
2ΩR+εI,(5)
dZ
dt=0=−2νP − 2αZ+ηI.(6)
We have here introduced the Reynolds shear stress R=
huxuyiand the palinstrophy P= (1/2)hk∇ωk2i. The
quantities εIand ηIare the injection rates of kinetic
energy and enstrophy, respectively, and are fully deter-
mined by the stochastic forcing. Because of the absence
of vortex stretching in 2D, the only source of enstrophy in
(6) is forcing. This implies that, contrarily to 3D, energy
is not dissipated by viscosity but is transferred to large
scales by an inverse cascade (see, e.g., [29]). Still, vor-
ticity gradients, and thus palinstrophy, can become very
large in 2D, leading to a direct cascade of enstrophy.
The two injection rates define a forcing length scale
`f= (εI/ηI)1/2. Similarly, by balancing the different
dissipative rates in (5) and (6), one can introduce a fric-
tion scale `α= (E/Z)1/2and a viscous dissipation scale
`ν= (Z/P)1/2. These length scales define two Reynolds
numbers: an outer-scale Reynolds number Rα=`α/`f
which measures the ratio between inertial and frictional
forces, and a viscous Reynolds number Rν= (`f/`ν)2,
which balances inertial and viscous forces. These two
numbers prescribe the extensions of the inverse energy
cascade and of the direct enstrophy cascade, respectively.
We here focus on the direct cascade of enstrophy and we
prescribe `α&`f`ν, so that Rαis of the order of 1
and Rν1.
As seen from Fig. 2, a mean shear (Ω 6= 0) develops
anisotropies in the flow and the energy budget is affected
by the Reynolds stress. The importance of shear has to
be measured by non-dimensionalizing it with a character-
istic time scale of the flow. Still, the flow timescales are
themselves modified by shear, so this choice cannot be
made a priori. In our protocol, the only time scale that
is prescribed by the simulation setup is the forcing time
scale τf= (`2
f/εI)1/3=η−1/3
I. The influence of shear on
the energy and enstrophy budget is then measured by the
non-dimensional shear rate parameter σ=3
2Ωτf. The
three illustrating values shown in Fig. 2 are (a) σ= 0,
(b) σ= 5.7, and (c) σ= 14.2. In the developed regime
attained once σhas been prescribed, the shear needs to
be compared to the typical dynamical timescale of the
direct cascade, namely τω= (2Z)−1/2=hω2i−1/2.
Direct numerical simulations are performed using a
pseudo-spectral solver. To construct periodic solutions
that account for the mean flow, we follow [30–33] and
integrate (4) on a distorting frame defined by x0=x,
y0=y+3
2Ωt x. The integration domain is the peri-
odic rectangle (x0, y0)∈[0,2π[×[0,8π[, with a sufficiently
large aspect ratio in order to limit spurious geometri-
cal effects at large shears. The distorted grid is regu-
larly shifted back to the Cartesian grid at times mul-
tiple of the shear time 2
3Ω−1. Such methods are now
standard for local numerical simulations of astrophysi-
cal disks [34, 35] where they are coined as the “shear-
ing box” approach. In our two-dimensional settings, we
make use of the vorticity formulation (4), together with
the Biot–Savard law to obtain the fluctuating velocity u
as a function of the vorticity ω. The stochastic forcing is
approximated as shot noise with spatial power spectrum
concentrated over wavenumbers kkk= 4. Time marching
uses a second-order Runge–Kutta method, which is ex-
plicit for the non-linear term and implicit for the friction
and viscous terms.
We have performed several numerical experiments in
which we varied the rotation rate Ω while keeping con-
stant the forcing mechanisms, and thus the injection rates
εIand ηI. In all cases, transients are followed by the es-
tablishment of a statistical steady state associated with a
direct cascade of enstrophy. These states are character-
ized by given values of E,Z,Pand Rthat balance differ-
ently in the budgets. Figure 3 shows the terms entering in
the energy budget (5) as a function of σ=3
2Ωτf. It pro-
vides information about the large scales. The viscous dis-
sipation, which is here proportional to Z, decreases with
shear. The contribution from shear through the Reynolds
摘要：

Clustersofheavyparticlesintwo-dimensionalKeplerianturbulenceFabiolaA.GerosaandHeloseMeheutUniversiteC^oted'Azur,ObservatoiredelaC^oted'Azur,CNRS,LaboratoireLagrange,Nice,FranceJeremieBecUniversiteC^oted'Azur,Inria,CNRS,Sophia-Antipolis,FranceMinesParis,PSLUniversity,CNRS,Cemef,Sophia-Antipo...

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