Figure 4: Temporal evolution of the magnetic flux function with respect to
radius in a non-ideal simulation (top, _Gressel et al._ 2020) and in an ideal
(bottom, from _Jacquemin-Ide et al._ 2021), both taken in the disk midplane.
The isosurface represent the footpoint of magnetic field lines. Field lines
diffuse outwards in the non-ideal regime of protoplanetary disks
($R>1\,\mathrm{AU}$) but are advected inwards in the ideal MHD regime.

###  4.4 Instability transition zones and interaction with the wind/dust

Since the astonishing observation of HL Tau in 2015 by the ALMA telescope we
now know that protoplanetary disks usually display rings and gaps, and
sometimes nonaxisymmetric structures such as spirals. Since then, many studies
investigated for the nature of these perturbations which are seen both in the
gas and the dust component. An important role is played here by the transition
zones between different types of hydro or magneto-hydrodynamical
instabilities, each of them leading to a different level of turbulence and so
different levels of accretion stress. These variations lead to perturbations
in the gas and are often accompanied by secondary instabilities like the
Rossby Wave Instability (RWI, _Lovelace et al._ , 1999). These perturbations
in the gas surface density, lead to even larger perturbations for the solid
material, especially for intermediate grain sizes whose radial drift is
strongly modulated by the gas-pressure perturbations. In the following, we
will review the most prominent transition zones in protoplanetary disks for
which we expect long-lived perturbations in the gas and dust.

Figure 5: 3D global radiation non-ideal MHD simulations of the inner dead-zone
edge for a disk around a Herbig star (_Flock et al._ , 2017a) including
irradiation heating and dust sublimation. Left: 3D volume rendering of the
dust density. Right: 3D contour plot of the toroidal magnetic field. Annotated
are the location of the dust sublimation (0.47 au), the inner dead-zone edge
(0.8 au), and the location of dust trapping (1au). On the right, the domain’s
top half is cut away to show the magnetic fields on the midplane. The inner
disk shows fully established turbulence by the magneto-rotational instability.

####  4.4.1 MRI Dead-zone edges

The most prominent zone is the transition between the MRI-active and MRI-
damped regions in protoplanetary disks. This transition region could lead to
variations in the stress-to-pressure ratio $\alpha$ between 10 to 1000
dependent on the magnetic field strength, the ionization degree and the level
of the hydrodynamical turbulence (_Turner et al._ , 2014; _Lyra et al._ ,
2015; _Flock et al._ , 2017a). The resulting perturbation in the gas surface
density often leads to secondary instabilities like the RWI which can generate
vortices at these transition zones. Such vortex formations were observed at
the inner dead-zone edge (_Lyra and Mac Low_ , 2012; _Faure et al._ , 2015;
_Flock et al._ , 2017a) at the outer dead-zone edge (_Regály et al._ , 2012;
_Lyra et al._ , 2015; _Flock et al._ , 2015) and even for pure hydrodynamical
instability transition zones (_Flock et al._ , 2020). Those perturbations in
the surface density and in the pressure, are ideal places for planet formation
(_Chatterjee and Tan_ , 2014) as they lead to the reduction of the radial
drift or even directly to the trapping of dust grains (_Pinilla et al._ ,
2016; _Ueda et al._ , 2019; _Jankovic et al._ , 2021, 2022). This
concentration of dust grains in a ring structure could be even followed by
further dust concentration along the azimuthal direction inside the vortices
caused by the RWI (_Ruge et al._ , 2016; _Miranda et al._ , 2017). Fig. 5
shows the dust density and the magnetic field structure around the dead-zone
inner edge for a protoplanetary disk around a Herbig type star.

The detailed location of the dead-zone inner edge lies between 0.05 and 0.1 au
for a young T Tauri like star (_Mohanty et al._ , 2018; _Flock et al._ , 2019)
while it can reach 1 au for a Herbig star (_Flock et al._ , 2016) due to the
greater stellar luminosity. In certain situations when the disk is very
optically thick, the dead-zone inner edge could be even further outside, e.g.
in the case of large accretion rate and strong accretion heating due to high
optical thickness (_Mohanty et al._ , 2018; _Schobert et al._ , 2019;
_Jankovic et al._ , 2021).

The exact location of the outer dead-zone edge is more difficult to determine
as it depends on the external ionization sources, the amount of dust and gas,
and the dust size and porosity. Dependent on these parameters, the dead-zone
outer edge usually lies between 10 to 100 au (_Turner et al._ , 2014; _Lyra et
al._ , 2015; _Flock et al._ , 2015; _Pinilla et al._ , 2016; _Delage et al._ ,
2022).

####  4.4.2 Wind and magneto/hydro instabilities

Further radial variations in the surface density can be caused by the
transition zones of the different hydrodynamical instabilities (§2). In the
following, we want to emphasize the interaction between the wind-dominated
upper layers of the protoplanetary disks and the magneto/hydrodynamical
dominated midplane regions in the disk.

3D stratified simulations of the MRI showed both, vertical transport of energy
(Poynting flux) and vertical transport of gas material induced by buoyant
motions (_Io and Suzuki_ , 2014; _Suzuki and Inutsuka_ , 2014) which could
support the mass loading at the wind base. In such simulations it was found
that the MRI causes a magnetized corona region in the disk with fast radial
inflow motions, which can influence the structure of the overall disk wind
(_Leung and Ogilvie_ , 2020). This magnetized corona region together with a
wind layer above was also found in recent 3D stratified ideal MHD simulations
(_Zhu and Stone_ , 2018; _Jacquemin-Ide et al._ , 2021).

If magnetized winds dominate angular momentum transport in non-ideal regions,
and therefore gas accretion, it is very likely that the disk spontaneously
organizes into rings and gaps regularly spaced because of the non-linear
dependence of the wind torque on the surface density (_Béthune et al._ , 2017;
_Suriano et al._ , 2019; _Riols et al._ , 2020; _Cui and Bai_ , 2021, see also
chapter on structure formation in the same book). In these regions, the VSI
(§2.1) could also be at play. Here, the VSI is very efficient in the vertical
mixing of gas and dust. In these non-ideal regimes, the VSI could be important
for the mixing of gas and dust material into the wind layers (_Cui and Bai_ ,
2020).

##  5 PRESENT & FUTURE TESTS OF THEORETICAL MODELS

Figure 6: Spatial map showing the dominant physical process for the dynamics
and evolution of protoplanetary disks. We assume a shallow power law surface
density profile in the non-self gravitating regime with
$M_{\mathrm{disk}}\sim$ a few % of $M_{\mathrm{star}}$. The ZVI, COS, and the
VSI have distinct requirements on the disk structure and cooling times (§2.4).
Thus their occurrence and distribution within the dead zone can be model-
dependent.

###  5.1 Occurrence diagram

Fig. 6 presents the dominant physical processes which we currently believe
control the dynamics and evolution of protoplanetary disks. Dependent on the
local density of the gas and dust, the total dust surface area, the stellar
and external irradiation, and the amount of magnetic flux, the spatial extent
and the strength of these individual physical processes vary. As we have noted
in §4.4 these instabilities interact with each other. The MRI can mix and
transport gas and dust material into the wind layers. Between these different
instability zones we expect pressure perturbation due to the different
strength of the turbulence level. In the following we will discuss in more
detail the structure of the different gas and dust dynamics shown in Fig. 6.

####  5.1.1 Disk layer $H/R\lessapprox 4$

In the inner disk regions, the conditions for MRI activity are met between the
disk co-rotation radius and right behind the silicate sublimation zone. We
expect a second MRI-active region in the outermost disk regions which are
ionized by high energy cosmic rays and or non-thermal radiations. Vertically,
the MRI-active regions extend to several $H$, and is found to be more vigorous
in the disk corona layers above 2 $H$ (§4.2). Behind the silicate sublimation
radius, where temperatures decrease below 900 K, the gas ionization fraction
drops, damping the MRI activity as the influence of the MHD non-ideal terms
increases (§4.1). In this regime of high optical thickness, hydrodynamical
instabilities, more specifically the ZVI, the COV and the VSI could become
active (§2.3, §2.2, §2.1). The radial and vertical spatial extent of these
hydrodynamical instabilities depend strongly on the local opacity which sets
the timescale of the radiative transfer and so the heating and cooling in the
disk (§2.4). We highlight that the opacity and the ionization degree depend
both on the local amount and size of the dust material embedded in the gas
disk. Dust-drag instabilities occur mostly in regions with a local dust-to-gas
mass ratio of order unity. One promising candidate of these dust drag
instabilities is the streaming instability, operating close to the disk
midplane between the silicate sublimation zone and the outer disk regions
undergoing strong vertical mixing (§3.2). The vertical spatial extent of the
SI is focused on a region around one-tenth of a pressure scale height above
the midplane, and its overall activity assumes a modest to small amount of
vertical mixing ($\alpha_{\mathrm{S}}\leq 10^{-4}$) by the hydrodynamical
instabilities. We expect modulations of the surface density between the
different instability regions because of variations of the accretion stress.
Depending on the strength of these surface density modulations, we foresee
dust concentrations to occur due to the reduction of the radial drift velocity
(6). In these regions, we expect an enhanced SI activity or even gravitational
instability of the dust layer (§3.2, §3.3) . The surface density modulations
can also trigger the growth of the Rossby-Wave-Instability leading to vortex
formation and further dust concentration (§4.4).

####  5.1.2 Wind layer $H/R\gtrapprox 5$

MHD driven wind activity starts at the upper disk layers, in which the
coupling between the gas and the magnetic field becomes stronger (FUV layer)
and for which the magnetic field strength controls the motion of the gas
($\beta\sim 1$, §4.3.2). The radial extent of the MHD wind could in principle
cover the whole disk, depending on the strength and distribution of the
magnetic flux. The strength of the photoevaporative driven wind can be
stronger or weaker, dependent on the X-Ray, FUV and EUV fluxes (§4.3.1). The
photoevaporative effects become important outside 1 astronomical unity (for
approximately a 1 $M_{\odot}$ star), where the influence of the stellar
gravitational potential can be easier be overcome. The magneto- and
hydrodynamical instabilities, the photoevaporative and magnetically-driven
winds all together shape the evolution of the protoplanetary disk, while their
individual strength and significance might vary during the lifetime of the
disk.

###  5.2 Turbulence

The notion of gas disks being rendered turbulent has recently been tested with
a number of independent observational efforts. The most direct observational
constraints on turbulence come from the turbulent broadening of molecular gas
emission lines. Most recently, a series of studies quantified the strength of
turbulence at large radial distances ($>30$ AU) in nearby protoplanetary disks
using multiple molecular species to probe different heights above the mid-
plane (see Pinte et al. chapter for details). These observations generally
constrain $\alpha_{\mathrm{S}}<10^{-4}$–$10^{-3}$ for four out of five sources
for which this analysis was carried out (_Flaherty et al._ , 2015; _Teague et
al._ , 2016; _Flaherty et al._ , 2017, 2018). In the remaining source (DM
Tau), $\alpha_{\mathrm{S}}\sim 0.01$–0.5 (_Guilloteau et al._ , 2012;
_Flaherty et al._ , 2020). A comparison of a disk with a weak turbulence level
throughout (HD 163296) and the strongly turbulent DM Tau disk is shown in Fig.
7

While these recent studies were largely constrained to the outer disk
($>30$AU), earlier work by _Carr et al._ (2004) used band head emission of CO
lines originating from upper disk regions and in the inner $0.1$AU of SVS 13.
They found a normalized turbulent velocity $v_{\rm turb}/c_{\rm s}\sim 1$ in
these regions, in agreement with theoretical predictions (_Gammie_ , 1996;
_Fromang and Nelson_ , 2006; _Simon et al._ , 2011, 2015)222A constraint on
$\alpha_{\mathrm{S}}$ cannot be made here because $\alpha_{\mathrm{S}}$ is
effectively the height-integrated turbulence, and these observations probed
only the upper disk layers..

While the most direct measurement of disk turbulence is line broadening, there
are two other approaches to independently quantify the level of turbulence in
disks. The first is quantifying the diffusion of dust grains due to
turbulence; since dust grains are (at least somewhat) coupled to gas motions,
this approach provides an indirect, yet independent constraint. Assuming a
standard dust settling model, _Pinte et al._ (2016) constrained
$\alpha_{\mathrm{S}}$ to be a few $10^{-4}$ in HL Tau. Focusing instead on the
radial structure of the rings _Dullemond et al._ (2018) found a lower limit of
$\alpha_{\mathrm{S}}\approx 10^{-4}$ for a handful of sources, though this
precise value is dependent on the poorly constrained grain sizes and width of
the gaseous pressure bumps. _Ohashi and Kataoka_ (2019) conducted radiative
transfer modeling that included self-scattering polarization to constrain the
grain size and distribution in HD 163296; they found that while
$\alpha_{\mathrm{S}}\lesssim 1.5\times 10^{-3}$ in one of the gaps (at 50 AU),
$\alpha_{\mathrm{S}}$ could be as large as 0.3 in the 90 AU gap.

These constraints are generally consistent with modeling the SED and comparing
with those from a sample of Herbig stars, T Tauri stars, and brown dwarfs, as
was done by _Mulders and Dominik_ (2012). They found $\alpha_{\mathrm{S}}\sim
10^{-4}$ assuming a standard grain size distribution (as well as a constant
$\alpha_{\mathrm{S}}$ throughout the disk), though their observations were
also consistent with $\alpha_{\mathrm{S}}\sim 10^{-2}$ if the grain size
distribution had fewer small grains or a lower gas-to-dust ratio.

Since radial angular momentum transport is likely driven by non-thermal
motions (though, these do not have to be turbulent per se; e.g., spiral
density waves), the degree of viscous spreading of disks can serve as another
indirect constraint on turbulence. The primary quantification of viscous
spreading is a measurement of the outer edge of the disk as a function of disk
age and there have been several attempts to put constraints on
$\alpha_{\mathrm{S}}$ based on such a measurement, both in terms of the dust
disk (e.g. _Rosotti et al._ 2019) and the gas disk (e.g., _Ansdell et al._
2018; _Trapman et al._ 2020).

(_Najita and Bergin_ , 2018) examined a number of disk size measurements from
the literature, and found that generally, younger Class I disks were more
compact than their Class II counterparts, lending support to the notion of
viscous angular momentum transport. _Ansdell et al._ (2018) measured the
radius that contains 90% of the CO J=2-1 flux in Lupus and found a large range
of $\alpha_{\mathrm{S}}$ values, ranging from $3\times 10^{-4}$ to 0.09.
_Trapman et al._ (2020) followed up on this work and concluded that using the
90% CO flux technique leads to an overestimate of $\alpha_{\mathrm{S}}$
values, finding instead that disks in Lupus are consistent with viscous
spreading with $\alpha_{\mathrm{S}}\approx 10^{-4}$–$10^{-3}$.

There are a number of complications that warrant caution in interpreting these
results on viscous spreading. For instance, in the case of measuring disk
sizes from the dust, the radial drift of particles has to be accounted for,
particularly for mm-sized grains (such as observed with ALMA in the continuum)
as these grains are only marginally coupled to the gas at large radial
distances. Furthermore, as pointed out by _Rosotti et al._ (2019), it is
possible that existing surveys of dust disks lack the sensitivity to detect
viscous spreading. Finally, in the case of the gas disk, changes to abundances
for chemical tracers (such as CO) have to be accounted for as well as
processes, such as photoevaporation (_Winter et al._ , 2020).

Taken together, all of these studies are consistent with
$\alpha_{\mathrm{S}}\sim 10^{-4}$. However, caution is warranted here as well.
First, the precise value of $\alpha_{\mathrm{S}}$ is still subject to
considerable uncertainty, either due to variations from source to source
(e.g., DM Tau _Guilloteau et al._ 2012; _Flaherty et al._ 2020) or limitations
of the methods employed (e.g., _Rosotti et al._ 2019). Furthermore, it is
inaccurate to strictly equate turbulent velocities and $\alpha_{\mathrm{S}}$,
as is done in both the molecular line and dust diffusion measurements; both of
these approaches assume $\alpha_{\mathrm{S}}\propto v_{\rm turb}^{2}$ (see
Pinte et al. chapter for instance). However, for angular momentum to be
transported outward, the radial and azimuthal turbulent velocities must be
correlated. For instance, considering only the Reynolds stress,
$\alpha_{\mathrm{S}}\propto\delta v_{r}\delta v_{\phi}$, which is not in
general equivalent to $v_{\rm turb}^{2}$.333On the other hand, it is
appropriate to say that $\alpha_{\mathrm{D}}\propto v_{\rm turb}^{2}$, as this
definition quantifies the dust diffusion (see Section 1.3). However, we are
restricting ourselves to a discussion of the magnitude of turbulent angular
momentum transport in this section. Thus, a disk can be strongly turbulent,
while transporting angular momentum outward, inward, or not at all.

These uncertainties aside, $\alpha_{\mathrm{S}}=10^{-4}$ is too small for
angular momentum transport to be completely dominated by turbulent/viscous
processes by about two orders of magnitude (eq.3). Thus, at face value, this
suggests that some other mechanism (e.g., MHD winds, §4.3.2) are also
contributing to angular momentum transport, a conclusion that is consistent
with theoretical models and numerical simulations that state MHD winds should
dominate the accretion process (e.g., _Gressel et al._ 2015; _Béthune et al._
2017; _Bai_ 2017). Ultimately, however, with ALMA continuing its full
operations well into the coming decade, observational constraints on
turbulence in disks will only improve with time. Even resolving the large-
scale turbulent eddies directly might become possible with ALMA using
spatially and spectrally resolved line emission (_Barraza-Alfaro et al._ ,
2021). Such observations of the line emission will help to identify the
physical mechanism generating the turbulent motions in protoplanetary disks.

Figure 7: Left: Upper limits to turbulent velocities in HD 163296 as a
function of radius $R$ and mid-plane height $z$. The colors denote the species
from which there is a majority of emission; the CO transitions are at large
$z$ whereas C18O and DCO+ are from lower in the disk. Also included on the
plot are lines of constant $\Sigma$, $H$ and 3$H$ ($H$ being the gas scale
height) and where CO is ice. In this source, turbulence is at most a few
percent of the sound speed. From _Flaherty et al._ (2017). Right: The
turbulent velocity as a function of radius as measured from CO emission in DM
Tau (hashed lines), compared to the results from previous work using CS
(_Guilloteau et al._ 2012; grey shaded region). As opposed to HD 163296, DM
Tau exhibits strong turbulence, consistent with theoretical predictions
_Flaherty et al._ (2020).

###  5.3 Winds

Disk winds have been detected through small blue-shifts (a few km/s) in a
number of emission lines from molecular, atomic, low-ionization species (see
the chapter by Pascucci et al and _Ercolano and Pascucci_ (2017)) and there
have been several theoretical efforts aiming at modeling the observations to
constrain wind properties and infer their driving mechanisms (e.g. _Ercolano
and Owen_ , 2010, 2016; _Weber et al._ , 2020; _Nemer et al._ , 2020; _Gressel
et al._ , 2020). Despite the focussed observational and theoretical effort in
this field in the last five years, several important questions remain open, in
particular, the relative occurrence of thermal (photoevaporative) and MHD-
driven winds is still unknown.

The determination of the mass loss rate and radial profile is crucial to
constrain the nature of the disk wind, as thermal winds are not expected to be
launched very close to the star, where gravity is strong and the escape
temperatures are high. The determination of the emission radius of a
(blueshifted) line profile by assumption of Kepler broadening is problematic,
given the velocity gradients within the wind (_Weber et al._ , 2020). For this
reason, alternatives are currently being explored to directly observe and test
disk winds. Two promising avenues are provided by (i) scattered light
observations of high inclination objects and (ii) future high-angular
resolution radio observations (e.g. with the nGVLA).

Several theoretical efforts have focused on the entrainments of dust grains in
thermal and MHD disk winds (_Owen et al._ , 2011; _Hutchison et al._ , 2016a,
b; _Miyake et al._ , 2016; _Franz et al._ , 2020; _Hutchison and Clarke_ ,
2021; _Booth and Clarke_ , 2021). These models predict significant differences
in the grain sizes that could be entrained in thermal and MHD winds as well as
in the spatial extent of the wind that would be populated by grains of
different sizes. These might be observable signatures that may help
identifying the nature of the winds, particularly for objects with high
inclinations. _Franz et al._ (2020) conclude that the detection of winds in
scattered light might be challenging with current instrumentation, but it may
still be possible for disks where small grains are well mixed with the gas in
the disk atmosphere. Indeed a possible detection of a dusty disk wind may have
occurred in the scattered light observations of My Lup presented by _Avenhaus
et al._ (2018).

Spatially-resolved observations of free-free emission at cm wavelengths, which
could be obtained by future radio telescopes, such as the Next Generation Very
Large Array (ngVLA), present a novel and exciting avenue to constrain the
nature of disk winds. _Ricci et al._ (2021) demonstrate that for the case of
TW Hydra, theoretical models of X-ray photoevaporation (_Picogna et al._ ,
2019) and simple analytical MHD wind models (_Weber et al._ , 2020) predict
very different radial profiles for the cm emission, which could be resolved by
the ngVLA.

Disks with inner cavities, so-called ”Transition disks”, have also been
studied intensively, as they may be objects on the verge of dispersal through
photoevaporative winds, particularly in the case where the measured accretion
rate on the central object is smaller than $10^{-8}M_{\odot}/yr$. Disks with
large cavities that are still vigorously accreting, are difficult to be
explained by photoevaporative models (_Ercolano and Pascucci_ , 2017) and
could be the result of planet-disk interactions or formed by magnetic winds
(_Combet and Ferreira_ , 2008; _Wang and Goodman_ , 2017). This last scenario
is appealing since large cavity transition disks tend to exhibit accretion
rates comparable to standard disks, despite the presence of a large and deep
cavity (_van der Marel et al._ , 2018), indicating transonic accretion
velocities in the cavity. In the future, deep ALMA observations will allow the
analysis of bright molecular lines that are optically thick and originate in
the warm surface layers of disks. This will allow measuring $v(R)$ using the
1st velocity moment of spectral datacubes (e.g. _Casassus et al._ , 2021) and
explore the kinematical properties of the intracavity gas of these objects.

###  5.4 Solar system bodies

Planetesimals in the Solar System – remnants of the planet formation stage
more than 4.5 billion years ago – provide an extensive ground for comparing
simulations of turbulence and planetesimal formation to the observed outcome
in the Solar System. The asteroid belt is dominated in mass by the largest
bodies (Ceres, Vesta and Pallas). These may in fact already be considered to
be protoplanets that underwent some degree of accretional growth after their
formation (_Johansen et al._ , 2015). Vesta differentiated and separated its
iron core from a rocky mantle, while Ceres formed later and experienced only
heating enough to melt the ice to form phyllosilicates.

Asteroids with a characteristic radius of approximately 100 km constitute a
second mass peak in the asteroid belt. _Bottke et al._ (2005) first suggested
that this size could be the peak of the birth size distribution of
planetesimals and that smaller asteroids are fragments of this birth
population, while the larger sizes are the result of collisional growth
(_Morbidelli et al._ , 2009). The Initial Mass Function (IMF) of planetesimals
formed by the streaming instability was evolved numerically by mutual
collisions and pebble accretion in _Johansen et al._ (2015). This study showed
indeed that the asteroids between 100 and 500 km in size may result from a
combination of pebble accretion and mutual collisions within a birth size
distribution peaking at 100 km. The accretion mode transitions to become
dominated by pebble accretion once the bodies reach 1000 km in diameter. The
smallest of these bodies are akin to Vesta and Ceres which could thus
represent bodies on the threshold to undergoing rapid growth towards planetary
bodies.

It has been proposed that small asteroids are completely dominated by
fragments of larger asteroids, with no discernible contribution from
primordial planetesimals for sizes less than 45 km _Delbo’ et al._ (2017).
This observation is in some tension with the IMF of planetesimals which
appears to follow a power law towards smaller masses (_Johansen et al._ ,
2015; _Simon et al._ , 2016, 2017). However, more recently _Li et al._ (2019)
used very high resolution simulations to demonstrate that the IMF may turn
down from a power law at small sizes, so that the IMF approaches more a log-
normal shape with tapering both at the small and at the large (_Schäfer et
al._ , 2017) end. _Schreiber and Klahr_ (2018) proposed that turbulence caused
by the streaming instability itself suppresses fragmentation on scales below
the characteristic scale of the filament. Future simulations at very high
resolution or using adaptive mesh refinement techniques will be needed to shed
light on the shape of the IMF for bodies down to 10 km.

The size distribution of the Kuiper belt has a power law similar to the
asteroid belt for small sizes, while the largest bodies in the cold classical
belt have a much steeper size distribution (_Fraser et al._ , 2014). The
latter could be the original exponential tapering from the birth process
(_Abod et al._ , 2019).

The binarity of planetesimals is another imprint of the formation process. The
cold classicals have a binary fraction of at least 20% (_Noll et al._ , 2008)
and another 20% are contact binaries (_Thirouin and Sheppard_ , 2019).
_Nesvorný et al._ (2010) showed in N-body simulations of collapsing pebble
clouds how such clouds would contract to form binary planetesimals (see
Section 3.4 for more details). This appears to be a unique prediction of
planetesimal formation from overdense clumps of pebbles (which could form by
the streaming instability or in a pressure bump). The formation of contact
binaries is poorly understood but could be a result of gas drag and Kozai-
Lidov cycles for initially highly inclined binary orbits (_Naoz et al._ ,
2010; _McKinnon et al._ , 2020; _Lyra et al._ , 2021). The cold classical
Kuiper belt contact binary Arrokoth was visited by the New Horizons mission in
2019 and the perfect shape of the two components indicates collision speeds
lower than a few m/s (_McKinnon et al._ , 2020).

Another prediction of planetesimal formation by the streaming instability is
that planetesimals that avoid heating and catastrophic disruption should
survive as piles of primordial pebbles from the solar protoplanetary disk
(_Wahlberg Jansson and Johansen_ 2014, 2017; see Section 3.4 for more
details). The Rosetta mission to the comet 67P/Churyumov-Gerasimenko found a
number of indications that this is the case: the interior structure is
homogeneous on scales less than a few meters while the density is low
consistent with porosity at the pebble scale _Lorek et al._ (2016), (2) the
size distribution of dust at the surface appears to peak in the mm-cm size
range and the outflow of material from the surface has a broader distribution
from mm sizes (interpreted to be primordial pebbles) to 10 m chunks of likely
consolidated pebble-rich material (_Blum et al._ , 2017b).

Evidence from debris disks is harder to interpret. Classical cascade models
start typically with small planetesimals that fragment faster _Krivov and
Wyatt_ (2021). However, 100-km-sized pebble piles may also be consistent with
the observed time-scales of dust production _Krivov et al._ (2018).

##  6 CONCLUSIONS AND PERSPECTIVES

In this chapter, we reviewed the multifaceted theories aimed at describing
planet-forming disks, from the clumping of tiny dust grains to the launching
of large-scale winds. We covered various instabilities and feedback processes
whose implications for planet formation are only beginning to become clear.

The innermost region of the disk near the star is well-ionized and must be
strongly turbulent because of the MRI (§4.2), but this is not necessarily the
case for the outer ($R\gtrsim 1\mathrm{AU}$) regions (§4.1), which are
spatially-resolved for instance with ALMA. For these outer regions, predicting
the dynamics and the strengths of the MRI and other instabilities requires
knowing how the ionization state varies with position. Even if the MRI is
suppressed, several thermally-driven instabilities have been established as
potential sources of hydrodynamic turbulence and structure formation in PPDs
(§2). Their basic properties, including the underlying physical mechanisms and
onset requirements, are understood. The main outstanding issues are whether
and how these instabilities operate under realistic PPD conditions, and what
kinds of turbulence or other flows they trigger (in terms of
$\alpha_{\mathrm{S}}$ and $\alpha_{\mathrm{D}}$ for instance). To this end,
radiation hydrodynamic simulations should be developed, treating heating and
cooling processes consistently with the disk structure – key factors that
determine whether the instabilities will operate and how they will evolve.
This endeavor should be carried out paying close attention to observational
constraints, which now include direct probes of turbulence in the disk bulk
(§5.2).

Turbulence and other flows in the gas in turn impact dust coagulation, since
planetesimal formation now appears to involve an interplay between linear
instability, nonlinear saturation, and triggers of strong clumping of solids
(§3). Much progress has been made on the conditions for triggering strong
clumping, which depend on dust size, solid-to-gas column density ratio, and
radial pressure gradient (§3.2). There has been progress also on the resulting
planetesimal mass functions with self-gravity included in the numerical
simulations (§3.4). While earlier studies focused on mono-disperse dust size
distributions, investigation of the streaming instability with a spread of
dust sizes and of the interaction of planetesimal formation with external
turbulence have just begun, yet already offer some surprises such as the
quenching of the linear streaming instability in some cases (§3.2.1). In the
near future, global simulations should be one of the major fronts in advancing
this topic.

Global-scale magnetic fields are also expected in PPDs as relicts from the
systems’ formation (§4.1). These can dominate the large-scale gas dynamics by
launching thermo-MHD disk winds (4.3.1-§4.3.2). In contrast to earlier wind
models, these can operate with weak global magnetic fields while accounting
for the accretion rates observed in class 2 objects, even where the disks are
otherwise non-turbulent. A major uncertainty here is the field strength, which
is not predicted by any complete theory, and for which we have few empirical
constraints. Modeling the long-term evolution of the large-scale field and
better measuring it, either through astrophysical observations or by analyzing
meteorites from the early solar system, should be focuses of future research.

The combined outcome of these diverse, dynamic processes is likely to be a
disk that is no longer smooth and monotonically-varying with distance from the
star. Radial variations in the accretion stress, due to transitions in the
ionization state, the main heating and cooling processes, and the magnetic
field transport will inevitably lead to structure in the gas surface density
profile and so the pressure (§4.4). Local pressure variations naturally have
outsized impacts on the dust surface density profile through the spatial
gradients they induce in the dust-gas drift speed (§1.3). From the apparent
ubiquity and complexity of these interactions, it is clear that models of disk
evolution, planet formation, and planet-disk interaction can advance only if
we move away from viscous, alpha-type disk models, which attempt to gloss over
fundamental physical and chemical ingredients. The next generation of models
will properly treat turbulence, dust transport, and wind-driven accretion,
either explicitly or through sub-grid or effective models. Such sub-grid
models may be based on simulations at higher resolution or including more
processes over a limited time or space domain. Modeling efforts furthermore
should be expanded to more closely consider disks at various evolutionary
stages around stars of various masses, to advance our understanding of how
disks’ diversity shapes the planet population across our Galaxy.

## ACKNOWLEDGMENTS

GL acknowledges support from the European Research Council (ERC) under the
European Union Horizon 2020 research and innovation program (Grant agreement
No. 815559 (MHDiscs)). BE acknoweldges support from DFG Research Unit
”Transition discs” (FOR 2634/1, ER 685/8-2) and from the Excellence Cluster
ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2094 -
390783311. MF acknowledges funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (grant
agreement No. 757957). MKL is supported by the Ministry of Science and
Technology of Taiwan (grants 107-2112-M-001-043-MY3, 110-2112-M-001-034-,
110-2124-M-002-012-, 111-2124-M-002-013-) and an Academia Sinica Career
Development Award (AS-CDA-110-M06). CCY acknowledges the support from NASA via
the Emerging Worlds program (grant number 80NSSC20K0347) and via the
Astrophysics Theory Program (grant number 80NSSC21K0141). JB and PM
acknowledge grant NSF AST-1510708 and computational resources from multiple
XSEDE awards. JJG was supported by NASA grant 17-ATP17-0094. AJ acknowledges
funding from the European Research Foundation (ERC Consolidator Grant
724687-PLANETESYS), the Knut and Alice Wallenberg Foundation (Wallenberg
Scholar Grant 2019.0442), the Swedish Research Council (Project Grant
2018-04867), the Danish National Research Foundation (DNRF Chair Grant
DNRF159) and the Göran Gustafsson Foundation. WL, JBS, OMU, CCY, and ANY
acknowledge the support from NASA via the Theoretical and Computational
Astrophysics Networks program (grant number 80NSSC21K0497). WL acknowledges
grant NSF AST-2007422, and computational resources from XSEDE through
allocation TG-AST140014. JBS acknowledges support from NASA under Emerging
Worlds through grant 80NSSC20K0702. NJT contributed at the Jet Propulsion
Laboratory, California Institute of Technology, under contract with NASA and
with the support of NASA grant 17-XRP17_ 2-0081. JS acknowledges support from
the Royal Society Te Apārangi, New Zealand through Rutherford Discovery
Fellowship RDF-U001804. OMU acknowledges NASA’s New Horizons Kuiper Belt
Extended Mission and NASA ISFM.

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