Astronomy & Astrophysics manuscript no. main ©ESO 2022January 6, 2022

Dust entrainment in magnetically and thermally driven disk windsP. J. Rodenkirch1 and C. P. Dullemond1

Institute for Theoretical Astrophysics, Zentrum für Astronomie, Heidelberg University, Albert-Ueberle-Str. 2, 69120 Heidelberg,Germany

January 6, 2022

ABSTRACT

Context. Magnetically and thermally driven disk winds have gained popularity in the light of the current paradigm of low viscositiesin protoplanetary disks that nevertheless present large accretion rates even in the presence of inner cavities. The possiblity of dustentrainment in these winds may explain recent scattered light observations and constitutes a way of dust transport towards outerregions of the disk.Aims. We aim to study the dust dynamics in these winds and explore the differences between photoevaporation and magneticallydriven disk winds in this regard. We quantify maximum entrainable grain sizes, the flow angle, and the general detectability of suchdusty winds.Methods. We used the FARGO3D code to perform global, 2.5D axisymmetric, nonideal MHD simulations including ohmic andambipolar diffusion. Dust was treated as a pressureless fluid. Synthetic observations were created with the radiative transfer codeRADMC-3D.Results. We find a significant difference in the dust entrainment efficiency of warm, ionized winds such as photoevaporation andmagnetic winds including X-ray and extreme ultraviolet (XEUV) heating compared to cold magnetic winds. The maximum entrain-able grain size varies from 3 µm − 6 µm for ionized winds to 1 µm for cold magnetic winds. The dust flow angle decreases rapidlywith increasing grain size. Dust grains in cold magnetic winds tend to flow along a shallower angle compared to the warm, ionizedwinds. With increasing distance to the central star, the dust entrainment efficiency decreases. Larger values of the turbulent viscosityincrease the maximum grain size radius of possible dust entrainment. Our simulations indicate that diminishing dust content in theouter regions of the wind can be mainly attributed to the dust settling in the disk. The Stokes number along the wind lauching frontstays constant in the outer region. In the synthetic images, the dusty wind appears as a faint, conical emission region which is brighterfor a cold magnetic wind.

Key words. protoplanetary disks - hydrodynamics - radiative transfer - Magnetic fields - Magnetohydrodynamics (MHD) - ISM:jets and outflows

1. Introduction

The limited lifetime of a few million years (Haisch et al., 2001;Mamajek et al., 2004; Ribas et al., 2015) of protoplanetary diskshas sparked interest in various mechanisms that transport and ex-tract angular momentum from the disk. One possibility to driveaccretion onto the central star is viscous turbulence, usually re-ferred to as α-viscosity in the context of disks (Shakura & Sun-yaev, 1973). Turbulence might be generated by a multitude ofphysical instabilities in gas and dust. The well studied magne-torotational instability (MRI) (Balbus & Hawley, 1991) presentin an ionized, differentially rotating disk threaded by a weakmagnetic field can lead to values of α roughly ranging from 10−3

to 10−2 and it was studied in a local shearing box (Fromang &Papaloizou, 2007; Fromang et al., 2007; Bai, 2011) as well as inglobal simulations (Dzyurkevich et al., 2010; Flock et al., 2011).The MRI can only operate if the ionization fraction of gas is suf-ficiently high, which is not necessarily the case in the midplaneof the disk (Igea & Glassgold, 1999), where nonideal magne-tohydrodynamic (MHD) effects such as ohmic diffusion dom-inate. These so-called dead zones prevent magnetically drivenaccretion in the midplane, whereas layered accretion in the ion-ized upper part of the disk atmosphere is still possible (Gammie,1996). In the more dilute outer regions of the disk, the ionizationfraction rises again and the edge between the inner dead zone andthe outer MRI-active part might cause jumps in the density pro-

file and therewith enable dust or pebble traps (Lyra et al., 2009;Dzyurkevich et al., 2010; Dzyurkevich et al., 2013). Purely hy-drodynamic instabilities such as the global baroclinic instability(Klahr & Bodenheimer, 2003) and the convective overstability(Klahr & Hubbard, 2014), on the other hand, are alternative tur-bulence driving mechanisms that can also operate in the weaklyionized parts of the disk. In the presence of short gas coolingtimes, the vertical shear instability (VSI) causes vertical oscilla-tions which are able to transport small dust grains towards theupper layers of the disk (Stoll & Kley, 2014, 2016; Flock et al.,2020; Blanco et al., 2021). Vortices triggered by the VSI in com-bination with the Rossby wave instability (Lovelace et al., 1999)may act as long-lived dust traps (Manger & Klahr, 2018; Pfeil &Klahr, 2021).Since recent observation hint towards small viscosity and thussmall values of α on the order of 10−4 to 10−3 (Flaherty et al.,2015, 2017), the effect of disk winds might be a possible expla-nation for the finite disk lifetimes. In the context of photoevapo-ration, ionizing radiation either from the central star or the fromthe stellar environment heats the surface layers of the disk anddrives winds due to the thermal pressure gradient. Dependingon the dominant type of radiation, the wind mass loss rates areon the order of 10−10 M yr−1 with EUV radiation (Hollenbachet al., 1994; Font et al., 2004), and 10−8 M yr−1 to 10−7 M yr−1

with FUV radiation (Adams et al., 2004; Gorti & Hollenbach,

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2009) and X-ray radiation (Ercolano et al., 2009; Owen et al.,2010, 2012; Picogna et al., 2019; Ercolano et al., 2021; Picognaet al., 2021).An alternative way to produce strong disk winds would be byconsidering the effect of magnetic fields. In the presence of aglobal magnetic field the differential shearing motion of the diskgenerates a magnetic pressure gradient, accelerating matter andthereby driving the outflow. In the strong field limit, the gas fol-lows the magnetic field lines and is magneto-centrifugally ac-celerated, as described in Blandford & Payne (1982). Contraryto photoevaporation, magnetically driven winds extract angularmomentum from the disk and drive accretion. The accretion flowhas to pass by the mostly vertically aligned magnetic field inthe disk which only allows stationary solutions if the couplingbetween gas and magnetic field is weak, thus if significant dif-fusive nonideal MHD effects are present (Wardle & Koenigl,1993; Ferreira & Pelletier, 1993; Ferreira & Pelletier, 1995). Themagnetic launching mechanism is considered to be responsiblefor high-velocity jet outflows and has been studied extensivelyin global resistive simulations (Casse & Keppens, 2002; Zanniet al., 2007; Tzeferacos et al., 2009; Sheikhnezami et al., 2012;Fendt & Sheikhnezami, 2013). Stepanovs & Fendt (2014) andStepanovs & Fendt (2016) determined the transition betweenmagnetocentrifugally driven and magnetic pressure-driven jetsat β ≈ 100 where β is the ratio of thermal over magnetic pres-sure.With the growing interest in winds in the context of proto-planetary disks, numerous studies have been performed overthe last decade, first in the form of local shearing box simula-tions (Suzuki & Inutsuka, 2009; Suzuki et al., 2010; Fromanget al., 2013; Bai & Stone, 2013; Bai, 2013). Global nonidealMHD simulations have been carried out by Gressel et al. (2015)including ohmic and ambipolar diffusion finding MRI-channelmodes to develop in the upper layers of the disk. They further-more stress the importance of ambipolar diffusion enabling amore laminar wind flow and decreased mass loss rates. Béthuneet al. (2017) find asymmetric wind flows, nonaccreting configu-rations and self-organizing structures such as rings. The resultsof Suriano et al. (2018) corroborate the ones by Béthune et al.(2017) such that rings emerge in ambipolar diffusion dominateddisks by reconnection events. A combination of both photoe-vaporation and magnetically driven winds, also called magne-tothermal winds were studied by Wang et al. (2019) and Ro-denkirch et al. (2020) in the regime of magnetic pressure sup-ported winds. With increasing magnetic field strengths, pho-toevaporative winds smoothly transition into magnetic windsdriven by the magnetic pressure gradient in the upper layersof the disk (Rodenkirch et al., 2020). In contrast to the resultsof Bai (2017a); Suriano et al. (2018); Wang et al. (2019) andRodenkirch et al. (2020), simulations in Gressel et al. (2020)demonstrate magneto-centrifugal winds mainly driven by mag-netic tension forces.Dust entrainment in protoplanetary disk winds has been mostlyexamined in photoevaporative winds. Low gas densities in thewind limit the maximum grain size to a few µm. For EUV-photoevaporation Owen et al. (2011) determined a grain sizelimit of ≈ 2.2 µm. Hutchison et al. (2016b) performed smallscale smoothed particle hydrodynamics simulations of a dustyEUV-driven wind and formulated an analytical expression forthe maximum entrainable dust grain size. They conclude thattypically the limiting size is less than 4 µm for typical T Tauristars and they also argue that dust settling may lower this limitfurther. Franz et al. (2020) simulated dust entrainment in post-processed XEUV-wind flows of Picogna et al. (2019) and found

a larger maximum dust size of 11 µm. Dust entrainment in mag-netically driven winds was tested by vertical 1D models in thework of Miyake et al. (2016), concluding that grains in the rangeof 25 µm to 45 µm can float up to 4 gas scale heights above thedisk mid plane. Giacalone et al. (2019) used a semianalyticalMHD-wind description to compute the dust transport and reporta maximum dust size of 1 µm. Considering the effect of radiationpressure, submicron dust grains can be blown away from the diskand support clearing out remaining dust (Owen & Kollmeier,2019).In this paper we present fully dynamic, multifluid, global, az-imuthally symmetric, nonideal MHD simulations of protoplane-tary disk winds including XEUV-heating to model photoevapo-rative flows. We aim to study dust entrainment in magneticallyand thermally driven disk winds in a dynamic fashion that alsoconsiders the vertical dust distribution due to turbulent diffusionof dust grains. We thus extend the previous study in Rodenkirchet al. (2020) by including dust as a pressureless fluid. We fur-thermore postprocess the dust density maps to examine observa-tional signatures of such dusty winds.In Sec. 2 we describe the theoretical and computational conceptsof the model. The results are presented in Sec. 3 and Sec. 4 dis-cusses limitations of the model as well as the comparison to otherstudies. Finally, in Sec. 5 we give concluding remarks.

2. Model

The simulations were carried out with the FARGO3D codeBenitez-Llambay & Masset (2016); Benitez-Llambay et al.(2019) in a 2.5D axisymmetric setup using a spherical mesh withthe coordinates (R, θ, ϕ). Throughout this work the cylindrical ra-dius will be referred to as r, whereas the spherical radius will bedenoted as R =

√r2 + z2. Observational inclinations, flow an-

gles and viewing angles will be expressed in terms of latitudeequivalent to θ ranging from 90° to −90°.

2.1. Basic equations

The FARGO3D code solves the following set of equations:

∂tρg + ∇ ·(ρg ug

)= 0 , (1)

ρg

(∂tug + ug · ∇ug

)= −∇P + ∇ ·Π − ρg∇Φ (2)

+1

4π(∇ × B) × B , (3)

∂tρd,i + ∇ ·(ρd,i ud,i + ji

)= 0 , (4)

ρd,i[∂tud,i +

(ud,i · ∇

)ud,i

]= −ρd,i∇Φ +

∑i

ρifi , (5)

where Eqs. 1 and 4 describe the mass conservation of gas anddust with ρg and ρd,i being the gas and dust density, respectively.The suffix i denotes the corresponding dust species. Eqs. 2 and5 represent the momentum equations of gas and dust. We notethat we did not consider the dust backreaction onto the gas inEq. 2 since the dust-to-gas mass ratio is assumed to be low. Thesymbol P refers to the thermal gas pressure, B the magnetic fluxdensity vector, u the velocity vectors, Φ = −GM∗

R the gravitationalpotential and fi the drag forces between dust and gas.Turbulent mixing of dust grains caused by the underlying turbu-lent viscosity in the gas was considered by defining the diffusionflux ji (Weber et al., 2019)

ji = −Di

(ρg + ρd,i

)∇

(ρd,i

ρg + ρd,i

), (6)

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P. J. Rodenkirch and C. P. Dullemond: Dust entrainment in disk winds

Table 1. Simulations and key parameters

Label β0 α X-ray heating rin [au] rout [au] rc εion Resolution Simulation timeb5c2 105 10−4 3 1 200 2 103 500 x 276 2000 orbitsb4c2 104 10−4 3 1 200 2 103 500 x 276 2000 orbitsb4c2np 104 10−4 7 1 200 2 103 500 x 276 1000 orbitsb4c2a3 104 10−3 3 1 200 2 103 500 x 276 1000 orbitsb4c2eps2 104 10−3 3 1 200 2 102 500 x 276 1000 orbitsphc2 [n/a] 10−4 3 1 200 2 103 500 x 276 2000 orbitsphc10 [n/a] 10−4 3 1 200 10 103 500 x 276 2000 orbits

Notes. The numerical resolution is given in number of cells in radial and polar direction. The simulation time is measured in units of orbits at theinner radius rin.

where the diffusion coefficient Di depends on the kinematic vis-cosity ν and the Stokes number St (Youdin & Lithwick, 2007):

Di = ν1 + St2(1 + St2

)2 . (7)

The dimensionless Stokes number can be expressed in terms ofthe stopping time ts as follows:

St = ΩKts = ΩKρmat ai

ρg vth, (8)

with the Keplerian angular frequency ΩK =√

GM∗/r3, the mate-rial density of the dust grain ρmath and the mean thermal velocityof the gas:

vth =

√8kBTg

µmpπ. (9)

The Boltzmann constant is expressed as kB, the gas temperatureas T , the mean molecular weight as µ and the proton mass as mp.The viscosity stress tensor is given by

Π = ρν

[∇v + (∇v)T −

23

(∇ · v) I], (10)

where I is the unit tensor, and the energy equation can be statedas

∂e∂t

+ ∇ · (ev) = −P∇ · v . (11)

The adiabatic equation of state is given by

P = (γ − 1)e , (12)

where γ = 5/3 is the adiabatic index. The induction equa-tion including ohmic and ambipolar diffusion reads as follows(Béthune et al., 2017):

∂B∂t− ∇ ×

(u × B −

4πcηohm · J + ηam · J × b × b

)= 0 , (13)

where b = B/|B| the normalized magnetic field vector, J theelectric current density vector, ηohm the ohmic diffusion coeffi-cient and ηam the ambipolar diffusion coefficient.

2.2. Disk model

2.2.1. Initial conditions of the gas in the disk

The vertically integrated surface density profile Σ(r) was as-sumed to be a power law Σ(r) = Σ0

(rr0

)−pwith the reference

surface density Σ0 at the characteristic radius r0 of the systemand the volumetric gas density takes the form

ρ(r, z) = ΘH(r − rc)Σ0

√2πH(r)

(rr0

)p

exp[

1h2

(r

√r2 + z2

− 1)],

(14)

where h = H/r is the local aspect ratio and H = cs/ΩK thegas scale height with the local isothermal sound speed cs =√

kBT (r)/(µmp). Within a certain cutoff radius rc, we set the den-sity to zero with a sharp transition and we thus mimiced a diskwith an inner cavity of size rc. The transition is initially takencare of by the Heaviside function ΘH(r− rc) in Eq. 14. The sharptransition quickly softens and relaxes to the shape of a smoothinner rim during the simulation run after a few orbits. The cavityin the model was applied for both gas and dust.In a simple blackbody disk being irradiated by stellar light witha grazing angle φ onto its surface, the temperature scales withradius as (Chiang & Goldreich, 1997)

Tg(r) =

(φ

2

) 14(R∗

r

) 12

T∗ =

(φL∗

8πr2σSB

)1/4

, (15)

where R∗, T∗ and L∗ are the stellar radius, temperature and lumi-nosity, respectively. The Stefan-Boltzmann constant is denotedby σSB. Thus, we assumed a radial power law profile for thetemperature with T (r) = T0 r−q and the gas scale height scaleswith radius as H(r) ∝

√T (r) ∝ r(3−q)/2.

The initial values of the azimuthal gas velocity are slightly sub-keplerian (Nelson et al., 2013):

vφ(r, z) = ΩKr√

h2(p + q) + (q + 1) −qr

√r2 + z2

. (16)

We assumed no initial velocities in the radial and polar direction,that is vr = vθ = 0.

2.2.2. Coronal gas structure

The hydrostatically stable coronal gas structure with densityρc(R) and pressure Pc(R) serves as a density floor throughout thesimulation and prevents excessively low initial densities whichprevents numerical instabilities. We used the same prescription

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0 2 4 6 8 10au

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

au

Photoevaporation

0 2 4 6 8 10au

= 105

0 2 4 6 8 10au

= 104

19

18

17

16

15

14

13

12

11

19

18

17

16

15

14

13

12

11

19

18

17

16

15

14

13

12

11

g/cm

3

Fig. 1. Gas density maps of the simulations phc2, b5c2 and b4c2 after 500 orbits at 1 au. Magnetic field lines are represented by the white linesin the corresponding panels. The color map scaled logarithmically annotated by the exponent to a base of ten and represents the volume density ofthe gas in g/cm3.

0 200 400 600 800t [T0]

10 14

10 12

10 10

10 8

Mas

s los

s [M

yr1 ]

Photoevaporation

gasdust (0.1 m)dust (1 m)

0 200 400 600 800t [T0]

10 14

10 12

10 10

10 8

= 105

0 200 400 600 800t [T0]

10 14

10 12

10 10

10 8

= 104

0 200 400 600 800t [T0]

10 14

10 12

10 10

10 8

= 104 (cold)

Fig. 2. Total wind mass loss rates of gas and dust for the simulation runs phc2, b5c2, b4c2 and b4c2np. Results from 10 µm grains are excludeddue to the negligible outflow and the lack of entrainment in the wind for all runs shown above.

as used in the previous models in Rodenkirch et al. (2020) basedon Sheikhnezami et al. (2012).The density was set to

ρc(R) = ρc0(ri)( ri

R

) 1γ−1

, (17)

where ri is the radius at the inner boundary of the simulationdomain and γ the adiabatic index. The corresponding pressureprofile is given by:

Pc(R) = ρc0(ri)γ − 1γ

GMri

( ri

R

) γγ−1

. (18)

The density at the inner boundary ρc0(ri) is related to the disk’sdensity at the midplane and ri with the density contrast δ =

ρc0(ri)/[Σ(ri)/

(√2πH

)]. Depending on Σ0, we set the density

contrast to values ranging from δ = 10−7 to 5 · 10−7. In absenceof X-ray / EUV heating, the gas temperature was directly set tothe coronal temperature.

2.2.3. Dust structure

Vertical dust settling towards the midplane becomes more effec-tive in the upper layers of the disk and the overall vertical thick-ness of the dust layer is over-estimated in models of a dust scale

height uniquely depending on midplane values as in Dubrulleet al. (1995) (Dullemond & Dominik, 2004). We therefore usedthe following vertical dust distribution

ρd(r, z) = εdg ρ(r, z)

St(r, z = 0)α

[exp

(z2

2H2

)− 1

]−

z2

2H2

(19)

given in Fromang & Nelson (2009). The dust-to-gas mass ratioof the dust fluids is denoted by εdg and the dimensionless diffu-sion coefficient follows the α-viscosity prescription ν = αcsH(Shakura & Sunyaev, 1973) with the kinematic viscosity ν. Weassumed a constant value for the Schmidt number Sc = 1.The initial azimuthal velocities were set to vd,φ = ΩK

√r2 + z2.

Similar to the gas, the initial radial and polar velocities of thedust fluids were set to zero.

2.2.4. Ionization rate

In all of the following models X-ray radiation was considered in-cluding both a radial and scattered component. The parametriza-tion was based on the prescription of Igea & Glassgold (1999)

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P. J. Rodenkirch and C. P. Dullemond: Dust entrainment in disk winds

0 2 4 6 8 10au

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0au

Photoevaporation

0 2 4 6 8 10au

= 105

0 2 4 6 8 10au

= 104

0 2 4 6 8 10au

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

au

Photoevaporation

0 2 4 6 8 10au

= 105

0 2 4 6 8 10au

= 104

0 2 4 6 8 10au

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

au

Photoevaporation

0 2 4 6 8 10au

= 105

0 2 4 6 8 10au

= 104

21

20

19

18

17

16

15

14

13g/

cm3

21

20

19

18

17

16

15

14

13

g/cm

3

21

20

19

18

17

16

15

14

13

g/cm

3

Fig. 3. Dust density maps for the simulations phc2, b5c2 and b4c2 after 500 orbits at 1 au. Dust velocity streamlines are annotated by the whitearrowed lines. The upper, center and lower three panels visualize the flow of grains with size 0.1 µm, 1 µm and 10 µm, respectively.

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0 2 4 6 8 10au

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0au

0.1 m

0 2 4 6 8 10au

1 m

0 2 4 6 8 10au

10 m

21

20

19

18

17

16

15

14

13

g/cm

3

Fig. 4. Dust density maps for the simulation b4c2np after 500 orbits at 1 au. Dust velocity streamlines are annotated by the white arrowed lines.

and Bai & Goodman (2009):

ζX =ζX,sca

exp(−

Σtop

Σ1

)0.65

+ exp(−

Σbot

Σ1

)0.65 ( Rau

)−2

(20)

+ ζX,rad exp(Σrad

Σ2

)0.4 ( Rau

)−2

.

Both coefficients were multiplied by a factor of 20 comparedto the cited values to model a fiducial X-ray luminosity of2 × 1030 erg s−1 and were set to ζX,rad = 1.2 × 10−10 s−1 andζX,sca = 2 × 10−14 s−1. The critical column densities accountedto Σ1 = 7 × 1023 cm−2 and Σ2 = 1.5 × 1021 cm−2.The luminosity increase was chosen to be compatible with thephotoevaporation model of Ercolano et al. (2008); Picogna et al.(2019) where the luminosity LX = 2 × 1030 erg s−1 was used asthe fiducial value. Taking into account the mass-luminosity rela-tion presented in Flaischlen et al. (2021), Fig. 1 based on Getmanet al. (2005), a luminosity of LX = 1 × 1029 erg s−1 would corre-spond in average to a M∗ ≈ 0.15 M star. The increased value isthus better suited for the T Tauri star with M∗ = 0.7 M assumedin our model.Additionally, the cosmic ray ionization rate ζcr was modeled bythe parametrization of Umebayashi & Nakano (2009) where acharacteristic column density of 96 g cm−2 was chosen. The sym-bols Σbot and Σtop refer to the column densities integrated alongthe polar direction, starting from the lower boundary (suffix bot)]and the upper boundary (suffix top). The radial column densitiesΣrad were computed starting from the radial inner boundary.

ζcr =5 × 10−18 s−1 exp(−

Σtop

Σcr

) 1 +

(Σtop

Σcr

)3/4−4/3

(21)

+ exp(−

Σbot

Σcr

) 1 +

(Σbot

Σcr

)3/4−4/3

To account for short-lived radio nuclides in the disk, an ion-ization rate of ζnuc = 7 × 10−19 s−1 caused by the 26Al isotope(Umebayashi & Nakano, 2009) acts as an ionization rate floorvalue. The resulting total ionization rate then simply sums up toζ = ζX + ζcr + ζnuc.

2.2.5. Ionization fraction

We computed the ionization fraction of the gas via a semiana-lytical model for fractal dust aggregates elaborated by Okuzumi(2009). The main assumption here is an approximately Gaussiancharge distribution in the dust grains. This approximation is rea-sonably well fulfilled if the dust aggregate consists of at least400 monomers Dzyurkevich et al. (2013). In this approach, theionization equilibrium was described by the following equationsOkuzumi (2009):

11 + Γ

−

siui

seueexp(Γ) +

1Θ

Γg(Γ)√1 + 2g(Γ) − 1

= 0 , (22)

with the parameter Θ:

Θ =ζnge2

siuiσan2dkBT

. (23)

The function g(Γ) is evaluated as:

g(Γ) =2αζng

siuiseue(σnd)2

exp(Γ)1 + Γ

. (24)

After solving Eq. 22 numerically for Γ with a simple Newton-Raphson scheme, finally the ion and electron densities ni and necan be calculated:

ni =ζng

siuiσnd

√1 + 2g(Γ) − 1(1 + Γ)g(Γ)

, (25)

ne =ζng

seueσnd

√1 + 2g(Γ) − 1

exp(−Γ)g(Γ). (26)

The sticking coefficients si and se were set to 1 and 0.3 respec-tively, as described in Okuzumi (2009). The thermal ion andelectron velocities ui,e follow:

ui,e =

√8kBTπmi,e

, (27)

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P. J. Rodenkirch and C. P. Dullemond: Dust entrainment in disk winds

where me describes the electron mass and the ion mass is set tomi = 24 mp, corresponding to magnesium.With the assumption of fractal dust aggregate with a dimensionof D = 2, the dust aggregate size becomes a = a0N1/2 with thenumber of monomers N of size a0. Similarly, the geometric crosssection σ of these aggregates can be expressed as σ ≈ πa2

0N. Thedust number density nd can be calculated as nd = εion ng/( 4π

3 a30N)

with the dust-to-gas ratio εion used in the ionization model.In the limit of a vanishing dust-to-gas mass ratio the ionizationfraction xe approaches

xe =ne

ng=

√ζ

αn, (28)

where ζ is the total ionization rate, α = 3 × 10−6 cm3s−1/√

Tthe dissociative recombination rate coefficient and n the neutralnumber density of the molecular species (Oppenheimer & Dal-garno, 1974; Ilgner & Nelson, 2006).In the same manner as described in Bai (2017a), we increased theionization fraction in the wind region as a proxy for the effect ofFUV-radiation:

xfuv = 2 · 10−5exp

− (Σrad

Σfuv

)4

+0.3Σfuv

Σrad + 0.03Σfuv

. (29)

2.2.6. Magnetic field

The magnetic field was set such that the initial plasma beta β atthe midplane is independent of the radius r. The poloidal fieldwas initialized from the vector potential Aφ

Aφ(r, θ) =

√8πPβ

(rr0

) 2p+q+14 [

1 + m · tan (θ)−2]− 5

8 (30)

described in Bai (2017a) and exhibits an outward-bent, hourglassshape depending on the parameter m where m = 0 would corre-spond to a completely vertical field. In order to avoid excessivelysmall time steps in the low density regions close to the rotationaxis, we limited the local Alfvén velocity vA = B/

√4πρg to

18vK(r0) by increasing the density in the cell accordingly, similarto Riols & Lesur (2018). This procedure does not affect the bulkof the wind flow.

2.2.7. Nonideal MHD

Using the electron fraction xe, the local ohmic diffusion coeffi-cient ηohm can be written as (Blaes & Balbus, 1994):

ηohm =c2mnme

4πe2

〈σv〉exe≈

234 T12

xe

cm3

s, (31)

where mn and me are the neutral and electron mass respectively.The electron-neutral collision frequency was set to 〈σv〉e = 8.28·10−10 T−1/2cm3s−1 (Draine et al., 1983). The ambipolar diffusioncoefficient ηam becomes:

ηam =B2

4π〈σv〉in2gxi

. (32)

The ion-neutral collision rate was set to 〈σv〉i = 1.9 ·10−9cm3s−1

(Draine et al., 1983). Since large diffusivities severely limitedthe simulation time step a super time stepping scheme was im-plemented and more details are given in Appendix C. We fur-thermore limited the magnetic Reynolds number for ambipolardiffusion Rm = cs H/ηam to Rm ≥ 0.05, similar to Gressel et al.(2015), confirming that this approximation does not significantlyalter the simulation outcome.

2.2.8. Heating and cooling

We employed the same temperature presciption as described inRodenkirch et al. (2020) based on Picogna et al. (2019) to mimicthe outflow due to photoevaporation in the simulation runs mod-eling a warm (magneto-) photoevaporative wind. According totheir model, XEUV heating is significant up to radial columndensities of Σr,crit = 2.5 × 1022 cm−2. In these regions the temper-atures Tphoto were updated to the values given by Eq. 1 in Picognaet al. (2019). Beyond the critical column density Σr,crit the tem-perature was cooled to the original gas temperature T0 = Tg.In the case of the cold magnetic wind model b4c2np (see Tab. 1),the temperature in the corona was only slightly increased by 20% compared to the bulk temperature in the disk. We used a sim-ple β-cooling recipe in the whole simulation domain to exponen-tially damp the temperatures to T0 on a time scale βcool

T (tn + ∆t) = T (tn) +(T (tn) − T0

)exp

(∆tβcool

). (33)

2.3. Boundary conditions

We employed simple symmetric boundary conditions for most ofthe variables and boundaries where the values of the active cellsare copied to the ghost cells. At the inner radial boundary, theradial velocity followed an outflow boundary condition wherevr = 0 in the ghost zone if vr > 0 in the active cell. In the con-trary case, vr was copied to the ghost cell. The same procedurewas applied to the outer radial boundary with the opposite sign.With this formulation mass influx from outside of the domainwas avoided.At the θmin and θmax boundaries vφ, vθ, Bφ, Bθ and the EMF inradial direction change sign in the ghost zone to mimic a polarboundary. In MHD disk models the radial inner boundary con-dition can lead to spurious effects in the simulation domain. Thesame applies to the outer radial boundary where, if ill defined,artificial collimation affects the flow (Ustyugova et al., 1999).To avoid these difficulties we added a zone with additional artifi-cial ohmic diffusion, so that the coupling between the gas and themagnetic field is weak enough that spurious effects were dampedand the simulation domain remained largely unaffected. A simi-lar approach was used at the inner boundary in Cui & Bai (2021)to stabilize the inner region of the domain.Dimensionless values of the artificial ohmic resistivity wareηohm,in = 2 × 10−4 and ηohm,out = 1 × 101 at the inner and outerdamping zone, respectively. The extent of the damping zones is10% of the inner and outer radius. With the choice of these val-ues we found that the simulation domain and the timestep arelargely unaffected by the boundary conditions. Adding an innercavity with greatly reduced amount of gas and dust furthermoreensures negligible impact of the inner boundary condition on thewind flow.

2.4. Radiative transfer model and post-processing

We post-processed the dust density outputs with the radiativetransfer code RADMC-3D (Dullemond et al., 2012). In order toobtain three dimensional dust density distributions the axisym-metric simulation outputs were extended in the azimuthal direc-tion by repeating the densities in 300 cells in φ. For the thermalMonte-Carlo step and the image reconstruction with scatteredlight nphot = 106 and nphot_scat = 108 photon packages were used,respectively.The thermal Monte-Carlo calculation was based on a modified

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version of the Bjorkman-Wood method (Bjorkman & Wood,2001) and the resulting temperatures were used in the syntheticimages instead of the temperature profile prescribed in the MHDsimulations. Every synthetic observation was created with onlyone dust species since the limited number of fluid would lead toartificial discrete step-like features in the image. Only consider-ing one dust species at a time may lead to unrealistic temperatureprofiles but after checking the output of the thermal calculationof RADMC-3D the temperatures agreed with the dust tempera-ture estimated computed in Appendix A in the relevant regions.We further emphasize that the synthetic images were producedin the H-band at a wavelength of 1.6 µm where scattered lighteffects dominate and thermal emission from the dust grains atthese low temperatures is negligible.The dust opacities were computed with optool (Dominik et al.,2021) and are similar to the DIANA opacities (Woitke et al.,2016) with a material mixture of 87 % pyroxene, 13 % carbon(by mass) but with an increased porosity of 64 % to be consis-tent with a dust material density of ρmat = 1 g/cm3 in the simula-tions. Concerning the dust grain sizes, we computed the opacitiesfor a1 = 0.1 µm, a2 = 1 µm and a3 = 10 µm corresponding tothe dust sizes in the simulations. Anisotropic scattering effectswere included by using the Henyey-Greenstein phase functionapproximation (Henyey & Greenstein, 1941). A classical T Tauristar (CTT) was assumed with a temperature of T∗ = 4000 K anda radial extent of R∗ = 2.55 R at a distance of 150 pc whichis in the range of typical properties of observed CTTs (Alcaláet al., 2021). The resulting images were convolved with a gaus-sian beam size of 75 mas to account for typical observational res-olution limits using VLT / SPHERE (Beuzit et al., 2019; Aven-haus et al., 2018).

3. Results

Sec. 3.1 describes the relevant parameters and simulation runs.In the following subsections we present the simulation resultsstarting with a description of the wind flow structure emergingfrom the disk in Sec. 3.2 and continue with the dust dynamics inSec. 3.3. The effectiveness of dust entrainment and the outflowangle of the grains is addressed in Sec. 3.4, Sec. 3.5 and Sec. 3.6.Finally, in Sec. 3.7 we compare synthetic observations computedfrom the simulation output with existing observations.

3.1. Parameters & normalization

Tab. 1 summarizes the key parameters of the different simulationruns presented in this study. The first part of the label describeswether a magnetic wind (for instance b4 with β = 104) or a pho-toevaporative wind without magnetic fields was modeled (withlabel ph). Subsequently, the letter “c” refers to the cavity loca-tion (such as c2 for a cavity at 2 au). The sublabels np, a3 andeps2 refer to “no photoevaporation”, α = 10−3 and εion = 10−2,respectively.In the code all quantities were treated in dimensionless units thatare scaled as follows:

v0 = r0 ΩK(r0) , (34)

ρ0 = M/r30 , (35)

P0 = ρ0v20 , (36)

where the length unit was set to r0 = 5.2 au, the semi-majoraxis of Jupiter as a natural scale of stellar systems. The ini-tial density profile was scaled to reach a surface density of

Σ = 200 g/cm2 at r = 1 au and decreases radially with power lawslope of p = 1. The central star mass was set to 0.7 M and thedisk follows a moderate aspect ratio of H/r0 = 0.055 (r/r0)1/4

which corresponds to a flared disk with a temperature slope ofq = 1/2. These parameters correspond to a stellar luminos-ity of L∗ = 1.5 L with the assumption of a grazing angle ofφ = 0.02. The luminosity is in line with the stellar parameters ofT∗ = 4000 K and R∗ = 2.55 R mentioned in Sec. 2.4.A temperature floor of Tmin = 10 K and a cooling time scale ofβcool = 10−4 was applied. The fiducial value of the inner cav-ity size was chosen to be rc = 2 au. Further parameters are theadiabatic index of γ = 5/3 corresponding to an ideal atomic gasand the mean molecular weight µ = 1.37125, consistent with thevalue chosen in Picogna et al. (2019) for the composition of theISM (Hildebrand, 1983).In Sec. 2.2.5 the model necessitates an assumption of the dustcontent in the disk. In this work the ionization model wasparametrized with a static value of the dust-to-gas ratio εion =10−3 and a monomer size of a0 = 5 µm that constitute aggregateswith 400 particles, resulting in a grain size of 100 µm. Increas-ing the dust-to-gas ratio by one order of magnitude did not alterthe ionization fraction profile by a great amount, as describedin Appendix B. In order to simplify the model, the dynamicallysimulated dust fluids were not coupled to the ionization modeland the relative insensitivity to the chosen parameters in the mid-plane supports this approximation.All simulation runs were carried out with three dust species ofthe sized a = 0.1 µm, 1 µm and 10 µm, intended to represent thethree cases of full entrainment, slow decoupling, and fast decou-pling from the gas in the wind. The material density of the grainswas set to ρmat = 1 g/cm3.The computational grid adopts a spherical symmetry and con-tains 500 logarithmically spaced cells in radial direction whereas276 cells were used in the polar (θ) direction. In order to increasethe vertical resolution closer to the disk mid plane, the polar cellswere arranged in a stretched grid, with a cell size at the polarboundaries that was twice as larger compared to the mid plane.With this approach a resolution of 7 cells per H in polar direc-tion at r = r0, z = 0 was reached. To avoid numerical issues atthe polar boundaries the grid limits in this direction were chosensuch that θmin ≈ 87° and θmax ≈ −87°. The radial domain spansfrom 1 au to 200 au as indicated in Tab. 1.

3.2. Gas flow structure

Fig. 1 visualizes the gas wind flow for the simulation runsphc2, b5c2 and b4c2. The latter two include magnetic fieldswhereas the former only includes the photoevaporation recipe.Comparing both magnetic winds in Fig. 1 the magnetic fieldstructure consists mostly of straight field lines for β = 104,whereas the weaker field shown in the central panel exposes amore irregular field structure towards the inner region of thedisk. For this field strength and configuration the wind is in thetransition between a thermally dominated and a magneticallydominated wind. We note that these are simulation snapshotsand not averaged wind flows.In terms of wind total mass loss rates all of these three runsdisplay similar results, as shown in Fig. 2. Generally, the windconverges to a steady-state flow after a time scale about 200orbits at 1 au and the simulation time chosen for these runs isthus sufficient to explore relevant wind dynamics in the innerregion.The sharp cutoff of the disk at r = 2 au quickly relaxes towardsa stable equilibrium after a few orbits. The choice of the initial

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P. J. Rodenkirch and C. P. Dullemond: Dust entrainment in disk winds

101 102

r [au]

10 9

10 8

Mas

s los

s [M

yr1 ]

gas

phc2b5c2b4c2b4c2a3

101 102

r [au]10 12

10 11

10 10

10 9 dust (0.1 m)

101 102

r [au]10 12

10 11

10 10

10 9 dust (1 m)

Fig. 5. Cumulative mass loss rates of gas and dust for the simulation runs phc2, b5c2, b4c2 and b4c2a3 starting from 3 au. The cylindrical radiusor foot point of the loss is obtained by tracing backwards the corresponding streamline from the outer radial boundary down to 5 scale heightsabove the disk mid plane. The flow field is averaged over 50 orbits at 1 au starting at 600 orbits. The vertical dashed lines mark the limit of theregion where 99 % of the mass loss occurred.

shape of the inner rim is thus not relevant for the wind flow.We observe accretion of gas through the inner cavity whena magnetic field is present. This effect, which can be bestobserved in the right panel of Fig. 1, is more pronounced forlarger magnetic field strengths and is caused by wind-drivenaccretion.The photoevaporation gas mass loss rates reach≈ 2.83 × 10−8 M yr−1, in good agreement with Picogna et al.(2019). The mass loss rates in the simulation runs b5c2 andb4c2 are ≈ 3.13 × 10−8 M yr−1 and ≈ 3.03 × 10−8 M yr−1,respectively. In the case of a purely magnetically drivenwind without any heating by ionizing radiation of modelb4c2np the wind mass loss rates are similar with a value of3.19 × 10−8 M yr−1. The mass loss rates were computed at 95% of the outer radial simulation boundary. The region withinfive gas scale heights of the disk was excluded from the massloss rate to avoid the corruption of the results from circulationwithin the disk.

3.3. Dust flow structure

In Fig. 3 the dust flows are shown for the grain sizes 0.1 µm,1 µm and 10 µm. Generally, 10 µm sized grains are hardly en-trained with the wind flow since the drag force is too weak tocounter act the gravitational pull. 0.1 µm grains are well coupledto the gas flow and can be considered as tracers of the wind.Throughout the analysis we refer to the flow angle or inclinationangle, meaning the angle enclosed between the dust flow direc-tion and this disk mid plane axis at z = 0. The inclinations θd ofthe dust flows are smaller in terms of latitude for the thermallydriven wind shown on the left-hand side compared to the mag-netically driven wind on the right-hand side. The magnetic fieldis well coupled to the gas in the wind region where the ioniza-tion fraction xe,i surpasses 10−4 and the field topology forces thewind flow to be more inclined compared to the thermally drivenwind.The same phenomenon also appears for 1 µm grains shown in thesecond row of Fig. 3. Due to the weaker coupling and the lack ofpressure support the dust flow is less inclined with respect to thegaseous wind compared to smaller grains. The dust flow exhibitswave-like patterns in the case of β = 105 and becomes morestable for larger magnetic field strengths. These wave structurescoincide with the irregular magnetic field patterns displayed inFig. 1. We furthermore point out that the dust flow consisting of1 µm grains is efficiently launched in the inner part of the disk,whereas beyond several au the wind significantly weakens. The

dusty wind cannot be sustained in the lower density regions fur-ther out and up in the disk since the Stokes number significantlyincreases.An example of a dust flow in a cold magnetically driven wind isgiven in Fig. 4 for an initial β = 104. The inclination with respectto the mid plane is smaller compared to the warm magnetother-mal or photoevaporative winds. Grains with a size of 1 µm areentrained at a rather shallow inclination angle and similar to thewarm winds, dust entrainment is negligible considering 10 µmdust particles. While material in the winds including photoevap-oration is ejected with a velocity of ≈ 15 km s−1 the outflow ve-locity of the cold magnetic wind in b4c2np only reaches val-ues of ≈ 5 km s−1. The decreased wind speed causes a smallerdrag force between gas and thus lowers the dust entrainment ef-ficiency. Given that the mass loss rate is approximately equal tothe one of the heated winds, the gas and dust densities in thewind region are significantly higher.In Fig. 5 the cumulative mass loss rate depending on the cylin-drical radius of the foot point of the wind is shown. The windstreamlines were traced backwards from the cells close to theouter radial boundary and the foot point was registered when thestreamlines crosses the surface at 5 gas pressure scale heights.Only streamlines with foot point rf ≥ 3 au were considered sincethe wind flow structure and the limited resolution did not in ev-ery case ensure a successful construction of traceable stream-lines close to the inner cavity. The mass loss rates were obtainedfrom a velocity and density field averaged over 50 orbits at 1 austarting from 600 orbits.Warm magnetic winds lead to more dust entrainment and astronger magnetic field increases the dust mass loss rate signif-icantly. The dashed lines in Fig. 5 mark the radius within fromwithin 99 % of the mass loss occurs, which illustrate less effi-cient dust entrainment for larger grains further out in the disk.Numerical values are 32.3 au, 34.5 au, 43 au and 88.8 au for thesimulations phc2, b5c2, b4c2, b4c2a3 and 0.1 µm grains, re-spectively. Analogously, the limits for 1 µm grains are 14.6 au,12.7 au, 14.6 au and 44.3 au. The general trend is that the dustgrains are preferably entrained starting from the inner regionsof the disk. The limiting radius decreases with increasing dustsize and hence we expect a much larger ejection region of dustfor the smallest grain size. Considering the results of the simu-lation b4c2a3 with an increased viscosity of one order of mag-nitude (α = 10−3) the limiting radius lies significantly furtheroutward. Increased turbulent diffusion effectively transports dustgrains towards the wind launching front at larger radii.Going back to Fig. 2 the mass loss rate for the two relevant dustspecies stabilizes over a few hundred orbits in the case of a ther-

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mal wind. In the case of magnetically driven winds however, dustmass loss rates reach a quasi-steady state but are slowly increas-ing throughout the simulation. In the presence of a cold mag-netic wind, the mass loss rates of larger micron-sized dust grainsare lower compared to the ionized winds since the flow angleis rather shallow and the entrainment efficiency suffers from thedecreased wind speed.

3.4. Maximum grain size and flow angle

In order to quantify the maximum entrained grain size and theentrainment angle, we chose the approach to numerically inte-grate streamlines of various dust species based on the gas veloc-ity field from the simulations. The following system of equationwas solved numerically with the given velocity field of the cor-responding simulation output:

vd,r = vd,φ −GM∗(

r2 + z2) 32

r −vd,r − vg,r

ts, (37)

vd,z = −GM∗(

r2 + z2) 32

z −vd,z − vg,z

ts. (38)

The numerical integration was carried out with the VODE solverBrown et al. (1989). We verified that the streamlines were com-parable to the ones in the dust velocity field output from the sim-ulation. No significant deviations were visible.The results are shown in Fig. 6 for 20 logarithmically distributedgrain sizes ranging from 0.1 µm to 10 µm. The resulting trajec-tories match the corresponding dust streamlines from the actualsimulation. As the starting point we chose r = 8 au, z = 2.5 auwhich is situated well above the wind launching front. As ex-pected from the simulation outputs 10 µm grains fall back to-wards the disk surface in the case of winds including photo-evaporation (models phc2, b5c2 and b4c2). Generally, grainsare lifted more efficiently in the case of warm, magnetic winds.Grains larger than ≈ 3 µm to 4 µm are lifted further but eventu-ally fall back onto the disk at larger radii in the presence of athermal wind.For warm, magnetically driven winds, the limiting grain sizeshifts towards ≈ 6 µm. In a cold, magnetic wind the picturechanges and only submicron dust grains are lifted far from thedisk surface and particles larger than 1 µm fall back onto the diskor are not entrained at all. The decreased wind speed is respon-sible for the lower dust entrainment efficiency as mentioned inSec. 3.3.Since the dust flow inclination angle is strongly dependent on

the grain size, we quantified the inclination by fitting linear func-tions to the dust trajectories presented in Fig. 6. We only consid-ered the final 25 % part of the cylindrical radius of the trajec-tory for the fits. The corresponding slopes and inclination an-gles based on the simulations phc2, b5c2, b4c2 and b4c2np areplotted in Fig. 7. For small grains and warm winds the differencein the inclination angle is rather small. The inclination of theflow of 1 µm grains ranges from roughly 67° to 69°. Consider-ing 3 µm grains however, the inclination angle is 36.1° assuminga thermal wind compared magnetic winds with 44.4° and 46.5°for β = 105 and β = 104, respectively. Generally, the inclinationangle is larger for warm, magnetic winds, especially consideringmicron sized grains.A significant difference in inclination is however observed ina cold magnetic wind. While 0.1 µm-sized dust grains are en-trained by an angle of 59.1°, the inclination drops down rapidlyto roughly 25° for micron-sized grains, as expected.

3.5. Wind launching surface

Since the dust entrainment efficiency seems to be dependent onthe dust reservoir at the location of the foot point of the wind,we therefore provide a more detailed analysis of the radial de-pendencies of the dust entrainment efficiency in this subsection.We only focus on winds including photoevaporation where thewind launching surface is marked by a sharp transition in the gasdensity. Fig. 8 and Fig. 9 visualize vertical slices of normalizedgas and dust densities as well as the respective Stokes numberprofile. As expected, in Fig. 8 the dust scale height of 10 µmsized grains narrows with increasing radius as the Stokes num-ber within the bulk of the disk increases due to the negative radialdensity slope. The kink at roughly 4.5 gas scale heights repre-sents the wind launching surface. Clearly, the dust scale heightis too small to lift a considerable amount of grains to the launch-ing surface, especially at larger radii.A comparison between the different dust species at 5 au is givenin Fig. 9. The trend of larger dust scale heights with smallerStokes numbers is readily visible. Weaker wind flows by sev-eral orders of magnitude appear for 1 µm sized grains comparedto 0.1 µm particles. Only the vertical distribution of the smallest0.1 µm grains that are well coupled to the gas appear to corre-spond the gas density distribution.In the following, we aim to examine the dust entrainment effi-ciency depending on the radial location in the disk. We thereforetraced various quantities, such as dust-to-gas ratio, Stokes num-ber, temperature and gas number density along the wind launch-ing surface. Numerically we defined this surface to be locatedabout 0.3 gas scale heights above the kink in the vertical gas den-sity profile in order to decrease the impact of fluctuations close tothe surface of the disk. The kink location was determined by nu-merically evaluating local extrema in the second derivative of thevertical gas density profile which we find to be a robust methodto find the launching surface.The corresponding quantities at these positions are displayedin Fig. 10 where thermally and magnetically driven winds with0.1 µm and 1 µm-sized grains are taken into account. Clearly, thedust entrainment efficiency, defined by the dust to gas ratio closeto the wind launching front shown in the upper most panel ofFig. 10, decreases significantly with increasing radius. At theinner part of the flow close to the cavity at 2 au, 0.1 µm- sizedgrains are basically perfectly entrained with the wind for bothphotoevaporative and magnetically driven outflows since the ini-tial dust-to-gas ratio is εdg = 10−2. In the outer part of the disk thedust-to-gas ratio in the wind decreased by one to two orders ofmagnitude. For 1 µm-sized grains the effect is more pronouncedwith a decline of six orders of magnitude. It becomes apparentthat generally the magnetically driven wind with β = 104, rep-resented by the dotted lines, leads to a higher dust-to-gas ratioin the wind flow compared to weaker field strengths or a purelythermally driven wind.One could argue that dust entrainment becomes less effectiveat larger radii because of the more diluted gas in the wind andthereby larger Stokes numbers. This is however not the case asshown in the second panel of Fig. 10. In the inner part of the diskup to ≈ 8 au to 9 au the Stokes number close to the launching sur-face increases until it stagnates and remains constant at the outerpart of the disk. The decrease in dust entrainment efficiency canthus not be attributed to a weaker coupling between gas and dustin these regions. The gas density in the wind close to the launch-ing surface indeed decreases with radius as shown in the bottompanel of Fig. 10. On the other hand, the gas temperature heatedby the ionizing radiation from the central star in turn also de-

Article number, page 10 of 18

P. J. Rodenkirch and C. P. Dullemond: Dust entrainment in disk winds

0 20 40 60 80 100au

0

20

40

60

80

100

au

Photoevaporation

0 20 40 60 80 100au

= 105

0 20 40 60 80 100au

= 104

0 20 40 60 80 100au

= 104 (cold)

0.1

0.2

0.3

0.5

0.9

1.4

2.3

3.8

6.2

10.0

Grai

n siz

e [

m]

Fig. 6. Numerically integrated dust trajectories for snapshots after 500 T0 of the simulation runs phc2, b5c2 and b4c2. The orange color maprepresents the gas density and the thin arrowed white lines denote the gas velocity streamlines. The thicker colored lines visualize the dusttrajectories for grain sizes ranging from 0.1 µm to 10 µm.

102030

40

50

60

70

Incli

natio

n [d

eg]

0 1 2 3 4 5 6 7 8Grain size [ m]

0.5

1.0

1.5

2.0

2.5

Slop

e

photoevaporation= 105

= 104

= 104 (cold)

Fig. 7. Asymptotic dust flow inclination depending on the grain size forthe simulation runs phc2, b5c2, b4c2 and b4c2np.

creases with radius. The peak in temperature is reached at ≈ 8 auto 9 au, agreeing with the stagnation point of the Stokes num-ber profile. By linear fitting the quantities at the wind launchingsurface in logarithmic space we determine the power law slopes.Beyond 8 au the thermal velocity vth decreases ∝ r−0.13 whereasthe gas density profile follows the power law ∝ r−1.41. Lookingat the definition of the Stokes number in Eq. 8, we expect an ap-proximately constant relation St ∝ ΩK/(ρg · vth) ∝ r0.04 with aKeplerian angular frequency following ΩK ∝ r−1.5.

3.6. Gas and dust streamlines

In Fig. 11 velocities, Stokes number and Alfvén Mach numberare plotted for a representative streamline starting at r = 8 au.Since the flow is close to a steady state we can consider thestreamline as a proxy for the actual trajectory of a dust and gasparcel moving in the wind. The radial velocities in the left-most

7.5 5.0 2.5 0.0 2.5 5.0 7.5Gas scale height

10 9

10 7

10 5

10 3

10 1

Norm

alize

d ga

s & d

ust d

ensit

y

5 au10 au20 au

10 5

10 4

10 3

10 2

10 1

100

101

Stok

es n

umbe

r

10 5

10 4

10 3

10 2

10 1

100

101

Stok

es n

umbe

r

10 5

10 4

10 3

10 2

10 1

100

101

Stok

es n

umbe

r

Fig. 8. Normalized vertical density slices of the phc2 at different radiallocations in the disk. The violet lines represent the normalized gas den-sity while the turquoise lines display the dust density of 10 µm grains.The gray lines denote the respective Stokes number profile.

panel are almost constant with the cylindrical radius whereas thevertical velocities in z-direction significantly increase up to val-ues on the order of 15 km s−1. The models including magneticeffects generally lead to slightly faster wind flows increasingwith the magnetic field strength. Depending on the dust grainsize the velocity can be significantly less compared to the gasvelocity, differing by roughly 5 km s−1. Dust velocities are alsoconsistently faster in the magnetic wind case. Looking at theStokes numbers of the two smallest dust species, the values up tounity are reached for micron-sized grains far from the disk. Theright-most panel demonstrates that the wind is only subalfvénicfor a small part close to the wind launching front. Even in thesimulation run b4c2 with an initial β = 104 the wind quicklyreaches superalfvénic speeds which in consequence representsa small magnetic lever arm. This picture is in line with recentmodels of magnetothermal winds such as Gressel et al. (2015);Bai (2017b); Rodenkirch et al. (2020).

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7.5 5.0 2.5 0.0 2.5 5.0 7.5Gas scale height

10 9

10 7

10 5

10 3

10 1

Norm

alize

d ga

s & d

ust d

ensit

y

0.1 m1 m10 m 10 6

10 5

10 4

10 3

10 2

10 1

100

Stok

es n

umbe

r

10 6

10 5

10 4

10 3

10 2

10 1

100

Stok

es n

umbe

r

10 6

10 5

10 4

10 3

10 2

10 1

100

Stok

es n

umbe

r

Fig. 9. Normalized vertical density slices and Stokes number profilesof the phc2 similar to Fig. 8 for the grain sizes 0.1 µm (straight lines),1 µm (dashed lines), 10 µm (dotted lines) at 5 au.

3.7. Synthetic observations

Given that the wind flow is rather thin and dust is not perfectlyentrained for larger grain sizes and distances from the centralstar, the observability of such dusty winds deserves a closer look.As a first simple estimate we computed the radial optical depthdepending on the latitude θ. The results are displayed in Fig. 12for three dust species ranging from 0.1 µm to 10 µm in the H-band. The optical depth τobs,i of dust species i was calculated by

τobs,i =

∫ rmax

rmin

κsca,i ρd,i (39)

with the scattering opacity κsca,i. The inner and outer radialboundaries are denoted by rmin and rmax, respectively. In Fig. 12the bulk of the disk appears to be optically thick for all dustspecies in the H-band. The wind itself however is optically thinthroughout the whole flow and its optical depth reaches values ofτobs ≈ 10−1 in the direction of the wind base. Naturally, the opti-cal depth decreases with larger viewing angles due to the thinnerdust and gas density in these regions.For 1 µm-sized grains the optical depth is comparable to the onefor 0.1 µm grains but tapers off at a flatter angle of 70° due to theless inclined dust flow. Finally, the larger 10 µm grains are almostirrelevant since dust entrainment is negligible and the densitiesare therefore approaching the dust density floor value. In Fig. 13and Fig. 14 synthetic observations are shown with inclinationangles of 20° and 0° in latitude, respectively. In comparison tothe disk brightness the wind structure can be clearly identified asan hourglass-shaped signature. Smaller 0.1 µm grains are con-fined to a narrower cone with diffuse emission towards the outerregions. A rather sharp transition between the region devoid ofdust close to the rotation axis and the inner most part of the dustywind is visible.The wind is not completely illuminated by the stellar light be-cause of the decreasing dust content towards the outer regions ofthe disk. Comparing the morphology of the scattered light sig-natures from 0.1 µm grains to emission of 1 µm grains, the largerdust particles lead to a conical shape with a wider opening anglewith respect to the rotation axis. This feature is expected fromFig. 7 where the increased grain size leads to weaker gas-dustcoupling and therefore causes a dust flow with a smaller inclina-tion. In addition, larger grains are entrained less efficiently in theouter regions of the disk compared to smaller grains, due to the

10 8

10 7

10 6

10 5

10 4

10 3

10 2

Dust

to g

as ra

tio

phot.ev.: 0.1 mphot.ev.: 1 m

= 105: 0.1 m= 105: 1 m= 104: 0.1 m= 104: 1 m

10 2

10 1

Stok

es n

umbe

r

3 × 103

4 × 103

5 × 103

6 × 103

Tem

pera

ture

[K]

phot.ev.= 105

= 104

103 4 5 6 7 8 9 20 30 40 50r [au]

106

107

Gas d

ensit

y [c

m3 ]

phot.ev.= 105

= 104

Fig. 10. Physical quantities at the wind launching surface of the simu-lation runs phc2 (straight lines), b5c2 (dashed lines) and b4c2 (dottedlines). In the top two panels the blue lines represent 0.1 µm-sized grains,whereas the green lines denote 1 µm-sized grains.

stronger dust settling towards the disk mid plane. In combinationwith the larger opening angle of the flow we therefore detect aless diffuse, conical shape with a sharp transition in brightnesstowards the outer regions of the disk.It is also visible that magnetic winds with stronger fields confinethe dust flow more towards the rotation axis and exhibit a larger

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20 40 60 80r [au]

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r [au]10 1

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101

102

103MA

Photoevaporation= 105

= 104

Fig. 11. Streamlines starting from r = 8 au, z = 2.5 au. Shown are poloidal velocities of gas and dust along the streamline (left two panels) and therespective Stokes numbers St as well as the Alfvén Mach number MA (right two panels).

7550250255075 [deg]

10 6

10 4

10 2

100

102

104

obs

0.1 m1 m10 m

Fig. 12. H-band radial optical depth τobs originating from the cen-tral star depending on the inclination expressed in latitude for threedust species with sizes 0.1 µm, 1 µm and 10 µm indicated by the colorscheme. The continuous lines refer to the photoevaporation simulationphc2, the dashed lines to the magnetic wind b5c2 with β = 105 and thedotted lines to the run b4c2 with β = 104.

inclination angle of the dust flow. The cold MHD wind model ex-hibits a brighter wind region compared to the models includingphotoevaporation. As expected, the opening angle of the 0.1 µmwind flow is larger compared to the warm wind models and re-sembles the emission of 1 µm if photoevaporative heating is in-cluded. In the cold wind model 1 µm-sized grains are entrainedin a shallow inclination angle and could be identified as a diskwith a larger aspect ratio than the thermodynamics would other-wise allow in the system. At a viewing angle of 20° the backsideof the disk almost shadows the dusty wind, making it difficultto identify such a dusty outflow on the basis of a low resolutionor a low signal-to-noise ratio. Generally, we find the brightnesscontrast between wind and disk to be less in the case of 1 µmgrains if the viewing angle is 20°. In the edge-on view shown inFig. 14 the wind signature brightness is comparable to the diskbrightness for both grain sizes.

4. Discussion

In the following we discuss limitations of our wind model andcompare the key results with other works in the literature. Anadvantage of the fluid description over a Lagrangian particle for-mulation of the dust dynamics in the simulations, is the abilityto resolve low density regions if a large density contrast betweenthe midplane and the corona as well as the wind is present, whichis the case in the disk-wind system shown here. The disadvantageis the lack of proper modeling of a dust grain size distribution.A single-size dust population is unlikely due to dust growth and

fragmentation. The synthetic observations thus only serve as ahint towards the detectability of dusty winds and the expectedshape of the flow assuming the corresponding predominant dustsize. Including multiple dust fluids in the synthetic observationswould lead to misleading structures in the emission because ofthe rather sharp transition between the inner dust free cone andthe wind region.Deducing the maximum dust grain size from the wind flow angleseems to be rather difficult since the underlying parameter spaceis degenerate. On the one hand, the opening angle of the dustywind increases with the grain size. On the other hand, the sameeffect occurs with colder magnetically driven winds. A distinc-tion could be made by probing the wind speed, which is smallerin the cold wind case, albeit being a specific example.If the wind is optically thin for the ionizing radiation of thecentral star, the difference between photoevaporative winds andwarm magnetically driven winds is minor. In future works, theparameter space could be extended by varying the luminosity ofthe central star and by using a more sophisticated photoevapora-tion model as presented in Ercolano et al. (2021); Picogna et al.(2021).Within the simulated time frame we do not observe any signifi-cant asymmetry between the upper and lower wind. Such asym-metric flows were observed in simulations by Béthune et al.(2017), Suriano et al. (2018) and Riols & Lesur (2018). Ringformation by magnetically driven winds does not occur in oursimulation runs compared to Suriano et al. (2018) and Riols &Lesur (2018) where ambipolar diffusion is prescribed to be thedominant nonideal MHD effect. In their work however, largerfield strength in the regime of β = 103 were probed. Riols &Lesur (2018) also observed ring formation for β = 104 and in-cluded fluid dust focusing on the dust dynamics in the mid plane.Given that the same photoevaporation recipe is used in this pa-per as the one in Franz et al. (2020) our results are compatiblein this regime. They conclude a maximum entrainable grain sizeof 11 µm. As shown in Fig. 6 our photoevaporative models allowa short-term lifting of 10 µm dust grains which eventually fallback onto the disk. In the complete dust fluid picture the smalldust grain scale height prevents any dust entrainment startingfrom the wind launching front. More recently, Franz et al. (2021)studied the observability of dust entrainment in photoevapora-tive winds based on their models in Franz et al. (2020). Theyconclude that the wind signature in scattered light is too faint tobe observed with SPHERE IRDIS, but might be detectable withthe JWST NIRCam under optical conditions. In their syntheticobservations a similar “chimney”-shaped wind emission is visi-ble compared to our photoevaporative models.Giacalone et al. (2019) found a positive correlation betweengas temperature and maximum dust grain size in their semi-analytical magneto-centrifugal wind model which corresponds

Article number, page 13 of 18

A&A proofs: manuscript no. main

1.0

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Fig. 13. Radiative transfer images in the H-Band (λ = 1.6 µm) of simulation runs with photoevaporation (left most column), β = 105 (center leftcolumn), β = 104 (center right column) and a cold magnetic wind with β = 104 (right most column). The upper panels are based on dust grainswith a = 0.1 µm whereas the lower panels demonstrate synthetic images with a = 1 µm. The object is modeled at a distance of 150 pc with a solartype star at an inclination of 20°.

1.0

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Fig. 14. Radiative transfer images in the H-Band (λ = 1.6 µm) of the same simulation runs as in Fig. 13 but with an edge-on view at an inclinationof 0°.

best to our cold magnetically driven wind model b4c2np. Ourresults corroborate their findings of a maximum entrainable dustsize in the submicron regime in a cold wind around a typical TTauri star. Our models including photoevaporation furthermoreagree with Hutchison et al. (2016a) where the maximum grainsize was determined to be < 10 µm depending on the disk radiusfor a typical T Tauri star. They additionally argue that dust set-

tling most certainly limits this maximum size further which weconfirm in our models, where the diminishing dust entrainmentefficiency can be attributed to the dust settling towards the midplane.In recent scattered light observations of RY Tau (Garufi et al.,2019) a broad dusty outflow obstructing the underlying diskwas detected. Both lobes are inclined by roughly 45°. The pres-

Article number, page 14 of 18

P. J. Rodenkirch and C. P. Dullemond: Dust entrainment in disk winds

ence of a jet in this system advocates a magnetic wind launch-ing mechanism that causes the features observed in the near-infrared. We do not observe an optically thick wind that wouldbe able to completely obstruct the disk, since the dust densitiesare too low, especially in the models including photoevapora-tion. A possibility might hence be a slow, cold, magnetic windat higher field strengths. The large opening angle in the obser-vation would suggest a wind flow containing dust grains of thesize of several µm if photoevaporation would be the significantwind mechanism. If a cold magnetically driven wind was thedominant factor, smaller submicron grains would be the solutioncompatible with our models.In this work we neglected the effect of radiation pressure thatmay be able to blow out small dust grains. Franz et al. (2020)argued that a photoevaporative wind can entrain grains of sizesroughly 20 times larger compared to a compatible radiation pres-sure model from Owen & Kollmeier (2019). Since we use simi-lar parameters for the photoevaporation model, the effect of radi-ation pressure should not significantly affect the dust dynamicsin the studied parameter space.In the photoevaporation models the gaseous part of the windreaches temperatures of up to 104 K. We verified that the dusttemperature stays below the dust sublimation temperature in Ap-pendix A.

5. Conclusion

We presented a suite of fully dynamic, global, multifluid simula-tions that model dust entrainment in photoevaporative as wellas magnetically driven disk wind using the FARGO3D code.We demonstrated that both types of winds are able to efficientlytransport dust in the wind flow and addressed the observabilityof such dusty winds with the help of the RADMC-3D radiativetransfer code. In the following we summarize the main results:

– The maximum entrainable dust grain size amax both dependson the type of the wind launching mechanism and the turbu-lent diffusivity of the disk. Whereas magnetic winds includ-ing XEUV-photoevaporation only show minor differences inamax, ranging from 3 µm to 6 µm, cold magnetically drivenwinds at comparable field strengths only entrain submicronsized grains efficiently.

– With increasing radius the dust to gas ratio in the wind dropsrapidly, mainly due to the smaller dust scale height in com-parison to the gas scale height. Dust grains are unable toreach the wind launching surface in the outer regions of thedisk and the dust content in the wind starting from these lo-cations hence decreases. In the case of warm, ionized windsthe Stokes number stays approximately constant with radiusbeyond 8 au. Generally, magnetically driven winds includ-ing photoevaporation are slightly more efficient in entrainingdust compared to pure photoevaporative flows.

– The dust flow angle θd decreases with increasing grain sizeto the weaker coupling between gas and dust. In warm, ion-ized winds the flow angle reaches values of 67° ≤ θd ≤ 69°for a = 1 µm sized grains. At a grain size of a = 3 µm a pho-toevaporative dust flow provides θd ≈ 36° whereas a warmmagnetic wind leads to a steeper dust flow with θd ≈ 45°. Ina cold magnetic wind the flow angle is significantly smaller,being θd ≈ 59° for a = 0.1 µm and θd ≈ 25° for micron-sizedgrains.

– Radiative transfer images in the H-band suggest a ratherweak, optically thin emission for photoevaporative winds.A more pronounced, conical shape is visible considering

1 µm grains. The larger dust flow angle θd leads to a nar-rower emission region in the case of warm, ionized mag-netic winds. Cold magnetic winds appear with a significantlylarger opening angle with comparable grain sizes. The emis-sion region of the wind appears brighter and more easily de-tectable due to the increased dust density of the slower mag-netic wind.

Our results presented in this paper might help to constrain themagnetic field strength as well as the dominating wind mecha-nism if such dust signatures will be detected in future observa-tions. The dust grain size could be determined via the inclina-tion angle of the flow. Although the parameter space is degener-ate, probing the wind speeds might help to differentiate betweenthermally and cold magnetically driven winds.Acknowledgements. We would like to thank R. Franz for helpful discussions. Wefurthermore thank the anonymous referee for helpful comments that further im-proved the manuscript. Authors Rodenkirch and Dullemond acknowledge fund-ing from the Deutsche Forschungsgemeinschaft (DFG, German Research Foun-dation) - 325594231 under grant DU 414/23-1. The authors acknowledge supportby the High Performance and Cloud Computing Group at the Zentrum für Daten-verarbeitung of the University of Tübingen, the state of Baden-Württembergthrough bwHPC and the German Research Foundation (DFG) through grantINST 37/935-1 FUGG. Plots in this paper were made with the Python librarymatplotlib (Hunter, 2007).

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Appendix A: Dust temperature estimate

The relatively high temperatures in the ionized wind models heatthe dust grains that are entrained in the gas flow. We show inthe following that dust particles do not reach the dust sublima-tion threshold and would thus remain solid along their trajec-tories. The expression of the heating rate per unit surface qd isdescribed in Gombosi et al. (1986), taking into account both con-tact and friction heating. We follow the approach of Stammler &Dullemond (2014) in this regard and evaluate the subsequent ex-pressions:

qd = ρgCH,d (Trec − Td) |vg − vd| , (A.1)

where the heat transfer coefficient CH,d is expressed as

CH,d =γ + 1γ − 1

kB

8µmps2

[s√π

exp(−s2

)+

(12

+ s2)

erf (s)], (A.2)

with the error function erf and the normalized difference be-tween dust and gas velocities:

s =|vg − vd|√

2kBTg/(µmp). (A.3)

The recovery temperature Trec is given by the expression (Gom-bosi et al., 1986):

Trec =Tg

γ − 1

2γ + 2(γ − 1)s2 −γ − 1

12 + s2 + s

√πexp

(−s2) erf−1 (s)

.(A.4)

The dust equilibrium temperature is then finally computed bysolving

qd +R∗r2 εdσSBT 4

∗ − 4σSBT 4d = 0 (A.5)

with the thermal cooling efficiency factor

εd =κP(T∗)κP(Td)

, (A.6)

using the Planck mean opacity κP with the corresponding tem-perature. Fig. A.1 provides the solutions of the equilibrium dusttemperatures in comparison with the gas temperatures in a pho-toevaporative flow including 0.1 µm grains. We assume a sun-like star in the computation and find maximum dust temperaturesof Td ≈ 345 K at a radial distance of r = 2 au. Due to the thin gasin the wind region the ambient heating has an almost negligibleimpact on the dust temperature. Without the flux of the centralstar the dust grains would reach Td ≈ 40 K only being heatedby the ionized gas. In the wind flows presented here dust tem-peratures are clearly below the dust sublimation threshold whichwould be ≈ 980 K assuming pyroxene as the grain material em-bedded in a thin medium of the density ρg = 1 × 10−16 g/cm3

(Pollack et al., 1994).

Appendix B: Ionization fraction

Since the ionization fraction depends on the dust-to-gas ratio εionin the disk a comparison between εion = 10−3 and εion = 10−2 ofthe simulation runs b4c2 and b4c2eps2 is shown in Fig. B.1.The ionization is relatively insensitive to the increased dust-to-gas ratio and only decreases by a small amount in the weaklyionized disk mid plane. In the dust density slice the effects arenegligible.

10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0Gas scale height

102

103

T [K

]

2 au3 au10 au

Fig. A.1. Vertical slices of gas and dust temperatures at different radiallocation in the disk and wind region of the photoevaporation simulationphc2. The gray lines represent the gas temperatures whereas the redlines display the equilibrium dust temperatures of the 0.1 µm grains.

7.5 5.0 2.5 0.0 2.5 5.0 7.5Gas scale height

10 9

10 7

10 5

10 3

10 1

Norm

alize

d du

st d

ensit

y

= 10 3

= 10 2

10 13

10 11

10 9

10 7

10 5

10 3

Ioni

zatio

n fra

ctio

n

10 13

10 11

10 9

10 7

10 5

10 3

Ioni

zatio

n fra

ctio

n

Fig. B.1. Normalized dust density (blue lines) and ionization fraction(gray lines) in the simulations b4c2 and b4c2eps2 with a dust-to-gasratio of ε = 10−3 and ε = 10−2, respectively. The snapshot is taken after1000 orbits at 1 au.

Appendix C: Super time stepping

This section describes the implementation of the Super Time-Stepping technique (STS) for parabolic or mixed advection-diffusion problems. Introduced by Alexiades et al. (1996), thefirst-order method divides one explicit super step ∆T in multi-ple substeps τ1, τ2, . . . τN. The algorithm ensures stability whilemaximizing ∆T =

∑Ni τi The optimized length of each substep

is based on Chevychev polynomials:

τj = ∆texp

[(ν − 1) cos

(2 j − 1

Nπ

2

)+ ν + 1

], (C.1)

where ∆texp is the explicit time step. The sum of all substeps thengives:

∆T = ∆texpN

2√ν

[(1 +

√ν)2N − (1 −

√ν)2N

(1 +√ν)2N + (1 −

√ν)2N

]. (C.2)

The parameter ν adopts values between 0 and 1. In the limitof ν → 0 the method is N times faster than the explicit in-tegration. Lower values of ν however can lead to oscillations

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0 2 4 6 8 10t

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0.02

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<B

2 z>

1/2

Fig. C.1. Decaying Alfvén wave test for 〈σv〉i = 1000. The dots corre-spond to the simulation results whereas the dashed line is the analyticalsolution.

and eventually to an unstable solution. In FARGO3D the mag-netic field is updated with the method of characteristics and con-strained transport (MOCCT) (Hawley & Stone, 1995) wherebythe constrained transport method described in Evans & Haw-ley (1988) is used (Benitez-Llambay & Masset, 2016). In thismethod the evolution of the magnetic field is determined bycomputing the electromotive forces (EMFs) Benitez-Llambay &Masset (2016):

E = v × B . (C.3)

We compute the contribution to the total EMFs of each substep jwith duration τj. The result is added to the EMFs, weighted by afactor of τj/∆T . The magnetic field updates at every substep areonly applied on a temporary buffer field and the main B-field isthen updated with the total EMFs, containing contributions fromnonideal terms and the ordinary MHD dynamics.In order to test the correctness of the implementation, a decayingstanding Alfvén wave test for ambipolar diffusion as describedin Choi et al. (2009) Sec. 4.2 is used. The time dependence ofthe first normal mode is given by Choi et al. (2009):

h(t) = h0 sin(ωRt)eωI t (C.4)

with ω = ωR + iωI . The terms ωR and ωI are the real and imag-inary parts of the angular frequency of the Alfvén wave. Thedispersion relation is given by Choi et al. (2009):

ω2 + ic2

Ak2

〈σv〉iniω + c2

Ak2 = 0 , (C.5)

where k is the real wave number and cA the characteristic Alfénspeed.In the test according to Choi et al. (2009) the magnetic field isset to B = B0x with B0 = 1. The density is uniformly set to 1.The wave is initialized in the x direction with an initial velocityof:

vy = v0cA sin(kx) . (C.6)

The amplitude of the perturbation is set to v0 = 0.1, the char-acteristic Alfén velocity to 1/

√2, the ion density to ρi = 0.1

and the wave number to k = 2π/L where L = 1 is the domainsize. The collision rate coefficient 〈σv〉i is varied from 100 to1000. Instead of an oblique wave test, the simulations are run ina one-dimensional configuration. In Fig. C.1 and Fig. C.2 thetest results are shown and the numerical results fit the analyticalsolution well. Stability is found to be fulfilled up to five substepswhich is sufficient for the simulation runs presented in this paper.An overall loss of accuracy in comparison to the solution with-out STS is also anticipated since this approach is only first-orderaccurate.

0 2 4 6 8 10t

0.00

0.01

0.02

0.03

0.04

0.05

<B

2 z>

1/2

Fig. C.2. Decaying Alfvén wave test for 〈σv〉i = 100 similar to Fig. C.1

Article number, page 18 of 18