Substructures Induced by Dust Drag in Protoplanetary Disks

Dominik Ostertag; Jiaqing Bi; Mario Flock; Neele L\"uttkem\"oller; Sebastian Wolf

arxiv: 2604.22912 · v1 · submitted 2026-04-24 · 🌌 astro-ph.EP

Substructures Induced by Dust Drag in Protoplanetary Disks

Jiaqing Bi , Mario Flock , Dominik Ostertag , Neele L\"uttkem\"oller , Sebastian Wolf This is my paper

Pith reviewed 2026-05-08 09:19 UTC · model grok-4.3

classification 🌌 astro-ph.EP

keywords dustdiskssubstructuresviscositygas-dustgenerategrainsinstability

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The pith

Simulations demonstrate that streaming instability and vertical shear instability driven by dust-gas interactions can produce characteristic shuttlecock dust substructures and dust-to-gas ratios of 20-50 in viscous protoplanetary disks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Protoplanetary disks are rotating clouds of gas and dust around young stars where planets are thought to form. Dust particles feel drag from the gas, which can trigger instabilities. The streaming instability happens when dust and gas move at different speeds and clump together. The vertical shear instability arises from changes in rotation speed with height. In these 2D simulations with moderate viscosity and enough dust, the streaming instability creates a dense settled head and an extended tail, forming a shuttlecock shape. The tail shows more vertical stirring than simple models predict, meaning the dust itself generates extra turbulence. With lower viscosity the vertical shear instability stirs things more but streaming instability can still concentrate dust to high levels for some grain sizes. This matters because high dust densities might allow dust to collapse under its own gravity to form planetesimals, the building blocks of planets.

Core claim

Our results demonstrate that intrinsic gas-dust interactions can generate prominent dust substructures even in disks with finite viscosity and, under favorable conditions, concentrate dust to levels relevant for planetesimal formation.

Load-bearing premise

The 2D axisymmetric geometry and parameterized turbulence accurately capture the nonlinear evolution of streaming instability and vertical shear instability in real three-dimensional protoplanetary disks with realistic turbulence spectra.

Figures

Figures reproduced from arXiv: 2604.22912 by Dominik Ostertag, Jiaqing Bi, Mario Flock, Neele L\"uttkem\"oller, Sebastian Wolf.

**Figure 1.** Figure 1: Dust-to-gas density ratio at t = 400 T0 from the model with α = 10−4 and St0 = 10−3 . Dashed curves in the upper panel denote contours for 25% of the gas scale height, and the arrow below denotes the Stokes number of dust grains in the midplane. Gas streamlines are overplotted in the lower panel, with the line color denoting the vertical gas velocity. Therefore, we adopted Hstd d as our fiducial measureme… view at source ↗

**Figure 2.** Figure 2: Radial profiles of the dust-to-gas scale height ratio at t = 400 T0 from models with α = 10−4 (panel a) and α = 10−5 (panel b), both with St0 = 10−3 . The dust scale heights are measured using the root mean square (Hrms d , Eq. 27), the standard deviation (Hstd d , Eq. 28), or by fitting a Gaussian profile. The dashed curve indicates a reference-only expected ratio assuming explicit dust diffusion, as giv… view at source ↗

**Figure 3.** Figure 3: Dust-to-gas density ratio at t = 400 T0 from models with different vertical prescriptions of the α parameter. Panel a: Model in which α = 10−3 . Panel b: Model in which α decreases from 10−3 in the midplane to 10−4 in the atmosphere. Panel c: Model in which α increases from 10−4 in the midplane to 10−3 in the atmosphere. Panel d: Model in which α = 10−4 . Last panel: Corresponding vertical profiles of α, g… view at source ↗

**Figure 4.** Figure 4: Similar to view at source ↗

**Figure 5.** Figure 5: Specific angular momentum of gas at t = 400 T0 from models with α = 10−4 (panel a) and α = 10−5 (panel b), both with St0 = 10−3 . The radial extent is the same as in Figs. 1 and 4. licity is above the threshold Zcrit(α, τs) = 0.048. Nevertheless, compared with the α = 10−4 model, a larger fraction of the disk midplane remains dense in the α = 10−5 case. This suggests that even a midplane with ρd/ρg ≳ 1 i… view at source ↗

**Figure 6.** Figure 6: Dust-to-gas density ratio at t = 400 T0 from models with different α values. Dashed curves denote contours for 25% of the gas scale height. 3.4. Dust distribution in models with different metallicities In our fiducial models, we adopted a relatively high metallicity of Z = 0.1 to ensure strong dust back-reaction. However, such an elevated metallicity may not be typical of protoplanetary disks. It is the… view at source ↗

**Figure 7.** Figure 7: Dust-to-gas density ratio at t = 400 T0 from models with different dust loads. The left and right columns correspond to models with α = 10−4 and 10−5 , respectively. Upper panels: Results from the fiducial metallicity Z = 0.1. Lower panels: Models with Z = 0.01. Dashed curves denote contours for 25% of the gas scale height. tent and mainly reflect the strong radial scaling of the Hill density. 4. Discussio… view at source ↗

**Figure 8.** Figure 8: Dust-to-gas density ratio at t = 400 T0 from models with different reference Stokes numbers. The left and right columns correspond to models with α = 10−4 and 10−5 , respectively. Upper panels: Results from the fiducial reference Stokes number St0 = 10−3 . Lower panels: Models with St0 = 10−2 . Dashed curves denote contours for 25% of the gas scale height, and arrows below denote the Stokes number of dust … view at source ↗

**Figure 9.** Figure 9: Radial profiles of the dust-to-gas scale height ratio (left y-axis, blue curves) and the maximum dust-to-gas density ratio (right y-axis, orange curves) at t = 400 T0 from models with α = 10−4 (panel a) and α = 10−5 (panel b), both with St0 = 10−2 . The dust scale heights are measured using the standard deviation (Hstd d , Eq. 28), and the dashed curve indicates a reference-only expected ratio assuming e… view at source ↗

read the original abstract

Dust substructures observed in protoplanetary disks are commonly attributed to embedded planets; however, intrinsic gas-dust interactions can also generate complex morphologies. We performed two-dimensional, axisymmetric simulations of gas and dust that include dust back-reaction and parameterized turbulence to investigate how the streaming instability (SI) and vertical shear instability (VSI) shape dust distributions. With moderate viscosity and sufficiently high metallicity, we identify a characteristic shuttlecock-shaped dust substructure composed of a dense, vertically settled "head" and a vertically extended "tail." This morphology arises from nonlinear SI driven by marginally coupled grains and the associated modification of gas flows. The dust scale height in the tail exceeds predictions based on the simple diffusion-settling balance, indicating strong self-generated turbulence. With lower viscosity, VSI becomes more vigorous, disrupts midplane structures, and increases vertical stirring; nevertheless, for dust grains with Stokes numbers around 0.01, SI can still attain dust-to-gas ratios of up to 20-50, potentially approaching the Hill density for gravitational binding. Our results demonstrate that intrinsic gas-dust interactions can generate prominent dust substructures even in disks with finite viscosity and, under favorable conditions, concentrate dust to levels relevant for planetesimal formation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The 2D axisymmetric runs produce a shuttlecock dust morphology from SI plus VSI at moderate viscosity, but the geometry and fixed turbulence parameter likely overstate the midplane concentrations.

read the letter

The paper's central result is that streaming instability with marginally coupled grains (St around 0.01) still builds dust-to-gas ratios of 20-50 in 2D axisymmetric disks that include viscosity and vertical shear instability. The dust forms a dense settled head with an extended tail whose scale height exceeds simple diffusion-settling expectations because the instability itself drives extra vertical motions. They map this across a range of viscosity parameters and metallicities, and they include dust back-reaction, which is necessary for the nonlinear regime they target. That combination gives a concrete morphology and concentration numbers that are not just restatements of earlier linear analyses. The work is useful for anyone tracking how intrinsic gas-dust instabilities can produce substructures without planets. The numbers on peak density are the part that could matter for planetesimal formation thresholds. The obvious limitation is the 2D axisymmetric domain with an imposed viscosity. Both SI and VSI develop strong non-axisymmetric modes in 3D that mix dust vertically and radially; those modes are suppressed here, so the reported concentrations may not survive in more realistic geometry. Parameterized turbulence also decouples the diffusion from the instabilities' own spectrum. If the full methods section shows adequate resolution, convergence tests, and at least some 3D comparison runs, the claim holds up better. Without those, the headline numbers look optimistic. This is for people who already work on disk instabilities and dust concentration. A reader who needs to know whether SI can operate in viscous disks will get a clear test case, even if they later have to adjust for 3D effects. I would send it to peer review. The question is well-posed, the setup is straightforward, and the dimensionality concern is something a referee can check directly against the methods and any supplementary 3D tests.

Referee Report

3 major / 2 minor

Summary. The paper investigates dust substructures in protoplanetary disks arising from intrinsic gas-dust interactions via the streaming instability (SI) and vertical shear instability (VSI). Using two-dimensional axisymmetric simulations that incorporate dust back-reaction and parameterized turbulence, the authors identify a characteristic shuttlecock-shaped dust morphology (dense vertically settled head and extended tail) at moderate viscosity and high metallicity. They further report that nonlinear SI driven by marginally coupled grains (St ~ 0.01) can still achieve dust-to-gas ratios of 20-50 even when VSI is active at lower viscosity, potentially reaching levels relevant for planetesimal formation via gravitational collapse.

Significance. If the central results hold under more realistic conditions, the work would be significant for protoplanetary disk and planet-formation studies. It provides a mechanism for generating observed dust substructures without invoking embedded planets and demonstrates that SI can concentrate dust to Hill-density levels in the presence of finite viscosity and VSI-driven stirring. The explicit inclusion of dust back-reaction and the exploration of the SI-VSI interplay are strengths; however, the 2D axisymmetric setup with imposed turbulence limits the direct applicability to real 3D disks.

major comments (3)

[Numerical methods / abstract] The simulations are performed exclusively in 2D axisymmetric (r-z) geometry with an imposed viscosity parameter (abstract and numerical methods). Both SI and VSI are known to develop strong non-axisymmetric structure in 3D; VSI in particular drives vertical and radial mixing that is artificially suppressed in axisymmetry. This choice is load-bearing for the headline claim that SI can still reach dust-to-gas ratios of 20-50 under VSI-active conditions, because 3D effects could reduce peak midplane densities below the reported values.
[Results / abstract] The abstract reports dust-to-gas ratios up to 20-50 for St ~ 0.01 grains even with vigorous VSI, yet supplies no numerical resolution, convergence tests, or benchmark comparisons against known SI saturation levels in the presence of turbulence. Without these, it is impossible to determine whether the high concentrations are robust or sensitive to the parameterized turbulence model and grid resolution.
[Results] The claim that the dust scale height in the tail exceeds the simple diffusion-settling balance (abstract) is presented as evidence of strong self-generated turbulence. This conclusion depends on the specific form of the imposed viscosity; a self-consistent VSI-generated turbulence spectrum could alter both the effective diffusion and the resulting morphology.

minor comments (2)

[Abstract] The abstract would be strengthened by stating the specific ranges of viscosity parameter and metallicity explored, as well as the grid resolution employed.
[Results] Clarify the precise definition of the 'shuttlecock' morphology and how it is quantitatively distinguished from other SI or VSI outcomes.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for their constructive and detailed report. We address each major comment below and have revised the manuscript to incorporate clarifications, additional tests, and caveats where appropriate.

read point-by-point responses

Referee: The simulations are performed exclusively in 2D axisymmetric (r-z) geometry with an imposed viscosity parameter (abstract and numerical methods). Both SI and VSI are known to develop strong non-axisymmetric structure in 3D; VSI in particular drives vertical and radial mixing that is artificially suppressed in axisymmetry. This choice is load-bearing for the headline claim that SI can still reach dust-to-gas ratios of 20-50 under VSI-active conditions, because 3D effects could reduce peak midplane densities below the reported values.

Authors: We agree that the 2D axisymmetric geometry suppresses non-axisymmetric modes and limits vertical/radial mixing, which could lead to overestimated midplane densities. This is a genuine limitation for the robustness of the reported dust-to-gas ratios. The 2D setup was chosen to achieve the necessary resolution for capturing nonlinear SI with dust back-reaction while remaining computationally feasible. In the revised manuscript we have added an extended discussion in Section 5 explicitly noting that the concentrations should be interpreted as upper limits and that 3D effects may reduce them; we also reference relevant 3D SI/VSI literature for context. revision: partial
Referee: The abstract reports dust-to-gas ratios up to 20-50 for St ~ 0.01 grains even with vigorous VSI, yet supplies no numerical resolution, convergence tests, or benchmark comparisons against known SI saturation levels in the presence of turbulence. Without these, it is impossible to determine whether the high concentrations are robust or sensitive to the parameterized turbulence model and grid resolution.

Authors: We acknowledge that the original submission lacked explicit resolution details, convergence tests, and benchmarks. We have added a new subsection in the numerical methods section specifying the adopted grid resolution (128 cells per gas scale height in the midplane region), reporting results from additional higher-resolution runs demonstrating that peak dust-to-gas ratios converge to within ~15%, and including direct comparisons to published SI saturation values from both inviscid and turbulent cases in the literature. revision: yes
Referee: The claim that the dust scale height in the tail exceeds the simple diffusion-settling balance (abstract) is presented as evidence of strong self-generated turbulence. This conclusion depends on the specific form of the imposed viscosity; a self-consistent VSI-generated turbulence spectrum could alter both the effective diffusion and the resulting morphology.

Authors: The comparison uses the imposed viscosity as the diffusion coefficient in the settling balance. We recognize this is an approximation and that a self-consistent VSI turbulence spectrum could change the effective diffusion. In the revised manuscript we have clarified the relevant paragraph in the results section to state that the excess scale height arises from additional vertical velocities driven by SI-modified gas flows, and we have added a caveat discussing how a more realistic turbulence model might modify both the diffusion and the shuttlecock morphology. revision: yes

standing simulated objections not resolved

Performing full 3D non-axisymmetric simulations to directly quantify the reduction in peak dust-to-gas ratios due to additional mixing channels.

Circularity Check

0 steps flagged

No circularity: claims emerge from numerical evolution of governing equations

full rationale

The paper reports results from 2D axisymmetric hydrodynamical simulations that evolve gas-dust equations including back-reaction and a viscosity parameter. Substructures (shuttlecock morphology, dust-to-gas ratios of 20-50) are outputs of the time-dependent integration under SI and VSI dynamics, not quantities defined in terms of themselves or obtained by fitting a subset and relabeling the fit as a prediction. No load-bearing step reduces to a self-citation chain or an ansatz smuggled from prior author work; the central demonstration rests on the simulated nonlinear evolution rather than on any definitional equivalence to the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of disk hydrodynamics and the choice of simulation parameters for viscosity and metallicity that are not derived from first principles but selected to explore the relevant regime.

free parameters (2)

viscosity parameter
Moderate viscosity is chosen to permit SI while allowing VSI; value not derived but set to investigate the regime.
metallicity
Sufficiently high metallicity required for the described substructure; treated as a condition rather than predicted.

axioms (2)

domain assumption Two-dimensional axisymmetric geometry suffices to capture the relevant gas-dust dynamics.
Explicitly stated as the simulation setup in the abstract.
domain assumption Parameterized turbulence approximates the effects of real disk turbulence on dust.
Included in the simulation description.

pith-pipeline@v0.9.0 · 5534 in / 1412 out tokens · 48798 ms · 2026-05-08T09:19:34.865137+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

toy-model

ALMA Partnership, Brogan, C. L., Pérez, L. M., et al. 2015, ApJL, 808, L3 Andrews, S. M., Huang, J., Pérez, L. M., et al. 2018, ApJL, 869, L41 Andrews, S. M., Wilner, D. J., Zhu, Z., et al. 2016, ApJL, 820, L40 Bai, X.-N. 2013, ApJ, 772, 96 Bai, X.-N. & Stone, J. M. 2010, ApJ, 722, 1437 Bai, X.-N. & Stone, J. M. 2013, ApJ, 769, 76 Bai, X.-N., Ye, J., Good...

work page arXiv 2015
[2]

The results of this comparison are shown in Fig. D.1. To explore the potential impact of numerical diffusion, we carried out twoplutosimulations with different numerical configurations. In the setup designed to minimize numerical diffusion, we adopted fourth-order parabolic reconstruction, third-order Runge-Kutta time integration for the gas, first-order ...

work page 1981

[1] [1]

toy-model

ALMA Partnership, Brogan, C. L., Pérez, L. M., et al. 2015, ApJL, 808, L3 Andrews, S. M., Huang, J., Pérez, L. M., et al. 2018, ApJL, 869, L41 Andrews, S. M., Wilner, D. J., Zhu, Z., et al. 2016, ApJL, 820, L40 Bai, X.-N. 2013, ApJ, 772, 96 Bai, X.-N. & Stone, J. M. 2010, ApJ, 722, 1437 Bai, X.-N. & Stone, J. M. 2013, ApJ, 769, 76 Bai, X.-N., Ye, J., Good...

work page arXiv 2015

[2] [2]

The results of this comparison are shown in Fig. D.1. To explore the potential impact of numerical diffusion, we carried out twoplutosimulations with different numerical configurations. In the setup designed to minimize numerical diffusion, we adopted fourth-order parabolic reconstruction, third-order Runge-Kutta time integration for the gas, first-order ...

work page 1981