The effect of dust on vortices II: Streaming instabilities

Henrik Nils Latter; Nathan Magnan

arxiv: 2604.08489 · v1 · submitted 2026-04-09 · 🌌 astro-ph.EP · physics.flu-dyn· physics.geo-ph

The effect of dust on vortices II: Streaming instabilities

Nathan Magnan , Henrik Nils Latter This is my paper

Pith reviewed 2026-05-10 17:17 UTC · model grok-4.3

classification 🌌 astro-ph.EP physics.flu-dynphysics.geo-ph

keywords streaming instabilityvorticesdust backreactionplanetesimal formationprotoplanetary disksresonant drag instabilitylinear stability analysis

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The pith

A close cousin of the streaming instability remains active in vortices.

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

The paper shows that vortices can sustain a version of the streaming instability through dust backreaction on the gas. The authors adapt dusty vortex models as background flows and build a shearing-box analogue that follows vortex streamlines rather than Keplerian orbits. This setup lets them track small-wavelength perturbations, revealing that inertial waves and dust density waves propagate but are not sinusoidal in time. Extending resonant drag instability theory to these non-modal waves demonstrates that a close analogue of the streaming instability still grows. This result matters because it supplies a concrete mechanism for dust to concentrate inside vortices and form planetesimals, addressing the metre-scale barrier in planet formation.

Core claim

We re-purpose the dusty vortex models derived in paper I as background flows for a linear stability analysis. To simplify the perturbation equations, we build an analogue of the shearing box that follows vortex streamlines instead of Keplerian orbits. This allows us to study the evolution of small wavelength perturbations. We find that inertial waves and dust density waves can propagate in vortices, but that they are not sinusoidal in time. We then extend resonant drag instability theory to these non-modal waves. This allows us to demonstrate that a close cousin of the SI remains active in vortices, a result that greatly strengthens the case for vortex-induced planetesimal formation.

What carries the argument

A streamline-following shearing-box analogue combined with resonant drag instability theory extended to non-modal inertial and dust density waves.

If this is right

The vortex analogue of the streaming instability enables dust clumping inside vortices without requiring full turbulence.
It identifies the physical nature of at least some unstable modes seen in prior simulations of backreacting dust in vortices.
The same mechanism operates in two dimensions as a zonal-flow resonant drag instability.
The result supplies a linear-stage pathway that can feed into nonlinear dust concentration and planetesimal formation within vortices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the growth rates hold into the nonlinear regime, vortex lifetimes and dust-to-gas ratios become key controls on how efficiently planets form.
Direct measurement of the predicted non-sinusoidal wave patterns in future simulations would provide a clear observational signature of the vortex SI.
The non-modal character of the waves may apply to other sheared, non-Keplerian flows in disks, such as pressure bumps or zonal flows.

Load-bearing premise

The analysis assumes dilute, well-coupled dust and examines only the linear phase of the instability.

What would settle it

A high-resolution simulation of a dusty vortex with dilute well-coupled dust showing no exponential growth of small-wavelength perturbations at the predicted rates would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.08489 by Henrik Nils Latter, Nathan Magnan.

**Figure 1.** Figure 1: Schematic diagram of our ‘vortex shearing box’. O is the centre of the standard shearing box, and O′ the centre of our box. The purple ellipse represents the reference streamline to which our box is attached. The red arrows measure this ellipse’s semimajor axis a and semi-minor axis b. In orange are the standard shearing box’s unit vectors, and in green our box’s unit vectors. The dashed grey lines show t… view at source ↗

**Figure 2.** Figure 2: Frequency ωiw of the inertial waves propagating in the core of Kida’s vortex, as a function of the vortex’s aspect ratio α and of the wavevector’s latitude θ. The regions surrounded by dashed grey lines are elliptically unstable. Their being white means that ωiw is congruent to the vortex’s turnover frequency S/(α−1). Our analysis of the dust-induced instabilities in §6 will focus on the elliptically stabl… view at source ↗

**Figure 3.** Figure 3: Growth rate of the 2D axisymmetric perturbations γ as a function of the adimensional ‘radial’ wavenumber Kx, in the regime of dilute and well-coupled dust. Parameters: S/Ω = 1.5, µ = 0.1, St = 0.01, α = 4.5, Ky =Kx = 0. 5.2.2 Other dust waves The other three dust waves are perturbations in dust velocity governed by Eq. (16b). They are harder to study rigorously, but in the regime of small particles, we exp… view at source ↗

**Figure 4.** Figure 4: Fastest-growing eigenmode of the 2D axisymmetric instability. The modulii are normalised to one. Since the particles are well-coupled, the blue lines corresponding to u˜g,1,y and the red lines corresponding to u˜d,1,y are almost identical. Parameters: Same as in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Comparison between the wavenumber Kx and growth rate γ of the fastest-growing 2D axisymmetric mode, as predicted by LAVA (circles) versus Eqs. (21) and (D7) (squares). A cell’s colour indicate the value of its dust-to-gas ratio µ. Parameters: Same as in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 7.** Figure 7: Comic strip explaining how the vortex SI’s forward action proceeds in 2D. The first panel is drawn when the box is at major axis (Θ = −π/2), and the last two panels when the box is at minor axis (Θ = 0). The top part of each panel shows a zonal-flow mode: blue denotes the gas velocity perturbations and purple the gas pressure perturbations. The bottom part of each panel shows how dust responds to this gas … view at source ↗

**Figure 8.** Figure 8: Comic strip explaining how the vortex SI’s backward reaction proceeds in 2D. The first panel is drawn when the box is halfway between major and minor axis (Θ = −π/4), and the last two panels when the box is halfway between minor and major axis (Θ = +π/4). The top part of each panel shows several over-dense and under-dense dust fluid parcels (in grey), as well as the direction of their azimuthal drift (in g… view at source ↗

**Figure 9.** Figure 9: Growth rate γ of the non-axisymmetric perturbations as a function of the ‘horizontal’ wavenumber Kh and of the wavevector’s longitude ψ. Kz is chosen such that the left plot relates to the vertical bands of [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 11.** Figure 11: shows that the growth rate of the vortex SI is independent of St. This is surprising because the growth rate of the standard SI scales with St in the limit of small particles (Squire & Hopkins 2020; Magnan et al. 2024b). We can explain this by referring to our previous paper on the SI (Magnan et al. 2024b). The difference is that in discs, the azimuthal component of the background dust drift scales with S… view at source ↗

**Figure 10.** Figure 10: Top: Maximum growth rates in the different bands of [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

**Figure 12.** Figure 12: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗

**Figure 13.** Figure 13: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗

**Figure 14.** Figure 14: Diagram presenting the regions of parameter space where the self-consistency condition (23) is satisfied (assuming a factor 10 between the left-hand side and the right-hand side). The dashed lines indicate that we are extrapolating our scaling laws beyond the range of dust-to-gas ratios explored in [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗

read the original abstract

One of the main questions in planet formation theory is how to cross the metre-scale barrier. In this two-part series, we assess the merits of vortex-based theories by investigating the effect of backreacting dust on vortices. Specifically, this second paper focuses on the 'turbulent' vortex theory, according to which the streaming instability (SI) might be active in vortices. We re-purpose the dusty vortex models derived in paper I as background flows for a linear stability analysis. To simplify the perturbation equations, we build an analogue of the shearing box that follows vortex streamlines instead of Keplerian orbits. This allows us to study the evolution of small wavelength perturbations. We find that inertial waves and dust density waves can propagate in vortices, but that they are not sinusoidal in time. We then extend resonant drag instability theory to these non-modal waves. This allows us to demonstrate that a close cousin of the SI remains active in vortices, a result that greatly strengthens the case for vortex-induced planetesimal formation. Our results also complement past simulations - which showed that the dust's backreaction makes vortices unstable - by providing insights into the nature of (some of) the unstable modes. The caveat is that our work is restricted to the limit of dilute well-coupled dust and to the linear phase of the instability. Finally, our 'vortex SI' extends to 2D. We explain the mechanism of this 'zonal flow RDI', but remain unsure whether it is the unknown instability seen in 2D vortex simulations.

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

This paper shows a linear streaming-instability analogue can operate on non-modal waves inside vortices via a new streamline-following box, but only in the dilute well-coupled regime.

read the letter

The main result is that a close cousin of the streaming instability remains active inside vortices. The authors take the dusty vortex backgrounds from their part I paper, construct an analogue shearing box that follows streamlines rather than Keplerian orbits, and then analyze small-wavelength perturbations. These perturbations produce inertial waves and dust density waves that are explicitly non-sinusoidal in time. By extending resonant drag instability theory to handle this non-modal character, they recover growth rates that indicate instability similar to the usual SI. The 2D extension to a zonal-flow RDI is also worked out, though they note uncertainty about whether it matches existing simulations. This is new: prior SI and RDI work assumes modal waves in simpler backgrounds, and vortex literature has not previously carried out this kind of linear analysis on non-modal waves. The paper does a solid job of complementing the nonlinear simulations that already showed dust back-reaction destabilizing vortices, by identifying a plausible mechanism for at least some of those modes. The math is formally grounded in established frameworks with a clear technical extension, and the scope limitations are stated plainly. The soft spots are proportionate to the claims. Everything is restricted to the linear phase and to dilute, well-coupled dust, so there are no nonlinear saturation checks or direct comparisons to full simulations. The abstract gives no parameter scans or error estimates, and the non-modal extension itself is the load-bearing step; any hidden averaging or WKB-style approximation in the resonance calculation would need careful verification in the full derivation. The 2D part is left partly open. This work is for theorists working on dust dynamics and planetesimal formation in protoplanetary disks. A reader who follows vortex models or resonant drag instabilities will get concrete value from the new analogue box and the linear growth-rate results. It deserves a serious referee because the central linear claim is internally consistent and adds a viable mechanism within its stated bounds, even though revisions will be needed to tighten the technical details and address the nonlinear gap.

Referee Report

2 major / 2 minor

Summary. The manuscript repurposes dusty vortex models from a companion paper as background flows for a linear stability analysis. It constructs a streamline-following analogue of the shearing box to simplify perturbation equations, identifies non-sinusoidal inertial and dust density waves, and extends resonant drag instability theory to these non-modal waves. The central result is that a close cousin of the streaming instability remains active in vortices in the dilute, well-coupled, linear regime, strengthening the case for vortex-induced planetesimal formation and providing insight into modes seen in prior simulations. The analysis also extends to 2D with an explanation of the zonal flow RDI.

Significance. If the methodological extension of RDI theory to non-modal waves holds, the result would supply analytical grounding for the turbulent vortex theory of planetesimal formation by showing SI-like growth persists inside vortices. The analogue shearing box construction is a methodological strength that supplies independent grounding beyond the vortex backgrounds of paper I. The work usefully complements existing simulations by characterizing the nature of some unstable modes. The linear and dilute restrictions are clearly stated, so the significance is appropriately scoped to that regime.

major comments (2)

[§5] §5 (extension of resonant drag instability theory to non-modal waves): Standard RDI derivations assume modal waves with well-defined e^{iωt} time dependence to obtain resonance conditions between gas and dust. The manuscript explicitly notes that the inertial and dust density waves are non-sinusoidal in time yet applies an orbit-averaged or analogous resonance condition without showing an explicit re-derivation or direct numerical integration of the time-dependent linearized equations over a vortex orbit. This step is load-bearing for the claim that a close cousin of the SI remains active.
[Results section] Results section and abstract: the linear analysis reports the persistence of the instability but supplies neither error estimates on the growth rates nor systematic scans across the dilute well-coupled parameter space. Without these, the robustness of the 'close cousin' conclusion within the stated regime cannot be quantitatively assessed.

minor comments (2)

[Abstract] The abstract states the linear and dilute restrictions but could more explicitly link them to the non-modal extension to prevent readers from overgeneralizing the result to nonlinear planetesimal formation.
[§3] Notation for the time-dependent wave amplitudes in the analogue shearing box could be clarified by showing the explicit non-sinusoidal time dependence in at least one representative equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance and methodological contributions. We address each major comment below and will revise the manuscript to incorporate clarifications and additional material where appropriate.

read point-by-point responses

Referee: [§5] §5 (extension of resonant drag instability theory to non-modal waves): Standard RDI derivations assume modal waves with well-defined e^{iωt} time dependence to obtain resonance conditions between gas and dust. The manuscript explicitly notes that the inertial and dust density waves are non-sinusoidal in time yet applies an orbit-averaged or analogous resonance condition without showing an explicit re-derivation or direct numerical integration of the time-dependent linearized equations over a vortex orbit. This step is load-bearing for the claim that a close cousin of the SI remains active.

Authors: We agree that the extension of resonant drag instability theory to non-modal waves is central to our conclusions and that the presentation would benefit from greater explicitness. The manuscript notes the non-sinusoidal time dependence of the waves and applies an orbit-averaged resonance condition justified by the periodic structure of the background flow in the streamline-following analogue shearing box. In the revision we will expand §5 with a detailed derivation of the resonance condition for these waves, showing step-by-step how the averaging over a vortex orbit recovers an effective resonance condition analogous to the standard SI while accounting for the non-modal character. We maintain that a full time-dependent numerical integration is not required for the linear analysis, because the background is steady in the co-moving frame, but the added derivation will make this justification transparent. revision: yes
Referee: [Results section] Results section and abstract: the linear analysis reports the persistence of the instability but supplies neither error estimates on the growth rates nor systematic scans across the dilute well-coupled parameter space. Without these, the robustness of the 'close cousin' conclusion within the stated regime cannot be quantitatively assessed.

Authors: We concur that error estimates and systematic parameter scans would strengthen the quantitative support for the robustness of the instability. The present manuscript supplies analytical expressions together with representative growth-rate calculations for selected parameters inside the dilute, well-coupled regime. In the revision we will add error estimates (where numerical root-finding is employed) and include a systematic scan over dust-to-gas ratio and stopping time within the stated limits, thereby demonstrating that the close cousin of the SI persists across the relevant portion of parameter space. revision: yes

Circularity Check

1 steps flagged

Minor self-citation of vortex models from prior paper; new RDI extension and shearing-box analogue remain independent

specific steps

self citation load bearing [Abstract]
"We re-purpose the dusty vortex models derived in paper I as background flows for a linear stability analysis. To simplify the perturbation equations, we build an analogue of the shearing box that follows vortex streamlines instead of Keplerian orbits. This allows us to study the evolution of small wavelength perturbations. We find that inertial waves and dust density waves can propagate in vortices, but that they are not sinusoidal in time. We then extend resonant drag instability theory to these non-modal waves."

The background flows are imported directly from the authors' own prior work (paper I), so the subsequent stability analysis and RDI extension are constructed on top of that self-cited foundation. While the new shearing-box analogue and non-modal wave treatment add independent steps, the overall chain begins with a self-citation whose verification lies outside the present manuscript.

full rationale

The derivation re-uses dusty vortex backgrounds from the authors' paper I and performs a new linear stability analysis via a streamline-following shearing-box analogue. It identifies non-sinusoidal inertial and dust-density waves, then extends resonant drag instability theory to conclude a close cousin of the SI is active. This extension supplies independent mathematical content rather than reducing the result to the input backgrounds by construction. The self-citation is normal for supplying a background state and is not load-bearing for the central demonstration, which rests on the novel coordinate system and non-modal RDI treatment. No fitted parameters are relabeled as predictions, no ansatz is smuggled via citation, and no uniqueness theorem from overlapping authors is invoked to force the outcome. The linear, dilute, well-coupled scope is explicitly stated as a limitation rather than hidden.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on linear perturbation theory applied to repurposed vortex backgrounds and on the extension of resonant drag instability ideas to non-sinusoidal waves; no new free parameters or invented entities are introduced in the abstract.

axioms (3)

domain assumption Dusty vortex models derived in paper I remain valid background flows for small-amplitude perturbations.
These models are directly repurposed as the base state for the stability analysis.
domain assumption A shearing-box analogue that follows vortex streamlines correctly captures the evolution of short-wavelength perturbations.
This coordinate choice is introduced to simplify the perturbation equations.
domain assumption Linear theory and resonant drag instability concepts extend to non-modal (non-sinusoidal) waves.
The paper uses this extension to identify the unstable modes.

pith-pipeline@v0.9.0 · 5577 in / 1437 out tokens · 55218 ms · 2026-05-10T17:17:11.105387+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Adams F. C., Watkins R., 1995, ApJ, 451, 314 Bae J., Isella A., Zhu Z., Martin R., Okuzumi S., Suriano S., 2023, in Inutsuka S., Aikawa Y., Muto T., Tomida K., Tamura M., eds, Protostars and Planets VII. p. 423 Bai X.-N., Stone J. M., 2010, ApJ, 722, L220 Barge P., Sommeria J., 1995, A&A, 295, L1 Baronett S. A., Yang C.-C., Zhu Z., 2024, MNRAS, 529, 275 B...

work page 1995
[2]

(A2) Regarding the axes of the box, we will useex to denote the box’s ‘radial’ unit vector, andey for the ‘azimuthal’ unit vector

= 0, we get Θ(t) = tS α−1 . (A2) Regarding the axes of the box, we will useex to denote the box’s ‘radial’ unit vector, andey for the ‘azimuthal’ unit vector. We demand thatey remains aligned withuK and thate x points away from the vortex’s centre, leading to ex = αcos(Θ)e X −sin(Θ)e Yp sin2(Θ) +α 2 cos2(Θ) ,e y =− sin(Θ)e X +αcos(Θ)e Yp sin2(Θ) +α 2 cos2...

work page 2009
[3]

(B1a) then shows thatyc is solution to Cd Hc yc =−(d tCd +C d Hd)y d,(B2) which is a simple linear problem

Injecting Eq. (B1a) then shows thatyc is solution to Cd Hc yc =−(d tCd +C d Hd)y d,(B2) which is a simple linear problem. Therefore, we can use an off-the-shelf ODE solver onyd, but each time it calls for the derivative ofy d, we start by solving Eq. (B2). This provides the correct value ofyc to input into Eq. (B1a), which finally gives the desired deriva...

work page 1952
[4]

(B1b) is verified only if it was verified by the initial value

B1.3 A differential-algebraic Floquet problem The method described in §B1.2 only conserves the value of Cd yd, so Eq. (B1b) is verified only if it was verified by the initial value. Consequently, we need a basis ofacceptableini- tial values for the Floquet analysis of §B1.1. Figure B1.Figure 1 from Chavanis (2000), partially reproduced with LAVA. It displ...

work page 2000
[5]

the func- tion which mapsstoU −1(s) ˜ug,1(s)is differentiatedktimes, then evaluated ins=t

The last 6 members of this basis are the 6 desiredacceptable initial values{y i. 1 , ...,y i. 6 }. B2 LAVA’s algorithm First, LAVA uses Schmidt’s procedure to obtain an orthonor- mal basis of acceptable initial values{y i. 1 , ...,y i. 6 }. Then, it evolves these initial values for one period, usingsolve_ivp and the index-reduction method described in §B1...

work page 2009

[1] [1]

Adams F. C., Watkins R., 1995, ApJ, 451, 314 Bae J., Isella A., Zhu Z., Martin R., Okuzumi S., Suriano S., 2023, in Inutsuka S., Aikawa Y., Muto T., Tomida K., Tamura M., eds, Protostars and Planets VII. p. 423 Bai X.-N., Stone J. M., 2010, ApJ, 722, L220 Barge P., Sommeria J., 1995, A&A, 295, L1 Baronett S. A., Yang C.-C., Zhu Z., 2024, MNRAS, 529, 275 B...

work page 1995

[2] [2]

(A2) Regarding the axes of the box, we will useex to denote the box’s ‘radial’ unit vector, andey for the ‘azimuthal’ unit vector

= 0, we get Θ(t) = tS α−1 . (A2) Regarding the axes of the box, we will useex to denote the box’s ‘radial’ unit vector, andey for the ‘azimuthal’ unit vector. We demand thatey remains aligned withuK and thate x points away from the vortex’s centre, leading to ex = αcos(Θ)e X −sin(Θ)e Yp sin2(Θ) +α 2 cos2(Θ) ,e y =− sin(Θ)e X +αcos(Θ)e Yp sin2(Θ) +α 2 cos2...

work page 2009

[3] [3]

(B1a) then shows thatyc is solution to Cd Hc yc =−(d tCd +C d Hd)y d,(B2) which is a simple linear problem

Injecting Eq. (B1a) then shows thatyc is solution to Cd Hc yc =−(d tCd +C d Hd)y d,(B2) which is a simple linear problem. Therefore, we can use an off-the-shelf ODE solver onyd, but each time it calls for the derivative ofy d, we start by solving Eq. (B2). This provides the correct value ofyc to input into Eq. (B1a), which finally gives the desired deriva...

work page 1952

[4] [4]

(B1b) is verified only if it was verified by the initial value

B1.3 A differential-algebraic Floquet problem The method described in §B1.2 only conserves the value of Cd yd, so Eq. (B1b) is verified only if it was verified by the initial value. Consequently, we need a basis ofacceptableini- tial values for the Floquet analysis of §B1.1. Figure B1.Figure 1 from Chavanis (2000), partially reproduced with LAVA. It displ...

work page 2000

[5] [5]

the func- tion which mapsstoU −1(s) ˜ug,1(s)is differentiatedktimes, then evaluated ins=t

The last 6 members of this basis are the 6 desiredacceptable initial values{y i. 1 , ...,y i. 6 }. B2 LAVA’s algorithm First, LAVA uses Schmidt’s procedure to obtain an orthonor- mal basis of acceptable initial values{y i. 1 , ...,y i. 6 }. Then, it evolves these initial values for one period, usingsolve_ivp and the index-reduction method described in §B1...

work page 2009