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Vertical Shear Instability in Thermally-Stratified Protoplanetary Disks: II. Hydrodynamic Simulations and Observability

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Pith's one-line read Thermally stratified protoplanetary disks drive vertical shear instability turbulence more than an order of magnitude stronger than isothermal disks, and the resulting ring-segment velocity residuals should be detectable at high…

desk verdict Solid, well-executed simulation study: thermal stratification boosts VSI turbulence an order of magnitude and changes its ALMA morphology, though the absolute alpha values are upper limits under instantaneous cooling. read the letter →

arxiv 2412.09930 v1 pith:J4VRIAOZ submitted 2024-12-13 astro-ph.EP

Han-Gyeol Yun , Woong-Tae Kim , Jaehan Bae , Cheongho Han This is my paper · ORCID
classification astro-ph.EP
keywords protoplanetarydisksverticalshearinstabilityhydrodynamicsimulationsthermalstratificationturbulencevelocityresidualssyntheticobservationsangularmomentumtransport
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Thermally stratified protoplanetary disks — with a hotter atmosphere above a cooler midplane — drive the vertical shear instability (VSI), a purely hydrodynamic source of turbulence, much harder than vertically isothermal disks do. In three-dimensional simulations, the VSI begins growing within a few orbits and saturates with a Reynolds stress-to-pressure ratio $\alpha_{R\phi} \approx 1\times10^{-3}$ (for an atmosphere twice as hot as the midplane) and $\approx 2\times10^{-3}$ (three times as hot), compared with $\approx 1\times10^{-4}$ in the isothermal case. The saturated turbulence is strongest near the disk surfaces, and after subtracting Keplerian rotation it appears in synthetic millimeter images as ring segments rather than full rings, with velocity residuals of roughly 50–100 m/s at 20° inclination and visibility up to 45°. The authors argue that this makes VSI turbulence detectable in real disks and that the height-dependent, longer-wavelength patterns observed in some disks are naturally explained by thermal stratification. They caution that the models assume instantaneous cooling, so these amplitudes are likely upper limits.

What carries the argument

The load-bearing object is the vertical shear parameter $q = -R \, \partial\ln\Omega/\partial Z$, the fractional change of orbital angular velocity with height. Realistic disks with a hotter atmosphere have a non-monotonic $q$ profile that peaks near $|Z| \approx Z_q/2$ (about 1.5 scale heights), stronger than the linearly growing $q$ of an isothermal disk; the simulations compare $n=1,2,3$, where $n = T_{\rm atm}/T_{\rm mid}$. This enhanced shear shortens the growth time, lengthens the preferred radial wavelength (the measured $\lambda_R/H$ grows from about 3.3 to 10.7 with $n$), and drives the two-stage growth of surface modes followed by body modes. Saturation is measured by $\alpha_{R\phi}$, the volume-averaged Reynolds stress to thermal pressure, and the observability claim rests on synthetic velocity residual maps $v_0 - v_{\rm mod}$ obtained by fitting and subtracting a Keplerian disk model from the line-of-sight velocity centroids.

What would settle it

A radiation-hydrodynamic rerun of the $n=2$ and $n=3$ models with finite cooling times that saturates $\alpha_{R\phi}$ below about $10^{-4}$, or an interferometric observation of a strongly stratified disk at $i=20°$ in 12CO that shows no 50–100 m/s ring segments, would falsify the central claim.

Watch

Extended reading notes

Core claim

The central claim is that vertical thermal stratification does not merely modify the vertical shear instability — it strengthens it. In the stratified models ($n \equiv T_{\rm atm}/T_{\rm mid} = 2$ and $3$), the instability grows earlier and faster than in the isothermal $n=1$ model, reaching a saturated turbulence level $\alpha_{R\phi}$ of about $1\times10^{-3}$ and $2\times10^{-3}$, respectively, more than an order of magnitude above the isothermal value of $1\times10^{-4}$. The perturbation velocities in the radial and azimuthal directions grow relative to the vertical ones as stratification increases, so the turbulence is less purely meridional. When the saturated velocity fields are put through synthetic line radiative transfer and a Keplerian model is subtracted, the residual maps show quasi-axisymmetric rings in the isothermal disk but fragmented ring segments in stratified disks; these segments survive up to an inclination of 45° and grow stronger when the tracer line is optically thick (12CO) because it probes the surface layers where the VSI is most active.

Load-bearing premise

The models treat the gas as locally isothermal with instantaneous cooling, the ideal limit for VSI growth; in real disks, finite cooling in optically thick regions damps the instability, so the simulated turbulence amplitudes are probably upper limits, as the authors state.

Editorial extensions

If this is right

  • In a thermally stratified disk, VSI-driven angular momentum transport reaches $\alpha_{R\phi} \gtrsim 10^{-3}$, enough to make the instability a significant driver of radial gas accretion and mixing, not just a source of small velocity noise.
  • VSI kinematics are height-dependent: optically thick tracers like 12CO show stronger, more readily detected residuals than optically thin tracers like C18O that probe the midplane.
  • The ring-segment morphology of stratified VSI turbulence remains visible at inclinations up to 45°, whereas isothermal rings become hard to detect at high inclination because they are dominated by vertical motion.
  • The longer radial wavelength and height dependence of stratified VSI can reconcile model kinematics with ring-like velocity perturbations observed in real disks that isothermal models could not explain.
  • Because the simulations use instantaneous cooling, the reported $\alpha$ and velocity amplitudes should be read as upper limits; finite cooling in optically thick regions is expected to weaken them.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If surface layers of real disks are this turbulent, dust settling and grain-growth timescales in the upper layers could differ from midplane values, changing predictions for where planetesimals form.
  • The fitted wavelength scaling ($\lambda_R/H$ increasing with $T_{\rm atm}/T_{\rm mid}$) could be turned around observationally: measuring the radial spacing of ring segments in residual maps would constrain the disk's vertical temperature stratification.
  • Comparing 12CO and C18O residual maps of the same disk could serve as a direct probe of the vertical shear profile, since the ratio of the two signals encodes where the instability operates.
  • A radiation-hydrodynamic version of these runs would show whether surface-dominated turbulence survives finite cooling; if it does, the observable ring segments might persist even when midplane VSI is suppressed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 11 minor

Summary. This paper presents three-dimensional hydrodynamic simulations of the vertical shear instability (VSI) in protoplanetary disks, extending Barraza-Alfaro et al. (2021) to vertically stratified temperature profiles and following the authors' linear analysis (Paper I). Three locally isothermal, inviscid models with thermal stratification ratio n = Tatm/Tmid = 1, 2, 3 (n = 1 being the isothermal baseline) are evolved with FARGO3D. The authors report that stratification enhances VSI growth and saturation: the volume-averaged Reynolds stress reaches alpha_Rphi ~ 1 x 10^-4 for n = 1 versus ~ 1 x 10^-3 and ~ 2 x 10^-3 for n = 2 and 3, with the saturated turbulence concentrated near the disk surfaces. Synthetic ALMA observations, produced with RADMC3D and the standard bettermoments/eddy/syndisk pipeline, show that after subtracting the fitted Keplerian model, VSI-induced velocity perturbations appear as quasi-axisymmetric rings in isothermal disks and as ring segments in stratified disks. Residual amplitudes are roughly 50-100 m/s at 20 degrees inclination, the ring segments remain visible up to 45 degrees in the n = 3 model, and the residual amplitude increases with tracer optical depth (12CO stronger than 13CO and C18O). The authors argue that these signatures may explain kinematic features such as the rings observed in MWC 480.

Significance. If the results hold, the paper makes two contributions. First, it establishes a quantitative, physically grounded link between vertical thermal stratification and VSI vigor in the idealized locally isothermal limit, with the n-dependence of the growth rates and wavelengths following the linear theory of Paper I. Second, and more robustly, it produces falsifiable kinematical predictions for ALMA: ring segments rather than full rings, visibility at inclinations up to 45 degrees, and a tracer-opacity dependence that a detection experiment can test. The computational work is careful and reproducible: public codes (FARGO3D, RADMC3D, syndisk, bettermoments, eddy) are used throughout; Appendix A presents resolution (15 vs 30 cells per scale height) and meridional boundary-condition convergence tests, showing the saturated alpha_Rphi to be insensitive to both; and the nonlinear state is emergent rather than fitted, so there is no circularity in the central results. The observability conclusions are the more durable part of the paper because the residuals are dominated by the surface layers, where finite cooling times are least damaging.

major comments (2)
  1. [Abstract; Section 4.4; Section 5 item 1] The headline quantitative claim, that "the turbulence stress reaches alpha_Rphi >= 10^-3, more than an order of magnitude stronger than the isothermal case," appears in the Abstract without qualification, and Section 5 item 1 repeats the values alpha_Rphi = 1 x 10^-4, 1 x 10^-3, and 2 x 10^-3 as results. The simulations, however, use a locally isothermal (instantaneous-cooling), inviscid equation of state, and Section 4.4 itself states that the VSI amplitudes are "likely to be upper limits compared to the ones in real disks." Because alpha_Rphi in Eq. (6) is a volume average that includes the midplane, and the cited radiation-hydrodynamic and two-temperature simulations (Fukuhara et al. 2023; Pfeil et al. 2023; Melon Fuksman et al. 2024; Zhang et al. 2024) find the VSI damped wherever the cooling time exceeds the critical value, the abstract's number will reasonably be read as a prediction for real disks unless the upper-limit status is stated there. Please move the qualification into the Abstract and Section 5, and preferably add a short quantitative estimate, for instance the mass or volume fraction of the computed disk in which the cooling time exceeds the local critical value using a cooling-time prescription from the cited literature, to indicate the expected reduction in the volume-averaged alpha_Rphi. The relative comparison between n = 1, 2, and 3 within the model family is unaffected and needs no change.
  2. [Section 3.2; Fig. 10; Section 5 item 2] The observability conclusions, that VSI signatures "can potentially be observable by ALMA" and "remain visible at disk inclinations as high as 45 degrees," rest on visual inspection of single-epoch synthetic maps with no quantitative detection criterion. The residual maps are produced by subtracting a 13-parameter fitted Keplerian model, and the turbulence itself is time-dependent: the stress profiles in Fig. 7 are averaged over t = 150-300 torb, while the images use only the epoch t = 300 torb. A matched-filter or SNR measurement of the ring-segment pattern in the noise-added cubes, ideally evaluated against a quiescent control disk and across several epochs in the saturated state, is needed to support the detectability claim. This is especially important at i = 45 degrees (Fig. 10) because the authors attribute the spurious radial features at i = 35 degrees (Fig. 9) to emission-surface fitting mismatches, an effect that should worsen with inclination.
minor comments (11)
  1. [Fig. 1 caption] The caption labels the models "Tatm/Tmid = 0, 1, 2"; this should read n = 1, 2, 3, since n = 0 is never defined in the text.
  2. [Fig. 11 caption and Section 4.1] The phrase "only vr, vphi, or vphi" duplicates vphi; the middle panel shows vtheta, so this should read "only vr, vtheta, or vphi."
  3. [Section 3.2 heading] "Synthethic Observations" is a misspelling of "Synthetic Observations."
  4. [Section 2] The isotope ratio is written as "[16C]/[18O] ~ 560"; 16C is not the relevant species, and this should be [16O]/[18O] to match Wilson & Rood (1994).
  5. [Section 3.2; Section 4.3] "Kelvin-Helomholtz-like parasitic instability" should be "Kelvin-Helmholtz-like," and "Rossyby instability" should be "Rossby instability."
  6. [Appendix A] The phrase "the models with n = 1, 2m and 3" contains a typo ("2m") and should read "n = 1, 2, and 3."
  7. [Section 4.4] The sentence "we expect that the amplitudes ... is likely to be upper limits" has a subject-verb agreement error and should read "the amplitudes ... are likely to be upper limits."
  8. [Abstract; Fig. 10] The claim that the ring segments "are visible at disk inclinations as high as 45 degrees" is demonstrated only for the n = 3 model; please qualify the statement as applying to n = 3 or show the i = 45 degrees case for n = 2 as well.
  9. [Appendix A.1] The resolution study spans only a factor of two (15 vs 30 cells per scale height), whereas Flores-Rivera et al. (2020) recommend at least 64 cells per scale height; the presented test does support convergence of the saturated alpha_Rphi, but the authors should state explicitly which diagnostics they regard as converged and why the factor-of-two test suffices for the observability claims, which depend on the resolved mode structure.
  10. [References] The Urpin & Brandenburg (1998) entry contains a duplicated DOI: "10.1111/j.1365-8711.1998.01118.x10.1111/j.1365-8711.1998.01118.x."
  11. [Section 3.1] The "time-averaged velocity perturbations" quoted for the midplane are not tied to a stated averaging window; please specify the interval (presumably t = 150-300 torb, as in Fig. 7).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: saturated turbulence and residual velocities are simulation outputs, not fitted inputs.

full rationale

The paper's central claims—saturated α_Rφ ≈ 1e-3 to 2e-3 in thermally stratified disks, enhanced velocity residuals, and ring-segment morphology—are measured from time-dependent 3D hydrodynamic simulations (Figures 1, 8, 9) rather than computed from any fitted target. Equation (6) defines α_Rφ as a volume-averaged Reynolds stress divided by thermal pressure, and the quoted numbers are the measured saturated values, not input parameters. The wavelength fits in Equation (5) are descriptive fits to simulation data, and their comparison with Equation (10) of Paper I is an interpretive step, not a premise of the calculation. The thermal stratification parameter n is imposed a priori in Equation (1), and the resulting shear profiles q in Equation (4) are computed, not tuned to match the turbulence outcome. Paper I (Yun et al. 2024) is self-cited for the linear mode structure and wavelength scalings used to interpret the nonlinear runs, but the simulations are fully specified in the present paper and do not require Paper I's results as an input. The synthetic observations use standard tools (RADMC3D, BETTERMOMENTS, EDDY) and subtract a fitted Keplerian model, which is the standard observational procedure and, if anything, tends to absorb rather than manufacture signal. The acknowledged caveat in Section 4.4—that the locally isothermal equation of state makes the quoted amplitudes upper limits in real disks—is a modeling limitation and correctness risk, not a circular step. No equation reduces to an input by construction, and no prediction is a renamed fit.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claims rest on an idealized hydrodynamic model with hand-chosen disk parameters and standard radiative transfer assumptions. No free parameter is fitted to an external target; the parameter n is a scan variable. The dominant caveat is the locally isothermal approximation, which the authors themselves state yields upper limits.

free parameters (2)
  • thermal stratification ratio n = Tatm/Tmid = 1, 2, 3
    Chosen by hand to represent isothermal and progressively hotter atmospheres; the central enhancement result depends on this scan (Section 2).
  • atmosphere height Zq = 3H
    Sets the height at which the atmosphere temperature begins; chosen following Dartois et al. (2003), affects the vertical shear profile (Section 2).
assumptions (5)
  • domain assumption The disk is inviscid and locally isothermal (temperature fixed, no cooling timescale).
    Stated in Section 2; maximizes VSI growth, and the authors flag in Section 4.4 that this makes amplitudes upper limits.
  • domain assumption The initial state satisfies vertical hydrostatic equilibrium and radial force balance (Eqs. 2 and 3).
    Sets the initial conditions from which VSI grows; standard for disk simulations.
  • domain assumption VSI is the only turbulence-driving mechanism considered; magnetic fields, non-ideal MHD, and buoyancy are neglected.
    Sections 1 and 4.4; the authors note these processes can stabilize or weaken the VSI.
  • domain assumption The synthetic observations assume standard CO abundances, isotope ratios, and LTE radiative transfer via RADMC3D.
    Section 3.2; adopted from literature, affects optical depth and velocity residual amplitudes.
  • domain assumption The Keplerian model subtracted from the velocity centroid has a fixed 13-parameter form (Eqs. 7 to 10) that does not absorb VSI signatures.
    Section 3.2; if the MCMC fit partially absorbs perturbations, the residual amplitudes are underestimated.

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Cite this review

Pith. "Pith review of Vertical Shear Instability in Thermally-Stratified Protoplanetary Disks: II. Hydrodynamic Simulations and Observability." pith.science (2026). https://pith.science/paper/J4VRIAOZ

@misc{pith2026241209930,
  author       = {Pith},
  title        = {Pith review of: Vertical Shear Instability in Thermally-Stratified Protoplanetary Disks: II. Hydrodynamic Simulations and Observability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4VRIAOZ}},
  note         = {Machine review of arXiv:2412.09930}
}
abstract

We conduct three-dimensional hydrodynamic simulations to investigate the nonlinear outcomes and observability of vertical shear instability (VSI) in protoplanetary disks. Our models include both vertically isothermal and thermally stratified disks, with the latter representing realistic conditions featuring a hotter atmosphere above the midplane. We find that the VSI grows more rapidly and becomes stronger in thermally stratified disks due to enhanced shear, resulting in higher levels of turbulence. At saturation, the turbulence stress reaches $\alpha_{R\phi}\gtrsim 10^{-3}$, more than an order of magnitude stronger than the isothermal case. The saturated turbulence is more pronounced near the disk surfaces than at the midplane. On synthetic velocity residual maps, obtained by subtracting the Keplerian rotational velocity, perturbations driven by the VSI manifest as axisymmetric rings in isothermal disks and as ring segments in thermally stratified disks. The latter are visible at disk inclinations as high as $45^\circ$ in thermally stratified disks. The amplitudes of these residual velocities range from $\sim 50$ to $\sim100$ $\mathrm{m\ s}^{-1}$ at a $20^\circ$ inclination, with larger values corresponding to greater thermal stratification. The magnitude of the observed velocity residual increases with the optical depth of the tracer used, as optically thick lines probe the regions near the disk surfaces.

Figures

Figures reproduced from arXiv: 2412.09930 by the authors.

Figure 1
Figure 1. Temporal variations of the perturbed kinetic en￾ergy δEK normalized by the initial kinetic energy EK(0) for models with Tatm/Tmid = 0, 1, 2. The initial increase in δEK results from the growth of surface modes, while the subse￾quent growth phase, indicated by an arrow in each model, is driven by body modes. reference radius. In each disk, the initial perturbations applied to vθ induce wave motions in the gas. The sy… view at source ↗
Figure 2
Figure 2. Snapshots of the meridional velocity vθ in the R–Z plane at selected times. Each row corresponds to the models with n = 1, 2, and 3 from top to bottom, respectively. The dashed lines in the first column marks the height with |Z| = Zq. In all models, VSI perturbations grow fastest near the disk atmospheres at smaller radii in the early stages. The VSI-unstable modes have higher growth rates and longer radial waveleng… view at source ↗
Figure 3
Figure 3. Radial wavelengths λR of the VSI measured at the times shown in the left panels of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Distributions of the perturbed quantities at the midplane at t/torb = 300. Each column represents the perturbed density δρ ≡ ρ − ρ(t = 0), radial velocity vr, meridional velocity vθ, and perturbed azimuthal velocity δvϕ ≡ vϕ − vϕ(t = 0) from left to right, respectively…
Figure 5
Figure 5. Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Distributions of the perturbations in the R–Z plane at ϕ = 0 when t/torb = 300. The panel layout is the same as in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 8
Figure 8. Figure 8: Temporal evolution of the Reynolds stress-to￾pressure ratio, αRϕ. At the end of the simulation, the ther￾mal stratification increases αRϕ by a factor of ∼ 10 and ∼ 20 in the disk with n = 2 and 3, respectively, compared to the isothermal disk. center, disk position ang…
Figure 9
Figure 9. Figure 9: Distributions of the residual velocity, v0 − vmod, of synthetic 12CO J = 2 − 1 lines, observed by a circular beam with FWHM of 0.10′′ after adding a root-mean-square noise of ∼ 1.5 mJy. Each row corresponds to the disk with n = 1, 2, 3 from left to right, while each co…
Figure 10
Figure 10. Figure 10: Velocity residual map for the n = 3 disk viewed at an inclination of i = 45◦ . Several ring segments generated by VSI perturbations are visible in the molecular line emis￾sion image. Barraza-Alfaro et al. (2021) demonstrated that the ve￾locity deviations in their velo…
Figure 11
Figure 11. Figure 11: Contributions of (left) vr, (middle) vθ, and (right) vϕ to the velocity residual, v0−vmod, for an inclination of i = 35◦ . Each row corresponds to the disk with n = 1, 2, 3 from top to bottom. The contributions of all three velocity components are comparable at this i…
Figure 12
Figure 12. Figure 12: Locations of the emission surface with an optical depth of unity, observed along the Z-axis, for 12CO (red), 13CO (green), and C18O (blue) lines in the disks with n = 1, 2, 3 from top to bottom. The solid black line denotes the simulation domain, while the dotted line…
Figure 13
Figure 13. Figure 13: Velocity residual maps for 12CO, 13CO, and C18O from left to right in the disks with n = 1 (top) and 3 (bottom) for an inclination of i = 20◦ . In the isothermal disk, the magnitudes of the velocity residuals remain small, irrespective of the tracer elements. In the n…
Figure 14
Figure 14. Figure 14: Comparison of αRϕ between models with different resolutions. The solid and dashed lines represent the models with 15 and 30 cells per scale height, respectively. While αRϕ varies with resolution at t/torb ≲ 250 in the vertically isothermal disk, it shows almost no dep…
Figure 15
Figure 15. Figure 15: Comparison of αRϕ from models with reflecting (solid) and zero-gradient (dashed) boundary conditions. Irrespective of vertical thermal stratification, saturated turbulence shows little sensitivity to the meridional boundary conditions at t/torb ≳ 200. suggests that th…

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.