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Vertical Shear Instability in Thermally-Stratified Protoplanetary Disks: I. A Linear Stability Analysis

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Thermally stratified protoplanetary disks grow the vertical shear instability faster and with more radial kinetic energy.

desk verdict A clean linear VSI analysis for stratified disks whose central new claim—the surface-mode bifurcation—is undermined by untested boundary placement. read the letter →

arxiv 2412.09924 v1 pith:7N7RBMVL 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 verticalshearinstabilityprotoplanetarydisksthermalstratificationlinearstabilityanalysissurfacemodesbodydiskturbulence
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

The paper tries to establish that the vertical shear instability (VSI), a purely hydrodynamic mechanism that can drive turbulence in protoplanetary disks, is substantially stronger in disks whose surfaces are hotter than their midplanes, as real irradiated disks are. Using a semi-global linear analysis (local in radius, global in height) of three disk models with atmosphere-to-midplane temperature ratios 1, 2, and 3, it finds that thermal stratification splits the previously known surface modes into two branches, raises their growth rates by up to a factor of about 2, and increases the share of kinetic energy in the radial direction. These changes matter because VSI turbulence regulates dust motions and planet formation, and because they give concrete predictions for what hydrodynamic simulations and molecular-line observations should see.

What carries the argument

The load-bearing object is a second-order linear ODE for the vertical structure of the density perturbation, Eq. (29): $$\frac{\$partial^{2}$ \hat{\Pi}}{\partial \$zeta^{2}$} + \left(\frac{\partial \bar{\Pi}}{\partial \zeta} + \frac{\partial \ln f}{\partial \zeta} + 2ik\bar{q}\right)\frac{\partial \hat{\Pi}}{\partial \zeta} - \$sigma^{2}$ $k^{2}$ \hat{\Pi} = 0,$$ with no-flow boundary conditions at $\zeta = \pm 5$. Here $\zeta$ is the scaled vertical coordinate, $f(\zeta) = T(\zeta)/T_{\rm mid}$ is the vertical temperature profile, $\bar{q}$ is the vertical shear parameter, $k$ is the scaled radial wavenumber, and $\sigma$ is the complex eigenfrequency whose real part is the growth rate. This equation turns the VSI into a vertical eigenvalue problem: the eigenfunctions classify modes by where their vertical structure is concentrated, and the non-monotonic $\bar{q}(\zeta)$ produced by stratification is what splits the surface modes into two branches.

What would settle it

Recompute the eigenvalues of Eq. (29) with the vertical boundaries moved from $\zeta = \pm 5$ to $\zeta = \pm 7$ or replaced by radiative boundary conditions; if the second surface-mode branch's growth rate changes substantially or the branch disappears, the two-branch result is an artifact of the no-flow walls.

Watch

Extended reading notes

Core claim

The central claim is that thermal stratification changes both the spectrum and the character of VSI eigenmodes. In an isothermal disk the unstable modes separate into surface modes, confined to regions of strongest vertical shear, and body modes that extend across the disk. When the disk is vertically stratified, the shear profile $q = -R \partial \ln \Omega / \partial Z$ develops a local maximum away from the surfaces, and the surface modes bifurcate into two branches: one localized near that interior shear peak (around $Z = 16$ au for the $n = 2$ and 3 models at $R_0 = 100$ au) with the larger growth rate, and one near the disk surfaces with growth nearly unchanged from the isothermal case. Stratification also increases body-mode growth at small radial wavenumbers and raises the ratio of radial to vertical kinetic energy; for $k = \pi/5$ the $n = 3$ disk's fundamental body mode has about three times the energy ratio of the $n = 1$ disk.

Load-bearing premise

The calculation assumes the gas cools infinitely fast (isothermal) and puts rigid walls at $\zeta = \pm 5$; the second surface-mode branch sits right at those walls, so its existence and growth rate may be set by the artificial boundary rather than by disk physics.

Editorial extensions

If this is right

  • In a stratified disk, the fastest-growing disturbances will be surface modes with large radial wavenumber $k_R$; body modes with small $k_R$ take over later, and the turnover happens earlier when the atmosphere is hotter.
  • The most unstable surface mode at $k = 5\pi$ grows about 1.4 times ($n = 2$) and 2.1 times ($n = 3$) faster than its isothermal counterpart, so turbulence should develop quicker in more stratified disks.
  • Thermal stratification raises the radial-to-vertical kinetic-energy ratio $R_{\rm KE}$; for the $k = \pi/5$ fundamental mode the $n = 3$ disk reaches roughly three times the $n = 1$ value, which alters the expected velocity signature of the turbulence.
  • Applying the local theory to the density-weighted mean shear predicts growth rates, radial wavelengths, and turbulent amplitudes in the $n = 3$ disk about 5–15 times those in the $n = 1$ disk, making stratification a first-order control on VSI outcomes.

Reading between the lines

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

  • If the vertical boundaries were moved from $\zeta = \pm 5$ outward (or replaced by open conditions), the second surface-mode branch, localized at $|\zeta| \approx 5$, might shift or vanish, which would mean the two-branch bifurcation is partly a box effect.
  • Solving the same eigenvalue problem with finite cooling time and the Brunt–Väisälä frequency from Eq. (6) would test how much of the stratification-driven growth survives where the disk cools too slowly for the isothermal assumption.
  • The preference for small-$k_R$ body modes in stratified disks implies that numerical experiments need wide radial domains; periodic radial boxes could suppress exactly the modes stratification favors.
  • A larger radial-to-vertical energy ratio should make VSI turbulence in flared irradiated disks more anisotropic, so high-resolution molecular-line observations may be able to distinguish stratified from isothermal disk models without resolving individual modes.
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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

3 major / 6 minor

Summary. This paper analyzes the linear stability of the vertical shear instability (VSI) in protoplanetary disks with vertical thermal stratification. The authors derive a semi-global eigenvalue equation (Eq. 29) for axisymmetric perturbations, adopting an isothermal equation of state and a background equilibrium with a vertically varying temperature profile. They solve the eigenvalue problem with Chebyshev collocation for isothermal (n=1) and stratified (n=2,3) disk models. They recover previously known surface and body modes in the isothermal limit, and for stratified disks they report a bifurcation of the surface modes into two branches, along with enhanced growth rates and an increased radial-to-vertical kinetic energy ratio. The paper concludes that VSI simulations will initially excite large-wavenumber surface modes, followed by small-wavenumber body modes, with the transition occurring earlier in more stratified disks.

Significance. If the findings are correct, the paper extends VSI linear theory to a more realistic disk temperature structure and makes concrete predictions about mode selection and kinetic energy partition that can be tested in upcoming simulations. The derivation of Eq. (29) is clear and reproduces the isothermal results of Nelson et al. (2013) and Barker & Latter (2015). The use of a public spectral code with a stated convergence check is a strength. However, the central novelty—the surface-mode bifurcation—depends on the placement of the computational boundary and on a vertical-coordinate scaling that is currently inconsistent between Eq. (22) and the text. These issues must be resolved before the main conclusions can be accepted.

major comments (3)
  1. [4.1, Eq. (22); Section 5] The vertical coordinate scaling is inconsistent with the physical disk height. With H0 = 10 au and h0 = 0.1, the definition ζ ≡ z/h0, where z = Z/H0, gives ζ = Z/(h0H0) = Z/(1 au). The computational domain |ζ| ≤ 5 therefore corresponds to |Z| ≤ 5 au, not to |Z| = 50 au as stated repeatedly (e.g., 'strong shear at the disk surfaces (Z = 50 au)' in Section 5). This discrepancy changes the interpretation of the surface modes: the second-branch modes at |ζ| ≈ 5 would lie at Z ≈ 5 au, inside the stratified region and far from the temperature-transition height Zq = 3H = 30 au. The authors must either correct the scaling definition or the mapping to physical units and re-examine whether the reported mode structure and growth rates remain the same.
  2. [4.2.2, Figures 5 and 6] The existence and growth rates of the second-branch surface modes have not been shown to be independent of the numerical boundary. These modes are localized at |ζ| ≈ 5, which is exactly the position of the imposed no-flow boundary condition ∂Π/∂ζ = 0. The convergence check with N = 400 and 600 grid points only establishes resolution convergence for a fixed domain; it does not test the sensitivity to the domain size. If the boundary is moved outward (e.g., |ζ| ≤ 10 or 20), the second-branch modes could disappear or change growth rate substantially if they are supported by the reflecting wall rather than by the disk physics. The authors should repeat the eigenvalue calculations for larger domains and show that the bifurcation of surface modes, including the growth rates and eigenfunctions of the second branch, converges as the boundary recedes. Without this test, the central claim of a surface-mode bifurcation remains unverified.
  3. [2, Eq. (5); Section 5] The isothermal equation of state is justified by the critical cooling time criterion of Eq. (5), but the paper does not evaluate whether the actual cooling time in the modeled disks satisfies tcool ≲ tcrit. The discussion in Section 5 notes that the isothermal assumption 'may not be valid at large radii in PPDs,' which is a relevant caveat, but it leaves open the possibility that the reported growth-rate enhancements for n = 2 and 3 are overestimated if tcool exceeds tcrit in the regions where the new surface modes reside. The authors should provide a quantitative estimate of tcool for their disk parameters (or at least for a representative dust model) and state explicitly that the results apply only where efficient cooling holds.
minor comments (6)
  1. [Section 5 title] The title contains a typo: 'DICUSSION' should be 'DISCUSSION'.
  2. [4.2.1] The statement 'surface modes disappear when the radial wavenumber k surpasses the threshold kcrit ∼ π/2' is internally inconsistent with the example k = π/5, which is smaller than π/2. The intended meaning is presumably that surface modes require k > kcrit; the wording should be corrected.
  3. [2, Eq. (2)] Equation (2) appears to have a rendering issue in the integral: 'Z Z 0' should be a definite integral from 0 to Z.
  4. [4.2.3] The sentence 'But, how does the ratio of the radial to vertical kinetic energy vary with the degree of thermal stratification.' is a fragment and should be completed or merged with the following sentence.
  5. [4.2.2] The critical wavenumbers kcrit for n = 2 and 3 are reported as π and 2π/3, but the method by which these values were determined is not described; please clarify the criterion used to define the disappearance of surface modes.
  6. [Figure 6] The x-axis tick labels in Figure 6 (e.g., '4 2 0 2 4') are ambiguous because the minus signs are not visible in the manuscript; please ensure the axis is clearly labeled with values such as -4, -2, 0, 2, 4.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the linear-stability calculation is self-contained and no fitted parameter or self-citation carries the central claims.

full rationale

The paper's derivation chain is self-contained. The background disks are constructed from externally adopted temperature prescriptions (Barraza-Alfaro et al. 2021; Dartois et al. 2003), hydrostatic equilibrium, and radial force balance, with the parameters n, Zq, h0, alpha_T, and alpha_rho chosen from observations or varied for exploration rather than tuned to reproduce the reported eigenvalues. The perturbation equation (29) is derived by linearizing the isothermal hydrodynamics equations and reducing them using the semi-global scaling of Nelson et al. (2013); no step in that reduction assumes the final growth rates or the two-branch surface-mode structure. The eigenvalues are then obtained by solving the resulting boundary-value problem with the DEDALUS spectral code, and the isothermal results are checked against the independent analytic benchmark of Barker & Latter (2015), Eq. (30), which is used only as a comparison for body modes, not as an input to the stratified calculation. The energy ratio RKE in Eq. (34) is a diagnostic computed from the already-obtained eigenfunctions, not a quantity fitted to the model. The only substantive concerns, namely the no-flow boundary at ζ = ±5 potentially affecting the second-branch surface modes localized near |ζ| ≈ 5 and the efficient-cooling/isothermal equation of state, are modeling assumptions and limitations rather than circular reductions; the authors explicitly acknowledge the isothermal limitation and the boundary placement is not hidden or retrofitted to force the bifurcation. Accordingly, no circular step of any enumerated kind is present.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on a standard semi-global reduction from Nelson et al. (2013), an isothermal efficient-cooling equation of state, and a no-flow vertical domain. No new physical entities are introduced. The free parameters are model choices from observations or exploration, not fitted to the target growth rates.

free parameters (6)
  • Thermal stratification ratio n = Tatm/Tmid = 1, 2, 3
    Model parameter controlling the strength of vertical thermal stratification; the central results are reported as functions of n.
  • Atmosphere transition height Zq = 3H
    Height where the temperature profile transitions from the midplane cosine form to the constant atmosphere; sets the location of the shear peak relative to the boundary.
  • Disk aspect ratio h0 = H0/R0 = 0.1
    Used in the asymptotic reduction and rescaling of the perturbation equations; controls the scale of the dimensionless wavenumber k.
  • Midplane temperature slope alpha_T = -1/2
    Adopted from Barraza-Alfaro et al. (2021) for the radial temperature profile; enters the isothermal qbar profile and the comparison limit.
  • Midplane density slope alpha_rho = -9/4
    Chosen to give a surface density profile Sigma ~ R^-1; a background model input, not fitted to the instability results.
  • Vertical domain boundary Zmax = 50 au (zeta = +-5)
    Location of the no-flow boundary conditions; only weakly justified and never tested for sensitivity, while several surface modes localize at this boundary.
assumptions (5)
  • domain assumption Isothermal equation of state with efficient cooling, so vertical buoyancy is neglected.
    Justified by the critical cooling time argument in Eq. (5), but the authors acknowledge the assumption may fail at large radii where cooling is inefficient.
  • domain assumption Semi-global ansatz: perturbations are local in the radial direction and global in the vertical direction, with terms of order h0^2 dropped.
    Inherited from Nelson et al. (2013) and Barker & Latter (2015); requires the radial wavelength to be much shorter than the radial disk scale.
  • domain assumption No-flow boundary conditions at zeta = +-5.
    Chosen to close the eigenvalue problem. The second-branch surface modes in stratified disks are localized near these boundaries, so the boundary placement may materially affect the result.
  • domain assumption Background equilibrium follows hydrostatic balance and radial force balance with a point-mass potential, neglecting self-gravity.
    Standard thin-disk setup. The disk mass is set to 5% of M_sun but self-gravity is ignored, which the authors state does not affect the linear results.
  • domain assumption Temperature prescription follows Dartois et al. (2003) with Zq = 3H and n = 1, 2, 3.
    A representative observational parameterization, not derived from first principles; all stratified results depend on this profile.

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Cite this review

Pith. "Pith review of Vertical Shear Instability in Thermally-Stratified Protoplanetary Disks: I. A Linear Stability Analysis." pith.science (2026). https://pith.science/paper/7N7RBMVL

@misc{pith2026241209924,
  author       = {Pith},
  title        = {Pith review of: Vertical Shear Instability in Thermally-Stratified Protoplanetary Disks: I. A Linear Stability Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7N7RBMVL}},
  note         = {Machine review of arXiv:2412.09924}
}
abstract

Vertical shear instability (VSI), driven by a vertical gradient of rotational angular velocity, is a promising source of turbulence in protoplanetary disks. We examine the semi-global stability of thermally stratified disks and find that the VSI consists of surface and body modes: surface modes are confined to regions of strong shear, while body modes extend perturbations across the disk, consistent with the previous findings. In thermally stratified disks, surface modes bifurcate into two branches. The branch associated with the strongest shear at mid-height exhibits a higher growth rate compared to the branch near the surfaces. Surface modes generally grow rapidly and require a high radial wave number $k_R$, whereas body mode growth rates increase as $k_R$ decreases. Thermal stratification enhances the growth rates of both surface and body modes and boosts VSI-driven radial kinetic energy relative to vertical energy. Our results suggest that simulations will initially favor surface modes with large $k_R$, followed by an increase in body modes with smaller $k_R$, with faster progression in more thermal stratified disks.

Figures

Figures reproduced from arXiv: 2412.09924 by the authors.

Figure 1
Figure 1. Vertical distributions of key quantities at the reference radius R0 = 100 au for disk models with differing thermal stratification (n ≡ Tatm/Tmid = 1, 2, 3): (a) the gas temperature T, (b) gas density ρ, (c) rotational velocity vϕ = RΩ, and (d) vertical shear q = −R∂ ln Ω/∂Z. In (a), various dashed lines depict the temperature profiles for some observed PPDs from the MAPS survey (Law et al. 2021). ρ0 = ρ(R0, 0): the… view at source ↗
Figure 2
Figure 2. Vertical distributions of (a) the normalzied Brunt-V¨ais¨al¨a frequency NZ and (b) the critical cooling time tcrit at the reference radius R0 for the disks with n = 1, 2, and 3. Urpin 2003) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Growth rates σr and oscillation frequencies σi for the VSI modes for (a) the radial wavenumber k = 5π (or, λ = 0.04H0) and (b) k = π/5 (or, λ = H0) in the isothermal disk with n = 1. Filled circles and triangles denote the body and surface modes, respectively. Dashed lines represent Equation (30) for the location of the body modes in the limit of an infinitely extended disk. The red circles indicate the most unstabl… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Eigenfunctions of the most unstable (a) surface mode with k = 5π, (b) body mode with k = 5π, and (c) body mode with k = π/5 in the n = 1 disk. The blue and red lines correspond to the real and imaginary parts of the eigenfunctions, respectively. Each eigenfunction is n…
Figure 5
Figure 5. Figure 5: Growth rates of the unstable modes vs. their oscillation frequencies for the thermally stratified disks with (upper) n = 2 and (lower) n = 3. The left and right panels correspond to the cases with k = 5π and k = π/5, respectively. Dashed lines represent Equation (30) f…
Figure 6
Figure 6. Figure 6: Eigenfunctions of the most unstable (a,e) first-branch surface mode with k = 5π, (b,f ) second-branch surface mode with k = 5π, (c,g) body mode with k = 5π, and (d,h) body mode with k = π/5. The upper and lower panels are for the thermally stratified disks with n = 2 a…
Figure 7
Figure 7. Figure 7: The ratio RKE of the kinetic energy densities in the radial and vertical directions for the modes shown in Figures 3 and 5. The top and bottom panels correspond to the modes with radial wavenumbers k = 5π and π/5, respectively. Each column represents the n = 1, 2, and …

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