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arxiv: 2604.12598 · v1 · submitted 2026-04-14 · 🌌 astro-ph.SR · physics.flu-dyn

Kelvin waves over a differentially rotating spherical shell

T. Boismard , M. Rieutord This is my paper

Pith reviewed 2026-05-10 14:45 UTC · model grok-4.3

classification 🌌 astro-ph.SR physics.flu-dyn
keywords Kelvin wavesdifferential rotationspherical shellgravito-inertial modescritical layerBe starsinstabilityshallow-water model
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The pith

Equatorial Kelvin waves on a spherical shell can be destabilized by radial differential rotation through a critical layer.

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

The paper examines whether equatorial Kelvin waves can become unstable in a rotating spherical shell, which could help explain episodic mass ejection in Be stars. Starting from analytical expressions for gravito-inertial modes in the shallow-water model, the authors numerically track how these modes behave as shell thickness, rotation, and differential rotation vary. They demonstrate that the waves persist in thick layers with weaker confinement and develop shear layers at low wavenumbers. When radial differential rotation is added, the waves destabilize for suitable values of shear and viscosity, with the growth rate peaking due to a critical layer where fluid speed matches the wave phase speed.

Core claim

Equatorial Kelvin waves still exist in a spherical shell of finite thickness, but their equatorial confinement is weaker. At low azimuthal wavenumbers, Kelvin waves enter the inertial waves frequency band and develop shear layers associated with singularities of the Poincaré equation. When a radial differential rotation is imposed, equatorial Kelvin waves can be destabilised provided that differential rotation and viscosity are in an appropriate range. The non-monotonic behaviour of the growth rate of the instability is traced back to the rise of a critical layer where the fluid azimuthal velocity equals the phase speed of the surface waves.

What carries the argument

The critical layer where the background azimuthal velocity matches the phase speed of the Kelvin waves, which enables instability growth when differential rotation and viscosity are tuned appropriately.

If this is right

  • Kelvin waves retain their identity in shells of finite thickness but exhibit weaker equatorial trapping.
  • Shear layers appear as new dissipative structures for low-wavenumber Kelvin waves.
  • Destabilization occurs only inside a bounded window of differential rotation strength and viscosity.
  • The resulting instability provides a possible wave-driven trigger for episodic excretion in Be stars.
  • Gravito-inertial waves in general can exhibit critical-layer instabilities under shellular rotation.

Where Pith is reading between the lines

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

  • Stellar evolution codes that include wave transport may need to add this instability channel when estimating surface mass-loss rates.
  • Laboratory rotating-tank experiments could reproduce the non-monotonic growth by varying the imposed shear and fluid viscosity.
  • The same critical-layer mechanism might operate in other astrophysical or geophysical flows that combine rotation and radial shear, such as planetary atmospheres.
  • Relaxing the shellular assumption to allow latitudinal variation in the background rotation could shift or enlarge the unstable parameter range.

Load-bearing premise

The background differential rotation remains strictly radial and shellular, with viscosity constant and small enough to allow a critical layer to form without being overwhelmed by damping.

What would settle it

A numerical experiment in which differential rotation and viscosity are set inside the predicted unstable window yet the Kelvin mode shows no growth, or direct visualization of the flow field showing no critical layer at the expected location.

Figures

Figures reproduced from arXiv: 2604.12598 by M. Rieutord, T. Boismard.

Figure 1
Figure 1. Figure 1: Dispersion relation of gravito-inertial modes in the shallow water system for Γ = 0.01. Left: Eigenfrequency of modes symmetric with respect to equator as a function of the azimuthal wavenumber m. Left : Same as right but for antisymmetric modes. Red dots show Poincaré modes, blue ones are for Rossby modes and green ones show Kelvin (left) and Yanai (right) modes. ζ = 2iµ ω √ Γ + mΓ wθ . (13) This leads to… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the eigenfrequency of the m = 1 Kelvin mode with the size of the core at γ = 1. In the shallow water and full sphere limits, the value given by their analytic expression, respectively (11) and (C.10), is found. 3.3. Properties of Kelvin Waves in a Thick Layer Kelvin waves do not exist without rotation, but they share several properties with surface gravity waves, which they are actually. They … view at source ↗
Figure 3
Figure 3. Figure 3: Top: Distribution of kinetic energy on a log10 scale in a meridional section of the spherical shell for the m = 10 Kelvin mode at λ = −1.31×10−5−12.31i. Bottom: Same for the m = 3 Kelvin mode at λ = −5.13×10−4 − 3.99i. In both cases η = 0.5, γ = 1, E = 10−7 , Nr = 200 and Lmax = 200. radius η = 0.7. This is explained by the fact that for such a high m, the eigenfunction has amplitude mainly close to the up… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the phase velocity, in the co-rotating frame, of Kelvin modes of various m as a function of the core size η for γ = 1, E = 10−3 with Nr =Lmax = 20. The red dashed line shows the phase velocity of the m = 10 Kelvin mode in the full sphere case. 0.0 0.2 0.4 0.6 0.8 1.0 1.4 1.3 1.2 1.1 1.0 m = 1 m = 9 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the frequency difference ∆ω = ωm+1 − ωm be￾tween two consecutive Kelvin modes as a function of the core radius. Red dots show the values of the full sphere case as de￾rived from (C.10), while the black dashed line shows the asymp￾totic shallow water case. Parameters are γ = 1, E = 10−3 with Nr =Lmax = 20. of the Kelvin wave takes a dependence on m and somehow weakens its dependence on Γ. When the c… view at source ↗
Figure 7
Figure 7. Figure 7: Top: Growth rate τ = ℜe(λ) of some Kelvin modes as a function of the Ekman number E, for Ωη = 2, Γ = 0.01 and η = 0.18. Bottom: same as top but as a function of differential rotation parametrized by Ωη, with E = 10−5 . Numerical resolu￾tion is Nr =Lmax = 100. A0 = Z V |u| 2 dV + 1 γ Z ∂V |r=1 |p − 2Eu′ r | 2 dS! > 0 (33) is a positive definite term. The first integral is the bulk vis￾cous dissipation, whic… view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of τ , the eigenvalue real part, of the m = 5 Kelvin mode as a function of Ωη, for three Ekman numbers. γ = 2, Nr=100 and Lmax=60. 1 2 3 4 5 6 7 8 0.150 0.125 0.100 0.075 0.050 0.025 0.000 (Ivis c o u s + Ic rp)/ E k E = 10 4 E = 5.5 × 10 4 E = 10 3 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the sum of the viscous integrals (I) and (II) in (32) as a function of Ωη for the m = 5 Kelvin mode, for three Ekman numbers. γ = 2, Nr=100 and Lmax=60. E, τ is positive when Ωη is large enough, however, remark￾ably, τ is negative again when Ωη is too large. Since τ is the sum of three integrals its behaviour can be explained by the dependence of these integrals with respect to Ωη. The bulk an… view at source ↗
Figure 12
Figure 12. Figure 12: Stability diagram of the m = 5 Kelvin mode at η = 0.18 and γ = 2. The purple region denotes the unstable domain in the explored parameter space, forming a single con￾nected region. The instability does not exist beyond a critical Ekman number E ≈ 5 × 10−4 . rotation strengthens, the critical layer moves towards the surface. Indeed, as shown by [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the growth rate τ of the m = 1 Kelvin mode as a function of Ekman number E. Frequency is ω ≃ −1.030303. Parameters are η = 0.35, γ −1 = 16.25, Ωη = 1.065 Nr=200 and Lmax=200. ity can also disappear if the Ekman number is low enough, as shown in the example of [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: Radial profiles of the coupling term in the equato￾rial plane θ = π/2, at longitude ϕ = 0, for an m = 5-Kelvin mode and for three differential rotations. Parameters are γ = 2, E = 1 × 10−5 , η = 0.18, Nr=100 and Lmax=100. Inertial frame eigenvalues are λ = −2.968 × 10−3 − 7.372i for Ωη = 2 implying rcri = 0.435. λ = 1.231 × 10−2 − 7.591i for Ωη = 5 implying and rcri = 0.704. λ = −4.083 × 10−2 − 7.929i, fo… view at source ↗
read the original abstract

Context. Be stars are presently viewed as B-type stars surrounded by a disc fueled by the star itself during episodicexcretion events. The origin of these events are poorly understood.Aims. This article aims to determine whether or not surface equatorial Kelvin waves can be unstable and therefore canplay a role in the triggering of the Be phenomenon.Methods. We first derive an analytical expression for gravito-inertial modes in the shallow-water framework. Then, weinvestigate numerically the evolution of equatorial Kelvin modes as system parameters vary. The study is extended tothick-layer configurations with a constant density fluid. We then analyze the stability of these modes under differentialrotation and viscous effects.Results. We show that equatorial Kelvin waves still exist in a spherical shell of finite thickness, but that their equatorialconfinement is weaker. At low azimuthal wavenumbers, Kelvin waves are in the inertial waves frequency band and thusget specificities of inertial waves like shear layers associated with singularities of the Poincar\'e equation. These shearlayers are new dissipative structures for Kelvin waves. When a radial (shellular) differential rotation is imposed, we showthat equatorial Kelvin waves can be destabilised provided that differential rotation and viscosity are in an appropriaterange. The non-monotonic behaviour of the growth rate of the instability is traced back to the rise of a critical layerwhere the fluid azimuthal velocity equals the phase speed of the surface waves.Conclusions. This study provides new insights into the behavior of equatorial Kelvin waves in astrophysics, particularlyin rapidly rotating stars. The results reinforce the idea that gravito-inertial waves, and more specifically the equatorialKelvin waves, can be unstable and thus be key parts in the mechanisms leading to the Be phenomenon.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript derives analytical expressions for gravito-inertial modes (including equatorial Kelvin waves) in the shallow-water framework on a spherical shell, then numerically evolves these modes under imposed shellular differential rotation and viscosity. It reports that Kelvin waves can be destabilized for appropriate ranges of differential rotation amplitude and viscosity, with non-monotonic growth rates attributed to the formation of a critical layer where the background azimuthal velocity matches the wave phase speed. The study is extended to finite-thickness constant-density layers, where Kelvin waves show weaker equatorial confinement and develop associated shear layers linked to Poincaré equation singularities.

Significance. If the central results hold, the work provides mechanistic insight into how gravito-inertial waves, specifically equatorial Kelvin waves, can become unstable in rapidly rotating stars and potentially contribute to the Be phenomenon. The combination of an analytical mode derivation with direct numerical time evolution of the linearized system, plus the explicit link between growth-rate behavior and critical-layer formation, strengthens the case for wave-driven surface dynamics in stellar interiors.

major comments (2)
  1. [Numerical evolution and stability analysis sections] The background differential rotation is imposed by hand as a fixed shellular profile throughout the evolution. The shallow-water equations with viscosity and wave-induced Reynolds stresses would in general cause this profile to diffuse or be modified; the manuscript does not demonstrate that the critical layer (where local azimuthal velocity equals the Kelvin-wave phase speed) can form and persist self-consistently before viscous damping or profile adjustment dominates. This assumption is load-bearing for the reported instability and non-monotonic growth-rate peak.
  2. [Results on growth rates and critical layer] The non-monotonic growth-rate behavior is traced to the critical layer, yet the manuscript provides no explicit quantitative comparison between the analytically predicted growth rates (from the derived dispersion relation) and the numerically measured values across the parameter space. Without this match, it remains unclear whether the instability is fully captured by the linear analysis or influenced by numerical resolution of the shear layers.
minor comments (2)
  1. [Extension to thick layers] The transition from the shallow-water limit to the thick-layer constant-density case would benefit from an explicit statement of the aspect-ratio parameter and how the equatorial confinement weakens with increasing thickness.
  2. [Abstract and parameter discussion] Notation for the viscosity coefficient and differential-rotation amplitude should be introduced once and used consistently when discussing the 'appropriate range' for instability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Numerical evolution and stability analysis sections] The background differential rotation is imposed by hand as a fixed shellular profile throughout the evolution. The shallow-water equations with viscosity and wave-induced Reynolds stresses would in general cause this profile to diffuse or be modified; the manuscript does not demonstrate that the critical layer (where local azimuthal velocity equals the Kelvin-wave phase speed) can form and persist self-consistently before viscous damping or profile adjustment dominates. This assumption is load-bearing for the reported instability and non-monotonic growth-rate peak.

    Authors: We agree that the fixed shellular profile constitutes an approximation within the linearised framework. This choice isolates the influence of a prescribed differential rotation on wave stability, which is standard for such analyses. We will add a dedicated paragraph in the revised manuscript comparing the instability growth timescale to the viscous diffusion timescale of the background profile, thereby delineating the parameter regimes in which the approximation remains valid. We also note that fully nonlinear evolution lies outside the present scope but would be a natural extension. revision: partial

  2. Referee: [Results on growth rates and critical layer] The non-monotonic growth-rate behavior is traced to the critical layer, yet the manuscript provides no explicit quantitative comparison between the analytically predicted growth rates (from the derived dispersion relation) and the numerically measured values across the parameter space. Without this match, it remains unclear whether the instability is fully captured by the linear analysis or influenced by numerical resolution of the shear layers.

    Authors: The analytical dispersion relation derived in the paper applies to the inviscid, uniformly rotating shallow-water system and yields the base frequencies and structures of the Kelvin waves. Growth rates arise only when differential rotation and viscosity are included, which are handled exclusively through numerical time integration; no closed-form analytical growth-rate expression exists for that regime. In the revision we will add a direct comparison of real frequencies (and thus phase speeds) between the analytical dispersion relation and the numerical results in the stable, non-diffusive limit to validate the code. We will further strengthen the critical-layer discussion with explicit diagnostics of the layer location and a brief resolution study confirming that the associated shear layers are adequately resolved. revision: partial

Circularity Check

0 steps flagged

No circularity: growth rates obtained from direct numerical evolution of linearized equations

full rationale

The paper first derives an analytical expression for gravito-inertial modes in the shallow-water framework, then performs numerical time evolution of the linearized system to diagnose the stability of equatorial Kelvin waves under an explicitly imposed shellular differential rotation and constant viscosity. The reported non-monotonic growth-rate behavior is traced to the emergence of a critical layer in the simulations; no equation reduces the growth rate to a quantity defined by the same fit or by self-citation. The background rotation profile is prescribed by hand as an input assumption rather than being derived from the wave dynamics, so the central claim does not collapse to a tautology. This is a standard linear stability analysis with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The central claim rests on the shallow-water approximation for the thin-shell case, the assumption of constant density in the thick-shell extension, and the imposition of a strictly radial (shellular) differential-rotation profile whose amplitude is treated as a free parameter.

free parameters (2)
  • differential rotation amplitude
    The range of differential rotation that produces positive growth rates is explored numerically and is not derived from first principles.
  • viscosity coefficient
    Viscosity must lie in an 'appropriate range' for the instability to appear; its value is chosen rather than predicted.
axioms (3)
  • domain assumption Shallow-water approximation remains valid for gravito-inertial modes near the equator
    Used to obtain the analytical expression for the modes.
  • domain assumption Background density is constant in the thick-layer extension
    Explicitly stated when the study is extended beyond the shallow-water limit.
  • domain assumption Differential rotation is purely radial (shellular)
    Imposed as the background flow whose effect on wave stability is examined.

pith-pipeline@v0.9.0 · 5610 in / 1568 out tokens · 32044 ms · 2026-05-10T14:45:12.866820+00:00 · methodology

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Works this paper leans on

2 extracted references · 2 canonical work pages

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    2021, Universe, 7, 97 Bryan, G

    Andersson, N. 2021, Universe, 7, 97 Bryan, G. 1889, Phil. Trans. R. Soc. Lond., 180, 187 Chandrasekhar, S. 1961, Hydrodynamic and hydromagnetic stability (Clarendon Press, Oxford) Chandrasekhar, S. 1964, ApJ, 139, 664 Chandrasekhar, S. 1969, Ellipsoidal figures of equilibrium (Yale Uni- versity Press) Chandrasekhar, S. & Lebovitz, N. R. 1963, ApJ, 138, 18...

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  2. [2]

    +O(ℓ −4)(A.7) This expression is the first-order-corrected Poincaré mode frequency in the limit of largeℓ. Appendix B: The stability of the uniformly rotating viscous spherical shell Ignoring self-gravity, perturbations of a uniformly rotating fluid in a spherical shell verify (18) together with boundary conditions (22) and (23). Taking the dot product of...

    work page 2015