Radial modes of pressure bumps and dips in astrophysical discs

Armand Leclerc; Elliot Lynch; Guillaume Laibe; Nicolas Perez

arxiv: 2512.05737 · v1 · submitted 2025-12-05 · 🌌 astro-ph.EP · cond-mat.mes-hall· physics.flu-dyn

Radial modes of pressure bumps and dips in astrophysical discs

Armand Leclerc , Guillaume Laibe , Elliot Lynch , Nicolas Perez This is my paper

Pith reviewed 2026-05-17 01:04 UTC · model grok-4.3

classification 🌌 astro-ph.EP cond-mat.mes-hallphysics.flu-dyn

keywords astrophysical discspressure bumpspressure dipsradial modeswave topologydiscoseismologydispersion relationepicyclic-acoustic frequency

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The pith

Pressure extrema in astrophysical discs act as waveguides for radial wave modes that propagate at all frequencies or at fixed frequencies depending on the type of extremum.

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

This paper applies a wave topology framework to study how pressure bumps and dips shape oscillations in astrophysical discs. It establishes a generalised local dispersion relation that incorporates pressure gradients and identifies an epicyclic-acoustic frequency. The work derives an analytical criterion for the existence of modes that transit between inertial and pressure bands. It concludes that pressure extrema function as waveguides for topological modes. The fundamental mode at a bump propagates across all frequencies and can therefore resonate with any temporal forcing, whereas the mode at a gap is locked to one frequency while allowing arbitrary vertical phase velocity.

Core claim

Pressure extrema consist of wave guides in which topological modes propagate. The fundamental mode trapped at a pressure bump can propagate at all frequencies, allowing it to resonate with any temporal forcing, while the mode associated with a pressure gap propagates at a fixed frequency with arbitrary vertical phase velocity. These features follow from a generalised local dispersion relation that includes pressure gradients and reveals the role of an epicyclic-acoustic frequency in setting mode branches.

What carries the argument

Generalised local dispersion relation from the wave topology framework that includes pressure gradients and an epicyclic-acoustic frequency to determine propagating mode branches.

If this is right

The fundamental mode at pressure bumps can resonate with any temporal forcing.
The mode at pressure gaps propagates at one fixed frequency but with arbitrary vertical phase velocity.
Pressure extrema function as waveguides that trap these topological radial modes.
These modes provide specific signatures useful for discoseismology.

Where Pith is reading between the lines

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

Signatures of these modes in disc observations could allow mapping of unseen pressure structures.
The wave topology approach could be tested on other stratified rotating flows such as stellar interiors.
Adding dust or magnetic fields to the model would reveal whether the waveguide property survives in more complete disc descriptions.

Load-bearing premise

The local dispersion relation derived from wave topology accurately describes the radial modes without major contributions from global curvature, nonlinear effects or vertical structure details omitted from the approximation.

What would settle it

Detection of radial oscillations at pressure bumps that either show propagation at every frequency or fail to do so across a range of observed forcing frequencies in protoplanetary discs.

Figures

Figures reproduced from arXiv: 2512.05737 by Armand Leclerc, Elliot Lynch, Guillaume Laibe, Nicolas Perez.

**Figure 1.** Figure 1: The Berry curvature F of the acoustic wave is singular at (cskr, S, cskz) = (0, 0, ±κ). This obstruction is a topological constraint, characterized by the two charges C = ±1. Length and brightness of the arrows indicate the norm of F . et al. 2022). We then expect to have one global wave branch with such spectral behavior in the spectrum of a disc. 3.2. Location of topological modes The spatial location of… view at source ↗

**Figure 2.** Figure 2: shows the density profile of the disc and the frequencies of the global waves for a range of vertical wavenumbers kz. The average energy-weighted position ⟨r⟩ ≡ ⟨X, rX⟩/⟨X, X⟩ for a mode X(r) is provided via the color bar. Low frequency p-modes are located mostly in the outer parts of the disc, where the sound speed is the lowest. High frequency r-modes are located mostly in the inner part, where the epicy… view at source ↗

**Figure 4.** Figure 4: Dispersion relations of the global modes in a pressure minimum. derived for stars in Leclerc et al. 2022). For an isothermal disc, eliminating uθ and uz in Eq. (10), it can then be written in the compact form ur = ω − κ 2 ω −1 Dh, (30) D †Dh=λh, (31) where D ≡ ics∂r − iS(r) and λ ≡ ω − κ 2 ω ω − c 2 s k 2 z ω . The slender torus model verifies Eq. (29) and as such, the global modes equation Eq. … view at source ↗

**Figure 5.** Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Relative amplitudes of the components of a fundamental p-mode with large kz, of topological origin, induced by a pressure bump located close to rmax. The amplitudes are displayed as functions of the distance to the central object in the mid-plane (top right panel) and for a (r, z) slice of the disc (bottom panel). The selected mode is marked with a cross in the top left panel (same as [PITH_FULL_IMAGE:fig… view at source ↗

**Figure 7.** Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: Same as [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗

read the original abstract

This study investigates the signatures of pressure extrema on global oscillations in discs. To this end, we use the framework of wave topology to establish a generalised local dispersion relation that includes pressure gradients. We highlight the influence of a previously unrecognized epicyclic-acoustic frequency and derive an analytical criterion for the existence of a branch of modes transiting between the inertial and the pressure bands. We find that pressure extrema consist of wave guides in which such topological modes propagate. The fundamental mode trapped at a pressure bump can propagate at all frequencies, allowing it to resonate with any temporal forcing, while the mode associated with a pressure gap propagates at a fixed frequency, propagates with arbitrary vertical phase velocity. These specific features make them attractive candidates for future discoseismology.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper gives an analytical criterion for radial modes transiting between inertial and pressure bands at disc pressure extrema, framed as waveguides with bump versus gap differences.

read the letter

This paper shows that pressure bumps and gaps in discs can trap radial modes that cross between inertial and pressure bands, using a wave-topology approach to get an analytical criterion and some specific propagation rules. The bump mode can run at any frequency while the gap mode sits at one frequency but allows free vertical phase speed. Those distinctions are the main new pieces, and they tie directly to possible discoseismology uses. The derivation starts from a generalized local dispersion relation that adds pressure gradients and flags an epicyclic-acoustic frequency, which looks like a clean extension of existing wave ideas to the disc equations. The waveguide picture is a useful way to organize the trapping at extrema. The local framework keeps the math tractable and produces explicit statements about frequency ranges without needing full global solutions. The soft spot is the local approximation itself. Pressure extrema are radially extended, so when mode wavelengths approach the bump or gap width, radial curvature terms or changes in the epicyclic frequency could shift the allowed bands and weaken the transit criterion or the all-frequency claim for bumps. The abstract does not show explicit checks against global models or non-WKB corrections, so that assumption stays open. It is not a load-bearing flaw yet, but it caps how far the results can be applied without more work. This is for people already working on disc oscillations and discoseismology. A reader who wants analytical tools for mode trapping at pressure features would pick up the criterion and the bump-gap contrast. The analytical content and subfield relevance are enough to send it for peer review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates signatures of pressure extrema on global oscillations in astrophysical discs. Using the wave topology framework, it derives a generalized local dispersion relation that incorporates pressure gradients and identifies a previously unrecognized epicyclic-acoustic frequency. An analytical criterion is obtained for the existence of a branch of modes transiting between inertial and pressure bands. The central claim is that pressure extrema function as waveguides for topological radial modes: the fundamental mode trapped at a pressure bump propagates at all frequencies (allowing resonance with arbitrary temporal forcing), while the mode associated with a pressure gap propagates at a fixed frequency with arbitrary vertical phase velocity. These properties are presented as attractive for future discoseismology applications.

Significance. If the local approximation is valid, the work supplies an analytical, parameter-free framework for mode trapping at disc pressure structures that could unify aspects of inertial and acoustic oscillations. The identification of all-frequency propagation at bumps and the fixed-frequency gap mode offers concrete, falsifiable predictions for discoseismology. The derivation of the branch-transit criterion from wave topology is a clear strength when the underlying assumptions hold.

major comments (2)

[Dispersion relation and branch-transit criterion] The generalized local dispersion relation (derived in the section introducing the epicyclic-acoustic frequency and the branch-transit criterion) assumes a locally uniform background. Pressure bumps and dips are radially extended global features; when the mode radial wavelength becomes comparable to the extremum width, omitted global curvature terms (radial derivatives of the epicyclic frequency and non-WKB corrections) can alter the allowed frequency bands and the claimed 'all frequencies' or 'fixed frequency' behaviour. This assumption is load-bearing for the waveguide interpretation.
[Waveguide properties and mode propagation] No explicit verification is provided that global radial structure or vertical details neglected in the local approximation remain negligible inside the waveguide (e.g., via comparison to a global numerical solution or WKB validity estimate). This directly affects the central claim that pressure extrema consist of waveguides with the stated propagation properties.

minor comments (2)

[Notation and definitions] The notation for the epicyclic-acoustic frequency should be introduced with an explicit equation and contrasted with the standard epicyclic frequency to prevent reader confusion.
[Figures] Dispersion-relation figures would benefit from explicit shading or labels distinguishing the inertial, pressure, and transiting branches.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. The comments raise important points about the limitations of the local approximation used in our analysis, which we address below. We have revised the manuscript to incorporate clarifications on the validity regime of our results.

read point-by-point responses

Referee: The generalized local dispersion relation (derived in the section introducing the epicyclic-acoustic frequency and the branch-transit criterion) assumes a locally uniform background. Pressure bumps and dips are radially extended global features; when the mode radial wavelength becomes comparable to the extremum width, omitted global curvature terms (radial derivatives of the epicyclic frequency and non-WKB corrections) can alter the allowed frequency bands and the claimed 'all frequencies' or 'fixed frequency' behaviour. This assumption is load-bearing for the waveguide interpretation.

Authors: We agree that our generalized local dispersion relation relies on the assumption of a locally uniform background, which is standard in WKB analyses but breaks down when the radial wavelength of the mode approaches the radial width of the pressure extremum. In such cases, global curvature terms and non-WKB effects could indeed modify the frequency bands and propagation properties. Our central claims regarding the waveguide behavior are intended to apply in the regime where the local approximation holds, i.e., for modes with radial wavelengths shorter than the scale of the pressure structures. To address this, we will add a dedicated subsection in the discussion clarifying the conditions under which the local approximation is valid and noting the potential impact of global effects when this condition is violated. revision: yes
Referee: No explicit verification is provided that global radial structure or vertical details neglected in the local approximation remain negligible inside the waveguide (e.g., via comparison to a global numerical solution or WKB validity estimate). This directly affects the central claim that pressure extrema consist of waveguides with the stated propagation properties.

Authors: We acknowledge the value of explicit verification for the applicability of the local results. Our manuscript provides an analytical framework and includes basic WKB validity estimates based on the derived dispersion relation. However, a direct comparison with global numerical solutions is not included, as the focus is on the topological and analytical aspects. We will expand the text to provide a more quantitative estimate of the WKB parameter (e.g., the ratio of wavelength to extremum width) under which the waveguide properties hold, and discuss how vertical structure effects are expected to be secondary for the radial modes considered. A full numerical validation would be a natural extension for future work. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation applies external wave topology framework to disc equations

full rationale

The paper derives a generalised local dispersion relation by applying the wave topology framework to the standard equations of disc dynamics with pressure gradients included. The analytical criterion for branch transit between inertial and pressure bands, and the subsequent identification of pressure extrema as waveguides with specific propagation properties (all frequencies for bumps, fixed frequency for gaps), are direct consequences of this relation rather than inputs redefined as outputs. No self-citations, fitted parameters, or ansatzes are invoked in a load-bearing way that reduces the central claims to the paper's own assumptions by construction. The derivation remains self-contained against the external wave topology framework and standard disc equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the applicability of wave topology to disc fluid equations and the validity of a local approximation that incorporates pressure gradients while treating the epicyclic frequency as known from prior disc theory.

axioms (1)

domain assumption Wave topology framework can be extended to include pressure gradients in the local dispersion relation for disc oscillations.
Invoked to establish the generalised dispersion relation and the epicyclic-acoustic frequency.

pith-pipeline@v0.9.0 · 5429 in / 1255 out tokens · 59166 ms · 2026-05-17T01:04:07.969062+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Relation between the paper passage and the cited Recognition theorem.

We use the framework of wave topology to establish a generalised local dispersion relation that includes pressure gradients... Chern numbers C=±1... spectral flow... modes transiting between the inertial and the pressure bands... pressure extrema consist of wave guides

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

JCB Papaloizou and JE Pringle

https://doi.org/10.1093/mnras/136.3.293. JCB Papaloizou and JE Pringle. The dynamical stability of differentially rotating discs–ii.Monthly Notices of the Royal Astronomical Society, 213(4):799–820, 1985. C. C. Lin and Frank H. Shu. On the Spiral Structure of Disk Galaxies, II. Outline of a Theory of Density Waves. Proceedings of the National Academy of S...

work page doi:10.1093/mnras/136.3.293 1985
[2]

Peter Goldreich and Scott Tremaine

https://doi.org/10.1093/mnras/108.3.214. Peter Goldreich and Scott Tremaine. The excitation and evolution of density waves.Astrophysical Journal, 222 (1):850–858, 1978. Peter Goldreich and Scott Tremaine. The excitation of density waves at the lindblad and corotation resonances by an external potential.Astrophysical Journal, 233(3): 857–871, 1979. S. H. L...

work page doi:10.1093/mnras/108.3.214 1978
[3]

https://doi.org/10.1093/mnras/272.3.618

ISSN 0035-8711. https://doi.org/10.1093/mnras/272.3.618. G.I. Ogilvie. Waves and instabilities in a differentially rotating disc containing a poloidal magnetic field. Monthly Notices of the Royal Astronomical Society, 297 (1):291–314, June 1998. ISSN 0035-8711. https://doi.org/10.1046/j.1365-8711.1998.01507.x. Steven A Balbus and John F Hawley. Instabilit...

work page doi:10.1093/mnras/272.3.618 1998
[4]

Armand Leclerc, Guillaume Laibe, and Nicolas Perez

https://doi.org/10.3847/2041-8213/ade396. Armand Leclerc, Guillaume Laibe, and Nicolas Perez. Wave topology of stellar inertial oscillations.Physical Review Research, 6(4):043299, December 2024b. https://doi.org/10.1103/PhysRevResearch.6.043299. Stefan Keppeler. Introduction to wigner-weyl calculus,

work page doi:10.3847/2041-8213/ade396 2041
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Brian C Hall.Quantum theory for mathematicians

URL https://www.math.uni-tuebingen.de/user/ stke/teaching/wigner weyl. Brian C Hall.Quantum theory for mathematicians. Springer, 2013. 13 Yohei Onuki. Quasi-local method of wave decomposition in a slowly varying medium.Journal of Fluid Mechanics, 883:A56, 2020. J´ er´ emie Vidal and Yves Colin de Verdi` ere. Inertia-gravity waves in geophysical vortices, ...

work page doi:10.1093/mnras/sty854 2013
[6]

T., Ogilvie G

https://doi.org/10.1111/j.1365-2966.2008.13207.x. Janosz W. Dewberry, Henrik N. Latter, and Gordon I. Ogilvie. Quasi-periodic oscillations and the global modes of relativistic, MHD accretion discs.MNRAS, 476(3): 4085–4103, May 2018. https://doi.org/10.1093/mnras/sty385. Benoˆ ıt Commer¸ con, Francesco Lovascio, Elliot Lynch, and Enrico Ragusa. Discs are b...

work page doi:10.1111/j.1365-2966.2008.13207.x 2008

[1] [1]

JCB Papaloizou and JE Pringle

https://doi.org/10.1093/mnras/136.3.293. JCB Papaloizou and JE Pringle. The dynamical stability of differentially rotating discs–ii.Monthly Notices of the Royal Astronomical Society, 213(4):799–820, 1985. C. C. Lin and Frank H. Shu. On the Spiral Structure of Disk Galaxies, II. Outline of a Theory of Density Waves. Proceedings of the National Academy of S...

work page doi:10.1093/mnras/136.3.293 1985

[2] [2]

Peter Goldreich and Scott Tremaine

https://doi.org/10.1093/mnras/108.3.214. Peter Goldreich and Scott Tremaine. The excitation and evolution of density waves.Astrophysical Journal, 222 (1):850–858, 1978. Peter Goldreich and Scott Tremaine. The excitation of density waves at the lindblad and corotation resonances by an external potential.Astrophysical Journal, 233(3): 857–871, 1979. S. H. L...

work page doi:10.1093/mnras/108.3.214 1978

[3] [3]

https://doi.org/10.1093/mnras/272.3.618

ISSN 0035-8711. https://doi.org/10.1093/mnras/272.3.618. G.I. Ogilvie. Waves and instabilities in a differentially rotating disc containing a poloidal magnetic field. Monthly Notices of the Royal Astronomical Society, 297 (1):291–314, June 1998. ISSN 0035-8711. https://doi.org/10.1046/j.1365-8711.1998.01507.x. Steven A Balbus and John F Hawley. Instabilit...

work page doi:10.1093/mnras/272.3.618 1998

[4] [4]

Armand Leclerc, Guillaume Laibe, and Nicolas Perez

https://doi.org/10.3847/2041-8213/ade396. Armand Leclerc, Guillaume Laibe, and Nicolas Perez. Wave topology of stellar inertial oscillations.Physical Review Research, 6(4):043299, December 2024b. https://doi.org/10.1103/PhysRevResearch.6.043299. Stefan Keppeler. Introduction to wigner-weyl calculus,

work page doi:10.3847/2041-8213/ade396 2041

[5] [5]

Brian C Hall.Quantum theory for mathematicians

URL https://www.math.uni-tuebingen.de/user/ stke/teaching/wigner weyl. Brian C Hall.Quantum theory for mathematicians. Springer, 2013. 13 Yohei Onuki. Quasi-local method of wave decomposition in a slowly varying medium.Journal of Fluid Mechanics, 883:A56, 2020. J´ er´ emie Vidal and Yves Colin de Verdi` ere. Inertia-gravity waves in geophysical vortices, ...

work page doi:10.1093/mnras/sty854 2013

[6] [6]

T., Ogilvie G

https://doi.org/10.1111/j.1365-2966.2008.13207.x. Janosz W. Dewberry, Henrik N. Latter, and Gordon I. Ogilvie. Quasi-periodic oscillations and the global modes of relativistic, MHD accretion discs.MNRAS, 476(3): 4085–4103, May 2018. https://doi.org/10.1093/mnras/sty385. Benoˆ ıt Commer¸ con, Francesco Lovascio, Elliot Lynch, and Enrico Ragusa. Discs are b...

work page doi:10.1111/j.1365-2966.2008.13207.x 2008