Revisiting barotropic instability from the perspective of wave evolution theory

Li, Yaokun

doi:10.5194/egusphere-2025-5427

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https://doi.org/10.5194/egusphere-2025-5427

Preprints

18 Dec 2025

| 18 Dec 2025

Revisiting barotropic instability from the perspective of wave evolution theory

Yaokun Li

Abstract. The instability of Rossby waves has been a long-standing topic in dynamical meteorology. The classic theoretical analysis had provided in–depth physical understanding of the problem. However, developing a systematic and quantitative comprehension of wave energy and amplitude evolution remains challenging. With an eye to such issues, this investigation provides a novel and practicable algorithm to solve the wave action conservation equation. Theoretical analysis establishes that wave packet energy evolves through two competing factors: direct proportionality to intrinsic frequency and inverse proportionality to group velocity magnitude. Energy density attains extremal values at turning points where group velocity magnitudes become extremized. To ensure a ray can be reflected by a turning point, zonal phase speed must be smaller than an upper limit determined by dispersion relation at the turning point. Crucially, a specific zonal phase speed range emerges below this maximum threshold where concurrent transient growth of both wave energy and amplitude occurs when a ray is moving toward the turning point, with the upper limit corresponding to optimal wave development conditions. Numerical experiments on a prototype westerly jet reveal distinct instability mechanisms: substantial transient growth–capable of triggering nonmodal instability–arises when rays moves toward turning points, while exponential amplification characteristic of modal instability develops at inflection points with positive energy growth rate. The derived zonal phase speed thresholds and transient growth metrics form a diagnostic framework applicable to observed atmospheric flows, enabling quantitative evaluation of both modal and nonmodal instability potentials. By unifying wave evolution dynamics with classical instability criteria, this work provides an operational bridge between theoretical predictions and real–world flow diagnostics.

Received: 03 Nov 2025 – Discussion started: 18 Dec 2025

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Journal article(s) based on this preprint

01 Jul 2026

Revisiting barotropic instability from the perspective of wave evolution theory

Yaokun Li

Weather Clim. Dynam., 7, 1073–1088, https://doi.org/10.5194/wcd-7-1073-2026,https://doi.org/10.5194/wcd-7-1073-2026, 2026

Short summary

Yaokun Li

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-5427', Anonymous Referee #1, 12 Feb 2026

The Abstract and main body are interesting and generally well written. The references are fairly complete, with one major exception and some minor ones noted below. The major omission is awareness of the Hurdle Theorem (see below), which establishes a sufficient condition for shear instability and is complementary to the necessary conditions for instability that the paper cites. The paper should be publishable after taking care of the following comments and minor issues.
Comments
Line 34-49: The emphasis on classical sufficient conditions for stability is outdated and needs to be rewritten to bring it up to date. In particular, a major sufficient condition for inviscid shear instability has been established, called the Hurdle Theorem. See Deguchi et al. (2024, J. Fluid Mech., 997, A25, doi:10.1017/jfm.2024.728) and Read and Dowling (2026, Encycl. Atmos. Sci. (3^rd Ed.) 4, 263—283, Academic Press, doi:10.1016/B978-0-323-96026-7.00211-3).
Ray tracing is heavily used, but without first establishing the context for which it is accurate and discussing general conditions when it is inaccurate, such as inhomogeneous environments and proximity to focusing points (caustics). This can be fixed by adding a short paragraph discussing the issues early in the introduction. There is a brief mention of such an issue on Line 132, another on Line 144-152, and a workaround on Line 191+ (Section 3). These could usefully be tied into an introductory short paragraph on the general strengths and shortcomings of ray tracing.
On a related note, it is not clear while reading this paper at what point or points in the evolution of an unstable shear flow the theory applies. When a shear flow is truly barotropically unstable, the end result often bears little resemblance to the initial conditions, a point made in e.g., the review by Read and Dowling (2026). It would help to add a few guideposts throughout the paper that indicate whether we are always teetering on marginal stability or not, and at what point does the evolution of unstable flow render the discussion moot.
It should be made clear early in the paper that the system under study has a flat bottom topography and thus misses out on an entire class of marginally stable cases that are especially relevant to Jupiter and Saturn. See Deguchi et al. (2024) for an extended discussion of this point. This limitation simply needs to be made explicit somewhere in the paper’s introduction, and ties into the conclusions and future work, for example the comment on Line 472 about “real-world atmospheric flows”.
The Hurdle Theorem (Deguchi et al. 2024) can be readily applied to the u = sech²(y) prototype (29) analysed in Section 4, which will significantly enhance the discussion, including the material shown in Fig. 1 c), which currently only illustrates sufficient-for-stability criteria and is lacking sufficient-for-instability criteria. Some of the main figures later in the paper are also ripe for addition of Hurdle Theorem regions.
Minor
Line 14: “by dispersion relation” -> “by the dispersion relation”
Line 17-18: the clause-isolating hyphens should be longer em-dashes, as in growth—capable
Line 26: “they may play” -> “they play”
Line 59: “packet, w to” needs to be fixed
Line 70: “Li et al.(2021a) -> “Li et al. (2021a)”
Line 91: The epsilon and Psi symbols are running into each other in (3). There are several spacing issues in the equations throughout the manuscript. Presumably these will at least get fixed in the typesetting.
Line 194 and (15): This transform needs to be better motivated and referenced in terms of how and why it is helpful and how old this strategy is.
Line 215-220: There are remarkably few citations in the beginning of Section 3. Please add some citations to similar work, or add emphasis that none exists.
Page 11: To this point there has been a noticeable lack of any figures. The effect is to make reading the paper more tedious than this important subject deserves. Please add a couple of examples to illustrate some of the key points being made, which will prop up the reader’s morale, particularly new students.
Line 438: It is arguable that the explicit physical understanding of Rossby wave instability in the context of the reciprocal Rossby-Mach number (e.g., Deguchi et al. 2024) has not been well known for several decades, which undermines the point of this sentence. This needs to be updated.

Citation: https://doi.org/10.5194/egusphere-2025-5427-RC1
- AC1: 'Reply on RC1', Yaokun Li, 25 Feb 2026
  
  Dear Reviewer,
  Thank you very much for your insightful comments and constructive suggestions on my manuscript. I have carefully considered all your feedback and made substantial revisions to address the issues raised. I believe these changes have significantly improved the quality and rigor of the paper. Please find my detailed responses to each comment below. I hope the revised manuscript meets the journal's standards and look forward to your further feedback.
  Sincerely,
  The author: Yaokun Li
  
  Citation: https://doi.org/10.5194/egusphere-2025-5427-AC1
RC2:
'Comment on egusphere-2025-5427', Anonymous Referee #2, 19 Mar 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-5427/egusphere-2025-5427-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2025-5427-RC2
- AC2:
  'Reply on RC2', Yaokun Li, 28 Mar 2026
  Dear Reviewer:
  Thank you very much for your insightful and constructive comments on my manuscript. I greatly appreciate the time and effort you have dedicated to reviewing my work, which has significantly helped improve the quality and clarity of the paper. I have carefully addressed all your concerns and revised the manuscript accordingly. Below is a summary of the key revisions made:
  I have compared the case discussed in the manuscript with the cases for the observed westerly jet to stress that it can represent the basic feature of the real world atmosphere.
  
  I have rewrittenSection 3.1 to enhance logical flow and readability. A new schematic diagram (shown in Response Figure 5) has been added to illustrate key concepts.
  
  I have added a new Appendix to introduce the detailed derivation for wave energy equation along rays.
  
  I have added a paragraphto explicitly discuss the ray tracing limitationsregarding wave superposition and tunneling, citing Harnik (2002) and Harnik & Heifetz (2007) to strengthen theoretical rigor.
  
  I have carefully checked the manuscript to avoid mistakes and typos.
  
  I believe these revisions have substantially improved the manuscript’s clarity, rigor, and accessibility. I hope the revised version meets your expectations and look forward to your further feedback.
  Sincerely,
  The author: Yaokun Li
  
  Citation: https://doi.org/10.5194/egusphere-2025-5427-AC2
EC1:
'Comment on egusphere-2025-5427', Gwendal Rivière, 23 Mar 2026

Dear author,
Please find some additional suggestions, comments directly in the pdf file.
Best regards,
Gwendal Riviere

Citation: https://doi.org/10.5194/egusphere-2025-5427-EC1
- AC3: 'Reply on EC1', Yaokun Li, 05 Apr 2026
  
  Dear Prof. Gwendal Rivière:
  Thank you for your careful review and valuable comments on my manuscript. In response to your questions and suggestions, I have carefully revised the manuscript and prepared a detailed point-by-point response.
  I have added an new Appendix to introduce the derivative of the maximum and minimum of the turning point curve. I have also explicitly illustrate the rationality in solving the divergence of group velocity. Following the reviewer’s comment, I have also added a new schematic figure to help understanding. I have rewritten Section 3.1. I have corrected the expressing mistakes and typos and carefully checked the manuscript to avoid similar mistakes.
  All revisions have been marked in blue at the corresponding positions in the revised manuscript, and the detailed responses are provided in the response file.
  Thank you again for your professional guidance. I hope the revised manuscript meets the journal's publication requirements.
  Sincerely,
  The author: Yaokun Li
  
  Citation: https://doi.org/10.5194/egusphere-2025-5427-AC3

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by Yaokun Li on behalf of the Authors (05 Apr 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (13 Apr 2026) by Gwendal Rivière

RR by Anonymous Referee #2 (16 May 2026)

For final publication, the manuscript should be

accepted as is

accepted subject to technical corrections

accepted subject to minor revisions

reconsidered after major revisions

rejected

Were a revised manuscript to be sent for another round of reviews:

I would be willing to review the revised manuscript.

I would not be willing to review the revised manuscript.

Suggestions for revision or reasons for rejection

Review of the revised version of: " Revisiting barotropic instability from the perspective of wave evolution theory" by Yaokun Li

After going over the revised paper, as well as the response, I better understand the main points of the paper. The most practical novelty for WCD readers, in my mind, is the detailed "dissection" of the range of possible wave geometry configurations that a barotropic jet can have, in the wavenumber-phase speed- jet parameter space, and the idea that this is significant for the spatial variations of wave energy. In particular, I found it interesting that the waveguide configuration is quite rare. A main remaining criticism I have is that there is something confusing and maybe a bit misleading in the way the topic is contextualized in the introduction and discussion. Specifically, the very broad statements about the central role of Rossby waves for weather and extremes in lines 27-30 almost seem to suggest that barotropic instability is the primary source of weather and extreme events. Also, the statement on line 31-32 suggests wave breaking excites instability, while typically it is the other way around (I think the Read paper referenced in this context also states that wave breaking acts to wipe out PV gradients and stabilize).
Also, lines 549-551 in the discussion: "For example, it provides a systematic assessment for Rossby wave evolution and instability in different basic flow profiles, especially in real–world atmospheric flows."
Reading through the paper one can get the impression that the author thinks the major instability of Rossby waves is baroclinic. There is a mention in the discussion that the work can be extended to baroclinic flows, but its minor place also contributes to the above impression. The relevance of the theory presented, in light of the fact that baroclinic instability is the major instability for Rossby waves, should be stated much explicitly.
I agree that barotropic wave dynamics is central for understanding realistic atmospheric flows, but barotropic instability is less commonly considered a main driver of weather.
I think the author should be a bit more explicit in how the understanding of barotropic amplification is relevant.

Looking at the papers mentioned in the first paragraph, Segalini et al (2024) discuss
waveguidability of stationary waves (Segalini et al 2024)- these waves are expluced from the current analysis. In fact, stationary waves are maybe quite unique in that they are in a WG state, especially if the winds are westerly and there are no critical surfaces. The uniqueness of the wave geometry of these waves is maybe one of the interesting easy to understand implications of this paper.

The paper by Woolinges et al discusses the turbulent vs wavy nature of the flow as a function of latitude with an emphasis on polar regions which this paper avoids discussing. The papers by Ali et al., 2022; Jiménez-Esteve et al., 2022; Kornhuber et al., 2019 on recurrent Rossby wave packets - these are short lived propagating waves which are phase locked. I am curios to hear how the author thinks the dynamics of this paper are relevant. An explicit statement on this will help the reader place this paper in context and understand possible uses of this methodology and thinking. The same goes for the tropical cyclone phase locking papers referenced - Aiyyer and Wade, 2021; Riboldi et al., 2019. Do you think your theory can illuminate some of this dynamics? These are papers where a subtropical anomaly is introduced - it is more relevant to the latitudes which you decided to look at.

More specific comments:
lines 97-98: this sentence confused me on two accounts- first, I think the fact that the atmosphere is baroclinic and baroclinic instability if the main unstable Rossby-wave process should also be mentioned if you are listing caveats. Second, for Earth, the main influence of topography is maybe setting the phase speed to be zero, no? This should also be discussed maybe. Also, the relevance of topography in the context of Venus and Jupiter can at first be confusing- I assume you mean that convection can act as a dynamic topography.

line 109 Read and Dowling (2026) are missing from the reference list.

line 134-135: "In the case of an almost plane Rossby wave.." - I am not sure this is always the case- if there is meridional reflection, the local wave is a superposition of poleward and equatorward waves. The wave is only a pure plane wave initially just after you turn on the wave source and before it reflects back from the turning surface (this is a the main point of Harnik, 2002).

line 164 - *background* absolute vorticity

line 176 - Harnik (2002) reference does not state this- remove. I would add some of Lindzen's over-reflection papers.

Figure 1- this figure helps but it is not clear what subplot b shows.

Figure 3- I assume you chose a specific value of k - maybe state what it is?

Hide

ED: Publish subject to revisions (further review by editor and referees) (26 May 2026) by Gwendal Rivière

AR by Yaokun Li on behalf of the Authors (29 May 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (15 Jun 2026) by Gwendal Rivière

RR by Anonymous Referee #2 (16 Jun 2026)

ED: Publish subject to technical corrections (17 Jun 2026) by Gwendal Rivière

AR by Yaokun Li on behalf of the Authors (20 Jun 2026) Author's response Manuscript

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-5427', Anonymous Referee #1, 12 Feb 2026

The Abstract and main body are interesting and generally well written. The references are fairly complete, with one major exception and some minor ones noted below. The major omission is awareness of the Hurdle Theorem (see below), which establishes a sufficient condition for shear instability and is complementary to the necessary conditions for instability that the paper cites. The paper should be publishable after taking care of the following comments and minor issues.
Comments
Line 34-49: The emphasis on classical sufficient conditions for stability is outdated and needs to be rewritten to bring it up to date. In particular, a major sufficient condition for inviscid shear instability has been established, called the Hurdle Theorem. See Deguchi et al. (2024, J. Fluid Mech., 997, A25, doi:10.1017/jfm.2024.728) and Read and Dowling (2026, Encycl. Atmos. Sci. (3^rd Ed.) 4, 263—283, Academic Press, doi:10.1016/B978-0-323-96026-7.00211-3).
Ray tracing is heavily used, but without first establishing the context for which it is accurate and discussing general conditions when it is inaccurate, such as inhomogeneous environments and proximity to focusing points (caustics). This can be fixed by adding a short paragraph discussing the issues early in the introduction. There is a brief mention of such an issue on Line 132, another on Line 144-152, and a workaround on Line 191+ (Section 3). These could usefully be tied into an introductory short paragraph on the general strengths and shortcomings of ray tracing.
On a related note, it is not clear while reading this paper at what point or points in the evolution of an unstable shear flow the theory applies. When a shear flow is truly barotropically unstable, the end result often bears little resemblance to the initial conditions, a point made in e.g., the review by Read and Dowling (2026). It would help to add a few guideposts throughout the paper that indicate whether we are always teetering on marginal stability or not, and at what point does the evolution of unstable flow render the discussion moot.
It should be made clear early in the paper that the system under study has a flat bottom topography and thus misses out on an entire class of marginally stable cases that are especially relevant to Jupiter and Saturn. See Deguchi et al. (2024) for an extended discussion of this point. This limitation simply needs to be made explicit somewhere in the paper’s introduction, and ties into the conclusions and future work, for example the comment on Line 472 about “real-world atmospheric flows”.
The Hurdle Theorem (Deguchi et al. 2024) can be readily applied to the u = sech²(y) prototype (29) analysed in Section 4, which will significantly enhance the discussion, including the material shown in Fig. 1 c), which currently only illustrates sufficient-for-stability criteria and is lacking sufficient-for-instability criteria. Some of the main figures later in the paper are also ripe for addition of Hurdle Theorem regions.
Minor
Line 14: “by dispersion relation” -> “by the dispersion relation”
Line 17-18: the clause-isolating hyphens should be longer em-dashes, as in growth—capable
Line 26: “they may play” -> “they play”
Line 59: “packet, w to” needs to be fixed
Line 70: “Li et al.(2021a) -> “Li et al. (2021a)”
Line 91: The epsilon and Psi symbols are running into each other in (3). There are several spacing issues in the equations throughout the manuscript. Presumably these will at least get fixed in the typesetting.
Line 194 and (15): This transform needs to be better motivated and referenced in terms of how and why it is helpful and how old this strategy is.
Line 215-220: There are remarkably few citations in the beginning of Section 3. Please add some citations to similar work, or add emphasis that none exists.
Page 11: To this point there has been a noticeable lack of any figures. The effect is to make reading the paper more tedious than this important subject deserves. Please add a couple of examples to illustrate some of the key points being made, which will prop up the reader’s morale, particularly new students.
Line 438: It is arguable that the explicit physical understanding of Rossby wave instability in the context of the reciprocal Rossby-Mach number (e.g., Deguchi et al. 2024) has not been well known for several decades, which undermines the point of this sentence. This needs to be updated.

Citation: https://doi.org/10.5194/egusphere-2025-5427-RC1
- AC1: 'Reply on RC1', Yaokun Li, 25 Feb 2026
  
  Dear Reviewer,
  Thank you very much for your insightful comments and constructive suggestions on my manuscript. I have carefully considered all your feedback and made substantial revisions to address the issues raised. I believe these changes have significantly improved the quality and rigor of the paper. Please find my detailed responses to each comment below. I hope the revised manuscript meets the journal's standards and look forward to your further feedback.
  Sincerely,
  The author: Yaokun Li
  
  Citation: https://doi.org/10.5194/egusphere-2025-5427-AC1
RC2:
'Comment on egusphere-2025-5427', Anonymous Referee #2, 19 Mar 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-5427/egusphere-2025-5427-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2025-5427-RC2
- AC2:
  'Reply on RC2', Yaokun Li, 28 Mar 2026
  Dear Reviewer:
  Thank you very much for your insightful and constructive comments on my manuscript. I greatly appreciate the time and effort you have dedicated to reviewing my work, which has significantly helped improve the quality and clarity of the paper. I have carefully addressed all your concerns and revised the manuscript accordingly. Below is a summary of the key revisions made:
  I have compared the case discussed in the manuscript with the cases for the observed westerly jet to stress that it can represent the basic feature of the real world atmosphere.
  
  I have rewrittenSection 3.1 to enhance logical flow and readability. A new schematic diagram (shown in Response Figure 5) has been added to illustrate key concepts.
  
  I have added a new Appendix to introduce the detailed derivation for wave energy equation along rays.
  
  I have added a paragraphto explicitly discuss the ray tracing limitationsregarding wave superposition and tunneling, citing Harnik (2002) and Harnik & Heifetz (2007) to strengthen theoretical rigor.
  
  I have carefully checked the manuscript to avoid mistakes and typos.
  
  I believe these revisions have substantially improved the manuscript’s clarity, rigor, and accessibility. I hope the revised version meets your expectations and look forward to your further feedback.
  Sincerely,
  The author: Yaokun Li
  
  Citation: https://doi.org/10.5194/egusphere-2025-5427-AC2
EC1:
'Comment on egusphere-2025-5427', Gwendal Rivière, 23 Mar 2026

Dear author,
Please find some additional suggestions, comments directly in the pdf file.
Best regards,
Gwendal Riviere

Citation: https://doi.org/10.5194/egusphere-2025-5427-EC1
- AC3: 'Reply on EC1', Yaokun Li, 05 Apr 2026
  
  Dear Prof. Gwendal Rivière:
  Thank you for your careful review and valuable comments on my manuscript. In response to your questions and suggestions, I have carefully revised the manuscript and prepared a detailed point-by-point response.
  I have added an new Appendix to introduce the derivative of the maximum and minimum of the turning point curve. I have also explicitly illustrate the rationality in solving the divergence of group velocity. Following the reviewer’s comment, I have also added a new schematic figure to help understanding. I have rewritten Section 3.1. I have corrected the expressing mistakes and typos and carefully checked the manuscript to avoid similar mistakes.
  All revisions have been marked in blue at the corresponding positions in the revised manuscript, and the detailed responses are provided in the response file.
  Thank you again for your professional guidance. I hope the revised manuscript meets the journal's publication requirements.
  Sincerely,
  The author: Yaokun Li
  
  Citation: https://doi.org/10.5194/egusphere-2025-5427-AC3

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by Yaokun Li on behalf of the Authors (05 Apr 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (13 Apr 2026) by Gwendal Rivière

RR by Anonymous Referee #2 (16 May 2026)

For final publication, the manuscript should be

accepted as is

accepted subject to technical corrections

accepted subject to minor revisions

reconsidered after major revisions

rejected

Were a revised manuscript to be sent for another round of reviews:

I would be willing to review the revised manuscript.

I would not be willing to review the revised manuscript.

Suggestions for revision or reasons for rejection

Hide

ED: Publish subject to revisions (further review by editor and referees) (26 May 2026) by Gwendal Rivière

AR by Yaokun Li on behalf of the Authors (29 May 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (15 Jun 2026) by Gwendal Rivière

RR by Anonymous Referee #2 (16 Jun 2026)

ED: Publish subject to technical corrections (17 Jun 2026) by Gwendal Rivière

AR by Yaokun Li on behalf of the Authors (20 Jun 2026) Author's response Manuscript

Journal article(s) based on this preprint

01 Jul 2026

Revisiting barotropic instability from the perspective of wave evolution theory

Yaokun Li

Weather Clim. Dynam., 7, 1073–1088, https://doi.org/10.5194/wcd-7-1073-2026,https://doi.org/10.5194/wcd-7-1073-2026, 2026

Short summary

Yaokun Li

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