A linear assessment of waveguidability for barotropic Rossby waves in different large-scale flow configurations

Segalini, Antonio; Riboldi, Jacopo; Wirth, Volkmar; Messori, Gabriele

doi:https://doi.org/10.5194/egusphere-2023-316

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https://doi.org/10.5194/egusphere-2023-316

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13 Mar 2023

| 13 Mar 2023

A linear assessment of waveguidability for barotropic Rossby waves in different large-scale flow configurations

Antonio Segalini, Jacopo Riboldi, Volkmar Wirth, and Gabriele Messori

Abstract. Topographically forced Rossby waves shape the upper-level waveguide over the midlatitudes, affecting the propagation of transient waves therein, and have been linked to multiple surface extremes. The complex interplay between the forcing and the background flow in shaping the Rossby wave response still needs to be elucidated in a variety of configurations. We propose here an analytical solution of the linearized barotropic vorticity equation to obtain the stationary forced Rossby wave resulting from arbitrary combinations of forcing and background zonal wind. While the onset of barotropic instability might hinder the applicability of the linear framework, we show that the nonlinear wave response can still be retrieved qualitatively from the linearized solution. Examples using single- and double-jet configurations are discussed to illustrate the method and study how the background flow can act as a waveguide for Rossby waves.

Received: 21 Feb 2023 – Discussion started: 13 Mar 2023

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Journal article(s) based on this preprint

06 Aug 2024

A linear assessment of barotropic Rossby wave propagation in different background flow configurations

Antonio Segalini, Jacopo Riboldi, Volkmar Wirth, and Gabriele Messori

Weather Clim. Dynam., 5, 997–1012, https://doi.org/10.5194/wcd-5-997-2024,https://doi.org/10.5194/wcd-5-997-2024, 2024

Short summary

Antonio Segalini, Jacopo Riboldi, Volkmar Wirth, and Gabriele Messori

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2023-316', Anonymous Referee #1, 24 Apr 2023
Summary
This manuscript offers a new method (to the best of my knowledge) for solving the barotropic vorticity equation linearized about a background solution, with a topographic forcing term. This method is based on expressing the solution as a superposition of Chebyshev polynomials and expressing the derivative operators using the Chebyshev basis. The authors argue that this method is computationally efficient, relative to numerical integrations, when examining the wave response to changes in the forcing or in the background flow. This method also allows for a calculation of the eigenvalues of the problem, given the background flow and regardless of the forcing, which allows for an examination of the stability of the flow to linear perturbations. The authors compare the solutions calculated using this method with numerical solutions of the fully nonlinear barotropic vorticity equation, with specific background flow configurations. The solutions are found to be quite similar, confirming the validity of the method proposed. The authors discuss some implications of this methods for estimating the waveguidability of different zonal flows.
I think the method and results presented in this manuscript are interesting and significant, and could be useful for future studies. I have major comments regarding the presentation of the manuscript. I personally had to read through the manuscript carefully twice before I had a sense of what it is about. I think the manuscript can be improved by making some modifications in the text, to make the ideas clearer.
Major comments
In my opinion, the introduction, the abstract and perhaps even the title do not express clearly what the paper is about. They give the impression that the paper is more about the physics of waveguides, whereas the main contribution of the paper, as I see it, is the computational method. The single and double jet cases shown in the paper are used as test cases for the computational method, rather than going deep into the dynamics of waveguides. In the last paragraph of the paper the authors write that “This study should be regarded as an introduction and explanation of the techniques, but possible applications of this approach could include systematic waveguidability assessments for different forcings and background zonal wind profiles.” I think this sentence should appear in the introduction, and the abstract should emphasize the main point of the paper.

Perhaps the authors could add some more motivation for specific choices in the derivation of the mathematical model, that could add to the clarity of the paper. One choice that wasn’t immediately clear to me was including a damping term in equation (1). It wasn’t clear what this damping term represents physically. It was also not clear to me what the motivation was for looking at the stationary solution in equation (13). Only after I finished reading it became clearer. I think that in section 2 it could help to explain better what the model is supposed to represent, before the derivation of the equations.

The authors present stationary solutions and time-dependent solutions, but it is not mentioned explicitly at which section each type of solution is examined, what each type of solution represents physically, what is the motivation for looking at each type of solution and what are the different methods for solving for a stationary or time-dependent solution. Each of these are explained somewhere, often after presenting the results, but it is not explained in an organized manner.

Section 4.3 analyzes the stability of the problem for different parameters of the jet profile. I was missing a discussion on the connection of this instability to linear barotropic instability, in the sense of the necessary conditions for instability including a change of sign of the PV gradient. Some more physical context would be useful.

The analysis of the double jet case shown is much shorter than that of the single jet case and does not include a calculation of waveguidability. Based on the last sentence of the abstract (“Examples using single- and double-jet configurations are discussed to illustrate the method and study how the background flow can act as a waveguide for Rossby waves”) I was expecting a comparison between the waveguidability of the two cases. If the authors choose not to include a calculation of waveguidability for the double jet case, they should otherwise motivate the choice for the analysis that is presented.

Minor comments
The term “analytical solution” used in the abstract and introduction was confusing for me. I expected to see a solution expressed as a mathematical function. It is true that the Chebyshev polynomials are analytical functions and in that sense the solution is analytical, but eventually there is a numerical calculation that leads to the solution, so perhaps a different terminology would describe the method more clearly. As I see it, the main difference between what is called a “numerical solution” and “analytical solution” by the authors is that the former is a time-integrated solution and the latter is an eigenvalue problem.

Lines 181-182: In what sense is it counterintuitive that the nonlinear solution is more dissipative?

Perhaps the appendix can include some more details of the computational method, such as the matrices D(1) and D(2), and how the time-dependent solution is calculated.
Citation: https://doi.org/10.5194/egusphere-2023-316-RC1
- AC1: 'Reply on RC1', Antonio Segalini, 23 May 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-316/egusphere-2023-316-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2023-316-AC1
RC2:
'Comment on egusphere-2023-316', Anonymous Referee #2, 25 Apr 2023

See uploaded pdf for review.

Citation: https://doi.org/10.5194/egusphere-2023-316-RC2
- AC2: 'Reply on RC2', Antonio Segalini, 23 May 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-316/egusphere-2023-316-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2023-316-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2023-316', Anonymous Referee #1, 24 Apr 2023
Summary
This manuscript offers a new method (to the best of my knowledge) for solving the barotropic vorticity equation linearized about a background solution, with a topographic forcing term. This method is based on expressing the solution as a superposition of Chebyshev polynomials and expressing the derivative operators using the Chebyshev basis. The authors argue that this method is computationally efficient, relative to numerical integrations, when examining the wave response to changes in the forcing or in the background flow. This method also allows for a calculation of the eigenvalues of the problem, given the background flow and regardless of the forcing, which allows for an examination of the stability of the flow to linear perturbations. The authors compare the solutions calculated using this method with numerical solutions of the fully nonlinear barotropic vorticity equation, with specific background flow configurations. The solutions are found to be quite similar, confirming the validity of the method proposed. The authors discuss some implications of this methods for estimating the waveguidability of different zonal flows.
I think the method and results presented in this manuscript are interesting and significant, and could be useful for future studies. I have major comments regarding the presentation of the manuscript. I personally had to read through the manuscript carefully twice before I had a sense of what it is about. I think the manuscript can be improved by making some modifications in the text, to make the ideas clearer.
Major comments
In my opinion, the introduction, the abstract and perhaps even the title do not express clearly what the paper is about. They give the impression that the paper is more about the physics of waveguides, whereas the main contribution of the paper, as I see it, is the computational method. The single and double jet cases shown in the paper are used as test cases for the computational method, rather than going deep into the dynamics of waveguides. In the last paragraph of the paper the authors write that “This study should be regarded as an introduction and explanation of the techniques, but possible applications of this approach could include systematic waveguidability assessments for different forcings and background zonal wind profiles.” I think this sentence should appear in the introduction, and the abstract should emphasize the main point of the paper.

Perhaps the authors could add some more motivation for specific choices in the derivation of the mathematical model, that could add to the clarity of the paper. One choice that wasn’t immediately clear to me was including a damping term in equation (1). It wasn’t clear what this damping term represents physically. It was also not clear to me what the motivation was for looking at the stationary solution in equation (13). Only after I finished reading it became clearer. I think that in section 2 it could help to explain better what the model is supposed to represent, before the derivation of the equations.

The authors present stationary solutions and time-dependent solutions, but it is not mentioned explicitly at which section each type of solution is examined, what each type of solution represents physically, what is the motivation for looking at each type of solution and what are the different methods for solving for a stationary or time-dependent solution. Each of these are explained somewhere, often after presenting the results, but it is not explained in an organized manner.

Section 4.3 analyzes the stability of the problem for different parameters of the jet profile. I was missing a discussion on the connection of this instability to linear barotropic instability, in the sense of the necessary conditions for instability including a change of sign of the PV gradient. Some more physical context would be useful.

The analysis of the double jet case shown is much shorter than that of the single jet case and does not include a calculation of waveguidability. Based on the last sentence of the abstract (“Examples using single- and double-jet configurations are discussed to illustrate the method and study how the background flow can act as a waveguide for Rossby waves”) I was expecting a comparison between the waveguidability of the two cases. If the authors choose not to include a calculation of waveguidability for the double jet case, they should otherwise motivate the choice for the analysis that is presented.

Minor comments
The term “analytical solution” used in the abstract and introduction was confusing for me. I expected to see a solution expressed as a mathematical function. It is true that the Chebyshev polynomials are analytical functions and in that sense the solution is analytical, but eventually there is a numerical calculation that leads to the solution, so perhaps a different terminology would describe the method more clearly. As I see it, the main difference between what is called a “numerical solution” and “analytical solution” by the authors is that the former is a time-integrated solution and the latter is an eigenvalue problem.

Lines 181-182: In what sense is it counterintuitive that the nonlinear solution is more dissipative?

Perhaps the appendix can include some more details of the computational method, such as the matrices D(1) and D(2), and how the time-dependent solution is calculated.
Citation: https://doi.org/10.5194/egusphere-2023-316-RC1
- AC1: 'Reply on RC1', Antonio Segalini, 23 May 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-316/egusphere-2023-316-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2023-316-AC1
RC2:
'Comment on egusphere-2023-316', Anonymous Referee #2, 25 Apr 2023

See uploaded pdf for review.

Citation: https://doi.org/10.5194/egusphere-2023-316-RC2
- AC2: 'Reply on RC2', Antonio Segalini, 23 May 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-316/egusphere-2023-316-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2023-316-AC2

Peer review completion

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

AR by Antonio Segalini on behalf of the Authors (17 Jul 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (18 Aug 2023) by Camille Li

RR by Anonymous Referee #1 (10 Sep 2023)

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

Summary
The authors have revised the manuscript following comments from two reviewers, including myself. The authors made considerable modifications, which improved the manuscript. They have addressed my main comment, regarding the gap between the main subject of the manuscript and the motivation, which didn’t seem to relate well in the previous version. In the current version, the motivation is clearer, and the connection between the new numerical method and the concept of waveguidability is clearer. I think the manuscript could fit for publication in WCD, after another round of revision.
My major comments regard the consistency of the terms used in the paper, some details of the methods that are missing, and issues with the interpretation of the nonlinear simulations. My other comments are minor.

Major comments
1) Consistency of terms.
a. There are 3 types of models used in this study. Their names, according to the legend in figure 2a are: Linear Chebyshev, Linear SHT and Nonlinear SHT, where “SHT” stands for spherical harmonics transform. However, these names are not used consistently throughout the paper. The linear Chebyshev method is often called “linear” (e.g., line 186, caption of figures 1 and 3), which may confuse it with the linear SHT method. The nonlinear SHT method is simply called “nonlinear”. The authors should choose a name for each method and use it consistently.
b. Waveguidability is sometimes called “normalized meridional enstrophy density”, though it is not made clear if these are two terms for the same physical variable. Specifically, is the expression in equation (17) the same as the waveguidability presented in figure 2?
c. The variable phi_0 is sometimes used to denote the latitude of the forcing (line 209) and sometimes – the latitude of the jet (caption of figure 3). The latitude of the forcing is also called “phi_F” (equation 15) and the latitude of the jet is also called “phi_J” (equation 16).
d. The term “temporal coefficient” (e.g, caption of figure 5, line 412) is used interchangeably with the term “principal component” (line 263) or simply “amplitude” (caption of figure 6). Please explain if these are all words to describe the same thing. If so – please be consistent with the terminology. If not – please explain what “temporal coefficient” means.
e. The bar (over-line) is used sometimes to demote the background flow (equation 1) and sometimes to demote the amplitude of a wave mode (line 124).
2) Missing details.
a. For most of the results presented (at least figures 3, 4, 5 and 8), it seems that the authors used the same latitude for the jet and for the forcing, however this is not mentioned explicitly and the reader is left to guess what the latitude of the forcing is. Specifically, it is confusing the phi_0 is initially used for denoting the forcing latitude, while the jet latitude is sometimes called phi_0 and sometimes phi_J (see comment 1c above). Also, it would help if the authors explain why they chose to use the same latitude for the jet and for the forcing (when this is the case).
b. It is not described how the EOF analysis is done exactly. Specifically, I would expect to find an exact explanation for how the “temporal coefficients” shown in the bottom panels of figure 5 were calculated. If these are the same as principle component time series of the EOFs, then it is surprising that the principle component of the first EOF oscillates around 1 and not around 0. Usually, an EOF analysis is performed after removing the trend from the time series. Is that the case here or not? Please explain.
c. According to the caption of figure 8a, the green line marks the “locus beyond which the temporal vorticity variance of the nonlinear simulation becomes 10 times larger than in the stable regimes”. While this is explained in the figure caption, this criterion is not mentioned explicitly in the text, when the “stability” of the nonlinear solution is considered (lines 291-294). I would expect a more detailed description and explanation for this criterion in the text. How is the temporal vorticity variance calculated? Which stable regime is it compared to?
3) Interpretation of the nonlinear simulations.
a. As far as I could understand, the nonlinear simulations solve equation (3). This equation includes wave-mean flow interactions and wave-wave interactions. In contrast, the linear equation (equation 7) includes the effect of the mean flow on the waves, but does not include the effect of the waves on the mean flow, therefore it neglects both the wave-mean flow interactions and the wave-wave interactions. In the discussions in the paper, where the nonlinear simulations are compared with the linear method results, the authors assume that the differences between the solutions arise from the inclusion of wave-wave interactions in the nonlinear model, while ignoring the wave-mean flow interactions (e.g., lines 187-188, 295-296, 387-389). I think this is a very fundamental issue. When wave-mean flow and wave-wave interactions are included, the equilibration occurs due to the combination of stabilization of the mean flow profile by the wave fluxes, and the dissipation by wave-wave interactions. While the authors mention the latter, they ignore the former, which could be very important.
b. The authors interpret the behavior of the nonlinear solution presented in figures 5 and 6 as evidence for a limit cycle (e.g. line 265, 294). I am skeptic about this interpretation. First, because in order to identify a limit cycle, a phase space should be defined and the existence of the limit cycle and its stability need to be shown in this phase space. I don’t see what is the phase space in which the authors find a limit cycle. Second, I disagree that the wave modes shown in figure 5 are not traveling waves. In lines 263-265 the authors argue that “The trajectories… behave as traveling waves: however, the linear stability analysis does not support this interpretation as the waves should grow exponentially in magnitude because they are unstable”. In this argument the authors ignore the fact that in the nonlinear simulation the mean flow is modified by the wave fluxes and therefore it can be stabilized. Additionally, the wave-wave interactions introduce a dissipative effect. So there is no reason to believe that these are not traveling waves (actually figure 6 shows exactly that these are traveling waves).
c. The comparison between the stationary solution of the linear equation and the time-mean solution of the nonlinear equation assumes that we should expect them to be similar (e.g. lines 250-253, 371-373). I think these two solutions capture a fundamentally different phenomenon. The linear stationary solution captures a stationary (i.e. zero-phase speed) wave, forced by the topography. The nonlinear time-mean solution captures a statistically steady state. The paragraph in lines 253-268 (as well as parts of the conclusions section) tries to explain the similarity between the stationary solution of the linear equation and the time-mean solution of the nonlinear equation in an unstable case. They argue that one should expect the nonlinear solution to diverge in the unstable conditions (e.g. lines 236-237). Their interpretation for the lack of instability in the nonlinear case is that the system reaches a limit cycle. I would argue instead that the system reaches a nonlinear statistically steady state, where the mean flow is stabilized by the waves (in the time-mean sense), and the waves are equilibrated in the sense that their life cycles give a net zero growth, when averaged over time. I would definitely expect to find traveling waves in such a solution. These traveling waves don’t necessarily need to be identical to the most unstable modes of the mean flow. They could be neutral modes in the linear sense, but they need to be able to maintain the mean flow in a profile that enables them to go through cycles of growth and decay (see for example DelSole 2004, Lachmy and Harnik 2016).
4) Interpretation of the stability analysis.
Figure 8a shows the maximum of the imaginary part of the linear eigenvalues (i.e., the linear growth rate) normalized by the damping time scale. The dashed line marks the locus where the absolute vorticity gradient changes sign (the Rayleigh stability criterion). The authors argue that the distance between the line where the linear growth rate is zero and the dashed line in figure 8a shows that “the Rayleigh criterion provides a necessary but not sufficient condition for the onset of instability” (lines 282-285). In the conclusions section (lines 382-384) they argue that the Rayleigh criterion was not capable of detecting the onset of barotropic instability. I disagree with this interpretation, because the linear stability criterion could easily be adapted to incorporate the effect of the damping term, by examining the line where the growth rate is equal to minus the damping time scale (i.e. where the growth rate is equal to -1 in figure 8a). Note that this line corresponds to the dashed line, meaning that it is consistent with the Rayleigh criterion of instability. This is not a coincidence. When the wave equation includes a linear damping term, the growth rate is expected to be the same as the linear growth rate of a model without damping, minus the damping time scale. Therefore, the results are consistent with the theory of barotropic instability, where the Rayleigh criterion marks the state where the linear growth rate of a model without damping is zero, and when linear damping is added, the growth rate is reduced by the damping time scale.
5) Section 6
This section doesn’t include a discussion of the implications of the results. I couldn’t understand the motivation for looking at the time-dependent solution and what the conclusions from this analysis are.

Minor comments
1) Line 84: Lambda is defined, but it is not used in equation (1).
2) Line 105: Since equation (9) includes variables with a “hat”, denoting the amplitude of the Fourier components, it would be better to define the Fourier components (Psi_hat(theta, t)exp(imx)) here, before the equation, or at least mention what the hat symbol means.
3) Line 118: Something in the wording is not correct, “achieved at regime” doesn’t sound right. What does it mean?
4) Line 124: The bar (over-line) was used before to denote the time-mean background solution, here it is the amplitude of the Fourier component in time.
5) Lines 137-138: This would be a good place to refer again to the appendix.
6) Line 141-142: The first sentence of this paragraph seems to belong to the previous paragraph.
7) Line 144: Please mention exactly which linear and nonlinear equations the SHT package solves. Are these equations 7 and 3?
8) Line 153: Bar (over-line) was used before to denote the time-mean, but here it is used to denote the zonal wind divided by cosine latitude.
9) Line 167: created by -> forced by.
10) Line 183: Delete the second “for the”.
11) Line 190: “between the forcing the the monitoring sector” – the first “the” after “forcing” should be replaced by “and”.
12) Caption of figure 2: It says that “N” is the number of latitude/longitude grid points for the Chebyshev simulations and the truncation number for the SHT method. Why not define N as the number of latitude grid points for all the cases, to be consistent?
13) Line 239: “…is also not be ruled out” – delete the “be”.
14) The authors use expressions related to time when referring to the behavior of the system as a function of the model parameters (the word “after” in line 299 and the words “started” and “later” in lines 387-388).
15) Line 300: “…the maximum growth rate in Fig.8b…” – figure 8b shows the wavenumber, not the growth rate.
16) Line 310: “jets” – change to “jet”.
17) Line 325: “…equally efficient waveguides” – why do you say they are equally efficient if their waveguidabilities are not equal?
18) Line 338: “…the waveguidability is reduced…”. I assume the authors mean that in the double jet case it is reduced compared to the single jet case. However, this should be mentioned explicitly and not left for the reader to guess.
19) Line 384-385: This sentence is not clear. Specifically, the phrasing of the part: ”…the condition of instability corresponds to a first increase of the…”.
20) Line 388: Please explain what were the signs of barotropic instability in the nonlinear simulations? Why do you consider them to be signs of barotropic instability?
21) Line 401: Delete the “that” after c).
22) Line 410: “these evidences” – change to “this evidence”.
23) Lines 412-413: The sentence “the temporal coefficients could be determined by solving a small set of nonlinear ordinary differential equations” is not clear. What do you mean by “temporal coefficients”? Are these the same as the principle component time series (see major comment 2b)? Which small set of equations are you referring to?

References
DelSole, T. (2004). Stochastic models of quasigeostrophic turbulence. Surveys in Geophysics, 25, 107-149.
Lachmy, O., & Harnik, N. (2016). Wave and jet maintenance in different flow regimes. Journal of the Atmospheric Sciences, 73(6), 2465-2484.

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RR by Anonymous Referee #2 (24 Sep 2023)

ED: Reconsider after major revisions (16 Oct 2023) by Camille Li

AR by Antonio Segalini on behalf of the Authors (24 Nov 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (31 Dec 2023) by Camille Li

RR by Anonymous Referee #1 (22 Jan 2024)

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

Summary

In the current version, the authors have addressed my comments regarding the consistency of terms and missing details. However, my comments regarding the interpretation of the results remain valid.
Following my comments, the authors added an appendix (B) to show that a limit cycle exists in the nonlinear simulations. This helped me understand what limit cycle they referred to in the previous version, but I see it as an over-complicated way of looking at a simple phenomenon. Traveling waves and the limit cycle are in fact the same thing, as I elaborate in major comment 4 below. In any case, I think appendix B is not necessary, and this discussion deviates from the main theme of the paper, at least as I see it. I also still think that the authors are missing a much simpler interpretation of the results of the nonlinear simulations, based on the role of wave-mean flow interactions. I elaborate on this in major comment 2 below.
In the bottom line, even though the paper does present a method that could be useful for future studies, and demonstrates its usefulness, I think that some parts of it are written in a way that will not be understandable to the potential readers, and are presented in an over-complicated way. I therefore recommend on another round of revisions, following the comments below.

Major comments

1) A general comment on section 4: It is hard to follow how this section is connected to the main theme of the paper. If I understand correctly, the purpose of this section is to compare the prediction for the steady state solution (and its stability properties) based on the linear model with the “real” solution of the nonlinear system. If this is indeed the case, that should be stated explicitly. I think that the analysis of the limit cycle in subsection 4.2 and appendix B is not necessary, because I don’t see how it contributes to the goal of this study. Perhaps all of subsection 4.2 is unnecessary. In any case, its title “dynamics of the unstable modes” does not describe the content well, because the modes of the nonlinear simulation are not unstable. If the authors choose to keep subsection 4.2, I think they can shorten it and remove the limit cycle analysis, for the reasons elaborated in comment 4 below.
2) The authors mention the role of wave-mean flow interactions only briefly and refer to it as a dissipative effect (lines 211-212). In other places it is ignored (lines 242, 282, 388-389). I commented on the previous version that a nonlinear simulation is expected to reach a statistically steady state with a mean flow that is stabilized (or neutralized) by the effect of the waves on the mean flow. It’s not a dissipative effect on the waves, it’s a stabilizing effect on the mean flow. In their response, the authors mentioned that “the linear model is linearized around the background state and not around the equilibrium state”. They say that “the remedy is to linearize around the time average instead” but claim that “the linear analysis with a nonzonal background flow is more computationally expensive”. I agree, but I don’t see why not take the time average of the zonal mean flow in the nonlinear simulation and perform the linear analysis on that. I would expect to see that it is more stable than the imposed background flow. If that is the case, it would show that the nonlinear solution does not diverge because the mean flow is stabilized by wave-mean flow interaction.
3) Lines 226-231: Even though the authors agreed with my comment on the relevance of the Rayleigh criterion for the interpretation of the stability analysis, they have not changed anything in this paragraph. It stayed the same as in the previous version. This paragraph is difficult to read, as it is phrased in a very confusing way. Actually, a much simpler interpretation can be given, as I explained in my previous review. The Rayleigh criterion predicts the borders of the stability region in the absence of dissipation. In order to adjust for the inclusion of dissipation, all you need to do is to subtract the dissipation time scale from the growth rate. The results in Figure 4 are consistent with this idea. Therefore, I don’t understand why the authors chose not to mention this simple explanation. This comment applies also to lines 385-386 in section 6.
4) Here is why I think the limit cycle interpretation is an over-complicated way of representing traveling waves: Equation B1 defines the coefficients of the EOF modes (that are called “temporal coefficients” in other places). The authors claim in their response that these coefficients represent the amplitudes of the Rossby waves in the nonlinear simulation. I disagree, and I will try to explain why. Looking, for example, at modes 2 and 3 in figure 5, I see that each one of them represents a different phase of the same mode with wavenumber 5. Let’s assume that this is a traveling wave. What would you expect the time trajectory to look like in the phase space defined by a2 and a3? It would look like a circle (as it does in figures 6 and B1a). This is because the traveling wave is equal to: a2(t)sin(m(x-x0))+a3(t)cos(m(x-x0)), where a2(t) = sin(omega*(t-t0)) and a3(t) = cos(omega*(t-t0)). Here I described, for simplicity, the wave shown in figure 52 as sin(m(x-x0)) and the wave in figure 53 as cos(m(x-x0)). The authors wrote in the response that “If the orbit depends on the initial condition a traveling wave is present”. But actually, what happens is that during the simulation (no matter with what initial conditions) the flow adjusts toward a statistically steady state that supports specific traveling waves. In any case, this discussion does not add much to the main argument of the paper, so I don’t think it is worth going into so much detail about the flow evolution. The way I see it, the nonlinear statistically steady state represents a mean flow that was neutralized by wave-mean flow interactions, with traveling waves that, on time average, do not grow or decay. This interpretation is consistent with the results presented in this paper and with previous studies (I mentioned two of them in my previous review, and you can add to that a few papers by Brian Farrell). Lines 275-282 should be revised according to this comment.
5) A general comment on section 5: It is not clear why the authors chose to compare between the waveguidability in the linear and nonlinear case only for the double-jet case (subsection 5.3), whereas for the single jet case (subsection 5.2) only the linear analysis is considered. The authors should at least provide a motivation for these choices.
6) A general comment on section 6: I think that organizing the paragraphs as a list (“Firstly… secondly… thirdly”) does not aid the reader to understand the overall theme of the paper. It would help to explain instead how each part is connected to the main goal and how all the parts combine to one story. For example, in line 382, instead of writing “Secondly, we elucidate some features of Rossby waves…”, it would be better to motivate it by explaining that the linear analysis is compared with the more realistic results of the fully nonlinear simulations, in order to assess its ability to capture what happens in the nonlinear simulations. The paragraph that starts with “Thirdly” does not connect to the subject of the previous paragraph – the comparison between the linear analysis and the nonlinear solution.

Minor comments

1) Lines 218-224: It’s hard to follow this paragraph, because the text sometimes refers to the nonlinear simulations and sometimes to the linear solution, and it is not clear which is which. The nonlinear solution is mentioned and then it points to figure 3a, which shows the linear solution.
2) Line 225: Is this sentence referring to the nonlinear simulations or to the linear solution?
3) Line 234: “the imaginary part of the most unstable eigenvalue” – why not use the term “maximal growth rate” here and in other places in the paper?
4) Line 235: “…has the same sign” – add after this “throughout the domain”, or change to “…has a uniform sign”.
5) Caption of figure 4: “(equivalent to the neutral curve in the nonlinear case)” – this is not the appropriate phrasing. You are using an arbitrary threshold for defining a “neutral curve”, so “equivalent” is not the right word. You could replace it with “(the chosen threshold for defining the neutral curve in the nonlinear case)”.
6) Line 236: What do you mean by “stability margin”? You mention the range 15-22 m s^(-1), but it’s not clear what feature in the figure the reader should look at to see this.
7) Lines 238-242: One gets the impression that the threshold of 2 m^2 s^(-2) represents an abrupt transition of the nonlinear simulations from low velocity variance to very high velocity variance (it says “increases drastically”). Is this really the case? If it is indeed an abrupt transition at this specific value, it would be good to show that in the figure. If not, then the phrasing should be changed to clarify that the threshold was chosen arbitrarily.
8) Line 251: “which is indeed proportional to cos(phi)” – but the equation says that L is constant. I suppose you mean to say that this gives a wavenumber that is proportional to cos(phi).
9) Line 252: “at different latitudes of the zonal jet” – add “of a given width”.
10) Line 253-255: It is not clear how the first sentence, that relates to the degree of instability, is related to the second sentence, that relates to the wavenumber. The word “indeed” seems not appropriate here.
11) Line 269: A reduced-order model of what?
12) Line 301: I assume the time averaging refers to the enstrophy and not to the vorticity anomaly. Please change the wording accordingly.
13) Line 302: The comment in the parenthesis is not clear. Why not use the time average? The statistically steady state represents fluctuations around a time-mean state, so an instantaneous state is not equal to the time average.
14) Line 307: It would be clearer if the words “when assessing how strong the jet stream is” were deleted.
15) Lines 314-315: It would help the reader if the authors point to the relevant features to look at in figure 8.
16) Line 331: “Increases with jet speed from 0 to around 90%” – The jet speed values should also be mentioned, otherwise this statement is meaningless.
17) Figure 8: Equation 19 is not expressed in percent, but in a dimensionless number between 0 and 1, but the variables in the figure are in percent. Please clarify this in the caption.
18) Line 343: Delete “once again”. Also, the location of the forcing should be mentioned.
19) Lines 350-351: It should be clear if this calculation is for the linear analysis or for the nonlinear simulations.
20) Lines 360-364: It is not clear where the forcing is located in each of the cases mentioned.

Language/typos

1) Line 75: “…in sections 4 and 5.3.” – delete the “3” after “5.”
2) Line 211: “The difference in waveguidability metric” – add “the” before “waveguidability”.
3) Line 220: “some eigenvalues have in fact positive imaginary part” – delete “in fact” and add “a” before “positive.
4) Line 238: delete the word “time” before “variance”.
5) Line 245: Change “associated to” -> “associated with”.
6) Line 295: the word “metric” is written twice.
7) Line 351: Change from “jets with same velocity” to “jets with the same velocity, but different latitudes”.

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ED: Publish subject to revisions (further review by editor and referees) (01 Feb 2024) by Camille Li

AR by Antonio Segalini on behalf of the Authors (08 Mar 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (05 Apr 2024) by Camille Li

RR by Anonymous Referee #1 (01 May 2024)

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

Summary
In this version, the authors have addressed all my comments. The manuscript is now clearer, with a coherent rationale linking the different sections. I find the manuscript now acceptable for publication in weather and climate dynamics. I have only a few minor comments.

Comments
1) Lines 230-233: This sentence was added following my comment, that it is not a coincidence that the Rayleigh criterion curve in figure 4a coincides with the growth rate of -1 (normalized by the damping rate, Xi). I think this point should be written in a more explicit way. The current text says that “… whenever the absolute vorticity gradient changes sign, the imaginary part of omega can be different than –Xi…”. But it is not mentioned that the dashed line in figure 4a shows exactly that the absolute vorticity gradient changes sign when the growth rate equals –Xi. This means that the Rayleigh criterion is relevant for the growth rate, since it gives exactly a zero growth rate if no damping is added, and the correction for the growth rate in the presence of damping is exactly the damping rate.
2) Line 234: “this results pinpoints” -> “this result pinpoints” (or “these results pinpoint”).
3) Line 246: “such as wave-wave interactions” – I suggest to add also wave-mean flow interactions, as explained in my previous review. The same comment applies to line 395, where I suggest to change to “…pointing to potential damping or stabilizing effects operated by nonlinear terms (e.g. wave-wave and wave-mean flow interactions)”.
4) Line 258: Following my comment the authors replaced “indeed” with “this is verified in…”, but the problem remains that the previous sentence is not connected to the next sentence, i.e. looking at the wavenumber of the most unstable mode does not verify the analysis of Gill (1982) that relates to the growth rate rather than the wavenumber.
5) Line 284-286: “The nonlinearity of the waves was further confirmed…” – I didn’t understand what this sentence is saying. Perhaps consider removing it.
6) Line 290: “similarly to what suggested by…” -> “similarly to what is suggested by”.
7) Lines 350-351: Is waveguidability assessed here around latitude 45? That’s what it sounds like here, but later in lines 366-369 the waveguidability is compared with its values calculated around the jet latitude. This sounds like a strange comparison, where in one case the jet latitude and the forcing latitude are different and waveguidability is calculated around the forcing latitude, while in the other case the forcing latitude and the jet latitude are the same. If I misunderstood it, then perhaps it should be explained more clearly in the text. Otherwise, please explain the choice to calculate the waveguidability around the forcing latitude (when it is different from the jet latitude) and to compare it with a case where the forcing latitude is at the jet latitude.

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ED: Publish subject to minor revisions (review by editor) (06 May 2024) by Camille Li

AR by Antonio Segalini on behalf of the Authors (08 May 2024) Author's response Author's tracked changes Manuscript

ED: Publish subject to minor revisions (review by editor) (16 May 2024) by Camille Li

AR by Antonio Segalini on behalf of the Authors (16 May 2024) Author's response Manuscript

EF by Anna Mirena Feist-Polner (10 Jun 2024) Author's tracked changes

ED: Publish as is (12 Jun 2024) by Camille Li

AR by Antonio Segalini on behalf of the Authors (12 Jun 2024) Manuscript

Journal article(s) based on this preprint

06 Aug 2024

A linear assessment of barotropic Rossby wave propagation in different background flow configurations

Antonio Segalini, Jacopo Riboldi, Volkmar Wirth, and Gabriele Messori

Weather Clim. Dynam., 5, 997–1012, https://doi.org/10.5194/wcd-5-997-2024,https://doi.org/10.5194/wcd-5-997-2024, 2024

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Antonio Segalini, Jacopo Riboldi, Volkmar Wirth, and Gabriele Messori

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Planetary Rossby waves are created by topography and evolve in time. In this work, an analytical solution of this classical problem is proposed under the approximation of linear wave dynamics. The theory is able to describe reasonably well the evolution of the perturbation and compares well with full nonlinear simulations. Several relevant cases with single and double zonal jets are assessed with the theoretical framework


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