Assessing Lagrangian Coherence in Atmospheric Blocking

Schoeller, Henry; Chemnitz, Robin; Koltai, Péter; Engel, Maximilian; Pfahl, Stephan

doi:https://doi.org/10.5194/egusphere-2024-2173

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https://doi.org/10.5194/egusphere-2024-2173

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30 Jul 2024

| 30 Jul 2024

Assessing Lagrangian Coherence in Atmospheric Blocking

Henry Schoeller, Robin Chemnitz, Péter Koltai, Maximilian Engel, and Stephan Pfahl

Abstract. Atmospheric blocking exerts a major influence on mid-latitude atmospheric circulation and is known to be associated with extreme weather events. Previous work has highlighted the importance of the origin of air parcels that define the blocking region, especially with respect to non-adiabatic processes such as latent heating. So far, an objective method of clustering the individual Lagrangian trajectories passing through a blocking into larger and, more importantly, spatially coherent air streams has not been established. This is the focus of our study.

To this end, we determine coherent sets of trajectories, which are regions in the phase space of dynamical systems that keep their geometric integrity in time, and can be characterized by robustness under small random perturbations. We approximate a dynamic diffusion operator on the available Lagrangian data and use it to cluster the trajectories into coherent sets. Our implementation adapts the existing methodology to the non-Euclidean geometry of Earth's atmosphere and its challenging scaling properties. The framework also allows statements about the spatial behavior of the trajectories as a whole. We discuss two case studies differing with respect to season and geographic location.

The results confirm the existence of spatially coherent feeder air streams differing with respect to their dynamical properties and, more specifically, their latent heating contribution. Air streams experiencing a considerable amount of latent heating occur mainly during the maturing phase of the blocking and contribute to its stability. In our example cases, trajectories also exhibit an altered evolution of general coherence when passing through the blocking region, which is in line with the common understanding of blocking as a region of low dispersion.

Received: 12 Jul 2024 – Discussion started: 30 Jul 2024

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Journal article(s) based on this preprint

27 Feb 2025

| Highlight paper

Assessing Lagrangian coherence in atmospheric blocking

Henry Schoeller, Robin Chemnitz, Péter Koltai, Maximilian Engel, and Stephan Pfahl

Nonlin. Processes Geophys., 32, 51–73, https://doi.org/10.5194/npg-32-51-2025,https://doi.org/10.5194/npg-32-51-2025, 2025

Short summary Executive editor

Henry Schoeller, Robin Chemnitz, Péter Koltai, Maximilian Engel, and Stephan Pfahl

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-2173', Anonymous Referee #1, 08 Sep 2024

This paper describes the use of a technique, developed by extending and adapting the procedures in Banish and Koltai (2017) to the analysis of atmospheric blocking. Two particular study cases are analysed in detail. The paper is interesting, well written, and represents a valuable contribution to the characterization and understanding of blocking events. Particularly interesting is the Lagrangian identification and three-dimensional characterization of warm conveyor belts, and its influence on blocking formation.
The methodology, as recognized in the paper, is certainly complex. Fortunately, the authors provide Python code implementing it, available via pypi.org/project/GeoCS/. In general, the procedures are described in detail and established with rigour. There are many tunable parameters for which choices have to be made (alpha, kappa, r, epsilon, …), which are justified by the authors. In my opinion, however, there is still one point in the methodology that remains weakly justified: One crucial point in the identification of coherent sets is the choice of the eigenvectors selected to perform the clustering. In all cases (e.g. Fig. 5a) the spectral gap after which eigenvalues are neglected is very weak, as the authors recognize. Thus, still there is some doubt about the extent on which the identified air masses can be called ‘coherent’, and what would be the difference if other choice of eigenvalues is done. I propose the authors to give a quantitative measure of the ‘coherence’ attained by all or of some of the detected sets (at least the most relevant, and for some choice of initial and final time) by assessing to which extent Eq. (1) is really satisfied by the sets (at least to some level of approximation). I imagine several more or less direct ways to check Eq. (1), although other quantitative assessments of coherence can also be given in terms of Cheeger ratios (Froyland, 2015) or other metrics (Froyland, 2013) that are expected to be optimized. In summary, I think the paper would be suitable for publication if the authors provide some extra evidence that the detected sets are really ‘coherent’ or at least significantly more ‘coherent’ than sets evolved, for example, from initial patches selected just by spatial proximity.
Other minor points:
- The bibliography is rather complete. However I think there are still a few papers related to this topic that consider either coherence or diabatic heating in blockings from the Lagrangian point of view and that can complete the reference list and being properly cited. Among them I suggest: Ehstand, N. et al.: Characteristic signatures of Northern Hemisphere blocking events in a Lagrangian flow network representation of the atmospheric circulation, Chaos 31, 093128 (2021). https://doi.org/10.1063/5.0057409. Zschenderlein, P. et al.: A Lagrangian analysis of upper-tropospheric anticyclones associated with heat waves in Europe, Weather Clim. Dynam., 1, 191–206 (2020). https://doi.org/10.5194/wcd-1-191-2020.
- page 7, line 191: please rewrite this sentence to clarify to which set the expression ‘this boundary set’ refers to.
- page 13, line 367: I think ‘placed in a regular three-dimensional grid’ should be rather ‘placed close to a regular three-dimensional grid’, since Sect. 3.2 states that some random displacements are applied.
- page 17, line 441: P_epsilon,t is substochastic, not stochastic.
- To help the readers, please indicate in the captions of Figures S3 and S4 the pertinence to the Canadian or to the European case of the different panels.

Citation: https://doi.org/10.5194/egusphere-2024-2173-RC1
- AC1: 'Reply on RC1', Henry Schoeller, 31 Oct 2024
  
  Thank you for your helpful advise. We agree that a measure of coherence of the sets identified by our method was lacking in the initial version of our paper. We have therefore developed such a metric that was derived in a natural way from our methodological setting. More specifically, we calculate time-averaged exit probabilties from individual sets based on transition probabilities in Q_epsilon. We compare the coherent sets found to test sets sampled randomly by spatial proximity and find the clusters identified are significantly more coherent.
  All of the minor issues have been addressed. In particular, we have made references to both of the suggested publications, which we agreed were relevant for our work. We have also made more clear the difference between the sets identified by the alpha shape algorithm (we call those "boundary sets") and the sets identified by k-means clustering containing mostly boundary points (we call those "residual sets").
  
  Citation: https://doi.org/10.5194/egusphere-2024-2173-AC1
RC2:
'Comment on egusphere-2024-2173', Anonymous Referee #2, 13 Sep 2024

The paper extends the diffusion map approach for the detection and characterization of coherent sets from tracer trajectories by Banisch & Koltai (2017) to the context of studying atmospheric blocking events. Two major blocking events are analyzed and the extracted coherent sets are shown to relate to warm conveyor belts, which are known to play a crucial part in the stabilization of the blocking regime. The proposed Lagrangian framework for the study of blocking events is novel and certainly of interest to the readers of NPG. The paper is very well written. Helpful graphics illustrate the careful computational studies, with all parameter choices well justified by the authors. However, I fully agree with the points raised by the other referee: In particular, a quantitative confirmation of the degree of coherence of the extracted flow patterns would considerably strengthen the study and should definitely be pursued.
Minor points:
-- Figure 1: It is unclear at this point what "2 pvu" means (pvu is only introduced later). Abbreviations should be explained in the caption.

What is the meaning of the black arrow?

-- Introduction: For readers unfamiliar with the underlying mechanisms leading to atmospheric blocking, some brief explanation on the meaning of negative potential vorticity anomaly would be helpful.

-- 281: "vertical distance of 7 hPa between 550 and 150 hPa". Can one say anything about the approximate altitude of these levels?

-- 291: capitalized R: "LAGRangian ANalysis TOol (LAGRANTO)“

-- 301: Definitions of u^i_t, v^i_t, w^i_t would be helpful.

-- Figure 2: The tropopause is difficult to distinguish from the continental borders. Please plot this in a different color. Insertion of the dateline could be helpful, because it is mentioned in the main text (334).

-- 377: "indicate" instead of "indicated"

-- Figure 5a): The position of the epsilon-legend is suboptimal as the dots are not really separated from the plotted eigenvalues

-- Figure 5b) Do the gray points correspond to the boundary set? This should be mentioned in the caption.

-- 441: "substochastic" instead of "stochastic". Is the spectral gap after the sixth instead or after the seventh eigenvalue (later the first six eigenvectors are used for clustering)?

-- Figures 5 + 7: Cluster coloring is different.

-- 523: "the brown": there is no brown cluster with number 0 in figure 7.

-- Figures 7-- : I find captions starting with "Same as Fig ..." very inconvenient. Please insert the details, no matter if this causes repetitions.

Citation: https://doi.org/10.5194/egusphere-2024-2173-RC2
- AC1: 'Reply on RC1', Henry Schoeller, 31 Oct 2024
  
  Thank you for your helpful advise. We agree that a measure of coherence of the sets identified by our method was lacking in the initial version of our paper. We have therefore developed such a metric that was derived in a natural way from our methodological setting. More specifically, we calculate time-averaged exit probabilties from individual sets based on transition probabilities in Q_epsilon. We compare the coherent sets found to test sets sampled randomly by spatial proximity and find the clusters identified are significantly more coherent.
  All of the minor issues have been addressed. In particular, we have made references to both of the suggested publications, which we agreed were relevant for our work. We have also made more clear the difference between the sets identified by the alpha shape algorithm (we call those "boundary sets") and the sets identified by k-means clustering containing mostly boundary points (we call those "residual sets").
  
  Citation: https://doi.org/10.5194/egusphere-2024-2173-AC1
- AC2: 'Reply on RC2', Henry Schoeller, 31 Oct 2024
  
  In addition to what has been mentioned in the answer to reviewer 1 that also applies to your comment, we have addressed all minor issues raised. In particular, we admit that there was a bit of confusion with respect to the spectral gap identified and the number of clusters searched for, due to a bug in our code. Thank you for letting us know. The issue has now been corrected and the approach is consistent. Though the number of clusters searched for has changed for the second and third application, the results and interpretations are not affected, since (as mentioned in the text) the clustering is quite robust against variation of epsilon or k.
  
  Citation: https://doi.org/10.5194/egusphere-2024-2173-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-2173', Anonymous Referee #1, 08 Sep 2024

This paper describes the use of a technique, developed by extending and adapting the procedures in Banish and Koltai (2017) to the analysis of atmospheric blocking. Two particular study cases are analysed in detail. The paper is interesting, well written, and represents a valuable contribution to the characterization and understanding of blocking events. Particularly interesting is the Lagrangian identification and three-dimensional characterization of warm conveyor belts, and its influence on blocking formation.
The methodology, as recognized in the paper, is certainly complex. Fortunately, the authors provide Python code implementing it, available via pypi.org/project/GeoCS/. In general, the procedures are described in detail and established with rigour. There are many tunable parameters for which choices have to be made (alpha, kappa, r, epsilon, …), which are justified by the authors. In my opinion, however, there is still one point in the methodology that remains weakly justified: One crucial point in the identification of coherent sets is the choice of the eigenvectors selected to perform the clustering. In all cases (e.g. Fig. 5a) the spectral gap after which eigenvalues are neglected is very weak, as the authors recognize. Thus, still there is some doubt about the extent on which the identified air masses can be called ‘coherent’, and what would be the difference if other choice of eigenvalues is done. I propose the authors to give a quantitative measure of the ‘coherence’ attained by all or of some of the detected sets (at least the most relevant, and for some choice of initial and final time) by assessing to which extent Eq. (1) is really satisfied by the sets (at least to some level of approximation). I imagine several more or less direct ways to check Eq. (1), although other quantitative assessments of coherence can also be given in terms of Cheeger ratios (Froyland, 2015) or other metrics (Froyland, 2013) that are expected to be optimized. In summary, I think the paper would be suitable for publication if the authors provide some extra evidence that the detected sets are really ‘coherent’ or at least significantly more ‘coherent’ than sets evolved, for example, from initial patches selected just by spatial proximity.
Other minor points:
- The bibliography is rather complete. However I think there are still a few papers related to this topic that consider either coherence or diabatic heating in blockings from the Lagrangian point of view and that can complete the reference list and being properly cited. Among them I suggest: Ehstand, N. et al.: Characteristic signatures of Northern Hemisphere blocking events in a Lagrangian flow network representation of the atmospheric circulation, Chaos 31, 093128 (2021). https://doi.org/10.1063/5.0057409. Zschenderlein, P. et al.: A Lagrangian analysis of upper-tropospheric anticyclones associated with heat waves in Europe, Weather Clim. Dynam., 1, 191–206 (2020). https://doi.org/10.5194/wcd-1-191-2020.
- page 7, line 191: please rewrite this sentence to clarify to which set the expression ‘this boundary set’ refers to.
- page 13, line 367: I think ‘placed in a regular three-dimensional grid’ should be rather ‘placed close to a regular three-dimensional grid’, since Sect. 3.2 states that some random displacements are applied.
- page 17, line 441: P_epsilon,t is substochastic, not stochastic.
- To help the readers, please indicate in the captions of Figures S3 and S4 the pertinence to the Canadian or to the European case of the different panels.

Citation: https://doi.org/10.5194/egusphere-2024-2173-RC1
- AC1: 'Reply on RC1', Henry Schoeller, 31 Oct 2024
  
  Thank you for your helpful advise. We agree that a measure of coherence of the sets identified by our method was lacking in the initial version of our paper. We have therefore developed such a metric that was derived in a natural way from our methodological setting. More specifically, we calculate time-averaged exit probabilties from individual sets based on transition probabilities in Q_epsilon. We compare the coherent sets found to test sets sampled randomly by spatial proximity and find the clusters identified are significantly more coherent.
  All of the minor issues have been addressed. In particular, we have made references to both of the suggested publications, which we agreed were relevant for our work. We have also made more clear the difference between the sets identified by the alpha shape algorithm (we call those "boundary sets") and the sets identified by k-means clustering containing mostly boundary points (we call those "residual sets").
  
  Citation: https://doi.org/10.5194/egusphere-2024-2173-AC1
RC2:
'Comment on egusphere-2024-2173', Anonymous Referee #2, 13 Sep 2024

The paper extends the diffusion map approach for the detection and characterization of coherent sets from tracer trajectories by Banisch & Koltai (2017) to the context of studying atmospheric blocking events. Two major blocking events are analyzed and the extracted coherent sets are shown to relate to warm conveyor belts, which are known to play a crucial part in the stabilization of the blocking regime. The proposed Lagrangian framework for the study of blocking events is novel and certainly of interest to the readers of NPG. The paper is very well written. Helpful graphics illustrate the careful computational studies, with all parameter choices well justified by the authors. However, I fully agree with the points raised by the other referee: In particular, a quantitative confirmation of the degree of coherence of the extracted flow patterns would considerably strengthen the study and should definitely be pursued.
Minor points:
-- Figure 1: It is unclear at this point what "2 pvu" means (pvu is only introduced later). Abbreviations should be explained in the caption.

What is the meaning of the black arrow?

-- Introduction: For readers unfamiliar with the underlying mechanisms leading to atmospheric blocking, some brief explanation on the meaning of negative potential vorticity anomaly would be helpful.

-- 281: "vertical distance of 7 hPa between 550 and 150 hPa". Can one say anything about the approximate altitude of these levels?

-- 291: capitalized R: "LAGRangian ANalysis TOol (LAGRANTO)“

-- 301: Definitions of u^i_t, v^i_t, w^i_t would be helpful.

-- Figure 2: The tropopause is difficult to distinguish from the continental borders. Please plot this in a different color. Insertion of the dateline could be helpful, because it is mentioned in the main text (334).

-- 377: "indicate" instead of "indicated"

-- Figure 5a): The position of the epsilon-legend is suboptimal as the dots are not really separated from the plotted eigenvalues

-- Figure 5b) Do the gray points correspond to the boundary set? This should be mentioned in the caption.

-- 441: "substochastic" instead of "stochastic". Is the spectral gap after the sixth instead or after the seventh eigenvalue (later the first six eigenvectors are used for clustering)?

-- Figures 5 + 7: Cluster coloring is different.

-- 523: "the brown": there is no brown cluster with number 0 in figure 7.

-- Figures 7-- : I find captions starting with "Same as Fig ..." very inconvenient. Please insert the details, no matter if this causes repetitions.

Citation: https://doi.org/10.5194/egusphere-2024-2173-RC2
- AC1: 'Reply on RC1', Henry Schoeller, 31 Oct 2024
  
  Thank you for your helpful advise. We agree that a measure of coherence of the sets identified by our method was lacking in the initial version of our paper. We have therefore developed such a metric that was derived in a natural way from our methodological setting. More specifically, we calculate time-averaged exit probabilties from individual sets based on transition probabilities in Q_epsilon. We compare the coherent sets found to test sets sampled randomly by spatial proximity and find the clusters identified are significantly more coherent.
  All of the minor issues have been addressed. In particular, we have made references to both of the suggested publications, which we agreed were relevant for our work. We have also made more clear the difference between the sets identified by the alpha shape algorithm (we call those "boundary sets") and the sets identified by k-means clustering containing mostly boundary points (we call those "residual sets").
  
  Citation: https://doi.org/10.5194/egusphere-2024-2173-AC1
- AC2: 'Reply on RC2', Henry Schoeller, 31 Oct 2024
  
  In addition to what has been mentioned in the answer to reviewer 1 that also applies to your comment, we have addressed all minor issues raised. In particular, we admit that there was a bit of confusion with respect to the spectral gap identified and the number of clusters searched for, due to a bug in our code. Thank you for letting us know. The issue has now been corrected and the approach is consistent. Though the number of clusters searched for has changed for the second and third application, the results and interpretations are not affected, since (as mentioned in the text) the clustering is quite robust against variation of epsilon or k.
  
  Citation: https://doi.org/10.5194/egusphere-2024-2173-AC2

Peer review completion

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

AR by Henry Schoeller on behalf of the Authors (01 Nov 2024) Author's response Author's tracked changes Manuscript

ED: Publish subject to technical corrections (17 Dec 2024) by Reik Donner

I need to express my sincere apologies to the authors for having needed more time than usual for rendering my final decision on their revised manuscript due to some unforeseen issues unrelated to the editorial process of their manuscript. I have carefully checked the revised version of this paper, the authors’ rebuttal letter to the two reviewers and also some of the related literature to finally conclude that the authors have indeed done an excellent job in preparing their revision, and that their paper can be accepted for publication in its present form.

I do have very few very minor technical/stylistic suggestions to be implemented in the final published version, which are listed below (line numbers refer to the tracked-changes version of the submitted revised manuscript):

• Line 88: “Section 3” (do not abbreviate Sect. at the beginning of the sentence)
• Lines 96-98: two subsequent sentences have the same form: “for a formal… introduction…, we refer to…”; some more variation in the phrasing might be good
• Lines 192 and 205: shouldn’t it be “(K+1)-th”?
• Line 216: “the discussion”
• Line 220: missing space before “For…”
• Line 305: “Boolean” should be capitalized (referring to George Boole)
• Line 347: Do you really mean “insignificant” or rather “insensitive” here?
• Line 383: “an initialization”
• Line 390: I am not sure that “degrees of freedom” is the most appropriate term here, maybe just use “parameters” instead?
• Line 482: “Figure 6” (do not abbreviate Fig. at the beginning of the sentence)
• Line 486: “Supplementary Fig. S5”
• Line 494: “Supplementary Fig. S3”
• Line 502: “suggests”?
• Line 532: typo “existing”
• Fig. 7, caption, line 4: “ratio between the number of…. clusters and the total…”
• Line 543: “Supplementary Fig. S4”
• Line 563: remove space in the velocity unit
• Lines 585-586: Does “lowest/highest” refer to “lowest/highest altitude”? I would suggest to write this out explicitly.
• Fig. 12, caption, line 1: “Coherent sets…” (do not capitalize “Sets”)
• Line 686: “Supplementary Fig. S4)
• Line 674: “are only shown”
• Line 683: “make it likely”
• Some of the references are not complete or not correctly formatted, please check (e.g. missing article ID of Allshouse & Peacock 2015; Froyland & Padberg-Gehle 2015, Pfahl & Wernli 2012; inconsistent abbreviation/capitalization of journal names, paper title of Schwierz et al. 2004).

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AR by Henry Schoeller on behalf of the Authors (06 Jan 2025) Manuscript

Journal article(s) based on this preprint

27 Feb 2025

| Highlight paper

Assessing Lagrangian coherence in atmospheric blocking

Henry Schoeller, Robin Chemnitz, Péter Koltai, Maximilian Engel, and Stephan Pfahl

Nonlin. Processes Geophys., 32, 51–73, https://doi.org/10.5194/npg-32-51-2025,https://doi.org/10.5194/npg-32-51-2025, 2025

Short summary Executive editor

Henry Schoeller, Robin Chemnitz, Péter Koltai, Maximilian Engel, and Stephan Pfahl

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https://doi.org/10.5194/egusphere-2024-2173-supplement

Henry Schoeller, Robin Chemnitz, Péter Koltai, Maximilian Engel, and Stephan Pfahl

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We identify spatially coherent air streams into atmospheric blockings, which are important weather phenomena. By adapting mathematical methods to the atmosphere, we confirm previous findings. Our work shows that spatially coherent air streams featuring cloud formation correlate with strengthening of the blocking. The developed framework also allows statements about the spatial behavior of the air parcels as a whole and indicates that blockings reduce the dispersion air parcels.


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