the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Jul 2024
Assessing Lagrangian Coherence in Atmospheric Blocking
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.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
Interactive discussion
Status: closed
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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
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AC1: 'Reply on RC1', Henry Schoeller, 31 Oct 2024
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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
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AC1: 'Reply on RC1', Henry Schoeller, 31 Oct 2024
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
-
AC1: 'Reply on RC1', Henry Schoeller, 31 Oct 2024
-
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
-
AC1: 'Reply on RC1', Henry Schoeller, 31 Oct 2024
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Henry Schoeller
Robin Chemnitz
Péter Koltai
Maximilian Engel
Stephan Pfahl
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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(9346 KB) - Metadata XML
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(4761 KB) - BibTeX
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