the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Sensitivity of Gyrescale Marine Connectivity Estimates to Finescale Circulation
Saeed Hariri
Sabrina Speich
Bruno Blanke
Marina Lévy
Abstract. We investigate the connectivity properties between different ocean stations in an idealized open ocean model of a western boundary current system separating two ocean gyres. We applied a Lagrangian framework to compute trajectories from various dynamical setups: a highresolution (1/9°) 3D velocity field reproducing a large range of the ocean finescale (i.e. mesoscale plus part of the submesoscale) dynamics, or a filtered velocity field on a coarseresolution (1°) grid, and one limited to the surface 2D velocities. As ocean connectivity has been assessed in the published literature using different definitions, in this work we compare different metrics such as the average values of transit time and arrival depth between specified sample stations as well as the probability density functions (PDFs) of transit times and betweenness for the different dynamical setups. Our results indicate that almost none of the PDFs show Gaussian behaviour. When the finescale dynamics are taken into account, the numerical particles move and connect pairs of stations faster (between 100 days to 300 days) than when it is absent. This is particularly true, along and near the jets separating the two gyres. Moreover, the connectivity is facilitated when 3D instead of 2D velocities are considered. Finally, our results suggest that western boundary currents are characterized by high betweenness centrality values, which confirms its key role in controlling the transfer of particles in the doublegyre configuration.
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Saeed Hariri et al.
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RC1: 'Comment on egusphere20221276', Anonymous Referee #1, 04 Dec 2022
In this manuscript the authors study Lagrangian connectivity across a set of stations within an oceanic basin using modelled velocity fields. They use approaches based on Lagrangian pdf and network theory and provide estimations of different connectivity metrics. The topic is relevant and the results presented seem interesting. However, in my opinion, some issues regarding the methodological framing of the study should be addressed before considering the manuscript suitable for publication. Please find below my argumentations that hopefully will be useful for the authors:
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Main issues:
==========================================1) The author define a set of 16 stations across the basin to evaluate connectivity among pairs of them. However, the choice of the number of stations and their location seems arbitrary. How much the analysis is sensitive to such choice? Did the authors tested different stations configurations? Why they did not consider a full covering of the domain instead than a few sparse stations?
2) A literature review would be useful since the authors did not discuss their work in the context of other similar approaches missing some key references (see specific comments for details).
3) Regarding the analysis of Lagrangian pdf the authors did not clearly explained when and where they use minimum connection times or connection probabilities and how the two quantities relates between them.
4) The authors seem to have misunderstood the concept of betweenness centrality confusing it with the concept of paths across a network (see specific comments).
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Linebyline comments on the manuscript:
==========================================(l. 3648) The author should also introduce other works where these concepts have been developed, for instance:
 Richter, DJ, et al. "Genomic evidence for global ocean plankton biogeography shaped by largescale current systems." Elife 11 (2022)
 Ward, BA, et al. "Selective constraints on global plankton dispersal." Proceedings of the National Academy of Sciences 118.10 (2021)
 Jacobi, Martin Nilsson, et al. "Identification of subpopulations from connectivity matrices." Ecography 35.11 (2012)(l. 9395) Connection time is just one possible option to characterise connectivity, see for instance different approaches based on fluid fractions (i.e. probabilities):
 Froyland, G, et al. "Almostinvariant sets and invariant manifolds—connecting probabilistic and geometric descriptions of coherent structures in flows." Physica D 238.16 (2009)
 SerGiacomi, E, et al. "Explicit and implicit network connectivity: Analytical formulation and application to transport processes." Physical Review E 103.4 (2021)(eq. 1) The formula is not explained sufficiently:
 please define the variable "a"
 how a pair of station for which the connectivity is calculated is specified in the equation?(l. 210) A general issue along the paper is that the authors did not clearly explained how the connectivity matrix used for network analysis is calculated. Is the matrix defined in terms of times or probabilities? Which algorithm they use to compute its elements?
(l.214215) Please note that betweenness centrality and pathsrelated analysis in fluid flow have been extensively introduced in:
 SerGiacomi, E, et al. "Most probable paths in temporal weighted networks: An application to ocean transport." Physical review E 92.1 (2015)
 Lindner, M et al. "Spatiotemporal organization of dynamics in a twodimensional periodically driven vortex flow: A Lagrangian flow network perspective." Chaos 27.3 (2017)(eq. 2) As commented before, how the matrix elements a_ij are defined? Please note that depending on the definition of the connectivity matrix the distance associated to each step of a path should be evaluated accordingly
(l. 236237) This seems an interesting feature? Why such separation is observed?
(l 253254) Why the shape of the pdf is changing qualitatively depending on the velocity fields and/or the pair of stations? Such different features of the pdf should reflect some dynamical proprieties of the advection pattern. Could the authors comment on this?
(l. 395405) This part should probably go to the Methods section and, again, it is not clear how the connectivity matrix is calculated
(l. 421) Please note that the betweenness metric that the authors are trying to calculate is a node and NOT a link propriety! Maybe the authors are confusing the concept of betweenness with the one of a path between a pair of nodes?
(l. 423424) No, Costa et al. did not improved the Dijkstra's algorithm.. They used a standard logarithmic transformation for the network links' weights, but they did not change anything of the algorithm.
(l. 440441) Again, maybe the authors are confusing the concept of betweenness with the one of path?
Citation: https://doi.org/10.5194/egusphere20221276RC1  AC1: 'Reply on RC1', Saeed Hariri, 27 Feb 2023

RC2: 'Comment on egusphere20221276', Terence Legrand, 14 Dec 2022
In this manuscript, the authors investigate the sensitivity of connectivity estimates regarding the resolution of the velocity fields used to implement Lagrangian particles trajectories. They employed three theoretical ocean circulation model setups: (i) a “1/9 °  every two days highresolution” 3D, (ii) a “1/9°  every two days highresolution” 2D and (iii) a “1°  every two days coarseresolution” 3D. On each circulation setup, they used the ARIANE Lagrangian model to track particles trajectories for a minimum time of 1 year from 15 release zones, called stations. They implemented stations regarding interesting ocean model features (i.e., main jet and secondary jet, location with low eddy kinetic energy) and model domain. At each station, particles were released from – 1 m to the top of the mixed layer depth for 3D setups, and on the surface only for the 2D consideration. They latter observed the position of particles at different tracking time intervals (7 d, 180 d, 540 d, 910 d) with PDF. They also compared the distribution of the transit time between selected stationpairs (i.e., stations 115 and stations 1012) and the minimum and median transit time among all stationpairs for the different circulation model setups. Overall, they found that the transit time (i) is lower for the highresolution 3D setup than for the coarseresolution 3D setup, (ii) is lower for the highresolution 2D setup than for the highresolution 3D setup.
The initial goal of investigating the sensitivity of connectivity estimates to finescale circulation is relevant and exciting for the biophysical modelling community investigating either demographic or genetic connectivity. Indeed, in this community, the limitation of Lagrangian simulations often refers to integrating properly larval behaviour (e.g., Leis, 2020), but rarely on the impact of meso to submeso scale dynamic features (e.g., Whitney et al., 2021). However, in my opinion, major issues should be addressed before considering this manuscript for publication in EGUsphere.
Main comments:
 Authors used 1/9°  every two days velocity fields as a highresolution setup. I do not believe that this resolution is precise enough to study the impact of finescale circulation on connectivity estimates. Most of the operational ocean models used in biophysical modelling studies are characterized by daily velocity fields with a higher resolution (e.g., 1/12° in SerGiacomi et al., 2020, Assis et al., 2022, Legrand et al., 2022). As such, it questions the utilisation of a theoretical ocean circulation model. Moreover, I wonder if 1° resolution is too coarse for a 3000 km * 2000 km domain. In this setup, it results on a domain of approximately 30 *20 velocity field grid cells, with stations which are only separated by ~ 45 grid cells (e.g., stations 1011, 118). Why not considering a 2 or 5times coarser setup rather than a ~ 10times?
 The stations are implemented in relation to flow features and model domain. I wonder how this impacts the results. Consequently, how are the results sensitive to a random implementation of stations?
 Results depicting the transit times between stations (section 3.2.1 to section 3.2.3 and Figure 5 to Figure 8) are only made on a station subset (e.g. stationpairs 115 and 1012 for section 3.2.1 and Figure 5). As such, how are the results sensitive to this station subset? Are the results similar when considering all the possible stations together?
 The Betweenness 2.2.4 Methods section is imprecise and muddled, and the results brought on make no sense. The authors mixed up between “betweenness centrality”, a node/edge measure, and “betweenness”, a link/vertices measure. Moreover, they have not specified how a_{ij} is computed to obtain betweenness results in section 3.3.1. Because of that, the comparison between betweenness value computed with the Costa et al., 2017 weight transformation and without is meaningless. Please consider rethinking all this section with a correct use of betweenness centrality measure.
References:
Leis, J. M. (2020). Perspectives on larval behaviour in biophysical modelling of larval dispersal in marine, demersal fishes. In Oceans (Vol. 2, No. 1, pp. 125). MDPI.
Whitney, J. L., Gove, J. M., McManus, M. A., Smith, K. A., Lecky, J., Neubauer, P., ... & Asner, G. P. (2021). Surface slicks are pelagic nurseries for diverse ocean fauna. Scientific reports, 11(1), 118
SerGiacomi, E., Legrand, T., HernandezCarrasco, I., & Rossi, V. (2021). Explicit and implicit network connectivity: Analytical formulation and application to transport processes. Physical Review E, 103(4), 042309.
Assis, J., Neiva, J., Bolton, J. J., Rothman, M. D., Gouveia, L., Paulino, C., ... & Serrão, E. A. (2022). Ocean currents shape the genetic structure of a kelp in southwestern Africa. Journal of Biogeography, 49(5), 822835.
Legrand, T., Chenuil, A., SerGiacomi, E., ArnaudHaond, S., Bierne, N., & Rossi, V. (2022). Spatial coalescent connectivity through multigeneration dispersal modelling predicts gene flow across marine phyla. Nature communications, 13(1), 112.
Costa, A., Petrenko, A. A., Guizien, K., & Doglioli, A. M. (2017). On the calculation of betweenness centrality in marine connectivity studies using transfer probabilities. PLoS One, 12(12), e0189021.Citation: https://doi.org/10.5194/egusphere20221276RC2 
RC3: 'Reply on RC2', Terence Legrand, 14 Dec 2022
Typo on "The authors mixed up between “betweenness centrality”, a node/edge measure, and “betweenness”, a link/vertices measure". The good sentence is "The authors mixed up between “betweenness centrality”, a node/vertex measure, and “betweenness”, a link/edge measure"
Citation: https://doi.org/10.5194/egusphere20221276RC3  AC4: 'Reply on RC3', Saeed Hariri, 27 Feb 2023
 AC2: 'Reply on RC2', Saeed Hariri, 27 Feb 2023

RC3: 'Reply on RC2', Terence Legrand, 14 Dec 2022

RC4: 'Comment on egusphere20221276', Anonymous Referee #3, 24 Dec 2022
In this manuscript, the authors aim to investigate how gyrescale marine connectivity is sensitive to resolved finescale flow features. This research question is relevant, as it can inform future connectivity study on whether finescale flow features should be resolved or whether coarse models provide an accurate enough representation of the flow. The authors run a coarsened version (1 degree) of a fine (1/9^{th} degree) hydrodynamic model and compare several metrics that are relevant to connectivity. They conclude that finescale features expediate connectivity in the basin and should be included in connectivity studies.
In my opinion, the manuscript has major flaws that should be addressed before it is suitable before publication. I will list these items individually below, followed by linebyline comments and minor/technical issues.
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Main issues
1) The definition and interpretation of betweenness and betweenness centrality is wrong in several places in the manuscript. Betweenness is claimed to be used to construct a connectivity matrix, but betweenness is a scalar measure of the number of shortest paths between pairs of nodes that pass through a given node (as correctly mentioned on 212213). It can be used to identify ‘bottlenecks’ in the flow (l134): regions through which a relatively large amount of transport occurs (SerGiacomi et al., 2021). However, on lines 100101 it is described as the number of shortest paths between nodes, suggesting that it is a measure defined in matrix form between i and j where it is defined a scalar for each node i. In section 3.3.1, betweenness centrality is wrongly used as a measure of transport probability between nodes, whereas the transport probability should simply be defined from the amount of particles that travels from node i to j (see e.g. Froyland et al., 2014).
Moreover, the paper from Costa et al. (2017) is wrongly interpreted as giving a new definition of betweenness, which, according to the author’s is different than that of Dijkstra (1959). Instead, Costa et al. use the textbook definition of betweenness centrality (see Newman 2010) and simply use a reweighting of the edges of the transition matrix that is used as the input graph that betweenness is used on. Dijkstra’s algorithm is simply a shortest path computation algorithm, which can still be used, next to Brandes’ algorithm for Betweenness computation (Brandes, 2001). So, the “Costa versus Dijkstra” distinction is wrong, but plays a quite central role in this paper.
Moreover, the concept of a ‘betweenness matrix’ in Figure 10 makes no sense, since betweenness centrality is defined per node, not between nodes. It is therefore unclear what these matrices represent, since it cannot be betweenness centrality. Perhaps transition matrices are really used instead, but then it is unclear which purpose the prior definition of betweenness centrality serves.
These misconceptions should be fully cleared up. This can be done by computing the correct betweenness values per station, which should be in interpreted as “how important is one station as a link between other stations?” and by computing transition matrices, which should be interpreted as “for a given station, what is the probability that it ends up at another station?”.
2) The manuscript uses an idealized twogyre model representative of a subtropical and subpolar gyre system as found for instance in the North Atlantic. A qualitative interpretation of how the dynamics differ between the HR and CR cases is useful. However, the results section is very lengthy with quantitative descriptions of connectivity properties of different (links between) stations. Since these exact details bear no relevance to the real ocean, the results section can be shortened and sharpened, as only the qualitative results are relevant with respect to our oceans.
3) The authors compare a highresolution flow field to a coarsened version of it. Then, particle trajectories are integrated on both flow fields. Naturally, particles in the HR case will experience dispersion on scales smaller than the 1degree grid, which leads to a divergence of the trajectories. I invite the authors to make a remark about the role of this subgridscale dispersion and on whether it may be simulated.
4) The authors refer to several studies that use Lagrangian PDFs, which usually are PDFs of particle velocities (e.g. Pope, 1985) or particle separation. Please make sure that the referenced papers discuss the type of PDFs used in this manuscript. This will sharpen the definition used. Perhaps using the name Transit Time PDF would already be clearer.
5) The manuscript could benefit from a clearer definition of ‘connectivity’. It can help to often plainly talk about transit times or betweenness (if correctly used), as to avoid confusion between the different concepts.
6) The introduction is unnecessarily lengthy and the discussion of specific papers from line 6189 is not relevant for the methods and analysis in this paper. For example, the paragraph between 7377 uses several sentences to mention a study that uses community detection, but community detection is not used in this paper. I see no reason to keep it, as examples of connectivity studies are already mentioned earlier.
7) The manuscript does not provide any specific hydrodynamic model configuration code, Lagrangian analysis configuration code, or analysis scripts, making it irreproducible. For example, it is unclear how the authors construct the graph/network on which betweenness metrics are computed. Readers would benefit from seeing the code, as the measures such as betweenness are heavily influenced by the way the network is constructed. This omission of code is not in line with the Open Science standards set by EGU journals. Please link to your NEMO model configuration code, Lagrangian simulation scripts, network generation code and analysis code.
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Line by line comments
 17: The authors mention that the PDFs are not Gaussian, but this is not a prior hypothesis: there is no reason to assume a Gaussian structure. It is not relevant to mention this, unless it has specific interpretations, which are not given.
 35: Connectivity is not just used in the context of species dispersal, but can also describe the exchange of water (properties) more generally, or that of plastic. See e.g. Froyland et al. (2014) or SerGiacomi et al. (2015)
 50: The authors mention that using an advectiondiffusion equation assumes uniformity in advection and diffusion coefficients. This is not an inherent assumption in using an advectiondiffusion equation, but simply an assumption that is used in the studies mentioned. After all, one can study connectivity using tracers in an OGCM, where advection nor diffusion need to be uniform.
 57: What is meant by an “ocean connection”?
 5960: “Population connectivity has mostly been studied using Lagrangian integration of surface ocean currents”. This is what the authors also do. Currently, the sentence hints at the manuscript providing an alternative method, which it does not.
 93: The concept of a connectivity time is illdefined and not used by all of the aforementioned examples, such as Rossi et al. (2014). Please provide a precise definition.
 101: This is an example of an incorrect interpretation of betweenness.
 103: This implies that Dijkstra’s algorithm is for betweenness computation, but Dijkstra (1959) simply concerns a shortest path algorithm.
 107109: Mention what the aim is of constructing such a matrix. How will it benefit your analysis?
 132: “all spatial scales of the modelled velocity”. Do other Lagrangian codes not integrate over all spatial scales and just some instead?
 132133: from “to better understand” is unnecessary.
 171: Please mention over what integration time particles are integrated.
 181: It is unclear how often particles are released. Is each particle released at a different initial timestep, or is this done in batches (of which size)?
 183: Are particles also released up to 150m deep if the mixed layer is shallower than that?
 187: The specification of the stations still seems arbitrary to me. How are they chosen exactly?
 193: “5 stations were used”. Which?
 194: “Note that stations 1, 3, 5, 8, 12, and 15 are important”. Please explain why.
 198: Taylor (1921) is seminal to the theory of eddy diffusion, but unrelated to Lagrangian PDFs.
 208: Please define the meaning of the ‘sample space variable’.
 214: The authors give a textbook definition of betweenness, which was not defined by Costa et al. (2017).
 216: sigma is the sum of shortest paths, not just the shortest path.
 220221: Costa et al. (2017) only proposed redefining the weights of the Lagrangian transition matrix. Furthermore, please explain what a_{ij }is, what d_{ij} is, and how these are used to compute betweennessin eq (2).
 224: This comparison is only in the supporting information. I suggest to either remove this sentence or to move the comparison to the main text.
 226250: Since the paper is mainly about a comparison between HR and CR, the section where transit times are just reported for HR seems irrelevant to me, especially since this quantitative assessment cannot be translated to the real ocean. Instead, qualitative assessments provide a more powerful analysis as these likely commute with the real ocean.
 232233: These conclusions cannot be drawn simply from Figure 4 alone, as this concerns only one release site.
 237: “not shown” > please show this
 244: “not shown” > please show this.
 244245: Do you have a hypothesis for why this behavior is so different from particles release in station one? After all, those particles cross the main jet too. Why are they not trapped in it?
 248250: This seems to be at odds with what happens for the particles in station 1, which can cross the main jet.
 252294: This section can be shortened. Details about the distributions are not relevant; only the comparison between the distributions among the CR and HR case are.
 253254: I don’t think Gaussian shapes should be expected in any case. See Van Sebille (2011) or O’Malley (2021) for similar transit time PDFs.
 285: specify “closer”
 330332: In the case that particles are assumed buoyant, the 2D assumption is still valid.Only for nonbuoyant particles, a simplification using only surface currents may be problematic.
 363: It is unclear how this figure is plotted. Is any smoothing used? Are all values statistically significant? Is there enough data in each region?
 371: Please also list the longest transit times associated to those stations.
 371373: The authors mention that for the mean arrival depth for the shortest and the longest arrival time differ by about 65 meters. Is this result generalizable? I.e. are short arrival times usually associated with shallower depths? If this is not a generalizable result, it can be left out, since it would be anecdotal.
 373375: Is this a general result or is it anecdotal for this case?
 403: The authors mention that graph theory is used to define hydrodynamic provinces, but this concept (Rossi et al., 2014) is not actually used in this paper.
 407408: Betweenness centrality is not at all a measure of transfer probabilities between two stations. That measure should simply be the amount of particles traveling between stations over time.
 412: Costa et al. (2014) do not have a special definition of betweenness.
 414416: This is not a different type of betweenness, but a different type of graph used to compute it.
 422: a_ij and log(1/a_ij) are not distances but weights.
 435437: Please show this claim.
 443: Renaming “connectivity matrix” to “transit time matrix” avoids confusion.
 465: “High connectivity” and “betweenness” are not the same
 485492: This section could use a stronger conclusion drawn. Currently the conclusion is simply that are differences in transit time between the two runs, but it remains unexplained what the precise reason for this is.
 490: Betweenness is not a ‘rate of connections’.
 495496: The authors wrongly claim novelty here about using HR flow fields to describe connectivity patterns in a largescale basin. Rossi et al. (2014) do the same for the Mediterranean and Reijnders et al. (2021) for the Arctic, which are both not idealized.
 503: See previous comment about Taylor not introducing Lagrangian PDFs.
 504: The authors mention that the PDFs are not Gaussian, but this is not a prior hypothesis: there is no reason to assume a Gaussian structure. It is not relevant to mention this, unless it has specific interpretations, which are not given.
 511: Please qualitatively describe the differences and draw a conclusion from it.
 533: Please mention the open and unsolved questions. These are currently not mentioned.
 Figure 3a: indicate the release location
 Figure S2 is not referenced in the main text.
––––
Minor/technical comments:
 51: change “not necessarily verified in” into “unrepresentative of”. The statement is currently too weak.
 61: “is” > “has become”
 6465: Please rewrite the sentence starting with “Based”. It is currently not a correct sentence.
 91: “graph theory” > “Community detection using graph theory”
 109: remove an unnecessary period.
 146: “lower” > “less”
 184: you cannot perform a property. Could you specify what is meant?
 230: Higher than what?
 336: “the deepest distance” > “deeper”. Deepest would suggest the deepest possible depth.
 355: “Mainly” what?
 493: This is not a full sentence.
 In general: The usage of ‘coarse resolution’ is wellchosen and more acurate than ‘low resolution’, but should be mirrored by ‘fine resolution’ rather than ‘high resolution’.
 Figure 3b: Either use ‘modulus of the annual mean velocity’ or ‘annual mean speed’
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References
Brandes, U. (2001). A faster algorithm for betweenness centrality*. The Journal of Mathematical Sociology, 25(2), 163–177. https://doi.org/10.1080/0022250X.2001.9990249
Dijkstra E.W. (1959). A note on two problems inconnexion with graphs. Numerische Mathematik; 1:269271. https://doi.org/10.1007/BF01386390
Froyland, G., Stuart, R. M., & van Sebille, E. (2014). How wellconnected is the surface of the global ocean? Chaos: An Interdisciplinary Journal of Nonlinear Science, 24(3), 033126. https://doi.org/10.1063/1.4892530
Newman, M. E. J. (2010). Networks. Oxford University Press.
O’Malley, M., Sykulski, A. M., LasoJadart, R., & Madoui, M.A. (2021). Estimating the travel time and the most likely path from Lagrangian drifters. Journal of Atmospheric and Oceanic Technology. https://doi.org/10.1175/JTECHD200134.1
Pope, S. (1985). PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci., 11(2), 119 – 192, https://doi.org/10.1016/03601285(85)900024
Reijnders, D., van Leeuwen, E. J., & van Sebille, E. (2021). Ocean Surface Connectivity in the Arctic: Capabilities and Caveats of Community Detection in Lagrangian Flow Networks. Journal of Geophysical Research: Oceans, 126(1), e2020JC016416. https://doi.org/10.1029/2020JC016416
Rossi, V., SerGiacomi, E., López, C., & HernándezGarcía, E. (2014). Hydrodynamic provinces and oceanic connectivity from a transport network help designing marine reserves. Geophysical Research Letters, 41(8), 2883–2891. https://doi.org/10.1002/2014GL059540
SerGiacomi, E., Baudena, A., Rossi, V., Follows, M., Clayton, S., Vasile, R., López, C., & HernándezGarcía, E. (2021). Lagrangian betweenness as a measure of bottlenecks in dynamical systems with oceanographic examples. Nature Communications, 12(1), 4935. https://doi.org/10.1038/s41467021251559
SerGiacomi, E., Rossi, V., López, C., & HernándezGarcía, E. (2015). Flow networks: A characterization of geophysical fluid transport. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(3), 036404. https://doi.org/10.1063/1.4908231
van Sebille, E., Beal, L. M., & Johns, W. E. (2011). Advective Time Scales of Agulhas Leakage to the North Atlantic in Surface Drifter Observations and the 3D OFES Model. Journal of Physical Oceanography, 41(5), 1026–1034. https://doi.org/10.1175/2011JPO4602.1
Citation: https://doi.org/10.5194/egusphere20221276RC4  AC3: 'Reply on RC4', Saeed Hariri, 27 Feb 2023
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