the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 25 Apr 2024
Persistence and Robustness of Lagrangian Coherent Structures
Abstract. Lagrangian coherent structures (LCS) are transient features in ocean circulation that describe particle transport, revealing information about transport barriers and accumulation or dispersion regions. Various methods exist to infer LCS from surface current fields provided by ocean circulation models. Generally, Lagrangian trajectories as well as LCS analysis inherit the uncertainty from the underlying ocean model, bearing substantial uncertainties as a result of chaotic and turbulent flow fields. In addition, velocity fields and resulting LCS evolve rapidly. In this study, finite time Lyapunov exponents (FTLE) are used to detect LCSs in surface current predictions from a regional ocean forecast system. We investigate the uncertainty of LCS at any given time using an ensemble prediction system (EPS) to propagate velocity field uncertainty into the LCS analysis. We evaluate variability of FTLE fields in time and across the ensemble at fixed times. Averaging over ensemble members can reveal robust FTLE ridges, i.e. FTLE ridges that exist across ensemble realisations. Time averages reveal persistent FTLE ridges, i.e. FTLE ridges that occur over extended periods of time. We find that LCS are generally more robust than persistent. Large scale FTLE ridges are more robust and persistent than small scale FTLE ridges. Averaging of FTLE field is effective at removing chaotic, short-lived and unpredictable structures and may provide the means to employ LCS analysis in forecasting applications that require to separate uncertain from certain flow features.
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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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RC1: 'Comment on egusphere-2024-1171', Anonymous Referee #1, 30 May 2024
This paper applies Finite-Time Lyapunov Exponent (FTLE) calculations to a region of the Barents Sea based of surface (Eulerian) velocity field measurements in early 2023. The authors' main conclusion seems to be that averaging the FTLE fields spatially and temporally provides robust measures in "removing short-lived and unpredictable structures" which can help in forecasting by "separat[ing] uncertain from certain flow structures" [quotations are from the abstract].
I do not recommend the article be accepted in its current format, due to a range of reasons which I outline in detail in my comments.
1. The term "Lagrangian Coherent Structures" (henceforth LCSs) is used prominently in the title, and throughout the article. This term was coined by Haller in the early 2000s, and it is the continuing work of this group that the authors almost exclusively cite in this paper. However, the work of this group now very clearly defines LCSs in very specific ways, which are outlined clearly in Haller (Annu Rev Fluid Mech, 2015). These definitions, stated within the two-dimensional context in which the current paper is situated, relate to attempting to find curves towards/from there is extremal attraction/repulsion -- so-called "hyperbolic LCSs." It is these, and some allied entities, that are LCSs according to Haller and the group's definition, and so citing those papers and not using those definitions does not make sense. Moreover, Haller and collaborators in this and a range of other papers make clear that FTLEs are not necessarily these LCSs, and so prominently using the term LCSs in this paper is incongruous at best. Since the paper deals exclusively with FTLEs (and their averaging -- not even their ridges which some other authors may refer to as a type of LCS), the title and the paper should exclusively use the term FTLEs. Of course, one should position FTLEs in the context of LCSs -- which to a range of other authors represent an entire suite of techniques devised to extract coherent structures (based on differing definitions and intuition) from genuinely unsteady data. There are many recent reviews of this range of different methods for LCSs in this broader context in the literature: Balasuriya et al (Physica D, 2018), Hadjighasem et al (Chaos, 2017), Shadden (in: Transport and Mixing in Laminar Flows, Wiley, 2011, pp59). The authors are advised to consult these in positioning their work (as in line 85 when they say "Various methods have been proposed for LCS detection") and deciding whether the term LCS is actually appropriate.
2. The main conclusion seems to be that taking FLTEs and averaging them gives a better diagnostic of "robust" coherent structures. In doing such an averaging, there are some scientific issues related to time-parametrization and that I will come back to in a later point. However, I have some comments with respect to the issue of averaging FTLE fields to smooth out what the authors seem to think of as non-robustness, and thereby extracting robust structures. First, the FTLE fields are definitely associated with a time-of-flow, and since the authors use 24 hours exclusively here, they will be identifying local exponential stretching rates over the past 24 hours. If the authors were *not* interested in stretching over that time-scale but rather over, say, a one-month period, then rather than taking a 24 time-window, they should take a one-month window. This will definitely smear over ephemeral stretching. (By the way, when the authors use the term "ephemeral" this depends on the context. It seems that this means structures which do not persist over a longer time-scale, say a week or two. Within this time-frame, perhaps 24 hours is ephemeral -- but the authors seem to have decided to use 24 hours in the FTLE calculation by choice, and so it's to be expected that features related to 24-hour time scales are what will be revealed in the FTLE field.) Second, the discussion seems to indicate that the authors want to find stationary structures which persist over a longer time. In other words, to think of the velocity field as predominantly steady, with unsteady variations, and the goal would be to find structures which are stable with respect to the unsteadiness (which is assumed smaller). Well, if so, there are better ways to approach this. Rather than calculating FTLE fields from the full unsteady Eulerian data, they can obtain a dominant steady part of the Eulerian velocity. One way to do this would be to average the Eulerian data over an explicit time period -- this has the added advantage of being able to specify a time-scale (the time of averaging) over which the Eulerian field is assumed "mostly" steady, and hence one can ascribe a time-scale to one's conclusions. Another alternative would be to use some smoothening technique -- and again, if using something like a spatial filter associated with an explicit length-scale, one can ascribe a scale (a length-scale in this instance) of "accuracy" of the processed data to enable any conclusions to be stated in relation to that. These methods work directly on the Eulerian velocity field, rather than applying techniques such at FLTEs on data which to all intents and purposes (based on the conclusions reached) seem to be viewed as "noisy." Remove the noise first to avoid amplification of inaccuracies when doing additional computations. Thirdly, if stationary objects are sought, it seems that the authors want to look at the dominant steady component, and if one has a steady velocity field, Lagrangian methods are irrelevant. Lagrangian issues only make sense if there is unsteadiness, and one needs to follow the flow. If steady, simple techniques (drawing streamlines, Okubo-Weiss criterion, etc) on the Eulerian velocity field give perfectly good information on what's going on.
3. Based on the conclusions that the authors seem to be reaching, there seems to be an incomplete understanding of what the FTLEs represent. The FTLE field $ \sigma_{t_0}^t $ represents a field at time $ t_0 $, associated with the stretching rate experienced by infinitesimal fluid parcels beginning at time $ t_0 $ and flowing until time $ t $. This rate is converted to an exponential one via the logarithm, and the time-of-flow $ T = |t-t_0| $ is used to time-average this quantity so that it's the average exponential rate over the time period considered. Note that there is absolutely no mention here of anything being a flow barrier, or a "coherent structure." Thus, the FLTE identifies regions in the time $ t_0 $ based on stretching rates over the time-of-flow (in this case, 24 hours in the past). When one time-averages the FTLE field, what exactly does that mean? Presumably (and this is not clearly stated), FTLE fields are generated for differing $ t_0 $s but the same time of flow (this is sometimes called time-windowing in the coherent structure community), but then are the authors averaging over $ t_0 $? (Give an explicit formula for the averaging, so that this is clear.) If so, they are taking scalar fields which are defined over different times $ t_0 $, each of which is associated with flow over a different time-window $ (t_0-T,t_0) $, and averaging them. If this is what is done, it needs to be explained clearly. But then, the interpretation of this needs to be carefully stated, since one is averaging over different initial times, and what one gets cannot be associated with a particular time instance (unlike one calculation of an FLTE field, which gives a field at time $ t_0 $). Of course, one might argue that calculationally the averaging tells us of how 24-hour motion calculated over several initial times (say 1 January to 31 January 2023) is averaged to give an "average exponential rate of motion in January 2023 when a time-scale of 24 hours is considered for the exponential rate". This would need to be explained, because it's quite awkward and hard to interpret.
4. The above point related to one aspect of time-parametrization: the $ t_0 $ in the field $ \sigma_{t_0}^t $. Another time-parametrization issue is the $ t $, and thus the time-of-flow. Everything in this paper has used a time-of-flow of 24 hours. Hence, everything is slaved to this time-scale -- the exponential rate is computed based on time-of-flow for this time-scale. This issue is buried in the paper, with the multitude of plots not mentioning this explicitly If a time-of-flow of 48 hours were chosen instead, how do things change? Basically, the calculation of an FTLE field explicitly picks out a time-scale, and this is not something which the authors have clarified. The results are explicitly associated with this time-scale, and no other. If the results are to be used in forecasting, why is this the correct time-scale? Or is this method robust to changing the time-scale? How does the time-scale interact with the time associated with the time-averaging as discussed in my previous point? Basically, the issue of TIME (initial plus time-of-flow in calculating the FTLE and the appropriate interpretation of the FTLE field, the times of computation chosen for averaging) is crucial, and needs to be carefully explained, interpreted, and robustness evaluated (if appropriate).
5. Returning to the definition of the FTLE mentioned earlier. It represents a field at time $ t_0 $, associated with the average exponential rate over flow from time $ t_0 $ to $ t $. Note that there is absolutely no mention here of anything being a flow barrier, or a "coherent structure." Yes, there are papers in the literature which seem to indicate such a connection, but the reality is that it is unjustified. The early papers in this area used STEADY toy models which had saddle points with one-dimensional stable and unstable manifolds emanating from them, and since these manifolds are associated with exponential decay rates, came up with the idea that FTLE ridges had something to do with stable and unstable manifolds. And these manifolds were flow barriers in some way. However, this argument does not hold water, since there are examples such as in Haller (Physica D, 2011) which show that the stable/unstable manifold interpretation sometimes fails even in infinite-time flow. (And, getting back to a previous point related to hyperbolic LCSs, the fact that repelling/attracting do not necessarily occur as expected are also shown.) Real data is much worse: it is unsteady, and finite-time. Finite-time aspects are awkward for inferring exponential rates of growth, since any function over a finite-time can be bounded by an exponential. Unsteadiness is yet another problem, because (even in the infinite-time context) saddle points generalize to hyperbolic trajectories, and their stable/unstable manifolds move around (another reason why time-averaging is questionable). Furthermore, it is not clear what an "FTLE ridge" is -- one never gets a genuinely one-dimensional curve which is well-defined, but rather gets regions of larger FTLE values. Balasuriya et al (J Fluid Mech, 2016) provide an assessment and interpretation of what the FTLE means, with an emphasis on fluid motion, which helps understand these issues. In particular, for finite-time, unsteady data sets, using FTLEs and their ridges cannot easily reach conclusions regarding flow barriers and coherence. FTLEs explicitly look at exponential growth rates with respect to the time-of-flow considered, and that's about it. So when one takes FTLE fields, as done in this paper, and tries to reach conclusions regarding coherent structures such as eddies (as the authors do towards the end of the paper), this needs to be treated with suspicion, because it is not on any firm scientific grounds. The idea that eddies can be demarcated by FTLE ridges -- notwithstanding my earlier comments on finite-time and unsteadiness -- may go back to work in the 1990s (such as del-Castillo-Negrete, Knobloch, Pierrehumbert) who analyzed perturbed toy models. In these cases, the unperturbed models were explicit and steady, and had saddle points with stable and unstable manifolds. In some cases, the geometry was such that these manifolds encircled an eddy. Since the manifolds many be discoverable using FTLE ridges (again subject to various caveats), in such cases ONLY, one might think of an eddy as being found using FTLE fields. However the interior of the eddy does NOT typically contain exponential stretching, and hence FTLE fields by themselves cannot be used to reliably identify eddies as the authors here are doing in their later figures. Actually, in the standard fluid-mechanical dichotomy between stretching and rotation, the eddies have the opposite of stretching, and thus the FTLE is exactly the wrong thing to use (Okubo-Weiss and related criteria may help; however as noted by many authors unsteadiness is a problem in using such Eulerian characteristics). Basically, statements such as "two additional regions are identified and considered as robust in Fig.7 " (line 211 in the paper) are, in this vein, questionable. Indeed, high FTLE values indicate greater separation, and hence LESS certainty!
6. The impression given throughout is that the authors are examining robustness of LCSs. I've already talked about why it's actually FLTEs and not LCSs, but in this point I want to question whether robustness is the right thing. There are many recent papers which examine robustness of FTLE fields (Balasuriya, J Comp Dyn, 2020; Guo et al, IEEE Trans Visual Comp Graphics, 2016; Raben, Exp Fluids, 2014), but this paper is not one of them. (There are a few other papers, from oceanographic situations, which the authors speak to in lines 339-344.) By "robustness," the authors seem to mean smearing over small time scale motion, in other words looking for entities which persist over longer times. This needs to be made clear throughout the manuscript. (I've talked about the time-parametrization issue previously; to smear over smaller time scales, one needs to simply choose appropriate times for the FTLE which are relevant to what one is looking for.) In Section 4.3, also, the word "uncertainty" is questionable because of this same reason -- the authors have no calculated any uncertainty (i.e., have not evaluated anything to do with uncertainty in the input data).
7. There are several statements around Equation (2) and (3) which are incorrect. The statements in line 101 are all incorrect: $ \delta x $ is not the final location, but $ x $ is, and the (1,1) term in Equation (2) then represents the partial derivatives of $ x $ with respect to $ x_0 $, for example. The integral bounds in (3) are unclear and inconsistent with the previous equation. Presumably the authors mean something like $$ x(t) = x_0 + \int_{t_0}^t u \left( x(\tau), t(\tau) \right) \mathrm{d} \tau $$, and then one needs to also clarify that $ x(\tau),y(\tau) $ are the evolving trajectory locations. However, this integral formulation may not be the most natural. The differential form with $ \dot{x} = u \left( x(t), y(t) \right) $, $ \dot{y} = v \left( x(t), y(t) \right) $ and the initial condition $ \left( x(t_0), y(t_0) \right) = \left( x_0, y_0 \right) $ connects better with the discussion, perhaps.
8. The discussion at the end of Section 2.3 is fraught. What is a "2D curve" [line 126]? The statement that "averaging smooths the ridges into fields of attraction/repulsion" [line 128] is incorrect because, as mentioned previously, claiming that FTLEs have anything to do with attraction/repulsion is questionable. "Making these ridges more certain" [line 130] relates to an earlier comment that what is being done here has nothing to do with robustness or certainty. The authors comment that the time- and ensemble-averaged FTLE fields are not transport barriers (which is correct -- but neither is the FTLE field -- and again it's not clear to me how a field can be a barrier), but then talk about things being barriers over larger regions. Figure 2 is strange. One does not usually get FTLE ridges (even in idealized steady toy models in 2D) which intersect and pile on each other like this. Intersections in such cases occur at saddle points. If unsteady, intersections that one gets, analogous to intersections between stable and unstable manifolds, can relate to chaotic motion -- but these are between forward-time FTLE ridges and backwards-time FTLE ridges, rather than self-intersections within one of these.
9. The fact that a large standard derivation of the time-averaged FTLE usually is close to where the FTLE is large [line 169-171] is no surprise. Large FTLE relates to larger uncertainty in the results, because any errors (based on interpolation to a grid, say) increase exponentially. Hence, the values one assigns to the FTLE tend to be less certain. This issue is well-known, and described in some of the papers I've mentioned previously on robustness of FTLEs.
10. For the discussion on Section 3.4, I can't quite understand what the $ k $ in the figures is. Since in 2D, one has two wavenumbers -- say $ l $ and $ m $, associated with the eastward and northwards coordinates. I don't understand the discussion [lines 230 onwards] about averaging over rows and columns (over $ l $ and $ m $?). Is $ k = \sqrt{l^2 + m^2} $? The spectral plots in Figure 9 are used to infer robustness in some way, based on the fact that one gets decay with $ k $ in the bottom figures, say. Any smoothening process will of course get rid of the smaller wavenumbers. Similarly one expects more smoothness when averaged over more and more days.
Citation: https://doi.org/10.5194/egusphere-2024-1171-RC1 - AC1: 'Reply on RC1', Mateusz Matuszak, 22 Jul 2024
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RC2: 'Comment on egusphere-2024-1171', Anonymous Referee #2, 24 Jun 2024
- AC2: 'Reply on RC2', Mateusz Matuszak, 22 Jul 2024
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2024-1171', Anonymous Referee #1, 30 May 2024
This paper applies Finite-Time Lyapunov Exponent (FTLE) calculations to a region of the Barents Sea based of surface (Eulerian) velocity field measurements in early 2023. The authors' main conclusion seems to be that averaging the FTLE fields spatially and temporally provides robust measures in "removing short-lived and unpredictable structures" which can help in forecasting by "separat[ing] uncertain from certain flow structures" [quotations are from the abstract].
I do not recommend the article be accepted in its current format, due to a range of reasons which I outline in detail in my comments.
1. The term "Lagrangian Coherent Structures" (henceforth LCSs) is used prominently in the title, and throughout the article. This term was coined by Haller in the early 2000s, and it is the continuing work of this group that the authors almost exclusively cite in this paper. However, the work of this group now very clearly defines LCSs in very specific ways, which are outlined clearly in Haller (Annu Rev Fluid Mech, 2015). These definitions, stated within the two-dimensional context in which the current paper is situated, relate to attempting to find curves towards/from there is extremal attraction/repulsion -- so-called "hyperbolic LCSs." It is these, and some allied entities, that are LCSs according to Haller and the group's definition, and so citing those papers and not using those definitions does not make sense. Moreover, Haller and collaborators in this and a range of other papers make clear that FTLEs are not necessarily these LCSs, and so prominently using the term LCSs in this paper is incongruous at best. Since the paper deals exclusively with FTLEs (and their averaging -- not even their ridges which some other authors may refer to as a type of LCS), the title and the paper should exclusively use the term FTLEs. Of course, one should position FTLEs in the context of LCSs -- which to a range of other authors represent an entire suite of techniques devised to extract coherent structures (based on differing definitions and intuition) from genuinely unsteady data. There are many recent reviews of this range of different methods for LCSs in this broader context in the literature: Balasuriya et al (Physica D, 2018), Hadjighasem et al (Chaos, 2017), Shadden (in: Transport and Mixing in Laminar Flows, Wiley, 2011, pp59). The authors are advised to consult these in positioning their work (as in line 85 when they say "Various methods have been proposed for LCS detection") and deciding whether the term LCS is actually appropriate.
2. The main conclusion seems to be that taking FLTEs and averaging them gives a better diagnostic of "robust" coherent structures. In doing such an averaging, there are some scientific issues related to time-parametrization and that I will come back to in a later point. However, I have some comments with respect to the issue of averaging FTLE fields to smooth out what the authors seem to think of as non-robustness, and thereby extracting robust structures. First, the FTLE fields are definitely associated with a time-of-flow, and since the authors use 24 hours exclusively here, they will be identifying local exponential stretching rates over the past 24 hours. If the authors were *not* interested in stretching over that time-scale but rather over, say, a one-month period, then rather than taking a 24 time-window, they should take a one-month window. This will definitely smear over ephemeral stretching. (By the way, when the authors use the term "ephemeral" this depends on the context. It seems that this means structures which do not persist over a longer time-scale, say a week or two. Within this time-frame, perhaps 24 hours is ephemeral -- but the authors seem to have decided to use 24 hours in the FTLE calculation by choice, and so it's to be expected that features related to 24-hour time scales are what will be revealed in the FTLE field.) Second, the discussion seems to indicate that the authors want to find stationary structures which persist over a longer time. In other words, to think of the velocity field as predominantly steady, with unsteady variations, and the goal would be to find structures which are stable with respect to the unsteadiness (which is assumed smaller). Well, if so, there are better ways to approach this. Rather than calculating FTLE fields from the full unsteady Eulerian data, they can obtain a dominant steady part of the Eulerian velocity. One way to do this would be to average the Eulerian data over an explicit time period -- this has the added advantage of being able to specify a time-scale (the time of averaging) over which the Eulerian field is assumed "mostly" steady, and hence one can ascribe a time-scale to one's conclusions. Another alternative would be to use some smoothening technique -- and again, if using something like a spatial filter associated with an explicit length-scale, one can ascribe a scale (a length-scale in this instance) of "accuracy" of the processed data to enable any conclusions to be stated in relation to that. These methods work directly on the Eulerian velocity field, rather than applying techniques such at FLTEs on data which to all intents and purposes (based on the conclusions reached) seem to be viewed as "noisy." Remove the noise first to avoid amplification of inaccuracies when doing additional computations. Thirdly, if stationary objects are sought, it seems that the authors want to look at the dominant steady component, and if one has a steady velocity field, Lagrangian methods are irrelevant. Lagrangian issues only make sense if there is unsteadiness, and one needs to follow the flow. If steady, simple techniques (drawing streamlines, Okubo-Weiss criterion, etc) on the Eulerian velocity field give perfectly good information on what's going on.
3. Based on the conclusions that the authors seem to be reaching, there seems to be an incomplete understanding of what the FTLEs represent. The FTLE field $ \sigma_{t_0}^t $ represents a field at time $ t_0 $, associated with the stretching rate experienced by infinitesimal fluid parcels beginning at time $ t_0 $ and flowing until time $ t $. This rate is converted to an exponential one via the logarithm, and the time-of-flow $ T = |t-t_0| $ is used to time-average this quantity so that it's the average exponential rate over the time period considered. Note that there is absolutely no mention here of anything being a flow barrier, or a "coherent structure." Thus, the FLTE identifies regions in the time $ t_0 $ based on stretching rates over the time-of-flow (in this case, 24 hours in the past). When one time-averages the FTLE field, what exactly does that mean? Presumably (and this is not clearly stated), FTLE fields are generated for differing $ t_0 $s but the same time of flow (this is sometimes called time-windowing in the coherent structure community), but then are the authors averaging over $ t_0 $? (Give an explicit formula for the averaging, so that this is clear.) If so, they are taking scalar fields which are defined over different times $ t_0 $, each of which is associated with flow over a different time-window $ (t_0-T,t_0) $, and averaging them. If this is what is done, it needs to be explained clearly. But then, the interpretation of this needs to be carefully stated, since one is averaging over different initial times, and what one gets cannot be associated with a particular time instance (unlike one calculation of an FLTE field, which gives a field at time $ t_0 $). Of course, one might argue that calculationally the averaging tells us of how 24-hour motion calculated over several initial times (say 1 January to 31 January 2023) is averaged to give an "average exponential rate of motion in January 2023 when a time-scale of 24 hours is considered for the exponential rate". This would need to be explained, because it's quite awkward and hard to interpret.
4. The above point related to one aspect of time-parametrization: the $ t_0 $ in the field $ \sigma_{t_0}^t $. Another time-parametrization issue is the $ t $, and thus the time-of-flow. Everything in this paper has used a time-of-flow of 24 hours. Hence, everything is slaved to this time-scale -- the exponential rate is computed based on time-of-flow for this time-scale. This issue is buried in the paper, with the multitude of plots not mentioning this explicitly If a time-of-flow of 48 hours were chosen instead, how do things change? Basically, the calculation of an FTLE field explicitly picks out a time-scale, and this is not something which the authors have clarified. The results are explicitly associated with this time-scale, and no other. If the results are to be used in forecasting, why is this the correct time-scale? Or is this method robust to changing the time-scale? How does the time-scale interact with the time associated with the time-averaging as discussed in my previous point? Basically, the issue of TIME (initial plus time-of-flow in calculating the FTLE and the appropriate interpretation of the FTLE field, the times of computation chosen for averaging) is crucial, and needs to be carefully explained, interpreted, and robustness evaluated (if appropriate).
5. Returning to the definition of the FTLE mentioned earlier. It represents a field at time $ t_0 $, associated with the average exponential rate over flow from time $ t_0 $ to $ t $. Note that there is absolutely no mention here of anything being a flow barrier, or a "coherent structure." Yes, there are papers in the literature which seem to indicate such a connection, but the reality is that it is unjustified. The early papers in this area used STEADY toy models which had saddle points with one-dimensional stable and unstable manifolds emanating from them, and since these manifolds are associated with exponential decay rates, came up with the idea that FTLE ridges had something to do with stable and unstable manifolds. And these manifolds were flow barriers in some way. However, this argument does not hold water, since there are examples such as in Haller (Physica D, 2011) which show that the stable/unstable manifold interpretation sometimes fails even in infinite-time flow. (And, getting back to a previous point related to hyperbolic LCSs, the fact that repelling/attracting do not necessarily occur as expected are also shown.) Real data is much worse: it is unsteady, and finite-time. Finite-time aspects are awkward for inferring exponential rates of growth, since any function over a finite-time can be bounded by an exponential. Unsteadiness is yet another problem, because (even in the infinite-time context) saddle points generalize to hyperbolic trajectories, and their stable/unstable manifolds move around (another reason why time-averaging is questionable). Furthermore, it is not clear what an "FTLE ridge" is -- one never gets a genuinely one-dimensional curve which is well-defined, but rather gets regions of larger FTLE values. Balasuriya et al (J Fluid Mech, 2016) provide an assessment and interpretation of what the FTLE means, with an emphasis on fluid motion, which helps understand these issues. In particular, for finite-time, unsteady data sets, using FTLEs and their ridges cannot easily reach conclusions regarding flow barriers and coherence. FTLEs explicitly look at exponential growth rates with respect to the time-of-flow considered, and that's about it. So when one takes FTLE fields, as done in this paper, and tries to reach conclusions regarding coherent structures such as eddies (as the authors do towards the end of the paper), this needs to be treated with suspicion, because it is not on any firm scientific grounds. The idea that eddies can be demarcated by FTLE ridges -- notwithstanding my earlier comments on finite-time and unsteadiness -- may go back to work in the 1990s (such as del-Castillo-Negrete, Knobloch, Pierrehumbert) who analyzed perturbed toy models. In these cases, the unperturbed models were explicit and steady, and had saddle points with stable and unstable manifolds. In some cases, the geometry was such that these manifolds encircled an eddy. Since the manifolds many be discoverable using FTLE ridges (again subject to various caveats), in such cases ONLY, one might think of an eddy as being found using FTLE fields. However the interior of the eddy does NOT typically contain exponential stretching, and hence FTLE fields by themselves cannot be used to reliably identify eddies as the authors here are doing in their later figures. Actually, in the standard fluid-mechanical dichotomy between stretching and rotation, the eddies have the opposite of stretching, and thus the FTLE is exactly the wrong thing to use (Okubo-Weiss and related criteria may help; however as noted by many authors unsteadiness is a problem in using such Eulerian characteristics). Basically, statements such as "two additional regions are identified and considered as robust in Fig.7 " (line 211 in the paper) are, in this vein, questionable. Indeed, high FTLE values indicate greater separation, and hence LESS certainty!
6. The impression given throughout is that the authors are examining robustness of LCSs. I've already talked about why it's actually FLTEs and not LCSs, but in this point I want to question whether robustness is the right thing. There are many recent papers which examine robustness of FTLE fields (Balasuriya, J Comp Dyn, 2020; Guo et al, IEEE Trans Visual Comp Graphics, 2016; Raben, Exp Fluids, 2014), but this paper is not one of them. (There are a few other papers, from oceanographic situations, which the authors speak to in lines 339-344.) By "robustness," the authors seem to mean smearing over small time scale motion, in other words looking for entities which persist over longer times. This needs to be made clear throughout the manuscript. (I've talked about the time-parametrization issue previously; to smear over smaller time scales, one needs to simply choose appropriate times for the FTLE which are relevant to what one is looking for.) In Section 4.3, also, the word "uncertainty" is questionable because of this same reason -- the authors have no calculated any uncertainty (i.e., have not evaluated anything to do with uncertainty in the input data).
7. There are several statements around Equation (2) and (3) which are incorrect. The statements in line 101 are all incorrect: $ \delta x $ is not the final location, but $ x $ is, and the (1,1) term in Equation (2) then represents the partial derivatives of $ x $ with respect to $ x_0 $, for example. The integral bounds in (3) are unclear and inconsistent with the previous equation. Presumably the authors mean something like $$ x(t) = x_0 + \int_{t_0}^t u \left( x(\tau), t(\tau) \right) \mathrm{d} \tau $$, and then one needs to also clarify that $ x(\tau),y(\tau) $ are the evolving trajectory locations. However, this integral formulation may not be the most natural. The differential form with $ \dot{x} = u \left( x(t), y(t) \right) $, $ \dot{y} = v \left( x(t), y(t) \right) $ and the initial condition $ \left( x(t_0), y(t_0) \right) = \left( x_0, y_0 \right) $ connects better with the discussion, perhaps.
8. The discussion at the end of Section 2.3 is fraught. What is a "2D curve" [line 126]? The statement that "averaging smooths the ridges into fields of attraction/repulsion" [line 128] is incorrect because, as mentioned previously, claiming that FTLEs have anything to do with attraction/repulsion is questionable. "Making these ridges more certain" [line 130] relates to an earlier comment that what is being done here has nothing to do with robustness or certainty. The authors comment that the time- and ensemble-averaged FTLE fields are not transport barriers (which is correct -- but neither is the FTLE field -- and again it's not clear to me how a field can be a barrier), but then talk about things being barriers over larger regions. Figure 2 is strange. One does not usually get FTLE ridges (even in idealized steady toy models in 2D) which intersect and pile on each other like this. Intersections in such cases occur at saddle points. If unsteady, intersections that one gets, analogous to intersections between stable and unstable manifolds, can relate to chaotic motion -- but these are between forward-time FTLE ridges and backwards-time FTLE ridges, rather than self-intersections within one of these.
9. The fact that a large standard derivation of the time-averaged FTLE usually is close to where the FTLE is large [line 169-171] is no surprise. Large FTLE relates to larger uncertainty in the results, because any errors (based on interpolation to a grid, say) increase exponentially. Hence, the values one assigns to the FTLE tend to be less certain. This issue is well-known, and described in some of the papers I've mentioned previously on robustness of FTLEs.
10. For the discussion on Section 3.4, I can't quite understand what the $ k $ in the figures is. Since in 2D, one has two wavenumbers -- say $ l $ and $ m $, associated with the eastward and northwards coordinates. I don't understand the discussion [lines 230 onwards] about averaging over rows and columns (over $ l $ and $ m $?). Is $ k = \sqrt{l^2 + m^2} $? The spectral plots in Figure 9 are used to infer robustness in some way, based on the fact that one gets decay with $ k $ in the bottom figures, say. Any smoothening process will of course get rid of the smaller wavenumbers. Similarly one expects more smoothness when averaged over more and more days.
Citation: https://doi.org/10.5194/egusphere-2024-1171-RC1 - AC1: 'Reply on RC1', Mateusz Matuszak, 22 Jul 2024
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RC2: 'Comment on egusphere-2024-1171', Anonymous Referee #2, 24 Jun 2024
- AC2: 'Reply on RC2', Mateusz Matuszak, 22 Jul 2024
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Model code and software
mateuszmatu/LCS: FTLE computation software release for article Mateusz Matuszak https://doi.org/10.5281/zenodo.10797134
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