the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 22 Jan 2026
Sandy beaches' chaos: shoreline-sandbar coupling inferred from observational time series
Abstract. Sandy shoreline–sandbar systems exhibit complex variability arising from the interplay between hydrodynamic forcing and intrinsic morphological feedbacks. Using long-term satellite-derived shoreline and sandbar observations, we applied global polynomial modeling to reconstruct low-dimensional deterministic dynamics for four contrasting coastal sites. The resulting autonomous models reproduce key morphodynamic features, including self-sustained shoreline oscillations, shoreline–sandbar coupling, and intermittent transitions between quasi-stable configurations. Nonlinear stability analyses reveal that these systems behave as chaotic oscillators, characterized by locally divergent yet globally bounded trajectories. Energetic episodes correspond to rapid shoreline–sandbar exchanges, whereas long low-energy states reflect stable attractor confinement. Together, these results demonstrate that sandy coasts are governed by deterministic but chaotic dynamics, in which internal coupling and self-organization control both variability and finite predictability. The proposed framework offers a physically consistent and data-driven approach to characterize and compare coastal morphodynamics within a unified nonlinear dynamical perspective.
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Notice on discussion status
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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Preprint
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Supplement
(1070 KB)
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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.
- Preprint
(6904 KB) - Metadata XML
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Supplement
(1070 KB) - BibTeX
- EndNote
- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2026-154', Anonymous Referee #1, 03 Feb 2026
- AC1: 'Response to RC1 & RC2', marius Aparicio, 24 Mar 2026
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RC2: 'Comment on egusphere-2026-154', Caterina Mosto, 05 Mar 2026
This manuscript investigates shoreline–sandbar coupling using long satellite-derived time series and the Global Polynomial Modeling (GPoM) framework to reconstruct dynamical equations from the data. The work addresses an interesting question and proposes a dynamical-systems perspective on coastal morphodynamics that stimulates further discussion in the field.
The remarks below aim to complement the comments already posted by Reviewer 1.
- Methodological context
It could be useful to situate the Global Polynomial Modeling approach within the broader set of methods that aim to infer dynamical equations directly from data. In recent years, several equation-discovery approaches have been proposed in the dynamical systems community, including Sparse Identification of Nonlinear Dynamics (SINDy) and related sparse regression frameworks (e.g. Brunton et al., 2016; Rudy et al., 2017). A short mention of these approaches may help place the present work within the wider methodological landscape of data-driven dynamical modeling.
- Topological approaches and references
In the section discussing topological analysis methods, the manuscript currently cites Charó et al., (2021). However, the reference that appears most relevant in the context of deterministic dynamical systems and homology-based analysis is Charó et al., (2022), whereas the 2021 paper concerns stochastic differential equations and addresses a different framework (random attractors).
In the same paragraph, the text can lead to the conclusion that the color tracer mapping method enables analyses that are not accessible through other topological approaches, while the so-called templex theory (Sciamarella & Charó, 2024) has addressed higher-dimensional attractors as well as toroidal chaos (e.g. Mosto et al., 2024).
- Relation to simplified morphodynamic models
Since the reconstructed systems are explicit autonomous ODE models, it may be useful to clarify how their dynamical structure relates to the simplified dynamical models historically proposed to describe shoreline–sandbar systems in coastal morphodynamics. In particular, a brief comparison between the phase-space or topological structure of the reconstructed systems and that of earlier conceptual models (e.g., Dean, 1991; Yates et al., 2009; Davidson et al., 2013) could help clarify what new dynamical insight is obtained from the data-driven reconstruction.
- Lyapunov exponents
Lyapunov exponents are reported as evidence of chaotic dynamics. It would be helpful to indicate the uncertainty associated with these estimates, for instance by providing confidence intervals or by discussing the sensitivity of the calculation to methodological choices (e.g., smoothing, embedding, or time-series length).
- Dimension of the reconstructed system
The manuscript reconstructs dynamical systems of dimension three or four directly from the time series. It would be useful to clarify how the dimensionality of the reconstructed system was determined. For instance, embedding diagnostics such as false nearest neighbours or related delay-embedding approaches are commonly used to estimate the minimal embedding dimension before fitting dynamical models. A brief discussion of this point would help assess whether the chosen model dimension is sufficient to represent the underlying dynamics.
Additional remarks
Figure 3: 3D graphics instead of 2D phase portraits might allow for a slightly better visual inspection of the geometric aspects.
Section 3.1.2: In connection with the remarks made by Reviewer 1 on the validation of the reconstructed dynamics, it may be worth recalling that different dynamical systems can share similar spectra while differing in geometry, invariant measures, or instability properties.
Figure 6: The bibliography corresponding to the fluence diagrams seems to be missing.
Line 545: The PCA performed to analyze the Poincaré section of Duck’s attractor would benefit from a brief explanation of how it is carried out in combination with the methodology used.
Line 548: Could the authors provide justification for the choice of the position of the Poincaré section(s)? What could happen if the Poincaré section was misplaced or had fewer components than necessary?
Line 569: Could the authors clarify why the bifurcation diagram at Ensenada cannot be reconstructed?
Line 608: The problem of overfitting is mentioned but not discussed in detail.
Typos
The references inserted in the text are, for the most part, not in chronological or alphabetical order when more than one work is cited.
Line 127: “≈” instead of “=”.
Line 269: extra space before “;”.
Line 570: “dynamics” instead of “dynamic”.
Figure 6 caption: missing space in “(Table 1)”.
Table 3: unlike the other tables, the entire first sentence appears in bold.
References:
Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences (PNAS), 113(15), 3932–3937. https://doi.org/10.1073/pnas.1517384113
D. Charó, C. Letellier, and D. Sciamarella, “Templex: A bridge between homologies and templates for chaotic attractors,” Chaos 32, 083108 (2022). https://doi.org/10.1063/5.0092933
Mosto C, Charó G. D., Letellier C., Sciamarella D. (2024). Templex-based dynamical units for a taxonomy of chaos. Chaos 34, 113111. https://doi.org/10.1063/5.0233160
Rudy, S. H., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2017). Data-driven discovery of partial differential equations. Science Advances, 3(4), e1602614. https://doi.org/10.1126/sciadv.1602614
Sciamarella and G. D. Charó, “New elements for a theory of chaos topology,” in Topological Methods for Delay and Ordinary Differential Equations, Advances in Mechanics and Mathematics, edited by P. Amster and P. Benevieri (Springer Birkhäuser, Cham, 2024). https://doi.org/10.1007/978-3-031-61337-1_9
Citation: https://doi.org/10.5194/egusphere-2026-154-RC2 - AC1: 'Response to RC1 & RC2', marius Aparicio, 24 Mar 2026
- AC1: 'Response to RC1 & RC2', marius Aparicio, 24 Mar 2026
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2026-154', Anonymous Referee #1, 03 Feb 2026
This manuscript presents a data-driven dynamical systems model of sandbar migration. The manuscript covers a lot of ground and substantial work was done behind the scenes. The figures are of very good quality, and the text (while a bit over-reliant on acronyms) is generally sound. I think a version of this manuscript could appear in NPG but the revisions I feel are necessary are conceptual, and so will take a bit of thought. They are eminently doable, and I look forward to reading a revised version. There are several issues to address
1. The conclusion that the model has been validated is based on spectra, when phase space plots rather clearly show a mismatch. At the very least language regarding strength of conclusions needs to be toned down considerably.
2. There is a lot of dynamical systems verbiage throughout (granted a lot of is standard). I think this should be balanced out by a discussion of the physical system. One example is “dissipation”. What are we talking about here turbulent dissipation in the overlying fluid? Grain on grain friction that leads to dissipation? I am guessing “neither”, and I think this needs to be clarified.
3. The fundamental structure of the model is as an unforced system. Much is made of the theorems of dynamical systems theory for autonomous systems. But the the physical sandbar system has forcing and explicitly non-autonomous terms (e.g. a strong storm passing through is likely to shift the sandbar a lot). This requires clear discussion.
4. The sandbars that are modelled are not measured but are inferred. I am a theorist/numericist but I want to note that the comment of a satellite data expert would be very helpful on the uncertainty in the inference.
Perhaps a suitable way to conclude is to ask: If I have a site for which climate change leads to an increase in storm incidence and hence sandbar movement, how can the model account for this? By refitting? Is there a way to assert standard causality or for intuition based on science? Indeed what is the role for physics in this modelling exercise?
Detailed comments:
1. The Introduction overplays the role of strongly chaotic models; most fluid physicists do not rely on things like the Lorenz model to understand geophysical fluid mechanics.
2. In the Introduction the only “field” examples quote are self-citation. Can the work of others be included, or is this so novel only the authors have done it? If it is the latter, say so explicitly.
3. The term validation is a bit of a gimmick, since no actual validation in an engineering sense is done. The model does as well as it can (and this would be considered “not very good” in most engineering applications). The real questions are I) What is the accuracy limit of these methods (e.g. would a higher order polynomial fit do better?), II) what would make a better model? In physics based models this is usually thought of as either a data problem, or a parametrization problem. I am not sure what it means here.
Citation: https://doi.org/10.5194/egusphere-2026-154-RC1 - AC1: 'Response to RC1 & RC2', marius Aparicio, 24 Mar 2026
-
RC2: 'Comment on egusphere-2026-154', Caterina Mosto, 05 Mar 2026
This manuscript investigates shoreline–sandbar coupling using long satellite-derived time series and the Global Polynomial Modeling (GPoM) framework to reconstruct dynamical equations from the data. The work addresses an interesting question and proposes a dynamical-systems perspective on coastal morphodynamics that stimulates further discussion in the field.
The remarks below aim to complement the comments already posted by Reviewer 1.
- Methodological context
It could be useful to situate the Global Polynomial Modeling approach within the broader set of methods that aim to infer dynamical equations directly from data. In recent years, several equation-discovery approaches have been proposed in the dynamical systems community, including Sparse Identification of Nonlinear Dynamics (SINDy) and related sparse regression frameworks (e.g. Brunton et al., 2016; Rudy et al., 2017). A short mention of these approaches may help place the present work within the wider methodological landscape of data-driven dynamical modeling.
- Topological approaches and references
In the section discussing topological analysis methods, the manuscript currently cites Charó et al., (2021). However, the reference that appears most relevant in the context of deterministic dynamical systems and homology-based analysis is Charó et al., (2022), whereas the 2021 paper concerns stochastic differential equations and addresses a different framework (random attractors).
In the same paragraph, the text can lead to the conclusion that the color tracer mapping method enables analyses that are not accessible through other topological approaches, while the so-called templex theory (Sciamarella & Charó, 2024) has addressed higher-dimensional attractors as well as toroidal chaos (e.g. Mosto et al., 2024).
- Relation to simplified morphodynamic models
Since the reconstructed systems are explicit autonomous ODE models, it may be useful to clarify how their dynamical structure relates to the simplified dynamical models historically proposed to describe shoreline–sandbar systems in coastal morphodynamics. In particular, a brief comparison between the phase-space or topological structure of the reconstructed systems and that of earlier conceptual models (e.g., Dean, 1991; Yates et al., 2009; Davidson et al., 2013) could help clarify what new dynamical insight is obtained from the data-driven reconstruction.
- Lyapunov exponents
Lyapunov exponents are reported as evidence of chaotic dynamics. It would be helpful to indicate the uncertainty associated with these estimates, for instance by providing confidence intervals or by discussing the sensitivity of the calculation to methodological choices (e.g., smoothing, embedding, or time-series length).
- Dimension of the reconstructed system
The manuscript reconstructs dynamical systems of dimension three or four directly from the time series. It would be useful to clarify how the dimensionality of the reconstructed system was determined. For instance, embedding diagnostics such as false nearest neighbours or related delay-embedding approaches are commonly used to estimate the minimal embedding dimension before fitting dynamical models. A brief discussion of this point would help assess whether the chosen model dimension is sufficient to represent the underlying dynamics.
Additional remarks
Figure 3: 3D graphics instead of 2D phase portraits might allow for a slightly better visual inspection of the geometric aspects.
Section 3.1.2: In connection with the remarks made by Reviewer 1 on the validation of the reconstructed dynamics, it may be worth recalling that different dynamical systems can share similar spectra while differing in geometry, invariant measures, or instability properties.
Figure 6: The bibliography corresponding to the fluence diagrams seems to be missing.
Line 545: The PCA performed to analyze the Poincaré section of Duck’s attractor would benefit from a brief explanation of how it is carried out in combination with the methodology used.
Line 548: Could the authors provide justification for the choice of the position of the Poincaré section(s)? What could happen if the Poincaré section was misplaced or had fewer components than necessary?
Line 569: Could the authors clarify why the bifurcation diagram at Ensenada cannot be reconstructed?
Line 608: The problem of overfitting is mentioned but not discussed in detail.
Typos
The references inserted in the text are, for the most part, not in chronological or alphabetical order when more than one work is cited.
Line 127: “≈” instead of “=”.
Line 269: extra space before “;”.
Line 570: “dynamics” instead of “dynamic”.
Figure 6 caption: missing space in “(Table 1)”.
Table 3: unlike the other tables, the entire first sentence appears in bold.
References:
Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences (PNAS), 113(15), 3932–3937. https://doi.org/10.1073/pnas.1517384113
D. Charó, C. Letellier, and D. Sciamarella, “Templex: A bridge between homologies and templates for chaotic attractors,” Chaos 32, 083108 (2022). https://doi.org/10.1063/5.0092933
Mosto C, Charó G. D., Letellier C., Sciamarella D. (2024). Templex-based dynamical units for a taxonomy of chaos. Chaos 34, 113111. https://doi.org/10.1063/5.0233160
Rudy, S. H., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2017). Data-driven discovery of partial differential equations. Science Advances, 3(4), e1602614. https://doi.org/10.1126/sciadv.1602614
Sciamarella and G. D. Charó, “New elements for a theory of chaos topology,” in Topological Methods for Delay and Ordinary Differential Equations, Advances in Mechanics and Mathematics, edited by P. Amster and P. Benevieri (Springer Birkhäuser, Cham, 2024). https://doi.org/10.1007/978-3-031-61337-1_9
Citation: https://doi.org/10.5194/egusphere-2026-154-RC2 - AC1: 'Response to RC1 & RC2', marius Aparicio, 24 Mar 2026
- AC1: 'Response to RC1 & RC2', marius Aparicio, 24 Mar 2026
Peer review completion
Journal article(s) based on this preprint
Data sets
Satellite-derived shoreline and sandbar positions (six sites) Salomé Frugier and Marcan Graffin https://doi.org/10.5281/zenodo.18220531
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Sylvain Mangiarotti
Salomé Frugier
Laurent Lacaze
Marcan Graffin
Rafael Almar
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(6904 KB) - Metadata XML
-
Supplement
(1070 KB) - BibTeX
- EndNote
- Final revised paper
This manuscript presents a data-driven dynamical systems model of sandbar migration. The manuscript covers a lot of ground and substantial work was done behind the scenes. The figures are of very good quality, and the text (while a bit over-reliant on acronyms) is generally sound. I think a version of this manuscript could appear in NPG but the revisions I feel are necessary are conceptual, and so will take a bit of thought. They are eminently doable, and I look forward to reading a revised version. There are several issues to address
1. The conclusion that the model has been validated is based on spectra, when phase space plots rather clearly show a mismatch. At the very least language regarding strength of conclusions needs to be toned down considerably.
2. There is a lot of dynamical systems verbiage throughout (granted a lot of is standard). I think this should be balanced out by a discussion of the physical system. One example is “dissipation”. What are we talking about here turbulent dissipation in the overlying fluid? Grain on grain friction that leads to dissipation? I am guessing “neither”, and I think this needs to be clarified.
3. The fundamental structure of the model is as an unforced system. Much is made of the theorems of dynamical systems theory for autonomous systems. But the the physical sandbar system has forcing and explicitly non-autonomous terms (e.g. a strong storm passing through is likely to shift the sandbar a lot). This requires clear discussion.
4. The sandbars that are modelled are not measured but are inferred. I am a theorist/numericist but I want to note that the comment of a satellite data expert would be very helpful on the uncertainty in the inference.
Perhaps a suitable way to conclude is to ask: If I have a site for which climate change leads to an increase in storm incidence and hence sandbar movement, how can the model account for this? By refitting? Is there a way to assert standard causality or for intuition based on science? Indeed what is the role for physics in this modelling exercise?
Detailed comments:
1. The Introduction overplays the role of strongly chaotic models; most fluid physicists do not rely on things like the Lorenz model to understand geophysical fluid mechanics.
2. In the Introduction the only “field” examples quote are self-citation. Can the work of others be included, or is this so novel only the authors have done it? If it is the latter, say so explicitly.
3. The term validation is a bit of a gimmick, since no actual validation in an engineering sense is done. The model does as well as it can (and this would be considered “not very good” in most engineering applications). The real questions are I) What is the accuracy limit of these methods (e.g. would a higher order polynomial fit do better?), II) what would make a better model? In physics based models this is usually thought of as either a data problem, or a parametrization problem. I am not sure what it means here.