# An Optimized-Parameter Spectral Clustering Approach to Coherent Structure Detection in Geophysical Flows

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Conventional Spectral Clustering Method

#### 2.2. Challenges of Conventional Spectral Clustering

#### 2.3. Improved Optimized-Parameter Spectral Clustering

#### 2.4. Noise-Based Cluster Coherence Metric

#### 2.5. Optimized-Parameter Spectral Clustering Algorithm Summary

- r-sweep: compute the normalized eigengap for variable sparsification radii r, from ${r}_{min}=10$× (trajectory grid spacing) to ${r}_{max}=$ (the domain size), keeping the fixed offset value ($w=max\left({w}_{ij}\right)$×${10}^{7}$ worked well in all examples), and identify all local maxima. This choice of the upper limit, ${r}_{max}$, is the most inclusive as it allows detecting clusters that are as large as the entire domain. This choice of the lower limit, ${r}_{min}$, ensures that clusters contain at least 10 nodes (to avoid detecting of very small clusters that might be numerically unstable.)
- w-sweep: for all local maxima, verify the convergence in the normalized eigengap by varying $w=max\left({w}_{ij}\right)$ × ${10}^{n}$, $n=[1,\dots ,10]$. In all our examples, convergence was observed at $n=7$, but we cannot guarantee that this is true for all flows. If the normalized eigengap still varies between $n=9$ and $n=10$, increase n past the value of 10 to ensure convergence.
- identify coherent sets corresponding to all local eigengap maxima and compute noise-based coherence metrics for the resulting clusters.

#### 2.6. Finite-Time Lyapunov Exponents and Poincaré Sections

## 3. Results

#### 3.1. Bickley Jet

#### 3.2. Asymmetric Duffing Oscillator

#### 3.3. Geophysical Example: Flow Around No Man’s Land Island

#### 3.3.1. 2017 Experiment

#### 3.3.2. 2018 Experiment

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FTLE | Finite-Time Lyapunov Exponent |

FCM | Fuzzy C-Means |

LCS | Lagrangian Coherent Structure |

## Appendix A. Verification of Parameter Convergence

#### Appendix A.1. Bickley Jet

**Figure A1.**(

**a**) Step 1 of the spectral clustering protocol for the Bickley Jet example: normalized eigengap as a function of r with the average distance function. (

**b**) Step 2 of the spectral clustering protocol for the Bickley Jet: sweep of offset coefficients ${10}^{n}$ for the gap ratio peaks in (

**a**) at $r=0.90$, $r=1.25$ and $r=2.00$ with the average distance function.

#### Appendix A.2. Asymmetric Duffing Oscillator

**Figure A2.**Steps 1 and 2 of the spectral clustering protocol for the asymmetric Duffing oscillator. (

**a**) Sweep of r parameters with offset coefficient ${10}^{7}$ for the average distance function. (

**b**) Sweep of offset coefficients ${10}^{n}$ for average distance function and the normalized eigengap peak at $r=1.0$.

#### Appendix A.3. 2017 No Man’s Land Experiment

**Figure A3.**Steps 1 and 2 of the spectral clustering protocol for the 2017 No Man’s Land experiment for the 15:51–21:51 window. (

**a**) Sweep of r parameters with offset coefficient ${10}^{7}$ for the average distance function. (

**b**) Sweep of offset coefficients ${10}^{n}$ for average distance function and the normalized eigengap peaks in (

**a**).

#### Appendix A.4. 2018 No Man’s Land Experiments

**Figure A4.**Steps 1 and 2 of the spectral clustering protocol for the 2018 No Man’s Land 16:00–22:00 experiment. (

**a**) Sweep of r parameters with offset coefficient ${10}^{7}$ for the average distance function. (

**b**) Sweep of offset coefficients ${10}^{n}$ for average distance function and the normalized eigengap peaks from (

**a**).

**Figure A5.**Steps 1 and 2 of the spectral clustering protocol for the 2018 No Man’s Land 20:00–02:00 experiment. (

**a**) Sweep of r parameters with offset coefficient ${10}^{7}$ for the average distance function. (

**b**) Sweep of offset coefficients ${10}^{n}$ for average distance function and the normalized eigengap peaks from (

**a**).

**Figure A6.**Steps 1 and 2 of the spectral clustering protocol for the 2018 No Man’s Land 04:00–10:00 experiment. (

**a**) Sweep of r parameters with offset coefficient ${10}^{7}$ for the average distance function. (

**b**) Sweep of offset coefficients ${10}^{n}$ for average distance function and the normalized eigengap peaks from (

**a**).

## Appendix B. Duffing Oscillator with Increased Asymmetry

**Figure A7.**Poincaré map for a second example of the asymmetric Duffing oscillator with 100 periods of perturbation T${}_{pert}$.

**Figure A8.**Step 1 of the spectral clustering protocol for the asymmetric Duffing oscillator shown in Figure A7: sweep of r parameters with offset coefficient ${10}^{7}$ for the average distance function.

**Figure A9.**Step 3 of the spectral clustering protocol for the asymmetric Duffing oscillator shown in Figure A7 for (

**a**) $r=1.125$, (

**b**) $r=2.0$ and (

**c**) $r=3.75$. All individual clusters are color-coded by their coherence metrics.

## Appendix C. Comparisons between Real and Simulated Drifter Trajectories

#### Appendix C.1. 2017 No Man’s Land Experiment

**Figure A10.**Comparison between real and simulated drifter trajectories for the 2017 No Man’s Land experiment. The trajectories of CODE drifters are plotted with thick lines. The corresponding numerical trajectories, simulated with the same initial conditions as their corresponding CODE drifter, are plotted with thin lines.

#### Appendix C.2. 2018 No Man’s Land Experiments

**Figure A11.**Comparison between real and simulated drifter trajectories for the 2018 No Man’s Land experiment between 16:00 and 22:00 UTC. The trajectories of CODE drifters are plotted with thick lines. The corresponding numerical trajectories, simulated with the same initial conditions as their corresponding CODE drifter, are plotted with thin lines.

**Figure A12.**Comparison between real and simulated drifter trajectories for the 2018 No Man’s Land experiment between 20:00 and 02:00 UTC. The trajectories of CODE drifters are plotted with thick lines. The corresponding numerical trajectories, simulated with the same initial conditions as their corresponding CODE drifter, are plotted with thin lines.

**Figure A13.**Comparison between real and simulated drifter trajectories for the 2018 No Man’s Land experiment between 04:00 and 10:00 UTC. The trajectories of CODE drifters are plotted with thick lines. The corresponding numerical trajectories, simulated with the same initial conditions as their corresponding CODE drifter, are plotted with thin lines.

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**Figure 1.**Example spectral clustering results for the Bickley Jet with parameters identical to those from HA16. The originally chosen value was $r=3.0$ in panel (

**b**). Three other values of r yielded a higher eigengap, however, as seen in panels (

**a**,

**c**,

**d**). Moreover, to follow the rule of thumb of 5–10% sparsification, $r=4.5$ in panel (

**d**) should be chosen instead. This value fails to detect the individual vortices, instead grouping them in pairs. It does, however, detect the meandering jet as an individual coherent structure. Looking at the maximum eigengap, the value of $r=1.5$ in panel (

**c**) should be chosen.

**Figure 2.**(

**a**) Poincaré map for the periodic Bickley Jet flow, computed over 1000 perturbation periods. (

**b**) Forward- and (

**c**) Backward-FTLE field, computed over 30 peturbation periods.

**Figure 3.**Coherent clusters color-coded by their coherence metrics resulting from the optimized-parameter spectral clustering for the Bickley Jet flow. The initial (${t}_{0}$—

**left**) and final (${t}_{f}$—

**right**) positions are shown.

**Figure 4.**The asymmetric Duffing oscillator. (

**a**) Poincaré map with 20 periods of perturbation T${}_{pert}$. (

**b**) Forward- and (

**c**) Backward- FTLE for 10T${}_{pert}$.

**Figure 5.**Coherent clusters, color-coded by their coherence metrics, resulting from the optimized-parameter spectral clustering for the asymmetric Duffing oscillator flow. The initial (${t}_{0}$—

**left**) and final (${t}_{f}$—

**right**) positions are shown. For comparison, black curves show the FTLE ridges in forward-time at ${t}_{0}$ and in backward time at ${t}_{f}$. The spectral clustering was done for 30T${}_{pert}$; FTLE ridges were computed for 10T${}_{pert}$.

**Figure 6.**Bathymetry of the 200-m resolution model domains around Martha’s Vineyard, extending south towards the continental shelf break. Depths in meters. Data from MSEAS. Large island is Martha’s Vineyard; small island near 70.82W and 41.25N is No Man’s Land.

**Figure 7.**LCS, numerical tracers and experimental drifter positions at the start (15:51 UTC) and end (21:51 UTC) of the 2017 experiment on August 14. (

**a**) Forward and (

**b**) backward FTLE. (

**c**,

**d**) Spectral clusters with coherence metric. The drifters in c-d are color-coded according to the cluster to which they belong, with red crosses if their final positions were outside of their initial cluster.

**Figure 8.**LCS, numerical tracers and experimental drifter positions at the start (16:00 UTC) and end (22:00 UTC) of the 2018 experiment on August 7. (

**a**) Forward and (

**b**) backward FTLE. (

**c**,

**d**) Spectral clusters with coherence metric. The drifters are color-coded according to the cluster to which they belong, with red crosses if their final positions were outside their initial cluster.

**Figure 9.**LCS, numerical tracers and drifter positions at the start (20:00 UTC) and end (02:00 UTC) of the 2018 experiment on August 7–8. (

**a**) Forward and (

**b**) backward FTLE. (

**c**,

**d**) Spectral clusters with coherence metric. The drifters are color-coded according to the cluster to which they belong, with red crosses if their final positions were outside their initial cluster.

**Figure 10.**LCS, numerical tracers and drifter positions at the start (04:00 UTC) and end (10:00 UTC) of the 2018 experiment on August 8. (

**a**) Forward and (

**b**) backward FTLE. (

**c**,

**d**) Spectral clusters with coherence metric. The drifters are color-coded according to the cluster to which they belong, with red crosses if their final positions were outside their initial cluster.

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**MDPI and ACS Style**

Filippi, M.; Rypina, I.I.; Hadjighasem, A.; Peacock, T.
An Optimized-Parameter Spectral Clustering Approach to Coherent Structure Detection in Geophysical Flows. *Fluids* **2021**, *6*, 39.
https://doi.org/10.3390/fluids6010039

**AMA Style**

Filippi M, Rypina II, Hadjighasem A, Peacock T.
An Optimized-Parameter Spectral Clustering Approach to Coherent Structure Detection in Geophysical Flows. *Fluids*. 2021; 6(1):39.
https://doi.org/10.3390/fluids6010039

**Chicago/Turabian Style**

Filippi, Margaux, Irina I. Rypina, Alireza Hadjighasem, and Thomas Peacock.
2021. "An Optimized-Parameter Spectral Clustering Approach to Coherent Structure Detection in Geophysical Flows" *Fluids* 6, no. 1: 39.
https://doi.org/10.3390/fluids6010039