Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T16:47:08.496Z Has data issue: false hasContentIssue false
  • English
  • Français

Detecting Lagrangian coherent structures from sparse and noisy trajectory data

Published online by Cambridge University Press:  06 September 2022

Saviz Mowlavi ,
Mattia Serra Open the ORCID record for Mattia Serra [Opens in a new window] ,
Enrico Maiorino  and
L. Mahadevan Open the ORCID record for L. Mahadevan [Opens in a new window]
Saviz Mowlavi
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Mattia Serra*
Affiliation:
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA Department of Physics, University of California, San Diego, CA 92093, USA
Enrico Maiorino
Affiliation:
Channing Division of Network Medicine, Brigham and Women's Hospital and Harvard Medical School, Boston, MA 02115, USA
L. Mahadevan*
Affiliation:
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA Department of Physics, Harvard University, Cambridge, MA 02138, USA
*
†Email addresses for correspondence: mserra@ucsd.edu, lmahadev@g.harvard.edu
†Email addresses for correspondence: mserra@ucsd.edu, lmahadev@g.harvard.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Many complex flows such as those arising from the collective motion of ocean plastics in geophysics or motile cells in biology are characterized by sparse and noisy trajectory datasets. We introduce techniques for identifying Lagrangian coherent structures (LCSs) of hyperbolic and elliptic nature in such datasets. Hyperbolic LCSs, which represent surfaces with maximal attraction or repulsion over a finite amount of time, are computed through a regularized least-squares approximation of the flow map gradient. Elliptic LCSs, which identify regions of coherent motion such as vortices and jets, are extracted using DBSCAN – a popular data clustering algorithm – combined with a systematic parameter selection strategy. We deploy these methods on various benchmark analytical flows and real-life experimental datasets ranging from oceanography to biology and show that they yield accurate results, despite sparse and noisy data. We also provide a lightweight computational implementation of these techniques as a user-friendly and straightforward Python code.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Chaos Nonlinear Dynamical Systems: Pattern formation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 948 , 10 October 2022 , A4
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allshouse, M.R. & Peacock, T. 2015 Lagrangian based methods for coherent structure detection. Chaos 25 (9), 097617.CrossRefGoogle ScholarPubMed
Banisch, R. & Koltai, P. 2017 Understanding the geometry of transport: diffusion maps for Lagrangian trajectory data unravel coherent sets. Chaos 27 (3), 035804.CrossRefGoogle ScholarPubMed
Budišić, M. & Mezić, I. 2012 Geometry of the ergodic quotient reveals coherent structures in flows. Physica D 241 (15), 1255–1269.CrossRefGoogle Scholar
del Castillo-Negrete, D. & Morrison, P.J. 1993 Chaotic transport by rossby waves in shear flow. Phys. Fluids A: Fluid 5 (4), 948–965.CrossRefGoogle Scholar
Dombre, T., Frisch, U., Greene, J.M., Hénon, M., Mehr, A. & Soward, A.M. 1986 Chaotic streamlines in the abc flows. J. Fluid Mech. 167, 353–391.CrossRefGoogle Scholar
Ester, M., Kriegel, H.-P., Sander, J. & Xu, X. 1996 A density-based algorithm for discovering clusters in large spatial databases with noise. In Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (KDD-96). pp. 226–231. AIAA.Google Scholar
Everitt, B.S., Landau, S., Leese, M. & Stahl, D. 2011 Cluster Analysis. John Wiley.CrossRefGoogle Scholar
Falk, M.L. & Langer, J.S. 1998 Dynamics of viscoplastic deformation in amorphous solids. Phys. Rev. E 57 (6), 7192.CrossRefGoogle Scholar
Filippi, M., Rypina, I.I., Hadjighasem, A. & Peacock, T. 2021 An optimized-parameter spectral clustering approach to coherent structure detection in geophysical flows. Fluids 6 (1), 39.CrossRefGoogle Scholar
Fortunato, S. 2010 Community detection in graphs. Phys. Rep. 486 (3–5), 75–174.CrossRefGoogle Scholar
Froyland, G. 2013 An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems. Physica D 250, 1–19.CrossRefGoogle Scholar
Froyland, G. & Junge, O. 2015 On fast computation of finite-time coherent sets using radial basis functions. Chaos 25 (8), 087409.CrossRefGoogle ScholarPubMed
Froyland, G. & Padberg, K. 2009 Almost-invariant sets and invariant manifolds—connecting probabilistic and geometric descriptions of coherent structures in flows. Physica D 238 (16), 1507–1523.CrossRefGoogle Scholar
Froyland, G. & Padberg-Gehle, K. 2012 Finite-time entropy: a probabilistic approach for measuring nonlinear stretching. Physica D 241 (19), 1612–1628.CrossRefGoogle Scholar
Froyland, G. & Padberg-Gehle, K. 2015 A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data. Chaos 25 (8), 087406.CrossRefGoogle ScholarPubMed
Froyland, G., Rock, C.P. & Sakellariou, K. 2019 Sparse eigenbasis approximation: multiple feature extraction across spatiotemporal scales with application to coherent set identification. Commun. Nonlinear Sci. Numer. Simul. 77, 81–107.CrossRefGoogle Scholar
Froyland, G., Santitissadeekorn, N. & Monahan, A. 2010 Transport in time-dependent dynamical systems: finite-time coherent sets. Chaos 20 (4), 043116.CrossRefGoogle ScholarPubMed
Froyland, G., Stuart, R.M. & van Sebille, E. 2014 How well-connected is the surface of the global ocean? Chaos 24 (3), 033126.CrossRefGoogle ScholarPubMed
Getscher, T. 2021 Observing and quantifying kinematic properties and Lagrangian coherent structures of ocean flows using drifter experiments. MSc thesis, Massachusetts Institute of Technology.Google Scholar
Hadjighasem, A., Farazmand, M., Blazevski, D., Froyland, G. & Haller, G. 2017 A critical comparison of Lagrangian methods for coherent structure detection. Chaos 27 (5), 053104.CrossRefGoogle ScholarPubMed
Hadjighasem, A., Karrasch, D., Teramoto, H. & Haller, G. 2016 Spectral-clustering approach to Lagrangian vortex detection. Phys. Rev. E 93 (6), 063107.CrossRefGoogle ScholarPubMed
Haller, G. 2001 Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149 (4), 248–277.CrossRefGoogle Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137–162.CrossRefGoogle Scholar
Haller, G., Aksamit, N. & Encinas-Bartos, A.P. 2021 Quasi-objective coherent structure diagnostics from single trajectories. Chaos 31 (4), 043131.CrossRefGoogle ScholarPubMed
Haller, G. & Yuan, G. 2000 Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147 (3–4), 352–370.CrossRefGoogle Scholar
Hamilton, P., Bower, A., Furey, H., Leben, R. & Pérez-Brunius, P. 2016 Deep circulation in the Gulf of Mexico: a Lagrangian study. Tech. Rep. OCS Study BOEM 2016-081, p. 289. Bureau of Ocean Energy Management.Google Scholar
Hogan, B.L.M. 1999 Morphogenesis. Cell 96 (2), 225–233.CrossRefGoogle ScholarPubMed
Kaipio, J. & Somersalo, E. 2006 Statistical and Computational Inverse Problems, vol. 160. Springer.Google Scholar
Kelley, D.H. & Ouellette, N.T. 2011 Separating stretching from folding in fluid mixing. Nat. Phys. 7 (6), 477–480.CrossRefGoogle Scholar
Lekien, F. & Ross, S.D. 2010 The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds. Chaos 20 (1), 017505.CrossRefGoogle ScholarPubMed
Lumpkin, R. & Pazos, M. 2007 Measuring surface currents with surface velocity program drifters: the instrument, its data, and some recent results. In Lagrangian Analysis and Prediction of Coastal and Ocean Dynamics, pp. 39–67. Cambridge University Press.CrossRefGoogle Scholar
von Luxburg, U. 2010 Clustering stability: an overview. Found. Trends Mach. Learn. 2 (3), 235–274.Google Scholar
Mancho, A.M., Wiggins, S., Curbelo, J. & Mendoza, C. 2013 Lagrangian descriptors: a method for revealing phase space structures of general time dependent dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18 (12), 3530–3557.CrossRefGoogle Scholar
Marchetti, M.C., Joanny, J.-F., Ramaswamy, S., Liverpool, T.B., Prost, J., Rao, M. & Simha, R.A. 2013 Hydrodynamics of soft active matter. Rev. Mod. Phys. 85 (3), 1143.CrossRefGoogle Scholar
Merzkirch, W. 2012 Flow Visualization. Elsevier.Google Scholar
Miron, P., Beron-Vera, F.J., Olascoaga, M.J., Froyland, G., Pérez-Brunius, P. & Sheinbaum, J. 2019 Lagrangian geography of the deep Gulf of Mexico. J. Phys. Oceanogr. 49 (1), 269–290.CrossRefGoogle Scholar
Morozov, A. 2017 From chaos to order in active fluids. Science 355 (6331), 1262–1263.CrossRefGoogle ScholarPubMed
Mundel, R., Fredj, E., Gildor, H. & Rom-Kedar, V. 2014 New Lagrangian diagnostics for characterizing fluid flow mixing. Phys. Fluids 26 (12), 126602.CrossRefGoogle Scholar
Nolan, P.J., Serra, M. & Ross, S.D. 2020 Finite-time Lyapunov exponents in the instantaneous limit and material transport. Nonlinear Dyn. 100 (4), 3825–3852.CrossRefGoogle Scholar
Padberg-Gehle, K. & Schneide, C. 2017 Network-based study of Lagrangian transport and mixing. Nonlinear Process. Geophys. 24 (4), 661–671.CrossRefGoogle Scholar
Pedregosa, F., et al. 2011 Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830.Google Scholar
Provenzale, A. 1999 Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31 (1), 55–93.CrossRefGoogle Scholar
Rozbicki, E., Chuai, M., Karjalainen, A.I., Song, F., Sang, H.M., Martin, R., Knölker, H.-J., MacDonald, M.P. & Weijer, C.J. 2015 Myosin-II-mediated cell shape changes and cell intercalation contribute to primitive streak formation. Nat. Cell Biol. 17 (4), 397–408.CrossRefGoogle ScholarPubMed
Rypina, I.I., Brown, M.G., Beron-Vera, F.J., Koçak, H., Olascoaga, M.J. & Udovydchenkov, I.A. 2007 On the Lagrangian dynamics of atmospheric zonal jets and the permeability of the stratospheric polar vortex. J. Atmos. Sci. 64 (10), 3595–3610.CrossRefGoogle Scholar
Rypina, I.I., Getscher, T.R., Pratt, L.J. & Mourre, B. 2021 Observing and quantifying ocean flow properties using drifters with drogues at different depths. J. Phys. Oceanogr. 51 (8), 2463–2482.Google Scholar
Rypina, I.I., Llewellyn Smith, S.G. & Pratt, L.J. 2018 Connection between encounter volume and diffusivity in geophysical flows. Nonlinear Process. Geophys. 25 (2), 267–278.CrossRefGoogle Scholar
Rypina, I.I. & Pratt, L.J. 2017 Trajectory encounter volume as a diagnostic of mixing potential in fluid flows. Nonlinear Process. Geophys. 24 (2), 189–202.CrossRefGoogle Scholar
Rypina, I.I, Scott, S.E., Pratt, L.J. & Brown, M.G. 2011 Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures. Nonlinear Process. Geophys. 18 (6), 977–987.CrossRefGoogle Scholar
Sander, J., Ester, M., Kriegel, H.-P. & Xu, X. 1998 Density-based clustering in spatial databases: the algorithm GDBSCAN and its applications. Data Min. Knowl. Disc. 2 (2), 169–194.CrossRefGoogle Scholar
Schall, P., Weitz, D.A. & Spaepen, F. 2007 Structural rearrangements that govern flow in colloidal glasses. Science 318 (5858), 1895–1899.CrossRefGoogle ScholarPubMed
Schlueter-Kuck, K.L. & Dabiri, J.O. 2017 a Coherent structure colouring: identification of coherent structures from sparse data using graph theory. J. Fluid Mech. 811, 468–486.CrossRefGoogle Scholar
Schlueter-Kuck, K.L. & Dabiri, J.O. 2017 b Identification of individual coherent sets associated with flow trajectories using coherent structure coloring. Chaos 27 (9), 091101.CrossRefGoogle ScholarPubMed
Schneide, C., Pandey, A., Padberg-Gehle, K. & Schumacher, J. 2018 Probing turbulent superstructures in Rayleigh–Bénard convection by Lagrangian trajectory clusters. Phys. Rev. Fluids 3 (11), 113501.CrossRefGoogle Scholar
Schubert, E., Sander, J., Ester, M., Kriegel, H.P. & Xu, X. 2017 DBSCAN revisited, revisited: why and how you should (still) use DBSCAN. ACM Trans. Database Syst. 42 (3), 1–21.CrossRefGoogle Scholar
Ser-Giacomi, E., Rossi, V., López, C. & Hernández-García, E. 2015 Flow networks: a characterization of geophysical fluid transport. Chaos 25 (3), 036404.CrossRefGoogle ScholarPubMed
Serra, M. & Haller, G. 2016 Objective Eulerian coherent structures. Chaos 26 (5), 053110.CrossRefGoogle ScholarPubMed
Serra, M., Sathe, P., Beron-Vera, F. & Haller, G. 2017 Uncovering the edge of the polar vortex. J. Atmos. Sci. 74 (11), 3871–3885.CrossRefGoogle Scholar
Serra, M., Sathe, P., Rypina, I., Kirincich, A., Ross, S.D., Lermusiaux, P., Allen, A., Peacock, T. & Haller, G. 2020 a Search and rescue at sea aided by hidden flow structures. Nat. Commun. 11 (1), 2525.CrossRefGoogle ScholarPubMed
Serra, M., Streichan, S., Chuai, M., Weijer, C.J. & Mahadevan, L. 2020 b Dynamic morphoskeletons in development. Proc. Natl Acad. Sci. USA 117 (21), 11444–11449.CrossRefGoogle ScholarPubMed
Shadden, S.C. 2012 Lagrangian coherent structures. In Transport and Mixing in Laminar Flows: From Microfluidics to Oceanic Currents, pp. 59–89.Google Scholar
Shadden, S.C., Lekien, F. & Marsden, J.E. 2005 Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212 (3–4), 271–304.CrossRefGoogle Scholar
Stern, C.D. 2004 Gastrulation: From Cells to Embryo. CSHL.Google Scholar
Tallapragada, P. & Ross, S.D. 2013 A set oriented definition of finite-time Lyapunov exponents and coherent sets. Commun. Nonlinear Sci. Numer. Simul. 18 (5), 1106–1126.CrossRefGoogle Scholar
Vieira, G.S., Rypina, I.I. & Allshouse, M.R. 2020 Uncertainty quantification of trajectory clustering applied to ocean ensemble forecasts. Fluids 5 (4), 184.CrossRefGoogle Scholar
Wichmann, D., Kehl, C., Dijkstra, H.A. & van Sebille, E. 2020 Detecting flow features in scarce trajectory data using networks derived from symbolic itineraries: an application to surface drifters in the North Atlantic. Nonlinear Process. Geophys. 27 (4), 501–518.CrossRefGoogle Scholar
Wichmann, D., Kehl, C., Dijkstra, H.A. & van Sebille, E. 2021 Ordering of trajectories reveals hierarchical finite-time coherent sets in Lagrangian particle data: detecting agulhas rings in the South Atlantic Ocean. Nonlinear Process. Geophys. 28 (1), 43–59.CrossRefGoogle Scholar
Wiggins, S. 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer.Google Scholar
Williams, M.O., Rypina, I.I. & Rowley, C.W. 2015 Identifying finite-time coherent sets from limited quantities of Lagrangian data. Chaos 25 (8), 087408.CrossRefGoogle ScholarPubMed

Mowlavi et al. Supplementary Movie 1

Temporal evolution of particle positions in the Bickley jet, colored according to the coherent groups identified in Figure 5(e). Top: instantaneous positions of all particles. Bottom: trajectories up to current time of one particle in each coherent group, and one particle classified as noise.
Download Mowlavi et al. Supplementary Movie 1(Video)
Video 22.5 MB

Mowlavi et al. Supplementary Movie 2

Temporal evolution of particle positions in the ABC flow, colored according to the coherent groups identified in Figure 7(c). Left: instantaneous positions of all particles. Right: trajectories up to current time of one particle in the coherent red, orange and brown groups, and one particle classified as noise.
Download Mowlavi et al. Supplementary Movie 2(Video)
Video 10.2 MB