Lagrangian descriptors in geophysical flows: a survey

SeMA Journal https://doi.org/10.1007/s40324-025-00382-y Lagrangian descriptors in geophysical flows: a survey Jezabel Curbelo1,2,3 Received: 8 September 2024 / Accepted: 18 February 2025 © The Author(s) 2025 Abstract This survey focuses on the application of Lagrangian descriptors to reveal the geometry of phase space structures that determine transport in dynamical systems. We present diverse formulations of the method and examine various applications of Lagrangian descriptors in geophysicalfluids,suchasatmosphericflowsandoceaniccurrents.ThemethodofLagrangian Descriptors has proven to be a powerful tool for characterizing transport and mixing in these contexts, demonstrating how these tools have enhanced our understanding of complex fluid dynamics in critical environments. Keywords Lagrangian descriptors ·Lagrangian coherent structures ·Phase space · Geophysical flows Mathematics Subject Classification 37-02 ·86-02 ·37N10 ·37C10 ·37C60 ·76F20 · 76F25 ·76U60 ·86A05 ·86A10 1 Introduction In dynamical systems theory, understanding the global and long-term behavior of complex systems often hinges on identifying the structures that organize their dynamics [1]. These dynamical structures, such as periodic orbits, invariant manifolds, homoclinic orbits, and invariant tori, serve as the backbone for constructing a comprehensive picture of the system’s evolution. By performing local analyses, such as linearization and the computation of normal forms around these structures, researchers can begin to piece together how they influence the broader dynamics. In fluid dynamics, a full understanding of any transitional fluid flow requires the identification and analysis of underlying Coherent Structures that govern the system’s behavior [2–5]. According to [6], coherent structures are large-scale, connected regions of fluid characterized by phase-correlated vorticity or other macroscopic quantities. These structures appear in a wide array of fluid flow problems, and their identification and accurate computation are BJezabel Curbelo [email protected] 1Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avda. Diagonal, 647, 08028 Barcelona, Catalunya, Spain 2IMTech, Institute of Mathematics of UPC-BarcelonaTech, 08028 Barcelona, Catalunya, Spain 3Centre de Recerca Matemàtica, Bellaterra, Spain 123 J. Curbelo crucial because they play a central role in the global transport of mass, heat, and momentum. Despite the inherent complexity of turbulent fluid flows, which often self-organize into coherent patterns of varying spatio-temporal complexity, there is a pressing question: Is it possible to accurately capture the complicated behavior of fluid flow using a finite number of coherent structures with known interaction rules? In the study of fluid dynamics, two fundamental approaches are used to describe the motion of fluid: the Eulerian and Lagrangian descriptions. The Eulerian perspective focuses on specific locations within the fluid domain, observing how the fluid properties change over time at these fixed points. This approach is often used in traditional fluid mechanics, where the velocity field and other quantities are measured as functions of position and time. In contrast, the Lagrangian description follows individual fluid particles as they move through space and time, tracking their trajectories and how their properties evolve. This perspective is particularly useful for understanding the transport and mixing of substances within the fluid, as it provides a direct view of how particles and tracers are advected by the flow. In large-scale geophysical flows, where advection-the transport of tracers by the fluid-is often the dominant process, the Lagrangian approach becomes essential for capturing the behavior of these systems. Geometric structures in phase space, as pioneered by Poincarè [7], have been crucial in characterizing the global behavior of dynamical systems. This perspective offers insights into the evolution of different trajectory classes and has also been highly informative for understanding fluid transport and mixing, especially given the formalequivalencebetweentheequationsforincompressiblefluidmotion(withoutmolecular diffusion) and Hamilton’s equations, where the stream function acts as the Hamiltonian, and the physical space corresponds to the phase space [8]. When advection is the primary driver, as is often the case in geophysical flows, Lagrangian Coherent Structures (LCSs) (concept introduced by [9]) become the critical finite-time structuresthatdictate thedeformationof thefluid and,consequently,theevolutionof anyadvective tracer fields. These structures include attracting/repelling hyperbolic LCSs, which are material lines that evolve with the flow, acting as regions that maximally attract/repulsion fluid. Around these LCSs, the fluid is stretched in one direction and compressed in the perpendicular direction, leading to the formation of filamentous tracer patterns. Therefore, LCSs serve as key material surfaces that shape global transport, acting as barriers that influence the movement and mixing of substances in the fluid. The identification and study of coherent structures, particularly LCSs, have become major challenges in geophysical fluid dynamics and are increasingly relevant in applied mathematics, physics, and engineering. This is especially true given their significant role in understanding and predicting natural phenomena [9–12] such as climate processes-like monsoons [13], atmospheric rivers [14], tropical cyclones[15], sudden stratospheric warmings [16–19], and ozone depletion [19–21]—and their importance in analyzing environmental disasters, including oil spills [22–27], transport pollution [27–29], volcanic eruptions [30, 31] os dust plumes [32]. This challenge has led to the development of powerful mathematical tools for identifying and analyzing these structures, which offer a more insightful view into the underlying dynamics of fluid flows. Several recent review articles [33–37] offer comprehensive descriptions and references to the multitude of methods proposed for LCS detection. Apart from Lagrangian Descriptors, other methods are Finite time Lyapunov exponents [38, 39], finite size Lyapunov exponents [40,41], clustering algorithms [42,43]; spectral colouring techniques [44], methods based on the idea of complexity of isolated trajectories[45], line integral convolution [46,47], Distinguished hyperbolic trajectories [48], encounter volumes [49], variational LCSs methods [50], Lagrangian averaged vorticity deviation (LAVD) [51], stochastic sensitivity [52], coherent set detection [53–55]. 123 Lagrangian descriptors in geophysical flows: a survey This review article provides a comprehensive overview of the Lagrangian Descriptor method for studying and detecting Lagrangian Coherent Structures (LCS). It does not engage in comparative analyses of various detection and analysis methods, as these have been thoroughly discussed in the existing literature (e.g., [37]). Instead, the focus is on surveying the literature on Lagrangian Descriptors, highlighting the diverse formulations of the methods (detailed in Sect.2) and emphasizing their extensive applications in geophysical flows Sect.3. Special attention is given to their use in atmospheric (Sect.3.1) and oceanic studies (Sect.3.2), demonstrating how these tools have enhanced our understanding of complex fluid dynamics in these critical environments. 2 Method The Lagrangian descriptors (LD) is a dynamical system tool which is able to highlight geometrical objects in phase portraits of dynamical systems with a general time dependence. The concept of Lagrangian descriptors was first introduced by [56], where it was applied to oceanic flows with general time dependence, using altimetric data sets. However, in the work by [57] the general methodology for constructing Lagrangian descriptors was fully presented. In that study, a “heuristic argument” was discussed, explaining the effectiveness of this method in revealing geometrical structures within the phase space of a dynamical system. It was demonstrated that Lagrangian descriptors could provide a comprehensive dynamical picture of geometric structures for flows with arbitrary time dependence. Notably, this tool is capable of identifying the primary “organizing centers” in the flow, such as hyperbolic trajectories along with their stable and unstable manifolds, as well as elliptic regions. In this section, we will review the precise meaning of the term “Lagrangian descriptor,” explore the heuristic justification for their effectiveness, and discuss the process of computing them along with the various theories developed around them. Neglecting diffusion, the motion of passive fluid particles is governed by the ordinary differential equation ˙ x=v(x,t). (1) In this equation, v(x,t)represents a sufficiently smooth, time-dependent velocity field, where x∈Xis the state, t∈Ris time, and X⊂Rn(n=2,3,...) is a compact subset representingthephysicaldomain.Lagrangian particletrajectoriesarethesolutionsx(t)ofthis differential equation, with coherence described by the behavior of groups of these Lagrangian trajectories. The Lagrangian descriptor or function Mis defined as the Euclidean arc length of a curve in phase space, representing the path of a trajectory governed by Eq. (1): M(x0,t0,τ)=t0+τ t0−τ v(x(t), t)dt,(2) where ·is the Euclidean norm, and (x1(t), x2(t),...,xn(t)) are the components of the trajectory x(t)in Rn. The trajectory starts from the point x0at time t=t0and is observed over the time interval [t0−τ,t0+τ]. Therefore, it is clear that M=M(x0,t0,τ), i.e., depends on the starting point (x0,t0)and the time interval given by τ. For example, in Fig.1we analyze the linear saddle which velocity field given by: ˙x=λx, ˙y=−λy,λ>0,(3) 123 J. Curbelo Fig. 1 (Left panels) Mmap for the Linear saddle point (Eq. (3)) with λ=1andτ=1andτ=10. The dark color represents short trajectories, while the light color represents long trajectories. If τis sufficiently large, the norm of the gradient ∇ Mhighlights the stable and unstable manifolds associated with the hyperbolic point (right panels) The flow generated by this velocity field is: x(t,x0)=x0eλt,(4) y(t,y0)=y0e−λt,λ>0.(5) where the origin, (x,y)=(0,0), is a hyperbolic fixed point with stable and unstable manifolds given by: Ws(0,0)={(x,y)∈R2|x=0,y= 0},(6) Wu(0,0)={(x,y)∈R2|y=0,x= 0}.(7) Figure1a shows that for small τ,Mis smooth, and for increasing τ(panel b), the plot displays the manifolds by means of large values of the ∇ M(right panel). At this point, it is helpful to explain why Lagrangian descriptors are effective for revealing geometric structures in the phase space of (1). Over the time interval of interest [t0−τ,t0+τ], trajectories with initial conditions that remain close throughout this period are expected to have similar Mvalues. However, at the boundaries between regions where trajectories behave differently, the arc-lengths of trajectories starting on either side of the boundary are expected to differ. These boundaries, or singular features, are indicated by abrupt changes, meaning the derivative of the Lagrangian descriptors across these boundaries is discontinuous. In a 123 Lagrangian descriptors in geophysical flows: a survey Fig. 2 Maps of M(a)and∇ M(b) at 600 K on January 29, 2008, using τ=20 days. The yellow areas in panel (a) correspond to initial conditions with long trajectories during the corresponding time interval, while dark blue areas correspond to short trajectories. In panel (b), the black lines indicate large values of ∇ M, highlighting the singular features of the function M, which approximates the locations of the manifolds hyperbolic region of phase space, these boundaries correspond to the stable and unstable manifolds of hyperbolic trajectories (see Fig.1). These features are easy to visualize when plotting the results of the method and are easy to highlight by considering the gradient of the Mfunction [16,58]. See an example in Fig.2,whichshowsamapofMand the norm of its gradient using τ=20 in an atmospheric dataset. Furthermore, there is a rigorous mathematical connection between these “singular features,” which arise from the discontinuity of the Lagrangian descriptors or its derivative, and the stable and unstable manifolds of hyperbolic points [57]. This connection was first established for two-dimensional flows in [59], extended to three-dimensional systems in [60] or for normally hyperbolic invariant manifolds in Hamiltonian systems with two or more degrees of freedom [61–63]. The (2) integral can be divided into two parts: M(x0,t0,τ) =Mback(x0,t0,τ)+Mfwd(x0,t0,τ) =t0 t0−τ v(x(t), t)dt +t0+τ t0 v(x(t), t)dt,(8) where the forward and backward time integration components can be used to detect the stable and unstable manifolds separately. The arc length of a trajectory segment, defined by (2), is calculated by integrating the Euclidean velocity valong the backward and forward trajectory over time. However, this approach can also be applied to other positive scalar functions that represent intrinsic physical or geometrical properties of trajectories, integrated along the trajectory over the time interval (t0−τ,t0+τ), with the only requirement being that these integrals are well-defined. Therefore, a general form of the Lagrangian Descriptors can be expressed as MF(x0,t0,τ)=t0+τ t0−τ |F(x(t), t)|dt,(9) where |F(x,t)|is a bounded, positive scalar function depending on x0,t0representing the physical or geometrical property of the velocity field of interest. For instance, Mancho et al. [57] presents some examples of bounded, positive scalar functions to define MF,such 123 J. Curbelo Table 1 Different formulations of the Lagrangian descriptors that exist in the literature Function F(x,t)Norm Some references Velocity: v(x,t)Euclidean norm [57] Velocity: v(x,t)p-norm or p-quasi norm [57,59] Acceleration: a(x,t)Euclidean norm [57] Time derivative of acceleration: (da/dt) Euclidean norm [57] Function of curvature Euclidean norm [57] Vorticity: ∇×v(x,t)Modulus [65] The arclength of a trajectory projected on the configuration space [66] as acceleration (a), the modulus of the time derivative of acceleration (da/dt), or the modulus of combinations of v,a,andda/dt. Another possibility in the generalization of this methods is consider other norm instead of the euclidean one, i.e. where the positive scalar function accumulated along the trajectories of the system is the p-norm of the vector field that determines the flow [57,59], i.e. Mp(x0,t0,τ)=t0+τ t0−τ n  i=1 |˙xi(t;x0)|pdt, where p∈(0,1]is a freely chosen parameter, and the overdot symbol represents the derivative with respect to time. A theoretical mathematical framework for this alternative definition is provided in [59]. Over the years, new formulations of the method have been developed to enhance the visualization of phase space structures using Lagrangian Descriptors (LD). For instance, some formulations in the literature consider varying values of τ(see [64]) (unlike the classical Lagrangiandescriptor (2)whereτisthesameforallinitial pointsx0), allowingtheintegration to stop once a trajectory exits a specific phase space region, revealing only the relevant structures. It is also possible to define Lagrangian descriptors based on the definition of curvature [57], which is an intrinsic property of curves: Mκ(x0,t0,τ)=t0+τ t0−τ 1 |κ|+εdt,(10) where ε>0 is introduced to avoid singularities, and the curvature is defined as a positive quantity that combines both velocity and acceleration, κ=(v·v)(a·a)−(v·a)2 (v·v)3/2. Some examples of |F(x,t)|, norms, and formulations are summarized in Table 1. To summarize, this methodology assigns a positive number Mto each initial condition in phase space. This number is computed by integrating a predefined positive function along the trajectory as the system evolves both forward and backward over a specified time interval. While the positive function used to define different types of Lagrangian Descriptors (LD) can have geometrical or physical significance, this is not essential for the method’s application [63]. 123 Lagrangian descriptors in geophysical flows: a survey And, in a more general case, Lagrangian descriptors encompass a family of methods based on averages along trajectories [37]: A(x0,t0,τ)=1 |t0−τ|τ t0 g(F(x(t), t)dt (11) where the A-field is an average along the Lagrangian trajectories x(t)that start at x0at time t0. The choice of the function gand the observable Fdetermines what exactly is being averaged. For example, F=vand g(x)=xrecover the classical lagrangian descriptor (2) but taking g(x)=x,A(x0,t0,τ)is the average lagrangian velocity [67]. Taking g◦F=|t0−τ||ω(x(t), t)−¯ω(t0)|,(12) where ωis the vorticity and ¯ωis its average over the entire domain at each time, therefore (11) becomes the Lagrangian Averaged Vorticity Deviation (LAVD) proposed by [51]. Alternative approaches involve integrating the instantaneous eigenvector field of the variational equation [68] or employing a set of functions to quantify a trajectory’s ergodicity defect [45]. The formulations presented in this section share the underlying principle that trajectories within coherent regions display consistent behavior, reflected in similar averaged values. In contrast, trajectories across different coherent regions exhibit distinct characteristics. The LD approach offers significant improvements over other techniques and has been effectively applied to geophysical flow transport, including oceanic and atmospheric dynamics, which will be reviewed in detail in the next two sections. Additionally, LDs have found applications in other areas, such as magnetohydrodynamics [69], transition state theory in chemistry [70–73], in the study of cardiovascular flows [74] or billiard dynamics [75]tonameafew. 2.1 Importance of the election of time integration  The choice of τ, the time interval over which the trajectory is integrated, plays a crucial role in the effectiveness and accuracy of Lagrangian Descriptors (LD). This parameter directly influences the resolution of the underlying dynamical structures and the ability to capture relevant features of the system. A well-chosen τallows the LD to effectively reveal stable and unstable manifolds, hyperbolic structures, and other important features of the phase space. If τis too small, the method may not capture enough of the trajectory’s evolution to reveal these structures, leading to an incomplete or blurred representation of the dynamics. Conversely, if τis too large, the integration may average out fine details, potentially missing local structures or introducing noise from unrelated dynamics. Increasing the τvalues adds more Lagrangian detail to the figures, revealing structures that may not be visible with smaller τ. However, these differences often involve filamentous structures that are difficult to follow and compare. Therefore, careful consideration of the choice of τbased on the specific dynamical system and the phenomena of interest is essential for the successful application of this method. For example, in geophysical flows like ocean currents or atmospheric dynamics, τmight need to match the time scales of relevant physical processes, such as the lifespan of a cyclone or the time it takes for a rotation around the Earth. In other fields, such as chemical reaction dynamics, τshould be chosen to reflect the time scale of the transition states of interest. Furthermore, the value of τalso affects the computational cost. Larger τvalues require longer integration times, increasing the computational load. Therefore, it is important to 123 J. Curbelo select τsuch that it balances the need for detailed resolution of phase space structures with the available computational resources. 2.2 Computation of Lagrangian descriptors in dataset The general procedure to compute Lagrangian descriptors utilizing any of the different formulations described in the Sect.2typically begins with calculating a scalar or vector field over the phase space, followed by the extraction of key features such as ridges (locally maximizing curves), valleys (locally minimizing curves), or curves with the highest gradient of the M-field. In specific cases, such as with the Lagrangian Averaged Vorticity Deviation (LAVD), this also includes identifying outermost convex closed contours [51,76]. Another approach could involves identifying coherent regions as areas characterized by plateaus of nearly constant Mvalues. Although the connection between these coherent structures and the ridges or plateaus of Mis not always rigorously defined, interesting correspondences have been observed in application and for various flow scenarios. While the task of computing Lagrangian descriptors across different formulations is relatively straightforward, tools like the open-source Python software package [77]have simplifiedtheanalysis.Thissoftwareoffersmodulestailoredforexaminingphasespacestructures in both continuous and discrete two-dimensional nonlinear systems, accommodating both deterministic and stochastic settings through the use of Lagrangian descriptors. Nevertheless, certain adaptations are necessary when extending these computations to three-dimensional systems or when applying them in geographic coordinate systems, as we will elaborate in the following sections. 2.2.1 Geographical coordinates The computation of the Lagrangian Descriptors (LD) function Min geographical coordinates (latitude, longitude, and height) for the analysis of 3D flow over the Earth requires a series of methodical steps. These steps ensure that the data is processed and interpolated correctly to account for the complexities of spherical coordinates andthe large datasets typically involved in geophysical studies. Below is an outline of the process that we have followed in previous studies [16,17,19,78,79], as described in [58], with additional details provided for clarity. •Step 1: Data acquisition, conversion and preparation. The initial step involves downloading and preparing the necessary data from the corresponding dataset. In general, to compute (2), we only need the velocity field. Fluid velocity data can come from model simulations (based on equations and physical models) and/or reanalysis products, which refer to datasets created by combining past observational data with numerical weather prediction (NWP) models to generate a consistent, comprehensive, and long-term record of atmospheric and climate conditions. For instance, in atmospheric studies, we could use datasets such as the Whole Atmosphere Community Climate Model (WACCM), developed at the National Center for Atmospheric Research (NCAR) and based on the Community Earth System Model (CESM) with state-of-the-art chemistry [80,81]. Other sources include ERA-5, the fifth-generation ECMWF atmospheric reanalysis of the global climate by the Copernicus Climate Change Service (C3S) [82], which provides comprehensive data on various atmospheric variables, and the NCEP/NCAR reanalysis [83], provided by the NOAA-OAR-ESRL PSL, Boulder, Colorado, USA, among many others. In some cases, data preprocessing is required before use because they are provided in 123 Lagrangian descriptors in geophysical flows: a survey different reference systems. For instance, ERA-5 data are provided on sigma levels and need to be converted to specified height levels for analysis. There are many datasets available, and it is impossible to explain the peculiarities of each one of them here. •Step 2: Vertical velocity. In geophysical flows,fully computing the evolving 3D velocity field is challenging. The vertical velocity, w, which is generally much smaller than the horizontal velocities, is often estimated as a diagnostic quantity rather than prognostically solved as part of the equations of motion like the horizontal velocity components [82,84]. Consequently, it is less reliable. It is therefore tempting to ignore win the computation of Lagrangian Coherent Structures, assuming that the flow motion occurs in 2D. In particular, in atmospheric flows, a widely accepted approximation assumes that parcels remain on isentropic surfaces during their trajectories [19,79]. This approximation is justified because, over the timescale of most atmospheric events (τ∼10 days), stratospheric flows are adiabatic and frictionless, meaning that fluid particles and their trajectories are constrained to remain on surfaces of constant potential temperature. Therefore, the data is preprocessed to use potential temperature θinstead of pressure pas the vertical coordinate. The potential temperature θis defined as the temperature that a parcel of dry air at pressure pand temperature Twould acquire if it were expanded or compressed adiabatically to the reference pressure ps=1000 hPa, i.e., θ=Tp psR/cp where Ris the gas constant for dry air, and cpis the specific heat at constant pressure, with R/cp≈2/7 in the atmosphere. Once constrained to an isentropic surface, we can consider the flow as being essentially 2D and dismiss the vertical component of wind velocity. However, in particular cases, vertical velocity and vertical shear can significantly affect the computation of LCS, requiring a 3D analysis [29]. To work in 3D, the vertical velocity wis necessary. The ECMWF provides the vertical velocity ωin Pa/s, with negative values corresponding to upward motion. To compute the vertical velocity in meters per second, we use the hydrostatic approximation, which assumes that the horizontal scale is much larger than the vertical one, i.e., ω=−ρgw where ρis the density, gis gravity, and wis the vertical velocity in m/s. The density is related to pressure pand temperature Tthrough the equation of state for ideal gases, p=RρT, with R=287.058 m2/s2K−1. This procedure computes w, which is essential for obtaining the 3D velocity field in the atmosphere. •Step 3: Transformation to cartesian coordinates. To avoid complications at the poles during trajectory calculations in spherical coordinates, the velocity components u,v, and ware transformed into Cartesian coordinates using the following equations (adapted from [85]): vx=wcosλcosφ−usin λ−vcos λsin φ, vy=wsin λcos φ+ucosλ−vsin λsin φ, vz=wsin φ+vcos φ, 123 J. Curbelo the time-evolving nature of fluid flows, making them particularly well-suited for studying time-dependent geophysical processes. One of the primary advantages of LDs is their ability to identify Lagrangian Coherent Structures (LCSs), which are key to understanding barriers to transport and regions of enhanced mixing in fluid flows. These structures are often hidden within the complex velocity fields typical of geophysical systems, yet LDs can effectively reveal them by computing scalar fields that highlight regions of interest. This capability has been demonstrated across a wide range of applications, from oceanographic studies to atmospheric dynamics, where LDs have provided new insights into the mechanisms driving transport and mixing. The LD method can also be extended to study higher-dimensional systems. However, the application of LDs is not without challenges. Factors such as the choice of integration time, the quality and resolution of velocity data, and the specific LD metric used can influence the method’s effectiveness. Additionally, while generally robust, LDs may require careful tuning and validation in complex, real-world scenarios where assumptions may be challenged by factors like turbulence, data sparsity, or non-stationary flow conditions. The versatility and effectiveness of LDs have made them a critical tool in advancing our understanding of fluid transport processes in various geophysical contexts. While the method faces challenges, ongoing research is likely to address these issues, enhancing the robustness and applicability of LDs in more complex scenarios. As the field of geophysical fluid dynamics continues to evolve, LDs are poised to play an increasingly important role in unraveling the intricacies of transport and mixing in natural systems. Acknowledgements This paper originated from a talk given at the “XXVI Congreso de Ecuaciones Diferenciales y Aplicaciones” and the “XVI Congreso de Matemática Aplicada,” titled “Lagrangian Methods to Characterize Transport and Mixing in Geophysical Flows.” During this event, I was honored to receive the 2020 Antonio Valle Prize from the Sociedad Española de Matemática Aplicada (SeMA). I would like to thank SeMA and the organizers of the CEDYA conference for providing such a great opportunity. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. We acknowledge the support of the RyC Grant RYC2018-025169, 2020/2021, the Spanish Grants PID2020-114043GB-I00, PID2021-122954NB-I00, CEX2020-001084-M, CNS2023-144360 funded by MICIU/AEI/10.13039/501100011033 and “European Union NextGenerationEU/PRTR” , the Ramón Areces Foundation and the “2022 Leonardo Grant for Researchers and Cultural Creators”, BBVA Foundation. Data availability The data sets used to generate figures 2and 3are publicly available: ERA5, Copernicus Climate Change Service (C3S) operated by ECMWF on behalf of the European Commission. https://doi. org/10.24381/cds.bd0915c6. They were obtained from https://cds.climate.copernicus.eu/datasets/reanalysisera5-pressure-levels (registration required). Declarations Conflict of interest Author has no conflict of interest to declare. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 123 Lagrangian descriptors in geophysical flows: a survey References 1. Wiggins, S.: The dynamical systems approach to Lagrangian transport in oceanic flows. Annu. Rev. Fluid Mech. 37, 295–328 (2005) 2. Wiggins, S.: Coherent structures and chaotic advection in three dimensions. J. Fluid Mech. 654, 1–4 (2010) 3. Jones, C., Winkler, S.: Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere. In: Handbook of Dynamical Systems vol. 2, pp. 55–92. North-Holland, Amsterdam (2002) 4. Mancho, A.M., Small, D., Wiggins, S.: A tutorial on dynamical systems concepts applied to Lagrangian transport in oceanic flows defined as finite time data sets: Theoretical and computational issues. Phys. Rep. 237(3–4), 55–124 (2006) 5. Samelson, R., Wiggins, S.: Lagrangian Transport in Geophysical Jets and Waves: The Dynamical Systems Approach. Springer, New York (2006) 6. Hussain, A.K.M.F.: Coherent structures-reality and myth. The Physics of Fluids 26(10), 2816–2850 (1983) 7. Poincaré, H.: Sur le problème des trois corps et les équations de la dynamique. Acta Mathematica 13, 1–270 (1890) 8. Aref, H.: Stirring by chaotic advection. J. Fluid Mech. 143, 1–21 (1984) 9. Haller, G., Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D: Nonlinear Phenomena 147(3–4), 352–370 (2000) 10. Rypina, I.I., Scott, S.E., Pratt, L.J., Brown, M.G., Hackett, E.E.: Transport pathways in the gulf of mexico from finite-time lyapunov exponents. Journal of Physical Oceanography 40(5), 1041–1056 (2010) 11. Haller, G.: Lagrangian coherent structures. Annual Review of Fluid Mechanics 47, 137–162 (2015) 12. Harvey, V.L., Datta-Barua, S., Pedatella, N.M., Wang, N., Randall, C.E., Siskind, D.E., van Caspel, W.E.: Transport of nitric oxide via lagrangian coherent structures into the top of the polar vortex. Journal of Geophysical Research: Atmospheres 126(11), 2020–034523 (2021) 13. Niang, C., Mancho, A.M., Garcia-Garrido, V.J., Rodriguez-Fonseca, B., Mohino, E., Curbelo, J.: Transport pathways across the west african monsoon as revealed by Lagrangian Coherent Structures. Scientific Reports 10, 12543 (2020) 14. Garaboa-Paz, D., Eiras-Barca, J., Huhn, F., Pérez-Munuzuri, V.: Lagrangian coherent structures along atmospheric rivers. Chaos: An Interdisciplinary Journal of Nonlinear Science 25(6) (2015) 15. Rutherford, B., Dangelmayr, G., Montgomery, M.T.: Lagrangian coherent structures in tropical cyclone intensification. Atmospheric Chemistry and Physics 12(12), 5483–5507 (2012) 16. Curbelo, J., Mechoso, C.R., Mancho, A.M., Wiggins, S.: Lagrangian study of the final warming in the southern stratosphere during 2002. Part I: the vortex splitting at upper levels. Climate Dynamics 53, 2779–2792 (2019) 17. Curbelo, J., Mechoso, C.R., Mancho, A.M., Wiggins, S.: Lagrangian study of the final warming in the southern stratosphere during 2002. Part II: 3D structure. Climate Dynamics 53, 1277–1286 (2019) 18. Lekien, F., Ross, S.D.: The computation of finite-time lyapunov exponents on unstructured meshes and for non-euclidean manifolds. Chaos: An Interdisciplinary Journal of Nonlinear Science 20(1), 017505 (2010) 19. Curbelo, J., Chen, G., Mechoso, C.R.: Lagrangian analysis of the northern stratospheric polar vortex split in April 2020. Geophysical Research Letters 48(16), 2021–093874 (2021) 20. Manney, G.L., Lawrence, Z.D.: The major stratospheric final warming in 2016: Dispersal of vortex air and termination of Arctic chemical ozone loss. Atmospheric Chemistry and Physics Discussions 2016, 1–40 (2016) 21. Manney, G.L., Lawrence, Z.D., Santee, M.L., Livesey, N.J., Lambert, A., Pitts, M.C.: Polar processing in a split vortex: Arctic ozone loss in early winter 2012/2013. Atmospheric Chemistry and Physics 15(10), 5381–5403 (2015) 22. García-Garrido, V.J., Ramos, A., Mancho, A.M., Coca, J., Wiggins, S.: A dynamical systems perspective for a real-time response to a marine oil spill. Marine Pollution Bulletin 112(1), 201–210 (2016) 23. García-Sánchez,G.,Mancho,A.M.,Ramos,A.G.,Coca,J.,Pérez-Gómez,B., Álvarez-Fanjul,E.,Sotillo, M.G., García-León, M., García-Garrido, V.J., Wiggins, S.: Very high resolution tools for the monitoring and assessment of environmental hazards in coastal areas. Frontiers in Marine Science 7, 605804 (2021) 24. García-Sánchez, G., Mancho, A.M., Ramos, A.G., Coca, J., Wiggins, S.: Structured pathways in the turbulence organizing recent oil spill events in the eastern mediterranean. Scientific Reports 12, 3662 (2022) 25. Duran, R., Nordam, T., Serra, M., Barker, C.H.: Horizontal transport in oil-spill modeling. In: Marine Hydrocarbon Spill Assessments, pp. 59–96. Elsevier, (2021) 123 J. Curbelo 26. Olascoaga, M.J., Beron-Vera, F.J., Haller, G., Trinanes, J., Iskandarani, M., Coelho, E., Haus, B.K., Huntley, H., Jacobs, G., Kirwan, A., Jr., et al.: Drifter motion in the gulf of mexico constrained by altimetric lagrangian coherent structures. Geophysical Research Letters 40(23), 6171–6175 (2013) 27. Olascoaga, M.J., Haller, G.: Forecasting sudden changes in environmental pollution patterns. Proceedings of the National Academy of Sciences 109(13), 4738–4743 (2012) 28. Curbelo,J.,Rypina, I.I.: A three dimensional lagrangiananalysisof the smoke plumefromthe 2019/2020 australian wildfire event. Journal of Geophysical Research: Atmospheres 128, 2023–039773 (2023) 29. J. Curbelo, I. I. Rypina. A 3D Lagrangian analysis of certain aspects of atmospheric circulation, with application to the 2019-2020 wildfire event. Mathematisches Forschungsinstitut Oberwolfach. Report No. 31/2023. Transport and Scale Interactions in Geophysical Flows, 31, 1745-1748 (2023) 30. Dacre, H.F., Harvey, N.J., Webley, P.W., Morton, D.: How accurate are volcanic ash simulations of the 2010 eyjafjallajökull eruption? Journal of Geophysical Research: Atmospheres 121(7), 3534–3547 (2016) 31. Peng, J., Peterson, R.: Attracting structures in volcanic ash transport. Atmospheric Environment 48, 230–239 (2012) 32. Jarvis,A.,Mardi,A.H.,Foroutan,H., Ross,S.D.: Atmospherictransportstructuresshapingthe “godzilla” dust plume. Atmospheric Environment, 120638 (2024) 33. Badza, A., Mattner, T.W., Balasuriya, S.: How sensitive are Lagrangian Coherent Structures to uncertainties in data? Physica D: Nonlinear Phenomena 444, 133580 (2023) 34. Allshouse, M.R., Peacock, T.: Lagrangian based methods for coherent structure detection. Chaos: An Interdisciplinary Journal of Nonlinear Science 25(9) (2015) 35. Hadjighasem, A., Farazmand, M., Blazevski, D., Froyland, G., Haller, G.: A critical comparison of lagrangian methods for coherent structure detection. Chaos: An Interdisciplinary Journal of Nonlinear Science 27(5) (2017) 36. Rypina, I.I., Getscher, T., Pratt, L.J., Ozgokmen, T.: Applying dynamical systems techniques to real ocean drifters. Nonlinear Processes in Geophysics 29(4), 345–361 (2022) 37. Balasuriya, S., Ouellette, N.T., Rypina, I.I.: Generalized lagrangian coherent structures. Physica D: Nonlinear Phenomena 372, 31–51 (2018) 38. Haller, G.: Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10(1), 99–108 (2000) 39. Shadden, S.C., Lekien, F., Marsden, J.E.: Definition and properties of Lagrangian Coherent Structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D: Nonlinear Phenomena 212, 271–304 (2005) 40. Cencini, M., Vulpiani, A.: Finite size lyapunov exponent: review on applications. Journal of Physics A: Mathematical and Theoretical 46(25), 254019 (2013) 41. Aurell, E., Boffetta, G., Crisanti, A., Paladin, G., Vulpiani, A.: Growth of noninfinitesimal perturbations in turbulence. Physical review letters 77(7), 1262 (1996) 42. Froyland, G., Padberg-Gehle, K.: A rough-and-ready cluster-based approach for extracting finitetime coherent sets from sparse and incomplete trajectory data. Chaos: An Interdisciplinary Journal of Nonlinear Science 25(8) (2015) 43. Filippi, M., Rypina, I.I., Hadjighasem, A., Peacock, T.: An optimized-parameter spectral clustering approach to coherent structure detection in geophysical flows. Fluids 6(1), 39 (2021) 44. Schlueter-Kuck, K.L., Dabiri, J.O.: Coherent structure colouring: identification of coherent structures from sparse data using graph theory. Journal of Fluid Mechanics 811, 468–486 (2017) 45. Rypina, I.I., Scott, S.E., Pratt, L.J., Brown, M.G.: Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures. Nonlinear Processes in Geophysics 18, 977–987 (2011) 46. Shi, K., Theisel, H., Weinkauf, T., Hege, H.-C., Seidel, H.-P.: Visualizing transport structures of timedependent flow fields. IEEE Computer Graphics & Applications, 24–36 (2008) 47. Cabral, B., & Leedom, L. C. (1993). Imaging vector fields using line integral convolution. In Proceedings of the 20th annual conference on Computer graphics and interactive techniques (pp. 263-270) 48. Ide, K., Small, D., Wiggins, S.: Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets. Nonlinear Processes in Geophysics 9(3/4), 237–263 (2002) 49. Rypina, I.I., Pratt, L.J.: Trajectory encounter volume as a diagnostic of mixing potential in fluid flows. Nonlinear Processes in Geophysics 24(2), 189–202 (2017) 50. Haller, G.: A variational theory of hyperbolic lagrangian coherent structures. Physica D: Nonlinear Phenomena 240(7), 574–598 (2011) 51. Haller, G., Hadjighasem, A., Farazmand, M., Huhn, F.: Defining coherent vortices objectively from the vorticity. Journal of Fluid Mechanics 795, 136–173 (2016) 123 Lagrangian descriptors in geophysical flows: a survey 52. Balasuriya, S.: Stochastic sensitivity: A computable lagrangian uncertainty measure for unsteady flows. SIAM Review 62(4), 781–816 (2020) 53. Froyland, G., Santitissadeekorn, N., Monahan, A.: Transport in time-dependent dynamical systems: Finite-time coherent sets. Chaos: An Interdisciplinary Journal of Nonlinear Science 20(4) (2010) 54. Froyland, G.: Dynamic isoperimetry and the geometry of lagrangian coherent structures. Nonlinearity 28(10), 3587 (2015) 55. Froyland, G., Kwok, E.: A dynamic laplacian for identifying lagrangian coherent structures on weighted riemannian manifolds. Journal of Nonlinear Science 30, 1889–1971 (2020) 56. Mendoza, C., Mancho, A.M.: The hidden geometry of ocean flows. Phys. Rev. Lett. 105(3), 038501 (2010) 57. Mancho, A.M., Wiggins, S., Curbelo, J., Mendoza, C.: Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems. Communications in Nonlinear Science and Numerical Simulations 18(12), 3530–3557 (2013) 58. Curbelo,J., García-Garrido, V.J., Mechoso,C.R.,Mancho, A.M., Wiggins,S., Niang,C.:Insights into the three-dimensional lagrangian geometry of the antarctic polar vortex. Nonlinear Processes in Geophysics 24(3), 379–392 (2017) 59. Lopesino, C., Balibrea-Iniesta, F., García-Garrido, V.J., Wiggins, S., Mancho, A.M.: A theoretical framework for lagrangian descriptors. International Journal of Bifurcation and Chaos. 27(01), 1730001 (2017) 60. García-Garrido, V.J., Curbelo, J., Mechoso, C.R., Mancho, A.M., Wiggins, S.: The application of lagrangian descriptors to 3d vector fields. Regular and Chaotic Dynamics 23(5), 547–564 (2018) 61. Demian, A.S., Wiggins, S.: Detection of periodic orbits in hamiltonian systems using lagrangian descriptors. International Journal of Bifurcation and Chaos 27(14), 1750225 (2017) 62. Naik, S., García-Garrido, V.J., Wiggins, S.: Finding nhim: Identifying high dimensional phase space structures in reaction dynamics using lagrangian descriptors. Communications in Nonlinear Science and Numerical Simulation 79, 104907 (2019) 63. Agaoglou, M., Aguilar Sanjuan, B., García-Garrido, V. J., Gonzalez Montoya, F., Katsanikas, M., Krajˇnák, V., Naik, S., & Wiggins, S. R. (2020). Lagrangian Descriptors: Discovery and Quantification of Phase Space Structure and Transport. Zenodo. https://doi.org/10.5281/zenodo.3958985 64. Montoya, F.G., Wiggins, S.: Revealing roaming on the double morse potential energy surface with lagrangian descriptors. Journal of Physics A: Mathematical and Theoretical 53(23), 235702 (2020) 65. Vortmeyer-Kley, R., Gräwe, U., Feudel, U.: Detecting and tracking eddies in oceanic flow fields: a lagrangian descriptor based on the modulus of vorticity. Nonlinear Processes in Geophysics 23(4), 159–173 (2016) 66. Craven, G.T., Hernandez, R.: Deconstructing field-induced ketene isomerization through lagrangian descriptors. Physical Chemistry Chemical Physics 18(5), 4008–4018 (2016) 67. Mezi´c, I., Loire, S., Fonoberov, V.A., Hogan, P.: A new mixing diagnostic and gulf oil spill movement. Science 330(6003), 486–489 (2010) 68. Rutherford, B., Dunkerton, T., Montgomery, M.: Lagrangian vortices in developing tropical cyclones. Quarterly Journal of the Royal Meteorological Society 141(693), 3344–3354 (2015) 69. Rempel, E.L., Chian, A.C.-L., Brandenburg, A., Muñoz, P.R., Shadden, S.C.: Coherent structures and the saturation of a nonlinear dynamo. Journal of Fluid Mechanics 729, 309–329 (2013) 70. Junginger, A., Hernandez, R.: Lagrangian descriptors in dissipative systems. Physical Chemistry Chemical Physics 18(44), 30282–30287 (2016) 71. Naik, S., Wiggins, S.: Detecting reactive islands in a system-bath model of isomerization. Physical Chemistry Chemical Physics 22(32), 17890–17912 (2020) 72. Nagahata, Y., Borondo, F., Benito, R.M., Hernandez, R.: Identifying reaction pathways in phase space via asymptotic trajectories. Physical Chemistry Chemical Physics 22(18), 10087–10105 (2020) 73. Junginger, A., Craven, G.T., Bartsch, T., Revuelta, F., Borondo, F., Benito, R.M., Hernandez, R.: Transition state geometry of driven chemical reactions on time-dependent double-well potentials. Physical Chemistry Chemical Physics 18(44), 30270–30281 (2016) 74. Darwish, A., Norouzi, S., Di Labbio, G., Kadem, L.: Extracting lagrangian coherent structures in cardiovascular flows using lagrangian descriptors. Physics of Fluids 33(11) (2021) 75. Carlo, G.G., Montes, J., Borondo, F.: Lagrangian descriptors for the bunimovich stadium billiard. Physical Review E 105(1), 014208 (2022) 76. Abernathey, R., Haller, G.: Transport by lagrangian vortices in the eastern pacific. Journal of Physical Oceanography 48(3), 667–685 (2018) 77. Aguilar-Sanjuan, B., García-Garrido, V.J., Krajˇnák, V., Naik, S., Wiggins, S.: LDDS: Python package for computing and visualizing lagrangian descriptors for dynamical systems. Journal of Open Source Software 6(65), 3482 (2021) 123 J. Curbelo 78. Bruera, R., Curbelo, J., Garcia-Sanchez, G., Mancho, A.M.: Mixing and geometry in the north atlantic meridional overturning circulation. Geophysical Research Letters 50, 2022–102244 (2023) 79. Curbelo, J., & Mechoso, C. R. (2024). Characterizing the spatial distribution of mixing and transport in the northern middle atmosphere during winter. Journal of Geophysical Research: Atmospheres, 129, e2023JD040666 80. Garcia, R.R., Smith, A.K., Kinnison, D.E., de la Cámara, Á., Murphy, D.J.: Modification of the Gravity Wave Parameterization in the Whole Atmosphere Community Climate Model: Motivation and Results. J. Atmos. Sci. 74(1), 275–291 (2017) 81. Marsh, D.R., Mills, M.J., Kinnison, D.E., Lamarque, J.-F., Calvo, N., Polvani, L.M.: Climate Change from 1850 to 2005 Simulated in CESM1(WACCM). Journal of Climate 26(19), 7372–7391 (2013) 82. Hersbach, H., Bell, B., Berrisford, P., Horányi, A., Sabater, J.M., Nicolas, J., Radu, R., Schepers, D., Simmons, A., Soci, C., et al.: Global reanalysis: goodbye ERA-interim, hello ERA5. ECMWF newsletter 159, 17–24 (2019) 83. Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., Iredell, M., Saha, S., White, G., Woollen, J., Zhu, Y., Chelliah, M., Ebisuzaki, W., Higgins, W., Janowiak, J., Mo, K.C., Ropelewski, C., Wang, J., Leetmaa, A., Reynolds, R., Jenne, R., Joseph, D.: The NCEP/NCAR 40-Year Reanalysis Project. Bulletin of the American Meteorological Society 77(3), 437–472 (1996) 84. Donald Ahrens, C., Henson, R.: Meteorology today: an introduction to weather, climate and the environment (2015) 85. de la Cámara, A., Mancho, A.M., Ide, K., Serrano, E., Mechoso, C.R.: Routes of transport across the Antarctic polar vortex in the southern spring. J. Atmos. Sci. 69(2), 753–767 (2012) 86. García-Garrido, V.J., Curbelo, J., Mechoso, C.R., Mancho, A.M., Wiggins, S.: A simple kinematic model for the lagrangian description of relevant nonlinear processes in the stratospheric polar vortex. Nonlin. Proc. Geophys. 34, 265–278 (2017) 87. de la Cámara, A., Mechoso, R., Mancho, A.M., Serrano, E., Ide, K.: Isentropic transport within the Antarctic polar night vortex: Rossby wave breaking evidence and Lagrangian structures. J. Atmos. Sci. 70, 2982–3001 (2013) 88. Guha, A., Mechoso, C.R., Konor, C.S., Heikes, R.P.: Modeling rossby wave breaking in the southern spring stratosphere. J. Atmos. Sci. 73(1), 393–406 (2016) 89. Smith, M.L., McDonald, A.J.: A quantitative measure of polar vortex strength using the function m. J. Gephys. Res. Atmos. 119, 5966–5985 (2014) 90. Balasuriya, S., Kalampattel, R., Ouellette, N.T.: Hyperbolic neighbourhoods as organizers of finite-time exponential stretching. Journal of Fluid Mechanics 807, 509–545 (2016) 91. McIntyre, M.E., Palmer, T.N.: Breaking planetary waves in the stratosphere. Nature 305, 593–600 (1983) 92. Bowman, K.P.: Rossby wave phase speeds and mixing barriers in the stratosphere. part i: Observations. Journal of the atmospheric sciences 53(6), 905–916 (1996) 93. Davis, R.E.: Observing the general circulation with floats. Deep Sea Research Part A. Oceanographic Research Papers 38, 531–571 (1991) 94. LaCasce, J.H.: Statistics from lagrangian observations. Progress in Oceanography 77(1), 1–29 (2008) 95. Huang, C.S.Y., Nakamura, N.: Local finite-amplitude wave activity as a diagnostic of anomalous weather events. J. Atmos. Sci. 73(1), 211–229 (2015) 96. Curbelo, J., Linz, M.: Lagrangian coherent structures to examine mixing in the stratosphere. EGUSPHERE 2024-1348, 1–21 (2024). https://doi.org/10.5194/egusphere-2024-1348 97. Nakamura, N.: Two-Dimensional Mixing, Edge Formation, and Permeability Diagnosed in an Area Coordinate. J. Atmos. Sci. 53(11), 1524–1537 (1996) 98. Shuckburgh, E., Norton, W., Iwi, A., Haynes, P.: Influence of the quasi-biennial oscillation on isentropic transport and mixing in the tropics and subtropics. Journal of Geophysical Research: Atmospheres 106(D13), 14327–14337 (2001) 99. Abalos, M., Legras, B., Shuckburgh, E.: Interannual variability in effective diffusivity in the upper troposphere/lower stratosphere from reanalysis data. Quarterly Journal of the Royal Meteorological Society 142(697), 1847–1861 (2016) 100. Gupta, A., Linz, M., Curbelo, J., Pauluis, O., Gerber, E.P., Kinnison, D.E.: Estimating the meridional extent of adiabatic mixing in the stratosphere using age-of-air. Journal of Geophysical Research: Atmospheres 128(4), 2022–037712 (2023) 101. Rivoire, L. and Curbelo, J.: Tracking the jets as Lagrangian objects, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-14165, https://doi.org/10.5194/egusphere-egu24-14165, 2024 102. Mancho,A.M.,Curbelo,J.,Wiggins,S.,Garcia-Garrido,V.J.,Mendoza,C.:BeautifulGeometriesUnderlying Ocean Nonlinear Processes. Chapter in the Book. A Voyage Through Scales. Eds. Günter Blöschl, 123 Lagrangian descriptors in geophysical flows: a survey Hans Thybo, Hubert Savenije, Lois Lammerhuber. Publishers European Geophysical Union & Edition Lammerhuber (2015) 103. Madrid, J.A.J., Mancho, A.M.: Distinguished trajectories in time dependent vector fields. Chaos 19, 013111 (2009) 104. Balibrea-Iniesta, F., Xie, J., García-Garrido, V.J., Bertino, L., Mancho, A.M., Wiggins, S.: Lagrangian transportacrosstheupperarcticwatersin thecanadabasin.QuarterlyJournaloftheRoyalMeteorological Society 145(718), 76–91 (2019) 105. García-Sánchez, G., Mancho, A.M., Ramos, A.G., Coca, J., Jiménez Madrid, J.A.: Dynamical systems for remote validation of very high-resolution ocean models. Nonlinear Dynamics, 1–21 (2024) 106. Ramos, A., García-Garrido, V., Mancho, A., Wiggins, S., Coca, J., Glenn, S., Schofield, O., Kohut, J., Aragon, D., Kerfoot, J., et al.: Lagrangian coherent structure assisted path planning for transoceanic autonomous underwater vehicle missions. Scientific Reports 8(1), 4575 (2018) 107. Chen,N., Lunasin, E., Wiggins,S.:Launching drifter observationsin thepresenceof uncertainty. Physica D: Nonlinear Phenomena 460, 134086 (2024) 108. Bollt, E., Chen, N., & Wiggins, S. (2024). A causation-based computationally efficient strategy for deploying Lagrangian drifters to improve real-time state estimation. Physica D: Nonlinear Phenomena, 467, 134283 109. Chen, N., Lunasin, E., Wiggins, S.: Lagrangian descriptors with uncertainty. Physica D: Nonlinear Phenomena 467, 134282 (2024) 110. García-Sánchez, G., Mancho, A.M., Wiggins, S.: A bridge between invariant dynamical structures and uncertainty quantification. Communications in Nonlinear Science and Numerical Simulation 104, 106016 (2022) 111. Covington, J., Chen, N., Wiggins, S., & Lunasin, E. (2025). Probabilistic eddy identification with uncertainty quantification. Physica D: Nonlinear Phenomena, 134542 112. Mendoza, C., Mancho, A., Wiggins, S.: Lagrangian descriptors and the assessment of the predictive capacity of oceanic data sets. Nonlinear Processes in Geophysics 21(3), 677–689 (2014) 113. Lünsmann, B., Vortmeyer-Kley, R., Kantz, H.: An extended transfer operator approach for timeconsistent coherent set analysis. arXiv preprint arXiv:1903.05086 (2019) 114. Vortmeyer-Kley, R., Lünsmann, B., Berthold, M., Gräwe, U., Feudel, U.: Eddies: fluid dynamical niches or transporters?-A case study in the western baltic sea. Frontiers in Marine Science 6, 118 (2019) 115. Vortmeyer-Kley, R., Holtermann, P., Feudel, U., Gräwe, U.: Comparing eulerian and lagrangian eddy census for a tide-less, semi-enclosed basin, the baltic sea. Ocean Dynamics 69, 701–717 (2019) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123