# Templex for Lagrangian dynamics in the Southwestern Atlantic

Juan Cruz Bonel, Nicolás Bodnariuk, Gisela D Charó, Christophe Letellier, Christophe Guinet, Martín Saraceno, Denisse Sciamarella

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Juan Cruz Bonel, Nicolás Bodnariuk, Gisela D Charó, Christophe Letellier, Christophe Guinet, et al.  
Templex for Lagrangian dynamics in the Southwestern Atlantic. Chaos: An Interdisciplinary Journal  
of Nonlinear Science, 2025, 35 (10), pp.103137. 10.1063/5.0255611.  
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# Templex for Lagrangian dynamics in the Southwestern Atlantic

Juan Cruz Bonel,¹,², a) Nicolás Bodnariuk,³,², b) Gisela D. Charó,²,⁴, c) Christophe Letellier,⁵, d) Christophe Guinet,⁶, e) Martín Saraceno,¹,⁷,², f) and Denisse Sciamarella⁸,², g)

¹) CONICET – Universidad de Buenos Aires. Centro de Investigaciones del Mar y la Atmósfera (CIMA), C1428EGA CABA, Argentina.

²) CNRS - IRD - CONICET - UBA. Institut Franco-Argentin d'Études sur le Climat et ses Impacts (IRL 3351 IFAECI), C1428EGA CABA, Argentina.

³) CNRS - LMD. Laboratoire de Météorologie Dynamique, Paris, France (UMR8539).

⁴) CNRS - CEA - LSCE. Laboratoire des Sciences du Climat et de l'Environnement, Paris, France (UMR 8212).

⁵) Rouen Normandie University — CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France

⁶) CEBC - CNRS Centre d'Etudes Biologiques de Chize, France.

⁷) UBA – Universidad de Buenos Aires. Departamento de Ciencias de la Atmósfera y de los Océanos (DCAO), C1428EGA CABA, Argentina.

⁸) CNRS - Centre National de la Recherche Scientifique, 75016 Paris, France.

(Dated: 27 August 2025 to submit to Chaos)

This work presents the first application of the templex approach to observational datasets, using Lagrangian trajectories obtained from satellite altimetry in the ocean. The templex is a recent topological construct that extends classical ideas from template theory to higher-dimensional systems. Unlike other methods in Topological Data Analysis, which lack flow information, a templex encodes both the structure of phase space through a BRAMAH cell complex and the organization of flow cycles upon it through a directed graph defined on its highest-dimensional cells. As shown in earlier works, the description of flows in phase space indirectly enables a description of fluid flows in physical space, since particles sharing the same dynamics are known to move coherently. Particle sets in the Southwestern Atlantic Ocean are analyzed, revealing that the Lagrangian finite-time dynamics in this region can be related to those produced by a nonautonomous meandering jet model. We distinguished non-mixing (regular) islands from the chaotic sea. The results are also compared to those obtained from metric methods describing material transport in fluid flows and to the spatial organization of chlorophyll-a concentration. The seasonal variability of chaotic dynamics is also discussed.

Chaotic advection in the ocean is an effective mechanism for transporting plankton, larvae, and other biological tracers. Given the biological implications of fluid organization, researchers in geoscience have long endeavored to understand the dynamics of transport. Satellite observations provide velocity fields in the Eulerian frame, which describe how the flow evolves at each location, but not where fluid parcels actually go. To address this challenge, Lagrangian analysis in the ocean can benefit from methods of nonlinear science to describe the dynamics of advected particles. This can be achieved by constructing a templex from the time series of a particle's position. Such a representation allows one to describe the topological structure in phase space from time

series originating in theoretical models or in observations, and to compare them. Whether regular or chaotic, the behavior of particles moving together for a certain time is expected to have the same topological properties. The templex allows computing these properties and using them to construct a dynamical cartography, providing insight into the transport properties of fluid flows around meandering jets in general, and into the Southwestern Atlantic in particular. This cartography presents a good agreement with maps of chlorophyll-a concentration and with the results obtained with traditional Lagrangian coherence detection methods.

## I. INTRODUCTION

In the history of the dynamical systems theory, it was Henri Poincaré who first recognized the crucial role of topology, laying the foundation for a new approach to studying the qualitative behavior of systems.¹⁻⁴ Following his legacy, the search for topological invariants to classify and distinguish different types of flows in state space has continued to this day.⁵⁻⁸

<footnote>a) Electronic mail: juan.cruz.bonel@cima.fcen.uba.ar</footnote>

<footnote>b) Electronic mail: nicolas.bodnariuk@lmd.ipsl.fr</footnote>

<footnote>c) Electronic mail: gisela.charo@cima.fcen.uba.ar</footnote>

<footnote>d) http://www.atomosyd.net/spip.php?article1; Electronic mail: christophe.letellier@coria.fr</footnote>

<footnote>e) Electronic mail: christophe.guinet@cebc.cnrs.fr</footnote>

<footnote>f) Electronic mail: martin.saraceno@cima.fcen.uba.ar</footnote>

<footnote>g) Electronic mail: denisse.sciamarella@cnrs.fr</footnote>
During the last decades, improvements in observational measurements have propelled major advances in geoscience. In the case of oceanography, the deployment of drifting buoys,⁹,¹⁰ profilers,¹¹ satellites,¹² and even elephant seals equipped with sensors¹³⁻¹⁶ has produced increasingly large amounts of data. Interpreting these observations in terms of theoretical models of geofluid dynamics poses new challenges.

Several methodologies have been developed to detect coherent patterns in Lagrangian dynamics. Table I, adapted from Allshouse and Peacock,¹⁷ organizes these approaches into four major categories: geometric, probabilistic, clustering, and topological. While geometric and probabilistic methods rely on velocity fields, and clustering on trajectory statistics, topological methods analyze qualitative features such as braiding or knots in trajectory data. More recent approaches have introduced tools from topological data analysis, such as persistent homology, to study image series of flow field patterns¹⁸,¹⁹ or to count loops in point clouds formed by intersections of particle trajectories with the midplane of a cylindrical steady flow.²⁰

A distinct line of work focuses on the state space, whose dimensionality grows with the number of state variables. In this setting, it is possible to detect invariant manifolds that organize the vector fields, as in Mancho et al.,²¹ Another possibility is to characterize the structure of the state-space trajectories—or the points in state space reconstructed from a set of time series. This can be done using a BRAMAH cell complex.²²,²³ BRAMAH is the acronym for *Branched Manifold Analysis through Homologies*—it is an algebraic topology approach specifically designed to extract the topological features of a branched manifold from point sets in a generic state space, without dimensional restrictions. The approach has been recently extended through the introduction of the *templex*, which captures the double topology of a flow in state space: that of the structure—already approximated by the cell complex—and that of the paths allowed by the flow around it.²⁴,²⁵ This extension enables the extraction of the strips defined in the classical template approach introduced in the 1990s,⁸,²⁶⁻²⁸ while overcoming its dimensional limitations. Whereas templates (as knotholders) are restricted to three-dimensional systems, complexes generalize the analysis to arbitrary dimension. Other interesting methods have also contributed to the characterization of unstable periodic orbits beyond three dimensions,²⁹⁻³¹ including the color method,³² which has been applied to epidemic processes, notably in the context of the Ebola virus outbreak in West Africa.³³ In the case of fluid systems, and as shown by Charó et al.,³⁴, Charó, Artana, and Sciamarella³⁵,³⁶, the holes, torsions, weak boundaries and other properties of the BRAMAH complex can be related to transport processes, following the principles discussed by Kelley, Allshouse, and Ouellette³⁷.

In this work we analyze particle trajectories in time-dependent settings using first, a two-dimensional mean-

dering jet model and second, particle advection within the velocity fields observed in the Southwestern Atlantic Ocean (SWAO).

The subsequent part of this paper is organized as follows. Section II explains how topology can be extracted from time series using the BRAMAH and templex framework. Section III presents the finite-time topological analysis of particle trajectories in the meandering jet model. Section IV describes the reconstruction of the state space from oceanic data and its topological analysis. Conclusions are given in Section V, followed by technical appendices.

## II. TOPOLOGY FROM TIME SERIES

The notion of continuous deformation is important in state space, where continuity and determinism constrain the shape of a flow. A time evolution of a given autonomous dynamical system is

$$ \frac{dx}{dt} \equiv \dot{x} = f(x) \qquad (1) $$

where $t$ is the time, $x(t) \in \mathbb{R}^n$ is the state vector and $f$ is locally Lipschitz. A solution of it takes the form of a flow $\phi(t, x_0)$ where $x_0$ is the initial state at $t=0$. When $f$ is deterministic, the trajectories in the state space cannot self-intersect. In other words, the solutions of an autonomous system (with constant parameter values) are structured by a given topology. This is no longer the case when the system is non autonomous, i.e., when there is an explicit dependence on time in the vector field. The dynamics can thus be characterized by different topologies for different time windows.

In nonautonomous systems, the topological analysis may require some specific strategies. If the time dependence of the governing equations can be removed by adding some equations governing this time evolution, the topological analysis can be performed in a state space whose dimension is accordingly increased.³⁸ For a state space reconstructed from a time series, this requirement can be met with some caveats, to be discussed further below. We assume that the analysis starts from time series with some characteristics in terms of sampling rate, length, and noise level. According to Gilmore,⁸ one looks for time series with more than twenty quasi-cycles (average peak-to-peak interval), and with more than fifty samples per quasi-cycle. When these characteristics are not readily available, as in many datasets in the SWAO region, one must carry out some interpolation and/or smoothing to render the time series tractable, or design a different way for exploring the state space, as later developed in this work.

Typically, topological analysis is performed in a state space reconstructed from a measured time series using delay coordinates. In that case, one assumes that the measured variable provides the required information about the original state space, although this is not always the
<table><caption>TABLE I: Lagrangian coherence detection methods categorized by their principles and data requirements, as described by Allshouse and Peacock, 2015, Ref. [17] and references therein.</caption><thead><tr><td>Category</td><td>Principle</td><td>Data Required</td><td>Example Methods</td></tr></thead><tbody><tr><td>Geometric Methods</td><td>Detects material lines and surfaces based on stretching or deformation.</td><td>Velocity field</td><td>Finite-Time Lyapunov Exponents (FTLE), Lagrangian Coherent Structures (LCS)</td></tr><tr><td>Probabilistic Methods</td><td>Identifies influential regions via probabilistic transport or uncertainty measures.</td><td>Velocity field</td><td>Transfer Operators, Dynamic Laplacian</td></tr><tr><td>Clustering Methods</td><td>Partitions flow into regions of coherence based on trajectory clustering.</td><td>Trajectory data</td><td>Spectral Clustering, K-means Clustering</td></tr><tr><td>Topological Methods</td><td>Analyzes topological patterns of trajectories, such as braiding or knots.</td><td>Trajectory data</td><td>Braid Analysis, Topological Data Analysis</td></tr></tbody></table>

case, as demonstrated by assessing the observability of the state space provided by a given variable.39 In Lagrangian problems, the most commonly available variable is one of the particle coordinates. As in a previous study,36 we select the coordinate that explores the direction of stronger variability of the flow. In zonal jets with particles moving along the $x$ direction, the $y$ coordinate is thus the best choice, particularly because it discriminates the two ‘halves’ of the domain.35 To reconstruct a time-delay embedding, the dimension $n$ of the reconstructed state space can be estimated using the false nearest neighbors technique.40–42 Once $n$ is set, the value of the time delay $\tau$ is assessed by visual inspection, knowing that when the reconstructed trajectory is too squeezed along the first (second) bisecting line, $\tau$ is too small (large). A point cloud in a space $\mathbb{R}^n$ is thus obtained.

A successful state-space reconstruction for Lagrangian flow depends on whether the tracked particle remains within the coherent Lagrangian set for a sufficiently long time to enable topological analysis. Unfortunately, this is not always the case for real oceanographic data, where transport barriers may be partially permeable. To deal with short residence times, the proposed multiparticle strategy is to ‘trade time for space,’ that is, to use a set of particles sufficiently close together to improve the exploration of the corresponding state space. A similar strategy for reconstructing the phase space from aggregated datasets was previously employed by Mangiarrotti et al.43. An important advantage of the BRAMAH method is that the reconstruction of the branched manifold does not require the reconstruction of orbits. The requirements differ from those of global modeling44 or other orbit-based techniques.8,45 In the case of altimetry velocity fields downloaded from the Copernicus Marine Service, this is implemented by advecting an array of virtual particles. A set of nine particles centered on a given particle is used to create the point cloud. To construct the cell complex, the advection time must be sufficiently long to allow complete tracking of the loops involved in

the skeleton, and the radius of the circle must be chosen carefully: if it is too small, it effectively reduces to the central particle, offering little additional exploration; if it is too large, it may extend beyond the Lagrangian region of interest and cover a different type of dynamics.

From a time-delay embedding, cell complexes are constructed with “blocks” of different dimensions called $k$-cells which are glued together. A $k$-cell is defined as a topological space whose interior is homeomorphic to the $k$-dimensional disk $D_k = \{x \in \mathbb{R}^k : \|x\| < 1\}$ with the additional property that its limit or boundary must be divided into a finite number of lower-dimensional cells, called faces of the $k$-cell. Thus, a cell of dimension 0 is a point; a 1-cell is a line segment; a 2-cell is a polygon; a 3-cell is a solid polyhedron, with polygons, edges and vertices as faces; and so forth. The dimension of the cell complex is given by its highest dimensional cell: a complex approximating a surface is said to be a 2-complex or a complex of dimension 2. A cell complex is a finite collection $K$ of cells satisfying that (i) for any cell $\sigma$ from $K$, all its faces are also contained in $K$, and that, (ii) for any two cells $\sigma$ and $\gamma$ of $K$, the intersection $\sigma \cap \gamma$ is a cell which is a face of both $\sigma$ and $\gamma$. Defining algebraic operations on the complex requires orienting the cells. Note that such orientation has no connection with flow resulting from the governing equations, as we shall see later.

For complexes $K$ of dimension $\kappa$, a $k$-chain $C_k$ is defined as

$$C_k = \sum_i a_i \sigma_k^i$$

where $a_i \in \mathbb{Z}$ and $\sigma_k^i$ is a $k$-cell ($\forall i \in \mathbb{N}$). The group of all $k$-chains $C_k$ ($k = 1, \dots, \kappa$) in $K$ is an abelian group. A boundary operator

$$\partial_k : C_k \to C_{k-1}$$

is introduced in such a way that, for instance, the boundary of an oriented 2-cell is the chain formed by the 1-cells on its border, with a positive sign if the orientation
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0255611

of an edge is consistent with the direction of the 2-cell, and with a negative sign otherwise. This construction provides a formal algebraic framework for detecting cycles and identifying which ones are boundaries of higher-dimensional cells.

The homology groups are then defined by an equivalence relation among cycles: two $k$-cycles are considered equivalent if they differ on a boundary. In algebraic terms, this corresponds to quotienting the group of $k$-cycles by the group of $k$-boundaries. The homology groups are represented by nontrivial cycles that are obtained through this quotient. The boundaries of a 2-cell, for instance, are trivial 1-cycles, since cell boundaries are cyclic by definition. On the other hand, nontrivial cycles must contain an empty space or a hole. These nontrivial representative cycles are the generators of the homology groups. The number of generators, or $k$-holes, for a given dimension is called the Betti number and is denoted as $\beta_k$. At the 0th level, the nontrivial representative cycles describe the connected components of the cell complex, counted by $\beta_0$. A disjoint union of two circles, for instance, has $\beta_0 = 2$. Each circle has one independent loop (a 1-hole), so $\beta_1 = 1$. Contrary to this, a sphere has no independent 1-holes (since any loop can be contracted to a point), but it does have a 2-hole (the cavity of the sphere), so $\beta_2 = 1$.

Torsion groups can also be extracted from a uniformly oriented cell complex to describe properties related to orientability. An element $X$ in a group is said to have torsion if it is of finite order, namely, if there exists a positive integer $r$ such that $X^r = 1$ where 1 denotes the identity element. In homology, these are torsion elements of order $r$. For instance, a cylinder or standard strip has no torsion element in its homology. The Möbius strip, instead, has a torsion element in its homology generated by a single element of order 2, corresponding to the half-twist in the strip. In order to give a quick glance on how a few well-known surfaces can be distinguished from the properties of a cell complex, let us consider Table II. Notice that a standard strip is topologically equivalent to a cylindrical surface (without caps); if the cavity in the cylinder is closed by adding caps, the surface will be homeomorphic to a sphere.

<table><caption>TABLE II: Betti numbers $\beta_k$ ($k = 0, 1, 2$) and torsion groups for various surfaces. A trivial torsion group indicates no torsion elements.</caption><thead><tr><th>Cell complex</th><th>β<sub>0</sub></th><th>β<sub>1</sub></th><th>β<sub>2</sub></th><th>Torsion Group</th></tr></thead><tbody><tr><td>Sphere</td><td>1</td><td>0</td><td>1</td><td>trivial</td></tr><tr><td>Standard strip</td><td>1</td><td>1</td><td>0</td><td>trivial</td></tr><tr><td>Möbius strip</td><td>1</td><td>1</td><td>0</td><td>order 2</td></tr><tr><td>Klein bottle</td><td>1</td><td>2</td><td>0</td><td>order 2</td></tr><tr><td>Toroidal surface</td><td>1</td><td>2</td><td>1</td><td>trivial</td></tr></tbody></table>

Constructing a cell complex from a point cloud is a procedure that can be carried out in many different ways. For example, in a Rips complex,⁴⁶ we connect points with

edges (1-cells) if their distance is below a threshold $\epsilon$, then add 2-cells (triangles) for triples of points with pairwise distances less than $\epsilon$, and continue adding higher-dimensional cells in a similar fashion. If the point cloud under analysis lies on a branched manifold with a certain local dimension that the complex must preserve, the most suitable procedure is quite specific—see Appendix A for details.

Fig. 1 shows a BRAMAH complex for a point cloud corresponding to the Lorenz attractor. The complex consists of eight 2-cells labeled $\gamma_i$, with $i = 1, \dots, 8$, and twelve 0-cells labeled $1, \dots, 12$, yielding $\beta_0 = 1$, $\beta_1 = 2$, and $\beta_2 = 0$. The computations allow for counting the holes as well as identifying them. The two 1-holes are each associated with one wing of the Lorenz attractor: $h_1 = \langle 0, 3 \rangle + \langle 3, 7 \rangle - \langle 0, 7 \rangle$ and $h_2 = \langle 2, 6 \rangle + \langle 6, 12 \rangle - \langle 2, 12 \rangle$.

<figure><img src="image_2.png" /><figcaption>FIG. 1: BRAMAH complex $K_L$ for the Lorenz 63 attractor.⁴⁷ The two 1-holes are colored in red and green, and the two 1-cells of the joining locus are colored in magenta.</figcaption></figure>

A BRAMAH complex for a chaotic attractor always presents a joining locus.²⁴ For a 2-complex, the joining locus is formed by the 1-cells shared by at least three 2-cells. The 1-cells are called *joining 1-cells*, and the 2-cells are called *joining 2-cells*. For the Lorenz attractor, the joining lines are $\langle 0, 1 \rangle$ and $\langle 1, 2 \rangle$: they correspond to the line where the two wings are squeezed together. A refined description of the topology of the point cloud can be obtained by considering how the flow visits the cells of the BRAMAH complex. This information is encoded in a templex using a digraph. A templex $T \equiv (K, D)$ of dimension $m$ is made of a BRAMAH $m$-complex $K$ and a digraph $D$ such that the nodes are the $m$-cells of $K$ and the directed edges represent how the flow visits the $m$-cells.²⁴ We further classify the joining 2-cells into *ingoing* and *outgoing* cells: those 2-cells in the joining locus that receive the flow according to the orientation of the digraph are called *outgoing*, while the others are called *ingoing*. For the Lorenz attractor, the outgoing cells are $\gamma_1$ and $\gamma_8$, and the ingoing cells are: $\gamma_3, \gamma_4,$
$\gamma_5$, and $\gamma_6$. Closed paths in the cell complex can be expressed as digraph cycles. We are interested in cycles that include both *outgoing* and *ingoing* cells. Two cycles in $D$ that share the same set of ingoing and outgoing cells are considered equivalent and belong to the same class. From each class, we select one cycle, referred to as the *representative cycle*. Each representative cycle in $D$ corresponds to a sequence of cells in the complex $K$. This subcomplex, associated with the digraph cycle, is called a *generatex* $G$. When a generatex crosses the joining locus $p$ times, it is said to be of *order* $p$. For $p \ge 1$, the generatex can be decomposed into a sequence of cell paths, each involving exactly one pair of ingoing and outgoing cells. The subset of cells associated with each of these paths within the cycle is called a *stripex*. Further details on the definitions of the templex properties and the algebraic procedure to compute them can be found in Appendix B and in Refs. 24 and 47.

<figure><img src="image_3.png" /><figcaption>FIG. 2: Digraph $D_L$ of the templex $T_L = (K_L, D_L)$ for the Lorenz attractor. The nodes are the highest dimensional cells in $K_L$: $\gamma_1...\gamma_8$. The directed edges in $D_L$ indicate how the flow connects the nodes, the circles indicate the outgoing nodes and the squares the ingoing nodes.</figcaption></figure>

For the Lorenz attractor, the digraph $D_L$ (Fig. 2) has three generatexes as follows.

$$ 
\begin{aligned}
\mathcal{G}_1 &= \{\gamma_1 \to \gamma_2 \to \gamma_3 \to \gamma_1\} \\
\mathcal{G}_2 &= \{\gamma_8 \to \gamma_7 \to \gamma_6 \to \gamma_8\} \\
\mathcal{G}_3 &= \{\gamma_1 \to \gamma_2 \to \gamma_4 \to \gamma_8 \to \gamma_7 \to \gamma_5 \to \gamma_1\}
\end{aligned}
$$

Generatexes $\mathcal{G}_1$ and $\mathcal{G}_2$ are classified as order-1 generatexes, and their corresponding stripexes are $S_1$ and $S_2$. The generatex $\mathcal{G}_3$, being of order 2, can be naturally decomposed into two distinct stripexes: $S_3$, and $S_4$.

$$ 
\begin{aligned}
S_1 &= \{\gamma_1 \to \gamma_2 \to \gamma_3 \to \gamma_1\} \\
S_2 &= \{\gamma_8 \to \gamma_7 \to \gamma_6 \to \gamma_8\} \\
S_3 &= \{\gamma_1 \to \gamma_2 \to \gamma_4 \to \gamma_8\} \\
S_4 &= \{\gamma_8 \to \gamma_7 \to \gamma_5 \to \gamma_1\}.
\end{aligned}
$$

These four stripexes are the generalization of the four strips that can be distinguished in a template for the Lorenz attractor.⁴⁸

A stripex can be either twisted or not, depending on its specific configuration within the templex. It is twisted if the edges glued to the joining line change their relative positions before reinsertion. In the Lorenz attractor, the

stripexes $S_3$ and $S_4$ have local twists, while $S_1$ and $S_2$ do not. Local twists in a stripex of a templex are equivalent to strips with an uneven number of local torsions in a template.⁴⁸

The templex allows the extraction of detailed information regarding (i) the topology of the structure underlying the point cloud (homology group generators, torsion elements, joining loci) — encoded by the BRAMAH complex — and (ii) the topology of the flow along the structure (generatexes, stripexes, twists) — encoded by the digraph. The description of a chaotic attractor provided by persistent homologies, based solely on the computation of Betti numbers (the most persistent holes), is notably limited compared to the refined topological characterization provided by the templex approach.

### III. MEANDERING JET MODEL (MJM)

Strong currents or jet streams are commonly encountered in geophysical flows. In the ocean, examples include the Gulf Stream,⁴⁹,⁵⁰ the Kuroshio current,⁵¹,⁵² the Antarctic Circumpolar Current,⁵³,⁵⁴ and the Malvinas Current,⁵⁵,⁵⁶ among others. Jets in the ocean may be intermittent or persistent, such as jet-core structures reported in the Malvinas Current,⁵⁶,⁵⁷ which are subjected to the influence of trapped topographic waves and mesoscale perturbations developing in the open ocean. Irrespective of their persistence, their spatial structure generally varies largely in time: jets may exhibit complex wave-like patterns, meanders, occlusion, and even eddy shedding, as is the case in the Brazil-Malvinas Confluence Region.⁵⁸,⁵⁹

Basic models of zonal jets in the atmosphere and the ocean have been proposed in the literature⁶⁰ to represent geophysical flows observed in nature. Generally, such models are defined in a 2D domain, as the flow is restricted to isentropic surfaces—an assumption that builds on the conservative properties of the flow. The idealized model explored in this section is rooted in dynamic principles. It originates from the linearization of the quasi-geostrophic equations under the assumption that small-amplitude waves are advected by a mean flow.⁶¹ The mean flow employed is the Bickley jet profile,⁶² and the wave parameters are selected to ensure the waves remain neutral, i.e., neither amplifying nor decaying over time.⁶³

#### A. Model equations and parameters

Let us start with a streamfunction $\psi(x, y, t)$ representative of an incompressible flow along an isentropic surface in the ocean. The zonal and meridional velocities $u$ and $v$ are related to the streamfunction $\psi$ as

$$ \frac{dx}{dt} = -\frac{\partial \psi}{\partial y}, \quad \frac{dy}{dt} = \frac{\partial \psi}{\partial x}. \qquad (2) $$
In quasi-geostrophic dynamics on a β-plane, centered at $\phi_0$, the streamfunction can be rewritten

$$ \psi(x, y, t) = \psi_0(y) + \psi_1(x, y, t), \quad (3) $$

that is, as the sum of a zonal unperturbed jet and a time-varying anomaly endowed with spatial structure, which may represent meanders, eddies, among other flow organizing patterns characteristic of geophysical fluids.

For the purpose of this study, $\psi_0(y)$ is a steady zonal background flow (Bickley velocity profile), and $\psi_1(x, y, t)$ represents two traveling Rossby waves$^{61,64}$ as follows.

$$ \psi_0(y) = -U_0 L \tanh\left(\frac{y}{L}\right) \quad (4) $$

$$ \psi_1(x, y, t) = U_0 L \operatorname{sech}^2\left(\frac{y}{L}\right) \sum_{n=1}^{2} A_n \cos[k_n(x - c_n t)] \quad (5) $$

The proposed model is inspired by the atmospheric jet model developed by Rypina,$^{61}$ which is based on the wave stability analysis conducted by Lipps.$^{63}$ To define this model, we adimensionalize the system's streamfunction and the ordinary differential equation that relates the velocity field to the rate of change of position along trajectories. This approach enables us to establish a regime equivalent to the one shown in Fig. 4 (middle panel) in Ref. 61, but specifically adapted to oceanic conditions. This zonal jet model, unconstrained by boundaries in the West-East direction, can be viewed as a zero-order representation of the Antarctic Circumpolar Current in the Southern Hemisphere. Early numerical efforts to understand this dynamics use quasigeostrophic channel models,$^{65,66}$ which conceptualize the current primarily as a zonal jet.

The dimensionless parameters appearing in Eqs. (4) and (5) are (i) the ratios $c_1/U_0$ and $c_2/U_0$ of wave speeds to the characteristic velocity of the Bickley jet, (ii) the dimensionless amplitudes $A_1$ and $A_2$ of the waves imposed on the background field, and the dimensionless parameter $\beta L^2/U_0$ associated with the existence of neutral waves. We assume that the characteristic timescale $t^* = L/U_0$ is advective. Time dependence associated with one of the two Rossby waves in (5) may be suppressed by choosing a reference frame moving at the phase speed of that wave. Naturally, any wave may be absorbed into the background flow – our choice is arbitrary. In the reference frame moving at speed $c_1$, the streamfunction reads:

$$ \psi(x, y, t) = c_1 y - U_0 L \tanh\left(\frac{y}{L}\right) + A_1 U_0 L \operatorname{sech}^2\left(\frac{y}{L}\right) \cos(k_1 x) + A_2 U_0 L \operatorname{sech}^2\left(\frac{y}{L}\right) \cos(k_2 x - \sigma_2 t) \quad (6) $$

where $\sigma_2 = c_2 k_2 - c_1 k_2 = k_2(c_2 - c_1)$. Note that $\sigma_2$ is negative because in the new reference frame, the $n = 2$

wave lags behind the faster $n = 1$ wave and exhibits a westward-propagating phase.

The driving term $u = \cos \sigma_2 t$ may be rewritten as the solution to the second-order differential equation$^{34,38}$

$$ \ddot{u} + \sigma_2^2 u = 0 \quad (7) $$

or, equivalently, as the two ordinary differential equations

$$ \begin{aligned} \dot{u} &= v \\ \dot{v} &= -\sigma_2^2 u. \end{aligned} \quad (8) $$

The initial conditions are given by

$$ \left| \begin{array}{l} u(t=0) = 1 \\ v(t=0) = 0 \end{array} \right. \quad (9) $$

This driving process can be combined with the expression for $\psi(x, y, t)$ in equations (2) and (6) to get the four-dimensional dynamical system

$$ \left\{ \begin{aligned} \dot{x} &= -U_0 \operatorname{sech}^2\left(\frac{y}{L}\right) \left[ 2A_2 \tanh\left(\frac{y}{L}\right) (u \cos(k_2 x) - \frac{v \sin(k_2 x)}{\sigma_2}) \right] + 2A_1 \cos(k_3 x) \tanh\left(\frac{y}{L}\right) + 1 + c_1 \\ \dot{y} &= LU_0 \operatorname{sech}^2\left(\frac{y}{L}\right) \left[ A_2 k_2 \left( \frac{v \cos(k_2 x)}{\sigma_2} + u \sin(k_2 x) \right) + A_1 k_1 \sin(k_1 x) \right] \\ \dot{u} &= v \\ \dot{v} &= -\sigma_2^2 u \end{aligned} \right. \quad (10) $$

describing the evolution of a fluid particle when the streamfunction is determined by expression (6). The nonautonomous system is thus rewritten as an autonomous one. The state vector is entirely determined by the four variables $(x, y, u, v)$.

The parameter values selected for our topological analysis are $c_2/U_0 = 0.205$, $c_3/U_0 = 0.461$, $A_2 = 0.1$, $A_3 = 0.3$, $U_0 = 1$ m/s, and $L = 223$ km. The choice of $U_0$ reflects typical velocity values for intense ocean currents, while $L$ is roughly of the order of characteristic mesoscale motions in the ocean. We have selected a configuration where a meandering jet is flanked by eddies in the chaotic sea$^{67}$ (Fig. 3). Let us note that the Lagrangian dynamics and coherent sets found here match those encountered in the Bickley Jet model.$^{35}$ The non-mixing regions are embedded in the chaotic sea, as shown by the streakline patterns traced by fluid particles that have passed through specific injection locations $(x_{inj}, y_{inj})$.

Let us end this section with an important issue for observational data. Time series are often observed for a short duration, which may not sufficiently cover the attractor. However, it is frequently possible to simultaneously track a given number of particles. Here, we compare the construction of point clouds for the MJM model from a long (80 days) time series for a single particle and from a short (12.6 days) time series for a set of particles, with the state space reconstructed in both cases using
<figure><img src="image_4.png" /><figcaption>FIG. 3: Streaklines in the MJM at 36 days: the injection locations are at $(x_{inj}, y_{inj})$, where $x_{inj}$ ranges from $-2.4 \times 10^6$ m to $-1.3 \times 10^6$ in steps of $0.1375 \times 10^6$ m, and $y_{inj}$ ranges from $-2.7 \times 10^5$ m to $2.7 \times 10^5$ m in steps of $1.35 \times 10^5$ m. Zone (1) corresponds to the ocean current (jet), zone (2) to the core of the eddy, zone (3) to the periphery of the eddy, and zone (4) to the chaotic sea.</figcaption></figure>

a sampling time of 1000 s. The reconstructed portraits are shown in Fig.4, and we have verified that the same topological properties are obtained in both cases. All reconstructions hereafter are conducted using the multi-particle strategy, described in Section II.

## B. Templex analysis for the MJM

For each point cloud in the four-dimensional state space, a BRAMAH complex is built as detailed in Appendix A. When the particle set is initially located in the current (jet), e.g., at $p_1(0) = (-2.06, -0.012) \times 10^6$ m, the BRAMAH cell complex has no joining locus, and the Betti numbers correspond to a standard strip (see Table II): the behavior is a period-1 orbit. When the particle is near the vortex core of the eddy at $p_2(0) = (-2.41, -0.12) \times 10^6$ m, the Betti numbers correspond to those of a toroidal surface: the behavior is quasi-periodic motion. When the particle is in the periphery of the eddy at $p_3(0) = (-2.1, -0.22) \times 10^6$ m, a Möbius strip is obtained: the behavior is a period-2 orbit. These are regular dynamics, as expected for the non-mixing Lagrangian regions, separated by hermetic transport barriers from the particles in the chaotic sea. The three types of dynamics are reported in Fig. 5.

For particles initially located in the MJM chaotic sea, e.g., $p_4(0) = (-1.4, 0.25) \times 10^6$ m, the BRAMAH cell complex presents a joining locus; that is, it describes a branched manifold, characteristic of chaotic behavior. It is now useful to draw a directed graph indicating how the highest-dimensional cells are visited by the flow. The BRAMAH complex $K_{MJM}(p_4)$ is shown in Fig. 6(a) and a simplified version, $\tilde{K}_{MJM}(p_4)$, with nine 2-cells,

<figure><img src="image_5.png" /><figcaption>(a) For a single particle</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) For a few particles</figcaption></figure>

<caption>FIG. 4: Comparison between a state portrait reconstructed from (a) the 80-day time series for a single particle at $(-1.4, 0.25) \times 10^6$ m, and (b) a 12.6-day time series for a small set of particles centered at the same position.</caption>

in Fig. 6(b). The Betti numbers for this structure are $\beta_0 = 1$, $\beta_1 = 3$, and $\beta_2 = 0$, i.e., there is a singly connected component with three 1-holes and no cavities enclosed. The templex is presented in Fig. 7 with a schematic representation of $\tilde{K}_{MJM}(p_4)$ in (a) and the digraph in (b).

Following the definitions in Appendix B, the topological analysis is completed as follows. The joining locus has two components as drawn with the thick solid lines in Fig. 7(a). From the digraph [Fig. 7(b)], we can determine the closed paths (leaving and returning to the
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0255611

<figure><img src="image_7.png" /><figcaption>FIG. 5: The three types of regular particle dynamics in the MJM. Top panel: Three-dimensional portraits of the four-dimensional embeddings. Middle panel: Multiparticle time series. Bottom panel: BRAMAH cell complexes. First column: Jet dynamics (period-1 orbit). Second column: Core of the eddy (quasi-periodic motion). Third column: Periphery of the eddy (period-2 orbit). The sequence of 1-cells forming the homology group generators at level 1, the 1-holes, are identified using colors in the cell complex.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 6: BRAMAH complex for the chaotic sea in the MJM model: (a) original; (b) simplified – with a reduced number of cells. The four-dimensional point cloud corresponds to the time-delay embedding in Fig. 4(b), constructed using the multiparticle array centered at $p_4(0) = (-1.4, 0.25) \times 10^6$ m. The three 1-holes are identified using colors in the cell complex.</figcaption></figure>
joining locus) to identify the generatex set.

$$
\begin{align*}
\mathcal{G}_1 &= \{\gamma_6 \to \gamma_7 \to \gamma_8\}, \\
\mathcal{G}_2 &= \{\gamma_4 \to \gamma_1 \to \gamma_2 \to \gamma_3\}, \\
\mathcal{G}_3 &= \{\gamma_4 \to \gamma_1 \to \gamma_2 \to \gamma_5 \to \gamma_6 \to \gamma_9\}.
\end{align*}
$$

The third generatex is of order 2 and, consequently, is
made of two stripexes. The stripex set has thus four
elements as follows.

$$
\begin{align*}
S_1 &= \{\gamma_6 \to \gamma_7 \to \gamma_8\}, \\
S_2 &= \{\gamma_4 \to \gamma_1 \to \gamma_2 \to \gamma_3\}, \\
S_3 &= \{\gamma_4 \to \gamma_1 \to \gamma_2\}, \\
S_4 &= \{\gamma_5 \to \gamma_6 \to \gamma_9\}.
\end{align*}
$$

No local twists are found in these stripexes.

Through the templex analysis, we have shown that,
at a finite time, the chaotic trajectory produced by a
conservative system can be described by a templex of
dimension 2. The specificity, already observed in an-
other four-dimensional chaotic attractor,²⁴ in the Lorenz-
84 system,⁶⁸ and in a global model for the epidemic Ebola
virus diseases,³³ is the splitting component between the
cells γ₂ and the two cells γ₃ and γ₅: the splitting is spread
over the width of the flow.

C. Topological grid for the MJM

In order to map the topological results onto the geo-
graphical domain, we use a topological grid, provid-
ing a cartographic representation of the Lagrangian dy-
namics at a certain time instant. In the MJM, this has
been done using 450 time series (with a set of ten parti-
cles centered around each grid position of the 9 × 5 grid)
and performing the templex analysis.

The 450 time series fall into one of the four categories:
(1) the particles in the current (jet) zone are always asso-
ciated with a standard strip, (2) the particles in the eddy
core with the toroidal surface, (3) those in the periphery
of the eddy with the Möbius strip, and (4) those in the
chaotic sea with the templex properties of T<sub>MJM</sub>(p<sub>4</sub>).

Each grid point displays the label corresponding to the
topological class of a particle passing through that posi-
tion at that time. Fig. 8 shows the result of this proce-
dure, juxtaposed with the streakline snapshot shown in
Fig. 3. In the case of the real ocean, the topological grid
can be juxtaposed with measured oceanographic features,
offering reference points for visualizing the evolving flow
configuration.

IV. SOUTHWESTERN ATLANTIC OCEAN (SWAO)

The circulation in the SWAO can be conceptually di-
vided into the flow of the Argentinian Continental Sea

and that of the open ocean. In the former region, the
dynamics is largely controlled by the wind forcing and
tides.⁶⁹ The modulation of the shelf’s circulation by the
wind has been reported to occur across timescales: at
the synoptic, as well as seasonal, subseasonal, interan-
nual, and decadal scales.⁷⁰⁻⁷³ This is particularly true
for the inner shelf region, where large sea surface height
gradients may emerge due to an Ekman transport per-
pendicular to the coast. The relaxation of the pressure
gradient is associated with the propagation of trapped
waves that impact the regional dynamics.⁷⁴ In addition
to these, the resonance of tidal waves leads to the am-
plification of the signal on shallow regions, particularly
of the Patagonian Shelf. The mean circulation of the
Shelf is directed towards the Northeast according to both
observations⁷⁵ and numerical studies.⁶⁹ Typical non-tidal
velocities are close to 0.2 m/s in the middle shelf and may
increase to 0.3 m/s in the outer shelf, where the Malv-
inas Current impacts the Continental Shelf circulation.⁷⁶
This interaction with the slope current has been physi-
cally explained in terms of idealized models⁵⁷,⁷⁷ that rep-
resent the spreading of a slope jet over the neighboring
shelf through diffusion of the across-shelf pressure gra-
dient due to bottom friction. Note, however, that this
average picture is complemented by high-frequency vari-
ability through water intrusions.⁷⁸

The Malvinas Current transports subantarctic waters along the Patagonian Slope. It constitutes an equiva- lent barotropic branch of the Circumpolar Current that veers northward into the Atlantic Ocean.⁵⁵ This flow is largely constrained by potential vorticity conservation, more precisely, the steep bathymetric gradients of the region. The Malvinas Current meets the Brazil Current in the Confluence Region, at approximately 38°S. The Brazil Current flow constitutes a warm and salty current that transports subtropical waters poleward.⁷⁹ The confluence region is characterized by large eddy kinetic values, the formation of eddies, filaments, and also frontal regions. The spectrum of motion scales ranges from the large mean flow characteristic of the jet cores to the mesoscale and submesoscale structures that populate the regions. These are ubiquitous features of flow organization in regions exhibiting large velocity shear and temperature gradients.⁸⁰ Hydrodynamic instabilities of the mean flow account for the existence of a rich dynamical spectrum that encompasses many different spatio-temporal scales. Water masses meeting at the confluence region turn eastward into the Atlantic Ocean. This flow, termed the Mid-Atlantic Current, constitutes the southern branch of the South Atlantic Subtropical Gyre.⁸¹ A distinctive feature of the SWAO is the development of a Malvinas Current Retroflexion region, where it reverses direction after looping southeastward.⁸²,⁸³
<figure><img src="image_9.png" /><figcaption>FIG. 7: Templex $T_{MJM}(\mathbf{p}_4)$ for the chaotic sea particles in the MJM model: (a) schematic complex with solid (dashed) lines representing the components of the joining (splitting) loci; (b) digraph with outgoing nodes marked with circles and ingoing nodes with squares. This dynamic has a joining locus with two components and four stripexes.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 8: Topological grid obtained from 450 time series (ten particles per grid point) in the MJM model, juxtaposed with the streakline snapshot at 36 days. The topological classes are: 1 - standard strip (green), 2 - toroidal surface (blue), 3 - Möbius strip (yellow), and 4 - templex properties in $T_{MJM}(\mathbf{p}_4)$ (magenta).</figcaption></figure>

A. Templex analysis for the SWAO Datasets

The general two-dimensional oceanographic geostrophic velocity and absolute dynamic topography fields of the studied area are obtained by the Global Ocean Gridded L4 product provided by the Copernicus Marine Environment Monitoring Service (CMEMS, https://doi.org/10.48670/moi-00148). Particles are

advected using a MATLAB script which integrates the velocity fields mentioned above.

The datasets analyzed in this section involves 4400 time series per topological grid. Each grid has 20×11 grid points, and the time-delay embeddings are constructed with 20 particles per grid point. All these data can be classified into topological classes, which will be illustrated using just one multiparticle set per class. The point clouds are not identical, of course, but the topological properties of all of them fit into one of these categories.

For the set located at $\mathbf{p}_1(0) = (59.1^\circ W, 44^\circ S)$, the complex $K_{SWAO}(\mathbf{p}_1)$ corresponds to a standard strip [Fig. 9(a)], corresponding to a period-1 orbit. When the set of particles is located at $\mathbf{p}_2(0) = (56^\circ W, 42.4^\circ S)$, the complex $K_{SWAO}(\mathbf{p}_2)$ forms a torus corresponding to quasi-periodic motion [Fig. 9(b)]. As a third example, for the set of particles located at $\mathbf{p}_3(0) = (56.4^\circ W, 42.5^\circ S)$, the complex forms a Möbius strip, associated with a period-2 orbit [Fig. 9(c)]. We, therefore, observe the same types of regular behavior in the SWAO dataset as in the MJM. The recurrence of topological classes 1, 2, and 3 in both datasets reflects the presence of non-mixing regions—separated from the surrounding fluid by fairly hermetic transport barriers—which are known to exhibit regular dynamics (e.g., period-1, period-2, or quasi-periodic motion).

A fourth set of particles located at $\mathbf{p}_4(0) = (54^\circ W, 42^\circ S)$ belongs to the SWAO's chaotic sea [Fig. 10(a)]. The BRAMAH complex $K_{SWAO}(\mathbf{p}_5)$ is constructed, leading to Betti numbers $\beta_0 = 1$, $\beta_1 = 3$, and
<figure><img src="image_11.png" /><figcaption>FIG. 9: The three types of regular particle dynamics in the SWAO. Top panel: Three-dimensional portraits of the four-dimensional embeddings. Middle panel: Multiparticle time series. Bottom panel: BRAMAH cell complexes. First column: Jet dynamics (period-1 orbit). Second column: Core of the eddy (quasi-periodic motion). Third column: Periphery of the eddy (period-2 orbit). The sequence of 1-cells forming the homology group generators at level 1, the 1-holes, are identified using colors in the cell complex.</figcaption></figure>

$\beta_2 = 0$, i.e., a singly connected object, with three 1-holes shown in different colors [Fig. 10(b)] and no enclosed empty cavities. The Betti numbers coincide with the MJM, but for different reasons, as we shall see.

The joining locus has a single component. The digraph $D_{\text{SWAO}}(\mathbf{p}_5)$ is drawn in Fig. 11(c). Comparing it with digraph $D_{\text{MJM}}(\mathbf{p}_4)$ clearly shows the topological non-equivalence between the two chaotic dynamics. From digraph $D_{\text{SWAO}}(\mathbf{p}_5)$, the generatex set –or the stripex set since all generatexes are of order 1– can be written as follows.

$$G_1 = S_1 = \{\gamma_4 \to \gamma_5 \to \gamma_1 \to \gamma_2 \to \gamma_3\},$$

$$G_2 = S_2 = \{\gamma_4 \to \gamma_5 \to \gamma_6\},$$

$$G_3 = S_3 = \{\gamma_4 \to \gamma_7 \to \gamma_8\}.$$

This dynamics has only three stripexes, contrary to the chaos observed in the MJM model, which is characterized by four stripexes. The three stripexes are Möbius strips in isolation, i.e., each of them has a local twist.

Moreover, in complex $\tilde{K}_{\text{SWAO}}(\mathbf{p}_5)$, each stripex has a local twist, another property that differs from the chaos in the MJM. The SWAO 1-holes are coherent with the stripex width-splitting and the tearing observed at the components of the splitting locus [Figs. 11(a) and 11(b)].

The central oscillatory loop and the width-splitting make up two 1-holes. The third one is due to a tearing setting $\gamma_1$ and $\gamma_6$ apart, in accordance with their local twists.

Even though the chaotic dynamics differ between the MJM and the SWAO dataset, the complexity observed in the ocean's chaotic sea appears comparatively moderate. This is partly due to the nature of the altimetric velocity fields used in this study, which are not direct observations but reconstructions from assimilated products. These products smooth out small-scale fluctuations and limit the resolution of high-frequency variability. As a result, many small-scale processes are absent from the dynamics we analyze. By focusing on the mesoscale and using altimetry data sampled daily, periodicities such as inertia and tides are effectively filtered out.

To illustrate how the topological results for the chaotic regions in the ocean may differ in different regions or periods, let us consider another example case of the SWAO chaotic sea belonging to May 2018. Results are shown in Fig.12 and Fig.13.

The joining locus in this case has two components. The digraph $D_{\text{SWAO}}(\mathbf{p}_6)$ is drawn in Fig. 13(c). Comparing it with digraph $D_{\text{SWAO}}(\mathbf{p}_5)$ clearly shows the topological non-equivalence between the two chaotic dynamics. From digraph $D_{\text{SWAO}}(\mathbf{p}_6)$, the generatex set –or the
<figure><img src="image_12.png" /><figcaption>FIG. 10: State portrait reconstruction for a set of particles centered at the initial position $p_5(0) = (54^\circ W, 42^\circ S)$, corresponding to the chaotic sea. (b) BRAMAH complex $K_{SWAO}(p_5)$ with the homology group generators at level 1 (the 1-holes) marked in different colors.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>FIG. 11: Templex $T_{SWAO}(p_5)$ for the chaotic sea particles in the SWAO dataset. The simplified BRAMAH complex $\tilde{K}_{SWAO}(p_5)$ with eight 2-cells is shown in the standard (a) and schematic complex (b) representations with solid lines representing the components of the joining (splitting) loci, alongside the corresponding digraph with outgoing nodes marked with circles and ingoing nodes with squares. (c). This dynamics has a joining locus with a single component and only three stripexes, contrary to the chaos that is observed in the MJM model, which is characterized by four stripexes.</figcaption></figure>

stripex set since all generatees are of order 1- can be written as follows.

$$G_1 = \{\gamma_6 \to \gamma_7 \to \gamma_8 \to \gamma_9\},$$

$$G_2 = \{\gamma_4 \to \gamma_5 \to \gamma_6 \to \gamma_{10} \to \gamma_{11}\},$$

$$G_3 = \{\gamma_4 \to \gamma_5 \to \gamma_6 \to \gamma_{10} \to \gamma_{12} \to \gamma_{13} \to \gamma_{14}\},$$

$$G_4 = \{\gamma_4 \to \gamma_5 \to \gamma_6 \to \gamma_{10} \to \gamma_{12} \to \gamma_1 \to \gamma_2 \to \gamma_3\}.$$
<figure><img src="image_14.png" /><figcaption>FIG. 12: State portrait reconstruction for a set of particles centered at the initial position $p_6(0) = (54.9°W, 43.2°S)$, corresponding to the chaotic sea. (b) BRAMAH complex $K_{SWAO}(p_6)$ with the homology group generators at level 1 (the 1-holes) marked in different colors.</figcaption></figure>

The stripex set has five elements as follows.

$$ 
\begin{aligned}
S_1 &= \{\gamma_6 \to \gamma_7 \to \gamma_8 \to \gamma_9\}, \\
S_2 &= \{\gamma_4 \to \gamma_5\}, \\
S_3 &= \{\gamma_6 \to \gamma_{10} \to \gamma_{11}\}, \\
S_4 &= \{\gamma_6 \to \gamma_{10} \to \gamma_{12} \to \gamma_{13} \to \gamma_{14}\}, \\
S_5 &= \{\gamma_6 \to \gamma_{10} \to \gamma_{12} \to \gamma_1 \to \gamma_2 \to \gamma_3\}.
\end{aligned}
$$

There are no local twists, in agreement with the template-like representation of the complex in Fig. 13.

The seasonal variability of the chaotic dynamics can be understood in terms of the two major oceanic fronts present in the SWAO: the Subantarctic Front (SAF) and the Brazil Current Front (BCF), which serve as boundaries between distinct water masses. At the eastern boundary of the Overshoot region, the orientation of these fronts changes seasonally, shifting from a north-south direction in winter to a northwest-southeast direction in summer$^{80,82}$. This observation, combined with the fact that high values of mean eddy kinetic energy have been reported to the east of the Overshoot$^{84}$, suggests that the orientation of the fronts may influence the emergence of distinct dynamical regimes. In particular, it may promote more chaotic behavior when the passage of mesoscale eddies is enhanced by the front's orientation. This hypothesis could explain the appearance of different topologies in different seasons.

## B. Topological grids for the SWAO Datasets

We label the topologies distinguished in the SWAO datasets with the integers 1, 2, 3, 5, and 6, where 5 and 6 correspond to the chaotic-sea dynamics in different seasons. Labels 1, 2, and 3 remain unchanged, as the topological features of the different types of regular dynamics are consistent across datasets. The label 4, which was used to denote the topological class of the chaotic sea in the MJM, is intentionally skipped.

The sequence of topological grids obtained for October-November 2018 is shown in Fig. 14, where the dynamics of the chaotic region is labeled as class 5. Although these grids evolve slightly over time, the location of the different types of dynamics remains broadly stable, with the quasi-periodic domains exhibiting the greatest variability. A single representative grid for May 2018 is shown in Fig. 15, where the chaotic dynamics corresponds to a different flow topology, labeled as class 6.

## C. Spatial organization in the SWAO

The spatial organization of the circulation in the SWAO has been addressed in the literature based on different methodologies. Among these, Lagrangian Coherent Structure (LCS) theory$^{85,86}$ provides the background for the detection of material curves in the flow that are centerpieces of fluid organization. These curves are typically defined on the basis of a variational principle as extremizers of a given geometrical property (e.g., finding the least stretching curve). This methodology is geomet-
<figure><img src="image_15.png" /><figcaption>FIG. 13: Templex $T_{SWAO}(\mathbf{p}_6)$ for the chaotic sea particles in the SWAO dataset for May 2018. The simplified BRAMAH complex $\tilde{K}_{SWAO}(\mathbf{p}_6)$ with fourteen 2-cells is shown in the standard (a) and schematic complex (b) representations with solid lines representing the components of the joining (splitting) loci, alongside the corresponding digraph with outgoing nodes marked with circles and ingoing nodes with squares. (c). This dynamics has a joining locus with two components and five stripexes, contrary to the chaos that is observed in the SWAO dataset for November 2018, which is characterized by only three stripexes.</figcaption></figure>

rical in nature and builds on the calculation of the metric tensor defining the derivative of the map connecting an initial distribution of fluid elements to their final position.

The templex approach and the LCS approach are in fact complementary, offering non-overlapping insights into spatial organization of the flow. Since they are based on fundamentally different principles –intrinsic properties in state space versus magnitudes in physical space– comparisons between them are not direct. Yet, Fig.16 shows a strong spatial resemblance between the hyperbolic Lagrangian Coherent Structures (LCS), colored according to their attracting rate, and the distribution of numbers in the topological grid. When plotted on a map, attracting rate values allow the heuristic identification of material barriers in the flow (e.g. minima may be related to jet-core structures).

The topological class associated with the Malvinas Current is characterized by standard strips with a short first-return timescale. This jet-like dynamics (1) can be distinguished from the standard strips in the Shelf (1\*) and the open ocean.⁵⁶,⁵⁷ In these studies, jet cores are unveiled based on the computation of parabolic LCS, which are not shown but are implicitly and heuristically suggested by trenches observed in the attraction rate of the LCS. The jet-like behavior extends out of the Malvinas Current and into its retroflection in the Western Atlantic

Ocean. The distinction between first-return timescales, the continental shelf and the Malvinas Current has been abundantly discussed in the literature:⁷⁶ the boundary current is typically associated with large velocity values and characteristic timescales that are shorter than those exhibited by the sea shelf. With regard to the eddy field in the open ocean, toroidal surfaces are the topological class associated with particles describing swirling motion around mesoscale features. Indeed, Fig.16 shows that large values of the attraction rate of the LCS generally correspond to regions exhibiting dynamics corresponding to a Möbius strip.

To complement the previous discussion on the consistency of results, the topological grid for November 6, 2018, has been superimposed on the map of chlorophyll-a concentration—see Fig. 17. On a large scale in the open ocean, a spatial resemblance can be observed between the chlorophyll-a concentration and the presence of jets or eddy-like features. This suggests that the templex approach correctly distinguishes the qualitatively different behaviors of the flow. It is important to note that the biological variable does not strictly constitute a passive tracer, as its evolution in time depends not only on advection but also on the availability of nutrients (source term) and on the death rate (sink). On average, the intercomparison remains valid since, over two weeks, advection terms typically dominate the balance.⁸⁷
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0255611

<figure><img src="image_16.png" /><figcaption>FIG. 14: Topological grid obtained from 4400 time series per snapshot (twenty particles per grid point) in the SWAO dataset. The tags correspond to the topological class found at each grid point: 1 and 1\* - standard strips with different periods (green), 2 - toroidal surface (blue), 3 - Möbius strip (yellow), and 5 - templex properties in *T*<sub>SWAO</sub>(*p*<sub>5</sub>) (red). The tags are juxtaposed with ADT (Absolute Dynamic Topography) height values [metres] overlaid with the altimetric velocity fields (arrows).</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>FIG. 15: Topological grid obtained from 4400 time series (twenty particles per grid point) in the SWAO dataset for the period of May 2018. The tags for topological classes are juxtaposed with ADT (Absolute Dynamic Topography) height values [metres] overlaid with the altimetric velocity fields.</figcaption></figure>

V. CONCLUDING REMARKS

This work builds upon previous attempts to describe unsteady fluid flows in terms of qualitatively different particle dynamics using algebraic topology. Due to the relationship between types of dynamics and state-space topology,⁶ this kind of approach complements a variety of diagnostics associated with transport and mixing, known as Lagrangian coherence detection methods.¹⁷ As particles moving together share the same type of behavior,³⁷ classifying them in terms of the underlying finite-time topological structure in state space allows one to map how the different types of behavior are spatially distributed in the fluid domain.³⁵ This principle was successfully applied, using the BRAMAH approach, to fluid particles governed by nonautonomous kinematic models for the ocean or the atmosphere—such as Shadden’s Driven Double Gyre⁸⁸ or the Bickley jet⁶²—as well as to flow fields resulting from direct numerical integrations of the Navier–Stokes equations.³⁶

Here we make a significant step forward in two directions: on the one hand, we extend the application of the topological approach to observational datasets; on the other hand, we improve the description of the chaotic sea⁶⁷ using a recently introduced concept called templex.²⁴ The templex arises from two research lines on the topological structure of chaotic attractors: the template,⁸,²⁶,²⁷ which describes flows in three-dimensional state space based on knot theory,⁸⁹⁻⁹¹ and the BRAMAH complex,²²,²³ which describes the topology of the (branched) manifold supporting the trajectories. The templex approach goes beyond the tem-

plate because it is ‘orbitless’ and ‘knotless’:⁴⁵ it does not rely on reconstructing unstable periodic orbits to form knots, and its properties can be computed regardless of the state-space dimension. It also goes beyond the BRAMAH complex because a collection of glued cells cannot describe how the cells are visited by the flow. The templex thus enables the computation of algebraic invariants that capture the full set of topological features, simultaneously required to identify—and thereby correctly distinguish—different types of chaos.

By applying a simple yet nonautonomous meandering jet model⁶¹ (rescaled to apply to the ocean) together with observational (altimetric) data for the Southwestern Atlantic, this work demonstrates that the templex approach enhances Lagrangian analysis in the ocean.⁹² It enables not only the detection of finite-time coherent sets in motion and the characterization of regular dynamics in non-mixing regions, but also a full description of the dynamics of the chaotic sea. The topological description of regular dynamics—such as period-1 orbits, period-2 orbits, and quasi-periodic motion—is consistent between the theoretical model and the altimetric data. As expected, differences arise in the chaotic dynamics. In real oceanic flows, additional factors such as bathymetry, coastline geometry, and atmosphere-ocean coupling introduce non-stationarity and structural variability that are typically absent in idealized models. These aspects can lead to differences in the phase-space organization observed in satellite-derived data, even when the underlying dynamics have some points in common. The Betti numbers coincide between the model and the chaotic sea in one season, but differ by one 1-hole when another season is explored. Even in the case where the homological description coincides, the templex properties reveal differences in the flow topology, illustrating that the mechanisms responsible for the 1-holes in the underlying structures are not the same. This result exemplifies why homologies alone are not sufficient to determine whether two chaotic attractors are topologically equivalent.

The theoretical model is dynamically motivated but is not expected to reproduce the specific Lagrangian features around the Malvinas Current.⁵⁶,⁵⁷ While many processes are neglected in the idealized streamfunction, the altimetric data also has limited spatio-temporal resolution and does not capture inertial motions, tides, or other processes that can instead be resolved by observational platforms like drifters.

The level of detail provided by our analysis shows that the templex can, indeed, be used to compare models against different types of observational data, as well as to unravel the particular dynamical features of different geographic zones and seasons. The Southwestern Atlantic Ocean, known for its high primary productivity, remains the focus of intense scientific debate regarding the origins of this productivity.⁹³ In this context, the comparative assessment of the mechanisms of chaotic transport should help provide answers to longstanding scientific questions. Moreover, this methodology is expected to shed light on
data collected from elephant seals,¹³⁻¹⁶ providing further insights into their movement patterns and interactions with ocean currents.

Finally, the templex properties obtained for the chaotic-sea particles demonstrate that conservative chaos can be topologically characterized over finite time. This represents a significant first success of its kind in the literature.

<figure><img src="image_18.png" /><figcaption>FIG. 16: Topological grid superimposed with hyperbolic Lagrangian Coherent Structures colored according to their attracting rate.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIG. 17: Topological grid superimposed with chlorophyll-a concentration [mg m⁻³].</figcaption></figure>

## DATA AVAILABILITY

The WOLFRAM MATHEMATICA notebooks for the topological analysis of the SWAO datasets are available at https://git.cima.fcen.uba.ar/sciamarella/templex-properties/-/tree/main/examples/SWAO.

## ACKNOWLEDGMENTS

This work was funded by the ANR project TeMPlex ANR-23-CE56-0002 (D.S.), which financed G.D.C.'s postdoctoral position at LSCE and N.B.'s postdoctoral position at LMD. J.C.B. gratefully acknowledges his doctoral scholarship from MITI (Mission pour les initiatives transverses et interdisciplinaires) and his completion doctoral scholarship from CONICET. This study was also carried out in the framework of the EUMET-SAT/CNES project "SABIO" 2020.0027659.

## Appendix A: BRAMAH cell complex

Let $S(x_0) = \{x_i | i = 1, \dots, n_c\}$ be a subset of $n_c$ points of the $\mathcal{N}$ cloud, around a point $x_0$. To verify whether $S(x_0)$ can be approximated by an $m$-cell, we analyze whether the points of $S$ belonging to a ball with center $x_0$ and radius $r$ locally approximate an $m$-dimensional disk. The distribution of points in $S(x_0)$ can be analyzed by constructing the matrix $X \in \mathbb{R}^{n_c \times n}$

$$X_{i,j} = \frac{1}{n_c^{1/2}} (x_{i,j} - x_{0,j})$$

where $n_c$ is the number of points that are intended to be grouped into a single $m$-cell. The eigenvectors of $X^T X$ are known as the singular vectors of $X$, and the square root of each eigenvalue of $X^T X$ is called the singular value of $X$. The singular vectors provide a local coordinate system centered on $x_0$, and their singular values describe the distribution of points around $x_0$. If $S(x_0)$ approximates a disk of dimension $m$ in $\mathbb{R}^n$, there must exist $m$ singular values of $X$ that are linear functions of the number of points in $S(x_0)$ ordered in a sequence of increasing distance to $x_0$. The remaining singular values ($n - m$) will measure the deviation of the tangent space from the disk and will scale as $r^\ell$ with $\ell \ge 2$. If the set satisfies this requirement, $S(x_0)$ is assigned an $m$-cell sharing faces with neighboring cells to form a complex.

A complex constructed in this manner is called a BRAMAH complex since it is designed to carry out a Branched Manifold Analysis through Homologies. Notice that the cell complex needs not correspond to a chaotic attractor; it can also describe manifolds supporting regular (nonchaotic) flows, as well as finite-time dataset embeddings. The construction parameters are the minimum and maximum number of points in an $m$-cell, and the regression coefficient for the linear regression, which regulates the curvature that is tolerated. These values can
be chosen according to the minimal and maximal size of the *m*-cells desired. The size must be large enough to approximate an *m*-dimensional disk and small enough to capture the structure of the manifold supporting the point cloud. Visual inspection helps determine these values in low-dimensional cases, where projections give a hint of the structure's shape. Within this range, there is no significant sensitivity to the specific choice of thresholds: homologies are designed to yield the same result independently of the number (and size) of *m*-cells used to build the complex.

## Appendix B: Templex properties

A templex $T = (K, D)$ of dimension $m$ is made of a BRAMAH *m*-complex $K$ approximating the structure supporting a flow in state space, and a digraph $D$ such that the nodes are the *m*-cells of $K$ and the directed edges are the flow-allowed connections between the *m*-cells.<fnref n="24" />,⁴⁷

Closed flow paths in the cell complex can be expressed as digraph cycles: these cycles are of course different from the nontrivial cycles used in the computation of the homology groups. The homology cycles concern the underlying structure and the digraph cycles concern the flow upon the structure. The representative flow cycles can be computed from the combined algebraic handling of the cell complex and the associated digraph.

A generatex $G$ is a subtemplex in $T = (K, D)$, where the subgraph is a cycle in $D$, satisfying a number of conditions regarding the cell distribution in $K$ next to the joining locus. The generatex set contains the representative flow cycles, which exit and return to the joining locus by following the flow-compatible higher-dimensional cell connections indicated in the digraph.

The path corresponding to a generatex that passes through different components of the same joining locus more than once, say *p* times, is said to be degenerated. When the paths are of order *p*, we get the stripex set. The splitting locus corresponds to the cells that are not part of the joining locus and serve as points from which the flow can diverge to at least two distinct elements of the stripex set. A stripex may exhibit a twisted or untwisted configuration within the templex. It is classified as twisted if the edges attached to the joining line shift their relative alignment prior to reinsertion.

An algorithmic implementation of the computation of both the BRAMAH complex and the templex properties, applied to several examples, is provided at https://community.wolfram.com/groups/-/m/t/3079776.⁹⁴

<footnote n="1">¹H. Poincaré, "Analysis Situs," Journal de l'École Polytechnique 1, 1-121 (1895).</footnote>

<footnote n="2">²H. Poincaré, "Sur le problème des trois corps et les équations de la dynamique," Acta mathematica 13, 1-270 (1890).</footnote>

<footnote n="3">³H. Poincaré, *Les Méthodes Nouvelles de la Mécanique Céleste* III (Gauthier-Vilard, Paris, 1899).</footnote>

<footnote n="4">⁴F. Verhulst, "Henri Poincaré's inventions in dynamical systems and Topology," in *The Foundations of Chaos Revisited: From Poincaré to Recent Advancements*, edited by C. Skiadas (Springer International Publishing, Cham, 2016) pp. 1-25.</footnote>

<footnote n="5">⁵J. Birman and R. F. Williams, "Knotted periodic orbits in dynamical systems I. Lorenz's equations," *Topology* **22**, 47-82 (1983).</footnote>

<footnote n="6">⁶J. Birman and R. F. Williams, "Knotted periodic orbits in dynamical systems II. Knot holders for fibred knots," *Contemporary Mathematics* **20**, 1-60 (1983).</footnote>

<footnote n="7">⁷N. B. Tufillaro, T. Abbott, and J. Reilly, *An experimental approach to nonlinear dynamics and chaos* (Addison-Wesley, Redwood City, CA, 1992).</footnote>

<footnote n="8">⁸R. Gilmore, "Topological analysis of chaotic dynamical systems," *Reviews of Modern Physics* **70**, 1455-1529 (1998).</footnote>

<footnote n="9">⁹R. Lumpkin, T. Özgökmen, and L. Centurioni, "Advances in the application of surface drifters," *Annual review of marine science* **9**, 59-81 (2017).</footnote>

<footnote n="10">¹⁰S. Essink, V. Hormann, L. R. Centurioni, and A. & Mahadevan, "On characterizing ocean kinematics from surface drifters," *Journal of Atmospheric and Oceanic Technology* **39**, 1183-1198 (2022).</footnote>

<footnote n="11">¹¹A. P. Wong, S. E. Wijffels, Riser, P. S. C., S. S., Hosoda, D. Roemmich, and H. M. Park, "Argo data 1999-2019: Two million temperature-salinity profiles and subsurface velocity observations from a global array of profiling floats." *Frontiers in Marine Science* **7**, 700 (2020).</footnote>

<footnote n="12">¹²S. Martin, "An introduction to ocean remote sensing," *Cambridge University Press* (2014).</footnote>

<footnote n="13">¹³C. Guinet, J. V. Garcia, B. Picard, G. Bessigneul, Y. Lebras, A. C. Dragon, V. Morgane, J. P. Y. Arnould, and F. Bailleul, "Southern elephant seal foraging success in relation to temperature and light conditions: insight into prey distribution," *Marine Ecology Progress Series* **499**, 285-301 (2014).</footnote>

<footnote n="14">¹⁴A. Della Penna, S. De Monte, E. Kestenare, C. Guinet, and F. d'Ovidio, "Quasi-planktonic behavior of foraging top marine predators," *Nature: Scientific reports* **5** (2015).</footnote>

<footnote n="15">¹⁵N. Fonvieille, C. Guinet, M. Saraceno, B. Picard, M. Tournier, P. Goulet, C. Campagna, J. Campagna, and D. Nerini, "Swimming in an ocean of curves: A functional approach to understanding elephant seal habitat use in the Argentine basin," *Progress in Oceanography* **218**, 103120 (2023).</footnote>

<footnote n="16">¹⁶N. Fonvieille, C. Guinet, M. Saraceno, B. Picard, M. Tournier, P. Goulet, C. Campagna, J. Campagna, and D. Nerini, "Satellite and high-spatio-temporal resolution data collected by southern elephant seals allow an unprecedented 3d view of the Argentine continental shelf," *Remote Sensing* **15**, 5604 (2023).</footnote>

<footnote n="17">¹⁷M. R. Allshouse and T. Peacock, "Lagrangian based methods for coherent structure detection," *Chaos: An Interdisciplinary Journal of Nonlinear Science* **25**, 097617 (2015).</footnote>

<footnote n="18">¹⁸J. Skinner, A. Lawrence, and J. Callies, "Persistent Homology of Ocean Flows," in *American Geophysical Union, Ocean Sciences Meeting* (2024) pp. PL44B-2304.</footnote>

<footnote n="19">¹⁹M. Kramár, R. Levanger, J. Tithof, B. Suri, M. Xu, M. Paul, M. F. Schatz, and K. Mischaikow, "Analysis of kolmogorov flow and rayleigh-bénard convection using persistent homology," *Physica D: Nonlinear Phenomena* **334**, 82-98 (2016).</footnote>

<footnote n="20">²⁰J. Núñez, A. González, and E. Ramos, "Topological data analysis of lagrangian orbits in natural convection flows confined in a cylinder." *Physical Review Fluids* **7**, 123501 (2022).</footnote>

<footnote n="21">²¹A. M. Mancho, S. Wiggins, J. Curbelo, and C. Mendoza, "Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems," *Communications in Nonlinear Science and Numerical Simulation* **18**, 3530-3557 (2013).</footnote>

<footnote n="22">²²D. Sciamarella and G. Mindlin, "Topological structure of chaotic flows from human speech data," *Physical review letters* **82**, 1450 (1999).</footnote>

<footnote n="23">²³D. Sciamarella and G. Mindlin, "Unveiling the topological structure of chaotic flows from data," *Physical review E* **64**, 036209 (2001).</footnote>

<footnote n="24">²⁴G. D. Charó, C. Letellier, and D. Sciamarella, "A bridge between</footnote>
homologies and templates for chaotic attractors,” Chaos: An Interdisciplinary Journal of Nonlinear Science 32, 083108 (2022).

25C. Mosto, G. D. Charó, C. Letellier, and D. Sciamarella, “Templex-based dynamical units for a taxonomy of chaos,” Chaos: An Interdisciplinary Journal of Nonlinear Science 34 (2024).

26G. M. Mindlin and R. Gilmore, “Topological analysis and synthesis of chaotic time series,” Physica D 58, 229–242 (1992).

27C. Letellier, P. Dutertre, and B. Maheu, “Unstable periodic orbits and templates of the Rössler system: Toward a systematic topological characterization,” Chaos 5, 271–282 (1995).

28C. Letellier, L. L. Sceller, P. Dutertre, G. Gouesbet, Z. Fei, and J. L. Hudson, “Topological characterization and global vector field reconstruction from an experimental electrochemical system,” Journal of Physical Chemistry 99, 7016–7027 (1995).

29R. Gilmore and M. Lefranc, *The Topology of Chaos: Alice in Stretch and Squeezeland* (Wiley-VCH, 2003) pp. 397–420.

30M. Lefranc, “Alternative determinism principle for topological analysis of chaos,” Physical Review E 74, 035202 (2006).

31R. Martin and S. Mangiarotti, “Topological description of the Lorenz-84 chaotic attractor,” From the nonlinear dynamical systems theory to observational chaos, 20–25 (2023).

32S. Mangiarotti, “Modélisation globale et caractérisation topologique de dynamiques environnementales: de l’analyse des enveloppes fluides et du couvert de surface de la Terre à la caractérisation topolodynamique du chaos,” Habilitation to Supervise Researches (2014), Université de Toulouse, France.

33S. Mangiarotti, M. Peyre, and M. Huc, “A chaotic model for the epidemic of Ebola virus disease in West Africa (2013-2016),” Chaos 26, 113112 (2016).

34G. D. Charó, D. Sciamarella, S. Mangiarotti, G. Artana, and C. Letellier, “Observability of laminar bidimensional fluid flows seen as autonomous chaotic systems,” Chaos: An Interdisciplinary Journal of Nonlinear Science 29 (2019).

35G. D. Charó, G. Artana, and D. Sciamarella, “Topology of dynamical reconstructions from lagrangian data,” Physica D: Nonlinear Phenomena 405, 132371 (2020).

36G. D. Charó, G. Artana, and D. Sciamarella, “Topological colouring of fluid particles unravels finite-time coherent sets,” Journal of Fluid Mechanics 923, A17 (2021).

37D. H. Kelley, M. R. Allshouse, and N. T. Ouellette, “Lagrangian coherent structures separate dynamically distinct regions in fluid flows,” Physical Review E 88, 013017 (2013).

38O. Ménard, C. Letellier, J. Maquet, L. L. Sceller, and G. Gouesbet, “Analysis of a non synchronized sinusoidally driven dynamical system,” International Journal of Bifurcation & Chaos 10, 1759–1772 (2000).

39C. Letellier, I. Sendiña-Nadal, E. Bianco-Martinez, and M. S. Baptista, “A symbolic network-based nonlinear theory for dynamical systems observability,” Scientific Reports 8, 3785 (2018).

40M. B. Kennel, R. Brown, and H. D. I. Abarbanel, “Determining embedding dimension for phase-space reconstruction using a geometrical construction,” Physical review A 45, 3403 (1992).

41H. D. I. Abarbanel and M. B. Kennel, “Local false nearest neighbors and dynamical dimensions from observed chaotic data,” Physical Review E 47, 3057–3068 (1993).

42H. D. Abarbanel and J. P. Gollub, “Analysis of observed chaotic data,” Physics Today 49, 86 (1996).

43S. Mangiarotti, F. Le Jean, M. Huc, and C. Letellier, “Global modeling of aggregated and associated chaotic dynamics,” Chaos, Solitons & Fractals 83, 82–96 (2016).

44L. A. Aguirre and C. Letellier, “Modeling nonlinear dynamics and chaos: A review,” Mathematical Problems in Engineering 2009, 238960 (2009).

45M. A. Natiello et al., *The user's approach to topological methods in 3d dynamical systems* (World Scientific, 2007).

46E. Rips, “On the dimension of certain compact sets,” Proceedings of the American Mathematical Society 45, 122–124 (1974).

47D. Sciamarella and G. D. Charó, “New elements for a theory of chaos topology,” in *Topological Methods for Delay and Ordi-

nary Differential Equations: With Applications to Continuum Mechanics* (Springer, 2024) pp. 191–211.

48C. Letellier and R. Gilmore, “Covering dynamical systems: Two-fold covers,” Physical Review E 63, 016206 (2001).

49E. Johns, R. Watts, and T. Rossby, “A test of geostrophy in the Gulf Stream,” Journal of Geophysical Research 94, 3211–3222 (1989).

50A. Robinson, “A discussion on ocean currents and their dynamics - the Gulf Stream,” Journal of Mathematical, Physical and Engineering Sciences 270 (1971).

51K. Komatsu and Y. Hiroe, “Structure and impact of the Kuroshio nutrient stream,” Geophysical Monograph Series 243, 85–102 (2019).

52K. V. Koshel and S. V. Prants, “Chaotic advection in the ocean,” Russian Academy of Sciences 49, 1151–1178 (2006).

53D. Nowlin and J. Klinck, “The physics of the Antarctic Circumpolar Current,” Review of Geophysics 24, 469–491 (1986).

54S. Rintoul, C. Hughes, and D. Olbers, “The Antarctic Circumpolar Current System,” International Geophysics 77, 271–302 (2001).

55R. P. Matano, E. D. Palma, and A. R. Piola, “The influence of the Brazil and Malvinas Currents on the Southwestern Atlantic Shelf circulation,” Ocean Science 6, 983–995 (2010).

56F. J. Beron Vera, N. Bodnariuk, M. Saraceno, M. J. Olascoaga, and C. Simionato, “Stability of the Malvinas Current,” Chaos 30 (2020).

57N. Bodnariuk, M. Saraceno, L. A. Ruiz-Etcheverry, C. Simionato, F. Andrade-Canto, F. J. Beron-Vera, and M. J. Olascoaga, “Multiple Lagrangian Jet-Core Structures in the Malvinas Current,” Journal of Geophysical Research: Oceans 129, e2023JC020446 (2024).

58E. Mason, A. Pascual, P. Gaube, S. Ruiz, J. Pelegrí, and A. Delepoulle, “Subregional characterization of mesoscale eddies across the Brazil-Malvinas Confluence,” Journal of Geophysical Research: Oceans 122, 2643–3499 (2017).

59I. S. Robinson, “Mesoscale ocean features: Eddies,” in *Discovering the Ocean from Space: The unique applications of satellite oceanography* (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010) pp. 69–114.

60S. Prants, M. Budyansky, M. Y. Uleysky, and G. Zaslavsky, “Chaotic mixing and transport in a meandering jet flow,” Chaos: An Interdisciplinary Journal of Nonlinear Science 16 (2006).

61I. Rypina, M. G. Brown, F. J. Beron-Vera, H. Koçak, M. J. Olascoaga, and I. Udovydchenkov, “On the lagrangian dynamics of atmospheric zonal jets and the permeability of the stratospheric polar vortex,” Journal of the Atmospheric Sciences 64, 3595–3610 (2007).

62W. Bickley, “The plane jet,” Philosophical Magazine and Journal of Science 23, 727–731 (1937).

63F. B. Lipps, “The barotropic stability of the mean winds in the atmosphere,” Journal of Fluid Mechanics 12, 397–407 (1962).

64A. Hadjighasem, D. Karrasch, H. Teramoto, and G. Haller, “Spectral-clustering approach to lagrangian vortex detection,” Physical Review E 93, 063107 (2016).

65V. O. Ivchenko, K. J. Richards, and D. P. Stevens, “The Dynamics of the Antarctic Circumpolar Current,” Journal of Physical Oceanography 26, 753–774 (1996).

66S. E. Best, V. O. Ivchenko, K. J. Richards, R. D. Smith, and R. C. Malone, “Eddies in Numerical Models of the Antarctic Circumpolar Current and Their Influence on the Mean Flow,” Journal of Physical Oceanography 29, 328–350 (1999).

67M. Hénon and C. Heiles, “The applicability of the third integral of motion: some numerical experiments,” The astronomical Journal 69, 73–79 (1964).

68S. Mangiarotti, “Analyse topologique appliquée à deux cas de chaos faiblement dissipatif: le système de Lorenz-84 et le modèle cereal crops,” Rencontres du Non-Linéaires 18, 73–78 (2015).

69E. D. Palma, R. P. Matano, and A. R. Piola, “A numerical study of the Southwestern Atlantic Shelf circulation: Barotropic response to tidal and wind forcing,” Journal of Geophysical Re-
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0255611

search: Oceans **109**, C8014 (2004).

Research: Oceans **110**, C11016 (2005).

<footnote n="70">⁷⁰N. Bodnariuk, C. G. Simionato, and M. Saraceno, "Sam-driven variability of the Southwestern Atlantic Shelf Sea Circulation," Continental Shelf Research **212**, 104313 (2021).</footnote>

<footnote n="71">⁷¹M. Saraceno, C. G. Simionato, and L. A. Ruiz-Etcheverry, “Sea surface height trend and variability at seasonal and interannual time scales in the Southeastern South American continental shelf between 27°s and 40°s,” Continental Shelf Research **91**, 82–94 (2014).</footnote>

<footnote n="72">⁷²L. S. Lago, M. Saraceno, P. Martos, R. A. Guerrero, A. R. Piola, G. F. Paniagua, R. Ferrari, C. I. Artana, and C. Provost, “On the Wind Contribution to the Variability of Ocean Currents Over Wide Continental Shelves: A Case Study on the Northern Argentine Continental Shelf,” Journal of Geophysical Research: Oceans **124**, 7457–7472 (2019).</footnote>

<footnote n="73">⁷³L. S. Lago, M. Saraceno, A. R. Piola, and L. A. Ruiz-Etcheverry, “Volume transport variability on the Northern Argentine Continental Shelf from in situ and satellite altimetry data,” Journal of Geophysical Research: Oceans **126**, e2020JC016813 (2021).</footnote>

<footnote n="74">⁷⁴M. G. Dinápoli, C. G. Simionato, G. Alonso, N. Bodnariuk, and R. Saurral, “Negative storm surges in the Río de la Plata Estuary: mechanisms, variability, trends and linkage with the Continental Shelf dynamics,” Estuarine, Coastal and Shelf Science **305**, 108844 (2024).</footnote>

<footnote n="75">⁷⁵A. L. Rivas, “Current-meter observations in the Argentine Continental Shelf,” Continental Shelf Research **17**, 391–406 (1997).</footnote>

<footnote n="76">⁷⁶M. Saraceno, N. Bodnariuk, L. Ruiz-Etcheverry, M. Berta, C. Simionato, F. Beron-Vera, and M. Olascoaga, “Lagrangian characterization of the Southwestern Atlantic from a dense surface drifter deployment,” Deep Sea Research Part I: Oceanographic Research Papers **208**, 104319 (2024).</footnote>

<footnote n="77">⁷⁷R. P. Matano and E. D. Palma, “On the upwelling of downwelling currents,” Journal of Physical Oceanography **38**, 2482 – 2500 (2008).</footnote>

<footnote n="78">⁷⁸A. R. Piola, N. Martínez Avellaneda, R. A. Guerrero, F. P. Jardón, E. D. Palma, and S. I. Romero, “Malvinas-slope water intrusions on the Northern Patagonia Continental Shelf,” Ocean Science **6**, 345–359 (2010).</footnote>

<footnote n="79">⁷⁹C. Artana, C. Provost, J.-M. Lellouche, M.-H. Rio, R. Ferrari, and N. Sennéchael, “The Malvinas Current at the Confluence with the Brazil Current: Inferences from 25 years of Mercator Ocean Reanalysis,” Journal of Geophysical Research: Oceans **124**, 7178–7200 (2019).</footnote>

<footnote n="80">⁸⁰M. Saraceno, C. Provost, and A. R. Piola, “On the relationship between satellite-retrieved surface temperature fronts and chlorophyll a in the western South Atlantic,” Journal of Geophysical</footnote>

<footnote n="81">⁸¹L. Stramma and R. G. Peterson, “The South Atlantic Current,” Journal of Physical Oceanography **20**, 846 – 859 (1990).</footnote>

<footnote n="82">⁸²M. Saraceno, C. Provost, A. R. Piola, J. Bava, and A. Gagliardi, “Brazil Malvinas frontal system as seen from 9 years of advanced very high resolution radiometer data,” Journal of Geophysical Research: Oceans **109**, C05027 (2004).</footnote>

<footnote n="83">⁸³C. Artana, R. Ferrari, Z. Koenig, M. Saraceno, A. R. Piola, and C. Provost, “Malvinas Current variability from Argo floats and satellite altimetry,” Journal of Geophysical Research: Oceans **121**, 4854–4872 (2012).</footnote>

<footnote n="84">⁸⁴R. Ferrari, C. Artana, M. Saraceno, A. R. Piola, and C. Provost, “Satellite altimetry and current-meter velocities in the Malvinas Current at 418s: Comparisons and modes of variations.” J. Geophys. Res. **122**, C05027 (2017).</footnote>

<footnote n="85">⁸⁵G. Haller and G. Yuan, “Lagrangian coherent structures and mixing in two-dimensional turbulence,” Physica D: Nonlinear Phenomena **147**, 352–370 (2000).</footnote>

<footnote n="86">⁸⁶G. Haller, “Lagrangian coherent structures,” Annual Review of Fluid Mechanics **47**, 137–162 (2015).</footnote>

<footnote n="87">⁸⁷N. Barré, C. Provost, and M. Saraceno, “Spatial and temporal scales of the Brazil-Malvinas Current confluence documented by simultaneous MODIS Aqua 1.1-km resolution SST and color images,” Advances in Space Research **37**, 770–786 (2006).</footnote>

<footnote n="88">⁸⁸S. C. Shadden, F. Lekien, and J. E. Marsden, “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).</footnote>

<footnote n="89">⁸⁹P. G. Tait, “On knots I,” Transactions of the Royal Society of Edinburgh **28**, 145–190 (1877).</footnote>

<footnote n="90">⁹⁰P. G. Tait, “On knots II,” Transactions of the Royal Society of Edinburgh **32**, 327–342 (1884).</footnote>

<footnote n="91">⁹¹P. G. Tait, “On knots III,” Transactions of the Royal Society of Edinburgh **32**, 493–506 (1884).</footnote>

<footnote n="92">⁹²E. Van Sebille, S. M. Griffies, R. Abernathey, T. P. Adams, P. Berloff, A. Biastoch, B. Blanke, E. P. Chassignet, Y. Cheng, C. J. Cotter, et al., “Lagrangian ocean analysis: Fundamentals and practices,” Ocean modelling **121**, 49–75 (2018).</footnote>

<footnote n="93">⁹³V. Lutz, V. Segura, A. Dogliotti, V. Tavano, F. P. Brandini, D. L. Calliari, A. M. Ciotti, V. F. Villafañe, I. R. Schloss, F. M. P. S. Corrêa, H. Benavides, and D. V. Cantonnet, “Overview on Primary Production in the Southwestern Atlantic,” in *Plankton Ecology of the Southwestern Atlantic* (Springer International Publishing, 2018) pp. 101–126.</footnote>

<footnote n="94">⁹⁴D. Sciamarella, “Templex properties,” Wolfram Staff Picks (2023).</footnote>