Lagrangian coherent structures (LCS) are exceptional material surfaces that act as cores of observed tracer patterns in fluid flows (see  and  for reviews). For oceanic flows, the tracers of interest include salinity, temperature, contaminants, nutrients and plankton—quantities that play an important role in the ecosystem and even in climate. Fluxes of these quantities are typically dominated by advective transport over diffusion.
An important component of advective transport in the ocean is governed by mesoscale eddies, i.e., vortices of – km in diameter. While eddies also stir and mix surrounding water masses by their swirling motion, here we focus on eddies that trap and carry fluid in a coherent manner. Eddies of this kind include the Agulhas rings of the Southern Ocean. They are known to transport massive quantities of warm and salty water from the Indian Ocean into the Atlantic Ocean . Current limitations on computational power prevent that climate models resolve mesoscale eddies in their flow field. Since the effect of mesoscale eddies on the global circulation is significant , the correct parameterization of eddy transport is crucial for the reliability of these models. As a consequence, there is a rising interest in systematic and accurate eddy detection and census in large global data sets, as well as in quantifying the average transport of trapped fluid by all eddies in a given region [9, 38, 25].
This quantification requires (i) a rigorous method that provides specific coherent eddy boundaries, and (ii) a robust numerical implementation of the method on large velocity data sets.
A number of vortex definitions have been proposed in the literature, [15, 37], most of which are of Eulerian type, i.e., use information from the instantaneous velocity field. Typical global eddy studies [5, 9, 38, 25] are based on such Eulerian approaches. Evolving eddy boundaries obtained from Eulerian approaches, however, do not encircle and transport the same body of water coherently [17, 37]. Instead, fluid initialized within an instantaneous Eulerian eddy boundary will generally stretch, fold and filament significantly. Yet only coherently transported scalars resist erosion by diffusion in a way that a sharp signature in the tracer field is maintained. All this suggests that coherent eddy transport should ideally be analysed via Lagrangian methods that take into account the evolution of trajectories in the flow, such as, e.g., [27, 1, 19, 28, 31, 26]. Notably, however, none of these methods focuses on the detection of vortices and none provides an algorithm to extract exact eddy boundaries in unsteady velocity fields.
Only recently have mathematical approaches emerged for the detection of coherent Lagrangian vortices. These include the geometric approach [15, 17] and the set-oriented approach [14, 12, 13]. Here, we follow the geometric approach to coherent Lagrangian vortices, which defines a coherent material vortex boundary as a closed stationary curve of the averaged material strain . All solutions of this variational problem turn out to be closed material curves that stretch uniformly. Such curves are practically found as closed orbits of appropriate planar line fields .
In contrast to vector fields, line fields are special vector bundles over the plane. In their definition, only a one-dimensional subspace (line) is specified at each point, as opposed to a vector at each point. The importance of line field singularities in Lagrangian eddy detection has been recognized in, but has remained only partially exploited. Here, we point out a topological rule that enables a fully automated detection of coherent Lagrangian vortex boundaries based on line field singularities. This in turn makes automated Lagrangian eddy detection feasible for large ocean regions.
Based on the geometric approach, coherent Lagrangian vortices have so far been identified in oceanic data sets [3, 17], in a direct numerical simulation of the two-dimensional Navier–Stokes equations , in a smooth area-preserving map , in a kinematic model of an oceanic jet in , and in a model of a double gyre flow . With the exception of , however, these studies did not utilize the topology of line field singularities. Furthermore, none of them offered an automated procedure for Lagrangian vortex detection.
The orbit structure of line fields has already received considerable attention in the scientific visualization community (see [7, 33] for reviews). The problem of closed orbit detection has been posed in [7, Section 5.2.3], and was considered by , building on . In that approach, numerical line field integration is used to identify cell chains that may contain a closed orbit. Then, the conditions of the Poincaré–Bendixson theorem are verified to conclude the existence of a closed orbit for the line field. This approach, however, does not offer a systematic way to search for closed orbits in large data sets arising in geophysical applications.
This paper is organized as follows. In sec:Index_vf, we recall the index theory of planar vector fields. In sec:Index_lf, we review available results on indices for planar line fields, and deduce a topological rule for generic singularities inside closed orbits of such fields. Next, in sec:Application, we present an algorithm for the automated detection of closed line field orbits. We then discuss related numerical results on ocean data, before presenting our concluding remarks in sec:Conclusions.
2 Index theory for planar vector fields
Here, we recall the definition and properties of the index of a planar vector field . We denote the unit circle of the plane by , parametrized by the mapping , . In our notation, we do not distinguish between a curve as a function and its image as a subset of .
Definition 1 (Index of a vector field).
For a continuous, piecewise differentiable planar vector field and a simple closed curve , let be a continuous function such that is the angle between the -axis and . Then, the index (or winding number) of along is defined as
i.e., the number of turns of during one anticlockwise revolution along . Clearly, is well-defined only if there is no critical point of along , i.e., no point at which vanishes.
The index defined in def:index_vf has two important properties :
whenever , and are well-defined.
whenever can be obtained from by a continuous deformation (homotopy).
If encloses exactly one critical point of , then the index of with respect to ,
is well-defined, because its definition does not depend on the particular choice of the enclosing curve by homotopy invariance. Furthermore, the index of equals the sum over the indices of all enclosed critical points, i.e.,
provided all are isolated critical points. Finally, the index of a closed orbit of the vector field is equal to , because the vector field turns once along . Therefore, closed orbits of planar vector fields necessarily enclose critical points.
3 Index theory for planar line fields
We now recall an extension of index theory from vector fields to line fields . Let be the set of one-dimensional subspaces of , i.e., the set of lines through the origin . is sometimes also called the projective line, which can be endowed with the structure of a one-dimensional smooth manifold . This is achieved by parametrizing the lines via the -coordinate at which they intersect the horizontal line . The horizontal line is assigned the value .
Equivalently, elements of can be parametrized by their intersection with the upper semi-circle, denoted , with its right and left endpoints identified. This means that lines through the origin are represented by a unique normalized vector, pointing in the upper half-plane and parametrized by the angle between the representative vector and the -axis (Fig. 1). A planar line field is then defined as a mapping , with its differentiability defined with the help of the manifold structure of .
Line fields arise in the computation of eigenvector fields for symmetric, second-order tensor fields[8, 32]
. Eigenvectors have no intrinsic sign or length: only eigenspaces are well-defined at each point of the plane. Their orientation depends smoothly on their base point if the tensor field is smooth and has simple eigenvalues at that point. At repeated eigenvalues, isolated one-dimensional eigenspaces (and hence the corresponding values of the line field) become undefined.
Points to which a line field cannot be extended continuously are called singularities. These points are analogous to critical points of vector fields. Away from singularities, any smooth line field can locally be endowed with a smooth orientation. This implies the local existence of a normalized smooth vector field, which pointwise spans the respective line. Conversely, away from critical points, smooth vector fields induce smooth line fields when one takes their linear span pointwise.
Based on the index for planar vector fields, we introduce a notion of index for planar line fields following . First, for some differentiable line field and along some closed curve , pick at each point the representative upper half-plane vector from . This choice yields a normalized vector field along which is as smooth as , except where
crosses the horizontal subspace. At such a point, there is a jump-discontinuity in the representative vector from right to left or vice versa. To remove this discontinuity, the representative vectors are turned counter-clockwise by, , , i.e., the parametrizing angle is doubled. Thereby, the left end-point with angle is mapped onto the right end-point with angle . This representation permits the extension of the notion of index to planar line fields as follows.
Definition 2 (Index of a line field).
For a continuous, piecewise differentiable planar line field and a simple closed curve , we define the index of along as
The coefficient in this definition is needed to correct the doubling effect of . It also makes the index for a line field, generated by a vector field in the interior of , equal to the index of that vector field. Since def:index_lf refers to def:index_vf, the additional definitions and properties described in sec:Index_vf for vector fields carry over to line fields.
We call a curve an orbit of , if it is everywhere tangent to . The scientific visualization community refers to orbits of lines fields arising from the eigenvectors of a symmetric tensor as tensor (field) lines or hyperorbit (trajectories) [8, 32, 34].
By definition, the index of singularities of line fields can be a half integer, as opposed to the vector field case, where only integer indices are possible. Also, two new types of singularities emerge in the line field case: wedges (type ) of index , and trisectors (type ) of index [8, 32]. The geometry near these singularities is shown in Fig. 2.
Node, centre, focus and saddle singularities also exist for line fields, but these singularities turn out to be structurally unstable with respect to small perturbations to the line field .
In this paper, we assume that only isolated singularities of the generic wedge and trisector types are present in the line field of interest. In that case, we obtain the following topological constraint on closed orbits of the line field.
Let be a continuous, piecewise differentiable line field with only structurally stable singularities. Let be a closed orbit of , and let denote the interior of . We then have
where and denote the number of wedges and trisectors, respectively, in .
First, has index with respect to , i.e., . Second, its index equals the sum over all enclosed singularities, i.e.,
Since we consider structurally stable singularities only, these are isolated and of either wedge or trisector type. From (2), we then obtain the equality
from which Eq. (1) follows. ∎
Consequently, in the interior of any closed orbit of a structurally stable line field, there are at least two singularities of wedge type, and exactly two more wedges than trisectors. Thus, a closed orbit necessarily encircles a wedge pair, and hence the existence of such a pair serves as a necessary condition in an automated search for closed orbits in line fields. In Fig. 3, we sketch two possible line field geometries in the interior of a closed orbit.
4 Application to coherent Lagrangian vortex detection
Finding closed orbits in planar line fields is the decisive step in the detection of coherent Lagrangian vortices in a frame-invariant fashion [16, 3, 17]. Before describing the algorithmic scheme and showing results on ocean data, we briefly introduce the necessary background and notation for coherent Lagrangian vortices.
4.1 Flow map, Cauchy–Green strain tensor and –line field
We consider an unsteady, smooth, incompressible planar velocity field given on a finite time interval , and the corresponding equation of motion for the fluid,
We denote the associated flow map by , which maps initial values at time to their respective position at time . Recall that the flow map is as smooth as the velocity field . Its linearisation can be used to define the Cauchy–Green strain tensor field
which is symmetric and positive-definite at each initial value. The eigenvalues and eigenvectors of characterize the magnitude and directions of maximal and minimal stretching locally in the flow. We refer to these positive eigenvalues as , with the associated eigenspaces spanned by the normalized eigenvectors and .
As argued by , the positions of coherent Lagrangian vortex boundaries at time are closed stationary curves of the averaged tangential strain functional computed from . All stationary curves of this functional turn out to be uniformly stretched by a factor of under the flow map . These stationary curves can be computed as closed orbits of the –line fields , spanned by the representing vector fields
We refer to orbits of as –lines. In the special case of , the line field coincides with the shear line field defined in , provided that the fluid velocity field is incompressible.
We refer to points at which the Cauchy–Green strain tensor is isotropic (i.e., equals a constant multiple of the identity tensor) as Cauchy–Green singularities. For incompressible flows, only is possible at Cauchy–Green singularities, implying at these points. The associated eigenspace fields, and , are ill-defined as line fields at Cauchy–Green singularities, thus generically the line fields , and have singularities at these points. Conversely, the singularities of the line fields , and are necessarily Cauchy–Green singularities, as seen from the local vector field representation in Eq. (3).
Following [16, 17], we define an elliptic Lagrangian Coherent Structure (LCS) as a structurally stable closed orbit of for some choice of the sign, and for some value of the parameter . We then define a (coherent Lagrangian) vortex boundary as the locally outermost elliptic LCS over all choices of .
4.2 Index theory for –line fields
In regions where is not satisfied, is undefined. Such open regions necessarily arise around Cauchy–Green singularities, and hence does not admit isolated point-singularities. Consequently, the index theory presented in sec:Index_lf does not immediately apply to the –line field. We show below, however, that Cauchy–Green singularities are still necessary indicators of closed orbits of for arbitrary .
For , the set , on which is undefined, consists of open connected components. All Cauchy–Green singularities are contained in some of these -components. A priori, however, there may exist -components that do not contain Cauchy–Green singularities.
On the boundary , we have , and hence coincides with on , as shown in Fig. 4. Therefore, we may extend into by letting for all , thereby obtaining a continuous, piecewise differentiable line field, whose singularity positions coincide with those of the -singularities.
thm:topo_line applies directly to the continuation of the line field , and enables the detection of closed orbits lying outside the open set . In the case , the line field can similarly be extended in a continuous fashion into the interior of the set through the definition for all .
After its extension into the set , the line field inherits each Cauchy–Green singularity either from or from . A priori, the same Cauchy–Green singularity may have different topological types in the and line fields. By [7, Theorem 11], however, this is not the case: corresponding generic singularities of and share the same index and have the same number of hyperbolic sectors. Furthermore, the separatrices of the -singularity are obtained from the separatrices of the -singularity by reflection with respect to the singularity. In summary, -wedges correspond exactly to -wedges, and the same holds for trisectors. For the singularity type classification for , , we may therefore pick , irrespective of the sign of .
The singularity-type correspondence extends also to the limit case , i.e., to , as follows. Consider the one-parameter family of line-field extensions . By construction, the locations of point singularities coincide with those of the -singularities for any . Variations of correspond to continuous line-field perturbations, which leave the types of structurally stable singularities unchanged. Hence, the types of -singularities must match the types of corresponding -singularities, or equivalently of corresponding -singularities. To summarize, we obtain the following conclusion.
Any closed orbit of a structurally stable field necessarily encircles Cauchy–Green singularities satisfying Eq. (1).
4.3 A simple example: coherent Lagrangian vortex in the double gyre flow
We consider the left vortex of the double gyre flow , defined on the spatial domain by the ODE
We choose the parameters of the flow model as , , , , and .
In the –line field shown in Fig. 5(a), we identify a pair of wedge singularities. Any closed –line must necessarily enclose this pair by prop:eta-field. This prompts us to define a Poincaré section through the midpoint of the connecting line between the two wedges. For computational simplicity, we select the Poincaré section as horizontal. Performing a parameter sweep over –values, we obtain the outermost closed orbit shown in Fig. 5(a) for a uniform stretching rate of . Other non-closing orbits and the –line field are also shown for illustration. In addition, we show the line field topology around the wedge pair in the vortex core in Fig. 5(b).
In this simple example, the vortex location is known, and hence a Poincaré section could manually be set for closed orbit detection in the –line fields. In more complex flows, however, the vortex locations are a priori unknown, making a manual search unfeasible.
4.4 Implementation for vortex census in large-scale ocean data
Our automated Lagrangian vortex-detection scheme relies on prop:eta-field, identifying candidate regions in which Poincaré maps for closed –line detection should be set up. In several tests on ocean data, we only found the singularity configuration inside closed –lines. This is consistent with our previous genericity considerations. Consequently, we focus on finding candidate regions for closed –lines as regions with isolated pairs of wedges in the field. In the following, we describe the procedure for an automated detection of closed –lines.
1. Locate singularities.
Recall that Cauchy–Green singularities are points where. We find such points at subgrid-resolution as intersections of the zero level sets of the functions and , where denote the entries of the Cauchy–Green strain tensor.
2. Select relevant singularities.
We focus on generic singularities, which are isolated and are of wedge or trisector type. We discard tightly clustered groups of singularities, which indicate non-elliptic behavior in that region. Effectively, the clustering of singularities prevents the reliable determination of their singularity type. To this end, we select a minimum distance threshold between admissible singularities as , where denotes the grid size used in the computation of . We obtain the distances between closest neighbours from a Delaunay triangulation procedure.
3. Determine singularity type.
Singularities are classified as trisectors or wedges, following the approach developed in. Specifically, a circular neighbourhood of radius is selected around a singularity, so that no other singularity is contained in this neighbourhood. With a rotating radius vector of length , we compute the absolute value of the cosine of the angle enclosed by and , i.e., , with the eigenvector field interpolated linearly to 1000 positions on the radius circle around the singularity. The singularity is classified as a trisector, if is orthogonal to at exactly three points of the circle, and parallel to at three other points, which mark separatrices of the trisector (Fig. 2). Singularities not passing this test for trisectors are classified as wedges. Other approaches to singularity classification can be found in  and , which we have found too sensitive for oceanic data sets.
We discard wedge points whose closest neighbour is of trisector-type, because these wedge points cannot be part of an isolated wedge pair. We further discard single wedges whose distance to the closest wedge point is larger than the typical mesoscale distance of km. The remaining wedge pairs mark candidate regions for elliptic LCS (Fig. 6(a)).
5. Launch –lines from a Poincaré-section
We set up Poincaré sections that span from the midpoint of a wedge pair to a point apart in longitudinal direction (Fig. 6(b)). This choice of length for the Poincaré section captures eddies up to a diameter of km, an upper bound on the accepted size for mesoscale eddies. For a fixed –value, –lines are launched from 100 initial positions on the Poincaré section, and the return distance is computed. Zero crossings of the return distance function correspond to closed –lines. The position of zeros is subsequently refined on the Poincaré section through the bisection method. The outermost zero crossing of the return distance marks the largest closed –line for the chosen –value. To find the outermost closed –line over all –values, we vary from to in steps, and pick the outermost closed orbit as the Lagrangian eddy boundary. During this process, we make sure that eddy boundaries so obtained do enclose the two wedge singularities used in the construction, but no other singularities.
The runtime of our algorithm is dominated by the fifth step, the integration of –lines, as illustrated in Table 1 for the ocean data example in the next section. This is the reason why our investment in the selection, classification and filtering of singularities before the actual –line integration pays off.
4.5 Coherent Lagrangian vortices in an ocean surface flow
We now apply the method summarized in steps 1-5 above to two-dimensional unsteady velocity data obtained from AVISO satellite altimetry measurements. The domain of the data set is the Agulhas leakage in the Southern Ocean, represented by large coherent eddies that pinch off from the Agulhas current of the Indian Ocean.
Under the assumption of a geostrophic flow, the sea surface height serves as a streamfunction for the surface velocity field. In longitude-latitude coordinates , particle trajectories are then solutions of
where is the constant of gravity, is the mean radius of the Earth, and is the Coriolis parameter, with denoting the Earth’s mean angular velocity. For comparison, we choose the same spatial domain and time interval as in [3, 17]. The integration time is also set to days.
Fig. 6 illustrates the steps of the eddy detection algorithm. From all singularities of the Cauchy–Green strain tensor, isolated wedge pairs are extracted (Fig. 6(a)) and closed orbits are found by launching –lines from Poincaré sections anchored at those wedge pairs (Fig. 6(b)). Altogether, 14 out of the selected wedge pairs are encircled by closed orbits and, hence, by coherent Lagrangian eddy boundaries (Fig. 6(c)). The reduction to candidate regions consistent with prop:eta-field leads to significant gain in computational speed. This is because the computationally expensive integration of the –line field is only carried out in these regions (Table 1). For comparison, the computational cost on a single Poincaré section is already higher than the cost of identifying the candidate regions. Note also that two regions contain three wedges, which constitute two admissible wedge pairs. This explains how 78 wedges constitute 40 wedge pairs altogether.
|Runtime||Number of points|
|1. Localization||s||14,211 singularities|
|2. Selection||s||912 singularities|
|3. Classification||s||414 wedges|
|4. Filtering||s||78 wedges|
|5. Integration||s / wedge pair / –value||40 wedge pairs|
|End result||—||14 eddies|
We have discussed the use of index theory in the detection of closed orbits in planar line fields. Combined with physically motivated filtering criteria, index-based elliptic LCS detection provides an automated implementation of the variational results of  on coherent Lagrangian vortex boundaries. Our results further enhance the power of LCS detection algorithms already available in the Matlab toolbox LCS TOOL (cf. ).
Our approach can be extended to three-dimensional flows, where line fields arise in the computation of intersections of elliptic LCS with two-dimensional planes . Applied over several such planes, our approach allows for an automated detection of coherent Lagrangian eddies in three-dimensional unsteady velocity fields.
Automated detection of Lagrangian coherent vortices should lead to precise estimates on the volume of water coherently carried by mesoscale eddies, thereby revealing the contribution of coherent eddy transport to the total flux of volume, heat and salinity in the ocean. Related work is in progress.
The altimeter products used in this work are produced by SSALTO/DUACS and distributed by AVISO, with support from CNES (http://www.aviso.oceanobs.com). We would like to thank Bert Hesselink for providing Ref. , Xavier Tricoche for pointing out Refs. [35, 34], and Ulrich Koschorke and Francisco Beron-Vera for useful comments.
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