Transport and mixing processes in fluid flows are crucially influenced by coherent structures and the characterization of these Lagrangian objects is a topic of intense current research. While established mathematical approaches such as variational methods or transfer-operator-based schemes require full knowledge of the flow field or at least high-resolution trajectory data, this information may not be available in applications. Recently, different computational methods have been proposed to identify coherent behavior in flows directly from Lagrangian trajectory data, that is, numerical or measured time series of particle positions in a fluid flow. In this context, spatio-temporal clustering algorithms have been proven to be very effective for the extraction of coherent sets from sparse and possibly incomplete trajectory data. Inspired by these recent approaches, we consider an unweighted, undirected network, where Lagrangian particle trajectories serve as network nodes. A link is established between two nodes if the respective trajectories come close to each other at least once in the course of time. Classical graph concepts are then employed to analyze the resulting network. In particular, local network measures such as the node degree, the average degree of neighboring nodes, and the clustering coefficient serve as indicators of highly mixing regions, whereas spectral graph partitioning schemes allow us to extract coherent sets. The proposed methodology is very fast to run and we demonstrate its applicability in two geophysical flows – the Bickley jet as well as the Antarctic stratospheric polar vortex.

The notion of coherence in time-dependent dynamical systems is used to
describe mobile sets that do not freely mix with the surrounding regions in
phase space. In particular, coherent behavior has a crucial impact on
transport and mixing processes in fluid flows. The mathematical definition
and numerical study of coherent flow structures has received considerable
scientific interest for the last 2 decades. The proposed methods roughly fall
into two different classes, geometric and probabilistic approaches;
see

To overcome these problems, different computational methods have been
proposed to identify coherent behavior in flows directly from Lagrangian
trajectory data, such as obtained from particle tracking algorithms. One of
the earliest attempts is the braiding approach proposed by

Very recently, spatio-temporal clustering algorithms have been proven to be
very effective for the extraction of coherent sets from sparse and possibly
incomplete trajectory data

Inspired by these recent approaches, our aim is to design a reliable but
computationally inexpensive method for studying coherent behavior as well as
mixing processes directly from Lagrangian trajectory data. For this, we
consider an unweighted, undirected network, where Lagrangian particle
trajectories serve as network nodes. A link is established between two nodes
if the respective trajectories come close to each other at least once in the
course of time. This construction is similar in spirit to the concept of
recurrence networks

We use classical graph concepts and algorithms to analyze our
trajectory-based undirected and unweighted flow network. Local network
measures such as node degrees or clustering coefficients highlight regions of
strong or weak mixing. These and other quantities have been considered in
previous work on recurrence networks by

In addition to considering local network measures, we will apply spectral
graph partitioning schemes for the solution of a balanced cut problem

The paper is organized as follows. In Sect.

In the following, we assume that we have

By an appropriate choice of

Alternatively, the network might be set up by linking the

We note that the network depends on the time interval under consideration. While the study of different time intervals may reveal relevant information about the timescales and other inherent properties of the dynamics, this will not be the focus of our work here.

Here, we briefly discuss standard analysis concepts for networks

From the adjacency matrix

The non-normalized Laplacian is formed by

The normalized symmetric graph Laplacian

The degree of a node encodes how many other nodes are connected to it. In our
setting, it measures how many different trajectories come close to the
trajectory represented by the respective node, and thus it carries
information about fluid exchange. The node degree

Here one considers the average node degree of the neighbors of a node

Both

Here one considers the induced subgraph formed by the vertex

The simple local network measures reviewed here depend on the local
properties of the network and therefore, of course, on the choice of

Spectral graph partitioning aims at decomposing a network into components
with specific properties. In our setting, the network encodes how material is
transported by the flow, in both space and time. We are interested in
identifying coherent structures in the flow, which are known to be organizers
of fluid transport. From a spatio-temporal point of view, coherent sets are
formed by trajectories that stay close to each other

Network measures for high-resolution initial conditions (case i) in
the Bickley jet for

Network measures for 1000 random initial conditions (case ii) in the
Bickley jet for

Leading eigenvectors

Leading eigenvectors

As outlined above, the normalized symmetric graph Laplacian

As our first example we consider the Bickley jet proposed by

Initial conditions are chosen in the domain

12 200 points from a regular grid on

1000 random points uniformly distributed on

For the first high-resolution setting (i) we study different

For the sparse setting (ii), we start with

Leading eigenvalues of the generalized graph Laplacian eigenvalue problem Eq. (

Extraction of eight coherent sets based on a

Node degree

Eigenvector

In Fig.

In Fig.

Computation times (in s).

This study supports the local network measures being of course

In Fig.

In the low-resolution case (ii), the leading eigenvectors match those of the high-resolution data case, but in a slightly different
order (see Fig.

The 10 leading eigenvalues for case (i) and

The first spectral gap is related to the coherent behavior of the upper and
lower parts of the cylinder, delineated by the jet core. The second (and
larger) spectral gap indicates the existence of altogether eight coherent
sets. These can be extracted via a standard

Finally, we note that the proposed approach is computationally inexpensive,
with total run times of

As a second example we study the transport and mixing dynamics in the
stratospheric polar vortex over Antarctica. The coherent behavior of the
polar vortex has already been numerically studied using transfer-operator
methods

In Fig.

In Fig.

We repeat the study of the spectrum by considering a new network where the
trajectories are restricted to the time span before the bifurcation
(1–26 September 2002); see Fig.

We have proposed a very simple and inexpensive approach to analyzing coherent
behavior and thus transport and mixing phenomena in flows. It is based on
a network in which Lagrangian particle trajectories form the nodes. A link is
established between two nodes if the respective trajectories come close to
each other at least once in the course of time. The resulting network is
unweighted and undirected and can be represented by a binary adjacency
matrix. Classical local network measures such as node degree and clustering
coefficient highlight regions of strong mixing and regular motion,
respectively. While these network measures are

While in this paper we have only demonstrated our approach in examples that are volume-preserving and two-dimensional, the extensions to three-dimensional flows and also to dissipative systems are straightforward. In addition, although not illustrated here, our method can easily deal with incomplete trajectory data as only one-time encounters of trajectories are required for setting up the network. The approach is not restricted to connected networks, and in particular in the presence of attracting sets in non-volume-preserving systems, these might be worthwhile considering as well. We have studied unweighted networks throughout the paper. Counting the number of times a trajectory comes close to another is one option for choosing weights. Our own preliminary studies indicate that in this case the node degree and average node degree become less meaningful, as these cannot distinguish any more between repeated encounters (as in regular regions) and many different encounters (as in mixing regions). Clustering coefficients and subdominant eigenvectors of the Laplacian appear to continue to highlight coherent regions.

There are some direct relations to other recently proposed methodologies such
as the dynamic isoperimetry framework introduced by

The velocity field that we used in Sect. 4.2 is publicly available from the ECMWF website:

The authors declare that they have no conflict of interest.

This article is part of the special issue “Current perspectives in modelling, monitoring, and predicting geophysical fluid dynamics”. It is not affiliated with a conference.

We thank Gábor Drótos and the second anonymous reviewer for insightful comments and suggestions that helped improve and clarify this paper. This work is supported by the Priority Programme SPP 1881 Turbulent Superstructures of the Deutsche Forschungsgemeinschaft (PA 1972/3-1). Kathrin Padberg-Gehle also acknowledges funding from EU Marie-Skłodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-2014-ITN 643073 CRITICS). We thank Naratip Santitissadeekorn for sharing code for the Antarctic polar vortex computations. Publication is supported by the Office of Naval Research under grant no. N00014-16-1-2492. Edited by: Cristóbal López Reviewed by: Gábor Drótos and one anonymous referee