Since eddies play a major role in the dynamics of oceanic flows, it is of great interest to detect them and gain information about their tracks, their lifetimes and their shapes. We present a Lagrangian descriptor based on the modulus of vorticity to construct an eddy tracking tool. In our approach we denote an eddy as a rotating region in the flow possessing an eddy core corresponding to a local maximum of the Lagrangian descriptor and enclosed by pieces of manifolds of distinguished hyperbolic trajectories (eddy boundary). We test the performance of the eddy tracking tool based on this Lagrangian descriptor using an convection flow of four eddies, a synthetic vortex street and a velocity field of the western Baltic Sea. The results for eddy lifetime and eddy shape are compared to the results obtained with the Okubo–Weiss parameter, the modulus of vorticity and an eddy tracking tool used in oceanography. We show that the vorticity-based Lagrangian descriptor estimates lifetimes closer to the analytical results than any other method. Furthermore we demonstrate that eddy tracking based on this descriptor is robust with respect to certain types of noise, which makes it a suitable method for eddy detection in velocity fields obtained from observation.

Transport of particles and chemical substances mediated by hydrodynamic flows
are important components in the dynamics of ocean and atmosphere. For this
reason, there is an increasing interest in identifying particular structures
in the flows such as eddies or transport barriers to understand their role in
transport and mixing of the fluid as well as their impact on marine biology
for instance. Of particular interest in oceanography are eddies, which can be
responsible for the confinement of plankton within them and, hence, important
for the development of plankton blooms

Algorithms for finding eddies in fluid flows are applied in very different
fields of science such as in atmospheric science

Based on dynamical systems theory, one can search for Lagrangian coherent
structures (LCSs) which describe the most repelling or attracting manifolds in
a flow

Despite the discussion about objectivity (cf. Haller's short comment SC2 in
the discussion of this paper, Mancho's editor comment EC1 and

In the recent years there has been some effort to derive Eulerian quantities
which can be used to draw conclusions about Lagrangian transport phenomena

In oceanography, one of the most popular methods with which to identify eddies is based
on the Okubo–Weiss parameter

In this paper we develop an eddy detection and tracking tool based on the
method of the Lagrangian descriptor introduced by Mancho and co-workers

The paper is organized as follows: Sect.

The dynamics of a fluid can be characterized employing two different concepts: the Eulerian and the Lagrangian view. While the Eulerian view uses quantities describing different properties of the velocity field, the Lagrangian view provides quantities from the perspective of a moving fluid particle. Out of the variety of different Eulerian and Lagrangian methods mentioned in the Introduction, we recall here briefly only those concepts which will be important for our development of a measure to identify eddies in a flow.

A Eulerian method to describe the circulation density of a velocity field in
hydrodynamics is vorticity

Another Eulerian quantity is the Okubo–Weiss parameter OW. It weights the
strain properties of the flow against the vorticity properties and thus
distinguishes strain-dominated areas from the vorticity-dominated one. The
Okubo–Weiss parameter is defined as

A Lagrangian view of the dynamics of the velocity field is given by the
Lagrangian descriptor developed by Mancho and co-workers

The Lagrangian descriptor

For each instant of time

Because the Lagrangian descriptor

The local maxima and the singular lines of

In the case of

Colour-coded representation of the modulus of vorticity

To compare the performance of the proposed Lagrangian
descriptor based on the modulus of vorticity to the others, two test cases – a
convection flow consisting of four counter-rotating eddies and a model of a
vortex street – are used. The four counter-rotating eddies are employed to
show that different methods detect different aspects of the eddies.
Additionally, we discuss how the displayed structure depends on the chosen

Colour-coded representation of the Lagrangian descriptor

To give a complete view of the advantages and disadvantages, the results of the different test cases are interpreted in a coherent discussion after presenting all results.

The equations of motion of fluid particles in a convection flow of four
counter-rotating eddies are given by

The model of the vortex street consists of two eddies that emerge at two
given positions in space, travel a distance

Modulus of vorticity

These two test cases reveal the following characteristics of the properties
of coherent structures in a flow: Eulerian as well as Lagrangian methods
display eddy cores as local maxima (modulus of vorticity,

To characterize Lagrangian coherent structures in a flow, not only
do distinguished trajectories surrounded by an elliptic region in the sense of

How detailed the displayed fine structure of the Lagrangian descriptors

From these properties, distinction between DHTs and eddy cores and
identification of manifolds, we can conclude that the Lagrangian descriptor

The mean oceanic flow is superimposed by many eddies of different sizes which emerge at some time instant, persist for some time interval and disappear. Consequently, an eddy tracking tool has to detect them at the instance of emergence, track them over their lifetime and detect their disappearance. To classify the different eddies, some information about their size is needed too. This way one can finally obtain the time evolution of a size distribution function of eddies.

In this section we apply the modulus-of-vorticity-based Lagrangian descriptor

First we check how well

The idea of the tracking inspired by

In order to check the accuracy of the eddy tracking algorithm, we use the
dimensionless model of the vortex street presented in Sect.

Eddy lifetime estimated with Okubo–Weiss (OW, violet); the modulus of
vorticity (absVorticity, cyan);

Time of birth of an eddy estimated with Okubo–Weiss (OW, violet);
the modulus of vorticity (absVorticity, cyan);

In all cases independent of the vortex strength, the results obtained with

Velocity fields describing ocean flows either have a finite resolution when obtained by simulations or contain measurement noise when retrieved from observational data. For this reason, an eddy tracking method has to be robust with respect to fluctuations of the velocity field. Therefore, we explore how the detected eddy lifetime depends on noise added to the velocity data.

Measured lifetime of an eddy obtained by means of

To test the influence of noise in a manageable test setup where we know all
parameters, e.g. eddy lifetime (here

Measured median lifetime obtained by different methods (Okubo–Weiss
(OW, violet), the modulus of vorticity (absVorticity, cyan),

The different noise types and their motivation are as follows:

Type 1: we add white Gaussian noise

Type 2: we add white Gaussian noise

Type 3: we add white Gaussian noise

Measured median lifetime obtained by different methods (Okubo–Weiss
(OW, violet), the modulus of vorticity (absVorticity, cyan),

The three types of noise illustrate different advantages and disadvantages of

In the case of type 2 noise,

Measured median lifetime obtained by different methods (Okubo–Weiss
(OW, violet), the modulus of vorticity (absVorticity, cyan),

In the case of type 3 noise,

Also for the other methods noise of type 3 affects strongly the identification of the eddy core because the weakening of the correlation between neighbouring points disturbs the key signal of an eddy core (a local minimum or maximum in a certain domain). The error in estimating the lifetime increases with increasing noise level. In all cases the number of outliers in the box plot (not shown here) increases with the noise level.

As a consequence, none of the methods performs in an optimal way when the
noise displaces the eddy cores during their motion. This disadvantage will
lead to deviations in the lifetime statistics for eddy tracking based on
observational data. However, the error in georeferencing of satellite images
(which is mimicked by type 3 noise) is mostly small. For special
applications, a georeferencing error of smaller than

In summary,

Besides its lifetime an eddy is characterized by its size. In the following we
will estimate the eddy size and shape using the the Lagrangian descriptor

Eddy boundaries detected with the method based on

As mentioned in Sect.

The Lagrangian descriptor

The ETTB by

The comparison of the different views on the eddy size and shape is presented
in Fig.

In a real oceanic flow, eddies of different lifetime, size and shape will
occur simultaneously. As an example of how different eddy shapes and sizes can
be detected in real oceanic flow fields, we apply our approach to a velocity
field of the western Baltic Sea for May 2009. The Baltic Sea is a good
test bed, since the tides there are negligible and the entire eddy dynamics
are driven by baroclinic instabilities, frontal dynamics and the interaction with
topography. Extended eddy statistics in the central Baltic Sea based on

A triple-nested circulation model was used to simulate the flow fields in the
western Baltic Sea. The innermost model domain was discretized in the
horizontal with a spatial resolution of

We have calculated

Figure

The black dots in Fig.

Close to or within eddies 1, 2 and 3 detected by the tracking based on

A general problem which arises when using surface velocity fields is that
this velocity field is not divergence-free. Although we have checked that
the vertical velocity is small compared to the horizontal ones, there is
still a finite residual left. However, we still assume that the velocities
are two-dimensional. Applying the ETTB by

In summary, the method based on the Lagrangian descriptor

Nevertheless, one has to take into account that the detection of eddy shapes
by the method based on the Lagrangian descriptor

The method to detect shapes should be chosen based on which type of shapes one is interested in, and the results of the method should be handled with care.

We have shown that the Lagrangian descriptor

To test all those properties in practice, we have first used some velocity
fields which are constructed in such a way that the lifetimes of eddies are
given analytically. It turns out that the Lagrangian descriptor

A general problem of any Lagrangian descriptor including

The example of the velocity field of the western Baltic Sea shows that eddy
tracking based on

In general, the choice of the detection method depends on the questions
asked. If one is only interested in tracking eddy cores, Eulerian methods are
a good choice. By contrast, Lagrangian methods give a more detailed view of
the dynamics and provide a more physical estimate of the eddy size.
Especially this feature, which describes the fluid volume trapped in an eddy,
promises to be more useful for applications that consider the growth of
plankton populations in oceanic flows. For the latter it has been shown that
eddies can act as incubators for plankton blooms due to the confinement of
plankton inside the eddy

Rahel Vortmeyer-Kley developed the idea of the eddy tracking tool based on
the Lagrangian descriptor

Rahel Vortmeyer-Kley would like to thank the Studienstiftung des Deutschen Volkes for a doctoral fellowship. The financing of further developments of the Leibniz Institute of Baltic Sea Research monitoring programme and adaptations of numerical models (STB-MODAT) by the federal state government of Mecklenburg-Vorpommern is greatly acknowledged by Ulf Gräwe.

The authors would like to thank Jan Freund, Ana Mancho, Matthias Schröder, Wenbo Tang, Tamás Tél and Alfred Ziegler for stimulating discussions. Edited by: A. M. Mancho Reviewed by: K. McIlhany and two anonymous referees