Introduction
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
. Such eddies possess a large
variety of sizes and lifetimes. To tackle the problem of recognizing such
eddies in aperiodic flows, different approaches have been developed: on the
one hand, there are several methods available which are inspired by dynamical
systems theory (, and references therein); on the
other hand, numerical software for automated eddy detection has been
developed in oceanography based on either physical quantities of the flow
or geometric measures .
Algorithms for finding eddies in fluid flows are applied in very different
fields of science such as in atmospheric science , celestial
mechanics , biological oceanography
and the dynamics of swimmers
. The largest field of application is oceanography, since
oceanic flows contain a large number of mesoscale eddies of size 100–200 km,
which are important components of advective transport. Their emergence and
lifetime influence the transport of pollutants or plankton blooms
. There
is an increasing number of eddy-resolving datasets available provided either
by observations or by numerical simulations
. Consequently there is a growing interest in
the census of eddies, their size and lifetimes depending on the season. This
task requires robust algorithms for the computation of eddy boundaries as
well as the precise detection of their appearance and disappearance in time
based on numerical velocity fields
as well as altimetry data
. However, the huge amount of available
data poses a challenge to data analysis. As pointed out in
, mesoscale and submesoscale eddies cannot be extracted
from a turbulent flow without a suitable definition and a competitive
automatic identification algorithm. Several such algorithms have been
developed based on the various concepts mentioned above. In the following we
will briefly discuss several of those algorithms.
Based on dynamical systems theory, one can search for Lagrangian coherent
structures (LCSs) which describe the most repelling or attracting manifolds in
a flow . The time evolution of these invariant manifolds
makes up the Lagrangian skeleton for the transport of particles in fluid
flows. LCSs can be considered as the organizing centres of hydrodynamic flows.
Their computation is based on the search for stationary curves of shear in
the case of hyperbolic or parabolic LCSs. Elliptic LCSs like eddies are computed as
stationary curves of averaged strain or Lagrangian-averaged vorticity
deviation . Other methods to determine whether an eddy
can be identified in the flow employ average Lagrangian velocities
or burning invariant manifolds .
The latter were originally introduced to track fronts in reaction
diffusion systems but have recently been extended to
the detection of eddies . A completely different
approach which connects geometric properties of a flow with probabilistic
measures utilizes transfer operators to identify LCSs .
Another approach is the computation of distinguished hyperbolic trajectories (DHTs)
and their stable and unstable manifolds to identify Lagrangian coherent
structures in a flow. DHTs can be considered as a generalization of
stagnation points of saddle type and their separatrices to general
time-dependent flows . DHTs and their
manifolds can be computed using Lagrangian descriptors, which integrate
intrinsic physical properties for a finite time and thereby reveal the
geometric structures in phase space . Stable and unstable
manifolds can also be calculated using the ridges of finite-time or finite-size
Lyapunov exponents (FTLEs or FSLEs) using the idea that
initially nearby particles in a flow will move apart in stretching regions,
while they will move closer to each other in contracting regions.
Despite the discussion about objectivity (cf. Haller's short comment SC2 in
the discussion of this paper, Mancho's editor comment EC1 and
) the method of Lagrangian descriptors is very appealing
and is appropriate to gain insight into oceanographic flows. It has already
been successfully applied to compute Lagrangian coherent structures in the
Kuroshio current (), in the polar
vortex , in the north-western Mediterranean Sea
and for analysing the possible dispersion of debris
from the Malaysian Airlines flight MH370 airplane in the Indian Ocean
.
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 . This method
relies on the strain and vorticity of the velocity field and has been
applied to both numerical ocean model output and satellite data
. Often, the underlying velocity
field is derived from altimetric data under the assumption of geostrophic
theory. In this approach two limitations can appear. First, the derivation of
the velocity field can induce noise in the strain and vorticity field. This
is usually reduced by applying a smoothing algorithm, which might, in turn,
remove physical information. Secondly, show that eddies
can have a significant ageostrophic contribution. Thus, the detection might
fail when relying on geostrophic theory. A slightly different approach was
developed by and , who used the
signature of eddies in the sea surface temperature (SST) to detect them. The
partially sparse coverage of satellite SST data limits the application of this method.
developed a tracking algorithm that is based on the
flow geometry. The assumption is that eddies can be defined as features
characterized by circular or spiral streamlines around the core of an eddy.
The streamlines are derived from the velocity field. Additionally, the change
of direction of the segments that compose the streamline (winding angle) is
computed for each streamline. applied this winding-angle
approach to a dataset of the South Pacific. Moreover, they compared
the winding-angle method to the Okubo–Weiss approach and concluded that the
former is more successful in detecting eddies and more important with a much
smaller excess of detection errors. A further method based on geometric
properties is proposed by . The underlying idea is that
within an eddy the velocity field changes its direction in a unique way.
Moreover, the relative velocity in the eddy core should vanish and should be
enclosed by closed streamlines. This detection and tracking algorithm was
successfully applied by in the Southern California Bight. In
addition, the detection algorithm of has the advantage
that its application is not limited to surface fields
. Thus, it is possible
to track eddies in the interior of the ocean, without any surface signature.
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
. For the purpose of automated eddy detection
we propose to use the modulus of the vorticity as the scalar quantity to be
computed along a trajectory instead of using the arc length of trajectories.
We compare our method to four others, namely the original Lagrangian
descriptor using the arc length , an
oceanographic method based on geometric properties of the flow field
and detection tools which employ the Okubo–Weiss
parameter and the modulus of vorticity itself.
The paper is organized as follows: Sect. briefly reviews the
Eulerian concepts of vorticity and the Okubo–Weiss parameter, as well as the Lagrangian
descriptors M based on the arc length and MV based on the modulus of
vorticity. To compare the performance of the two Lagrangian descriptors and
the Eulerian concepts, we use two simple velocity fields: the model of four
counter-rotating eddies and a modified van Kármán vortex street in
Sect. . In Sect. we describe the
implementation of the Lagrangian descriptor based on the modulus of vorticity
as a tracking tool identifying eddy lifetimes (Sect. ) and
compare the results again with the other aforementioned methods. In
Sect. we study the performance of the method in cases where we
corroborate the velocity fields with noise to test the robustness of the
method if applied to velocity fields obtained from observational data.
Finally in Sect. we compare the Eulerian and the Lagrangian
view of the eddy shape with application to the modified van Kármán
vortex street and to a velocity field from oceanography describing the
dynamics of the western Baltic Sea . We conclude the
paper with a discussion in Sect. .
From Eulerian to Lagrangian methods
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 W(x, t), defined as the curl of the
velocity field v(x, t). The vorticity associates a vector with each
point in the fluid representing the local axis of rotation of a fluid
particle. It displays areas with a large circulation density like eddies as
regions of large vorticity and eddy cores as local maxima.
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
OW=sn2+ss2-ω2,
where the normal strain component sn, the shear strain component ss and
the relative vorticity ω of a two-dimensional velocity field
v = (u, v) are defined as
sn=∂u∂x-∂v∂y,ss=∂v∂x+∂u∂yandω=∂v∂x-∂u∂y.
Eddies are areas that have a negative Okubo–Weiss parameter with a local minimum
at the eddy core because here the vorticity component outweighs the strain
component, while strain-dominated areas are characterized by a positive
Okubo–Weiss parameter.
A Lagrangian view of the dynamics of the velocity field is given by the
Lagrangian descriptor developed by Mancho and co-workers
. A more general definition of the Lagrangian descriptor
is outlined in . Here we focus on the Lagrangian descriptor
based on the arc length of a trajectory, defined as
Mx*,t*v,τ=∫t*-τt*+τ∑i=1ndxi(t)dt21/2dt,
with x(t) = (x1(t), x2(t) … xn(t)) being the trajectory of a fluid
particle in the velocity field v that is defined in the time interval
[t* - τ, t* + τ] and going through the point x* at time t*.
The Lagrangian descriptor M yields singular features that can be
interpreted as time-dependent “phase space structures” like (time-dependent
or moving) elliptic or hyperbolic “fixed” points (denoted as distinguished
elliptic or hyperbolic trajectories, DET and DHT respectively, in
) and their time-dependent stable and unstable manifolds
. The reason for the singular features of M
is that M accumulates different values of the arc length depending on the
dynamics in the region. Trajectories that have a similar dynamical evolution
yield similar values of M. When the dynamics changes abruptly, M will
change too. This is the case for DHTs
and their stable and unstable manifolds. Trajectories on both sides of the
manifold have a different behaviour compared to the behaviour of the
trajectories on the manifold. Either they approach the manifold very fast or
they move away from the manifold very fast. In both cases they accumulate
larger values of M in a given time interval than trajectories on the
manifold. Therefore, the singular line of M in a colour-coded plot of M
can be interpreted as corresponding to a manifold. If a trajectory stays in a
region or at one point, it accumulates a small or zero value of M and M
becomes a local minimum. While DHTs have been extensively studied,
distinguished trajectories possessing an elliptic type are less understood.
However, such trajectories can also be identified as singular features of M
being surrounded by an elliptic region in the sense of .
For an extensive discussion about the notion of hyperbolic and elliptic
regions in flows we refer to .
For each instant of time t* the colour-coded plots of M can be
interpreted as a “snapshot” of the phase space, where the minima correspond
to one point of a DHT or a distinguished trajectory surrounded by an elliptic
region. When t* changes, M reveals the time evolution of the phase
space and, loosely speaking, distinguished hyperbolic trajectories can be
considered as “moving saddle points”, and distinguished trajectories surrounded
by an elliptic region in the sense of as “moving elliptic
points”. Due to the arbitrary time dependence of the flow, both the DHTs
and the distinguished trajectories surrounded by an elliptic region are
time-dependent and exist in general only for a finite time in a
time-dependent flow. Hyperbolicity in the case of DHT refers to the fact that
along those trajectories Lyapunov exponents are positive or negative but not
zero except for the direction along the trajectory .
Because the Lagrangian descriptor M would display minima in both cases,
i.e. DHT and distinguished trajectories surrounded by an elliptic region, a
second criterion is needed to distinguish them properly. To avoid such an
additional distinction criterion, we suggest a Lagrangian descriptor based on
the modulus of vorticity to simplify the automated eddy detection. We
emphasize that it has already been pointed out by that
any intrinsic physical or geometrical property of trajectories can be used to
construct a Lagrangian descriptor by integrating this property along
trajectories over a certain time interval. Therefore, we introduce a
vorticity-based Lagrangian descriptor MV in which the physical quantity is
the modulus of the vorticity W of a velocity field v(x, t):
W(x,t)=|∇×v(x,t)|.
We define the Lagrangian descriptor MV based on the modulus of vorticity as
MVx*,t*τ=∫t*-τt*+τ(W(x,t))1/2dt.
The Lagrangian descriptor MV based on the modulus of vorticity measures
the Eulerian quantity modulus of vorticity along a trajectory (Lagrangian
view) passing through a position x* at time t* in a time interval
[t* - τ, t* + τ]. Within this time interval trajectories
accumulate different values of MV. Similar to the arc-length-based Lagrangian
descriptor M, the Lagrangian descriptor MV based on the modulus of
vorticity displays singular features such as lines or local minima or maxima. in the case
of local maxima, a trajectory does not leave the region of large values
of the modulus of vorticity. Such regions are typical for the inner part of an
eddy. Therefore, a local maximum corresponds to the eddy core and can be
interpreted as a snapshot of the distinguished trajectory at time t*
surrounded by an elliptic region in the sense of . By
contrast, local minima of MV arise if a trajectory stays in a region of
small values of the modulus of vorticity. In analogy with the singular lines in the
case of M, singular lines of MV can be interpreted as the boundaries of
regions of different dynamical behaviour. In this sense they can be
understood as manifolds of the DHTs.
The local maxima and the singular lines of MV will be used to construct an
eddy tracking tool based on the following concept of an eddy: we denote an
eddy as being bounded by pieces of stable and unstable manifolds of DHTs
(according to , and ) surrounding an
area in which the flow is rotating. The manifolds correspond to singular
lines in MV which are used to describe the eddy boundaries. The eddy core
is considered to be a local maximum of MV within this bounded region and
can be interpreted as one point of a distinguished trajectory surrounded by
an elliptic region.
In the case of MV as well as in the case of M the resolution of these structures
depends on the choice of the parameter τ that gives the length of the
time interval. Structures that live shorter than 2τ cannot be resolved.
Even structures that live longer than 2τ can only be resolved if τ
is chosen large enough. The choice of τ depends on the structure and the
timescale of the flow field considered. Within the range of the timescale
of the problem that should be resolved some variation of τ is needed
until the optimal τ for a given problem is found.
Colour-coded representation of the modulus of vorticity (a)
and the Okubo–Weiss parameter (b) for the eddy field in Eq. ().
All plots are normalized to the maximum value.
Eddies in a flow: comparing Eulerian and Lagrangian methods
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 τ.
The vortex street is particularly used to test how suitable our
method is to detecting and tracking eddies in comparison to other methods and how
well they all estimate eddy lifetimes and shapes. This way we gain insight
into performance, advantages and disadvantages of the proposed method
compared to the others.
Colour-coded representation of the Lagrangian descriptor MV (a)–(c)
and the Lagrangian descriptor M (d)–(f) for the eddy field in
Eq. () with τ chosen as 0.5 (a, d),
25 (b, e) and 100 (c, f). All plots are normalized to the maximum
value.
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
u=x˙=sin(2π⋅x)⋅cos(2π⋅y)v=y˙=-cos(2π⋅x)⋅sin(2π⋅y).
We compute the four different quantities, the modulus of vorticity, the Okubo–Weiss
parameter and the two Lagrangian descriptors M and MV in a spatial
domain [0, 1] × [0, 1]. To this end, the spatial domain is decomposed
into a discrete grid (201 × 201), and the different methods are
calculated for each grid point. The results are presented in Figs. and .
The model of the vortex street consists of two eddies that emerge at two
given positions in space, travel a distance L in positive x direction and
fade out. The two eddies are counter-rotating. They emerge and die out
periodically with a time shift of half a period. The model is adapted from
and , with the difference that the
cylinder as the cause of eddy formation and its impact on the flow field due
to its shade are neglected. In this sense, the eddies emerge non-physically
out of nowhere, but all quantities like lifetime and radius to be estimated
by means of eddy tracking are then given analytically and make up a perfect
test scenario. A detailed description of the model can be found in the
Supplement to this article. Again all methods are applied to this
velocity field using a (302 × 122) grid. Unless otherwise stated, the
time interval τ for the Lagrangian methods is set to 0.15 times the
lifetime of an eddy. The results are presented in Fig. .
Modulus of vorticity (a), Okubo–Weiss parameter (b), Lagrangian
descriptor M (c) and Lagrangian descriptor MV (d) for the hydrodynamic
model of a vortex street at t = 0.151 with τ = 0.15, normalized to the
maximum value. Blue colours indicate small values, and red large values of the
depicted quantity. The dark blue regions in (c) and (d) are regions where the
trajectories have left the region of interest.
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, MV) or local
minima (Okubo–Weiss, M) of the respective quantity
(Figs.–). Local minima of the
Lagrangian methods correspond to DHTs (Fig. e and f). For
the Lagrangian descriptor M the core of the eddy and the DHT are
indistinguishable since they are both displayed as local minima of M. The
Lagrangian descriptor MV based on the modulus of vorticity can clearly
distinguish between the core of an eddy and a DHT (Fig. a–c).
For this reason, Eulerian methods and the Lagrangian
descriptor MV are more appropriate than the Lagrangian descriptor M for
an automated identification of eddies, since no further criteria are needed.
To characterize Lagrangian coherent structures in a flow, not only
do distinguished trajectories surrounded by an elliptic region in the sense of
associated with eddy cores and DHTs have to be identified;
the stable and unstable manifolds associated with the latter have to be identified as well to find
eddy boundaries according to the concept of an eddy in Sect. .
Those manifolds are visible as singular lines in the colour-coded plot of the
Lagrangian descriptor M (Figs. d–f and c)
and the Lagrangian descriptor MV (Figs. a–c
and d).
How detailed the displayed fine structure of the Lagrangian descriptors M
and MV is represented depends on the chosen value of the time interval
τ. It ranges from no clear structure for small τ (Fig. a
and d) to a detailed structure for large τ (Fig. c and f).
From these properties, distinction between DHTs and eddy cores and
identification of manifolds, we can conclude that the Lagrangian descriptor MV
is a suitable method for an automated search for eddies in
oceanographic flows. Out of the four considered quantities, MV best allows for a
clear identification of eddy cores and the stable and unstable manifolds of
DHTs necessary to get more insight into the size of eddies with the least
smallest number of check criteria. For this reason we suggest using MV as the
basis for an eddy tracking tool. How these properties of MV are
implemented into an eddy tracking tool is explained for the eddy core in
Sect. and the eddy size and shape in Sect. .
The Lagrangian descriptor MV as an eddy tracking tool
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 MV
to the hydrodynamic model of a vortex street to test its performance as
an eddy tracking tool. We use the local maxima of MV for an automated
search for eddy cores and, in addition, the area enclosed by the singular
lines of MV associated with the manifolds of the DHTs as a measure of the
size of the eddies.
Eddy birth and lifetime
First we check how well MV detects the birth of an eddy and its lifetime
and compare the results to the oceanographic eddy tracking tool box (ETTB) by
, as well as Eulerian quantities like the Okubo–Weiss
parameter and the modulus of vorticity.
The idea of the tracking inspired by is to search for
local maxima (MV and modulus of vorticity) or local minima
(Okubo–Weiss and velocity-based method by ) surrounded by a region of
gradient towards the maximum or minimum in a given search window. The size of
the search window determines which maximal eddy size can be detected. The
eddy is tracked from one time step to the next by searching for an eddy core
with the same direction of rotation within a given distance. The choice of
this distance depends on the velocity field. It should be in the range of the
maximal distance a particle could travel in the time span of interest. The
position of an eddy is logged in a track list for each eddy at each time
step. A track list that is shorter than a given threshold number of time
steps is deleted to focus on eddies which exist longer than this minimum time
interval. A detailed description of the algorithms can be found in the
Supplement to this article.
In order to check the accuracy of the eddy tracking algorithm, we use the
dimensionless model of the vortex street presented in Sect. ,
since the time instant of birth of the eddies and their
lifetimes are given analytically. We measure both quantities for different
dimensionless lifetimes Tc and dimensionless vortex strengths of 50, 100
and 200 for the eddy that arises at time Tc/2. The rationale behind varying
the vortex strength is to estimate how weak an eddy could be to be still
reliably detected by the methods. For MV, τ was chosen as
0.15 ⋅ Tc. The results are presented in Figs. and .
Eddy lifetime estimated with Okubo–Weiss (OW, violet); the modulus of
vorticity (absVorticity, cyan); MV (red); and the eddy tracking tool box
(ETTB, blue) by for vortex strength w 50, 100 and 200.
The black diagonal depicts the analytical lifetime.
Time of birth of an eddy estimated with Okubo–Weiss (OW, violet);
the modulus of vorticity (absVorticity, cyan); MV (red); and the eddy
tracking tool box (ETTB, blue) by for vortex strength
w 50, 100 and 200. The black diagonal depicts the analytical time of birth
of an eddy.
In all cases independent of the vortex strength, the results obtained with
MV are close to the analytical Tc (Fig. ) or the
analytical time instant of birth (Fig. ). All other methods
underestimate Tc and overestimate the time instant of birth. Especially in the
case of the ETTB the estimated times depend heavily on the vortex strength. For
that method it becomes more and more difficult to detect the eddy as its
rotation speed decreases. The reason for the good estimates provided by MV
lies in its construction, which makes use of the history of the eddy (past and
future). Hence it can detect eddies earlier than they arise by taking into
account the future or detect them longer than they actually exist by looking
into the past. MV is not restricted to the information about the velocity
field at one instant of time like the other methods. However, the performance
of MV depends on the chosen value of τ (Fig. ). If
τ gets too large in relation to Tc, the estimate of the lifetime
deviates from the analytical one because the trajectories contain too much of
the history of the eddy. There exists a small range of optimal τ for a
certain class of eddies. In our case the range is between about 15 and
18 % of the eddy lifetime. We have chosen 15 % of the eddy lifetime,
because larger τ values increase the computational costs for MV, too.
The range of the optimal τ depends crucially on the application. Other
applications might need a larger or smaller τ or a τ that is a
compromise between structures with very different lifetimes. It is also
advisable to vary τ to detect different size and lifetime spectra of eddies.
Robustness of the lifetime detection with respect to noise
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 MV (red line)
versus the chosen τ (analytical lifetime Tc = 1 (blue line); vortex
strength: 200).
To test the influence of noise in a manageable test setup where we know all
parameters, e.g. eddy lifetime (here Tc = 1) or vortex strength (here
w = 200), we use the velocity components u(x, y, t) and v(x, y, t) of the
vortex street mentioned in Sect. and add three different
types of noise to it mimicking measurement noise that can arise in
observations. The result are noisy velocity components uN(x, y, t) and
vN(x, y, t) for which we calculate Okubo–Weiss, the modulus of vorticity and
MV and then apply the different eddy tracking methods. The noise is
realized as white Gaussian noise in form of a matrix of normally distributed
random numbers of the grid size for each time step multiplied by a factor
that is referred to as noise level or noise strength. The noise level is
given dimensionless, because the noise is applied to the dimensionless model
of the vortex street presented in Sect. .
Measured median lifetime obtained by different methods (Okubo–Weiss
(OW, violet), the modulus of vorticity (absVorticity, cyan), MV (red) and
the eddy tracking tool box (ETTB, blue) by ) depending on
the noise level. The computations have been performed in a velocity field
mimicking a vortex street with type 1 noise
(1000 noise realizations). The error bars indicate the whiskers of the
distribution in the box plot (not shown here) corresponding to approximately
±2.7σ.
The different noise types and their motivation are as follows:
Type 1: we add white Gaussian noise ξ(x, y, t) of different noise
strength σ between 0.05 and 0.95 to the velocity components u(x, y, t)
and v(x, y, t) of the vortex street. The noise is uncorrelated in space and time.
The resulting velocity components uN(x, y, t) = u(x, y, t) + σ ⋅ ξu(x, y, t)
and vN(x, y, t) = v(x, y, t) + σ ⋅ ξv(x, y, t) in this case are still periodic
but noisy. This type of noise mimics the effect of computing derivatives of
observed velocity fields (e.g. by satellites or high-frequency (HF) radar).
Type 2: we add white Gaussian noise ξ(x, y, t) of different noise
strength σ between 0.05 and 0.95 to the velocity components u(x, y, t)
and v(x, y, t) of the vortex street but take into account that the actual noise
depends on the velocity itself by taking the maximum of it over the whole spatial
grid. The motivation is that the strength of noise depends on the signal-to-noise
ratio. If we have a strong current, it is easy to detect this by a
satellite, since the signal strength is high. This is the opposite for slow currents,
where the noise level is much higher. Thus, we add white noise that is inversely
proportional to the current speed. The noisy velocity components are given as
uN(x, y, t) = u(x, y, t) + σ ⋅ ξu(x, y, t)/(1 + maxx,y(u(x, y, t)))
and vN(x, y, t) = v(x, y, t) + σ ⋅ ξv(x, y, t)/(1 + maxx,y(v(x, y, t))).
Type 3: we add white Gaussian noise ξ(t) of different noise
strength σ between 0.05 and 0.5 to the y component of the eddy centres'
movement. The equations of the unperturbed velocity field contain a part that
describes the movement of the eddy centres (see Supplement). The
motion of the y components of the eddy centres in the unperturbed velocity
field (u, v) is given by y1(t) = y0 = -y2(t), where the index 1 or 2 refers to
the two eddies. The movement of the eddy centres in the case of noise is given by
y1N(t) = y0 + σ ⋅ ξ(t) and y2N(t) = -y0 + σ ⋅ ξ(t). This
type of noise can be observed if the velocity fields have to rely on georeferencing.
For instance, satellite-generated velocity fields have to be mapped on a
longitude–latitude grid, since the satellite is moving. During this postprocessing
step a shift in the georeference is possible, leading to translational shifts
and thus to type 3 noise. However, a high noise level of type 3 is not very likely.
If one deals with typical geophysical applications, which have a grid resolution
of the order 1 to 10 km, the georeferencing errors are mostly small
compared to the grid cell size. For this reason, the considered noise levels
for type 3 noise are smaller than for type 1 and 2.
To explore the impact of noise systematically, we have used different noise
strengths. For each noise strength σ 1000 realizations of the white
Gaussian noise were calculated. In the resulting velocity fields we estimated
the lifetime of each eddy that undergoes a whole life cycle within the
simulation time. The plotted eddy lifetimes obtained with all different
tracking methods are medians of the distributions of the lifetimes for the
1000 realizations per noise strength (Figs. –).
Measured median lifetime obtained by different methods (Okubo–Weiss
(OW, violet), the modulus of vorticity (absVorticity, cyan), MV (red) and
the eddy tracking tool box (ETTB, blue) by ) depending on
the noise level. The computations have been performed in a velocity field
mimicking a vortex street with type 2 noise (1000 noise realizations). The
error bars indicate the whiskers of the distribution in the box plot (not shown
here) corresponding to approximately ±2.7σ.
The three types of noise illustrate different advantages and disadvantages of
MV compared to the other methods. In the case of type 1 noise, MV gives the
best estimate of the lifetime compared to all other methods independent of
the increasing noise level. The reason why the error of the estimate in the case
of MV does not increase with increasing noise level is that MV is a
measure that is based on an integral. Integrating over accumulated,
uncorrelated noise along the trajectory from past to future can be considered
as a smoothing process. Also the ETTB by gives a good
result independent of the increasing noise level, because the signal-to-noise
ratio is small. The minimum of the velocity that is the key signal for
determining the eddy core in their method remains a local minimum in the
contour plot of the velocity. However, with increasing noise level we find an
increase of outliers for the ETTB by and MV (box plot
not shown here). The performance of the modulus of vorticity and the Okubo–Weiss
parameter decreases as expected with increasing noise level, while the
distribution increases in width (Fig. ). The reason is that
the noise gets so large that it increasingly disturbs the key signal for an
eddy core until no distinct eddy core can be identified anymore.
In the case of type 2 noise, MV and the ETTB show a similar behaviour to the case
of type 1 noise. Both yield good results independent of the noise level. This
is again due to the smoothing process in the case of MV. The modulus of
vorticity performs even better than MV in the case of small noise levels, but
its performance drops below the results of MV with increasing noise level
(Fig. ). The reason is that the key signal for determining an
eddy core using the modulus of vorticity is stronger in the case of small noise
levels and gets disturbed by the noise with increasing noise level. As
expected, the performance of Okubo–Weiss decreases with increasing noise
level. In contrast to type 1 noise, Okubo–Weiss can identify eddy cores even
in the case of strong noise, because the key signal for an eddy core is less disturbed.
Measured median lifetime obtained by different methods (Okubo–Weiss
(OW, violet), the modulus of vorticity (absVorticity, cyan), MV (red) and
the eddy tracking tool box (ETTB, blue) by ) depending on
the noise level. The computations have been performed in a velocity field
mimicking a vortex street with type 3 noise (1000 noise realizations). The
error bars indicate the whiskers of the distribution in the box plot (not shown
here) corresponding to approximately ±2.7σ.
In the case of type 3 noise, MV yields an estimate of the lifetime with the
largest error (Fig. ). In this case noisy trajectories that
start close to each other diverge fast, while the ones with no noise have a
similar dynamical evolution. This divergence due to noise leads to a loss of
structure in space that can be interpreted as a weakening of the correlation
between neighbouring trajectories. This effect is strongest in the case of MV
because it integrates over time, and so neighbouring trajectories that have
similar values of MV in the case of no noise yield very different values of
MV due to the divergence of the trajectories. As a consequence no clear
structure in MV can be identified. This effect increases with the noise level.
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 1/50 pixel is achievable
. show that with reasonable effort
a mapping error smaller than 0.5 pixel is possible if fixed landmarks
(coastlines, islands) are in the images. With the increase in Earth-orbiting
satellites and thus the increase in available images, it can be assumed that
this error will drop even more . If numerically generated
velocity fields are used, noise of type 3 is completely absent. Here the
evolution of neighbouring trajectories is smooth and correlated.
In summary, MV can be used for the detection of eddies and the estimate of
eddy lifetimes for velocity fields with and without noise, and it yields good
results independent of the noise level in the case of type 1 and 2 noise.
However, one has to take into account that the velocity field should not be
too noisy and that one has to choose a τ that fits the problem. The
Lagrangian descriptor MV has an additional advantage in detecting arising
eddies earlier than other methods due to collecting information along the
trajectory from past to future. This can be useful in the identification of
regions that will be eddy-dominated in the further evolution of the flow.
Detecting eddy sizes and shapes
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 MV
based on the modulus of vorticity and compare the results to the size
detected by the ETTB by . In this way, we demonstrate the
differences between the Eulerian and Lagrangian point of view of the eddy
size and shape.
Eddy boundaries detected with the method based on MV (red line)
and with the eddy tracking tool by (black line) at
t = 0.201. (a) MV without noise, (b) MV with type 1 noise of noise
level 0.95, (c) MV with type 2 noise of noise level 0.95, (d) MV
with type 3 noise of noise level 0.5. The τ value is chosen as
0.15 Tc with Tc = 1. The dark blue regions are regions where the trajectories
have left the region of interest. All plots are normalized to the maximum
value.
As mentioned in Sect. the estimation of the eddy shape and size
from the Lagrangian point of view is based on the idea that the boundaries of
the eddy are linked to manifolds of DHTs that surround the eddy
. These manifolds cannot be crossed by
any trajectories; therefore, trajectories starting inside the manifolds
are trapped in the eddy. Defining the boundaries in this way, one can estimate
the trapping region or volume that is transported by an eddy.
The Lagrangian descriptor MV displays singular lines that correspond to
manifolds. Therefore, the shape detection algorithm searches for the largest
closed contour line of MV for which MV is an extremum and which
surrounds an eddy core found with MV. This contour line, extracted from
MV with the MATLAB function contourc and along which the gradient of MV
is large, should be a line on or very close to a singular line displayed by
MV corresponding to a manifold and will give an estimate of the eddy boundary.
The ETTB by gives a Eulerian view of the eddy shape by
defining the eddy boundaries as the largest closed streamline of the
streamfunction around the eddy centre where the velocity still increases
radially from the centre. The contour lines as well as the streamlines are
extracted in a given search window which is centred on the eddy core.
MV for the western Baltic Sea for 11 May 2009 at 01:00 LT with
τ = 36 h normalized to the maximum value of MV. The red lines
are the eddy boundaries, and red dots the eddy cores detected with the method
based on MV. The black lines are the eddy boundaries
detected with the ETTB by on 11 May 2009 at 01:00 LT. The
black dots are the eddy cores detected with the ETTB by
within the time interval of 11 May 2009 at 01:00 LT ±36 h. The dark blue
regions are areas where the trajectories have left the domain of interest;
light grey regions indicate land.
The comparison of the different views on the eddy size and shape is presented
in Fig. for the vortex street without (Fig. a) and with noise of
type 1, 2 and 3 (Fig. b–d). The size detected with the ETTB by
is much smaller than the size based on the Lagrangian
view (Fig. a–c). Additionally, the evolution of the eddy
is captured by both methods even in the case of strong type 1 and 2 noise
(Fig. b and c). Here, the eddy boundaries in the case of noise show
small irregularities due to the noise. In general, the eddy boundary computed
based on MV is detected earlier and shows more growing and shrinking
during the evolution of the eddy than the eddy boundary extracted by the
ETTB. This is due to the conceptual idea of MV that contains the history
of the trajectories. As shown in Sect. , this leads to
problems in the case of a velocity field with type 3 noise (although significant
type 3 noise levels are very unlikely). If the noise level is too large, no
structure – neither a clear eddy core nor a clear eddy boundary – can be detected
(Fig. d) within MV. But if an eddy core can be detected
as in the case of the left eddy in Fig. d, the eddy shape
detection based on MV gives an idea of the size and the noisy eddy boundary.
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
MV will be the content of further research.
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 1/3 nautical mile (∼ 600 m).
The model domain covers the Danish straits and the western
Baltic. The open boundaries are located in the Kattegat and at the eastern
rim of the Bornholm Basin. In the vertical 50 terrain-following adaptive
layers with a zooming toward stratification were used. The setup is
identical to the one used by or .
There, a detailed description and validation of the present setup can be
found. At the open boundaries of the model domain, the water elevations,
depth-averaged currents, and salinity and temperature profiles are
prescribed. This external forcing was taken from a model of the North
Sea–Baltic Sea with a horizontal resolution of 1 nautical mile and
50 vertical layers. To account for large-scale variations and remotely generated
storm surges, the North Sea–Baltic Sea model was nested into a depth-averaged
storm surge model of the North Atlantic with a resolution of 5 nautical
miles. The atmospheric forcing was derived from the operational model of the
German Weather Service with a spatial resolution of 7 km and
temporal resolution of 3 h. A more detailed description of the model
system is given by . The flow fields for May 2009 were
taken out of a running simulation covering the period 1948–2015. The velocity
field was interpolated to an equidistant spacing of 1 m and finally
averaged over the upper 10 m to produce a “quasi”-two-dimensional
field. The temporal resolution was set to 1 h to resolve, for instance,
inertial oscillations.
We have calculated MV for 11 May 2009 at 01:00 LT with τ = 36 h and
applied the eddy tracking based on MV. A τ value of 36 h
corresponds to 15 % of an eddy lifetime of approximately 10–12 days, which was
reported previously by . In contrast to the test case of
the vortex street, we do not expect that the eddies are perfectly circular.
To account for deformed and distorted eddies, we had to introduce a threshold
for the convexity deficiency to eliminate contours that are only made out of
filaments and are not an eddy in the sense of oceanography. We set the threshold
to an 11 % difference between the area of the convex hull of the points that
form the boundary and the area enclosed by the boundary itself normalized to
the area enclosed by the boundary. This definition of convexity deficiency is
according to . Please note that we still allow detecting
contours that cover eddy merging and decay processes, which are characterized
by filaments.
Figure shows the eddy boundaries detected with the method based
on MV (red) and the ETTB by (black) on 11 May 2009
at 01:00 LT for the same search window size. There are several differences
between the number and shapes of eddies which must be explained. One hundred fifty eddies
can be detected with the method based on MV, whereas the ETTB detects only
24 eddies at the same instant of time. One reason for the differences is that
MV contains the information of the velocity field of a time interval,
namely 11 May 2009 at 01:00 LT ±36 h. Each eddy that exists, starts
to arise, merges with another eddy or dies within this time interval leaves a
footprint in MV like the many small eddies visible in MV. How
visible this footprint is in MV depends on the choice of τ.
Therefore, the number of eddies detected with the method based on MV has
to be compared with the number of eddies detected with the ETTB by
in the whole time interval that is covered by MV.
The black dots in Fig. are the eddy cores detected with the
ETTB by within the time interval of 11 May 2009 at 01:00 LT
±36 h. In total, 339 eddies are detected which exist between <1 h
and 72 h. For some eddies we will discuss exemplarily why
they are detected by one of the methods and not by the other to illustrate
which different problems have to be taken into account if one interprets the
results of the different methods.
Close to or within eddies 1, 2 and 3 detected by the tracking based on
MV there are several eddy cores detected by the ETTB by if
one takes into account the whole time interval. At 01:00 LT on 11 May 2009 the
ETTB does not detect eddies 1, 2 and 3 because they are too weak or do not exist
yet. By contrast, the eddy detection method based on MV detects them due
to the construction of MV as an integral over time. For eddy 4 only a few
eddy cores are detected by the ETTB by for the whole
time interval; probably the eddy is too weak and lives too briefly to be seen as
a structure in MV. In the case of eddy 5 the method based on MV does not
detect an eddy, although the ETTB by detects several eddy
cores in the region. One reason could be that the eddy arises, moves a lot
and dies within the time interval such that MV only captures a blurred
structure of the eddy that does not fulfil the convexity criterion. Eddy 6 is
not detected by the method based on MV, although the eddy boundary is
obvious in the structure of MV. The reason is that the choice of the
search window size for the eddy core detection determines if an eddy core is
detected or not. An enlarged search window could solve this problem for eddy 6,
but a larger search window influences the number of detected eddies. A
solution could be an eddy core search independent of the search window size.
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 to these
quasi-2-D fields does not cause difficulties, since the algorithm works
on an instantaneous snapshot – a frozen velocity field. Thus, the error made
by the 2-D assumption is small. The situation changes when employing a
Lagrangian descriptor. During the integration interval [t* - τ t* + τ],
MV accumulates these residuals. Therefore, MV can show
structures that seems to be eddies but are regions of a stronger vertical
velocity or Lagrangian divergence . Therefore, the number
of eddies of both methods will include false positives.
In summary, the method based on the Lagrangian descriptor MV can be used
for the detection of eddy boundaries that act as boundaries of a
trapping region. Comparing the latter to boundaries detected with the ETTB by
leads to large differences in the shape and in the size.
Those deviations are due to the difference in the definition of the boundary.
In the case of the vortex street the eddy sizes detected by the ETTB by are
much smaller than the sizes detected by the method based on the Lagrangian descriptor MV. Another advantage of the method
based on the Lagrangian descriptor MV is that it even shows filament
structures of the eddy boundary in contrast to the ETTB by
visible in the example of the western Baltic Sea. These
filaments can be linked to the dynamics of the eddy, e.g. as it starts
interacting, merging or repelling with other eddies or fading out. Though
these filament shapes of eddies might not be eddies according to a
stricter mathematical definition of an eddy boundary as in
and , they are still important structures in
the flow from an oceanographic point of view and should be considered in a
census of eddies.
Nevertheless, one has to take into account that the detection of eddy shapes
by the method based on the Lagrangian descriptor MV is restricted by the
choice of τ. In highly dynamical velocity fields like the example of the
Baltic Sea not all structures can be resolved by the same τ, which leads
to a compromise for τ. This choice of τ influences if an eddy can
be detected by the method based on MV and not by the ETTB by
or the other way round.
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.
Discussion and conclusion
We have shown that the Lagrangian descriptor MV based on the modulus of
vorticity provides good insight into the structure of a hydrodynamic flow.
It can be used to identify eddy cores as well as distinguished hyperbolic
trajectories. Eddy cores can be found as local maxima of MV, while DHTs
correspond to minima of MV. Hence, compared to the Lagrangian descriptor
M based on the arc length, it does not need an additional criterion to
distinguish between eddy cores and DHTs. Similar to any other Lagrangian
descriptor, it displays singular lines that can be linked to the stable and unstable
manifolds of the DHTs, which allows for a simultaneous estimate of the
boundaries of the eddies to get an assessment of their size and shape. These
features make the quantity MV suitable for designing an eddy tracking
tool which should be able to detect eddy cores; to track them over time; and
additionally to provide information about the eddies' lifetime, size and
shape. Moreover, the eddy tracking should be robust with respect to velocity
fields corroborated with errors when the velocity field is extracted from observations.
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 MV is
superior in estimating lifetimes compared to the other considered methods.
This is due to its definition as an integral which takes the history into
account. Eulerian methods like Okubo–Weiss or the ETTB by
detect eddies too late and underestimate their lifetime. The formulation of
MV as an integral is also beneficial in the case of different types of noise.
However, none of the tested methods can deal in a convincing way with type 3
noise which mimics errors to shifts in georeferencing.
A general problem of any Lagrangian descriptor including M and MV is
that the resolution of the structures to be detected depends on the chosen
time τ. Structures that live too short in relation to the chosen τ
cannot be resolved and will be missed. Hence the choice of τ contains a
decision as to which timescale and consequently which eddy lifetime will be resolved.
The example of the velocity field of the western Baltic Sea shows that eddy
tracking based on MV is able to detect the essential eddies that are
visible in the velocity field and also detected by the ETTB by
. Furthermore, it detects eddies that cannot be detected
by the ETTB at this instant of time t* but was or will be detected by the
ETTB at an earlier or later instant of time within the time interval
[t* - τ t* + τ]. Nevertheless, one has to be aware that both the ETTB and
the eddy tracking based on MV give false positives. The reason could be
that structures of strong vertical velocity are identified as eddies. On the
other hand false negatives can arise if (i) the eddies are too weak,
(ii) the chosen τ value is too large or too small or (iii) the search window
is too large or too small.
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 .