# Relative Fluid Stretching and Rotation for Sparse Trajectory Observations

Nikolas O. Aksamit1<EMAIL_ADDRESS>Alex P. Encinas-Bartos2  George
Haller2  David E. Rival3  1Department of Mathematics and Statistics, UiT - The
Arctic University of Norway, Forskningsparken 1, Tromsø 9037, Norway
2Institute for Mechanical Systems, Swiss Federal Institute of Technology (ETH)
- Zürich, Leonhardstrasse 21, Zürich 8092, Switzerland 3Institute of Fluid
Mechanics, Technische Universität Braunschweig, Hermann-Blenk-Strasse 37,
Braunschweig 38108, Germany

###### Abstract

As most mathematically justifiable Lagrangian coherent structure detection
methods rely on spatial derivatives, their applicability to sparse trajectory
data has been limited. For experimental fluid dynamicists and natural
scientists working with Lagrangian trajectory data via passive tracers in
unsteady flows (e.g. Lagrangian particle tracking or ocean buoys), obtaining
material measures of fluid rotation or stretching is currently only possible
for trajectory concentrations that are often out-of-reach. To facilitate
frame-indifferent investigations in unsteady and sparsely sampled flows, we
present a novel approach to quantify fluid stretching and rotation via
relative Lagrangian velocities. This technique provides a formal objective
extension of recent quasi-objective metrics by accounting for mean flow
behavior. We also provide a solution for extremely sparse experimental data
where fluid structures are significantly undersampled, and the mean flow
behavior is difficult to quantify. We use data from multiple numerical and
experimental flows to show that our methods can identify structures beyond
existing limits of sparse, frame-indifferent diagnostics, and exhibit improved
interpretability over common frame-dependent diagnostics.

##  1 Introduction

Experimental methods for spatially and temporally resolving fluid velocities
in both large and natural flow domains have improved significantly in recent
years. For a wide range of turbulent flows, ground truth measurements of fluid
velocity can be measured using high-speed imaging of advected particles and
any number of particle image velocimetry (PIV) and Lagrangian particle
tracking (LPT) algorithms. For the largest spatial scales and for the sparsest
trajectory data, only LPT is suitable, and provides a Lagrangian framework
with which to extract transport features in the flow. At oceanographic and
atmospheric circulation length scales, GPS-tracking of buoys and balloons
typically replaces PIV and LPT imaging approaches. To date, Lagrangian data
has provided great insights in our study of sea ice and ocean dynamics, as
well as for meteorologists studying atmospheric behaviors with drifting
weather-balloon measurements (Businger et al., 2006; Leppäranta, 2011; van
Sebille et al., 2018). Coherent structure identification from sparse data,
however, has typically relied on a grab-bag of techniques, often tailored to
each individual flow, with no unifying metrics that work in all domains.

Historically, lab-based measurement techniques have evolved hand-in-hand with
systematic technological advances such as integrated circuits, the laser, and
most recently the CMOS chip. Starting out from intrusive, probe-based
extraction of Eulerian data (fixed-point statistics) through to time-averaged
planar field measurements, e.g. PIV, and then most recently to dense time-
resolved particle tracking in three dimensions, the availability of said tools
has influenced the choice of metrics used to describe the flow in question. As
a case in point, the use of Reynolds stresses to describe shear flows has
dominated the community since the days of hot-wire anemometry even though such
stresses are only a proxy to the coherent structures driving the turbulent
processes on hand.

Commonly used approaches for identifying structures in experimental flows are
typically frame-dependent. That is, the extracted features revealed depend on
the choice of reference frame of the experimentalist, and thus violate a
fundamental requirement from continuum mechanics for describing material fluid
behavior. Material behavior can be thought of as the features in a flow
revealed in a tracer visualization experiment (e.g. dye, smoke, etc.). While
our physical intuition around Lagrangian velocities may be strong, e.g., one
can easily imagine a leaf floating downstream on a river surface, extracting
physically meaningful diagnostics that describe the material deformation of
the surrounding fluid is much more difficult. Indeed many common and intuitive
trajectory metrics are frame-dependent, such as the Lagrangian velocity,
looping (Lumpkin, 2016), curvature (Bristow et al., 2023), complexity measures
(Rypina et al., 2011), and network-based approaches (Iacobello & Rival, 2023),
as well as many diagnostics from gridded velocity data such as vorticity
(Bernard & Thomas, 1993), and swirling strength (Zhou et al., 1999).

While choosing a single common reference frame may appear logical to study
flows we understand well, this approach quickly loses its foundation when
encountering flows where no _a priori_ knowledge of the relevant structures or
a canonical reference frame are available. Furthermore, truly material
behavior does not depend on the reference frame, regardless of whether a user-
preferred frame exists or not (Truesdell & Noll, 2004). Thus frame-
indifference is a fundamental litmus test for structure identification schemes
to actually identify material features even if one always conducts their
research in the same reference frame.

Methods that can effectively reveal fluid structures at both high and low
trajectory densities, in a frame-independent manner, and for the widest
variety of flows would provide a wealth of information useful for both
understanding fluids in observational environments and as a common ground for
comparison with numerical simulations. In order to test sparse diagnostics in
very different flows, we will quantify the sparsity of a dataset by
normalizing the number of trajectories by the square or cube of a
characteristic length scale of the flow $\ell$, for 2D and 3D examples,
respectively.

Frame-indifferent (objective) diagnostics that identify Lagrangian coherent
structures have been extensively developed over the last two decades (see
Haller, 2023) but their implementation relies on spatial derivatives that are
difficult to accurately compute from sparse or unstructured data. To account
for this, other approaches have been developed to adapt these diagnostics to
sparse data (Lekien & Ross, 2010; Rypina et al., 2021; Mowlavi et al., 2022)
or perform Green’s theorem approximations of rate-of-strain metrics from
arrays of trajectories (Kwok et al., 1990). Most notably, Mowlavi et al.
(2022) compared multiple sparse methods for identifying hyperbolic
(stretching) and elliptic (rotating) Lagrangian coherent structures in the
Bickley jet and ABC flow. When initializing particles on a structured grid,
Mowlavi et al. (2022) were able to accurately cluster particles into elliptic
LCS at particle concentrations of 38$\ell^{-2}$ and 421$\ell^{-3}$ for the
Bickley jet and ABC flow ($\ell=\pi$ for both), respectively. They were also
able to identify hyperbolic structures from randomly initialized particles at
concentrations of 157$\ell^{-2}$, and 3750$\ell^{-3}$ for the Bickley jet and
ABC flow, respectively.

The squared relative dispersion ($d^{2}$) is an outlier diagnostic as it is
suitable for sparse and randomly oriented trajectories, and is also objective
(Haller & Yuan, 2000). This fluid stretching metric has been used for a number
of years, particularly to understand dispersion and mixing by oceanographers
(LaCasce, 2008). The ability of $d^{2}$ to identify coherent structures,
however, is limited as one is forced to initially choose particle pairs, whose
relative motion inevitably becomes uncorrelated at an _a priori_ unknown
temporal horizon (Haller et al., 2021).

While progress is being made in developing physically meaningful sparse
trajectory diagnostics, current approaches still rely on either a relatively
dense field of particles (for a comparison, see, e.g., Mowlavi et al., 2022),
or well-behaved trajectory array geometries (Lindsay & Stern, 2003). In light
of the unstructured nature of trajectory data, it is more common for
experimental LPT data to be bin-averaged or assimilated into gridded products
prior to structure analysis (Schröder & Schanz, 2023). This approach, however,
neglects potentially critical transient flow behavior that can be gained from
the underlying Lagrangian data.

Complementing these sparse approaches is the recent development of quasi-
objective coherent structure diagnostics, the trajectory rotation average
(TRA) and trajectory stretching exponent (TSE) (Haller et al., 2021). TSE and
TRA calculate stretching and rotation, respectively, for individual particle
trajectories with no requirement of nearby velocity or trajectory data. The
authors mathematically proved that under suitable conditions (slowly-varying,
relatively small mean vorticity) TRA and TSE approximate objective measures of
rotation and stretching. Given these flow conditions, TRA and TSE have proven
to be advantageous in several extremely sparse geophysical buoy experiments,
accurately identifying algae trapping eddies in the ocean (Encinas-Bartos et
al., 2022), as well as predicting Arctic sea ice stretching and breakup events
that were missed by other approaches (Aksamit et al., 2023).

In the present research, we introduce relative stretching and rotation metrics
for individual trajectories that incorporate knowledge of the average
translation and rotation of the sampled fluid. By utilizing bulk behavior of
concurrent trajectories in a given experiment, we can obtain objective
diagnostics of stretching and rotation in unsteady flows with much of the same
flexibility of a true single-trajectory method. Similar velocity deformation
fields have proven effective at isolating important flow features in Eulerian
velocity data (Kaszás et al., 2023).

In the following, we develop the theory of relative stretching and rotation
and provide several examples of performance. The robustness of relative
Lagrangian stretching and rotation is displayed by effective structure
boundary extraction at low densities in 2D and 3D flows, in instantaneous and
trajectory-averaged forms, and their use in frame-indifferent flow statistics.
The enhanced performance over other methods is shown for numerical simulation
data and in real world observations of ocean buoys from the global drifter
database and a large-scale LPT wind tunnel experiment.

##  2 Methods

###  2.1 Background

Consider a fluid flow with time-varying velocity field
$\mathbf{v}(\mathbf{x},t)$. Infinitesimal fluid particle trajectories can be
generated as solutions to the differential equation
$\dot{\boldsymbol{x}}(t)=\mathbf{v}(\boldsymbol{x}(t),t)$ from some initial
position $\boldsymbol{x}(t_{0})=\boldsymbol{x}_{0}$. The flow map $\mathbf{F}$
maps fluid particles from their time $t_{0}$ positions to their position at a
time $t$, along the trajectory $\boldsymbol{x}(t;t_{0},\boldsymbol{x}_{0})$,

$\mathbf{F}_{t_{0}}^{t}:\boldsymbol{x}_{0}\mapsto\boldsymbol{x}(t;t_{0},\boldsymbol{x}_{0}).$

The measurement of a scalar quantity coupled with a fluid particle
$P(\boldsymbol{x}(t))$, such as its temperature, is objective (frame-
indifferent) under Euclidean transformations of the form

$\mathbf{x}=\mathbf{Q}(t)\mathbf{y}+\mathbf{b}(t),$ (1)

where $\mathbf{Q}(t)\in\mathrm{SO}(n)$ (for $n=2$ or 3) and $\mathbf{b}(t)$ is
a time-varying translation vector, if in the two reference frames, the scalar
quantity remains the same. That is,
$\tilde{P}(\boldsymbol{y})=P(\boldsymbol{x})$, where $P$ and $\tilde{P}$ are
measured in the original and translated frames, respectively.

Similarly, one can also define an objective Lagrangian vector
$\boldsymbol{\xi}$ as one that transforms under eq (1) as

$\tilde{\boldsymbol{\xi}}(\boldsymbol{y}(t,\boldsymbol{y}_{0}))=\mathbf{Q}^{T}(t)\boldsymbol{\xi}(\boldsymbol{x}(t,\boldsymbol{x}_{0})).$

In this context, the primary issue surrounding objectivity of single
trajectory diagnostics is that one can always pass to the frame of the
particle, and in that reference frame, the particle is not moving. In the
following, we derive the experiment-relative Lagrangian velocity of a fluid
particle and show that it is an objective vector. With this, we can
objectively define the relative stretching and rotation of fluid from sparsely
sampled experimental data, with no a priori knowledge of the structures being
investigated.

###  2.2 Relative Lagrangian Velocity

Suppose we have a collection of Lagrangian particle trajectory observations
$\\{\boldsymbol{x}_{i}(t)\\}$, with corresponding Lagrangian velocities
$\dot{\boldsymbol{x}}_{i}(t)=\boldsymbol{v}_{i}(t)$. By linearity of the time-
derivative, the velocity of the average position of these particles at time
$t$ is equivalent to the average of their Lagrangian velocities,

$\overline{\boldsymbol{x}}(t)=\frac{1}{n}\sum_{i=1}^{n}\boldsymbol{x}_{i}(t)$
(2)

$\boldsymbol{v}(t)=\dot{\boldsymbol{x}}(t),\qquad\overline{\boldsymbol{v}}(t)=\frac{1}{n}\sum_{i=1}^{n}\boldsymbol{v}_{i}(t)=\dot{\overline{\boldsymbol{x}}}(t)$
(3)

Furthermore, define the time-varying moment-of-inertia tensor

$\Theta(t)=\overline{(|\boldsymbol{x}_{i}-\overline{\boldsymbol{x}}|^{2}I-(\boldsymbol{x}_{i}-\overline{\boldsymbol{x}})\otimes(\boldsymbol{x}_{i}-\overline{\boldsymbol{x}})}$

where averaging is over the trajectory index $i$. From our sparse
trajectories, we define the average experiment vorticity as

$\overline{\boldsymbol{\omega}}(t)=\Theta^{-1}\overline{(\boldsymbol{x}_{i}-\overline{\boldsymbol{x}})\times(\boldsymbol{v}_{i}-\overline{\boldsymbol{v}})}.$
(4)

To avoid cumbersome notation, we will drop the subscript $i$ from hereon when
it is clear we are referring to a specific trajectory
$\boldsymbol{x}(t;t_{0},\boldsymbol{x}_{0})$. Having now calculated the
average rotation and translation of the sampled fluid in our experiment, we
define the relative Lagrangian velocity as

$\boldsymbol{v}_{d}(\boldsymbol{x}(t))=\boldsymbol{v}(\boldsymbol{x}(t))-\overline{\boldsymbol{v}}(t)-\overline{\boldsymbol{\omega}}(t)\times(\boldsymbol{x}-\overline{\boldsymbol{x}}).$
(5)

We show in Appendix A that $\boldsymbol{v}_{d}(\boldsymbol{x}(t))$ is an
objective Lagrangian vector. That is, by removing the mean translation and
average total rotation of our sample of fluid particles, we can study
objective properties of dynamic fluid structures in our sampled domain.

This is a Lagrangian-observation-based implementation of the objective
Eulerian deformation velocity $\mathbf{v}_{d}$ original derived by Kaszás et
al. (2023). In our case, we have replaced their use of full spatial averages
with trajectory averages to accommodate limitations when performing Lagrangian
measurements. Trajectories calculated in the deformation velocity field
$\mathbf{v}_{d}$ do not necessarily trace material flow features, but
$\mathbf{v}_{d}$ can be used to objectivize frame-dependent Eulerian
diagnostics. In contrast, the relative Lagrangian velocity
$\boldsymbol{v}_{d}$ provides a means to objectivize trajectory diagnostics
relative to the flow measurements available without a priori knowledge of the
underlying flow.

###  2.3 Stretching

For steady flows, Haller et al. (2021) showed that fluid particle trajectories
(streamlines) are material features and fluid stretching (or compression)
normal to the trajectory from time $t_{0}$ to time $t_{N}$ can be quantified
from the ratio of Lagrangian velocities at times $t_{N}$ and $t_{0}$. In
slowly-varying flows, the trajectory stretching exponent (TSE) also
approximates material deformation with the difference from true deformation
being a function of $\partial_{t}\mathbf{v}(\boldsymbol{x}(t),t)$. TSEs were
shown to highlight hyperbolic invariant manifolds and faithfully reproduce the
features of finite time Lyapunov exponent fields.

Here, the evolution of the magnitude of the relative Lagrangian velocity
vector provides a measure of stretching relative to the average behavior of
our fluid being observed. Formally, the relative trajectory stretching
exponents rTSE and $\mathrm{r\overline{TSE}}$ from time $t_{0}$ to time
$t_{N}$ can be written as

$\displaystyle\mathrm{rTSE}(\boldsymbol{x}_{i})_{t_{0}}^{t_{N}}$
$\displaystyle=\frac{1}{t_{N}-t_{0}}\log\frac{|\boldsymbol{v}_{d}({\boldsymbol{x}}_{i}(t_{N}))|}{|\boldsymbol{v}_{d}({\boldsymbol{x}_{i}}(t_{0}))|}$
(6)
$\displaystyle\mathrm{r\overline{TSE}}(\boldsymbol{x}_{i})_{t_{0}}^{t_{N}}$
$\displaystyle=\frac{1}{t_{N}-t_{0}}\sum_{j=0}^{N-1}\left|\log\frac{|\boldsymbol{v}_{d}({\boldsymbol{x}}_{i}(t_{j+1}))|}{|\boldsymbol{v}_{d}({\boldsymbol{x}}_{i}(t_{j}))|}\right|.$
(7)

$\mathrm{rTSE}$ provides the stretching exponent for the relative Lagrangian
velocity vector from time $t_{0}$ to $t_{N}$, whereas
$\mathrm{r\overline{TSE}}$ is a cumulative measure of all stretching and
contraction that occurs in the same time window. Given the objectivity of the
relative Lagrangian velocity $\boldsymbol{v}_{d}$ defined in eq (5), then
$\mathrm{rTSE}$ and $\mathrm{r\overline{TSE}}$ are also objective. That is,
because
$\hat{\boldsymbol{v}}_{d}(\boldsymbol{y}(t))=Q^{T}\boldsymbol{v}_{d}(\boldsymbol{x}(t))$,
then
$|\hat{\boldsymbol{v}}_{d}(\boldsymbol{y}(t))|=|\boldsymbol{v}_{d}(\boldsymbol{x}(t))|$
and rTSEs are frame-indifferent scalars that quantify the relative deviation
of sampled particle behavior from the mean fluid deformation. This provides a
formal objective extension of the TSEs from Haller et al. (2021) that works in
unsteady (time-varying) flows.

###  2.4 Rotation

For steady flows with negligible mean vorticity, Haller et al. (2021) also
derived the trajectory rotation average ($\overline{\mathrm{TRA}}$), which
provides a single trajectory rotation measure that identifies elliptic
Lagrangian coherent structures. Under these flow assumptions,
$\overline{\mathrm{TRA}}$ calculates the cumulative rotation of material
streamline tangent vectors. For highly unsteady, or strongly rotational flows
this approach is insufficient as streamlines no longer resemble material
lines. Instead we adapt their average rotation speed of material tangent
vectors for relative Lagrangian velocities as follows.

For a given relative Lagrangian velocity vector $\boldsymbol{v}_{d}$, we can
define the unit vector pointing in that direction:

$\mathbf{e}(t)=\frac{\boldsymbol{v}_{d}(t)}{|\boldsymbol{v}_{d}(t)|}.$

Then,

$\alpha_{t_{0}}^{t_{N}}=\frac{1}{t_{N}-t_{0}}\int_{t_{0}}^{t_{N}}\left|\dot{\mathbf{e}}(t)-\frac{1}{2}\bar{\boldsymbol{\omega}}(t)\times\mathbf{e}(t)\right|dt$
(8)

is an objective measure of the average rotation speed of $\boldsymbol{v}_{d}$
from time $t_{0}$ to $t_{N}$ (Haller et al., 2021). The instantaneous limit of
eq (8) also exists ($\alpha_{t_{0}}=\lim_{t_{N}\to
t_{0}}\alpha_{t_{0}}^{t_{N}}$) and provides the instantaneous rate of rotation
of the relative Lagrangian velocity vector with respect to the spatially-
averaged rotation.

###  2.5 Extremely Sparse Sampling

As we will see in the following sections, calculating
$\overline{\boldsymbol{v}}(t)$ and $\overline{\boldsymbol{\omega}}(t)$
directly from trajectory data can become problematic at extremely sparse
trajectory concentrations. By extremely sparse, we mean less than $1\ell^{-2}$
and $1\ell^{-3}$ where $\ell$ is a characteristic length of structures in the
flow (e.g., eddy diameter, obstruction dimensions). At this limit of
trajectory concentration, we are fundamentally undersampling the prominent
structures in the flow, and an accurate depiction of mean behavior is not
reasonably expected.

For these experiments, if additional sources of velocity measurements are
available, such as from anemometers, remotely sensed ocean currents, or wind
tunnel probes, $\boldsymbol{v}_{\mathrm{ref}}$ and
$\boldsymbol{\omega}_{\mathrm{ref}}$ can be calculated from those data
sources, and used to estimate the bulk reference frame. We can then write the
Lagrangian velocity relative to this prescribed reference frame as:

$\boldsymbol{v}_{d\mathrm{,ref}}(\boldsymbol{x}(t))=\boldsymbol{v}(\boldsymbol{x}(t))-\boldsymbol{v}_{\mathrm{ref}}(t)-\boldsymbol{\omega}_{\mathrm{ref}}(t)\times(\boldsymbol{x}-\boldsymbol{x}_{\mathrm{ref}}).$
(9)

where $\boldsymbol{v}_{\mathrm{ref}}$ and $\boldsymbol{\omega}_{\mathrm{ref}}$
are the velocity and rotation of the chosen reference frame relative to the
frame in which the measurements were made and $\boldsymbol{x}_{\mathrm{ref}}$
specifies the reference point around which the rotation occurs. When using
$\boldsymbol{v}_{d\mathrm{,ref}}$ in lieu of $\boldsymbol{v}_{d}$, eqs (6 \-
8) calculate trajectory stretching and rotation relative to this choice of
frame. Since we are explicitly keeping track of a given choice of reference
frame in eq (9), $\boldsymbol{v}_{d\mathrm{,ref}}$ remains objective (frame-
indifferent) to Euclidean frame changes. This in turn generalizes our original
definition where $\boldsymbol{v}_{\mathrm{ref}}=\overline{\boldsymbol{v}}(t)$,
$\boldsymbol{\omega}_{\mathrm{ref}}=\overline{\omega}(t)$ and
$\boldsymbol{x}_{\mathrm{ref}}(t)=\overline{\boldsymbol{x}}(t)$, but we now
have a less direct connection of fluid deviations from the true mean behavior.
Keeping track of the reference frame also connects $\mathrm{r}\alpha$ and
$\mathrm{rTSE}$ to the quasi-objective diagnostics TRA and TSE, as imposing
$\boldsymbol{v}_{\mathrm{ref}}=0$, and $\boldsymbol{\omega}_{\mathrm{ref}}=0$
then provides the definitions from Haller et al. (2021). Further, the
generalized relative Lagrangian velocity frameworks allows researchers to
transparently maintain objectivity if a given reference frame is preferred.

##  3 Results

We now provide examples of $r\alpha$ and $\mathrm{r\overline{TSE}}$ analysis
for numerical simulations and experimental observations. We compare against an
array of diagnostics suitable for different experimental flows, with each
example detailing a specific advantage of $r\alpha$ and/or
$\mathrm{r\overline{TSE}}$ over other approaches. We present these examples in
a systematic order so as to avoid overburdening the reader with exhaustive and
repetitive figures.

###  3.1 Numerical Simulations

####  3.1.1 Unsteady Bickley Jet

For our first example we consider the unsteady Bickley jet. This is a two-
dimensional geophysical model of a quasi-periodic zonal jet with adjacent
advected eddies whose Lagrangian dynamics have been studied in depth (see,
e.g., Rypina et al., 2007). This flow was also used as a benchmark for
Lagrangian coherent structure methods (Hadjighasem et al., 2017). Its time-
dependent stream function is given by

$\psi(x,y,t)=-UL\mathrm{tanh}\left(\frac{y}{L}\right)+UL\mathrm{sech}^{2}\left(\frac{y}{L}\right)Re\left[\sum_{n=1}^{3}\epsilon_{n}\exp(-ik_{n}c_{n}t)\exp(ik_{n}x)\right].$
(10)

We use the parameters from Rypina et al. (2007) found in Table 1.

U | L | $k_{n}$ | $\epsilon_{1}$ | $\epsilon_{2}$ | $\epsilon_{3}$ | $c_{3}$ | $c_{2}$ | $c_{1}$
---|---|---|---|---|---|---|---|---
62.66 $m$ $s^{-1}$ | 1770 $km$ | $2n/r_{0}$ | 0.0075 | 0.15 | 0.3 | $0.205U$ | $0.461U$ | $c_{3}+\frac{k_{2}(\sqrt{5}-1)}{2k_{1}}(c_{2}-c_{3})$
Table 1: Bickley Jet model parameters.

In figure 1 we compare $r\alpha$ and $r\overline{\mathrm{TSE}}$ to benchmarks
of material rotation and stretching, the Lagrangian averaged vorticity
deviation (LAVD) (Haller et al., 2016), and the finite time Lyapnuov exponent
(FTLE) (Haller, 2015) on a dense grid of trajectories (3600$\ell^{-2}$), for
an integration time corresponding to 30 days. We find that $r\alpha$ is able
to match prominent features of the LAVD field and effectively identify eddies
adjacent to the central jet as regions of strong rotation.
$r\overline{\mathrm{TSE}}$ is also able to identify the edges of the central
jet and edges of vortices as regions of significant stretching, similar to
FTLE. Deviations primarily exist in the centers of the vortex cores, the cause
of which is discussed in Appendix B.

While $\overline{\mathrm{TRA}}$ and $\overline{\mathrm{TSE}}$ have proven to
effectively reproduce the same features as LAVD and FTLE in slowly-varying
flows (Haller et al., 2021), in figure 1 we highlight how necessary this
slowly-varying assumption truly is. With the chosen parameters, the Bickley
jet is highly unsteady. For meaningful quasi-objective calculations, one
requires $|\mathbf{v}_{t}|/|\ddot{\boldsymbol{x}}|<<1$, but only $41\%$ of
trajectories experience $|\mathbf{v}_{t}|/|\ddot{\boldsymbol{x}}|<1$ and only
$0.03\%$ of particle paths show
$|\mathbf{v}_{t}|/|\ddot{\boldsymbol{x}}|<0.1$. This suggests that the
coherent structures in the flow are likely evolving or traveling much faster
than a Lagrangian particle is able to trace them out. In figure 1, the
advected eddies are actually measured as having relatively low rotation by
$\overline{\mathrm{TRA}}$. $\overline{\mathrm{TSE}}$ also suggests nearly
everything outside the central jet is undergoing significant stretching, in
contrast to FTLE. For this degree of unsteadiness and strong eddy advection,
$r\alpha$ and $r\overline{\mathrm{TSE}}$ are much better suited than
$\overline{\mathrm{TRA}}$ and $\overline{\mathrm{TSE}}$, but still have much
more relaxed data requirements than LAVD and FTLE.

Figure 1: Evaluation of material rotation (LAVD) and stretching (FTLE), for
the unsteady Bickley Jet. Note the distinct central jet with rotating eddies
on either side.

We now progressively and randomly downsample the number trajectories to show
the robustness of our objective diagnostics at low trajectory densities, well
beyond the bounds of Lekien & Ross (2010) and Mowlavi et al. (2022). In figure
2, we calculate $r\alpha$ and $r\overline{\mathrm{TSE}}$ as well as the
particle trajectory length (Mancho et al., 2013) and initial particle speed
$|\mathbf{v}|$. While trajectory length and particle speed are not objective,
they are either easily computed or commonly used diagnostics for experimental
LPT studies (see, e.g., Fu et al., 2015; Tauro et al., 2017; Rosi & Rival,
2018). After calculating each diagnostic, we reconstruct the full resolution
diagnostic field using radial basis functions, similar to Encinas-Bartos et
al. (2022).

At 150$\ell^{-2}$, $r\alpha$ looks relatively similar to its full resolution
sampling, but the thin hyperbolic structures identified by
$r\overline{\mathrm{TSE}}$ in the Bickley jet are already beginning to
disappear. This degradation of hyperbolic coherent structures
$r\overline{\mathrm{TSE}}$ is similar to the findings of Haller et al. (2021)
for $\overline{\mathrm{TSE}}$ and Mowlavi et al. (2022) for
$\overline{\mathrm{TSE}}$ and FTLE. The trajectory length reveals fluid
particles in the central jet have the longest trajectories over this time
window when compared with nearby particles in the adjacent eddies.
$|\mathbf{v}|$ is also largest in the center of flow domain, and suggests some
oscillating feature may be present. At 15$\ell^{-2}$, $r\alpha$ no longer
maintains such clear boundaries between eddies and jets, but its physical
definition enables a meaningful interpretation: multiple localized regions of
strong rotation are present in the flow. $r\overline{\mathrm{TSE}}$ no longer
provides meaningful information, and without _a priori_ knowledge of the flow,
neither does the trajectory length field. Of all metrics, $|\mathbf{v}|$ shows
the smallest change under downsampling.

At the lowest resolution (1.5$\ell^{-2}$), we find that
$r\overline{\mathrm{TSE}}$ still suggests five distinct local rotation maxima
corresponding to the five eddies. The trajectory length is now contradicting
features that were revealed with 100 times the number of trajectories, and
$|\mathbf{v}|$ is still relatively unchanged. While the resilience of
$|\mathbf{v}|$ under downsampling is notable, the structures suggested by the
diagnostic field are in fact misleading, even at full resolution.

Figure 2: Comparison of relative rotation ($r\alpha$) and stretching
($\mathrm{r\overline{TSE}}$) with non-objective diagnostics trajectory length
and particle velocity magnitude. Each diagnostic was progressively and
randomly downsampled, with the original resolution diagnostic recreated using
radial basis functions.

We examine the ability of the particle velocity to accurately extract coherent
structures in Figure 3. Figure 3a shows a section of the full resolution
velocity magnitude field at time $t=0$. Figure 3b shows the discrete
probability histogram of $|\mathbf{v}|$. The fast moving central core is
present as a peak on the right side of the histogram, with the slow moving
surrounding fluid creating a probability peak as well. The vertical red line
in figure 3b separates the two potentially distinct features at the
$|\mathbf{v}|$ value corresponding to the distribution minimum, $s_{\delta}$.
The $|\mathbf{v}|=s_{\delta}$ level-set contour corresponding to this value is
drawn in red in figure 3c, as well as other velocity contours in blue, on top
of the FTLE field.

Examination of figure 3c show that the jet and eddy structures suggested by
the FTLE field have boundaries that are actually transverse to all of the
velocity contours. In figure 3d we plot the final position of all fluid
particles in the velocity core ($|\mathbf{v}|\geq s_{\delta}$), colored by
their $|\mathbf{v}(t_{0})|$ value, after the same 30 day advection used in
figure 2. Particles corresponding to the red boundary have been advected as
well. It is clear that the velocity-based jet is not actually a coherent
structure as the proposed feature has been stretched inside and around the
eddies, as well as down the eastward jet, with initial particle speed features
playing no role in determining where the particles end up.

Figure 3: Comparison of particle speed, FTLE, and the evolution of fluid
particles advected from inside a given velocity level set.

For comparison, we also show the evolution of $r\alpha$ structures, those that
were similarly resilient to downsampling. In figure 4a we plot $r\alpha$, with
rotationally coherent structure boundaries identified as the outermost closed
convex contours of $r\alpha$, as has been previously used in other elliptic
Lagrangian coherent structure methods (Haller et al., 2016). Figure 4b shows
the location of the same $r\alpha$-colored fluid particles after 30 days of
advection. The $r\alpha$ eddy boundaries have minimally deformed after being
advected downstream more than 20 times their diameter. This strong coherence
can be attributed to the mathematical definition of $r\alpha$ as a measure of
relative fluid rotation and suggest we have effectively and objectively
identified the true rotationally coherent structures in our flow field.

Figure 4: Outermost closed convex $r\alpha$ level-set contours overlaid on the
$r\alpha$ field. Advected position of $r\alpha$ contours and surrounding
fluid, colored by initial $r\alpha$ values.

####  3.1.2 AVISO

We now consider the identification of elliptic Lagrangian flow structures from
a two-dimensional ocean satellite altimetry data provided by AVISO which has
been the focus of several coherent structure studies (see, e.g., Haller et
al., 2016, 2021). The zonal and meridional component
$\mathbf{v}=(\mathrm{v}_{1},\mathrm{v}_{2})$ of the ocean currents are derived
from the sea-surface height profile

$\displaystyle f\mathrm{v}_{2}(\mathbf{x},t)$
$\displaystyle=\dfrac{1}{\rho}\dfrac{\partial}{\partial x}p(\mathbf{x},t)$
(11) $\displaystyle f\mathrm{v}_{1}(\mathbf{x},t)$
$\displaystyle=\dfrac{1}{\rho}\dfrac{\partial}{\partial y}p(\mathbf{x},t)$
(12) $\displaystyle-g$ $\displaystyle=\dfrac{1}{\rho}\dfrac{\partial}{\partial
z}p(\mathbf{x},t),$ (13)

where $p(\mathbf{x},t)$ is the pressure, $\rho$ is the fluid density, $f$ is
the Coriolis parameter and $g$ is the earth’s acceleration. The daily-gridded
velocity data is freely available from the Copernicus Marine Environment
Monitoring Service. Our analysis focuses on the North Atlantic Gulf Stream
between longitudes $70^{\circ}$W and $55^{\circ}$W, and latitudes
$30^{\circ}$N and $45^{\circ}$N, spanning September and October 2006. We start
with an initial grid of trajectories consisting of $150\times 150$ points.
Using a characteristic length scale of mesoscale eddies ($\ell=100$ km), this
density roughly corresponds to $100\ell^{-2}$.

Figure 5: Overview of rotation diagnostics for the AVISO data in the Gulf
Stream at full resolution ($100\ell^{-2}$).

Elsewhere in the Atlantic, Haller et al. (2021) and Encinas-Bartos et al.
(2022) have shown that the AVISO velocity field is slowly varying, with a
vanishing spatially-averaged vorticity. Furthermore, these authors showed that
$\mathrm{\overline{TRA}}$ effectively identifies ocean eddies, and outperforms
other sparse trajectory rotation diagnostics. In figure 5 we compare the
$r\alpha$ (5a) with $\mathrm{\overline{TRA}}$ (5b), using LAVD as ground truth
of material rotation (5c). As seen before in slowly-varying ocean flows,
$\mathrm{\overline{TRA}}$ faithfully approximates the LAVD field, identifying
the meanders of the Gulf Stream, and adjacent eddies. In figure 5a, we find
that $r\alpha$ can match the ability of $\mathrm{\overline{TRA}}$ at this
trajectory density.

We compare the sparse data performance of $r\alpha$ against
$\mathrm{\overline{TRA}}$ in figure 6 by progressively and randomly
subsampling as in section 3.1.1. For trajectory densities $10\ell^{-2}$ and
$1\ell^{-2}$, $\mathrm{r\alpha}$ is practically indistinguishable from
$\mathrm{\overline{TRA}}$ with the mesoscale LAVD eddies in figure 5c being
visible as local maxima in both diagnostic fields. For a trajectory
concentration of $0.1\ell^{-2}$, $\mathrm{\overline{TRA}}$ is able to capture
most of the eddies in the region, with missing features arising because no
position data (black dots) were sampled inside particular features. While
$r\alpha$ is able to capture the leftmost Lagrangian eddy, it introduces
unexpected nearshore rotational features that do not exist at full resolution.
These false positives in $r\alpha$ at extremely low resolution directly result
from the error induced by approximating the mean flow properties using a very
sparse trajectory ensemble.

We show the effect of undersampling on bulk rotation and translation,
$\boldsymbol{\overline{\omega}}(t)$ and $\boldsymbol{\overline{v}}(t)$,
respectively, in Figure 7. At $100\ell^{-2}$ the mean flow properties are
nearly constant over 15 days. This is consistent with the expectation from
earlier studies that Eulerian features do not change significantly over such
relatively small time scales in the ocean. As we progressively reduce the
trajectory density, however, $\boldsymbol{\overline{\omega}}(t)$ and
$\boldsymbol{\overline{v}}(t)$ start displaying strong oscillations in time.
At extremely low trajectory densities ($<1\ell^{-2}$), the averages are no
longer representative of the underlying mean flow properties (dashed red
curves in Figure 7). These oscillations contribute unphysical fluctuations in
$\boldsymbol{v}_{d}$ and hinder our interpretation of $\mathrm{r\alpha}$.

In this extremely sparse sampling situation, we instead utilize a pre-defined
reference frame to compute $\mathrm{r\alpha}$. To do this, we replace the
time-dependent bulk values
$\boldsymbol{\overline{\omega}}(t),\boldsymbol{\overline{v}},\boldsymbol{\overline{x}}$
from eqs (2-4) with their temporal averages over the integration window

$\displaystyle\boldsymbol{v}_{\mathrm{ref}}$
$\displaystyle=\dfrac{1}{t_{1}-t_{0}}\int_{t_{0}}^{t_{1}}\boldsymbol{\overline{v}}(t)dt,$
(14) $\displaystyle\boldsymbol{\omega}_{\mathrm{ref}}$
$\displaystyle=\dfrac{1}{t_{1}-t_{0}}\int_{t_{0}}^{t_{1}}\boldsymbol{\overline{\omega}}(t)dt,$
(15) $\displaystyle\boldsymbol{x}_{\mathrm{ref}}$
$\displaystyle=\dfrac{1}{t_{1}-t_{0}}\int_{t_{0}}^{t_{1}}\boldsymbol{\overline{x}}(t)dt.$
(16)

The relative trajectory rotation computed with respect to
$\boldsymbol{v}_{\mathrm{ref}},\boldsymbol{\omega}_{\mathrm{ref}},\boldsymbol{x}_{\mathrm{ref}}$
is denoted as $\mathrm{r\alpha}_{\mathrm{ref}}$ in Figure 6. All mean values
used to calculate relative Lagrangian velocities are still computed from the
sparse pseduo-buoy data, but the mean values no longer vary in time. Using a
general knowledge of the true slowly-varying nature of the flow (e.g., figure
7), we know eqs (14-15) are good approximations of
$\boldsymbol{\overline{\omega}}(t),\boldsymbol{\overline{v}},\boldsymbol{\overline{x}}$,
and we are able to extend our objective rotation measure to extremely sparse
sampling. In this way, we are able to remove spurious near-shore features and
match the excellent performance of $\overline{\mathrm{TRA}}$ for slowly-
varying ocean flows (figure 6c,i). Combined with the enhanced performance of
r$\alpha$ over $\overline{\mathrm{TRA}}$ for the unsteady Bickley jet, we find
r$\alpha$ is a promising diagnostic with a broader range of applicability than
$\overline{\mathrm{TRA}}$.

Figure 6: Comparison for the AVISO data between objective rotational sparse
data-diagnostic $\mathrm{r\alpha}$, $\mathrm{r\alpha_{\mathrm{ref}}}$ and its
quasi-objective single trajectory counterpart $\mathrm{\overline{TRA}}$ under
varying trajectory densities ($10\ell^{-2},1\ell^{-2},0.1\ell^{-2}$). The
black markers denote the trajectory endpoints. Figure 7: Data-based estimates
for bulk translation velocity $|\boldsymbol{\overline{v}}(t)|$ and rotation
$\boldsymbol{\overline{\omega}}(t)$ under varying trajectory density
($100\ell^{-2},10\ell^{-2},1\ell^{-2},0.1\ell^{-2}$).

####  3.1.3 Stationary Concentrated Vortex

Our third example considers the identification of distinct features inside the
stationary concentrated vortex model (SCVM) of Onishchenko et al. (2021). The
SCVM represents finite-size cyclones in the Earth’s atmosphere as steady
axially-symmetric solutions to the Euler equations. Both the vertical and
radial extent of the vortex is controlled by model parameters. The flow
consists of inner and outer regions, an internal upward motion and external
downward motion. A distinct central torus was also identified by Aksamit
(2023) as a Lagrangian coherent structure. The cyclone is further divided
vertically into a centripetal and centrifugal flow with upward moving fluid
recirculated from the top of the vortex back down.

One can write the SVCM velocity field in cylindrical coordinates as

$\displaystyle\mathrm{v}_{r}$
$\displaystyle=-\mathrm{v}_{0}\frac{r}{L}\left(1-\frac{z}{L}\right)e^{-\frac{z}{L}-\frac{r^{2}}{r_{0}^{2}}}$
$\displaystyle\mathrm{v}_{\theta}$
$\displaystyle=\pm\mathrm{v}_{\theta_{0}}\frac{z}{L}e^{-\frac{z}{L}-\frac{r^{2}}{r_{0}^{2}}}$
$\displaystyle\mathrm{v}_{z}$
$\displaystyle=2\mathrm{v}_{0}\frac{z}{L}\left(1-\frac{r^{2}}{r_{0}^{2}}\right)e^{-\frac{z}{L}-\frac{r^{2}}{r_{0}^{2}}},$

where $r_{0}$ is the radial extent separating inner and outer vortex
structures, $L$ is the height of maximum vertical velocity at $r=0$, and
$\mathrm{v}_{0}$ and $\mathrm{v}_{\theta_{0}}$ are characteristic velocities.
For our example, we use $\mathrm{v}_{0}=1$, $L=10$, $r_{0}=1$ and
$\mathrm{v}_{\theta_{0}}=9$. We consider our characteristic length of the flow
to be equal to the separation of inner and outer motions, $\ell=r_{0}=1$, and
the approximate radius of the main central torus.

This numerical model test is distinguished from the previous two examples as
the SCVM is a recirculation flow that has zero mean velocity, no advected
vortices, and no time dependence. In figure 8 we compare the objective
rotational diagnostics $r\alpha$ and $\mathrm{LAVD}$ advected for $\Delta
t=500$ with the widely used $\lambda_{2}$ criterion (Jeong & Hussain, 1995)
and vorticity magnitude, $|\boldsymbol{\omega}|$, at high spatial resolution
($2000\ell^{-3}$). Stationarity of the flow suggests $\overline{\mathrm{TRA}}$
would also be a suitable comparison, but in light of the strong performance of
r$\alpha$ against $\overline{\mathrm{TRA}}$ in the previous two examples, we
have left it out for compactness of presentation.

A two-dimensional slice along the $y=0$ plane is shown for each field in
figure 8, with vortex boundaries in $r\alpha$, $\mathrm{LAVD}$,
$|\boldsymbol{\omega}|$ extracted as the set of outermost closed convex
contours (level-set curves) in white. Following the derivation of the
$\lambda_{2}$ criterion, we use $\lambda_{2}<0$ as the designation of strong
rotational motion (Jeong & Hussain, 1995). We find that $r\alpha$ and
$\mathrm{LAVD}$ contours reveal very similar structures, both suggesting the
same location of a central torus vortex surrounded by recirculating flow.
$\lambda_{2}$ and $|\boldsymbol{\omega}|$ fields are also qualitatively
similar, suggesting a central vortex positioned over $x=0$, and two outer
features. The $\lambda_{2}=0$ contour, however, encloses a much larger region
of the flow than for $r\alpha$, $\mathrm{LAVD}$, or $|\boldsymbol{\omega}|$
structures. As an additional comparison, we include r$\overline{\mathrm{TSE}}$
and FTLE, both of which show strong spiraling into the center of the vortices
identified by $r\alpha$ and $\mathrm{LAVD}$.

We can test the performance of each coherent structure diagnostic by
evaluating which fluid structures maintain coherence under the flow. We select
fluid particles inside the boundaries of the distinguished features for each
rotation diagnostic on the $y=0$ plane, and advect them over an integration
time of $\Delta t=1000$. These fluid particle paths are shown in figure 9 with
trajectories colored by the initial diagnostic field value. We include the
corresponding figure 8 diagnostic field on the $y=0$ plane.

The stationary coherent structures identified by $r\alpha$ and $\mathrm{LAVD}$
maintain coherence over time, with seemingly no deviation of fluid from the
domain identified as the central torus vortex. Considering that $r\alpha$ does
not rely on the same three-dimensional velocity gradient data as
$\mathrm{LAVD}$, $r\alpha$ shows exceptional performance in identifying this
Lagrangian coherent structure. Fluid initially inside $\lambda_{2}$ and
$|\boldsymbol{\omega}|$ structures have instead maintained no coherence, and
are sufficiently dispersed in the flow. While the $|\boldsymbol{\omega}|$
vortices are now mixed throughout the inner SVCM ”beehive” domain, the
$\lambda_{2}<0$ fluid is now recirculating in both the inner and outer regions
of the SVCM. It is also worth noting the beehive structure seen in figure 9c,d
is also identified visible as a structure in $r\alpha$ and
r$\overline{\mathrm{TSE}}$, though not in any other diagnostic.

The ability of $r\alpha$ to identify these same Lagrangian coherent structures
at increasing sparsity is shown in Figure 10. Here we utilize a random set of
initial particle trajectories in three dimensions and extract isosurfaces from
a reconstructed $r\alpha$ field using natural neighbor interpolation. We
utilize the same $r\alpha$ value for 3D level-set extraction as the outermost
closed convex contour in figures 8 and 9. Plots of reconstructed $r\alpha$
fields on the $y=0$ plane (figure 10a-c) reveal that small deviations from
this range would not significantly change the topology of the surfaces in
figure 10d-f. Note the difference of $z$-value ranges between figure 10a-c and
figure 10d-f.

In figure 10, $r\alpha$ at $10\ell^{-3}$ looks only moderately different from
$2000\ell^{-3}$. Two distinct intersections of the central torus with the
$y=0$ plane are still evident in white, and the annulus shape is clearly
visible in the associated isosurface. At $1\ell^{-3}$, the ”donut hole” above
$x=0$ becomes less obvious, but the horizontal and vertical extent of the
central torus is still accurate. For $0.1\ell^{-3}$, we use eqs (14-15) to
implement the same bulk time-averaged reference values for
$\boldsymbol{v}_{\mathrm{ref}},\boldsymbol{\omega}_{\mathrm{ref}},\boldsymbol{x}_{\mathrm{ref}}$.
At this extremely sparse sampling, the correct extent of strong central
rotation is still evident, as is a suggestion of the inner beehive feature,
but the vortex surface is now sufficiently deformed that it no long resembles
a torus.

Figure 8: Comparison of rotation diagnostics $r\alpha$, $\mathrm{LAVD}$,
$\lambda_{2}$, and $|\boldsymbol{\omega}|$ for the stationary concentrated
vortex model in the $y=0$ plane. Outermost closed convex contours for
$r\alpha$, $\mathrm{LAVD}$, and $|\boldsymbol{\omega}|$, and $\lambda_{2}=0$
contours, identify potential coherent rotational features. Note the similarity
between $r\alpha$ and $\mathrm{LAVD}$ features, and $|\boldsymbol{\omega}|$,
and $\lambda_{2}$ features. As a reference, we show several of the same
stretching features are indicated in both $r\mathrm{\overline{TSE}}$ and
$\mathrm{FTLE}$ fields.

Figure 9: Trajectories corresponding to coherent structures identified by the
four diagnostics, calculated for the same integration times Figure 10: Top
Row: Comparison of rotation diagnostics $r\alpha$ and
$\mathrm{\overline{TRA}}$ on the $y=0$ plane reconstructed from randomly
selected trajectories at decreasing particle density. The inner vortex is
extracted at each concentration as the closed contours corresponding to the
level-set value from Figure 8. Bottom Row: Comparison of the inner vortex
surface as extracted from $r\alpha$ and $\mathrm{\overline{TRA}}$ fields
reconstructed from randomly selected trajectories at decreasing particle
density. The inner vortex is extracted in each scenario as closed contours
corresponding to the level-set value from Figure 8.

####  3.1.4 Turbulent Cylinder

Our last numerical example involves a large turbulent three-dimensional direct
numerical simulation of the time-varying wake behind a cylinder at $Re=3900$.
This dataset is publicly available as a test bed for both LPT algorithms and
studies of turbulence diagnostics. A full explanation of the dataset is
provided by Khojasteh et al. (2022). Our study will focus on approximately
200,000 time-resolved Lagrangian trajectories with random initial positions in
an inner domain closest to the cylinder.

A single Eulerian snapshot of the velocity field in this computational domain
is nearly 13 Gb in raw .txt format. Advecting Lagrangian particles in such
large datasets quickly becomes a significant computational challenge. For
researchers interested in understanding Lagrangian coherent structures,
analyzing a subsample of Lagrangian data that is computed during the DNS
solver becomes much more computationally practical. For example, 350 time
steps of position and velocity of 200,000 randomly initialized particle
trajectories from the same simulation is only 8 Gb.

A simple initial visual analysis of the flow can be performed by computing
$\mathrm{r\alpha}$ and $\mathrm{r\overline{TSE}}$ for a random selection of
trajectories. In figure 11 we show 200 randomly selected particle paths
colored by $\mathrm{r\alpha}$ and $\mathrm{r\overline{TSE}}$ calculated over
their full trajectories. Using the cylinder diameter as the characteristic
length, 200 trajectories corresponds with a concentration of $1\ell^{-3}$ in
this volume. The trajectory data are not all the same length (ranging from 4
to 350 time steps) but this is not problematic at this concentration. The wake
is immediately evident in figure 11 as a region of large $\mathrm{r\alpha}$,
whose dimensions grow as you move downstream from the cylinder. We can also
see large shear in $\mathrm{r\overline{TSE}}$ for fluid that deforms around
the cylinder, as well as a clear separation between the turbulent wake, and
the much more linear and accelerating trajectories outside the wake.

To complement the previous examples, we can also refine our qualitative
description of the flow to much shorter experimental observations by looking
at the instantaneous limit r$\alpha_{t_{0}}$. In figure 12 we examine the
instantaneous behavior of fluid particles at model time $t=75$ in a $0.1d$
thick slice surrounding the $z=3$ plane. In figure 12a we have colored fluid
particles by instantaneous vorticity deviation (IVD) calculated from the full
Eulerian dataset. IVD is a frame-indifferent Eulerian measure of fluid
rotation (Haller et al., 2016) that relies on spatial derivatives of the three
velocity components, but provides us with a ground truth of the instantaneous
vortical features in the wake. There is a clear agreement between the
meandering turbulent wakes defined by objective IVD with $\mathrm{r\alpha}$ in
figure 12b, as well as the dissipation in rotational strength as you move
downstream and laterally away from the wake.

Figure 12c-d provides two additional instantaneous turbulence diagnostics, the
particle speed and the fluctuating component of the streamwise velocity around
the bulk average. Both diagnostics suggest some meandering slowdown behind the
cylinder, but as discussed in section 3.1.1, these velocity level-set
structures are frame-dependent and do not guarantee any specific material
fluid evolution.

Similar to the SVCM, we can also generate gridded $\mathrm{r\alpha}$ data from
randomly positioned particle trajectories, and extract level-set surfaces to
identify the meandering wake. In figure 13 we overlay nested
$\mathrm{r\alpha}_{t_{0}}$ surfaces for a range of values
($\mathrm{r\alpha}_{t_{0}}\in\\{0.05,0.065,0.08\\}$) in the region $3<z<6$. To
generate the wake surfaces in figure 13, we interpolate approximately $5\times
10^{4}$ instantaneous values ($415\ell^{-3}$) to a grid of $5\times 10^{6}$
points using radial-basis functions. For comparison, we also map the IVD field
from figure 12 at $z=3$. The complexity of the turbulent wake and its
meandering structure are clearly evident in the reconstructed
$\mathrm{r\alpha}$ field.

Lastly, we test the ability of $\mathrm{r\alpha}_{t_{0}}$ to identify wake
structures against the instantaneous diagnostics, $\lambda_{2}$ and
$|\boldsymbol{\omega}|$, for progressively sparser observations at model time
$t=75$. To estimate $\lambda_{2}$ and $|\boldsymbol{\omega}|$ from
unstructured particle positions we used interpolated particle velocity grids
to calculate the necessary partial derivatives. In Figure 14 we compare
reconstructed $\mathrm{r\alpha}_{t_{0}}$, $\lambda_{2}$, and
$|\boldsymbol{\omega}|$ fields restricted to the $z=3$ plane at concentrations
of $50\ell^{-2}$, $5\ell^{-2}$, $0.5\ell^{-2}$, and $0.25\ell^{-2}$. As
before, for $0.5\ell^{-2}$, and $0.25\ell^{-2}$, we utilize a
$\boldsymbol{v}_{d\mathrm{,ref}}$ using a reference frame determined by the
bulk spatio-temporal averages of all trajectories at each concentration. We
also mark the location of fluid particles used for calculations for the
$0.5\ell^{-2}$, and $0.25\ell^{-2}$ scenarios.

The strong rotation immediately behind the cylinder is evident at all
resolutions in $\mathrm{r\alpha}_{t_{0}}$. The bend of the wake is also
accurately represented with low rotation zones accurately positioned around
high rotation zones at $0.5\ell^{-2}$, and $0.25\ell^{-2}$. Approximating the
velocity gradient for $\lambda_{2}$ and $|\boldsymbol{\omega}|$ works with
decreasing accuracy as sparsity increases. The $\lambda_{2}$ field shows some
turbulent action behind the wake, but the dimensions and topology are hard to
interpret, especially when considering only the strict $\lambda_{2}<0$
criteria designates vortical regions. Additionally, $\lambda_{2}$ values vary
over two orders of magnitude depending on the trajectory concentration. The
vorticity magnitude does a better job at consistently resolving the same wake
structures across all concentrations, but at the lowest two concentrations
local extrema appear outside of the wake region that is clearly defined at
$50\ell^{-2}$.

As shown in the previous section, distinct $\lambda_{2}$ and
$|\boldsymbol{\omega}|$ features do not necessarily represent structurally
coherent fluid, but they only require instantaneous velocity fields, not
Lagrangian trajectories. The instantaneous limit $\mathrm{r\alpha}_{t_{0}}$ is
an objective Eulerian alternative for temporally-limited experimental data
with a meaningful physical connection to Lagrangian $\mathrm{r\alpha}$
computations, which can identify material and temporally coherent rotating
features. Our relative Lagrangian velocity and $r\alpha$ thus provide useful
coherent structure identification from sparse trajectories in both Eulerian
and Lagrangian frameworks.

Figure 11:  Trajectory-integrated $r\alpha$ and $\mathrm{r\overline{TSE}}$ for
200 randomly selected trajectories ($1\ell^{-3}$) from the turbulent cylinder
DNS. Figure 12: Comparison of instantaneous $r\alpha$, particle speed, and
stream wise particle velocity fluctuation values for all tracked particles in
a $0.1d$ neighborhood of the $z=3$ plane (84$\ell^{-2}$). Instantaneous
vorticity deviation on the $z=3$ plane at the same time. The high rotation in
the wake is clearly captured by both IVD and $r\alpha$ as is a velocity
deficit, to varying degrees. Figure 13:  Instantaneous volumetric level-sets
of $r\alpha$ ($415\ell^{-3}$) compared with IVD on the $z=3$ plane. Figure 14:
Effect of reducing the number of sampled volumes on instantaneous values of
$r\alpha$ compared with Eulerian diagnostics $\lambda_{2}$ and
$|\boldsymbol{\omega}|$ on the $z=3$ plane. Particle sample locations are
marked by white x’s for the two lowest observation densities.

###  3.2 Experimental Data

In the previous section we investigated the ability of $r\alpha$ and
$r\overline{\mathrm{TSE}}$ to accurately represent elliptic and hyperbolic
Lagrangian coherent structures at increasing sparsity in numerical data. In
this section, we show how the sparse capabilities of $r\alpha$ apply to real
experimental data for both a two-dimensional and a three-dimensional flow.

####  3.2.1 Global Drifter Database

For our first experimental data set we consider a set of drifters from the
Global Drifter Program (GDP) located in the North Atlantic Ocean. The GDP data
set contains more than 40000 drifters spanning the last 4 decades. Roughly
1200 drifters are currently active worldwide and report their position every 6
hours.

Specifically, we focus here on a subset of drifters active over 15 days from
20 September to 4 October 2006 in the Gulf Stream (Figure 15). This region
overlaps with the temporal-spatial Gulf Stream domain investigated with
geostrophic current data in section 3.1.2. In figure 15, the drifter density
corresponds to $0.63\ell^{-2}$, with $\ell=100\mathrm{km}$. We are thus in an
extremely sparse data setting ($<1\ell^{-2}$) and we again compute $r\alpha$
by replacing the time-dependent mean flow properties with AVISO-informed
$\boldsymbol{\omega}_{\mathrm{ref}},\boldsymbol{v}_{\mathrm{ref}}$ and
$\boldsymbol{x}_{\mathrm{ref}}$, as already done in section 3.1.2.

Following the benchmark analysis by Encinas-Bartos et al. (2022), we compare
two sparse drifter-based rotation diagnostics
$\mathrm{r\alpha_{\mathrm{ref}}}$, $\overline{\mathrm{TRA}}$ with the kinetic
energy of a drifter averaged along a trajectory $\mathrm{KE}$ (figure 15).
Additionally, we extract looping segments from the drifter trajectories as
proposed by the frame-dependent algorithm from Lumpkin (2016). The white
trajectory segments indicate anticyclonic loopers which frequently arise in
connection with ocean eddies (see, e.g., Griffa et al., 2008; Dong et al.,
2011).

Both $\mathrm{r\alpha_{\mathrm{ref}}}$ and $\overline{\mathrm{TRA}}$ are able
to detect rotational features surrounding the jet stream from this extremely
sparse drifter data. Anticyclonic loopers are also observed in the regions
surrounding the principal local maxima of $\mathrm{r\alpha_{\mathrm{ref}}}$
and $\overline{\mathrm{TRA}}$ . The leftmost eddy contains two looping
drifters that display high rotational values (figure 15a-b). In contrast, the
$\mathrm{\overline{KE}}$ fails to clearly identify the leftmost eddy since it
displays sharp gradients between the two drifters inside the eddy (figure
15c). Both $\mathrm{r\alpha_{\mathrm{ref}}}$ and $\mathrm{\overline{TRA}}$
display a weaker rotational feature at roughly $32^{\circ}$N latitude and
$64^{\circ}$W longitude. No loopers are identified in this region, but the
yellow trajectories in the high $\mathrm{r\alpha}$ and
$\overline{\mathrm{TRA}}$ zones show greater circulating behavior with respect
to the surrounding green trajectories (figure 15, bottom insets), suggesting
stronger rotational motion in the surface currents. There is no feature that
resembles any vortical structure in the $\mathrm{\overline{KE}}$ field in this
region.

In this ocean buoy experiment, we were able to identify eddies adjacent to the
Gulf Stream in both $\mathrm{\overline{TRA}}$ and $\mathrm{r\alpha}$ fields
that were not identified using existing sparse data diagnostics. In contrast
with the $\mathrm{\overline{TRA}}$, which requires additional assumptions on
the frame of reference, the $\mathrm{r\alpha}$ returns structures that are
valid for all observers.

Figure 15: Drifter based $\overline{\mathrm{TRA}}$ (a), $r\alpha$ (b),
$\mathrm{\overline{KE}}$ (c), and LAVD (d) in the North Atlantic Gulf Stream.
The black triangles indicated drifter positions.

####  3.2.2 Large-scale LPT Experiment

Our last example involves a novel, large-volume three-dimensional Lagrangian
particle tracking velocimetry dataset from an industrial-scale wind tunnel
facility (Hou et al., 2021). Utilizing glare-point spacing on large soap
bubbles, Hou et al. (2021) were able to investigate the vortical truck wake
behind a tractor-trailer model at a $9^{\circ}$ yaw angle. For details on the
methodology, see Kaiser & Rival (2023). We compare two wind-tunnel
experiments, one with the tractor-trailer model obstructing the flow, and a
uniform flow without the tractor-trailer.

Each experiment provides around 300 s of trajectory data, sampled at 150 Hz.
The bubble generator and camera setup generating time-varying particle
concentrations at the low end of our spectrum. For measurements of the
vortical wake, trajectory concentrations oscillate between 0.2 and
1.5$\ell^{-3}$. The uniform flow measurements have a slightly higher
resolution, ranging from 0.5 to 3$\ell^{-3}$. To avoid significant
oscillations in relative Lagrangian velocities, we again utilize a reference
frame determined by the experimental setup. In this case, we set
$\boldsymbol{v}_{\mathrm{ref}}$ to the free stream velocity of the wind
tunnel, and set $\boldsymbol{\omega}_{\mathrm{ref}}$ to zero.

Even at this low concentration, and with this experimental methodology, we are
able to identify distinguished rotating features in flow diagnostic
visualizations and bulk diagnostic statistics. We select particles that are
tracked for at least 10 frames, and integrate $r\alpha$ over their entire
trajectory. This provides trajectories with between 10 and 70 datapoints. In
figure 16 we plot all particle positions colored by $r\alpha$ spanning the
entire experiments from two perspectives. Strongly rotating trajectories are
visible near the center of the tip vortex, and at the outer regions in the
highly turbulent flow (figure 16a, c). In contrast, for the uniform flow, the
densely sampled center of the flow actually shows the lower $r\alpha$ values.
The distribution of $r\alpha$ for each experiment is shown in figure 16e. Here
the tractor-trailer wake clearly generates a broader range of $r\alpha$, fewer
small values, and a higher average rate of rotation. The uniform flow has a
single concentrated $r\alpha$ peak corresponding to the lower end of the
tractor-trailer wake $r\alpha$ distribution, quantitatively confirming more
homogeneous flow conditions with less fluid rotation.

We also harness the spatial distribution of the instantaneous relative
rotation $r\alpha_{t_{0}}$ to identify the strongly rotating core of the tip
vortex in figure 17. By projecting bubble positions onto the $y-z$ plane (e.g.
figure 16c, d), and calculating their radial distance from the center of the
flow, we can quantify the gradient of the rate of rotation gradient away from
the vortex core. Figure 17a shows 30 bin-averages of $r\alpha_{t_{0}}$ for the
tractor-trailer wake (red line) overlaid on all instantaneous measurements
shown (red dots). Peak rotation is evident in the first $0.4$ m surrounding
the tractor-trailer vortex core, which gradually increases again at the outer
edge of the measurement domain. The uniform flow in figure 17b has no such
rotating core, with only a gentle increase in rotation on the outer edge of
the measurement domain. This outer domain increase may be a result of
turbulent boundary layer structures in the wind tunnel, or relatively fewer
bubbles in a region with larger single-camera measurement errors.

Lastly, we compare discrete probability histograms of several frame-dependent
metrics that are easy to compute from LPT data. In figure 18, we show particle
speed, trajectory length, and fluctuating streamwise Lagrangian velocity
spanning the two experiments. The tractor-trailer wake shows a slight speed up
of the flow (figure 18a), but also shorter bubble trajectories. The
fluctuating streamwise components of bubble velocities are nearly identical
for the two flows. None of these Lagrangian diagnostics present clear a
picture of wake dynamics, and $\mathbf{v}_{x}^{\prime}$ cannot distinguish the
two flows.

Figure 16: Comparison of r$\alpha$ for the tractor-trailer wake vortex and
object-free uniform flow LPT experiments at $\sim 1\ell^{-3}$. Figure 17:
Comparison of instantaneous TRA values, and their binned averages (30 bins),
as functions of radial distance from center of observation volume projected on
the $y-z$ plane for the two experiments. Note the peak rotation near the
center of the truck wake vortex that is not present in the uniform flow.
Figure 18: Comparison of discrete probability histograms of frame-dependent
metrics for the truck wake vortex and object-free uniform flow experiments at
$\sim 1\ell^{-3}$.

##  4 Conclusion and Outlook

Objective coherent structures identification tools for sparse trajectory data
are valuable for researchers in a wide range of scientific and industrial
fields. In this work, we have mathematically derived novel frame-independent
diagnostics of trajectory rotation and stretching well-suited for coherent
structure identification in sparse experimental trajectory data. Through
multiple systematic comparisons with existing Lagrangian flow diagnostics, we
have shown that:

  * •

$r\mathrm{\overline{TSE}}$ and $r\alpha$ are objective metrics that can
identify elliptic and hyperbolic Lagrangian coherent structures in highly
unsteady flows, and that work equally well in flows with large or small bulk
translation and rotation;

  * •

$r\mathrm{\overline{TSE}}$ and $r\alpha$ work well in many of the situations
where the quasi-objective diagnostics $\mathrm{\overline{TSE}}$ and
$\mathrm{\overline{TRA}}$ have previously been successful, including ocean
drifter applications;

  * •

$r\alpha$ can accurately identify coherent structures as a Lagrangian
diagnostic, and as a fixed-time Eulerian diagnostic in turbulent flows;

  * •

$r\alpha$ can identify 2D and 3D coherent structures exceptionally well in
both computational and real experimental observations, showing resilience and
interpretability at extremely sparse sampling when compared with common frame-
dependent diagnostics; and

  * •

$r\mathrm{\overline{TSE}}$ and $r\alpha$ provide an avenue for meaningful LPT
analysis and post-processing that does not require data interpolation to an
Eulerian grid, but interpolation does allow explicit identification of
individual structure boundaries.

Being able to identify coherent flow features using $r\mathrm{\overline{TSE}}$
and $r\alpha$ for the analysis of unsteady flows in a physically meaningful,
frame-indifferent and Lagrangian manner will help advance our study of fluid
dynamics in large and natural domains. Further applications of these methods,
such as the similar-time-series analytic techniques used by Aksamit et al.
(2023), is expected to open the door for an even wider range of experiments
over a vast range of scales that can harness the Lagrangian nature of
turbulent structures.

[Funding]This research received no specific grant from any funding agency,
commercial or not-for-profit sectors

[Declaration of interests]Declaration of Interests. The authors report no
conflict of interest.

[Data availability statement]The LPT wind tunnel data that support the
findings of this study are available from the authors directly. The turbulent
DNS data can be found at https://doi.org/10.15454/GLNRHK. AVISO ocean surface
current data can be found at https://doi.org/10.48670/moi-00145. Ocean drifter
data can be accessed through https://www.aoml.noaa.gov/phod/gdp/index.php.

[Author ORCIDs]N. Aksamit, https://orcid.org/0000-0002-2610-7258; A. Encinas-
Bartos, https://orcid.org/0000-0002-4203-9128; G. Haller,
https://orcid.org/0000-0003-1260-877X; D. Rival,
https://orcid.org/0000-0001-7561-6211.

##  Appendix A Objectivity of $\boldsymbol{v}_{d}$

We are interested in how $\boldsymbol{v}_{d}$ changes under Euclidean
transformations of the form

$\displaystyle\boldsymbol{x}(t)=Q(t)\boldsymbol{y}(t)+\boldsymbol{b}(t).$ (17)

Under this transformation
$\boldsymbol{y}(t)=Q^{T}(t)(\boldsymbol{x}(t)-\boldsymbol{b}(t))$. Then

$\displaystyle\hat{\boldsymbol{v}}(t)=Q^{T}(t)[\boldsymbol{v}(t)-\dot{\boldsymbol{b}}(t)-\dot{Q}(t)Q^{T}(t)(\boldsymbol{x}(t)-\boldsymbol{b}(t))]$
(18)

and

$\displaystyle\hat{\boldsymbol{v}}(t)-\hat{\overline{\boldsymbol{v}}}(t)=Q^{T}(t)[\boldsymbol{v}(t)-\overline{\boldsymbol{v}}(t)-\dot{Q}(t)Q^{T}(t)(\boldsymbol{x}(t)-\overline{\boldsymbol{x}}(t))]$
(19)

In the new reference frame, $\Theta$ transforms as

$\displaystyle\begin{split}\hat{\Theta}&=\overline{|Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})|^{2}I-Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})\otimes
Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})]}\\\
&=\overline{|(\boldsymbol{x}-\overline{\boldsymbol{x}})|^{2}I-Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})(\boldsymbol{x}-\overline{\boldsymbol{x}})^{T}Q}\\\
&=\overline{Q^{T}|(\boldsymbol{x}-\overline{\boldsymbol{x}})|^{2}IQ-Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})(\boldsymbol{x}-\overline{\boldsymbol{x}})^{T}Q}\\\
&=Q^{T}[\overline{|(\boldsymbol{x}-\overline{\boldsymbol{x}})|^{2}I-(\boldsymbol{x}-\overline{\boldsymbol{x}})\otimes(\boldsymbol{x}-\overline{\boldsymbol{x}})}]Q\\\
&=Q^{T}\Theta Q\end{split}$ (20)

We also have the useful identities

$\dot{Q}Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})=\dot{\boldsymbol{q}}\times(\boldsymbol{x}-\overline{\boldsymbol{x}})$
(21)

and

$\displaystyle\begin{split}\Theta\dot{\boldsymbol{q}}&=\overline{|\boldsymbol{x}-\overline{\boldsymbol{x}}|^{2}I-(\boldsymbol{x}-\overline{\boldsymbol{x}})\otimes(\boldsymbol{x}-\overline{\boldsymbol{x}})}\dot{\boldsymbol{q}}\\\
&=\overline{|\boldsymbol{x}-\overline{\boldsymbol{x}}|^{2}\dot{\boldsymbol{q}}-(\boldsymbol{x}-\overline{\boldsymbol{x}})(\boldsymbol{x}-\overline{\boldsymbol{x}})^{T}\dot{\boldsymbol{q}}}\\\
&=\overline{|\boldsymbol{x}-\overline{\boldsymbol{x}}|^{2}\dot{\boldsymbol{q}}-(\boldsymbol{x}-\overline{\boldsymbol{x}})((\boldsymbol{x}-\overline{\boldsymbol{x}})\cdot\dot{\boldsymbol{q}})}\\\
\end{split}$ (22)

Under the frame change, we find
$\hat{\overline{\boldsymbol{\omega}}}=\hat{\Theta}^{-1}\overline{(\boldsymbol{y}-\overline{\boldsymbol{y}})\times(\hat{\boldsymbol{v}}-\hat{\overline{\boldsymbol{v}}})}.$
So that

$\displaystyle\begin{split}\hat{\Theta}\hat{\overline{\boldsymbol{\omega}}}&=\overline{(\boldsymbol{y}-\overline{\boldsymbol{y}})\times(\hat{\boldsymbol{v}}-\hat{\overline{\boldsymbol{v}}})}\\\
&=\overline{Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})\times
Q^{T}(\boldsymbol{v}-\overline{\boldsymbol{v}}-\dot{Q}Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}}))}\\\
&=Q^{T}\overline{(\boldsymbol{x}-\overline{\boldsymbol{x}})\times(\boldsymbol{v}-\overline{\boldsymbol{v}})-(\boldsymbol{x}-\overline{\boldsymbol{x}})\times\dot{\boldsymbol{q}}\times(\boldsymbol{x}-\overline{\boldsymbol{x}})}\qquad\text{by
(5)}\\\
&=Q^{T}\overline{(\boldsymbol{x}-\overline{\boldsymbol{x}})\times(\boldsymbol{v}-\overline{\boldsymbol{v}})}-Q^{T}\overline{\dot{\boldsymbol{q}}|\boldsymbol{x}-\overline{\boldsymbol{x}}|^{2}+(\boldsymbol{x}-\overline{\boldsymbol{x}})(\dot{\boldsymbol{q}}\cdot(\boldsymbol{x}-\overline{\boldsymbol{x}}))}\\\
&=Q^{T}\overline{(\boldsymbol{x}-\overline{\boldsymbol{x}})\times(\boldsymbol{v}-\overline{\boldsymbol{v}})}-Q^{T}\Theta\dot{\boldsymbol{q}}\qquad\text{from
(6)}\\\
&=Q^{T}[\Theta\overline{\boldsymbol{\omega}}-\Theta\dot{\boldsymbol{q}}]\end{split}$
(23)

Since $\hat{\Theta}=Q^{T}\Theta Q$, we have $Q^{T}\Theta
Q\hat{\overline{\boldsymbol{\omega}}}=Q^{T}[\Theta\overline{\boldsymbol{\omega}}-\Theta\dot{\boldsymbol{q}}]$
and

$\displaystyle\hat{\overline{\boldsymbol{\omega}}}=Q^{T}[\overline{\boldsymbol{\omega}}-\dot{\boldsymbol{q}}]$
(24)

These identities all lead to

$\displaystyle\begin{split}\hat{\boldsymbol{v}}_{d}(\boldsymbol{y}(t))&=Q^{T}[\boldsymbol{v}-\overline{\boldsymbol{v}}-\dot{Q}Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})]-\hat{\overline{\boldsymbol{\omega}}}\times(Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}}))\\\
&=Q^{T}[\boldsymbol{v}-\overline{\boldsymbol{v}}-\dot{Q}Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})]-Q^{T}[\overline{\boldsymbol{\omega}}-\dot{\boldsymbol{q}}]\times(Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}}))\qquad\text{by
(8)}\\\
&=Q^{T}[\boldsymbol{v}-\overline{\boldsymbol{v}}-\dot{Q}Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})]-Q^{T}[\overline{\boldsymbol{\omega}}\times(\boldsymbol{x}-\overline{\boldsymbol{x}})]+Q^{T}\dot{\boldsymbol{q}}\times(\boldsymbol{x}-\overline{\boldsymbol{x}}))\\\
&=Q^{T}[\boldsymbol{v}-\overline{\boldsymbol{v}}-\dot{Q}Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}})-\overline{\boldsymbol{\omega}}\times(\boldsymbol{x}-\overline{\boldsymbol{x}})+\dot{Q}Q^{T}(\boldsymbol{x}-\overline{\boldsymbol{x}}))]\\\
&=Q^{T}[\boldsymbol{v}-\overline{\boldsymbol{v}}-\overline{\boldsymbol{\omega}}\times(\boldsymbol{x}-\overline{\boldsymbol{x}})]\\\
&=Q^{T}\boldsymbol{v}_{d}(\boldsymbol{x}(t))\end{split}$ (25)

##  Appendix B Material Evolution of $\boldsymbol{v}_{d}$

It is also common in the fluid dynamics community to forgo notions of frame-
indifference, instead preferring to make calculations in a particular
reference frame, such as that of a stationary observer standing next to a
laboratory experiment. In this situation, the relative frame choice is
$\boldsymbol{v}_{\mathrm{ref}}=0$, and $\boldsymbol{\omega}_{\mathrm{ref}}=0$.
The question then remains which is the best frame to choose. If one seeks to
understand the actual material stretching and rotation of the fluid, as it
deforms in a flow, we can answer that question definitively. For quantifying
material deformation, we want to use a relative reference frame in which
$\boldsymbol{v}_{\mathrm{ref}}$ evolves as a material vector. That is, if
$\boldsymbol{v}_{\mathrm{ref}}(t)$ is tangent to a material feature in the
flow (such as dye in a fluid), $\boldsymbol{v}_{\mathrm{ref}}(t+\delta t)$
will also be tangent to that material feature after it has deformed from time
$t$ to $t+\delta t$. Mathematically, this requires
$\boldsymbol{v}_{\mathrm{ref}}$ to evolve as

$\mathbf{\dot{v}}_{\mathrm{ref}}(t)=\mathbf{\nabla
v}\mathbf{v}_{\mathrm{ref}}(t)$ (26)

We now seek conditions under which $\mathbf{v}_{d}$ approximates the evolution
of a material vector. In order to avoid cumbersome notation, we write:

$2\mathbf{\overline{W}}(t)(\mathbf{x}-\mathbf{\overline{x}})=\mathbf{\overline{\omega}}(t)\times(\mathbf{x}-\mathbf{\overline{x}}),$
(27)

with $\mathbf{\overline{W}}(t)$ simply being a skew-symmetric matrix
($\mathbf{\overline{W}}(t)=-\mathbf{\overline{W}}^{T}(t)$). We start by
differentiating the deformation velocity $\boldsymbol{v}_{d}(t)$ with respect
to time which yields

$\displaystyle\mathbf{\dot{v}}_{d}$
$\displaystyle=\mathbf{\dot{v}}-\mathbf{\dot{\overline{v}}}-2\dfrac{d}{dt}[\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}})]$
(28) $\displaystyle=\mathbf{\nabla vv}+\dfrac{\partial}{\partial
t}\mathbf{v}-\mathbf{\dot{\overline{v}}}-2\dfrac{d}{dt}[\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}})]$
(29) $\displaystyle=\mathbf{\nabla v}\mathbf{v}_{d}+\mathbf{\nabla
v}[\mathbf{\overline{v}}+2\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}})]+\dfrac{\partial}{\partial
t}\mathbf{v}-\mathbf{\dot{\overline{v}}}-2\dfrac{d}{dt}[\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}})]$
(30) $\displaystyle=\mathbf{\nabla v}\mathbf{v}_{d}+\mathbf{\nabla
v}[\mathbf{\overline{v}}+2\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}})]+\dfrac{\partial}{\partial
t}\mathbf{v}-\dfrac{d}{dt}[\mathbf{\overline{v}}+2(\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}}))]$
(31) $\displaystyle=\mathbf{\nabla v}\mathbf{v}_{d}+\mathbf{\nabla
v}[\mathbf{\overline{v}}+2\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}})]-\dfrac{d}{dt}[\mathbf{\overline{v}}+2(\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}}))]+\dfrac{\partial}{\partial
t}\mathbf{v}$ (32)

Therefore, $\mathbf{v}_{d}$ evolves nearly materially as long as

$\displaystyle|\mathbf{\nabla
v}[\mathbf{\overline{v}}+2\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}})]-\dfrac{d}{dt}[\mathbf{\overline{v}}+2[\mathbf{\overline{W}}(\mathbf{x}-\mathbf{\overline{x}}))]]+\dfrac{\partial}{\partial
t}\mathbf{v}|\ll|\mathbf{\dot{v}}_{d}|,$ (33)
$\displaystyle|\dfrac{d}{dt}[\boldsymbol{v_{d}}-\boldsymbol{v}]-\boldsymbol{\nabla
v}[\boldsymbol{v_{d}}-\boldsymbol{v}]+\dfrac{\partial}{\partial
t}\boldsymbol{v}|\ll|\boldsymbol{\dot{v}}_{d}|,$ (34)

where we have used $\boldsymbol{\dot{v}}=\boldsymbol{\nabla
vv}+\dfrac{\partial}{\partial t}\boldsymbol{v}$.

## References

  * Aksamit (2023) Aksamit, Nikolas O. 2023 Mapping the shape and dimension of three-dimensional Lagrangian coherent structures and invariant manifolds. Journal of Fluid Mechanics 958, A11.
  * Aksamit et al. (2023) Aksamit, Nikolas O., Scharien, Randall K, Hutchings, Jennifer K & Lukovich, Jennifer V. 2023 A quasi-objective single buoy approach for Lagrangian coherent structures and sea ice fracture events. The Cryosphere pp. 1545–1566.
  * Bernard & Thomas (1993) Bernard, Peter S. & Thomas, James M. 1993 Vortex Dynamics and the Production of Reynolds Stress. Journal of Fluid Mechanics 253, 385–419.
  * Bristow et al. (2023) Bristow, Nathaniel, Li, Jiaqi, Hartford, Peter, Guala, Michele & Hong, Jiarong 2023 Imaging-based 3D particle tracking system for field characterization of particle dynamics in atmospheric flows. Experiments in Fluids 64 (4), 1–14, arXiv: 2210.07106.
  * Businger et al. (2006) Businger, S., Johnson, R. & Talbot, R. 2006 Scientific insights from four generations of Lagrangian smart balloons in atmospheric research. Bulletin of the American Meteorological Society 87 (11), 1539–1554.
  * Dong et al. (2011) Dong, Changming, Liu, Yu, Lumpkin, Rick, Lankhorst, Matthias, Chen, Dake, McWilliams, James C & Guan, Yuping 2011 A scheme to identify loops from trajectories of oceanic surface drifters: An application in the kuroshio extension region. Journal of Atmospheric and Oceanic Technology 28 (9), 1167–1176.
  * Encinas-Bartos et al. (2022) Encinas-Bartos, Alex P., Aksamit, Nikolas O. & Haller, George 2022 Quasi-objective eddy visualization from sparse drifter data. Chaos 32 (11), 113143.
  * Fu et al. (2015) Fu, Sijie, Biwole, Pascal Henry & Mathis, Christian 2015 Particle tracking velocimetry for indoor airflow field: A review. Building and Environment 87, 34–44.
  * Griffa et al. (2008) Griffa, Annalisa, Lumpkin, Rick & Veneziani, Milena 2008 Cyclonic and anticyclonic motion in the upper ocean. Geophysical Research Letters 35 (1).
  * Hadjighasem et al. (2017) Hadjighasem, Alireza, Farazmand, Mohammad, Blazevski, Daniel, Froyland, Gary & Haller, George 2017 A critical comparison of Lagrangian methods for coherent structure detection. Chaos 27 (5), 1–25, arXiv: 1704.05716.
  * Haller (2015) Haller, G. 2015 Lagrangian Coherent Structures. Annu. Rev. Fluid Mech. 47, 137–162, arXiv: arXiv:1407.4072v1.
  * Haller (2023) Haller, G. 2023 Transport Barriers in Flow Data: Advective, Diffusive, Stochastic and Active Methods. Cambridge, UK: Cambridge University Press.
  * Haller et al. (2021) Haller, George, Aksamit, Nikolas O. & Bartos, Alex P. Encinas 2021 Quasi-Objective Coherent Structure Diagnostics from Single Trajectories. Chaos 31, 043131–1–17, arXiv: 2101.05903.
  * Haller et al. (2016) Haller, G., Hadjighasem, A., Farazmand, M. & Huhn, F. 2016 Defining coherent vortices objectively from the vorticity. Journal of Fluid Mechanics 795, 136–173, arXiv: 1506.04061.
  * Haller & Yuan (2000) Haller, George & Yuan, G 2000 Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys. D Nonlinear Phenom. 147 (3-4), 352–370.
  * Hou et al. (2021) Hou, Jianfeng, Kaiser, Frieder, Sciacchitano, Andrea & Rival, David E. 2021 A novel single-camera approach to large-scale, three-dimensional particle tracking based on glare-point spacing. Experiments in Fluids 62 (5), 1–10.
  * Iacobello & Rival (2023) Iacobello, Giovanni & Rival, David E. 2023 Identifying dominant flow features from very-sparse Lagrangian data: a multiscale recurrence network-based approach. Experiments in Fluids 64 (10), 1–14.
  * Jeong & Hussain (1995) Jeong, Jinhee & Hussain, Fazle 1995 On the Identification of a vortex. Journal of Fluid Mechanics 285, 69–94.
  * Kaiser & Rival (2023) Kaiser, F. & Rival, D. E. 2023 Large-scale volumetric particle tracking using a single camera: analysis of the scalability and accuracy of glare-point particle tracking. Experiments in Fluids 64 (9), 1–12.
  * Kaszás et al. (2023) Kaszás, Bálint, Pedergnana, Tiemo & Haller, George 2023 The objective deformation component of a velocity field. European Journal of Mechanics, B/Fluids 98, 211–223.
  * Khojasteh et al. (2022) Khojasteh, Ali Rahimi, Laizet, Sylvain, Heitz, Dominique & Yang, Yin 2022 Lagrangian and Eulerian dataset of the wake downstream of a smooth cylinder at a Reynolds number equal to 3900. Data in Brief 40, 107725.
  * Kwok et al. (1990) Kwok, Ronald, Curlander, John C., Pang, Shirley S. & Mcconnell, Ross 1990 An Ice-Motion Tracking System at the Alaska SAR Facility. IEEE Journal of Oceanic Engineering 15 (1), 44–54.
  * LaCasce (2008) LaCasce, J. H. 2008 Statistics from Lagrangian observations. Progress in Oceanography 77 (1), 1–29.
  * Lekien & Ross (2010) Lekien, Francois & Ross, Shane D. 2010 The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds. Chaos 20 (1).
  * Leppäranta (2011) Leppäranta, Matti 2011 The Drift of Sea Ice, 2nd edn. Springer-Verlag.
  * Lindsay & Stern (2003) Lindsay, Ronald W. & Stern, H. L. 2003 The RADARSAT Geophysical Processor System: Quality of sea ice trajectory and deformation estimates. Journal of Atmospheric and Oceanic Technology 20 (9), 1333–1347.
  * Lumpkin (2016) Lumpkin, Rick 2016 Global characteristics of coherent vortices from surface drifter trajectories. Journal of Geophysical Research: Oceans 121 (2), 1306–1321.
  * Mancho et al. (2013) Mancho, A. M., Wiggins, S., Curbelo, J. & Mendoza, C. 2013 Lagrangian descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems. Commun. Nonlinear Sci. 18, 3530–3557.
  * Mowlavi et al. (2022) Mowlavi, Saviz, Serra, Mattia, Maiorino, Enrico & Mahadevan, L. 2022 Detecting Lagrangian coherent structures from sparse and noisy trajectory data. Journal of Fluid Mechanics 948, 1–34, arXiv: 2110.10884.
  * Onishchenko et al. (2021) Onishchenko, Oleg, Fedun, Viktor, Horton, Wendell, Pokhotelov, Oleg, Astafieva, Natalia, Skirvin, Samuel J. & Verth, Gary 2021 The Stationary Concentrated Vortex Model. Climate 9 (3), 39.
  * Rosi & Rival (2018) Rosi, Giuseppe A. & Rival, David E. 2018 A Lagrangian perspective towards studying entrainment. Experiments in Fluids 59 (1), 1–17.
  * Rypina et al. (2007) Rypina, I. I., Brown, M. G., Beron-Vera, F. J., Koçak, H., Olascoaga, M. J. & Udovydchenkov, I. A. 2007 On the Lagrangian dynamics of atmospheric zonal jets and the permeability of the stratospheric polar vortex. Journal of the Atmospheric Sciences 64 (10), 3595–3610.
  * Rypina et al. (2021) Rypina, Irina I., Getscher, Timothy R., Pratt, Lawrence J. & Mourre, Baptiste 2021 Observing and Quantifying Ocean Flow Properties Using Drifters with Drogues at Different Depths. Journal of Physical Oceanography 51 (8), 2463–2482.
  * Rypina et al. (2011) Rypina, I. I., Scott, S. E., Pratt, L. J. & Brown, M. G. 2011 Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures. Nonlinear Processes in Geophysics 18 (6), 977–987.
  * Schröder & Schanz (2023) Schröder, Andreas & Schanz, Daniel 2023 3D Lagrangian Particle Tracking in Fluid Mechanics. Annual Review of Fluid Mechanics 55, 511–540.
  * van Sebille et al. (2018) van Sebille, Erik, Griffies, Stephen M., Abernathey, Ryan, Adams, Thomas P., Berloff, Pavel, Biastoch, Arne, Blanke, Bruno, Chassignet, Eric P., Cheng, Yu, Cotter, Colin J., Deleersnijder, Eric, Döös, Kristofer, Drake, Henri F., Drijfhout, Sybren, Gary, Stefan F., Heemink, Arnold W., Kjellsson, Joakim, Koszalka, Inga Monika, Lange, Michael, Lique, Camille, MacGilchrist, Graeme A., Marsh, Robert, Mayorga Adame, C. Gabriela, McAdam, Ronan, Nencioli, Francesco, Paris, Claire B., Piggott, Matthew D., Polton, Jeff A., Rühs, Siren, Shah, Syed H.A.M., Thomas, Matthew D., Wang, Jinbo, Wolfram, Phillip J., Zanna, Laure & Zika, Jan D. 2018 Lagrangian ocean analysis: Fundamentals and practices. Ocean Modelling 121 (October), 49–75.
  * Tauro et al. (2017) Tauro, F., Piscopia, R. & Grimaldi, S. 2017 Streamflow Observations From Cameras: Large-Scale Particle Image Velocimetry or Particle Tracking Velocimetry? Water Resources Research 53 (12), 10374–10394.
  * Truesdell & Noll (2004) Truesdell, C. & Noll, W. 2004 The Non-Linear Field Theories of Mechanics. The Non-Linear Field Theories of Mechanics pp. 1–579, arXiv: arXiv:1011.1669v3.
  * Zhou et al. (1999) Zhou, J., Adrian, R. J., Balachandar, S. & Kendall, T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. Journal of Fluid Mechanics 387, 353–396.