# Quasi-Objective Coherent Structure Diagnostics
from Single Trajectories

George Haller, Nikolas Aksamit and Alex P. Encinas Bartos
Institute for Mechanical Systems
ETH Zürich
<EMAIL_ADDRESS>
###### Abstract

We derive measures of local material stretching and rotation that are
computable from individual trajectories without reliance on other trajectories
or on an underlying velocity field. Both measures are quasi-objective: they
approximate objective (i.e., observer-independent) coherence diagnostics in
frames satisfying a certain condition. This condition requires the trajectory
accelerations to dominate the angular acceleration induced by the spatial mean
vorticity. We illustrate in examples how quasi-objective coherence diagnostics
highlight elliptic and hyperbolic Lagrangian coherent structures even from
very sparse trajectory data.

> Summary
>
> A vast amount of discrete tracer trajectory data is available for fluid
> flows in the laboratory and in nature. All features of tracer trajectories,
> however, including their shape, velocities and looping, depend on the
> reference frame of the observer even though the coherent flow structures one
> often tries to infer from tracers are observer indifferent. Here we develop
> quasi-objective coherent structure diagnostics that can be computed from
> individual single tracer trajectories without reliance on other
> trajectories. A quasi-objective diagnostic approximates an objective
> diagnostics in all frames of reference that satisfy certain conditions we
> derive. We show how the single-trajectory quasi-objective scalar fields
> highlight material coherent structures even from sparse trajectory data sets
> on which multi-trajectory objective diagnostics tend to perform poorly.

##  1 Introduction

We seek objective (i.e. observer-indifferent) diagnostics for material
stretching and rotation that can be evaluated on individual trajectories
without reliance on other trajectories or on the underlying velocity field.
Applications calling for such diagnostics include the analysis of
observational drifter data, balloon data and particle tracks from particle
tracking velocimetry (PTV).

All elementary features of particle paths are non-objective, i.e., depend on
the observer. For instance, in a frame traveling with any given particle, the
particle becomes just a fixed point. All features of the trajectory, such as
velocity, acceleration, looping number, curvature and trajectory length,
therefore, vanish in this frame. In contrast, Lagrangian coherent structures
(LCS), the persistent features of the material deformation field of the fluid
as a moving continuum, are objective by definition [24]. Accordingly, any
self-consistent identification of LCS should be carried out using objective
quantities. Indeed, objectivity as a minimal requirement for flow-feature
identification was already identified in the 1970’s [9, 10, 4, 31] and a
number of recent approaches conform to this requirement (see [19, 24, 35, 26,
15] for reviews).

Objective geometric quantities used in LCS identification are, however, based
on the gradient of the flow map, whose computation requires a large number of
trajectories to be released from an initial grid. Examples of such quantities
include the finite-time Lyapunov exponent (or FTLE, [18]) for hyperbolic LCS
and the polar rotation angle (or PRA, [11]) for elliptic LCS in two
dimensions. The computation of these diagnostics is only feasible if the
velocity field is known over, or can at least can be interpolated onto, a
regular grid over a full flow domain [24]. For the special case of closed (or
elliptic) LCSs, alternative probabilistic methods have been developed that
seek regions enclosed by such LCSs as coherent sets [12, 13]. These objective
methods do not technically require differentiation of the flow map, but their
implementations utilize box-counting methods that only converge for large
numbers of densely spaced trajectories. More recent clustering methods for
coherent set detection are, in principle, applicable to sparse trajectory
data, but still require a reasonably high number of evenly placed trajectories
to give meaningful results [16, 14, 40]. None of these methods offer a
coherence indicator that is computable on a single trajectory. Rather, the
coherence measures assigned to individual trajectories by probabilistic and
statistical methods will change if the position and number of the other
trajectories used in the algorithm change–even though the underlying fluid
flow and its coherent structures remain the same.

Notable exceptions to this trend are two objective indicators of the
complexity of individual trajectories: the correlation dimension and the
ergodicity defect proposed in [39] for LCS extraction. While these objective
diagnostics can be computed from single trajectories, their connection to
material attributes of fluid trajectory motion, such as material stretching
and rotation, are unknown. Absolute dispersion [36], trajectory length [33]
and maximal trajectory extent [34] are also attractively simple single-
trajectory diagnostics, but they are not objective and their relationship to
material stretching and rotation is also unknown. In contrast, the Lagrangian-
averaged vorticity deviation (LAVD) is an objective single-trajectory
diagnostic [22] but requires the knowledge of the full velocity field for the
purposes of computing the vorticity along trajectories. A further non-
objective, single-particle diagnostics for LCS identification is the
Lagrangian spin parameter [43], which relies on the calibration of an
additional stochastic model for the velocity field. Yet another approach
adapts the wavelet ridge analysis method of [7] to Lagrangian trajectory
analysis [29, 30]. While physically insightful, this technique is non-
objective either, as it fits a time-varying ellipse model to identify
signatures of coherent eddies based on the looping characteristics of
trajectories.

While all identifiable features of individual trajectories are non-objective,
trajectory data is often the only available information about fluid flows. It
is therefore essential to assess what trajectory-based quantities (if any) can
still be tied mathematically, at least in qualifying frames, to established
and objective LCS diagnostics. Answering this question would enable, for
instance, a mathematically justifiable and physically self-consistent
extraction of Lagrangian jets, fronts and eddies from NOAA’s Global Drifter
Program, which comprises more than 20,000 trajectories [32]. Such an
extraction should then lead to a systematic assessment of the performance of
promising nonobjective trajectory-based methods, such as [30], in avoiding
false positives and negatives in elliptic LCS detection.

Here we address this challenge by deriving objective coherence diagnostics for
material stretching and rotation that are computable solely from single-
particle trajectory data. Each diagnostic approximates a related objective,
material coherence measure in frames that satisfy certain computable
conditions. We refer to the approximate coherence diagnostics developed here
as quasi-objective __ under those conditions. Physically, the quasi-
objectivity condition we obtain requires the trajectory accelerations to
dominate the angular acceleration induced by the spatial mean vorticity in the
given frame. We illustrate the power of quasi-objective stretching and
rotation measures on two- and three-dimensional examples.

##  2 Objective Lagrangian stretching and rotation measures

Consider particle motions generated by a differential equation

$\dot{\mathbf{x}}=\mathbf{v}(\mathbf{x},t),$ (1)

with a continuously differentiable velocity field $\mathbf{v}$ defined for
times $t\in[t_{0},t_{N}]$ on a spatial domain $U\subset\mathbb{R}^{n}$ of
dimension $n\geq 1$. Let $\mathbf{x}(t;t_{0},\mathbf{x}_{0})$ be a trajectory
with initial condition
$\mathbf{x}(t_{0};t_{0},\mathbf{x}_{0})=\mathbf{x}_{0}$. The flow map induced
by trajectories is
$\mathbf{F}_{t_{0}}^{t}\colon\mathbf{x}_{0}\to\mathbf{x}(t;t_{0},\mathbf{x}_{0})$,
mapping initial conditions at time $t_{0}$ to their current positions at time
$t$. When $t_{0}$ and $\mathbf{x}_{0}$ are fixed and hence do not need to be
carried in our arguments, we will also use the shorthand notation
$\boldsymbol{x}(t):=\mathbf{x}(t;t_{0},\mathbf{x}_{0})$.

A Lagrangian scalar field $\mathcal{P}(\mathbf{x}_{0})$ is called objective
[42] if it remains invariant under all Euclidean observer changes of the form

$\mathbf{x}=\mathbf{Q}(t)\mathbf{y}+\mathbf{b}(t),$ (2)

with a proper orthogonal tensor $\mathbf{Q}(t)\in\mathrm{SO}(n)$ and a vector
$\mathbf{b}(t)$. Specifically, under all such transformations, the transformed
scalar field $\tilde{\mathcal{P}}$ in the $\mathbf{y}$-frame satisfies
$\tilde{\mathcal{P}}(\mathbf{y}_{0})\equiv\mathcal{P}(\mathbf{x}_{0}).$

Consider a material curve,
$\boldsymbol{\gamma}(t;s)=\mathbf{F}_{t_{0}}^{t}\left(\boldsymbol{\gamma}(t_{0};s)\right)\subset
U$, at time $t$, which has evolved from an initial curve
$\boldsymbol{\gamma}(t_{0};s)$ parametrized by the scalar parameter
$s\in\mathbb{R}$. Any tangent vector,

$\boldsymbol{\xi}(t;s)=\partial_{s}\boldsymbol{\gamma}(t;s),$ (3)

of $\boldsymbol{\gamma}$ then satisfies the evolution equation

$\displaystyle\dot{\boldsymbol{\xi}}(t;s)$
$\displaystyle=\partial_{t}\boldsymbol{\xi}(t;s)=\partial_{t}\partial_{s}\boldsymbol{\gamma}(t;s)=\partial_{s}\partial_{t}\boldsymbol{\gamma}(t;s)=\partial_{s}\partial_{t}\mathbf{F}_{t_{0}}^{t}\left(\boldsymbol{\gamma}(t_{0};s)\right)$
$\displaystyle=\partial_{s}\mathbf{v}(\boldsymbol{\gamma}(t;s),t)=\boldsymbol{\nabla}\mathbf{v}(\boldsymbol{\gamma}(t;s),t)\partial_{s}\boldsymbol{\gamma}(t;s).$

Therefore, the tangent vector $\boldsymbol{\xi}(t;s)$ of the material curve
$\boldsymbol{\gamma}(t;s)$ satisfies the classic equation of variations [3]

$\dot{\boldsymbol{\xi}}=\boldsymbol{\nabla}\mathbf{v}\left(\boldsymbol{x}(t),t\right)\boldsymbol{\xi}$
(4)

along the trajectory $\boldsymbol{x}(t)$ with
$\boldsymbol{x}(t_{0})=\boldsymbol{\gamma}(t_{0};s).$

###  2.1 Objective stretching measures

We now fix the parameter $s$ along the material curve
$\boldsymbol{\gamma}(t;s)$ such that
$\boldsymbol{\gamma}(t_{0};s)=\mathbf{\boldsymbol{x}}_{0}$. We then seek to
characterize the time-evolution of the tangent vector
$\left|\boldsymbol{\xi}(t)\right|$ at $\boldsymbol{\gamma}(t;s)$ with an
exponent $\lambda$ as

$\left|\boldsymbol{\xi}(t)\right|\sim
e^{\lambda\left(t-t_{0}\right)}\left|\boldsymbol{\xi}_{0}\right|$ (5)

for $\boldsymbol{\xi}_{0}\neq\mathbf{0}$. This leads us to define the
_averaged stretching exponent_

$\lambda{}_{t_{0}}^{t_{N}}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{\xi}_{0}):=\frac{1}{t_{N}-t_{0}}\log\frac{\left|\boldsymbol{\xi}(t_{N})\right|}{\left|\boldsymbol{\xi}_{0}\right|}.$
(6)

Under any Euclidean observer change (2) we obtain

$\tilde{\mathbf{\boldsymbol{\xi}}}=\mathbf{Q}^{T}\boldsymbol{\xi},$

whose substitution into (6) shows that the Lagrangian scalar field
$\lambda_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0},\boldsymbol{\xi}_{0})$ is
objective, i.e., remains invariant under arbitrary observer changes of the
form (2). We recall that the maximum of $\lambda>0$ associated with the
initial point $\boldsymbol{x}\mathbf{}_{0}$ is the finite-time Lyapunov
exponent (FTLE) by definition [24].

If the instantaneous growth exponent satisfies $\lambda<0$ at time $t$ in a
two-dimensional incompressible flow, then a normal vector
$\boldsymbol{\xi}^{\perp}(t;s)$ to $\boldsymbol{\xi}(t;s)$ at the same point
of the trajectory will have a growth exponent equal to
$\left|\lambda\right|>0$. Similarly, in three-dimensional incompressible
flows, $\lambda<0$ implies that the instantaneous growth exponent of the
_area_ in a plane normal to the material line $\boldsymbol{\gamma}(t;s)$ is
equal to $\left|\lambda\right|>0$. Therefore, $\left|\lambda\right|$ is a
measure of the strength of hyperbolicity experienced by a trajectory in an
incompressible flow. To measure the temporal average of $\left|\lambda\right|$
along the trajectory $\boldsymbol{x}(t)$, we define the _averaged
hyperbolicity strength_

$\bar{\lambda}{}_{t_{0}}^{t_{N}}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{\xi}_{0}):=\frac{1}{t_{N}-t_{0}}\int_{t_{0}}^{t_{N}}\left|\frac{d}{dt}\log\frac{\left|\boldsymbol{\xi}(t)\right|}{\left|\boldsymbol{\xi}_{0}\right|}\right|\,dt,$
(7)

where $\boldsymbol{x}(t_{0})=\mathbf{\boldsymbol{x}}_{0}$. While the quantity
$\lambda{}_{t_{0}}^{t_{N}}$ measures the average net stretching (including
shrinkage) in the direction of $\boldsymbol{\xi}(t)$ over the time interval
$[t_{0},t_{N}]$ , the quantity $\bar{\lambda}{}_{t_{0}}^{t_{N}}$ accounts for
all (positive) stretching experienced either parallel or normal to
$\boldsymbol{\xi}(t)$. For this reason, $\bar{\lambda}{}_{t_{0}}^{t_{N}}$ is a
measure of the average strength of hyperbolicity experienced by a trajectory
of an incompressible flow. By the same considerations that we have applied to
$\lambda{}_{t_{0}}^{t_{N}}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{\xi}_{0})$,
the scalar field
$\bar{\lambda}{}_{t_{0}}^{t_{N}}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{\xi}_{0})$
is also objective.

###  2.2 Objective rotation measures

To introduce our second class of trajectory-based diagnostic tool we normalize
the material tangent vector $\boldsymbol{\xi}(t)$ defined in (3) to the unit
tangent vector

$\mathbf{e}(t)=\frac{\boldsymbol{\xi}(t)}{\left|\boldsymbol{\xi}(t)\right|}$

of the material line $\boldsymbol{\gamma}(t;s)$; we will denote by
$\mathbf{e}_{0}:=\mathbf{e}(t_{0})=\boldsymbol{\xi}_{0}/\left|\boldsymbol{\xi}_{0}\right|$
the initial position of $\mathbf{e}(t)$ at time $t_{0}$.

Under any Euclidean observer change (2) we then obtain

$\tilde{\mathbf{e}}(t)=\mathbf{Q}^{T}(t)\mathbf{e}(t).$ (8)

Over the time interval $[t_{0},t_{N}],$ the net rotation angle
$\alpha_{t_{0}}^{t_{N}}(\mathbf{x}_{0},\mathbf{e}_{0})$ experienced by
$\boldsymbol{\xi}_{0}$ along the trajectory starting from
$\mathbf{x}_{0}=\boldsymbol{\gamma}(t_{0};s)$ can be computed as

$\alpha_{t_{0}}^{t_{N}}(\mathbf{x}_{0},\boldsymbol{\xi}_{0})=\cos^{-1}\left\langle\mathbf{e}(t_{0}),\mathbf{e}(t_{N})\right\rangle.$
(9)

Formula (8) shows that $\alpha_{t_{0}}^{t}$ is not an objective scalar because
$\mathbf{e}_{0}=\mathbf{e}(t_{0})$ is rotated by $\mathbf{Q}^{T}(t_{0})$ under
an observer change (2), whereas $\mathbf{e}(t_{N})$ is rotated by
$\mathbf{Q}^{T}(t_{N}).$ Still, rotations commute in two dimensions, so if we
have
$\alpha_{t_{0}}^{t}(\mathbf{x}_{01},\mathbf{e}_{01})=\alpha_{t_{0}}^{t}(\mathbf{x}_{02},\mathbf{e}_{02})$
for two different initial positions $\mathbf{x}_{01}$ and $\mathbf{x}_{02}$
with corresponding initial material tangent vectors
$\mathbf{\boldsymbol{\xi}}_{01}$ and $\mathbf{\boldsymbol{\xi}}_{02}$, then we
also have
$\alpha_{t_{0}}^{t}(\mathbf{y}_{01},\mathbf{\tilde{\boldsymbol{\xi}}}_{01})=\alpha_{t_{0}}^{t}(\mathbf{y}_{02},\mathbf{\tilde{\boldsymbol{\xi}}}_{02})$
for their transformed counterparts. This is because the angle between
$\mathbf{\boldsymbol{\xi}}_{1}(t_{0})$ and
$\mathbf{\boldsymbol{\xi}}_{1}(t_{N})$ will be altered by the spatially
independent rotation tensors $\mathbf{Q}^{T}(t_{0})$ and
$\mathbf{Q}^{T}(t_{N})$ in the new frame by the same amount as the angle
between $\mathbf{\boldsymbol{\xi}}_{2}(t_{0})$ and
$\mathbf{\boldsymbol{\xi}}_{2}(t_{N})$ will be altered by the same rotation
tensors. We conclude that the set of level curves of (9) is objective (i.e.,
invariant under observer changes), even though the value of
$\alpha_{t_{0}}^{t}(\mathbf{x}_{0},\mathbf{e}_{0})$ on them changes from one
frame to the other. We stress that the same statement does not hold in three
dimensions, unless the vector field $\mathbf{v}$ itself is objective. These
properties mimic the objectivity properties of the PRA ([11]).

We now seek to construct a quantity that objectively characterizes the
rotation of unit vectors in both two and three dimensions. To this end, we now
consider the deviation of $\dot{\mathbf{e}}$ from the domain-average of the
local mean velocities of all unit material tangent vectors emanating from the
same point. We first recall the result from [20] that the local mean angular
velocity of material fibers at a point $\mathbf{x}$ at time $t$ equals

$\boldsymbol{\nu}(\mathbf{x},t)=\frac{1}{2}\boldsymbol{\omega}(\mathbf{x},t),$
(10)

with $\boldsymbol{\omega}=\boldsymbol{\nabla}\times\mathbf{v}$ denoting the
vorticity field. We also recall that under a frame change (2), vorticity
transforms into

$\tilde{\boldsymbol{\omega}}=\mathbf{Q}^{T}\left(\boldsymbol{\omega}-\dot{\mathbf{q}}\right),$
(11)

where $\dot{\mathbf{q}}$ is the vorticity of the frame change defined by the
identity

$\frac{1}{2}\dot{\mathbf{q}}\times\mathbf{e}=\dot{\mathbf{Q}}\mathbf{Q}^{T}\mathbf{e}$

for all vectors $\mathbf{e}\in\mathbb{R}^{3}.$

By eq. (10), the deviation of $\dot{\mathbf{e}}$ from the spatially averaged
local velocities of material unit vectors, or _relative tangent velocity_ , is

$\dot{\boldsymbol{\epsilon}}=\dot{\mathbf{e}}-\frac{1}{2}\bar{\boldsymbol{\omega}}(t)\times\mathbf{e}=\dot{\mathbf{e}}-\bar{\mathbf{W}}(t)\mathbf{e},$
(12)

with $\bar{\boldsymbol{\omega}}(t)$ denoting the spatial average of the
vorticity over the domain $U$, and with $\bar{\mathbf{W}}(t)$ denoting the
spatial average of the spin tensor

$\mathbf{W}=\frac{1}{2}\left[\boldsymbol{\nabla}\mathbf{v}-\left[\boldsymbol{\nabla}\mathbf{v}\right]^{T}\right]=-\mathbf{W}^{T}$
(13)

over the same domain. Under the frame change (2), formula (11) shows that
$\dot{\boldsymbol{\epsilon}}$ transforms as

$\displaystyle\dot{\tilde{\boldsymbol{\epsilon}}}$
$\displaystyle=\dot{\tilde{\mathbf{e}}}-\frac{1}{2}\bar{\tilde{\boldsymbol{\omega}}}\times\tilde{\mathbf{e}}=\mathbf{Q}^{T}\left[\dot{\mathbf{e}}-\dot{\mathbf{Q}}\mathbf{Q}^{T}\mathbf{e}\right]-\frac{1}{2}\mathbf{Q}^{T}\left(\bar{\boldsymbol{\omega}}-\dot{\mathbf{q}}\right)\times\mathbf{Q}^{T}\mathbf{e}$
$\displaystyle=\mathbf{Q}^{T}\left[\dot{\mathbf{e}}-\frac{1}{2}\dot{\mathbf{q}}\times\mathbf{e}\right]-\frac{1}{2}\mathbf{Q}^{T}\left(\bar{\boldsymbol{\omega}}-\dot{\mathbf{q}}\right)\times\mathbf{Q}^{T}\mathbf{e}$
$\displaystyle=\mathbf{Q}^{T}\left[\dot{\mathbf{e}}-\frac{1}{2}\dot{\mathbf{q}}\times\mathbf{e}-\frac{1}{2}\left(\bar{\boldsymbol{\omega}}-\dot{\mathbf{q}}\right)\times\mathbf{e}\right]$
$\displaystyle=\mathbf{Q}^{T}\left[\dot{\mathbf{e}}-\frac{1}{2}\bar{\boldsymbol{\omega}}\times\mathbf{e}\right]=\mathbf{Q}^{T}\dot{\boldsymbol{\epsilon}},$

implying that the relative tangent velocity $\dot{\boldsymbol{\epsilon}}$ of
$\boldsymbol{\xi}$ is an objective vector. Consequently,

$\bar{\alpha}_{t_{0}}^{t_{N}}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{\xi}_{0}):=\frac{1}{t_{N}-t_{0}}\int_{t_{0}}^{t_{N}}\left|\dot{\boldsymbol{\epsilon}}(t)\right|dt=\int_{t_{0}}^{t_{N}}\left|\dot{\mathbf{e}}(t)-\frac{1}{2}\bar{\boldsymbol{\omega}}(t)\times\mathbf{e}(t)\right|dt,$
(14)

with $\mathbf{e}(t)=$ $\boldsymbol{\xi}(t)/\left|\boldsymbol{\xi}(t)\right|$,
is an objective measure for the average rotation speed experienced by the
tangent vector $\boldsymbol{\xi}$ during its evolution along the trajectory
starting from $\mathbf{x}_{0}$. Indeed, this rate of change is entirely due to
the rotation of $\boldsymbol{\xi}$, unaffected by any change in the length of
$\boldsymbol{\xi}$.

##  3 Quasi-objective trajectory stretching and rotation in steady flows

We will call a Lagrangian scalar field $P(\mathbf{x}_{0})$ computed from
$\left\\{\boldsymbol{x}(t_{i})\right\\}_{i=0}^{N}$ _quasi-objective under a
condition_(A) if there exists an objective Lagrangian scalar field
$\mathcal{P}(\mathbf{x}_{0})$ such that

$P(\mathbf{x}_{0})\approx\mathcal{P}(\mathbf{x}_{0})$ (15)

in any $\mathbf{x}$-frame in which condition __(A) holds. The accuracy of the
approximation indicated by the symbol $\approx$ in (15) depends on the extent
to which the condition (A) is satisfied.

In other words, a frame-dependent scalar field $P(\mathbf{x}_{0})$ is quasi-
objective under a condition $(\mathbf{A})$ if it happens to approximate the
same objective scalar field $\mathcal{P}(\mathbf{x}_{0})$ in any frame in
which condition $(\mathbf{A})$ is satisfied. $P(\mathbf{x}_{0})$ will
generally not approximate an objective field in all frames, as condition
$(\mathbf{A})$ is generally frame-dependent and hence will not hold in all
frames. In the following sections, we will derive quasi-objective measures of
material stretching and rotation with computable formulas for their
corresponding quasi-objectivity condition.

Our first quasi-objectivity conditions for our upcoming derivations will be
the assumption that the velocity field is steady in the current frame, i.e.,

(A1) $\left|\partial_{t}\mathbf{v}(\mathbf{x},t)\right|=0,$

holds. We will remove this assumption in our later extension of our results to
unsteady flows.

Assumption (A1) enables us to rewrite (1) as the autonomous dynamical system

$\dot{\mathbf{x}}=\mathbf{v}(\mathbf{x}).$ (16)

Trajectories of two-dimensional steady flows ($n=2$) coincide with streamlines
and already provide detailed information about coherent structures without
further analysis. For this reason, the present discussion is mostly of
interest for three-dimensional flows ($n=3$). Although the flow map and
trajectories of the autonomous differential equation depend on the elapsed
time $t-t_{0}$ and hence $t_{0}=0$ could be fixed, we still keep the explicit
dependence on the current time $t$ and initial time $t_{0}$ in our formulas to
facilitate their later extension to the unsteady case in which (A1) will no
longer be assumed.

###  3.1 Trajectory-based approximations of stretching measures

Consider now the Lagrangian velocity vector

$\boldsymbol{v}(t):=\mathbf{v}(\boldsymbol{x}(t)),$

which satisfies the homogeneous linear evolution equation

$\dot{\mathbf{\mathbf{\boldsymbol{v}}}}=\boldsymbol{\nabla}\mathbf{v}(\boldsymbol{x}(t))\mathbf{\boldsymbol{v}}.$
(17)

Note that this formula relies heavily on our assumption (A1), otherwise the
extra term $\partial_{t}\mathbf{v}(\boldsymbol{x}(t),t)$ would appear on the
right-hand side, making (17) an inhomogeneous linear system of differential
equations.

A comparison of eqs. (4) and (17) shows that $\boldsymbol{v}(t)$ evolves as a
material tangent vector in the given coordinate frame. Therefore, for the
choice of the initial material tangent vector
$\boldsymbol{\xi}_{0}=\mathbf{\boldsymbol{v}}_{0}:=\mathbf{v}(\boldsymbol{x}_{0})\neq\mathbf{0}$,
the averaged stretching exponent and averaged hyperbolicity strength can be
written as

$\displaystyle\lambda{}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0},\mathbf{\boldsymbol{v}}_{0})=$
$\displaystyle\frac{1}{t_{N}-t_{0}}\log\frac{\left|\mathbf{v}\left(\boldsymbol{x}(t_{N})\right)\right|}{\left|\mathbf{\boldsymbol{v}}_{0}\right|},\qquad\bar{\lambda}{}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0},\mathbf{\boldsymbol{v}}_{0})=\frac{1}{t_{N}-t_{0}}\int_{t_{0}}^{t_{N}}\left|\frac{d}{dt}\log\frac{\left|\mathbf{v}\left(\boldsymbol{x}(t)\right)\right|}{\left|\mathbf{\boldsymbol{v}}_{0}\right|}\right|\,dt.$
(18)

The right-hand sides of the formulas in (18) give close approximation for the
objective expressions (6) and (7) with
$\boldsymbol{\xi}_{0}=\mathbf{\boldsymbol{v}}_{0}$ in the frame where
assumption (A1) holds for the velocity field. In other words,
$\lambda{}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0},\mathbf{\boldsymbol{v}}_{0})$
and
$\bar{\lambda}{}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0},\mathbf{\boldsymbol{v}}_{0})$
are quasi-objective under condition (A1).

If discretized trajectory data
$\left\\{\boldsymbol{x}(t_{i})\right\\}_{i=0}^{N}$ is available for the
trajectory $\boldsymbol{x}(t)$, then formulas (18) yield the following result,
with the trajectory velocities $\dot{\mathbf{\boldsymbol{x}}}(t_{j})$ computed
from finite differencing or spline interpolation.

######  Theorem 1.

Under assumption (A1)_,_ the trajectory stretching exponents __(TSEs), defined
as

$\mathrm{TSE}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0})=\frac{1}{t_{N}-t_{0}}\log\frac{\left|\dot{\mathbf{\boldsymbol{x}}}(t_{N})\right|}{\left|\dot{\mathbf{\boldsymbol{x}}}(t_{0})\right|},\qquad\overline{\mathrm{TSE}}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0})=\frac{1}{t_{N}-t_{0}}\sum_{i=0}^{N-1}\left|\log\frac{\left|\dot{\mathbf{\boldsymbol{x}}}(t_{i+1})\right|}{\left|\dot{\mathbf{\boldsymbol{x}}}(t_{i})\right|}\right|$
(19)

for all $\dot{\mathbf{\boldsymbol{x}}}(t_{i})\neq\mathbf{0}$, $i=0,\ldots,N$,
are quasi-objective measures of trajectory stretching and hyperbolicity
strength.

The following remarks are in order regarding Theorem 1:

  1. 1.

While two-dimensional steady flows can already be fully understood from
streamlines generated by trajectory plots, the analysis of three-dimensional
steady flows benefits from the $\mathrm{TSE}$ diagnostics.

  2. 2.

The $\mathrm{TSE}$ metrics are only quasi-objective, not objective. One
manifestation of this is that they depend on trajectory velocities which are
not objective. That said, the $\mathrm{TSE}$ metrics give the correct
stretching of velocity vectors in case those vectors evolve materially in a
given frame, which is guaranteed by condition (A1).

  3. 3.

If a steady vector field $\mathbf{v}(\mathbf{x})$ also happens to be objective
(unlike fluid velocity fields), then the $\mathrm{TSEs}$ are not only quasi-
objective but objective for $\mathbf{v}(\mathbf{x})$. Such objective vector
fields include the active barrier vector fields derived in [23]. Invariant
manifolds of these barrier fields define initial positions of objective
material barriers to vorticity and momentum in the underlying fluid flow.

  4. 4.

By its construction, the $\mathrm{TSE}$ field is expected to reproduce
features of the FTLE field but without the computational burdens associated
with the FTLE field. Indeed, unlike the FTLE, the $\mathrm{TSE}$ can be
computed even for a single trajectory.

  5. 5.

For $N=1$, ridges (or trenches) of the $\mathrm{TSE}$ yield single-trajectory-
based approximation for hyperbolic (or parabolic) objective Eulerian coherent
structures (or OECS, [41]), which are the instantaneous limits of hyperbolic
(or parabolic) LCSs.

###  3.2 Trajectory-based approximations of rotation measures

Consider now the normalized Lagrangian velocity vector

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

Equation 12 shows that if
$\frac{1}{2}\bar{\boldsymbol{\omega}}(t)\times\mathbf{e}$ is much smaller in
norm than $\dot{\mathbf{e}}$, then $\dot{\boldsymbol{\epsilon}}$ is
approximately equal to $\dot{\mathbf{e}}$. Therefore, if assumption (A1) holds
(thus $\mathbf{e}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t)$ evolves as a material
unit vector $\mathbf{e}(t))$ and, in addition,

(A2)
$\left|\frac{1}{2}\bar{\boldsymbol{\omega}}(t)\times\mathbf{\boldsymbol{e}_{v}}(t)\right|\ll\left|\dot{\mathbf{e}}_{\mathbf{v}}(t)\right|$

holds in the given frame, then the relative tangent velocity

$\dot{\boldsymbol{\epsilon}}_{\mathbf{\mathbf{\boldsymbol{v}}}}=\dot{\mathbf{e}}_{\mathbf{\mathbf{\boldsymbol{v}}}}-\frac{1}{2}\bar{\boldsymbol{\omega}}\times\mathbf{e}_{\mathbf{\mathbf{\boldsymbol{v}}}}$

of $\mathbf{\boldsymbol{v}}(t)$ satisfies

$\displaystyle\left|\dot{\boldsymbol{\epsilon}}_{\mathbf{\mathbf{\boldsymbol{v}}}}\right|$
$\displaystyle\approx\left|\dot{\mathbf{e}}_{\mathbf{\mathbf{\boldsymbol{v}}}}\right|.$

Note that assumption (A2) requires the Lagrangian accelerations to dominate
the angular acceleration of the trajectory induced by the spatial mean
vorticity. Consequently, if assumptions (A1) and (A2) hold in the frame in
which the trajectory data $\boldsymbol{x}(t)$ is given, then

$\bar{\alpha}_{t_{0}}^{t_{N}}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{v}_{0}):=\frac{1}{t_{N}-t_{0}}\int_{t_{0}}^{t_{N}}\left|\dot{\mathbf{e}}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t)\right|dt$
(20)

approximates the average of
$\left|\dot{\boldsymbol{\epsilon}}_{\mathbf{\mathbf{\boldsymbol{v}}}}\right|$.

To see the physical meaning of
$\bar{\alpha}_{t_{0}}^{t_{N}}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{v}_{0})$
more specifically, we set $t_{N}=t+\delta$ in formula (9) to obtain

$\cos\alpha_{t}^{t+\delta}=\left\langle\mathbf{e}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t),\mathbf{e}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t+\delta)\right\rangle$
(21)

for the angle $\alpha_{t}^{t+\delta}$ between
$\mathbf{e}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t)$ and
$\mathbf{e}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t+\delta)$. Differentiating
(21) with respect to $\delta$, expressing
$\frac{1}{\delta}\sin\alpha_{t}^{t+\delta}\frac{d\alpha_{t}^{t+\delta}}{d\delta}$
and taking its $\delta\to 0$ limit gives

$\lim_{\delta\to
0}\frac{1}{\delta}\sin\alpha_{t}^{t+\delta}\frac{d\alpha_{t}^{t+\delta}}{d\delta}=\lim_{\delta\to
0}\left|\frac{d\alpha_{t}^{t+\delta}}{d\delta}\right|^{2}=\left|\dot{\alpha}_{t}^{t}\right|^{2}=\lim_{\delta\to
0}\left[-\left\langle\mathbf{e}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t),\ddot{\mathbf{e}}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t+\delta)\right\rangle\right]=\left|\dot{\mathbf{e}}_{\mathbf{\mathbf{\boldsymbol{v}}}}(t)\right|^{2},$
(22)

where the last equality follows from differentiating the identity
$\left\langle\mathbf{e}(t),\mathbf{e}(t)\right\rangle\equiv 0$ twice with
respect to $t$. Formula (22) shows that

$\bar{\alpha}_{t_{0}}^{t_{N}}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{v}_{0})=\frac{1}{t_{N}-t_{0}}\int_{t_{0}}^{t_{N}}\left|\dot{\alpha}_{t}^{t}\right|dt$
(23)

is the average angular velocity of the Lagrangian velocity
$\boldsymbol{v}(t)$. If only discretized trajectory data
$\left\\{\boldsymbol{x}(t_{i})\right\\}_{i=1}^{N}$ is available, then a
discretized approximation of (23) leads to the following result.

######  Theorem 2.

Under assumptions (A1) and (A2), the trajectory rotation angle, defined as

$\overline{\mathrm{TRA}}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0})=\sum_{i=0}^{N-1}\cos^{-1}\frac{\left\langle\dot{\mathbf{\boldsymbol{x}}}(t_{i}),\dot{\mathbf{\boldsymbol{x}}}(t_{i+1})\right\rangle}{\left|\dot{\mathbf{\boldsymbol{x}}}(t_{i})\right|\left|\dot{\mathbf{\boldsymbol{x}}}(t_{i+1})\right|}$
(24)

for all $\dot{\mathbf{\boldsymbol{x}}}(t_{i})\neq\mathbf{0}$, $i=0,\ldots,N$,
is a quasi-objective measure of total trajectory rotation.

Indeed, the $i^{th}$ term of the sum in (24) measures the angle between the
velocity vectors $\dot{\mathbf{\boldsymbol{x}}}(t_{i})$ and
$\dot{\mathbf{\boldsymbol{x}}}(t_{i+1})$, returning always a positive value to
account for the modulus sign in (23). As already noted, the trajectory
velocities $\dot{\mathbf{\boldsymbol{x}}}(t_{i})$ can be computed from finite
differencing or spline interpolation performed on the subsequent trajectory
positions $\mathbf{\boldsymbol{x}}(t_{i})$.

We make the following remarks regarding Theorem 2:

  1. 1.

If a steady vector field $\mathbf{v}(\mathbf{x})$ is also objective (unlike
fluid velocity fields), then the unit vector field
$\mathbf{e}_{\boldsymbol{v}}(t)$ is already objective and hence one does not
need to use its modified version, $\boldsymbol{\epsilon}_{\boldsymbol{v}}(t)$.
As a consequence, assumption (A2) is no longer necessary for such vector
fields and $\mathrm{\mathrm{\overline{\mathrm{TRA}}}}$ is objective without
further assumptions (see [23] for examples of such vector fields). For any
$\mathbf{v}(\mathbf{x})$, the net rotation angle $\alpha_{t_{0}}^{t_{N}}$
defined in (9) is also an objective vector field and can be approximated from
trajectories as

$\mathrm{TRA}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0})=\cos^{-1}\frac{\left\langle\dot{\mathbf{\boldsymbol{x}}}(t_{0}),\dot{\mathbf{\boldsymbol{x}}}(t_{N})\right\rangle}{\left|\dot{\mathbf{\boldsymbol{x}}}(t_{0})\right|\left|\dot{\mathbf{\boldsymbol{x}}}(t_{N})\right|}$
(25)

in both two and three dimensions (see Section 5.3).

  2. 2.

Assumption (A2) will automatically hold for any trajectory with nonzero
acceleration if $\bar{\boldsymbol{\omega}}(t)\approx\mathbf{0}$. The mean
vorticity $\bar{\boldsymbol{\omega}}(t)$ has been found to vanish, up to a
very good approximation, in large enough ocean regions [20, 1, 6].

  3. 3.

By their construction, the $\overline{\mathrm{TRA}}$ and $\mathrm{TRA}$ fields
are expected to reproduce features of the PRA field but without the
computational challenges of the PRA field. Indeed, unlike the PRA, the
$\overline{\mathrm{TRA}}$ and $\mathrm{TRA}$ can be computed even for a single
trajectory.

  4. 4.

For $N=1$, closed level curves of the $\mathrm{\overline{\mathrm{TRA}}}$ yield
single-trajectory-based approximation for elliptic OECS [41], the
instantaneous limits of elliptic LCSs.

In order to verify assumption (A2), one can compute $\ddot{\boldsymbol{x}}(t)$
by finite-differencing as we already noted in the previous section. In
addition, one needs an estimate for the averaged vorticity
$\bar{\boldsymbol{\omega}}(t)$, which may either be generally known for the
underlying velocity field or needs to be estimated from the available
trajectory data (see, e.g., [37]).

##  4 Quasi-objective coherence measures for unsteady flows

We now extend the results of the previous section to unsteady flows. The idea
is to consider trajectory motion in the extended phase space of positions and
times, in which trajectories are always governed by a steady velocity field.
Each trajectory is a material curve in the extended phase space and one may
track the relative stretching and rotation of the tangent vectors of such
curves using the measures we introduced in the previous section for steady
flows.

The quantities $\mathbf{x}$, $t$ and $\mathbf{v}$ generally have physical
units but our upcoming discussion requires non-dimensionalized variables. To
this end, we select a dimensional characteristic length $L>0$ and a
dimensional characteristic time $T>0$ for the flow defined by (1), and
introduce the new quantities

$\mathbf{y}:=\frac{\mathbf{x}}{L},\quad\tau:=\tau_{0}+\frac{t-t_{0}}{T},\quad\mathrm{v}_{0}:=\frac{L}{T},\quad\mathbf{u}(\mathbf{y},\tau):=\mathbf{u}\left(L\mathbf{y},t_{0}+T\left(\tau-\tau_{0}\right)\right),$
(26)

where $\mathbf{y}$ is a non-dimensional position variable, $\tau$ is a non-
dimensional time, $\mathrm{v}_{0}$ is a characteristic (dimensional) velocity
for the velocity field $\mathbf{v}(\mathbf{x},t)$ and
$\mathbf{u}(\mathbf{y},\tau)$ is the non-dimensionalized velocity field. If
the variables in (1) were non-dimensional to begin with, then one can simply
set $L=T=\mathrm{v}_{0}=1$, $\tau=t$, $\mathbf{y=}\mathbf{x}$ and
$\mathbf{u}\equiv\mathbf{v}$ in all our formulas below.

The non-dimensionalized trajectories satisfy the differential equation

$\mathbf{y}^{\prime}=\mathbf{u}(\mathbf{y},\tau).$ (27)

with $\mathbf{y}\in V\subset\mathbb{R}^{n}$ and $\tau\in[\tau_{0},\tau_{N}].$
We introduce the extended phase space

$\mathcal{V}=\left\\{\mathbf{Y=}\left(\begin{array}[]{c}\mathbf{y}\\\
z\end{array}\right):\mathbf{y}\in V,\,\,z\in\mathbb{R}\right\\},$

on which (1) becomes an autonomous dynamical system of the form

$\mathbf{Y}^{\prime}=\mathbf{U}(\mathbf{Y}),\qquad\mathbf{U}(\mathbf{Y}):=\left(\begin{array}[]{c}\mathbf{u}(\mathbf{y},z)\\\
1\end{array}\right),\qquad\mathbf{Y\in}\mathcal{V}.$ (28)

Note that without non-dimensionalizing, different components of the extended
quantities $\mathbf{Y}$ and $\mathbf{U}$ would have different physical units.

The autonomous dynamical system (28) is of the general form (1) with $n=3$
(planar flows) or $n=4$ (three-dimensional flows). An extended material curve
in this system is of the form

$\boldsymbol{\Gamma}(\tau;s)=\left(\begin{array}[]{c}\boldsymbol{\gamma}(\tau;s)\\\
\gamma_{z}(s)\end{array}\right)=\left(\begin{array}[]{c}\mathbf{F}_{\tau_{0}}^{\tau}\left(\boldsymbol{\gamma}(\tau_{0};s)\right)\\\
s\end{array}\right).$ (29)

The initial tangent vector
$\boldsymbol{\Xi}_{0}=\partial_{s}\boldsymbol{\Gamma}(0;\mathbf{X}_{0})$ to
this material curve evolves under the extended equation of variations

$\boldsymbol{\Xi}^{\prime}=\boldsymbol{\partial}_{Y}\mathbf{V}(\mathbf{Y}(\tau))\boldsymbol{\Xi}.$
(30)

The relationship between the extended material tangent vector
$\boldsymbol{\Xi}(\tau)$ and the tangent vector $\boldsymbol{\xi}(t)$ of a
material curve $\boldsymbol{\gamma}(t;s)$ advected by the original velocity
field $\mathbf{v}(\mathbf{x},t)$ is

$\boldsymbol{\Xi}(\tau)=\left(\begin{array}[]{c}\frac{T}{L}\boldsymbol{\xi}(t)\\\
1\end{array}\right)=\frac{1}{\mathrm{v}_{0}}\left(\begin{array}[]{c}\boldsymbol{\xi}(t)\\\
\mathrm{v}_{0}\end{array}\right),$ (31)

with $v_{0}$ denoting the characteristic velocity introduced in the non-
dimensionalization (26).

We can apply the general Lagrangian stretching and rotation measures
introduced in Section 2 to track the stretching and rotation of
$\boldsymbol{\Xi}(\tau)$. As we have seen, these measures,
$\lambda{}_{\tau_{0}}^{\tau_{N}}(\boldsymbol{Y}\mathbf{}_{0},\boldsymbol{\Xi}_{0})$,
$\bar{\lambda}{}_{\tau_{0}}^{\tau_{N}}(\boldsymbol{Y}\mathbf{}_{0},\boldsymbol{\Xi}_{0})$
and
$\bar{\alpha}{}_{\tau_{0}}^{\tau_{N}}(\boldsymbol{Y}\mathbf{}_{0},\boldsymbol{\Xi}_{0})$,
will remain invariant under all time-dependent, formally extended Euclidean
transformations

$\mathbf{Y}=\boldsymbol{\mathcal{Q}}(\tau)\tilde{\mathbf{Y}}+\mathbf{B}(\tau),\quad\boldsymbol{\mathcal{Q}}(\tau)=\boldsymbol{\mathcal{Q}}^{-T}(\tau)\in
SO(n+1),$ (32)

of the extended phase space. In particular, the extended stretching and
rotation measures for $\boldsymbol{\Xi}(\tau)$ remain invariant under the
physically relevant subset of the transformations (32),

$\boldsymbol{\mathcal{Q}}(\tau)=\left(\begin{array}[]{cc}\mathbf{Q}(\tau)&\mathbf{0}\\\
\mathbf{0}^{T}&1\end{array}\right),\quad\mathbf{B}(\tau)=\left(\begin{array}[]{c}\mathbf{b}(\tau)\\\
0\end{array}\right),$

which represent physical observer changes for the non-dimensionalized system
(27).

Assumption (A1) is always satisfied for the autonomous extended velocity field
(28). To evaluate assumption (A2), we denote the normalized extended velocity
vector and extended averaged spin tensor, respectively, by

$\mathbf{E}_{\mathbf{U}}:=\frac{\mathbf{U}}{\left|\mathbf{U}\right|},\qquad\overline{\boldsymbol{\mathcal{W}}}:=\left(\begin{array}[]{cc}\overline{\mathbf{W}}_{\mathbf{u}}&\mathbf{0}\\\
\mathbf{0}^{T}&0\end{array}\right)=-\overline{\boldsymbol{\mathcal{W}}}^{T},\qquad\mathbf{W}_{\mathbf{u}}=\frac{1}{2}\left[\boldsymbol{\partial}_{\mathbf{y}}\mathbf{u}-\left(\boldsymbol{\boldsymbol{\partial}_{\mathbf{y}}}\mathbf{u}\right)^{T}\right]$

with bar denoting spatial averaging over the domain $V$. In terms of these
quantities, assumption (A2) applied to the extended system takes the form
$\left|\overline{\boldsymbol{\mathcal{W}}}\mathbf{E}_{\mathbf{U}}\right|\ll\left|\mathbf{\mathbf{E}_{\mathbf{U}}}^{\prime}\right|$,
or, equivalently,
$\left|\bar{\mathbf{W}}\mathbf{e}_{\mathbf{v}}\right|\ll\left|\dot{\mathbf{e}}_{\mathbf{v}}\right|$,
which is just the original form of assumption (A2) in dimensional coordinates.
Theorems 1 and 2 then become applicable in the present setting, without the
requirement (A1), but have to be stated for the extended Lagrangian velocity

$\mathbf{Y}^{\prime}(\tau_{i})=\left(\begin{array}[]{c}\frac{T}{L}\mathbf{v}(\mathbf{x}(t_{i}),t_{i})\\\
1\end{array}\right)=\left(\begin{array}[]{c}\frac{1}{\mathrm{v}_{0}}\mathbf{v}(\mathbf{x}(t_{i}),t_{i})\\\
1\end{array}\right)$

Substituting these fields into formulas (19) and (24), we can summarize the
main results for unsteady flows as follows.

######  Theorem 3.

(i) The extended trajectory stretching exponents __(TSEs), defined as

$\mathrm{TSE}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0})=\frac{1}{t_{N}-t_{0}}\log\sqrt{\frac{\left|\dot{\mathbf{x}}(t_{N})\right|^{2}+\mathrm{v}_{0}^{2}}{\left|\dot{\mathbf{x}}(t_{0})\right|^{2}+\mathrm{v}_{0}^{2}}},\qquad\overline{\mathrm{TSE}}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0})=\frac{1}{t_{N}-t_{0}}\sum_{i=0}^{N-1}\left|\log\sqrt{\frac{\left|\dot{\mathbf{x}}(t_{i+1})\right|^{2}+\mathrm{v}_{0}^{2}}{\left|\dot{\mathbf{x}}(t_{i})\right|^{2}+\mathrm{v}_{0}^{2}}}\right|$
(33)

are objective measures of trajectory stretching and hyperbolicity strength in
the extended phase space.

(ii) Under assumption (A2), the extended trajectory angular velocity
__($\mathrm{\overline{\mathrm{TRA}}}$), defined as

$\displaystyle\mathrm{\overline{\mathrm{TRA}}}_{t_{0}}^{t_{N}}(\boldsymbol{x}\mathbf{}_{0})$
$\displaystyle=\sum_{i=0}^{N-1}\cos^{-1}\frac{\left\langle\dot{\mathbf{\boldsymbol{x}}}(t_{i}),\dot{\mathbf{\boldsymbol{x}}}(t_{i+1})\right\rangle+\mathrm{v}_{0}^{2}}{\sqrt{\left|\dot{\mathbf{\boldsymbol{x}}}(t_{i})\right|^{2}+\mathrm{v}_{0}^{2}}\sqrt{\left|\dot{\mathbf{\boldsymbol{x}}}(t_{i+1})\right|^{2}+\mathrm{v}_{0}^{2}}},$
(34)

is a quasi-objective measure of total trajectory rotation in the extended
phase space.

Note that the TSE measures defined in Theorem 3 are objective without any
particular assumption but in the extended phase space. This means they always
approximate an objective measure of material stretching for the material curve
defined by the extended trajectory
$\left(\mathbf{\boldsymbol{x}}(t),t\right)$. The TSE measures, however, are
not objective in the original phase space of the $\mathbf{x}$ variable and
they are only quasi-objective in that phase space under assumption (A1), as we
discussed in Theorem 1. In contrast, the extended $\overline{\mathrm{TRA}}$
measure defined in Theorem 3 is quasi-objective on the extended phase under
assumption (A2) and quasi-objective on the original phase space under
assumptions (A1)-(A2), as we discussed in Theorem 2.

##  5 Examples

###  5.1 Two-dimensional, unsteady ocean surface velocity (AVISO) data set

We first evaluate the proposed quasi-objective, single-particle LCS
diagnostics on a two-dimensional, satellite-altimetry-derived ocean-surface
current product (AVISO) that has been the focus of several coherent structure
studies (see, e.g. [5, 22, 16, 2]). The geostrophic component
$\mathbf{v}=(v_{1},v_{2})$ of ocean currents are calculated from the sea-
surface height anomaly via the formulas

$fv_{2}=\frac{1}{\rho}\frac{\partial p}{\partial x},\qquad
fv_{1}=-\frac{1}{\rho}\frac{\partial p}{\partial
y},\qquad\frac{1}{\rho}\frac{\partial p}{\partial z}=-g,$ (35)

where $\rho$ is the density of water, $p$ is the pressure as calculated from
the sea-surface height, $g$ is the constant of gravity, and $f$ is the
Coriolis parameter. A global daily gridded version of this data is freely
available from the Copernicus Marine Environment Monitoring Service.

We first verify the minimal differences between the objective continuous and
the quasi-objective discrete formulations of the stretching and rotation
metrics developed in Section 4. Our analysis focuses on the Agulhas leakage
region

$U=\left\\{(x,y)\in\left[-40^{\circ},-25^{\circ}\right]\times\left[-5^{\circ},10^{\circ}\right]\right\\}$
(36)

where strong hyperbolic and elliptic coherent structures have been previously
identified. Our default rectangular initial grid for generating trajectories
in $U$ will contain $250\times 250$ points. Using an integration time of 25
days, we find the characteristic velocity of $U$

$\mathrm{v}_{0}=\frac{1}{NM}\sum_{j=1}^{M}\sum_{i=0}^{N-1}\left|\mathbf{v}(\boldsymbol{x}_{j},t_{i})\right|=0.2\text{
m$\cdot$s}^{-1}$ (37)

where $j$ is an index over all initial positions $\boldsymbol{x}_{0}\in U$. We
have found that $96.7\%$ of
$\overline{\mathrm{TSE}}{}_{0}^{25}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{v}_{0})$
values are within $1\%$ of
$\bar{\lambda}{}_{0}^{25}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{v}_{0})$ and
$99.9\%$ of
$\overline{\mathrm{TRA}}{}_{0}^{25}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{v}_{0})$
values are within $1\%$ of
$\bar{\alpha}_{0}^{25}(\mathbf{\boldsymbol{x}}_{0},\boldsymbol{v}_{0})$.
Hence, the discretized formulations approximate the underlying quasi-objective
fields closely.

As already noted in Remark 4 after Theorem 4, condition (A2) is expected to
hold for surface currents on large enough ocean domains. We nevertheless
verify (A2) on trajectory data generated by the AVISO velocity field. Along
individual trajectories originating in $U$, we compute the ratio

$\delta_{A2}(\boldsymbol{x}\mathbf{}_{0},t_{0},t_{N})=\frac{\int_{t_{0}}^{t_{1}}\left|\frac{1}{2}\bar{\boldsymbol{\omega}}(t)\times\frac{\mathbf{\boldsymbol{v}}(t)}{\left|\mathbf{\boldsymbol{v}}(t)\right|}\right|dt}{\int_{t_{0}}^{t_{1}}\left|\frac{d}{dt}\frac{\mathbf{\boldsymbol{v}}(t)}{\left|\mathbf{\boldsymbol{v}}(t)\right|}\right|dt},$
(38)

which reflects the trajectory-averages of the quantities featured in
assumption (A2). If $\delta_{A2}\ll 1$ holds, then (A2) is satisfied on
average along the trajectory data serving as a basis of our analysis. We find
that on the domain $U$ defined in (36),
$\delta_{A2}(\boldsymbol{x}\mathbf{}_{0},0,25)<0.01$ for $98.9\%$ of all
initial positions, and $\delta_{A2}(\boldsymbol{x}\mathbf{}_{0},0,25)<0.1$ for
all fluid particles.

Ridges of the forward-time FTLE provide an effective diagnostic for
identifying repelling LCSs [24], but their computation relies on spatial
differentiation of the flow map over a regular grid, which cannot be carried
out from randomly placed single trajectories. For a fair comparison, we will
restrict our calculations of the FTLE to the same grid resolution as our
underlying flow, i.e. we do not interpolate to a refined grid for spatial
differentiation. Another objective LCS diagnostic we will consider in our
comparison is the single point (squared) relative dispersion [17]

${d}^{2}(\boldsymbol{x}\mathbf{}_{i0},t_{0},t)=\frac{\left|\boldsymbol{x}\mathbf{}_{i}(t)-\boldsymbol{x}\mathbf{}_{-i}(t)\right|^{2}}{\left|\boldsymbol{x}\mathbf{}_{i0}-\boldsymbol{x}\mathbf{}_{-i0}\right|^{2}},$
(39)

where $\boldsymbol{x}{}_{i}$ and $\boldsymbol{x}{}_{-i}$ are trajectory pairs
with initial conditions $\boldsymbol{x}\mathbf{}_{i0}$ and
$\boldsymbol{x}\mathbf{}_{-i0}$ that are initially close at time $t_{0}$
(e.g.,
$\left|\boldsymbol{x}\mathbf{}_{i0}-\boldsymbol{x}\mathbf{}_{-i0}\right|=r_{0}\ll
1$). While dividing by the initial distance makes no difference for a uniform
grid of initial conditions, we still include this normalization to allow for
small deviations in $r_{0}$ under a random subsampling of the initial
conditions that mimics real-life sparse trajectory data. We note that temporal
behavior of ensemble averages of relative dispersion are commonly studied
statistics in oceanography [27].

To highlight the significant advantages of $\mathrm{TSE}$ and $\mathrm{TRA}$
(as defined in Theorem 3) on sparse and non-uniform data, we perform a
progressive random subsampling of the grid of initial positions to degrade the
resolution of trajectories in a manner typical of what is found in
experimental data. Computations of $d^{2}$ can accommodate this subsampling if
we choose an $r_{0}$ value for which every initial position
$\boldsymbol{x}\mathbf{}_{i0}$ has a suitable match
$\boldsymbol{x}\mathbf{}_{-i0}$. The FTLE, however, practically requires a
regular grid of initial conditions for the spatial differentiation of the flow
map. To this end, we mimic the standard FTLE computation over an irregular
gird by creating a $C^{1}$-interpolant of final positions, and then computing
the deformation gradient from those interpolations. Alternative methods for
unstructured meshes have been suggested for FTLE but are not the focus of this
study (see e.g. [28]).

The first row of Figure 1 shows the full-resolution computation of
$\mathrm{TSE}_{0}^{25}(\boldsymbol{x}\mathbf{}_{0})$,
${d}^{2}(\boldsymbol{x}\mathbf{}_{i0},0,25)$ and
$\mathrm{FTLE}_{0}^{25}(\boldsymbol{x}\mathbf{}_{0})$, with an inset zoom of a
front that separates two recirculation regions in the flow. Successive rows
gradually degrade from $10\%$ to $0.1\%$ of the full resolution. We replace
the stretching diagnostic $\mathrm{TSE}_{0}^{25}(\boldsymbol{x}\mathbf{}_{0})$
with the hyperbolicity strength diagnostic
$\mathrm{\overline{TSE}}_{0}^{25}(\boldsymbol{x}\mathbf{}_{0})$, which is a
more robust single-trajectory indicator for sparse data. The highlighted
repelling LCS is evident in the top row as a ridge in all three diagnostics.
The quality of FTLE quickly degrades as interpolations between final positions
of particles oversimplify the flow dynamics. The FTLE ridge of interest is
still evident at the $10\%$ subsampling, but it disappears below that
resolution along with many other previously distinguishable features. The
relative dispersion $d^{2}$ does a good job of approximating FTLE ridges at
full resolution, and provides a hint at some circulation patterns at lower
resolutions, but displays no clear separating feature reminiscent of a ridge
at any degree of subsampling.

Figure 1: Stretching metric comparisons for AVISO ocean surface current
fields. The top row shows computations at full resolution with the lower rows
having progressively reduced resolution by randomly subsampling with fewer
trajectories.

As TSE values at a given location do not rely on any nearby neighbors, their
pointwise values remain unchanged and their features are more resilient under
subsampling in comparison to those of multi-trajectory objective LCS metrics.
Boundaries of multiple vortical features in the flow, such as those shown in
the inset, can be tracked through multiple reductions in resolution as
transitions between blue elliptic features and the surrounding green turbulent
regions. In fact, at $0.1\%$ of the full resolution, there is still evidence
of several of the original major Lagrangian eddies in blue, as well as the
front in our inset.

Next we evaluate the ability of the $\overline{\mathrm{TRA}}$, as defined in
Theorem 3, to identify vortical features. We apply the same method of random
subsampling to compare $\mathrm{\overline{\mathrm{TRA}}}_{0}^{25}$ to the
polar rotation angle, $\mathrm{PRA}_{0}^{25}$, whose features (i.e., level
curves) are objective in two dimensions [11]. In order to compute the flow map
for the purposes of calculating PRA from sparse data, we employ the same
$C^{1}$ interpolant on final trajectory positions that we used for computing
$\mathrm{FTLE}_{0}^{25}$. No objective, sparse-data rotation-specific metric
is available, which prompts us to use $d^{2}$ instead in our comparison of
rotation metrics. We do expect $d^{2}$ to highlight elliptic LCSs as coherent
Lagrangian vortices, as observed first by [36].

Figure 2 focuses on the same time period and Agulhas region as Fig. 1. Inset
in each plot is a zoom on a region with a vortex previously identified by
exact mathematical methods as a black-hole-type elliptic LCS (see e.g. [21]).
In the first row of plots, at full resolution, all three metrics
($\mathrm{\overline{\mathrm{TRA}}}_{0}^{25}$,
${d}^{2}(\boldsymbol{x}\mathbf{}_{i0},0,25)$ and $\mathrm{PRA}_{0}^{25}$)
highlight this vortex, as well as several other elliptic LCSs in the flow. The
degree of detail provided by the objective, multi-trajectory diagnostic, PRA,
is not matched by the trajectory rotation metric
$\overline{\mathrm{TRA}}_{0}^{25}$, which nevertheless correctly highlights
the vortex.

Figure 2: Rotation metric comparisons for AVISO ocean surface current fields.
The top row shows computations at full resolution with the lower rows having
progressively reduced resolution by randomly subsampling with fewer
trajectories. We display $\sqrt{d^{2}}$ here for clarity.

The insensitivity of $\mathrm{\overline{\mathrm{TRA}}}_{0}^{25}$ to small
details becomes an advantage at lower resolutions, as shown by subsequent rows
of Fig. 2. Indeed, $\mathrm{\overline{\mathrm{TRA}}}_{0}^{25}$retains much of
its structure with progressive random subsamplings. As a result, it is still
possible to recognize three of its maxima on the right-hand-side,
corresponding to multiple vortices, even at the lowest resolution. Some strong
rotational features can be tracked below $10\%$ resolution in the PRA fields,
but the elliptic LCS quickly becomes indistinguishable. Similarly,
distinguishing coherent structures in the $\sqrt{d^{2}}$ plots becomes
unfeasible quickly under progressive subsampling,

###  5.2 Unsteady ABC Flow

As a second example, we consider the three-dimensional velocity field

$\mathbf{v}(\boldsymbol{x},t)=e^{-\nu t}\begin{pmatrix}A\sin x_{3}+C\cos
x_{2}\\\ B\sin x_{1}+A\cos x_{3}\\\ C\sin x_{2}+B\cos x_{1}\end{pmatrix},$
(40)

a viscous, unsteady version of the steady ABC flow, defined on the triply
periodic spatial domain $U=[0,2\pi]^{3}.$ As in ref. [11], we use the
parameter values $A=1$, $B=\sqrt{2/3}$, $C=\sqrt{1/3}$, with the added
viscosity value $\nu=0.01$.

The unsteadiness of the flow necessitates the use of Theorem 3. By the triple
periodicity of $\mathbf{v}(\boldsymbol{x},t)$ on $U$, we also have
$\bar{\boldsymbol{\omega}}(t)\equiv\mathbf{0}$ and hence condition (A2) is
also satisfied on $U$. We conclude that both TSE metrics are objective on the
extended phase space and the $\overline{\mathrm{TRA}}$ metric is quasi-
objective on the extended phase space in the frame of (40). As the ABC flow is
a dimensionless set of equations, we use $\mathrm{v}_{0}=1$.

In Fig. 3, we show plots of the TSE and $\overline{\mathrm{TRA}}$ metrics
computed from Theorem 3 over trajectories launched from an initial grid of
$200\times 200\times 200$ points, integrated from time $t_{0}=0$ to
$t_{N}=20$. Values of the FTLE and LAVD are also included for comparison. (We
did not compute PRA for this example as levels sets of the PRA are not
objective in three dimensions.) We conclude that quasi-objective, single-
trajectory diagnostics ($\mathrm{TSE_{0}^{50}}$,
$\mathrm{\overline{TSE}_{0}^{50}}$,
$\mathrm{\overline{\mathrm{TRA}}_{0}^{50}}$) perform just as well as the
multi-trajectory, objective metric, the $\mathrm{FTLE}_{0}^{50}$, and the
single-trajectory, objective, but velocity-field-reliant metric, the
$\mathrm{LAVD}_{0}^{50}$.

Figure 3: Elliptic and hyperbolic LCSs in the unsteady ABC flow. The plots
compare quasi-objective, single-trajectory metrics ($\mathrm{TSE_{0}^{50}}$
and $\mathrm{\mathrm{\overline{\mathrm{TRA}}{}_{0}^{50}}}$) with objective LCS
metrics ($\mathrm{FTLE}_{0}^{50}$ and $\mathrm{LAVD}_{0}^{50}$) that require
multiple neighboring trajectories or detailed knowledge of the velocity field.

To evaluate the performance of the quasi-objective metrics under a random
degradation of the initial grid, we employ the same random subsampling method
as in 5.1. Even though level sets of the PRA are not objective in three-
dimensions, we include PRA here in our comparison. This is because there is no
systematic or widely accepted procedure for accurately approximating vorticity
from sparse Lagrangian trajectories for LAVD calculations.

Figure 4 shows that $\mathrm{\overline{TSE}_{0}^{50}}$ and
$\mathrm{\mathrm{\overline{\mathrm{TRA}}{}_{0}^{50}}}$ continue to provide
information on dominant flow features even as trajectory data becomes
substantially sparse. Specifically, as we keep the bounds on the colormaps
constant, the separation between rotational regions remains clear from
$\mathrm{\overline{TSE}_{0}^{50}}$ and
$\mathrm{\mathrm{\mathrm{\overline{\mathrm{TRA}}{}_{0}^{50}}}}$ when compared
with $\mathrm{FTLE}_{0}^{50}$ and $\mathrm{PRA}_{0}^{50}$, down to a random
selection of trajectories consisting of $0.1\%$ of the original computation
grid. We conclude that even in three-dimensional unsteady flows, it is
possible to observe a consistent range of stretching and rotation rates with
single-trajectory, quasi-objective metrics from very sparse and randomly
positioned data. Furthermore, interpolating between randomly oriented
$\mathrm{TSE}$ and $\mathrm{TRA}$ metrics provides a diagnostic that
highlights time-varying coherent structures to an extent that is unreachable
by prior LCS methods.

Figure 4: Elliptic and hyperbolic LCS in the randomly subsampled unsteady ABC
flow. The plots compare quasi-objective single-trajectory metrics
($\mathrm{\overline{TSE}{}_{0}^{50}}$ and $\mathrm{\overline{TRA}_{0}^{50}}$)
with LCS metrics ($\mathrm{FTLE}_{0}^{50}$ and $\mathrm{PRA}_{0}^{50}$), whose
computation requires multiple neighboring trajectories. The initial condition
grid for trajectories is randomized and its density is gradually decreased to
$0.1\%$ of its initial value.

###  5.3 Momentum barriers in the unsteady ABC flow

As our third example, we consider the linear-momentum barrier field equations,
as defined in [23], for the unsteady ABC flow (40). As shown in [23], at a
time $t$, the instantaneous barriers to momentum transport are invariant
manifolds of the autonomous dynamical system

$\boldsymbol{x}^{\prime}=\nu\rho\Delta\boldsymbol{\mathbf{v}}(\boldsymbol{x},t),$
(41)

with prime denoting differentiation with respect to a fictitious time (barrier
time) parametrizing the curves forming the barrier surface. At time $t=0$, eq.
(41) simplifies to
$\boldsymbol{x}^{\prime}=-\nu\rho\mathbf{v}(\boldsymbol{x},0)$ in the case of
the unsteady ABC flow defined in (40). This autonomous dynamical system
provides a steady flow field on which we can compute coherent structure
diagnostics to highlight instantaneous barriers to the transport of momentum
in the underlying fluid flow. This allows us to compare the steady versions of
the quasi-objective single trajectory metrics
$\mathrm{\overline{TSE}{}_{0}^{50}}$, $\mathrm{TSE}{}_{0}^{50}$, and
$\mathrm{\overline{TRA}{}_{0}^{50}}$, as defined in Theorems 1 and 2, with the
objective LCS metrics $\mathrm{FTLE}_{0}^{50}$ and $\mathrm{LAVD}_{0}^{50}$.
As these active barrier vector fields are objective [23], we can now also
include a comparison with $\mathrm{TRA}{}_{0}^{50}$, as defined in (25). In
our comparison, we will use the same parameters as in the previous section for
the flow (40).

Figure 5 shows the ability of the single-trajectory metrics to accurately
represent the elliptic and hyperbolic structures in the active barrier vector
field (41) for trajectories with initial positions on the same rectangular
grid in the domain $U$ as in Section 5.2. When compared with
$\mathrm{FTLE}_{0}^{50}$, we find that the steady-version of
$\mathrm{TSE_{0}^{50}}$ provides a great degree of detail, highlighting
hyperbolic regions of the flow as before. In addition, $\mathrm{TRA_{0}^{50}}$
is able to provide a comparable amount of detail for elliptic features. As we
degrade the resolution with the same random subsampling of initial positions,
Figure 6 shows that $\overline{\mathrm{TSE}}_{0}^{50}$ and
$\overline{\mathrm{TRA}}_{0}^{50}$ are again robust and able to highlight
hyperbolic and elliptic transport barriers down to $0.1\%$ of the original
resolution. Calculations of $\mathrm{FTLE}_{0}^{50}$ and
$\mathrm{PRA}_{0}^{50}$ from sparse trajectory data on the steady barrier
field perform comparably to the unsteady flow example, showing a quick
deterioration in quality and interpretability.

Figure 5: Elliptic and hyperbolic barriers of instantaneous linear momentum
transport in the unsteady ABC flow. The plots compare quasi-objective single-
trajectory metrics ($\mathrm{TSE_{0}^{50}}$,
$\overline{\mathrm{TSE}}_{0}^{50}$, $\mathrm{TRA}{}_{0}^{50}$, and
$\overline{\mathrm{TRA}}{}_{0}^{50}$) with objective LCS metrics
($\mathrm{FTLE}_{0}^{50}$ and $\mathrm{LAVD}_{0}^{50}$) that require multiple
neighboring trajectories or detailed knowledge of the velocity field.  Figure
6: Elliptic and hyperbolic momentum transport barriers in the randomly
subsampled unsteady ABC flow. The plots compare quasi-objective single-
trajectory metrics ($\mathrm{\overline{TSE}{}_{0}^{50}}$ and
$\mathrm{\overline{TRA}_{0}^{50}}$) with LCS metrics ($\mathrm{FTLE}_{0}^{50}$
and $\mathrm{PRA}_{0}^{50}$), whose computation requires multiple neighboring
trajectories. The initial condition grid for trajectories is randomized and
its density is gradually decreased to $0.1\%$ of its initial value.

##  6 Conclusions

Coherent structures, such as fronts, jets and vortices, are best viewed as
material objects. Indeed, these structures are most reliably identified in
experiments and observations from their signatures in material tracer fields.
By definition, such material coherent structures are indifferent to the
observer and hence can only be self-consistently identified from objective
quantities. In contrast, flows of practical relevance are mostly known from
trajectory data, which is inherently non-objective. Specifically, all features
of trajectories (including their length, curvature, velocity and acceleration)
are dependent on the observer. The question, therefore, naturally arises: How
can one still use trajectory data to extract reliable information from
material coherent structures in a flow?

We have addressed this question here by introducing the notion of quasi-
objectivity. We call a scalar-, vector- or tensor-field quasi-objective under
a condition if the field approximates another, objective field in all frames
in which that condition is satisfied. We have then derived quasi-objective
measures of trajectory stretching ($\mathrm{TSE}$ and
$\mathrm{\overline{TSE}}$) that approximate objective material stretching
exponents in frames where the flow is steady, ie., satisfies condition (A1).
We have also derived quasi-objective measures of trajectory rotation
($\mathrm{TRA}$ and $\mathrm{\overline{TRA}}$) that approximate objective
material rotation under conditions (A1) and (A2). The latter condition
requires trajectory accelerations to dominate the angular acceleration induced
by the spatial mean vorticity. Viewing general unsteady flows as steady flows
in their extended phase space (space-time), we have also obtained objective
TSE metrics and quasi-objective TRA metrics under assumption (A2) on the
extended phase space. These extended metrics track the stretching and rotation
of tangent vectors to fluid trajectories in the extended phase space.

We have tested these quasi-objective diagnostics on two-dimensional mesoscale
ocean surface velocity data (AVISO) obtained from satellite altimetry.
Condition (A2) is clearly satisfied on regions large enough to contain several
mesoscale eddies. Accordingly, we have found the quasi-objective metrics to
perform very well. As a further test case, we have considered an unsteady
version of the classic ABC flow [23] which is an exact solution of the three-
dimensional unsteady Navier–Stokes equation. This flow satisfies condition
(A2) on its triply periodic domain of definition. We have also considered the
steady version, which satisfies assumption (A1).

On trajectories generated by these example velocity fields from sufficiently
dense grids, the trajectory stretching exponent ($\mathrm{TSE}$) faithfully
reproduced features of the FTLE field. This is notable because $\mathrm{TSE}$
is a single-trajectory-based diagnostic whose pointwise value is independent
from the number and location of other trajectories in the data set. For sparse
and irregularly spaced trajectory data, the trajectory hyperbolicity strength
($\mathrm{\overline{TSE}}$) returned distinct indications of coherent
structures even when FTLE and relative dispersion computations gave no
meaningful results. Similarly, we have found that on well-resolved trajectory
data from objective vector fields, the trajectory rotation angle
($\mathrm{TRA}$) accurately reproduced details of the PRA field, even though
$\mathrm{TRA}$ is a single-trajectory-based diagnostic. For sparse and
irregularly spaced trajectory data, the cancellation-free total rotation angle
($\mathrm{\overline{TRA}}$) gave clear indication of coherent structures even
when PRA computations were no longer feasible. All this suggest great
potential for $\mathrm{\overline{TSE}}$ and $\mathrm{\overline{TRA}}$ in
visualizing coherent structures in 3D particle tracking velocimetry
experiments, in which tracer concentrations are generally low (see, e.g., [25,
38]). In contrast, $\mathrm{TSE}$ and (for objective vector fields, such as
the barrier equations in [23] )the $\mathrm{TRA}$, are useful in reducing
computational costs relative to multi-trajectory LCS diagnostics when
trajectories from a well-resolved initial grid are available.

Further applications of the quasi-objective diagnostic involve the detection
of barriers to the transport of dynamically active vector fields, such as the
momentum and the vorticity [23]. These barriers are invariant manifolds of
appropriate steady, volume-preserving and objective vector fields, the barrier
vector fields. The computation of these barrier vector fields requires two
spatial differentiations of the underlying velocity field and hence their
analysis via FTLE and PRA is numerically challenging, given that the latter
two methods involve further differentiation. In this setting, condition (A1)
is always satisfied and (A2) can be waived without any loss of generality, as
we have pointed out. Consequently, the use of $\mathrm{TSE}$ and
$\mathrm{TRA}$ promises to bring a major improvement in computational
efficiency for the detection of active transport barriers in three-dimensional
unsteady flows.

Acknowledgment

The authors acknowledge financial support from Priority Program SPP 1881
(Turbulent Superstructures) of the German National Science Foundation (DFG).

Data Availability

The AVISO geostrophic current velocity product used in this study, “Global
Ocean Gridded L4 Sea Surface Heights and Derived Variables Reprocessed”, is
freely available and is hosted by the Copernicus Marine Environment Monitoring
Service (http://marine.copernicus.eu).

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