## 1. Introduction

In the paradigm of chaotic advection, the trajectories of passive tracers can be complex even when the velocity field of the flow is simple. This is the case, for example, for time-dependent two-dimensional flows or even steady three-dimensional flows, like the celebrated ABCflow [

1]. However, even flows with a complicated time-dependent structure allow for the formation of coherent patterns that influence the evolution of tracers. These structures are common in nature, appearing both at short and long time scales, as well as small and large spatial scales. Remarkable examples of these structures are eddies and jets in the ocean and atmosphere, the Gulf Stream current and ring clouds [

2]. These patterns can influence, for example, the evolution of nutrients, as well as oil spills and other pollutants. Furthermore, these coherent structures could act as a local inhibitor for the energy transfer between scales [

3]. They also appear for other planets and other astrophysical systems, such as the jets on the surface of Jupiter, Saturn and other gaseous planets and in the solar photospheric flows [

4]. These structures, often referred to as Lagrangian Coherent Structures (LCSs), can shed light on the mixing and transport properties of a particular system on a finite time interval. The term “Lagrangian” in their name is motivated by the fact that they evolve as material lines with the flow [

2]. A particular kind of LCS is called Hyperbolic LCS (HLCS) and can be seen as the locally most attracting or repelling material lines that characterize the dynamical system over a finite time interval. The term hyperbolic is just an analogy of the stability of fixed points in dynamical systems, since usually one wants to study non-autonomous systems for which entities such as fixed points or stable and unstable manifolds are not defined. Moreover, these systems are studied for finite time.

Early attempts to detect HLCSs were based on Finite Time Lyapunov Exponents (FTLEs), which measure the rate at which initial conditions (or equivalently, tracer particles) separate locally after a given interval of time [

5,

6,

7,

8,

9,

10,

11,

12]. One of the first rigorous definitions of the LCSs was based on ridges of the FTLEs [

10]. FTLEs and the so-defined LCSs were used to study for example Lagrangian dynamics of atmospheric jets [

13] and oceanic stirring by mesoscale eddies [

14,

15,

16,

17,

18,

19], describing for example the chaotic advection emerging by mixed layer instabilities and its sensitivity to the vertical shear [

20]. A similar method to detect coherent structures uses the Finite Size Lyapunov Exponents (FSLEs), which represent the separation rate of particles given a specific final distance [

21,

22,

23]. However, several counter examples are available for which both FTLE and FSLE ridges fail to characterize the LCSs [

24,

25]. Although the FTLE field remains a popular diagnostic of chaotic stirring, other methods now are available to detect LCSs, which include for example the so-called Lagrangian descriptors [

26,

27], which are based on integration along trajectories, for a finite time, of an intrinsic bounded positive geometrical and/or physical property of the trajectories themselves. Notice however that the method of Lagrangian descriptors is not objective [

28]; regarding the connection between the Perron–Frobenius operator and almost invariant coherent sets of non-autonomous dynamical systems defined over infinite times [

29], the use of braids [

30] and the extrema of trajectories [

31]. Other Fast Indicators (FIs) besides FTLEs and FSLEs, i.e., computational diagnostics that characterize chaos quickly and can be used to determine coherent structures, are the Smaller (SALI) and Generalized (GALI) Alignment Indices [

32], the Mean Exponential Growth rate of Nearby Orbits (MEGNO) [

33] and the Finite Time Rotation Number (FTRN) [

34,

35,

36].

Particularly promising for the detection of coherent structures is a variational theory that considers the extremum properties of a specific repulsion rate function [

25,

37,

38,

39,

40,

41,

42]. Using this theory, it has been further shown that HLCSs can be described using geodesics [

43]. In this way, HLCSs can be represented as minimizers of a material length, with specific boundary conditions for the variation function. Another geodesic theory describes HLCSs in terms of the shearless transport barrier that minimizes the average shear functional [

41,

44]. These theories are based on the computation of shrink/stretch-lines (tensor lines); thus, trajectories along the eigenvectors of the deformation tensor are also called the Cauchy Green Tensor (CGT). Geodesic theories are also able to detect two other kinds of LCSs called parabolic and elliptic, which are however of no interest for the present work.

All these methods such as FTLEs, FSLEs or the variational and geodesic theories aim to find particular structures on the flow, among these the most repelling, or attracting, structures in the flow on a finite time interval, the HLCS. Particle tracers around these most influential material structures in finite time are maximally repelled or attracted. However, changing the finite time interval under study also changes the dynamical system and the correspondent structures emerging from the flow. Recent effort has been made in the understanding the instantaneous most influential coherent structures, Objective Eulerian Coherent Structures (OECSs), using a method that is not based on a finite time interval of evolution [

45,

46,

47]. The tracer particles could be maximally repelled for a short time, but over a long time, they could have a different behavior. Is it natural to wonder what happens asymptotically to the tracer particles. Is it possible to find some coherent structures that suggest the asymptotic behavior of the tracer particles?

In this work, we thus propose an alternative method to detect coherent patterns emerging in chaotic advection, which is based on the Covariant Lyapunov Vectors (CLVs). CLVs were first introduced by Oseledec [

48] and Ruelle [

49], but for a long time, they have received very little attention due to the lack of an efficient algorithm to compute them. Only in the last decade has the computation of such vectors become possible [

50,

51,

52], and CLVs have been used to investigate, e.g., the motion of rigid disk systems [

53], convection [

54] and other atmospheric phenomena [

55,

56,

57,

58]. For other theoretical discussions or for reviews on CLVs, see, e.g., [

59,

60,

61,

62,

63]. Unlike the Lyapunov Exponents (LEs), which are time independent, the correspondent vectors, also known as forward and backward Lyapunov Vectors (LVs), do depend on time. The LVs are orthonormal, and their direction can thus give only limited information about the local structure of the attractor. LVs also depend on the chosen norm; they are not invariant under time inversion; and they are not covariant, where covariance is here defined as the property of the forward (backward) LVs to be mapped by the tangent dynamics in forward (backward) LVs at the image point [

61]. Differently from the LVs, CLVs are norm independent, invariant for temporal inversion and covariant with the dynamics, making them thus mapped by the tangent linear operator into another CLV bases during the evolution of the system. They are not orthonormal, and their directions can thus better probe the tangent structure of the system.

All this intrinsic information can be summarized using the angle between the CLVs, a scalar field that allows one to investigate the spatial structures of the system. The need to pay more attention to the directions between LVs, backward and forward, has also been suggested for the study of turbulence [

9] and for the definition of a diagnostic quantity for the study of mixing, Lyapunov’s diffusion [

64]. Using three simple examples, we show that the attracting and repelling barriers tend to align along the paths on which the CLVs are orthogonal. The directions of the CLVs along these maxima provide thus information on the attracting or repelling nature of the barriers and can be related to the geometry of the system. Using the CLVs, we can define structures, at a given time instant, that are asymptotically the most attractive or repelling. Furthermore, since these structures can be defined for every time instant, it is possible to follow the formation of coherent structures during the evolution of the flow.

In

Section 2, we discuss the theory behind the CLVs and suggest a definition for coherent structures that give asymptotic information. The strategy will be to make use of the scalar quantity defined by the angle between the CLVs to locally identify the structures that asymptotically are maximally attractive or repulsive. In

Section 3, we use the CLVs to identify particular patterns in three different systems, and we compare the results with FTLE fields. Finally, in

Section 4, we summarize the conclusions.

## 2. Covariant Lyapunov Vectors

In this section, we summarize the theory behind CLVs for two-dimensional flows. For a more general and detailed review, see for example [

61]. Let the open set

$D\subset {\mathbb{R}}^{2}$ be the domain of the flow,

$t\in \mathbb{R}$ the time and

$\mathbf{v}(\mathbf{x},t)$ a vector velocity field in

D. The dynamical system that describes the motion of a tracer advected by the flow is thus:

where

$\mathbf{x}({\mathbf{x}}_{0},{t}_{0},t)\in D$ is the trajectory of the tracer starting at the point

${\mathbf{x}}_{0}$ at time

${t}_{0}$. To Equation (1) is associated the flow map

${\mathit{\varphi}}_{{t}_{0}}^{t}\left({\mathbf{x}}_{0}\right)$:

that maps the initial position

${\mathbf{x}}_{0}$ at time

${t}_{0}$ to the position

$\mathbf{x}({\mathbf{x}}_{0},{t}_{0},t)$ at time

t. It should be noted that the dependence on the initial condition is very important here, since the vectors will be considered as a function of time and of the initial positions. In the following, the contracted form

$\mathbf{x}=\mathbf{x}({\mathbf{x}}_{0},{t}_{0},t)$ will be used.

At each point

$\mathbf{x}\in D$, we can identify the tangent space

${T}_{\mathbf{x}}D\subset {\mathbb{R}}^{2}$. Infinitesimal perturbations,

$\mathbf{u}\left(t\right)\in {T}_{\mathbf{x}}D$, to a trajectory of this system can be described by the linearized system:

where

$\mathbf{J}\left(t\right)\in {\mathbb{R}}^{2\times 2}$ is the Jacobian matrix composed by the derivatives of the vector field

$\mathbf{v}(\mathbf{x},t)$ with respect to the component of the vector

$\mathbf{x}$. Using the fundamental matrix

$\mathbf{M}\left(t\right)\in {\mathbb{R}}^{2\times 2}$, of Equation (3), that satisfies:

we define the so-called tangent linear propagator:

$\mathbf{F}({t}_{0},t)$ maps a vector in

${\mathbf{x}}_{0}$ at time

${t}_{0}$ into a vector in

$\mathbf{x}$ at time

t along the same trajectory of the starting system Equation (1), that is:

According to Equation (5), the propagator is always nonsingular. In terms of the flow map, the tangent linear propagator is:

Exploiting Oseledec’s Theorem [

48,

65], it is possible to characterize the system using quantities that are independent of

t or

${t}_{0}$. By virtue of this theorem, the far-future operator:

and the far-past operator:

are well-defined quantities. Note that the product

${\mathbf{F}}^{\top}({t}_{0},t)\mathbf{F}({t}_{0},t)$ determines the Euclidean norm of the tangent vectors in the forward-time dynamics (a similar role is played by

${\mathbf{F}}^{-\top}({t}_{0},t){\mathbf{F}}^{-1}({t}_{0},t)$ for the backward-time dynamics), in fact,

Operator Equations (8) and (9) probe respectively the future and past dynamics of a certain point and share the same eigenvalues:

which, assuming ergodicity, are independent of time and space. Each eigenvalue has multiplicity

${m}_{i}$ (

${m}_{1}+{m}_{2}=2$). Their logarithms correspond to the LEs of the dynamical system Equation (1). If the limits in Equations (8) and (9) are not considered, the resulting eigenvalues are time and space dependent and are called FTLEs.

The two operator Equations (8) and (9) can be evaluated at the same point in space at a given time t. The correspondent eigenvectors, $\{{\mathbf{l}}_{1}^{+}\left(t\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{l}}_{2}^{+}\left(t\right)\}$, $\{{\mathbf{l}}_{1}^{-}\left(t\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{l}}_{2}^{-}\left(t\right)\}$, will define thus the forward and backward Lyapunov basis computed at the same time. Conversely, the respective eigenvalues, whose bases are time dependent, depend on the chosen scalar product and are not invariant under time reversal. Furthermore, these vectors are always orthogonal and give thus limited information on the spatial structure of the configuration space.

To overcome these issues, one can build particular spaces, the backward and forward Oseledec subspaces, defined as [

48]:

and:

In the forward dynamics, the generic vector

${\mathbf{l}}_{i}^{+}\left(t\right)$ grows or decays exponentially with an average rate

${\lambda}_{i}$. If the system is evolved forward in time, by means of the tangent linear propagator, the evolution of the vector

${\mathbf{l}}_{1}^{+}\left(t\right)$ will have a non-zero projection inside the space generated by

${\mathbf{l}}_{1}^{+}\left({t}^{\prime}\right)$, but it will also have a non-zero projection onto the space generated by

${\mathbf{l}}_{2}^{+}\left({t}^{\prime}\right)$. On the other hand,

${\mathbf{l}}_{2}^{+}\left(t\right)$ will be transported onto the space generated by

${\mathbf{l}}_{2}^{+}\left({t}^{\prime}\right)$ and will have zero projection onto the space generated by

${\mathbf{l}}_{1}^{+}\left({t}^{\prime}\right)$. Repeating similar arguments for the backward Lyapunov Vectors leads to the observation that

${L}_{i}^{-}$ and

${L}_{i}^{+}$ are covariant subspaces,

Vectors that are covariant with the dynamics and invariant with respect to time reversal will be found at the intersection of the Oseledec subspaces, i.e., at:

These spaces, often referred to as Oseledec splitting [

48,

49,

51,

63], are not empty [

61], and their vectors, also called covariant Lyapunov vectors (CLVs), are covariant with the dynamics, i.e.,

It should be noted that CLVs posses thus the properties of a semi-group. These vectors have an asymptotic grow the or decay with an average rate

${\lambda}_{i}$, so that their asymptotic behavior can be summarized as:

where

$\tau =|{t}^{\prime}-t|$, which shows their invariance under time reversal. Note also that these vectors do not depend on any particular norm.

Equations (12), (13) and (15) imply a simple relation between CLVs and LVs in a two-dimensional system,

The first CLV corresponds to the first backward Lyapunov vector and the second CLV to the second forward Lyapunov vector.

Furthermore, if the Jacobian matrix appearing in Equation (3) is constant, the CLVs are not just covariant, but invariant with the dynamics. In this particular case, in fact, they correspond to the eigenvectors of the Jacobian, and the eigenvalues of this matrix coincide with the LEs.

Since the CLVs highlight particular expansion and contraction directions at each point of the coordinate space and these directions are not necessarily orthogonal, they can be used to understand the geometrical structure of the tangent space. This geometric information can be summarized by the scalar field of the angle

$\theta \left(t\right)$ between the CLVs. Because the CLVs identify asymptotically the expansion and contraction direction sets of the tangent space associated at every point of the domain, the

$\theta $ field represents a measure of the hyperbolicity of the system, that is a measure of the orthogonality between these two directions. Note that the orientation of the CLVs is defined as arbitrary. The angle between the CLVs,

is thus:

where

$\{{\mathbf{w}}_{1}\left(t\right),\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{2}\left(t\right)\}$ are the first and the second CLVs [

51]. It is interesting to point out that when

$\theta \left(t\right)=\pi /2$, the CLVs reduce to LVs in two dimensions, and the backward and forward Lyapunov bases coincide. If the computation of the angle is done at every point of the domain and if we consider the system Equation (1) in which the initial conditions are varied in such a way that all the domain is spanned, we can build a field of the orthogonality between the expansion and contraction directions of the system.

In the following section, we will show how CLVs, with their capability of probing the geometric structure of the tangent space, can be used to determine coherent structures that give asymptotic information on the tracer and, then, how the CLVs highlight the mixing template of the flow.

#### Hyperbolic Covariant Coherent Structures

Several frameworks are available to study passive scalar mixing. Although all these methods aim to describe the mechanism underlying the chaotic advection, they have significantly different approaches. However, most of them share a fundamental feature called objectivity. Objectivity is a fundamental requirement to define structures emerging from a flow. In practical applications, objectivity can be used, e.g., to move the reference frame into the reference frame of a coherent structure. In particular, a structure can be considered objective if it is invariant under a coordinate change of the form:

where

$\mathbf{Q}$ denotes a time-dependent orthogonal matrix and

$\mathbf{P}$ a time-dependent translation [

2]. The FTLEs, FSLEs and geodesic theory define objective quantities. Sala et al. [

62] have shown that, generally, also for a linear transformation of coordinates, the new angle non-linearly depends on the angle of the old reference system. However, for the particular class of transformation, we are interested in, Equation (21), the angle

$\theta $ is invariant. This means that structures highlighted by

$\theta $ are objective.

To study the asymptotically most attractive or repelling behavior of tracer particles near a particular structure identified at a given instant of time, one can consider the lines

$\mathbf{r}(s,t)$, with

s representing the length parameter, defined by:

where the primes indicate the derivative with respect to

s and characterized by

$\theta =\pi /2$ along their paths. In fact, in this case:

where

${\mathbf{l}}_{1}$ and

${\mathbf{l}}_{2}$ are Lyapunov vectors. In these circumstances, the backward and the forward bases are coincident. A sphere of initial conditions around a point of the path will be deformed in an ellipsoid whose axis is aligned with these Lyapunov vectors. If Equation (22) is aligned along a ridge of the hyperbolicity field

$\theta $, characterized by

$\theta =\pi /2$, then they are also pointwise the most attractive or repelling lines in terms of the asymptotic behavior of tracer particles. Consider in fact line Equation (22a) and its tangent CLV

${\mathbf{w}}_{1}$ as depicted in

Figure 1.

Along this curve, for any point

${P}_{1}$, we can choose a point

${P}_{2}$, which is on the normal vector to the curve in

${P}_{1}$,

$\mathbf{n}$. Because of Equation (11), the distance between these two points decreases as:

Consider now a point

${P}_{3}$ on a nearby curve and a point

${P}_{4}$ laying on the normal vector to the curve in

${P}_{3}$. The normal vector

$\mathbf{n}$ at

${P}_{3}$ can be written as a linear combination of

${\mathbf{w}}_{1}$ and

${\mathbf{w}}_{2}$, and considering again Equation (11), one has thus:

The previous computation can be repeated analogously for ${\mathbf{w}}_{2}$. The same reasoning holds for ${\mathbf{w}}_{2}$.

It must be remarked that Equation (22b) gives asymptotic information, but is defined for a particular instant of time. For every time instant, the line Equation (22), together with the orthogonality condition

$\theta \left(t\right)=\pi /2$ describe the so-called tensor lines and can be seen as the asymptotic version of shearless barriers [

44].

Taking into account these properties of the CLVs, we propose a definition to describe these asymptotic coherent patterns:

**Definition** **1** (Hyperbolic Covariant Coherent Structures (HCCSs)).

At each time t, a Hyperbolic Covariant Coherent Structure is an isoline of the hyperbolicity scalar field θ at the level $\theta =\pi /2$. Its attractive or repelling nature is determined by the CLVs aligned with it.

It is important to emphasize the difference between the information provided by FTLEs and that provided by CLVs. FTLEs are the finite time version of Lyapunov’s exponents and, as shown in the

Appendix, are calculated as a mean time of the logarithms of the separation of two trajectories that start from nearby points. The FTLEs therefore give the average information over the time interval considered. Moreover, if the time interval is long enough and the analyzed region can be considered ergodic, the dependence on initial conditions is lost and, therefore, the possibility of displaying possible structures. CLVs, on the contrary, as well as LVs, depend on a time instant and do not converge to the same value. CLVs therefore allow the definition of instant structures, which, however, give asymptotic information.

This approach presents some numerical challenges: CLVs are not computed as quickly as the FTLEs, and numerically, in order to identify the isolines, it is necessary to consider an interval of values for the angle. The HCCSs are thus computed here considering the isoregion of $\theta \in [\frac{\pi}{2}-\delta ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{\pi}{2}]$, where $\delta $ is $0.087$ rads.

## 4. Conclusions

We have here proposed a new definition and a new computational framework to determine hyperbolic structures in a two-dimensional flow based on covariant Lyapunov vectors. CLVs are covariant with the dynamics, invariant for temporal inversion and norm independent. These vectors are the natural mathematical entity to probe the asymptotic behavior of the tangent space of a dynamical system. All these properties allow an exploration of the spatial structures of the flow, which cannot be done using the Lyapunov vectors bases due to their orthogonality.

CLVs are related to the contraction and expansion directions passing through a point of the tangent space, and the angle between them can be thus considered as a measure of the hyperbolicity of the system. This information can be summarized in a scalar field, the angle $\theta $, between the CLVs referring to the initial grid conditions, and used to define hyperbolic structures. The structures identified with the isolines of this field, characterized by $\theta =\pi /2$, are called hyperbolic covariant coherent structures. These patterns are the most repelling or attracting pointwise, in terms of the asymptotic behavior of tracer particles, with respect to nearby structures at a given time. In terms of practical applications, this has important consequences, as it will provide an indicator for the long time transport of passive tracers such as for example oil spills in the ocean.

CLVs, and the correspondent $\theta $ field, have been computed for three numerical examples to compare how the behavior of the particle tracer near HLCSs, highlighted by the FTLEs, can change asymptotically in time. The three examples include a Hamiltonian autonomous system and two non-autonomous systems that are bounded or periodic. For all these examples, it is possible to compute CLVs, HCCSs and compare them with the HLCSs. Since the FTLEs tend to converge to LEs and lose their dependence on the initial conditions, the angle between the CLVs could give more detailed information about possible structures that can emerge from the flow. This feature has been highlighted in particular for the Hamiltonian autonomous system, in which $\theta $ is able to detect the central barrier in contrast with the FTLE field, and in the Bickley jet, in which the FTLEs converge to the same value in the jet region, and it is not possible to see any kind of particular finest structure. The use of $\theta $ provides information on the structures appearing at each time of the evolution of the flow, and the three examples underline that the HCCSs did not always correspond to HLCSs and vice versa. Therefore, particle tracers, such as chlorophyll or oil in water, can be maximally attracted or repelled by some HLCSs, but if we consider a different time interval and in particular the asymptotic behavior of these particles, we can obtain a distribution that is completely different. Note that, in practice, the asymptotic time length can be considered as the time taken by two random initial bases to converge to the same BLVs basis.

It should be noticed that, while no fluxes can be present across the HLCSs, the same does not necessarily hold for all the structures appearing characterized by $\theta =\pi /2$. HCCSs can be found for every instant of time, but they give asymptotic information about the behavior of the particles tracer near the HCCS at that instant of time, and for this reason, the zero flux requirement of the HLCSs is not necessary. HCCSs are not necessarily a barrier, and their meaning is different from the one of HLCSs.

For the three examples considered, we have also computed HLCSs with the geodesic theory using the LCS tool [

66] (not shown). The results are in agreement with the discussion above. Looking at asymptotic time, it is still possible to find particular structures emerging from flow, but clearly, these structures do not necessarily correspond to HLCSs computed for a particular time interval.

For the two non-autonomous systems, we have considered the evolution of the PDFs of the $\theta $ field and the evolution of its first four moments. For the Bickley jet, the probability distribution of the angle is stationary in time, and so its moments, but for the double gyre, it is possible to appreciate a small variation in time for the PDF. The information deriving from the evolution of the $\theta $ field, related to the variation of the strength of the hyperbolicity field, could be used to characterize the dynamical mixing of the system.

Finally, future studies will have to address the detection of hyperbolic structures beyond analytical systems, i.e., for two-dimensional turbulent flows. This will be particularly interesting for flows at the transition between balance and lack of balance [

69,

70,

71,

72], where the detection of HCCSs can shed light both on the structure of mixing and on the forward cascade of energy to dissipation, or the lack of thereof. In particular, the evolution of the moments of the PDF of

$\theta $ could be used to define an index of dynamical mixing for the system under study. Particularly interesting will be the behavior of the HCCSs in the presence of intermittency. We can conjecture that in particular cases, such as, e.g., the merging of two vortices, the instantaneous structures underlined by the HCCS can give reliable information about the asymptotic tracer dynamics, that is the dynamics after the merging event. This is however left for future studies.