A variational theory of hyperbolic Lagrangian Coherent Structures

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Abstract

We develop a mathematical theory that clarifies the relationship between observable Lagrangian Coherent Structures (LCSs) and invariants of the Cauchy–Green strain tensor field. Motivated by physical observations of trajectory patterns, we define hyperbolic LCSs as material surfaces (i.e., codimension-one invariant manifolds in the extended phase space) that extremize an appropriate finite-time normal repulsion or attraction measure over all nearby material surfaces. We also define weak LCSs (WLCSs) as stationary solutions of the above variational problem. Solving these variational problems, we obtain computable sufficient and necessary criteria for WLCSs and LCSs that link them rigorously to the Cauchy–Green strain tensor field. We also prove a condition for the robustness of an LCS under perturbations such as numerical errors or data imperfection. On several examples, we show how these results resolve earlier inconsistencies in the theory of LCS. Finally, we introduce the notion of a Constrained LCS (CLCS) that extremizes normal repulsion or attraction under constraints. This construct allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.

Research highlights

► Lagrangian Coherent Structures (LCSs) are cores of observed tracer patterns. ► LCS definition can be formulated mathematically as a variational principle. ► Variational LCS theory yields sufficient and necessary criteria for LCSs in flows. ► Theory of a constrained LCS yields unique weak unstable and center manifolds.

Introduction

This paper is concerned with the development of a self-consistent theory of coherent trajectory patterns in dynamical systems defined over a finite time-interval. Following Haller and Yuan [1], we use the term Lagrangian Coherent Structures (or LCSs, for short) to describe the core surfaces around which such trajectory patterns form.

As an example, Fig. 1 shows the formation of passive tracer patterns in a quasi-geostrophic turbulence simulation described in [1]. We seek to locate the dynamically evolving LCSs that form the skeleton of these patterns. Beyond offering conceptual help in interpreting and forecasting complex time-dependent data sets, LCSs are natural targets through which to control ensembles of trajectories.

As proposed in [1], repelling LCSs are the core structures generating stretching, attracting LCSs act as centerpieces of folding, and shear LCS delineate swirling and jet-type tracer patterns. In order to act as organizing centers for Lagrangian patterns, LCSs are expected to have two key properties:

(1) An LCS should be a material surface, i.e., a codimension-one invariant surface in the extended phase space of a dynamical system. This is because (a) an LCS must have sufficiently high dimension to have visible impact and act as a transport barrier and (b) an LCS must move with the flow to act as an observable core of evolving Lagrangian patterns.

(2) An LCS should exhibit locally the strongest attraction, repulsion or shearing in the flow. This is essential to distinguish the LCS from all nearby material surfaces that will have the same stability type, as implied by the continuous dependence of the flow on initial conditions over finite times.

Based on (1)–(2), a purely physical definition of an observable LCS can be given as follows (cf. [1]):

Definition 1 Physical Definition of Hyperbolic LCS

A hyperbolic LCS over a finite time-interval I=[α,β] is a locally strongest repelling or attracting material surface over I (cf. Fig. 2).

This definition does not favor any particular diagnostic quantity, such as finite-time or finite-size Lyapunov exponents, relative or absolute dispersion, vorticity, strain, measures of hyperbolicity, etc. Instead, it describes the main physical property of LCSs that enables us to observe them as cores of Lagrangian patterns. Ideally, a mathematical definition of an LCS should capture the essence of the above physical definition, and lead to computable mathematical criteria for the LCS. As we shall see below, however, such a mathematical definition and the corresponding criteria have been missing in the literature.

In particular, while LCSs have de facto become identified with local maximizing curves (ridges) of the Finite-Time Lyapunov Exponent (FTLE) field (see, e.g., [2], [3], [4], [5], and the recent review by Peacock and Dabiri [6]), simple counterexamples reveal conceptual problems with such an identification (see Section 2.3). Computational results on geophysical data sets also show that several FTLE ridges in real-life data sets do not repel or attract nearby trajectories (see, e.g., [7], [8]).

The present paper addresses this theoretical gap by providing a mathematical version of the above physical LCS definition, and by deriving exact computable criteria for LCSs in n-dimensional dynamical systems defined over a finite time-interval. Our focus is hyperbolic (repelling or attracting) LCSs; a similar treatment of shear LCSs will appear elsewhere.

Our analysis is based on a new notion of finite-time hyperbolicity of material surfaces. This hyperbolicity concept is expressed through the normal repulsion rate and the normal repulsion ratio that are finite-time analogues of the Lyapunov-type numbers introduced by Fenichel [9] for normally hyperbolic invariant manifolds. Unlike Fenichel’s numbers, however, the normal repulsion rate and ratio are smooth quantities that are computable for a given material surface and flow.

We employ a variational approach to locate LCSs as material surfaces that pointwise extremize the normal repulsion rate among all C1-close material surfaces. We also introduce the notion of a Weak LCS (WLCS), which is a stationary surface–but not necessarily an extremum surface–for our variational problem. As we show, the use of WLCS resolves notable counterexamples to the identification of LCSs with FTLE ridges.

Solving the variational problem leads to a necessary and sufficient LCS criterion that involves invariants of the inverse Cauchy–Green strain tensor (Theorem 7). Specifically, WLCSs at time t0 must be hypersurfaces in the phase space satisfying the equation λn(x0),ξn(x0)=0, where x0Rn is the phase space variable, λn(x0) denotes the largest eigenvalue of C(x0)=[Ft0t0+T(x0)]Ft0t0+T(x0), the Cauchy–Green strain tensor computed from the flow map Ft0t0+T between time t0 and t0+T; ξn(x0) is the eigenvector corresponding to the largest eigenvalue of C(x0). Condition (1) turns out to be equivalent to C1(x0)[ξn(x0),ξn(x0),ξn(x0)]=0, where C1(x0) is a three-tensor, the gradient of C1, evaluated on the vector ξn.

Theorem 7 also states that a material surface satisfying (1) at time t0 must be orthogonal to the ξn(x0) vector field in order to be a repelling LCS. This condition is non-restrictive for a long-lived LCS because all repelling material surfaces turn out to align at a rate ebTλn1/λn with directions normal to ξn(x0) (Theorem 4 and formula (30)). For small T, however, the orthogonality condition may not hold on any material surface, which underscores a fundamental limitation to identifying cores of Lagrangian patterns from short-term observations.

Finally, Theorem 7 requires a matrix L(x0,t0,T), defined in (31), to be positive definite on the zero set (1) in order for the underlying WLCS to be an LCS. A necessary condition for the positive definiteness of L is 2C1(x0)[ξn(x0),ξn(x0),ξn(x0),ξn(x0)]>0, with the four-tensor 2C1(x0), the second derivative of C1, evaluated on the strongest strain eigenvector field ξn(x0) (Proposition 8).

The LCSs we identify are robust under perturbations as long as ξn,2λnξn+λn,ξnξn0 holds along them (Theorem 11). The admissible perturbations to the underlying dynamical system need not be pointwise small as long as they translate to small perturbations to the flow map. For example, large amplitude but short-lived localized perturbations to the vector field governing the dynamics are admissible.

These general results allow us to examine the relevance of FTLE for LCS detection in rigorous terms. FTLE ridges turn out to mark the presence of LCSs under four conditions. First, along FTLE ridges, λn must be larger than one and of multiplicity one. Second, FTLE ridges have to be normal to the ξn(x0) field. Third, along FTLE ridges, the gradient of ξn(x0) in directions parallel to ξn(x0) must be small enough. Fourth, the FTLE ridge must be steep enough (cf. Proposition 14).

For LCSs marked by such FTLE ridges, the robustness criterion (4) takes the more specific form ξn,2λnξn+λn,ξnξn<0, as we show in Proposition 15. This again implies that FTLE ridges that are steep (i.e., ξn,2λnξn0), nearly flat (i.e., |λn|0), and lie in regions of moderately nonlinear strain (i.e., |ξnξn|1) are the most robust under perturbations.

The present approach also allows for the treatment of a Constrained LCS (or CLCS), which is a solution of the above maximum repulsion problem under constraints. In this paper, we explore two such constraints: (1) The constraint that the LCS be an invariant manifold in phase space, not just in extended phase space. (2) The constraint that the LCS be a level surface of a first integral.

The CLCS approach enables us to identify unique attracting and repelling LCSs in linear flows that have so far defied LCS extraction techniques. CLCSs also turn out to be useful in extracting unique weak unstable manifolds from finite-time data sets, even though such manifolds are nonunique in the classic theory of invariant manifolds.

We believe that these results establish the first rigorous link between a Lagrangian diagnostic tool, the Cauchy–Green stains tensor, and invariant coherent structures in a finite-time dynamical system. Notably, however, the recent work of Froyland et al. [10] provides a rigorous link between properties of the Perron–Frobenius operator and almost invariant coherent sets of non-autonomous dynamical systems defined over infinite times.

In Section 2, we fix our notation and review discrepancies between observable LCSs and their commonly assumed FTLE signature.

In Section 3, we develop the notion of finite-time hyperbolicity for material surfaces and show that normals of finite-time hyperbolic material surfaces align exponentially fast with the largest strain eigenvector of the Cauchy–Green strain tensor. In Section 4, we define Weak LCSs and LCSs as repelling material surfaces that are pointwise stationary surfaces and extrema, respectively, of a variational principle for the repulsion rate. We then solve this variational problem and obtain sufficient and necessary conditions for WLCSs and LCSs.

Section 5 discusses the robustness of LCSs obtained from our theory with respect to perturbations to the underlying dynamical system. Applying our general results, we examine the relevance of FTLE ridges in LCS detection in Section 6. In Section 7, we discuss implications for the numerical detection of LCSs.

In Section 8, we review the counterexamples of Section 2.3 and show how they are resolved by our main result, Theorem 7. Section 9 discusses constrained LCS problems with applications to linear flows and weak unstable manifolds. We present our conclusions and directions for future work in Section 10.

Section snippets

Set-up and notation

Consider a dynamical system of the form ẋ=v(x,t),xURn,t[α,β], with a smooth vector field v(x,t) defined on the n-dimensional bounded, open domain U over a time interval [α,β], and with the dot denoting differentiation with respect to the time variable t. This paper is primarily motivated by applications to fluid mechanics, in which case we have n=2 (planar flows) or n=3 (three-dimensional flows). The finite length of the time interval [α,β] reflects temporal limitations to the available

Hyperbolicity of material surfaces

In our setting, LCSs are only assumed to exist over a finite time-interval [α,β]. To describe their attracting and repelling properties, therefore, we cannot rely on classic notions of hyperbolicity that are inherently asymptotic in time (see, e.g., [9]).

The concept of uniform finite-time hyperbolicity for individual trajectories was apparently first introduced in [12], then elaborated on in [13], [1], [22], [2], [3], [14]. Extensions of these results to higher-dimensional and

Weak LCS and LCS

Consider now material surfaces that are small and smooth deformations of a normally repelling material surface M(t). All such deformed material surfaces will be normally repelling by the continuity of the inequalities (21) in their arguments.

For a repelling material surface M(t) to be a repelling LCS, we will require M(t) to be pointwise more repelling over [t0,t0+T] than any other nearby material surface. Specifically, at any point x0M(t0), perturbations to M(t0) along its normal n0Nx0M(t0)

Robustness of hyperbolic LCS

A major question about any coherent structure identification scheme is its robustness. If small data imperfections, numerical errors, or other sources of noise can significantly alter the identified structures, then the structures are of little practical use.

The only non-robust feature of an LCS turns out to be the exact alignment of its normals with the ξn field. For material surfaces that are normally repelling for long enough T, however, the alignment error becomes numerically undetectable

When does an FTLE ridge indicate a hyperbolic LCS?

First, we give a definition of an FTLE ridge that is somewhat weaker than the second-derivative ridge definition we used in analyzing the examples of Section 2.3 (cf. Appendix B).

Definition 12 FTLE Ridge

For a fixed time interval [t0,t0+T], we call a compact hypersurface M(t0)U an FTLE ridge if for all x0M(t0), we have λn(x0,t0,T)Tx0M(t0),ξn(x0,t0,T),2λn(x0,t0,T)ξn(x0,t0,T)<0.

This definition formulates the ridge conditions in terms of the λn field, which is equivalent to similar conditions on the FTLE field Λt0t0

Alignment of the LCS normal with the largest strain eigenvector

As we remarked earlier, requiring the zero set of λn,ξn to be orthogonal to the vector field ξn(x0,t0,T) appears to be a restrictive condition, yet it emerges as a necessary requirement for the existence of an LCS by Theorem 7.

In view of the alignment result in Theorem 4, this orthogonality requirement is no longer surprising. Specifically, for a material surface that is normally repelling over a long enough time interval, the strain eigenvectors ξn(x0,t0,T) align with the normals of the

Example 5: linear saddle flow

Consider again the linear strain flow ẋ=x,ẏ=y, analyzed in Example 1. The Cauchy–Green strain tensor for this example is simply Ct0t0+T(x0)=(e2T00e2T), with eigenvalues and eigenvectors λ1=e2T,λ2=e2T,ξ1=(01),ξ2=(10). Since λ2 is a constant field, condition (ii)/2 of Theorem 7 is violated, and hence system (70) admits no repelling LCSin the sense Definition 6.

As for WLCS, note that conditions (i)/1 and (i)/3 of Theorem 7 hold at any point x0 in the phase space. Then condition (i)/2 and (71)

Constrained LCS

So far, we have identified hyperbolic LCSs as solutions of an extremum problem. Here we restrict this general extremum problem to a constrained extremum problem: we seek to find the most attracting or repelling material surface out of a prescribed family of codimension-one surfaces.

An example where this approach is conceptually useful is the linear saddle flow ẋ=x,ẏ=y. Out of all the attracting WLCS we have identified for this system in Example 5 (i.e., all horizontal material lines), the x

Conclusions

We have developed a variational theory of hyperbolic Lagrangian Coherent Structures (LCSs) in finite-time dynamical systems of arbitrary dimension. Our objective was to identify observed cores of Lagrangian patterns rigorously without a priori favoring any particular Lagrangian diagnostic tool. Our analysis has yielded an exact relationship between observable LCSs and invariants of the Cauchy–Green strain tensor.

We have defined hyperbolic LCSs as locally the strongest normally repelling or

Acknowledgements

I am grateful to Harry Dankowitz, Mohammad Farazmand, Jim Meiss, Ronny Peikert, Gilead Tadmor, and Lai-Sang Young for their insightful questions and comments that have led to improvements in this manuscript. I am also thankful to Wenbo Tang for reviewing some of the proofs in detail. Finally, I would like to express my gratitude to Shawn Shadden for reading this work in detail, catching a number of misprints, and providing a number of thoughtful suggestions.

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