Lagrangian coherent structure

Hyperbolic LCS (attracting in red and repelling in blue) and elliptic LCS in a two-dimensional turbulence simulation.

Lagrangian coherent structures (LCSs) are distinguished sets of trajectories in a dynamical system that exert a major influence on nearby trajectories over a time interval of interest. The type of this influence may vary, but it invariably creates a coherent trajectory pattern for which the underlying LCS serves as a theoretical centerpiece. In observations of tracer patterns in nature, one readily identifies coherent features, but it is often the underlying structure creating these features that is of interest.

Indeed, individual tracer trajectories forming coherent patterns are generally sensitive with respect to changes in their initial conditions and the system parameters. In contrast, the LCSs creating these trajectory patterns turn out to be robust and provide a simplified skeleton of the overall dynamics of the system. Physical examples include floating debris, oil spills, surface drifters and chlorophyll patterns in the ocean; clouds of volcanic ash and spores in the atmosphere; and coherent crowd patterns formed by humans and animals.

While LCSs may exist in any dynamical system, their role in creating coherent patterns is perhaps most readily observable in fluid flows. The images below are examples of how different types of LCSs hidden in geophysical flows shape tracer patterns.

• 
  Spiral eddies:
  Hyperbolic and elliptic LCSs
  (Paul Scully-Power/NASA)
• Swimmers in a rip current Hyperbolic LCS (Joo Yong Lee/Sungkyunkwan University)
  Swimmers in a rip current
  Hyperbolic LCS
  (Joo Yong Lee/Sungkyunkwan University)
• 
  Sea surface temperature in Gulf Stream
  Parabolic LCSs
  (NASA)
• 
  Phytoplankton in Agulhas ring
  2D elliptic LCS
  (NASA/GSFC)
• 
  Tornado off the Florida Keys
  3D elliptic LCS (cylindrical)
  (Joseph Golden/NOAA)
• 
  A steam ring from Mount Etna
  3D elliptic LCS (toroidal)
  (Tom Pfeiffer [1])

General definitions

Material surfaces

Figure 1: An invariant manifold in the extended phase space, formed by an evolving material surface.

On a phase space ${\mathcal {P}}$ and over a time interval ${\mathcal {I}}=[t_{0},t_{1}]$ , consider a non-autonomous dynamical system defined through the flow map $F_{t_{0}}^{t}\colon x_{0}\mapsto x(t,t_{0},x_{0})$, mapping initial conditions $x_{0}\in {\mathcal {P}}$ into their position $x(t,t_{0},x_{0})\in {\mathcal {P}}$ for any time $t\in {\mathcal {I}}$. If the flow map $F_{t_{0}}^{t}$ is a diffeomorphism for any choice of $t\in {\mathcal {I}}$, then for any smooth set ${\mathcal {M}}(t_{0})$ of initial conditions in ${\mathcal {P}}$, the set

${\mathcal {M}}=\{(x,t)\in {\mathcal {P}}\times {\mathcal {I}}\,\colon [F_{t_{0}}^{t}]^{-1}(x)\in {\mathcal {M}}(t_{0})\}$

is an invariant manifold in the extended phase space ${\mathcal {P}}\times {\mathcal {I}}$. Borrowing terminology from fluid dynamics, we refer to the evolving time slice ${\mathcal {M}}(t)=F_{t_{0}}^{t}({\mathcal {M}}(t_{0}))$ of the manifold ${\mathcal {M}}$ as a material surface (see Fig. 1). Since any choice of the initial condition set ${\mathcal {M}}(t_{0})$ yields an invariant manifold ${\mathcal {M}}\in {\mathcal {P}}\times {\mathcal {I}}$, invariant manifolds and their associated material surfaces are abundant and generally undistinguished in the extended phase space. Only few of them will act as cores of coherent trajectory patterns.

LCSs as exceptional material surfaces

In order to create a coherent pattern, a material surface ${\mathcal {M}}(t)$ should exert a sustained and consistent action on nearby trajectories throughout the time interval ${\mathcal {I}}$. Examples of such action are attraction, repulsion, or shear. In principle, any well-defend mathematical property qualifies that creates coherent patterns out of randomly selected nearby initial conditions.

Most such properties can be expressed by strict inequalities. For instance, we call a material surface ${\mathcal {M}}(t)$ attracting over the interval ${\mathcal {I}}$ if all small enough initial perturbations to ${\mathcal {M}}(t_{0})$ are carried by the flow into even smaller final perturbations to ${\mathcal {M}}(t_{1})$. In classical dynamical systems theory, invariant manifolds satisfying such an attraction property over infinite times are called attractors. They are not only special, but even locally unique in the phase space: no continuous family of attractors may exist.

In contrast, in dynamical systems defined over a finite time interval ${\mathcal {I}}$, strict inequalities do not define exceptional (i.e., locally unique) material surfaces. This follows from the continuity of the flow map $F_{t_{0}}^{t}$ over ${\mathcal {I}}$ . For instance, if a material surface ${\mathcal {M}}(t)$ attracts all nearby trajectories over the time interval ${\mathcal {I}}$, then so will any sufficiently close other material surface.

Thus, attracting, repelling and shearing material surfaces are necessarily stacked on each other, e.g., occur in continuous families. This leads to the idea of seeking LCSs in finite-time dynamical systems as exceptional material surfaces that exhibit a coherence-inducing property more strongly than any of the neighboring material surfaces. Such LCSs, defined as extrema (or more generally, stationary surfaces) for a finite-time coherence property, will indeed serve as observed centerpieces of trajectory patterns.

Objectivity (frame-indifference) of LCSs

Assume that the phase space of the underlying dynamical system is the material configuration space of a continuum, such as a fluid or a deformable body. For instance, for a dynamical system generated by an unsteady velocity field

$v=v(x,t),\qquad x\in U\subset {\mathbb {R} }^{3},$

the open set $U$ of possible particle positions is a material configuration space. In this space, LCSs form the skeleton of the material response by the continuum, subject to external forces and constitutive properties of the material. A fundamental requirement for the self-consistent description of any material response in continuum mechanics is objectivity (material frame-indifference). Objectivity stipulates that the material response (and hence the shape and the type of LCSs framing the material response) must be invariant with respect to Euclidean changes of coordinates

$x=Q(t)y+b(t),$

where $y\in {\mathbb {R} }^{3}$ is the vector of the trasnsformed coordinates; $Q(t)$ is an arbitrary $3\times 3$ proper orthogonal matrix representing time-dependent rotations; and $b(t)$ is an arbitrary $3$-dimensional vector representing time-dependent translations. As a consequence, any self-consistent LCS definition or criterion should be expressible in terms of quantities that are frame-invariant. For instance, the rate of strain $S(x,t)$ and the vorticity tensor $W(x,t)$ defined as

$S(x,t)={\frac {1}{2}}\left(\nabla v(x,t)+(\nabla v(x,t))^{T}\right),\qquad W(x,t)={\frac {1}{2}}\left(\nabla v(x,t)-(\nabla v(x,t))^{T}\right),$

transform under Euclidean changes of frame into the quantities

${\tilde {S}}(y,t)=Q(t)^{T}S(x,t)Q(t),\qquad {\tilde {W}}(y,t)=Q(t)^{T}S(x,t)Q(t)-Q(t)^{T}{\dot {Q}}(t).$

A Euclidean frame change is, therefore, equivalent to a similarity transform for $S(x,t)$, and hence an LCS approach depending only on the eigenvalues and eigenvectors of $S(x,t)$ is automatically frame-invariant. In contrast, an LCS approach depending on the eigenvalues of $W(x,t)$ is not guaranteed to be frame-invariant.

A number of frame-independent quantities, such as $\nabla v(x,t)$, ${W}(y,t)$, $\nabla F_{t_{0}}^{t}$, as well as the averages or eigenvalues of these quantities, are routinely used in heuristic LCS detection. While such quantitiesmay effectively mark features of the instantaneous velocity field $v(x,t)$, the ability of these quantities to capture material mixing, transport, and coherence is limited and a priori unknown in any given frame. As an example, consider the linear unsteady fluid particle motion

${\dot {x}}=v(x,t)={\begin{pmatrix}\sin {4t}&2+\cos {4t}\\-2+\cos {4t}&-\sin {4t}\end{pmatrix}}x,$

which is an exact solution of the two-dimensional Navier--Stokes equations. The (frame-dependent) Okubo-Weiss criterion classifies the whole domain in this flow as elliptic (vortical) because $q={\frac {1}{2}}({\vert S\vert }^{2}-{\vert W\vert }^{2})<0$ holds, with $\vert \,\cdot \,\vert$ referring to the Euclidean matrix norm. As seen in Fig. 2, however, trajectories grow exponentially along a rotating line and shrink exponentially along another rotating line. In material terms, therefore, the flow is hyperbolic (saddle-type) in any frame.

Figure 2: Instantaneous streamlines and the evolution of trajectories starting from the interior of one of them in a linear solution of the Navier--Stokes equation. This dynamical system is classified as elliptic by a number of frame-dependent coherence diagnostics, such as the Okubo--Weiss criterion. (Image: Francisco Beron--Vera)

Hyperbolic LCS

Figure 3. Attracting and repelling LCSs in the extended phase space of a two-dimensional dynamical system.

Motivated by the above discussion, the simplest way to define an attracting LCS is by requiring it to be a locally strongest attracting material surface in the extended phase space ${\mathcal {P}}\times {\mathcal {I}}$ (see. Fig. 3) . Similarly, a repelling LCS can be defined as a locally strongest repelling material surface. Attracting and repelling LCSs together are usually referred to as hyperbolic LCSs, as they provide a finite-time genearalization of the classic concept of normally hyperbolic invariant manifolds in dynamical systems.

A simple approach to hyperbolic LCS: Finite-time Lyapunov eponents

Heuristically, one may seek initial positions ${\mathcal {M}}(t_{0})$ of repelling LCSs as set of initial conditions at which infinitesimal perturbations to trajectories starting from ${\mathcal {M}}(t_{0})$ grow locally at the highest rate relative to trajectories starting off of ${\mathcal {M}}(t_{0})$. The heuristic element here is that instead of constructing a highly repelling material surface, one simply seeks points of large particle separation. Such a separation may well be due to strong shear along the set of points so identified; this set is not at all guaranteed to exert any normal repulsion on nearby trajectories.

The growth of an infinitesimal perturbation ${\xi }(t)$ along a trajectory $x(t,t_{0},x_{0})$ is governed by the flow map gradient $\nabla F_{t_{0}}^{t}$. Let $\epsilon {\xi }(t_{0})$ be a small perturbation to the initial condition $x_{0}$, with $0<\epsilon \ll 1$, and with ${\xi }(t_{0})$ denoting an arbitrary unit vector in ${\mathbb {R} }^{n}$. This perturbation generally grows along the trajectory $x(t,t_{0},x_{0})$ into the perturbation vector ${\xi }_{\epsilon }(t_{1};x_{0})=\nabla F_{t_{0}}^{t_{1}}(x_{0})\epsilon {\xi }(t_{0})$. Then the maximum relative stretching of infinitesimal perturbations at the point $x_{0}$ can be computed as

$\delta _{t_{0}}^{t_{1}}(x_{0})=\lim _{\epsilon \to 0}{\frac {1}{\epsilon }}\max _{\left|\xi (t_{0})\right|=1}\left|\xi _{\epsilon }(t_{1};x_{0})\right|=\max _{\left|\xi (t_{0})\right|=1}{\sqrt {\left\langle \nabla F_{t_{0}}^{t_{1}}(x_{0})\xi (t_{0}),\nabla F_{t_{0}}^{t_{1}}(x_{0})\xi (t_{0})\right\rangle }}=\max _{\left|\xi (t_{0})\right|=1}{\sqrt {\left\langle \xi (t_{0}),\left[\nabla F_{t_{0}}^{t_{1}}(x_{0})\right]^{T}\nabla F_{t_{0}}^{t_{1}}(x_{0})\xi (t_{0})\right\rangle }}=\max _{\left|\xi (t_{0})\right|=1}{\sqrt {\left\langle \xi (t_{0}),C_{t_{0}}^{t_{1}}(x_{0})\xi (t_{0})\right\rangle }},$

where $C_{t_{0}}^{t_{1}}=\left[\nabla F_{t_{0}}^{t_{1}}\right]^{T}\nabla F_{t_{0}}^{t_{1}}$ denotes the right Cauchy--Green strain tensor. One then concludes that the maximum relative stretching experienced along a trajectory starting from $x_{0}$ is just $\delta _{t_{0}}^{t_{1}}(x_{0})={\sqrt {\lambda _{n}(x_{0})}}$. As this relative stretching tends to grow rapidly, it is more convenient to work with its growth exponent $(\log {\delta _{t_{0}}^{t_{1}}})/(t_{1}-t_{0})$, which is then precisely the finite-time Lyapunov exponent (FTLE)

$\mathrm {FTLE} _{t_{0}}^{t_{1}}(x_{0})={\frac {1}{2(t_{1}-t_{0})}}\log \lambda _{n}(x_{0}).$

Figure 4a. Attracting (red) and repelling (blue) LCSs extracted as FTLE ridges from a two-dimensional turbulence experiment (Image: Manikandan Mathur)

Therefore, one expects hyperbolic LCSs to appear as codimension-one local maximizing surfaces (or ridges) of the FTLE field. This expectation turns out to be justified in the majority of cases: time $t_{0}$ positions of repelling LCSs are marked by ridges of $\mathrm {FTLE} _{t_{0}}^{t_{1}}(x_{0})$. By applying the same argument in backward time, we obtain that time $t_{1}$ positions of attracting LCSs are marked by ridges of the backward FTLE field $\mathrm {FTLE} _{t_{1}}^{t_{0}}$.

The classic way of computing Lyapunov exponents is solving a linear differential equation for the linearized flow map $\nabla F_{t_{0}}^{t}(x_{0})$. A more expedient approach is to compute the FTLE field from a simple finite-difference approximation to the deformation gradient. For example, in a three-dimenisonal flow, we launch a trajectory $x(t;t_{0},x_{0})$ from any element $x_{0}$ of a grid of initial conditions. Using the coordinate representation $x=(x^{1},x^{2},x^{3})$ for the evolving trajectory $x(t;t_{0},x_{0})$, we approximate the gradient of the flow map as

$\nabla F_{t_{0}}^{t}(x_{0})\approx {\begin{pmatrix}{\frac {x^{1}(t;t_{0},x_{0}+\delta _{1})-x^{1}(t;t_{0},x_{0}-\delta _{1})}{\left|2\delta _{1}\right|}}&{\frac {x^{1}(t;t_{0},x_{0}+\delta _{2})-x^{1}(t;t_{0},x_{0}-\delta _{2})}{\left|2\delta _{2}\right|}}&{\frac {x^{1}(t;t_{0},x_{0}+\delta _{3})-x^{1}(t;t_{0},x_{0}-\delta _{3})}{\left|2\delta _{3}\right|}}\\{\frac {x^{2}(t;t_{0},x_{0}+\delta _{1})-x^{2}(t;t_{0},x_{0}-\delta _{1})}{\left|2\delta _{1}\right|}}&{\frac {x^{2}(t;t_{0},x_{0}+\delta _{2})-x^{2}(t;t_{0},x_{0}-\delta _{2})}{\left|2\delta _{2}\right|}}&{\frac {x^{2}(t;t_{0},x_{0}+\delta _{3})-x^{2}(t;t_{0},x_{0}-\delta _{3})}{\left|2\delta _{3}\right|}}\\{\frac {x^{3}(t;t_{0},x_{0}+\delta _{1})-x^{3}(t;t_{0},x_{0}-\delta _{1})}{\left|2\delta _{1}\right|}}&{\frac {x^{3}(t;t_{0},x_{0}+\delta _{2})-x^{3}(t;t_{0},x_{0}-\delta _{2})}{\left|2\delta _{2}\right|}}&{\frac {x^{3}(t;t_{0},x_{0}+\delta _{3})-x^{3}(t;t_{0},x_{0}-\delta _{3})}{\left|2\delta _{3}\right|}}\end{pmatrix}},$

with a small vector $\delta _{i}$ pointing in the $x^{i}$ coordinate direction. For two-dimensional flows, only the first $2\times 2$ minor matrix of the above matrix is relevant.

Figure 4b. Attracting (red) and repelling (blue) LCSs extracted as FTLE ridges from a two-dimensional simulation of a von Karman vortex street (Image: Jens Kasten)

Figure 5. FTLE ridges highlight both hyperbolic LCS and shearing material lines, such as the boundaries of a riverbed in a 3D model of the New River Inlet, Onslow, North Carolina (Image: Allen Sanderson).

FTLE ridges have proven to be a simple and efficient tool for the visualize hyperbolic LCSs in a number of physical problems, yielding intriguing images of initial positions of hyperbolic LCSs in different applications (see, e.g., Fig. 4). However, FTLE ridges obtained over sliding time windows $[t_{0}+T,t_{1}+T]$ do not form material surfaces. Thus, ridges of $\mathrm {FTLE} _{t_{0}+T}^{t_{1}+T}(x_{0})$ under varying $T$ cannot be used to define Lagrangian objects, such as hyperbolic LCSs. Indeed, a locally strongest repelling material surface over $[t_{0},t_{1}]$ will generally not play the same role over $[t_{0}+T,t_{1}+T]$ and hence its evolving position at time $t_{0}+T$ will not be a ridge for $\mathrm {FTLE} _{t_{0}+T}^{t_{1}+T}$. Nonetheless, evolving second-derivative FTLE ridges computed over sliding intervals of the form $[t_{0}+T,t_{1}+T]$ have been identified by some authors broadly with LCSs. In support of this identification. it is also often argued that the material flux over such sliding-window FTLE ridges should necessarily be small. This "FTLE ridge=LCS" identification would create several conceptual and mathematical inconsistencies:

• Second-derivative FTLE ridges are necessarily straight lines and hence do not exist in physical problems.
• FTLE ridges computed over sliding time windows $[t_{0}+T,t_{1}+T]$ with a varying $T$ are generally not Lagrangian and the flux though them may well be large
• FTLE ridges mark hyperbolic LCS positions, but also highlight surfaces of high shear. A convoluted mixture of both types of surfaces often arises in applications (see Fig. 5 for an example).
• There are several other types LCSs (elliptic and parabolic) beyond the hyperbolic LCSs highlighted by FTLE ridges

Local variational approach to hyperbolic LCS: shrink and stretch surfaces

Figure 6. Linearized flow geometry along an evolving material surface.

The local variational theory of hyperbolic LCSs builds on their original definition as strongest repelling, attracting or shearing material surfaces in the flow over the time interval $[t_{0},t_{1}]$. At an initial point $x_{0}$ , let $n_{0}$ denote a unit normal to an initial material surface ${\mathcal {M}}(t_{0})$ (cf. Fig. 6). By the invariance of material lines, the tangent space $T_{x_{0}}{\mathcal {M}}(t_{0})$ is mapped into the tangent space of $T_{x_{1}}{\mathcal {M}}(t_{1})$ by the linearized flow map $\nabla F_{t_{0}}^{t_{1}}(x_{0})$. At the same time, the image of the normal $m$ normal under $\nabla F_{t_{0}}^{t_{1}}(x_{0})$ generally does not remain normal to ${\mathcal {M}}(t_{1})$. Therefore, in addition to a normal component of length $\rho (x_{0,}n_{0})$, the advected normal also develops a tangential component of length $\sigma (x_{0},n_{0})$ (cf. Fig. 6).

If $\rho (x_{0},n_{0})>1$, then the evolving material surface ${\mathcal {M}}(t)$ strictly repels nearby trajectories by the end of the time interval $[t_{0},t_{1}]$. Similarly, $\rho (x_{0},n_{0})<1$ signals that ${\mathcal {M}}(t)$ strictly attracts nearby trajectories along its normal directions. A repelling (attracting) LCS over the interval $[t_{0},t_{1}]$ can be defined as a material surface ${\mathcal {M}}(t)$ whose net repulsion $\rho (x_{0},n_{0})$ is pointwise maximal (minimal) with respect to perturbations of the initial normal vector field $n_{0}$. As earlier, we refer to repelling and attracting LCSs collectively as hyperbolic LCSs.

Solving these local extremum principles for hyperbolic LCSs in two and three dimensions yields unit normal vector fields to which hyperbolic LCSs should everywhere be tangent. The existence of such normal surfaces also requires a Frobenius-type integrability condition in the three-dimensional case. All these results can be summarized as follows:

Hyperbolic LCS conditions from local variational theory in dimensions n=2 and n=3

LCS	Normal vector field of ${\mathcal {M}}(t_{0})$ for $n=2,3$	ODE for ${\mathcal {M}}(t_{0})$ for n=2	Frobenius-type PDE for ${\mathcal {M}}(t_{0})$ for n=3
Attracting	$\xi _{1}(x_{0})$	$x_{0}^{\prime }=\xi _{2}(x_{0})$ (stretch lines)	$\left\langle \nabla \times \xi _{1}(x_{0}),\xi _{1}(x_{0})\right\rangle =0$ (stretch surfaces)
Repelling	$\xi _{n}(x_{0})$	$x_{0}^{\prime }=\xi _{1}(x_{0})$ (shrink lines)	$\left\langle \nabla \times \xi _{3}(x_{0}),\xi _{3}(x_{0})\right\rangle =0$ (shrink surfaces)

Repelling LCSs are obtained as most repelling shrink lines, starting from local maxima of $\lambda _{2}(x_{0})$. Attracting LCSs are obtained as most attracting stretch lines, starting from local minima of $\lambda _{1}(x_{0})$. These starting points serve are initial positions of exceptional saddle-type trajectories in the flow. An example of the local variational computation of a repelling LCS is shown in FIg. 7. The computational algorithm is available in LCS Tool.

Figure 7. A repelling LCS visualized as an FTLE ridge (left) and computed exactly as a shrink line (right), i.e., a solution of the ODE $x_{0}^{\prime }=\xi _{1}(x_{0})$ starting from a global maximum of $\lambda _{2}(x_{0})$.

Relation to classical invariant manifolds

Classical invariant manifolds are invariant sets in the phase space ${\mathcal {P}}$ of an autonomous dynamical system. In contrast, LCSs are only required to be invariant in the extended phase space. This means that even if the underlying dynamical system is autonomous, the LCSs of the system over the interval $I$ will generally be time-dependent, acting as the evolving skeletons of evolving coherent patterns.

See also

• Turbulence
• Lagrangian mechanics
• Chaos theory
• Dynamical systems theory

Software packages for LCS

Software packages have been developed to perform Lagrangian coherent structures computations:

• ManGen (source code)
• LCS MATLAB Kit (source code)
• FlowVC (source code)
• LCS Tool
• cuda_ftle (source code)
• CTRAJ
• Newman (source code)
• FlowTK (source code)

Lists of software packages for Lagrangian coherent structures computations have been created by Jerrold E. Marsden and Steven K. Baum.