Lagrangian coherent structure

Lagrangian coherent structures are distinguished surfaces 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.
As illustrated on the right, 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. The robustness of this skeleton makes LCSs ideal tools for model validation, model comparison and benchmarking. LCSs can also be used for now-casting and even short-term forecasting of pattern evolution in complex dynamical systems.
Physical phenomena governed by LCSs 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. It has been used by underwater glider for efficient ocean navigation, and is hypothesized to be used by albatross for foraging.
While LCSs generally exist in any dynamical system, their role in creating coherent patterns is perhaps most readily observable in fluid flows.

General definitions

Material surfaces

On a phase space and over a time interval , consider a non-autonomous dynamical system defined through the flow map, mapping initial conditions into their position for any time. If the flow map is a diffeomorphism for any choice of, then for any smooth set of initial conditions in, the set
is an invariant manifold in the extended phase space. Borrowing terminology from fluid dynamics, we refer to the evolving time slice of the manifold as a material surface. Since any choice of the initial condition set yields an invariant manifold, 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 should exert a sustained and consistent action on nearby trajectories throughout the time interval. Examples of such action are attraction, repulsion, or shear. In principle, any well-defined 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 attracting over the interval if all small enough initial perturbations to are carried by the flow into even smaller final perturbations to. 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, strict inequalities do not define exceptional material surfaces. This follows from the continuity of the flow map over. For instance, if a material surface attracts all nearby trajectories over the time interval, then so will any sufficiently close other material surface.
Thus, attracting, repelling and shearing material surfaces are necessarily stacked on each other, i.e., 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 for a finite-time coherence property, will indeed serve as observed centerpieces of trajectory patterns. Examples of attracting, repelling and shearing LCSs are in a direct numerical simulation of 2D turbulence are shown in Fig.2a.

LCSs vs. classical invariant manifolds

Classical invariant manifolds are invariant sets in the phase space 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 will generally be time-dependent, acting as the evolving skeletons of observed coherent trajectory patterns. Figure 2b shows the difference between an attracting LCS and a classic unstable manifold of a saddle point, for evolving times, in an autonomous dynamical system.

Objectivity 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
the open set of possible particle positions is a material configuration space. In this space, LCSs are material surfaces, formed by trajectories. Whether or not a material trajectory is contained in an LCS is a property that is independent of the choice of coordinates, and hence cannot depend of the observer. As a consequence, LCSs are subject to the basic objectivity requirement of continuum mechanics. The objectivity of LCSs requires them to be invariant with respect to all possible observer changes, i.e., linear coordinate changes of the form
where is the vector of the transformed coordinates; is an arbitrary proper orthogonal matrix representing time-dependent rotations; and is an arbitrary -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 strain rate and the spin tensor defined as
transform under Euclidean changes of frame into the quantities
A Euclidean frame change is, therefore, equivalent to a similarity transform for, and hence an LCS approach depending only on the eigenvalues and eigenvectors of is automatically frame-invariant. In contrast, an LCS approach depending on the eigenvalues of is generally not frame-invariant.
A number of frame-dependent quantities, such as,,, as well as the averages or eigenvalues of these quantities, are routinely used in heuristic LCS detection. While such quantities may effectively mark features of the instantaneous velocity field, 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
which is an exact solution of the two-dimensional Navier–Stokes equations. The Okubo-Weiss criterion classifies the whole domain in this flow as elliptic because holds, with referring to the Euclidean matrix norm. As seen in Fig. 3, however, trajectories grow exponentially along a rotating line and shrink exponentially along another rotating line. In material terms, therefore, the flow is hyperbolic in any frame.
Since Newton's equation for particle motion and the Navier–Stokes equations for fluid motion are well known to be frame-dependent, it might first seem counterintuitive to require frame-invariance for LCSs, which are composed of solutions of these frame-dependent equations. Recall, however, that the Newton and Navier–Stokes equations represent objective physical principles for material particle trajectories. As long as correctly transformed from one frame to the other, these equations generate physically the same material trajectories in the new frame. In fact, we decide how to transform the equations of motion from an -frame to a -frame through a coordinate change precisely by upholding that trajectories are mapped into trajectories, i.e., by requiring to hold for all times. Temporal differentiation of this identity and substitution into the original equation in the -frame then yields the transformed equation in the -frame. While this process adds new terms to the equations of motion, these inertial terms arise precisely to ensure the invariance of material trajectories. Fully composed of material trajectories, LCSs remain invariant in the transformed equation of motion defined in the -frame of reference. Consequently, any self-consistent LCS definition or detection method must also be frame-invariant.

Hyperbolic LCSs

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 . 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 generalization of the classic concept of normally hyperbolic invariant manifolds in dynamical systems.

Diagnostic approach: Finite-time Lyapunov exponent (FTLE) ridges

Heuristically, one may seek initial positions of repelling LCSs as set of initial conditions at which infinitesimal perturbations to trajectories starting from grow locally at the highest rate relative to trajectories starting off of. 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 along a trajectory is governed by the flow map gradient. Let be a small perturbation to the initial condition, with, and with denoting an arbitrary unit vector in. This perturbation generally grows along the trajectory into the perturbation vector. Then the maximum relative stretching of infinitesimal perturbations at the point can be computed as
where denotes the right Cauchy–Green strain tensor. One then concludes that the maximum relative stretching experienced along a trajectory starting from is just. As this relative stretching tends to grow rapidly, it is more convenient to work with its growth exponent, which is then precisely the finite-time Lyapunov exponent
Therefore, one expects hyperbolic LCSs to appear as codimension-one local maximizing surfaces of the FTLE field.
This expectation turns out to be justified in the majority of cases: time positions of repelling LCSs are marked by ridges of. By applying the same argument in backward time,
we obtain that time positions of attracting LCSs are marked by ridges of the backward FTLE field.
The classic way of computing Lyapunov exponents is solving a linear differential equation for the linearized flow map. 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-dimensional flow, we launch a trajectory from any element of a grid of initial conditions. Using the coordinate representation for the evolving trajectory, we approximate the gradient of the flow map as
with a small vector pointing in the coordinate direction. For two-dimensional flows, only the first minor matrix of the above matrix is relevant.