Lagrangian coherent structures (LCSs) 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 P {\displaystyle {\mathcal {P}}} and over a time interval I = [ t 0 , t 1 ] {\displaystyle {\mathcal {I}}=[t_{0},t_{1}]} , consider a non-autonomous dynamical system defined through the flow map F t 0 t : x 0 ↦ x ( t , t 0 , x 0 ) {\displaystyle F_{t_{0}}^{t}\colon x_{0}\mapsto x(t,t_{0},x_{0})} , mapping initial conditions x 0 ∈ P {\displaystyle x_{0}\in {\mathcal {P}}} into their position x ( t , t 0 , x 0 ) ∈ P {\displaystyle x(t,t_{0},x_{0})\in {\mathcal {P}}} for any time t ∈ I {\displaystyle t\in {\mathcal {I}}} . If the flow map F t 0 t {\displaystyle F_{t_{0}}^{t}} is a diffeomorphism for any choice of t ∈ I {\displaystyle t\in {\mathcal {I}}} , then for any smooth set M ( t 0 ) {\displaystyle {\mathcal {M}}(t_{0})} of initial conditions in P {\displaystyle {\mathcal {P}}} , the set
M = { ( x , t ) ∈ P × I : [ F t 0 t ] − 1 ( x ) ∈ M ( t 0 ) } {\displaystyle {\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 P × I {\displaystyle {\mathcal {P}}\times {\mathcal {I}}} . Borrowing terminology from fluid dynamics, we refer to the evolving time slice M ( t ) = F t 0 t ( M ( t 0 ) ) {\displaystyle {\mathcal {M}}(t)=F_{t_{0}}^{t}({\mathcal {M}}(t_{0}))} of the manifold M {\displaystyle {\mathcal {M}}} as a material surface (see Fig. 1). Since any choice of the initial condition set M ( t 0 ) {\displaystyle {\mathcal {M}}(t_{0})} yields an invariant manifold M ∈ P × I {\displaystyle {\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 M ( t ) {\displaystyle {\mathcal {M}}(t)} should exert a sustained and consistent action on nearby trajectories throughout the time interval I {\displaystyle {\mathcal {I}}} . 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 M ( t ) {\displaystyle {\mathcal {M}}(t)} attracting over the interval I {\displaystyle {\mathcal {I}}} if all small enough initial perturbations to M ( t 0 ) {\displaystyle {\mathcal {M}}(t_{0})} are carried by the flow into even smaller final perturbations to M ( t 1 ) {\displaystyle {\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 I {\displaystyle {\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 {\displaystyle F_{t_{0}}^{t}} over I {\displaystyle {\mathcal {I}}} . For instance, if a material surface M ( t ) {\displaystyle {\mathcal {M}}(t)} attracts all nearby trajectories over the time interval I {\displaystyle {\mathcal {I}}} , 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 (or more generally, stationary surfaces) 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 P {\displaystyle {\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 {\displaystyle I} 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
v = v ( x , t ) , x ∈ U ⊂ R 3 , {\displaystyle v=v(x,t),\qquad x\in U\subset \mathbb {R} ^{3},}
the open set U {\displaystyle U} 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 (material frame-indifference) 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
x = Q ( t ) y + b ( t ) , {\displaystyle x=Q(t)y+b(t),}
where y ∈ R 3 {\displaystyle y\in \mathbb {R} ^{3}} is the vector of the transformed coordinates; Q ( t ) {\displaystyle Q(t)} is an arbitrary 3 × 3 {\displaystyle 3\times 3} proper orthogonal matrix representing time-dependent rotations; and b ( t ) {\displaystyle b(t)} is an arbitrary 3 {\displaystyle 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 strain rate S ( x , t ) {\displaystyle S(x,t)} and the spin tensor W ( x , t ) {\displaystyle W(x,t)} defined as
S ( x , t ) = 1 2 ( ∇ v ( x , t ) + ( ∇ v ( x , t ) ) T ) , W ( x , t ) = 1 2 ( ∇ v ( x , t ) − ( ∇ v ( x , t ) ) T ) , {\displaystyle 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
S ~ ( y , t ) = Q ( t ) T S ( x , t ) Q ( t ) , W ~ ( y , t ) = Q ( t ) T W ( x , t ) Q ( t ) − Q ( t ) T Q ˙ ( t ) . {\displaystyle {\tilde {S}}(y,t)=Q(t)^{T}S(x,t)Q(t),\qquad {\tilde {W}}(y,t)=Q(t)^{T}W(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 ) {\displaystyle S(x,t)} , and hence an LCS approach depending only on the eigenvalues and eigenvectors of S ( x , t ) {\displaystyle S(x,t)} is automatically frame-invariant. In contrast, an LCS approach depending on the eigenvalues of W ( x , t ) {\displaystyle W(x,t)} is generally not frame-invariant. A number of frame-dependent quantities, such as ∇ v ( x , t ) {\displaystyle \nabla v(x,t)} , W ( y , t ) {\displaystyle {W}(y,t)} , ∇ F t 0 t {\displaystyle \nabla F_{t_{0}}^{t}} , 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 v ( x , t ) {\displaystyle 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
x ˙ = v ( x , t ) = ( sin 4 t 2 + cos 4 t − 2 + cos 4 t − sin 4 t ) x , {\displaystyle {\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 = 1 2 ( | S | 2 − | W | 2 ) < 0 {\displaystyle q={\frac {1}{2}}({\vert S\vert }^{2}-{\vert W\vert }^{2})<0} holds, with | ⋅ | {\displaystyle \vert \,\cdot \,\vert } 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 (saddle-type) 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 x {\displaystyle x} -frame to a y {\displaystyle y} -frame through a coordinate change x = Q ( t ) y + b ( t ) {\displaystyle x=Q(t)y+b(t)} precisely by upholding that trajectories are mapped into trajectories, i.e., by requiring x ( t ) = Q ( t ) y ( t ) + b ( t ) {\displaystyle x(t)=Q(t)y(t)+b(t)} to hold for all times. Temporal differentiation of this identity and substitution into the original equation in the x {\displaystyle x} -frame then yields the transformed equation in the y {\displaystyle y} -frame. While this process adds new terms (inertial forces) 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 y {\displaystyle y} -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 P × I {\displaystyle {\mathcal {P}}\times {\mathcal {I}}} (see. Fig. 4) . 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 M ( t 0 ) {\displaystyle {\mathcal {M}}(t_{0})} of repelling LCSs as set of initial conditions at which infinitesimal perturbations to trajectories starting from M ( t 0 ) {\displaystyle {\mathcal {M}}(t_{0})} grow locally at the highest rate relative to trajectories starting off of M ( t 0 ) {\displaystyle {\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 ξ ( t ) {\displaystyle {\xi }(t)} along a trajectory x ( t , t 0 , x 0 ) {\displaystyle x(t,t_{0},x_{0})} is governed by the flow map gradient ∇ F t 0 t {\displaystyle \nabla F_{t_{0}}^{t}} . Let ϵ ξ ( t 0 ) {\displaystyle \epsilon {\xi }(t_{0})} be a small perturbation to the initial condition x 0 {\displaystyle x_{0}} , with 0 < ϵ ≪ 1 {\displaystyle 0<\epsilon \ll 1} , and with ξ ( t 0 ) {\displaystyle \xi (t_{0})} denoting an arbitrary unit vector in R n {\displaystyle \mathbb {R} ^{n}} . This perturbation generally grows along the trajectory x ( t , t 0 , x 0 ) {\displaystyle x(t,t_{0},x_{0})} into the perturbation vector ξ ϵ ( t 1 ; x 0 ) = ∇ F t 0 t 1 ( x 0 ) ϵ ξ ( t 0 ) {\displaystyle {\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 {\displaystyle x_{0}} can be computed as
δ t 0 t 1 ( x 0 ) = lim ϵ → 0 1 ϵ max | ξ ( t 0 ) | = 1 | ξ ϵ ( t 1 ; x 0 ) | = max | ξ ( t 0 ) | = 1 ⟨ ∇ F t 0 t 1 ( x 0 ) ξ ( t 0 ) , ∇ F t 0 t 1 ( x 0 ) ξ ( t 0 ) ⟩ = max | ξ ( t 0 ) | = 1 ⟨ ξ ( t 0 ) , C t 0 t 1 ( x 0 ) ξ ( t 0 ) ⟩ {\displaystyle {\begin{aligned}\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}),C_{t_{0}}^{t_{1}}(x_{0})\xi (t_{0})\right\rangle }}\\\end{aligned}}}
where C t 0 t 1 = [ ∇ F t 0 t 1 ] T ∇ F t 0 t 1 {\displaystyle 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 {\displaystyle x_{0}} is just δ t 0 t 1 ( x 0 ) = λ n ( x 0 ) {\displaystyle \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 δ t 0 t 1 ) / ( t 1 − t 0 ) {\displaystyle (\log {\delta _{t_{0}}^{t_{1}}})/(t_{1}-t_{0})} , which is then precisely the finite-time Lyapunov exponent (FTLE)
F T L E t 0 t 1 ( x 0 ) = 1 2 ( t 1 − t 0 ) log λ n ( x 0 ) . {\displaystyle \mathrm {FTLE} _{t_{0}}^{t_{1}}(x_{0})={\frac {1}{2(t_{1}-t_{0})}}\log \lambda _{n}(x_{0}).}
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 {\displaystyle t_{0}} positions of repelling LCSs are marked by ridges of F