Lagrangian coherent structure

Material surfaces 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_{t_0}\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_{t_0} 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_{t_0}]^{-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_{t_0}(\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-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 \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_{t_0} 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, 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 \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 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. is automatically frame-invariant. In contrast, an LCS approach depending on the eigenvalues of W(x,t) is generally not frame-invariant. A number of frame-dependent quantities, such as \nabla v(x,t), {W}(y,t), \nabla F^t_{t_0}, 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), 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) holds, with \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-frame to a y-frame through a coordinate change 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) to hold for all times. Temporal differentiation of this identity and substitution into the original equation in the x-frame then yields the transformed equation in the 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-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 \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, 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_{t_0}. Let \epsilon{\xi}(t_0) be a small perturbation to the initial condition x_0, with 0, and with \xi(t_0) denoting an arbitrary unit vector in \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_1}_{t_0}(x_0)\epsilon{\xi}(t_0). Then the maximum relative stretching of infinitesimal perturbations at the point x_0 can be computed as \begin{align} \delta^{t_1}_{t_0}(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{align} where C^{t_1}_{t_0}=\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 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}_{t_0}(x_0). A more expedient approach is to compute the FTLE field from a simple finite-difference approximation to the deformation gradient. 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. Issues with inferring hyperbolic LCSs from FTLE ridges 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., Figs. 5a-b). 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. The "FTLE ridge=LCS" identification, • FTLE ridges computed over sliding time windows [t_0+T,t_1+T] with a varying T are generally not Lagrangian and the flux through them is generally not small. • In particular, a broadly referenced material flux formula 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: D_{t_0}^{t_1}(x_0)=\frac{1}{2}\left[C_{t_0}^{t_1}(x_0)\Omega-\Omega C_{t_0}^{t_1}(x_0)\right], \qquad \Omega = \begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix} . Such null-geodesics can be proven to be tensorlines of the Cauchy–Green strain tensor, i.e., are tangent to the direction field formed by the strain eigenvector fields \xi_i(x_0). Specifically, repelling LCSs are trajectories of x_0^{\prime} =\xi_1(x_0) starting from local maxima of the \lambda_2(x_0) eigenvalue field. Similarly, attracting LCSs are trajectories of x_0^{\prime}=\xi_2(x_0) starting from local minims of the \lambda_1(x_0) eigenvalue field. This agrees with the conclusion of the local variational theory of LCSs. The geodesic approach, however, also sheds more light on the robustness of hyperbolic LCSs: hyperbolic LCSs only prevail as stationary curves of the averaged shear functional under variations that leave their endpoints fixed. This is to be contrasted with parabolic LCSs (see below), which are also shearless LCSs but prevail as stationary curves to the shear functional even under arbitrary variations. As a consequence, individual trajectories are objective, and statements about the coherent structures they form should also be objective. A sample application is shown in Fig. 9, where the sudden appearance of a hyperbolic core (strongest attracting part of a stretchline) within the oil spill caused the notable Tiger-Tail instability in the shape of the oil spill. == Elliptic LCSs ==

Elliptc LCSs are closed and nested material surfaces that act as building blocks of the Lagrangian equivalents of vortices, i.e., rotation-dominated regions of trajectories that generally traverse the phase space without substantial stretching or folding. They mimic the behavior of Kolmogorov–Arnold–Moser (KAM) tori that form elliptic regions in Hamiltonian systems. There coherence can be approached either through their homogeneous material rotation or through their homogeneous stretching properties. Rotational coherence from the polar rotation angle (PRA) As a simplest approach to rotational coherence, one may define an elliptic LCS as a tubular material surface along which small material volumes complete the same net rotation over the time intervall [t_0 ,t_1] of interest. A challenge in that in each material volume element, all individual material fibers (tangent vectors to trajectories) perform different rotations. To obtain a well-defined bulk rotation for each material element, one may employ the unique left and right polar decompositions of the flow gradient in the form \nabla F^{t_1}_{t_0}=R^{t_1}_{t_0}U^{t_1}_{t_0}=V^{t_1}_{t_0}R^{t_1}_{t_0}, where the proper orthogonal tensor R^{t_1}_{t_0} is called the rotation tensor and the symmetric, positive definite tensors U^{t_1}_{t_0},V^{t_1}_{t_0} are called the left stretch tensor and right stretch tensor, respectively. Since the Cauchy–Green strain tensor can be written as C^{t_1}_{t_0}=[\nabla F^{t_1}_{t_0}]^T\nabla F^{t_1}_{t_0}=U^{t_1}_{t_0}U^{t_1}_{t_0}=V^{t_1}_{t_0}V^{t_1}_{t_0}, the local material straining described by the eigenvalues and eigenvectors of C^{t_1}_{t_0} are fully captured by the singular values and singular vectors of the stretch tensors. The remaining factor in the deformation gradient is represented by R^{t_1}_{t_0}, interpreted as the bulk solid-body rotation component of volume elements. In planar motions, this rotation is defined relative to the normal of the plane. In three dimensions, the rotation is defined relative to the axis defined by the eigenvector of R^{t_1}_{t_0} corresponding to its unit eigenvalue. In higher-dimensional flows, the rotation tensor cannot be viewed as a rotation about a single axis. File:PRA for 2D turbulence.pdf|250px|right|thumb|Figure 10a. Elliptic LCSs revealed by closed level curves of the PRA distribution in a two-dimensional turbulence simulation. (Image: Mohammad Farazmand) Therefore, while R_{t_0}^{t_1} is the closest rotation tensor to \nabla F^{t_1}_{t_0} in the L^2 norm over a fixed time interval [t_0,t_1] , these piecewise best fits do not form a family of rigid-body rotations as t_0 and t_1 are varied. For this reason, rotations predicted by the polar rotation tensor over varying time intervals divert from the experimentally observed mean material rotation of fluid elements. of Jupiter. == Parabolic LCSs ==

Parabolic LCSs are shearless material surfaces that delineate cores of jet-type sets of trajectories. Such LCSs are characterized by both low stretching (because they are inside a non-stretching structure), but also by low shearing (because material shearing is minimal in jet cores). Diagnostic approach: Finite-time Lyapunov exponents (FTLE) trenches Since both shearing and stretching are as low as possible along a parabolic LCS, one may seek initial positions of such material surfaces as trenches of the FTLE field FTLE^{t_1}_{t_0}(x_0). A geophysical example of a parabolic LCS (generalized jet core) revealed as a trench of the FTLE field is shown in Fig. 14a. Global variational approach: Heteroclinic chains of null-geodesics In two dimensions, parabolic LCSs are also solutions of the global shearless variational principle described above for hyperbolic LCSs. As such, parabolic LCSs are composed of shrink lines and stretch lines that represent geodesics of the Lorentzian metric tensor D^{t_1}_{t_0}. In contrast to hyperbolic LCSs, however, parabolic LCSs satisfy more robust boundary conditions: they remain stationary curves of the material-line-averaged shear functional even under variations to their endpoints. This explains the high degree of robustness and observability that jet cores exhibit in mixing. This is to be contrasted with the highly sensitive and fading footprint of hyperbolic LCSs away from strongly hyperbolic regions in diffusive tracer patterns. Under variable endpoint boundary conditions, initial positions of parabolic LCSs turn out to be alternating chains of shrink lines and stretch lines that connect singularities of these line fields. These singularities occur at points where \lambda_1(x_0)=\lambda_2(x_0), and hence no infinitesimal deformation takes place between the two time instances t_0 and t_1. Fig. 14b shows an example of parabolic LCSs in Jupiter's atmosphere, located using this variational theory. The chevron-type shapes forming out of circular material blobs positioned along the jet core is characteristic of tracer deformation near parabolic LCSs. == Software packages for LCS computations ==

Particle advection and Finite-Time Lyapunov Exponent calculation: • ManGen (source code) • LCS MATLAB Kit (source code) • FlowVC (source code) • cuda_ftle (source code) • CTRAJ • Newman (source code) • FlowTK ([//github.com/FlowPhysics/FlowTK source code]) Jupyter notebooks that guide you through methods used to extract advective, diffusive, stochastic and active transport barriers from discrete velocity data. • TBarrier (source code) == See also ==