Lagrangian coherent structure

[1][2][3][4] 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.

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.

Physical phenomena governed by LCSs include floating debris, oil spills,[7] surface drifters[8][9] and chlorophyll patterns[10] in the ocean; clouds of volcanic ash[11] and spores in the atmosphere;[12] and coherent crowd patterns formed by humans[13] and animals.

While LCSs generally exist in any dynamical system, their role in creating coherent patterns is perhaps most readily observable in fluid flows.

In principle, any well-defined mathematical property qualifies that creates coherent patterns out of randomly selected nearby initial conditions.

In classical dynamical systems theory, invariant manifolds satisfying such an attraction property over infinite times are called attractors.

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.

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.

, the ability of these quantities to capture material mixing, transport, and coherence is limited and a priori unknown in any given frame.

Fully composed of material trajectories, LCSs remain invariant in the transformed equation of motion defined in the

[2][16] The heuristic element here is that instead of constructing a highly repelling material surface, one simply seeks points of large particle separation.

Therefore, one expects hyperbolic LCSs to appear as codimension-one local maximizing surfaces (or ridges) of the FTLE field.

A more expedient approach is to compute the FTLE field from a simple finite-difference approximation to the deformation gradient.

planes, then fitting a surface to the curve family so obtained yields a numerical approximation of a 2D repelling LCS.

The two-dimensional geodesic theory of LCSs seeks exceptionally coherent locations where this general trend fails, resulting in an order of magnitude smaller variability in shear or strain than what is normally expected across an

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.

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

In two-dimensions, therefore, (polar) elliptic LCSs are simply closed level curves of the PRA, which turn out to be objective.

[32] In three dimensions, (polar) elliptic LCSs are toroidal or cylindrical level surfaces of the PRA, which are, however, not objective and hence will generally change in rotating frames.

Coherent Lagrangian vortex boundaries can be visualized as outermost members of nested families of elliptic LCSs.

Two- and three-dimensional examples of elliptic LCS revealed by tubular level surfaces of the PRA are shown in Fig.

Unlike the classic polar decomposition, however, the dynamic rotation and stretch tensors are obtained from solving linear differential equations, rather than from matrix manipulations.

This result applies both in two- and three dimensions, and enables the computation of a well-defined, objective and dynamically consistent material rotation angle along any trajectory.

Outermost complex tubular level curves of the LAVD define initial positions of rotationally coherent material vortex boundaries in two-dimensional unsteady flows (see Fig.

(Exceptions are inviscid flows where such a global departure of LAVD level surfaces from a vortex is possible as fluid elements preserve their material rotation rate for all times[34]).

Remarkably, they are initial positions of material lines that are infinitesimally uniformly stretching under the flow map

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

[38][39] A geophysical example of a parabolic LCS (generalized jet core) revealed as a trench of the FTLE field is shown in Fig.

[36] The chevron-type shapes forming out of circular material blobs positioned along the jet core is characteristic of tracer deformation near parabolic LCSs.

Individual trajectories in a model flow generally show vastly different behavior from trajectories starting from the same initial condition of the real flow. This is due to the inevitable accumulation of errors and uncertainties, as well as sensitive dependence on initial conditions, in any realistic flow model. Yet an attracting LCS (such as the unstable manifold of a saddle point) is remarkably robust with respect to modelling errors and uncertainties. LCSs are, therefore, ideal tools for model validation and benchmarking
Figure 1: An invariant manifold in the extended phase space, formed by an evolving material surface.
Figure 2a: Hyperbolic LCS (attracting in red and repelling in blue) and elliptic LCS (boundaries of green regions) in a two-dimensional turbulence simulation. (Image: Mohammad Farazmand)
Fig. 2b: An attracting LCS is the locally most attracting material line (invariant manifold in the extended phase space of position and time), acting as the backbone curve of deforming tracer patterns over a finite time interval. In contrast, the unstable manifold of a saddle-type fixed point is an invariant curve in the phase space, acting as the asymptotic target for tracer patterns over infinite time intervals. Image: Mohammad Farazmand.
Figure 3: 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)
Figure 4. Attracting and repelling LCSs in the extended phase space of a two-dimensional dynamical system.
Figure 5a. Attracting (red) and repelling (blue) LCSs extracted as FTLE ridges from a two-dimensional turbulence experiment (Image: Manikandan Mathur) [ 17 ]
Figure 5b. Attracting (blue) and repelling (red) LCSs extracted as FTLE ridges from a two-dimensional simulation of a von Karman vortex street (Image: Jens Kasten) [ 19 ]
Figure 6. 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). [ 20 ]
Figure 7. Linearized flow geometry along an evolving material surface.
Figure 8. A repelling LCS visualized as an FTLE ridge (left) and computed exactly as a shrink line (right), i.e., a solution of the ODE starting from a global maximum of . [ 28 ] (Image: Mohammad Farazmand)
Figure 10a. Elliptic LCSs revealed by closed level curves of the PRA distribution in a two-dimensional turbulence simulation. (Image: Mohammad Farazmand) [ 32 ]
Figure 10b. Elliptic LCSs revealed by closed level curves of the PRA distribution in the steady ABC flow . (Image: Mohammad Farazmand) [ 32 ]
Figure 11a: Rotationally coherent mesoscale eddy boundaries in the ocean at time t0 = November 11, 2006, identified from satellite-based surface velocities, using the integration time t1-t0=90 days. The boundaries are identified as outermost closed contours of the LAVD with small convexity deficiency. Also shown in the background is the contour plot of the LAVD field for reference. (Image: Alireza Hadjighasem) [ 34 ]
Figure 11b: Materially advected rotationally coherent mesoscale eddy boundaries and eddy centers in the ocean, along with representative inertial particle trajectories initialised on the eddy boundaries. The eddy centers are obtained as local maxima of the LAVD field. As can be proven mathematically, heavy particles (cyan) converge to the centers of anti-cyclonic (clockwise) eddies. Light particles (black) converge to the centers of cyclonic (clockwise) eddies. (Movie: Alireza Hadjighasem) [ 34 ]
Figure 11c: A rotationally coherent mesoscale eddy (yellow) in the Southern Ocean State Estimate (SOSE) ocean model at t0 = May 15, 2006, computed as a tubular LAVD level surface over t1-t0=120 days. Also shown are nearby LAVD level surfaces to illustrate the rotational incoherence outside the eddy. (Image: Alireza Hadjighasem) [ 34 ]
Fig. 11c Material advection of a rotationally coherent Lagrangian vortex and its core in the 3D SOSE model data set. (Animation: Alireza Hadjighasem) [ 34 ]
Figure 11: An elliptic Lagrangian Coherent Structure (or LCS, in green, on the left) and its advected position under the flow map (on the right) of a chaotically forced ABC flow. Also shown in green is a circle of initial conditions placed around the LCS (on the left), advected for the same amount of time (on the right). Image: Daniel Blazevski.
Figure 13. Nested family of elliptic LCSs, obtained as -lines, forming transport barriers around the Great Red Spot (GRS) of Jupiter. These LCSs were identified in a two-dimensional, unsteady velocity field reconstructed from a video footage of Jupiter. [ 36 ] The color indicates the corresponding values of the parameter . Also shown is the perfectly coherent ( -line) bounding the core of the GRS, as well as the outermost elliptic LCS serving as the Lagrangian vortex boundary of the GRS. Image:Alireza Hadjighasem.
Figure 14b: Parabolic LCSs delineating unsteady Lagrangian jet cores in the atmosphere of Jupiter. [ 36 ] Also shown is the evolution of the elliptic LCS marking the boundary of the Great Red Spot. Video:Alireza Hadjighasem.