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# Lagrangian Coherent Structures and Finite-Time Lyapunov Exponents

The transport of a tracer in a fluid is closely related to emergent patterns
that are commonly referred to as coherent structures or *Lagrangian Coherent
Structures* (LCS) when they are defined using fluid trajectories. For time-
independant dynamic systems, they correspond to stable and unstable manifolds
of hyperbolic trajectories [1]. Coherent structures delimit regions of
whirls, stretching, or contraction of tracer [2]: contraction is observed
along stable manifolds whereas unstable manifolds correspond to divergent
directions along which the tracer is stretched. Stable and unstable manifolds
are material curves that act as transport barriers and that exhibits locally
the strongest attraction, repulsion or shearing in the flow over a finite-time
interval [3]. LCSs are usually indentified in a practicall manner as
maximizing ridges of Finite-Time Lyapunov Exponents (FTLE) field [4], [5],
[6], [7], [8].

FTLE is defined as the largest eigenvalue of the Cauchy-Green strain tensor of the flow map (see below for more details). The corresponding eigenvector is called Finite-Time Lyapunov Vector. FTLE and FSLE are widely used to characterise transport and mixing processes in oceanographic flows where the velocity field is only known as a finite data set [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Recent work also demonstrate the potential use of FTLE in tracer image assimilation in geophysical models [22], [23]. [24] which reviews the use of FTLE for detecting LCSs in other contexts.

## FTLE and FSLE: definition and properties

FTLE is a scalar local notion that represents the rate of separation of
initially neighboring particules over a finite-time window \([t, t+T]\).
FTLE at a point \(x\) is defined as the growth factor of the norm of the
perturbation delta started around \(x\) at time \(t\) and advected by
the flow after an advection time \(T\). Maximal stretching occurs when
delta is aligned with the eigenvector associated to the maximum eigenvalue
\(\lambda_{max}\) of the Cauchy-Green strain tensor: \(\Delta = M^T M\)
where \(M : x(t) \mapsto x(t+T)\) is the flow map of the advection system.
This eigenvector is referred to as the *forward* Finite-Time Lyapunov Vector.
The *forward* FTLE at the point \(x\) is then defined as:
\(FTLE(x) = \frac{log(\lambda max)}{2T}\).

FSLE are FTLE where T is not fixed a-priori but determined so that the particule separate from a given distance. FSLE at a point x and for a final separation distance d is defined as: \(FSLE(x, d) = \frac{log(\lambda max(d))}{2T(d)}\). Seeding a domain with particules initially located on a grid leads to the computation of a discretized FSLE and FSLV fields.

*Backward* FSLE-Vs are similarly defined, with the time direction being
inverted in the advection equation resolution. Ridges of BFTLE fields
approximate attracting LCSs [25].

Backward FTLE fields show contours that correspond reasonably well to the main
structures such as filaments, fronts and spirals that appear in geophysical
and bio-geochemical tracer fields [10], [14], [15], [26]. Also,
orientation of the gradient of a *passive* tracer converges to that of
backward FTLVs freely decaying 2D turbulence flow [27] but this behaviour has
been observed for realistic oceanic flows and tracers [20].

Furthermore [11], [28] showed, using real data, that properties of FSLE and FTLE remain valid with a mesoscale advection, i.e. when the resolution of the velocity field --- from which FTLE-V are computed --- is much lower than the resolution of the observed tracer field [20] ,[21]_.

## References

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[2] | Ottino, J. 1989. The kinematics of mixing: Stretching, chaos, and transport. Cambridge University Press, Cambridge. |

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