Lagrangian coherent structures

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.

While LCSs generally exist in any dynamical system, their role in creating coherent patterns is perhaps most readily observable in fluid flows. The images below are examples of how different types of LCSs hidden in geophysical flows shape tracer patterns.

Spiral eddies:
Hyperbolic and elliptic LCSs
(Paul Scully-Power/NASA)

Sea surface temperature in Gulf Stream
Parabolic LCSs
(NASA)

Phytoplankton in Agulhas ring
2D elliptic LCS
(NASA/GSFC)

Tornado off the Florida Keys
3D elliptic LCS (cylindrical)
(Joseph Golden/NOAA)

A steam ring from Mount Etna
3D elliptic LCS (toroidal)
(Tom Pfeiffer [1])

On a phase space ${\displaystyle {\mathcal {P}}}$ and over a time interval ${\displaystyle {\mathcal {I}}=[t_{0},t_{1}]}$ , consider a non-autonomous dynamical system defined through the flow map ${\displaystyle F_{t_{0}}^{t}\colon x_{0}\mapsto x(t,t_{0},x_{0})}$, mapping initial conditions ${\displaystyle x_{0}\in {\mathcal {P}}}$ into their position ${\displaystyle x(t,t_{0},x_{0})\in {\mathcal {P}}}$ for any time ${\displaystyle t\in {\mathcal {I}}}$. If the flow map ${\displaystyle F_{t_{0}}^{t}}$ is a diffeomorphism for any choice of ${\displaystyle t\in {\mathcal {I}}}$, then for any smooth set ${\displaystyle {\mathcal {M}}(t_{0})}$ of initial conditions in ${\displaystyle {\mathcal {P}}}$, the set

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