Lagrangian coherent structures (LCSs) are distinguished surfaces of
trajectories
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
in a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
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.
File:Spiral eddies.jpeg, ''Spiral eddies:''
Hyperbolic and elliptic LCSs
(Paul Scully-Power/NASA)
File:SST Gulf stream.jpeg, ''Sea surface temperature in Gulf Stream''
Parabolic LCSs
(NASA)
File:Agulhas ring.jpeg, ''Phytoplankton in Agulhas ring''
2D elliptic LCS
(NASA/GSFC)
File:Trombe.jpg, ''Tornado off the Florida Keys''
3D elliptic LCS (cylindrical)
(Joseph Golden/NOAA)
File:Steam ring.jpeg, ''A steam ring from Mount Etna''
3D elliptic LCS (toroidal)
(Tom Pfeiffe
General definitions
Material surfaces
On a
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
and over a time interval