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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
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.[1][2][3] 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.[3][4][5] 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,[6] surface drifters[7][8] and chlorophyll patterns[9] in the ocean; clouds of volcanic ash[10] and spores in the atmosphere;[11] and coherent crowd patterns formed by humans[12] and animals.
While LCSs generally exist in any dynamical system, their role in creating coherent patterns is perhaps most readily observable in fluid flows.
^Haller, G.; Yuan, G. (2000). "Lagrangian coherent structures and mixing in two-dimensional turbulence". Physica D: Nonlinear Phenomena. 147 (3–4): 352. Bibcode:2000PhyD..147..352H. doi:10.1016/S0167-2789(00)00142-1.
^Peacock, T.; Haller, G. (2013). "Lagrangian coherent structures: The hidden skeleton of fluid flows". Physics Today. 66 (2): 41. Bibcode:2013PhT....66b..41P. doi:10.1063/PT.3.1886.
^ abHaller, G. (2015). "Lagrangian Coherent Structures". Annual Review of Fluid Mechanics. 47 (1): 137–162. Bibcode:2015AnRFM..47..137H. doi:10.1146/annurev-fluid-010313-141322.
^Bozorgmagham, A. E.; Ross, S. D.; Schmale, D. G. (2013). "Real-time prediction of atmospheric Lagrangian coherent structures based on forecast data: An application and error analysis". Physica D: Nonlinear Phenomena. 258: 47–60. Bibcode:2013PhyD..258...47B. doi:10.1016/j.physd.2013.05.003.
^Bozorgmagham, A. E.; Ross, S. D. (2015). "Atmospheric Lagrangian coherent structures considering unresolved turbulence and forecast uncertainty". Communications in Nonlinear Science and Numerical Simulation. 22 (1–3): 964–979. Bibcode:2015CNSNS..22..964B. doi:10.1016/j.cnsns.2014.07.011.
^Olascoaga, M. J.; Haller, G. (2012). "Forecasting sudden changes in environmental pollution patterns". Proceedings of the National Academy of Sciences. 109 (13): 4738–4743. Bibcode:2012PNAS..109.4738O. doi:10.1073/pnas.1118574109. PMC 3323984. PMID 22411824.
^Nencioli, F.; d'Ovidio, F.; Doglioli, A. M.; Petrenko, A. A. (2011). "Surface coastal circulation patterns by in-situ detection of Lagrangian coherent structures". Geophysical Research Letters. 38 (17): n/a. Bibcode:2011GeoRL..3817604N. doi:10.1029/2011GL048815.
^Olascoaga, M. J.; Beron-Vera, F. J.; Haller, G.; Triñanes, J.; Iskandarani, M.; Coelho, E. F.; Haus, B. K.; Huntley, H. S.; Jacobs, G.; Kirwan, A. D.; Lipphardt, B. L.; Özgökmen, T. M.; h. m. Reniers, A. J.; Valle-Levinson, A. (2013). "Drifter motion in the Gulf of Mexico constrained by altimetric Lagrangian coherent structures". Geophysical Research Letters. 40 (23): 6171. Bibcode:2013GeoRL..40.6171O. doi:10.1002/2013GL058624.
^Huhn, F.; von Kameke, A.; Pérez-Muñuzuri, V.; Olascoaga, M. J.; Beron-Vera, F. J. (2012). "The impact of advective transport by the South Indian Ocean Countercurrent on the Madagascar plankton bloom". Geophysical Research Letters. 39 (6): n/a. Bibcode:2012GeoRL..39.6602H. doi:10.1029/2012GL051246.
^Peng, J.; Peterson, R. (2012). "Attracting structures in volcanic ash transport". Atmospheric Environment. 48: 230–239. Bibcode:2012AtmEn..48..230P. doi:10.1016/j.atmosenv.2011.05.053.
^Tallapragada, P.; Ross, S. D.; Schmale, D. G. (2011). "Lagrangian coherent structures are associated with fluctuations in airborne microbial populations". Chaos: An Interdisciplinary Journal of Nonlinear Science. 21 (3): 033122. Bibcode:2011Chaos..21c3122T. doi:10.1063/1.3624930. hdl:10919/24411. PMID 21974657.
^Ali, S.; Shah, M. (2007). "A Lagrangian Particle Dynamics Approach for Crowd Flow Segmentation and Stability Analysis". 2007 IEEE Conference on Computer Vision and Pattern Recognition. p. 1. CiteSeerX 10.1.1.63.4342. doi:10.1109/CVPR.2007.382977. ISBN 978-1-4244-1179-5. S2CID 8190391.
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