Performance of Lagrangian descriptors and their variants in incompressible flows. (English) Zbl 1378.37058
Summary: The method of Lagrangian Descriptors has been applied in many different contexts, specially in geophysical flows. In this paper, we analyze their performance in incompressible flows. We construct broad families of systems where this diagnostic fails in the detection of barriers to transport. Another aim of this manuscript is to illustrate the same deficiencies in the recent diagnostic proposed by Craven and Hernández.{
©2016 American Institute of Physics}
©2016 American Institute of Physics}
MSC:
| 37D10 | Invariant manifold theory for dynamical systems |
| 37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
References:
| [1] | Mendoza, C.; Mancho, A. M., The hidden geometry of ocean flows, Phys. Rev. Lett., 105, 038501 (2010) · doi:10.1103/PhysRevLett.105.038501 |
| [2] | Mancho, A. M.; Wiggins, S.; Curbelo, J.; Mendoza, C., Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 18, 3530-3557 (2013) · Zbl 1344.37031 · doi:10.1016/j.cnsns.2013.05.002 |
| [3] | Lopesino, C.; Balibrea, F.; Wiggins, S.; Mancho, A. M., Lagrangian descriptors for two dimensional, area preserving autonomous and nonautonomous maps, Commun. Nonlinear Sci. Numer. Simul., 27, 40-51 (2015) · Zbl 1457.37037 · doi:10.1016/j.cnsns.2015.02.022 |
| [4] | Mancho, A. M.; Curbelo, J.; Wiggins, S.; Garcia-Garrido, V. J.; Mendoza, C.; Blschl, G.; Thybo, H.; Savenije, H.; Lammerhuber, L., Beautiful geometries underlying ocean nonlinear processes, A Voyage Through Scales (2015) |
| [5] | Smith, M. L.; McDonalds, A. J., A quantitative measure of polar vortex strength using the function M, J. Geophys. Res.: Atmos., 119, 5966-5986 (2014) · doi:10.1002/2013JD020572 |
| [6] | Rempel, E. L.; Chian, A.; Brandenburg, A.; Munoz, P.; Shadden, S., Coherent structures and the saturation of a nonlinear dynamo, J. Fluid Mech., 729, 309-329 (2013) · Zbl 1291.76361 · doi:10.1017/jfm.2013.290 |
| [7] | Mendoza, C.; Mancho, A. M.; Wiggins, S., Lagrangian descriptors and the assesment of the predictive capacity of oceanic data sets, Nonlinear Processes Geophys., 21, 677-689 (2014) · doi:10.5194/npg-21-677-2014 |
| [8] | Wiggins, S.; Mancho, A. M., Barriers to tranport in aperiodically time-dependent two dimensional velocity fields: Nekhoroshev”s theorem and ‘Nearly Invariant’ Tori, Nonlinear Processes Geophys., 21, 165-185 (2014) · doi:10.5194/npg-21-165-2014 |
| [9] | de la Cámara, A.; Mechoso, R.; Mancho, A. M.; Serrano, E.; Ide, K., Quasi-horizontal transport within the Antarctic polar night vortex: Rossby wave breaking evidence and Lagrangian structures, J. Atmos. Sci., 70, 2982-3001 (2013) · doi:10.1175/JAS-D-12-0274.1 |
| [10] | Mendoza, C.; Mancho, A. M., The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current, Nonlinear Processes Geophys., 19, 449-472 (2012) · doi:10.5194/npg-19-449-2012 |
| [11] | de la Cámara, A.; Mancho, A. M.; Ide, K.; Serrano, E.; Mechoso, C. R., Routes of transport across the Antarctic polar vortex in the southern spring, J. Atmos. Sci., 69, 741-752 (2012) · doi:10.1175/JAS-D-11-0142.1 |
| [12] | Mendoza, C.; Mancho, A. M.; Rio, M.-H., The turnstile mechanism across the Kuroshio current: Analysis of dynamics in altimeter velocity fields, Nonlinear Processes Geophys., 17, 103-111 (2010) · doi:10.5194/npg-17-103-2010 |
| [13] | García-Garrido, V. J.; Mancho, A. M.; Wiggins, S.; Mendoza, C., A dynamical systems approach to the surface search for debris associated with the disappearance of flight MH370, Nonlinear Processes Geophys., 22, 701 (2015) · doi:10.5194/npg-22-701-2015 |
| [14] | Guha, A.; Mechoso, C. R.; Konor, C. S.; Heikes, R. P., Modeling Rossby wave breaking in the southern spring stratosphere, J. Atmos. Sci., 73, 393-406 (2016) · doi:10.1175/JAS-D-15-0088.1 |
| [15] | Vortmeyer-Kley, R.; Grave, U.; Feudel, U., Detecting and tracking eddies in oceanic flow fields: A Lagrangian Descriptor based on the modulus of vorticity, Nonlinear Processes Geophys., 23, 159 (2016) · doi:10.5194/npg-23-159-2016 |
| [16] | Garcia-Garrido, V. J.; Ramos, A.; Mancho, A. M.; Coca, J.; Wiggins, S., A dynamical system perspective for a real-time response to a marine oil spill, Mar. Popul. Bull., 112, 201 (2016) · doi:10.1016/j.marpolbul.2016.08.018 |
| [17] | Madrid, J. A.; Mancho, A. M., Distinguished trajectories in time dependent vector fields, Chaos, 19, 013111 (2009) · doi:10.1063/1.3056050 |
| [18] | Ruiz-Herrera, A., Some examples related to the method of Lagrangian descriptors, Chaos, 25, 063112 (2015) · doi:10.1063/1.4922182 |
| [19] | Palis, J. J.; De Melo, W., Geometric Theory of Dynamical Systems: An Introduction (2012) |
| [20] | Haller, G. |
| [21] | Peacock, T.; Froyland, G.; Haller, G., Introduction to focus issue: Objective detection of coherent structures, Chaos, 25, 087201 (2015) · doi:10.1063/1.4928894 |
| [22] | Craven, G. T.; Hernandez, R., Deconstructing field-induced ketene isomerization through Lagrangian descriptors, Phys. Chem. Chem. Phys., 18, 4008 (2016) · doi:10.1039/C5CP06624G |
| [23] | Craven, G.; Hernandez, R., Lagrangian descriptors of thermalized transition states on time-varying energy surfaces, Phys. Rev. Lett., 115, 148301 (2015) · doi:10.1103/PhysRevLett.115.148301 |
| [24] | Junginger, A.; Hernandez, R., Uncovering the geometry of barrierless reactions using Lagrangian descriptors, J. Phys. Chem. B, 120, 1720-1725 (2016) · doi:10.1021/acs.jpcb.5b09003 |
| [25] | Balibrea-Iniesta, F.; Curbelo, J.; Garcia-Garrido, V. J.; Lopesino, C.; Mancho, A. M.; Mendoza, C.; Wiggins, S., Response to: ‘Limitations of the method of Lagrangian descriptors’ [arXiv: 1510.04838] (2016) |
| [26] | Haller, G., Lagrangian coherent structures, Ann. Rev. Fluid Mech., 47, 137-162 (2015) · doi:10.1146/annurev-fluid-010313-141322 |
| [27] | Allshouse, M. R.; Peacock, T., Refining finite-type Lyapunov exponents ridges and the challenges of classifying them, Chaos, 25, 087410 (2015) · Zbl 1374.37117 · doi:10.1063/1.4928210 |
| [28] | Froyland, G., An analytical framework for identifying finite-time coherent sets in time dependent dynamical systems, Physica D, 250, 1-19 (2013) · Zbl 1355.37013 · doi:10.1016/j.physd.2013.01.013 |
| [29] | Karrasch, D.; Huhn, F.; Haller, G., Automated detection of coherent Lagrangian vortices in two dimensional unsteady flows, , A, 471, 20140639 (2014) · Zbl 1371.76044 |
| [30] | Froyland, G., Dynamic isoperimetry and the geometry of Lagrangian Coherent Structures, Nonlinearity, 25, 087409 (2015) · Zbl 1422.65448 |
| [31] | Peacock, T.; Haller, G., Lagrangian coherent structures: The hidden skeleton of fluid flows, Phys. Today, 66, 2, 41 (2013) · doi:10.1063/PT.3.1886 |
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