{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T11:46:29Z","timestamp":1768650389361,"version":"3.49.0"},"reference-count":124,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,20]],"date-time":"2023-10-20T00:00:00Z","timestamp":1697760000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["23-17-00068"],"award-info":[{"award-number":["23-17-00068"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["121021700341-2"],"award-info":[{"award-number":["121021700341-2"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002674","name":"Pacific Oceanological Institute","doi-asserted-by":"publisher","award":["23-17-00068"],"award-info":[{"award-number":["23-17-00068"]}],"id":[{"id":"10.13039\/501100002674","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002674","name":"Pacific Oceanological Institute","doi-asserted-by":"publisher","award":["121021700341-2"],"award-info":[{"award-number":["121021700341-2"]}],"id":[{"id":"10.13039\/501100002674","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the Lagrangian approach, the transport processes in the ocean and atmosphere are studied by tracking water or air parcels, each of which may carry different tracers. In the ocean, they are salt, nutrients, heat, and particulate matter, such as plankters, oil, radionuclides, and microplastics. In the atmosphere, the tracers are water vapor, ozone, and various chemicals. The observation and simulation reveal highly complex patterns of advection of tracers in turbulent-like geophysical flows. Transport barriers are material surfaces across which the transport is minimal. They can be classified into elliptic, parabolic, and hyperbolic barriers. Different diagnostics in detecting transport barriers and the analysis of their role in the dynamics of oceanic and atmospheric flows are reviewed. We discuss the mathematical tools, borrowed from dynamical systems theory, for detecting transport barriers in simple kinematic and dynamic models of vortical and jet-like flows. We show how the ideas and methods, developed for simple model flows, can be successfully applied for studying the role of barriers in oceanic and atmospheric flows. Special attention is placed on the significance of transport barriers in important practical issues: anthropogenic and natural pollution, advection of plankton, cross-shelf exchange, and propagation of upwelling fronts in coastal zones.<\/jats:p>","DOI":"10.3390\/sym15101942","type":"journal-article","created":{"date-parts":[[2023,10,20]],"date-time":"2023-10-20T07:22:32Z","timestamp":1697786552000},"page":"1942","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Transport Barriers in Geophysical Flows: A Review"],"prefix":"10.3390","volume":"15","author":[{"given":"Sergey","family":"Prants","sequence":"first","affiliation":[{"name":"V.I. Il\u2019ichev Pacific Oceanological Institute, Russian Academy of Sciences, 43 Baltiyskaya St., 690041 Vladivostok, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Arnold, V.I., Kozlov, V.V., and Neishtadt, A.I. (2006). Mathematical Aspects of Classical and Celestial Mechanics, Springer. [3rd ed.]. Encyclopaedia of Mathematical Sciences.","DOI":"10.1007\/978-3-540-48926-9"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wiggins, S. (1992). Chaotic Transport in Dynamical Systems, Springer. Interdisciplinary Applied Mathematics.","DOI":"10.1007\/978-1-4757-3896-4"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1103\/RevModPhys.64.795","article-title":"Symplectic maps, variational principles, and transport","volume":"64","author":"Meiss","year":"1992","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Prants, S.V., Uleysky, M.Y., and Budyansky, M.V. (2017). Lagrangian Oceanography: Large-Scale Transport and Mixing in the Ocean, Springer. Physics of Earth and Space Environments.","DOI":"10.1007\/978-3-319-53022-2"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/S0167-2789(01)00330-X","article-title":"Detecting barriers to transport: A review of different techniques","volume":"159","author":"Boffetta","year":"2001","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_6","first-page":"L17203","article-title":"Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents","volume":"31","year":"2004","journal-title":"Geophys. Res. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1018","DOI":"10.1134\/1.1842883","article-title":"Chaotic scattering, transport, and fractals in a simple hydrodynamic flow","volume":"99","author":"Budyansky","year":"2004","journal-title":"J. Exp. Theor. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1680","DOI":"10.1016\/j.physd.2012.06.012","article-title":"Geodesic theory of transport barriers in two-dimensional flows","volume":"241","author":"Haller","year":"2012","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1851","DOI":"10.1063\/1.1477449","article-title":"Lagrangian coherent structures from approximate velocity data","volume":"14","author":"Haller","year":"2002","journal-title":"Phys. Fluids"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1146\/annurev-fluid-010313-141322","article-title":"Lagrangian Coherent Structures","volume":"47","author":"Haller","year":"2015","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1016\/j.physd.2010.11.010","article-title":"A variational theory of hyperbolic Lagrangian Coherent Structures","volume":"240","author":"Haller","year":"2011","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0022112084001233","article-title":"Stirring by chaotic advection","volume":"143","author":"Aref","year":"1984","journal-title":"J. Fluid Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1063\/1.1458932","article-title":"The development of chaotic advection","volume":"14","author":"Aref","year":"2002","journal-title":"Phys. Fluids"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kozlov, V. (1995). Symmetries, Topology and Resonances in Hamiltonian Mechanics, UdGU. (In Russian).","DOI":"10.1007\/978-3-642-78393-7"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1134\/1.1383840","article-title":"A mechanism of chaotic mixing in an elementary deterministic flow","volume":"27","author":"Budyansky","year":"2001","journal-title":"Tech. Phys. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zaslavsky, G.M. (2005). Hamiltonian Chaos and Fractional Dynamics, Oxford University Press.","DOI":"10.1093\/oso\/9780198526049.001.0001"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1357\/002224011798765204","article-title":"Ellipsoidal vortex in a nonuniform flow: Dynamics and chaotic advections","volume":"69","author":"Zhmur","year":"2011","journal-title":"J. Mar. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"437","DOI":"10.5194\/npg-20-437-2013","article-title":"Diffusion-affected passive scalar transport in an ellipsoidal vortex in a shear flow","volume":"20","author":"Koshel","year":"2013","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5194\/npg-24-1-2017","article-title":"Parametric resonance in the dynamics of an elliptic vortex in a periodically strained environment","volume":"24","author":"Koshel","year":"2017","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"013107","DOI":"10.1063\/1.2835349","article-title":"Determination of optimal excitation frequency range in background flows","volume":"18","author":"Izrailsky","year":"2008","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1016\/j.fluiddyn.2008.03.001","article-title":"Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle","volume":"40","author":"Koshel","year":"2008","journal-title":"Fluid Dyn. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1134\/S0001433810020064","article-title":"Chaotic transport and mixing of a passive admixture by vortex flows behind obstacles","volume":"46","author":"Ryzhov","year":"2010","journal-title":"Izv. Atmos. Ocean. Phys."},{"key":"ref_23","unstructured":"Longhurst, A.R. (1981). Analysis of Marine Ecosystems, Academic Press Inc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3294","DOI":"10.1038\/ncomms4294","article-title":"Global heat and salt transports by eddy movement","volume":"5","author":"Dong","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1134\/S0001433821040216","article-title":"Trench Eddies in the Northwest Pacific: An Overview","volume":"57","author":"Prants","year":"2021","journal-title":"Izv. Atmos. Ocean. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"112245","DOI":"10.1016\/j.rse.2020.112245","article-title":"Global chlorophyll distribution induced by mesoscale eddies","volume":"254","author":"Zhao","year":"2021","journal-title":"Remote Sens. Environ."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1111","DOI":"10.1175\/JPO-D-21-0103.1","article-title":"Global Oceanic Mass Transport by Coherent Eddies","volume":"52","author":"Xia","year":"2022","journal-title":"J. Phys. Oceanogr."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"102955","DOI":"10.1016\/j.pocean.2022.102955","article-title":"Review of oceanic mesoscale processes in the North Pacific: Physical and biogeochemical impacts","volume":"212","author":"Ueno","year":"2023","journal-title":"Prog. Oceanogr."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1029\/JC087iC01p00385","article-title":"Compact, intrathermocline eddies in the Sargasso Sea","volume":"87","author":"Dugan","year":"1982","journal-title":"J. Geophys. Res. Oceans"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1960","DOI":"10.1175\/1520-0485(2002)032<1960:JESIE>2.0.CO;2","article-title":"Japan\/East Sea Intrathermocline Eddies","volume":"32","author":"Gordon","year":"2002","journal-title":"J. Phys. Oceanogr."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4811","DOI":"10.1002\/jgrc.20337","article-title":"Intrathermocline eddies in the coastal transition zone off central Chile (31\u201341\u2218 S)","volume":"118","author":"Hormazabal","year":"2013","journal-title":"J. Geophys. Res. Ocean."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1007\/s10236-013-0628-y","article-title":"Dynamics of intrathermocline vortices in a gyre flow over a seamount chain","volume":"63","author":"Sokolovskiy","year":"2013","journal-title":"Ocean Dyn."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Sokolovskiy, M.A., and Verron, J. (2014). Dynamics of Vortex Structures in a Stratified Rotating Fluid, Springer International Publishing. Atmospheric and Oceanographic Sciences Library.","DOI":"10.1007\/978-3-319-00789-2"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1017\/S0022112000001816","article-title":"Finite-core hetons: Stability and interactions","volume":"423","author":"Sokolovskiy","year":"2000","journal-title":"J. Fluid Mech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"045501","DOI":"10.1088\/0169-5983\/42\/4\/045501","article-title":"Baroclinic multipole formation from heton interaction","volume":"42","author":"Sokolovskiy","year":"2010","journal-title":"Fluid Dyn. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1023\/A:1013779219578","article-title":"Hydrodynamical modeling of oceanic vortices","volume":"22","author":"Carton","year":"2001","journal-title":"Surv. Geophys."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Cushman-Roisin, B., and Beckers, J.M. (2011). Introduction to Geophysical Fluid Dynamics, Physical and Numerical Aspects, Academic Press. [2nd ed.]. International Geophysics.","DOI":"10.1016\/B978-0-12-088759-0.00001-8"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1137\/S0036141094276913","article-title":"On Quasi-Periodic Perturbations of Elliptic Equilibrium Points","volume":"27","author":"Jorba","year":"1996","journal-title":"SIAM J. Math. Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1063\/PT.3.1886","article-title":"Lagrangian coherent structures: The hidden skeleton of fluid flows","volume":"66","author":"Peacock","year":"2013","journal-title":"Phys. Today"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1017\/jfm.2016.151","article-title":"Defining coherent vortices objectively from the vorticity","volume":"795","author":"Haller","year":"2016","journal-title":"J. Fluid Mech."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.pocean.2011.01.002","article-title":"Global observations of nonlinear mesoscale eddies","volume":"91","author":"Chelton","year":"2011","journal-title":"Prog. Oceanogr."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1109\/LGRS.2011.2155029","article-title":"An Automated Approach to Detect Oceanic Eddies From Satellite Remotely Sensed Sea Surface Temperature Data","volume":"8","author":"Dong","year":"2011","journal-title":"IEEE Geosci. Remote. Sens. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1175\/JTECH-D-17-0010.1","article-title":"Angular Momentum Eddy Detection and Tracking Algorithm (AMEDA) and Its Application to Coastal Eddy Formation","volume":"35","author":"Stegner","year":"2018","journal-title":"J. Atmos. Ocean. Technol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3944","DOI":"10.1002\/2015JC011620","article-title":"The life cycle of a coherent Lagrangian Agulhas ring","volume":"121","author":"Wang","year":"2016","journal-title":"J. Geophys. Res. Ocean."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"C11017","DOI":"10.1029\/2009JC005349","article-title":"Pattern and velocity of propagation of the global ocean eddy variability","volume":"114","author":"Fu","year":"2009","journal-title":"J. Geophys. Res. Ocean."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.asr.2021.01.022","article-title":"Altimetry for the future: Building on 25 years of progress","volume":"68","author":"Abdalla","year":"2021","journal-title":"Adv. Space Res."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"150028","DOI":"10.1038\/sdata.2015.28","article-title":"A daily global mesoscale ocean eddy dataset from satellite altimetry","volume":"2","author":"Faghmous","year":"2015","journal-title":"Sci. Data"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2081","DOI":"10.1002\/2017JC013610","article-title":"How Eddies Gain, Retain, and Release Water: A Case Study of a Hokkaido Anticyclone","volume":"123","author":"Prants","year":"2018","journal-title":"J. Geophys. Res. Ocean."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/s10236-022-01508-w","article-title":"Odyssey of Aleutian eddies","volume":"72","author":"Budyansky","year":"2022","journal-title":"Ocean Dyn."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"104129","DOI":"10.1016\/j.dsr.2023.104129","article-title":"A census and properties of mesoscale Kuril eddies in the altimetry era","volume":"200","author":"Udalov","year":"2023","journal-title":"Deep. Sea Res. Part I Oceanogr. Res. Pap."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.dsr.2012.06.001","article-title":"Eddy analysis in the subtropical zonal band of the North Pacific Ocean","volume":"68","author":"Liu","year":"2012","journal-title":"Deep. Sea Res. Part I Oceanogr. Res. Pap."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1765","DOI":"10.5194\/essd-15-1765-2023","article-title":"A global Lagrangian eddy dataset based on satellite altimetry","volume":"15","author":"Liu","year":"2023","journal-title":"Earth Syst. Sci. Data"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1175\/JPO-D-17-0102.1","article-title":"Transport by Lagrangian Vortices in the Eastern Pacific","volume":"48","author":"Abernathey","year":"2018","journal-title":"J. Phys. Oceanogr."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1016\/0021-9169(84)90063-1","article-title":"The \u2018surf zone\u2019 in the stratosphere","volume":"46","author":"McIntyre","year":"1984","journal-title":"J. Atmos. Terr. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"23013","DOI":"10.1029\/93JD02599","article-title":"Large-scale isentropic mixing properties of the Antarctic polar vortex from analyzed winds","volume":"98","author":"Bowman","year":"1993","journal-title":"J. Geophys. Res."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"165","DOI":"10.2151\/jmsj.85B.165","article-title":"Transport in the Middle Atmosphere","volume":"85","author":"Shepherd","year":"2007","journal-title":"J. Meteorol. Soc. Jpn. Ser. II"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1038\/nature10556","article-title":"Unprecedented Arctic ozone loss in 2011","volume":"478","author":"Manney","year":"2011","journal-title":"Nature"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1080\/03091929108227343","article-title":"Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves","volume":"58","author":"Pierrehumbert","year":"1991","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"2462","DOI":"10.1175\/1520-0469(1993)050<2462:GCMOIS>2.0.CO;2","article-title":"Global Chaotic Mixing on Isentropic Surfaces","volume":"50","author":"Pierrehumbert","year":"1993","journal-title":"J. Atmos. Sci."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1198","DOI":"10.1175\/1520-0469(2002)059<1198:RBKBSA>2.0.CO;2","article-title":"Relation between Kinematic Boundaries, Stirring, and Barriers for the Antarctic Polar Vortex","volume":"59","author":"Joseph","year":"2002","journal-title":"J. Atmos. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1175\/JAS-D-11-0142.1","article-title":"Routes of Transport across the Antarctic Polar Vortex in the Southern Spring","volume":"69","author":"Mancho","year":"2012","journal-title":"J. Atmos. Sci."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"687","DOI":"10.5194\/npg-19-687-2012","article-title":"Brief communication \u201cStratospheric winds, transport barriers and the 2011 Arctic ozone hole\u201d","volume":"19","author":"Olascoaga","year":"2012","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1175\/1520-0485(1991)021<0173:ASKMFM>2.0.CO;2","article-title":"A Simple Kinematic Mechanism for Mixing Fluid Parcels across a Meandering Jet","volume":"21","author":"Bower","year":"1991","journal-title":"J. Phys. Oceanogr."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1175\/1520-0485(1992)022<0431:FEAAMJ>2.0.CO;2","article-title":"Fluid Exchange across a Meandering Jet","volume":"22","author":"Samelson","year":"1992","journal-title":"J. Phys. Oceanogr."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.physd.2006.07.021","article-title":"Lobe dynamics in a kinematic model of a meandering jet. I. Geometry and statistics of transport and lobe dynamics with accelerated convergence","volume":"223","author":"Raynal","year":"2006","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"033117","DOI":"10.1063\/1.2229263","article-title":"Chaotic mixing and transport in a meandering jet flow","volume":"16","author":"Prants","year":"2006","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"215102","DOI":"10.1088\/1751-8113\/41\/21\/215102","article-title":"Genesis and bifurcations of unstable periodic orbits in a jet flow","volume":"41","author":"Uleysky","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.1070\/PU2006v049n11ABEH006066","article-title":"Chaotic advection in the ocean","volume":"49","author":"Prants","year":"2006","journal-title":"Phys.-Uspekhi"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"948","DOI":"10.1063\/1.858639","article-title":"Chaotic transport by Rossby waves in shear flow","volume":"5","author":"Morrison","year":"1993","journal-title":"Phys. Fluids A Fluid Dyn."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"3595","DOI":"10.1175\/JAS4036.1","article-title":"On the Lagrangian Dynamics of Atmospheric Zonal Jets and the Permeability of the Stratospheric Polar Vortex","volume":"64","author":"Rypina","year":"2007","journal-title":"J. Atmos. Sci."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"3640","DOI":"10.1175\/JAS4030.1","article-title":"Transport and Mixing in Kinematic and Dynamically Consistent Flows","volume":"64","author":"Haynes","year":"2007","journal-title":"J. Atmos. Sci."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"017202","DOI":"10.1103\/PhysRevE.81.017202","article-title":"Mechanism of destruction of transport barriers in geophysical jets with Rossby waves","volume":"81","author":"Uleysky","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1134\/S1063776110120174","article-title":"Chaotic transport across two-dimensional jet streams","volume":"111","author":"Uleysky","year":"2010","journal-title":"J. Exp. Theor. Phys."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1038\/337058a0","article-title":"Laboratory model of a planetary eastward jet","volume":"337","author":"Sommeria","year":"1989","journal-title":"Nature"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"1971","DOI":"10.1063\/1.858824","article-title":"Shear flow instabilities and Rossby waves in barotropic flow in a rotating annulus","volume":"5","author":"Solomon","year":"1993","journal-title":"Phys. Fluids A Fluid Dyn."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1175\/1520-0469(1949)006<0105:DIOTDN>2.0.CO;2","article-title":"Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere","volume":"6","author":"Kuo","year":"1949","journal-title":"J. Meteorol."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"056215","DOI":"10.1103\/PhysRevE.79.056215","article-title":"Detection of barriers to cross-jet Lagrangian transport and its destruction in a meandering flow","volume":"79","author":"Budyansky","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1143\/PTP.97.379","article-title":"The Breakup Condition of Shearless KAM Curves in the Quadratic Map","volume":"97","author":"Shinohara","year":"1997","journal-title":"Prog. Theor. Phys."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1063\/1.858052","article-title":"Chaos and mixing in a geostrophic flow","volume":"3","author":"Behringer","year":"1991","journal-title":"Phys. Fluids A Fluid Dyn."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/S0167-2789(00)00199-8","article-title":"Distinguished material surfaces and coherent structures in three-dimensional fluid flows","volume":"149","author":"Haller","year":"2001","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.ocemod.2011.02.008","article-title":"Lagrangian study of transport and mixing in a mesoscale eddy street","volume":"38","author":"Prants","year":"2011","journal-title":"Ocean Model."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.physd.2005.10.007","article-title":"Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows","volume":"212","author":"Shadden","year":"2005","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1142\/9789814699983_0007","article-title":"Oil Spills: Causes, Consequences, Prevention, and Countermeasures","volume":"Volume 1","author":"Crawley","year":"2016","journal-title":"Fossil Fuels"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1007\/s007730050012","article-title":"Simulation of the oil spill processes in the Sea of Japan with regional ocean circulation model","volume":"4","author":"Varlamov","year":"1999","journal-title":"J. Mar. Sci. Technol."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s10872-013-0182-8","article-title":"Particle-tracking simulation for the drift\/diffusion of spilled oils in the Sea of Okhotsk with a three-dimensional, high-resolution model","volume":"69","author":"Ono","year":"2013","journal-title":"J. Oceanogr."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1016\/j.scitotenv.2019.06.255","article-title":"Modelling oil trajectories and potentially contaminated areas from the Sanchi oil spill","volume":"685","author":"Qiao","year":"2019","journal-title":"Sci. Total. Environ."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.jmarsys.2003.11.023","article-title":"Particle tracking method in the approach for prediction of oil slick transport in the sea: Modelling oil pollution resulting from river input","volume":"48","author":"Korotenko","year":"2004","journal-title":"J. Mar. Syst."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/s00024-022-03197-4","article-title":"Lagrangian Oil Spill Simulation in Peter the Great Bay (Sea of Japan) with a High-Resolution ROMS Model","volume":"180","author":"Prants","year":"2023","journal-title":"Pure Appl. Geophys."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"4738","DOI":"10.1073\/pnas.1118574109","article-title":"Forecasting sudden changes in environmental pollution patterns","volume":"109","author":"Olascoaga","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"5218","DOI":"10.1038\/s41598-018-23121-y","article-title":"Extracting quasi-steady Lagrangian transport patterns from the ocean circulation: An application to the Gulf of Mexico","volume":"8","author":"Duran","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_91","doi-asserted-by":"crossref","unstructured":"Fifani, G., Baudena, A., Fakhri, M., Baaklini, G., Faug\u00c3\u0161re, Y., Morrow, R., Mortier, L., and d\u2019Ovidio, F. (2021). Drifting Speed of Lagrangian Fronts and Oil Spill Dispersal at the Ocean Surface. Remote Sens., 13.","DOI":"10.3390\/rs13224499"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1134\/S1028334X11080277","article-title":"Numerical simulation of propagation of radioactive pollution in the ocean from the Fukushima Dai-ichi nuclear power plant","volume":"439","author":"Prants","year":"2011","journal-title":"Dokl. Earth Sci."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.jenvrad.2011.10.007","article-title":"Distribution of oceanic 137Cs from the Fukushima Dai-ichi Nuclear Power Plant simulated numerically by a regional ocean model","volume":"111","author":"Tsumune","year":"2012","journal-title":"J. Environ. Radioact."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.dsr.2013.05.015","article-title":"Multi-decadal projections of surface and interior pathways of the Fukushima Cesium-137 radioactive plume","volume":"80","author":"Rossi","year":"2013","journal-title":"Deep Sea Res. Part I Oceanogr. Res. Pap."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.dsr.2014.09.007","article-title":"Role of mesoscale eddies in transport of Fukushima-derived cesium isotopes in the ocean","volume":"96","author":"Budyansky","year":"2015","journal-title":"Deep Sea Res. Part I Oceanogr. Res. Pap."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"453","DOI":"10.5194\/os-13-453-2017","article-title":"Lagrangian simulation and tracking of the mesoscale eddies contaminated by Fukushima-derived radionuclides","volume":"13","author":"Prants","year":"2017","journal-title":"Ocean Sci."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"113483","DOI":"10.1016\/j.marpolbul.2022.113483","article-title":"The impact of circulation features on the dispersion of radionuclides after the nuclear submarine accident in Chazhma Bay (Japan Sea) in 1985: A retrospective Lagrangian simulation","volume":"177","author":"Budyansky","year":"2022","journal-title":"Mar. Pollut. Bull."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s10512-020-00604-8","article-title":"Reconstruction of the Radioactive Contamination Occurring in the Environment in Primorskii Krai as a Result of a Nuclear Accident on a Submarine in Bukhta Chazhma","volume":"127","author":"Sarkisov","year":"2020","journal-title":"At. Energy"},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1146\/annurev-marine-010814-015717","article-title":"Cross-Shelf Exchange","volume":"8","author":"Brink","year":"2016","journal-title":"Annu. Rev. Mar. Sci."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1146\/annurev-marine-121211-172315","article-title":"High-Frequency Radar Observations of Ocean Surface Currents","volume":"5","author":"Paduan","year":"2013","journal-title":"Annu. Rev. Mar. Sci."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physd.2005.06.023","article-title":"Pollution release tied to invariant manifolds: A case study for the coast of Florida","volume":"210","author":"Lekien","year":"2005","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"L22603","DOI":"10.1029\/2006GL027800","article-title":"Persistent transport barrier on the West Florida Shelf","volume":"33","author":"Olascoaga","year":"2006","journal-title":"Geophys. Res. Lett."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"1221","DOI":"10.1016\/S0278-4343(99)00021-7","article-title":"Lagrangian circulation and forbidden zone on the West Florida Shelf","volume":"19","author":"Yang","year":"1999","journal-title":"Cont. Shelf Res."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"C12014","DOI":"10.1029\/2007JC004533","article-title":"Tracing the early development of harmful algal blooms on the West Florida Shelf with the aid of Lagrangian coherent structures","volume":"113","author":"Olascoaga","year":"2008","journal-title":"J. Geophys. Res. Ocean."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1134\/S0001433813010088","article-title":"Lagrangian analysis of mixing and transport of water masses in the marine bays","volume":"49","author":"Prants","year":"2013","journal-title":"Izv. Atmos. Ocean. Phys."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.csr.2018.04.016","article-title":"Evaluation of surface Lagrangian transport barriers in the Gulf of Trieste","volume":"167","author":"Enrile","year":"2018","journal-title":"Cont. Shelf Res."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1175\/2009JPO4116.1","article-title":"Evidence for Submesoscale Barriers to Horizontal Mixing in the Ocean from Current Measurements and Aerial Photographs","volume":"39","author":"Gildor","year":"2009","journal-title":"J. Phys. Oceanogr."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1080\/20026491051695","article-title":"The Effects of Harmful Algal Blooms on Aquatic Organisms","volume":"10","author":"Landsberg","year":"2002","journal-title":"Rev. Fish. Sci."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1086\/282171","article-title":"The Paradox of the Plankton","volume":"95","author":"Hutchinson","year":"1961","journal-title":"Am. Nat."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"13661","DOI":"10.1073\/pnas.240242797","article-title":"Chaotic flow: The physics of species coexistence","volume":"97","author":"Scheuring","year":"2000","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.1098\/rspb.2000.1212","article-title":"Mesoscale vortices and the paradox of the plankton","volume":"267","author":"Bracco","year":"2000","journal-title":"Proc. R. Soc. Lond. Ser. B Biol. Sci."},{"key":"ref_112","doi-asserted-by":"crossref","unstructured":"Alvain, S., Moulin, C., Dandonneau, Y., and Loisel, H. (2008). Seasonal distribution and succession of dominant phytoplankton groups in the global ocean: A satellite view. Glob. Biogeochem. Cycles, 22.","DOI":"10.1029\/2007GB003154"},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"18366","DOI":"10.1073\/pnas.1004620107","article-title":"Fluid dynamical niches of phytoplankton types","volume":"107","author":"Alvain","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.physd.2013.05.005","article-title":"Leaving flatland: Diagnostics for Lagrangian coherent structures in three-dimensional flows","volume":"258","author":"Sulman","year":"2013","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"3530","DOI":"10.1016\/j.cnsns.2013.05.002","article-title":"Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems","volume":"18","author":"Mancho","year":"2013","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"254022","DOI":"10.1088\/1751-8113\/46\/25\/254022","article-title":"Characterization of coherent structures in three-dimensional turbulent flows using the finite-size Lyapunov exponent","volume":"46","author":"Bettencourt","year":"2013","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.physd.2014.01.007","article-title":"Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows","volume":"273\u2013274","author":"Blazevski","year":"2014","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"2307","DOI":"10.1175\/2010JAS3176.1","article-title":"Lagrangian Coherent Structures near a Subtropical Jet Stream","volume":"67","author":"Tang","year":"2010","journal-title":"J. Atmos. Sci."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.ocemod.2012.04.004","article-title":"Oceanic three-dimensional Lagrangian coherent structures: A study of a mesoscale eddy in the Benguela upwelling region","volume":"51","author":"Bettencourt","year":"2012","journal-title":"Ocean Model."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1038\/ngeo2570","article-title":"Boundaries of the Peruvian oxygen minimum zone shaped by coherent mesoscale dynamics","volume":"8","author":"Bettencourt","year":"2015","journal-title":"Nat. Geosci."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"7433","DOI":"10.1002\/2017JC012700","article-title":"Characterization of the structure and cross-shore transport properties of a coastal upwelling filament using three-dimensional finite-size Lyapunov exponents","volume":"122","author":"Bettencourt","year":"2017","journal-title":"J. Geophys. Res. Ocean."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"379","DOI":"10.5194\/npg-24-379-2017","article-title":"Insights into the three-dimensional Lagrangian geometry of the Antarctic polar vortex","volume":"24","author":"Curbelo","year":"2017","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"013111","DOI":"10.1063\/1.3056050","article-title":"Distinguished trajectories in time dependent vector fields","volume":"19","author":"Madrid","year":"2009","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"102790","DOI":"10.1016\/j.pocean.2022.102790","article-title":"Marine life at Lagrangian fronts","volume":"204","author":"Prants","year":"2022","journal-title":"Prog. Oceanogr."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/10\/1942\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:10:12Z","timestamp":1760130612000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/10\/1942"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,20]]},"references-count":124,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["sym15101942"],"URL":"https:\/\/doi.org\/10.3390\/sym15101942","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,20]]}}}