Abstract
A simple model of fluid particle advection induced by the interaction of a point vortex and incident plane flow occurring near a curved boundary is analyzed. The use of the curved boundary in this case is aimed at mimicking the geometry of an isolated bay of a circular shape. An introduction of such a boundary to the model results in the appearance of retention zones, where the vortex can be permanently trapped being either stationary or periodically oscillating. When stationary, it induces a steady velocity field that in turn ensures regular advection of nearby fluid particles. When the vortex oscillates periodically, the induced velocity field turns unsteady leading to the manifestation of chaotic advection of fluid particles. We show that the size of the fluid region engaged into chaotic advection increases almost monotonically with the increased magnitude of the vortex oscillations provided the magnitude remains relatively small. The monotonicity is accounted for the fact that the frequency of the vortex oscillations incommensurable with the proper frequency of fluid particle rotations in the steady state. Another point of interest is that it is demonstrated that bounded regions, in which the vortex may be trapped, can appear even at a significant distance from the bay. Making use of a Lagrangian indicator, examples of fluid particle advection induced by the periodic motion of the vortex inside the bay are adduced.
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References
Allen J, Samelson RM, Newberger P (1991) Chaos in a model of forced quasi-geostrophic flow over topography: an application of Melnikov’s method. J Fluid Mech 226:511–547
Aref H (1984) Stirring by chaotic advection. J Fluid Mech 143:1– 21
Aref H (2002) The development of chaotic advection. Phys Fluids 14:1315–1325
Baines PG (1993) Topographic effects in stratified flows. Cambridge University Press, Cambridge
Baines PG, Smith RB (1993) Upstream stagnation points in stratified flow past obstacles. Dyn Atmos Oceans 18:105–113
Balasuriya S (2005) Optimal perturbation for enhanced chaotic transport. Physica D 202:155–176. https://doi.org/10.1016/j.physd.2004.11.018
Barbosa Aguiar AC, Peliz A, Carton X (2013) A census of meddies in a long-term high-resolution simulation. Prog Oceanogr 116:80–94. https://doi.org/10.1016/j.pocean.2013.06.016
Budyansky M, Uleysky M, Prants S (2004a) Chaotic scattering, transport, and fractals in a simple hydrodynamic flow. J Exp Theor Phys 99:1017–1027
Budyansky M, Uleysky M, Prants S (2004b) Hamiltonian fractals and chaotic scattering of passive particles by a topographical vortex and an alternating current. Physica D 195:369–378. https://doi.org/10.1016/j.physd.2003.11.013
Budyansky MV, Uleysky MY, Prants SV (2009) Detection of barriers to cross-jet Lagrangian transport and its destruction in a meandering flow. Phys Rev E 79:056,215. https://doi.org/10.1103/PhysRevE.79.056215
Chelton DB, Schlax MG, Samelson RM, de Szoeke RA (2007) Global observations of large oceanic eddies. Geophys Res Lett 34:L15,606
Chelton DB, Schlax MG, Samelson RM (2011) Global observations of nonlinear mesoscale eddies. Prog Oceanogr 91(2):167–216
del-Castillo-Negrete D, Morrison P (1993) Chaotic transport by rossby waves in shear flow. Phys Fluids 5:948–965
d’Ovidio F, Isern-Fontanet J, López C, Hernández-García E, García-Ladona E (2009) Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the algerian basin. Deep Sea Res 56:15–31. https://doi.org/10.1016/j.dsr.2008.07.014
Duran-Matute M, Velasco-Fuentes OU (2008) Passage of a barotropic vortex through a gap. J Phys Oceanogr 38:2817–2831. https://doi.org/10.1175/2008JPO3887.1
Gryanik VM, Doronina TN (1990) Advective transport of passive mixture by localized (point) geostrophic vortices in the atmosphere (ocean). Izv Atmos Ocean Phys 26:1011–1026
Haller G (2015) Lagrangian coherent structures. Annuv Rev Fluid Mech 47:137–162. https://doi.org/10.1146/annurev-fluid-010313-141322
Haller G, Poje A (1998) Finite time transport in aperiodic flows. Physica D 119:352–380. https://doi.org/10.1016/S0167-2789(98)00091-8
Haller G, Yuan G (2000) Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147:352–370. https://doi.org/10.1016/S0167-2789(00)00142-1
Izrailsky YG, Kozlov VF, Koshel KV (2004) Some specific features of chaotization of the pulsating barotropic flow over elliptic and axisymmetric sea-mounts. Phys Fluids 16:3173–3190
Izrailsky YG, Koshel KV, Stepanov DV (2008) Determination of optimal excitation frequency range in background flows. Chaos 18(1):013,107. https://doi.org/10.1063/1.2835349
Koshel KV, Izrailsky YG, Stepanov DV (2006a) Determining the optimal frequency of perturbation in the problem of chaotic transport of particles. Dokl Phys 51:219–222. https://doi.org/10.1134/S102833580604015X
Koshel KV, Prants SV (2006b) Chaotic advection in the ocean. Phys Usp 49(11):1151–1178. https://doi.org/10.1070/PU2006v049nl1ABEH006066
Koshel KV, Stepanov DV (2005) Boundary effect on the mixing and transport of passive impurities in a nonstationary flow. Tech Phys Lett 31:135–137. https://doi.org/10.1134/1.1877626
Koshel KV, Sokolovskiy MA, Davies PA (2008) Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle. Fluid Dyn Res 40:695–736. https://doi.org/10.1016/j.fluiddyn.2008.03.001
Koshel KV, Ryzhov EA, Zhmur VV (2015) Effect of the vertical component of diffusion on passive scalar transport in an isolated vortex model. Phys Rev E 92:053,021. https://doi.org/10.1103/PhysRevE.92.053021
Kozlov VF, Koshel KV (1999) Barotropic model of chaotic advection in background flows. Izv Atmos Ocean Phys 35:638–648
Kuznetsov L, Zaslavsky GM (1998) Regular and chaotic advection in the flow field of a three-vortex system. Phys Rev E 58:7330–7349
Kuznetsov L, Zaslavsky GM (2000) Passive particle transport in three-vortex flow. Phys Rev E 61:3777–3792
Lee WK, Taylor PH, Borthwick AGL, Chuenkhum S (2010) Vortex-induced chaotic mixing in wavy channels. J Fluid Mech 654:501–538. https://doi.org/10.1017/S0022112010000674
Lichtenberg A, Lieberman M (1983) Regular and stochastic motion. Springer, New York
Lichtenberg AJ, Lieberman MA (1992) Regular and chaotic dynamics, 2 edn. Springer, Berlin
Lin C (1941) On the motion of vortices in two dimensions. I. Existence of the Kirchhoff–Routh function. Proc Nat Acad Sci 27:570–575
Lipphardt BL, Small D, Kirwan AD, Wiggins S, Ide K, Grosch CE, Paduan JD (2006) Synoptic Lagrangian maps: application to surface transport in monterey bay. J Mar Res 64:221–247. https://doi.org/10.1357/002224006777606461
Milne–Thomson L (1968) Theoretical hydrodynamics. Macmillan, London
Noack BR, Mezic I, Tadmor G, Banaszuk A (2004) Optimal mixing in recirculation zones. Phys Fluids 16:867–888. https://doi.org/10.1063/1.1645276
Pierrehumbert RT, Yang H (1993) Global chaotic mixing on isentropic surfaces. J Atmos Sci 50:2462–2480
Polvani LM, Wisdom J (1990) Chaotic Lagrangian trajectories around an elliptical vortex patch embedded in a constant and uniform background shear flow. Phys Fluids A 2:123–126. https://doi.org/10.1063/1.857814
Prants SV (2013) Dynamical systems theory methods to study mixing and transport in the ocean. Phys Scr 87:038,115. https://doi.org/10.1088/0031-8949/87/03/038115
Prants SV (2014) Chaotic lagrangian transport and mixing in the ocean. Eur Phys J Spec Top 223:2723–2743. https://doi.org/10.1140/epjst/e2014-02288-5
Prants SV (2015) Backward-in-time methods to simulate chaotic transport and mixing in the ocean. Phys Scr 90:074,054. https://doi.org/10.1088/0031-8949/90/7/074054
Prants SV, Budyansky MV, Uleysky MY (2017a) Lagrangian simulation and tracking of the mesoscale eddies contaminated by Fukushima-derived radionuclides. Ocean Sci 13:453–463. https://doi.org/10.5194/os-13-453-2017
Prants SV, Uleysky MY, Budyansky MV (2017b) Lagrangian oceanography: large-scale transport and mixing in the ocean. Springer International Publishing. https://doi.org/10.1007/978-3-319-53022-2
Rom-Kedar V, Leonard A, Wiggins A (1990) An analytical study of transport, mixing and chaos in an unsteady vortical flow. J Fluid Mech 214:347–394
Rypina II, Scott SE, Pratt LJ, Brown MG (2011) Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures. Nonlinear Process Geophys 18:977–987
Ryzhov EA (2011) On changing the size of the atmosphere of a vortex pair embedded in a periodic external shear flow. Phys Lett A 375:3884–3889
Ryzhov EA, Koshel KV (2010) Chaotic transport and mixing of a passive admixture by vortex flows behind obstacles. Izv Atmos Ocean Phys 46(2):184–191
Ryzhov EA, Koshel KV (2011a) The effects of chaotic advection in a three-layer ocean model. Izv Atmos Ocean Phys 47(2):241–251
Ryzhov EA, Koshel KV (2011b) Estimating the size of the regular region of a topographically trapped vortex. Geophys Astrophys Fluid Dyn 105:536–551. https://doi.org/10.1080/03091929.2010.511205
Ryzhov EA, Koshel KV (2011c) Ventilation of a trapped topographic eddy by a captured free eddy. Izv Atmos Ocean Phys 47(2):780–791
Ryzhov EA, Koshel KV (2015) Global chaotization of fluid particle trajectories in a sheared two-layer two-vortex flow. Chaos 25:103,108. https://doi.org/10.1063/1.4930897
Ryzhov EA, Koshel KV (2016a) Resonance phenomena in a two-layer two-vortex shear flow. Chaos 26:113,116. https://doi.org/10.1063/1.4967805
Ryzhov EA, Koshel KV (2016b) Steady and perturbed motion of a point vortex along a boundary with a circular cavity. Phys Lett A 380:896–902. https://doi.org/10.1016/j.physleta.2015.12.043
Ryzhov EA, Sokolovskiy MA (2016) Interaction of a two-layer vortex pair with a submerged cylindrical obstacle in a two layer rotating fluid. Phys Fluids 28:056,602. https://doi.org/10.1063/1.4947248
Ryzhov E, Koshel K, Stepanov D (2010) Background current concept and chaotic advection in an oceanic vortex flow. Theor Comput Fluid Dyn 24:59–64. https://doi.org/10.1007/s00162-009-0170-1
Ryzhov EA, Izrailsky YG, Koshel KV (2014) Vortex dynamics of a fluid near a boundary with a circular cavity. Iz Atmos Ocean Phys 50:420–425. https://doi.org/10.1134/S0001433814040203
Saffman PG (1992) Vortex dynamics. Cambridge University Press, Cambridge
Samelson RM (1992) Fluid exchange across a meandering jet. J Phys Oceanogr 22:431–440. https://doi.org/10.1175/1520-0485(1992)022〈0431:FEAAMJ〉2.0.CO;2
Shagalov SV, Reutov VP, Rybushkina GV (2010) Asymptotic analysis of transition to turbulence and chaotic advection in shear zonal flows on a beta-plane. Izv Atmos Ocean Phys 46:95–108. https://doi.org/10.1134/S0001433810010135
Sokolovskiy MA, Zyryanov VN, Davies PA (1998) On the influence of an isolated submerged obstacle on a barotropic tidal flow. Geophys Astrophys. Fluid Dyn 88:1–30
Sulman MHM, Huntley HS, Lipphardt BL, Kirwan AD (2013) Leaving flatland: diagnostics for Lagrangian coherent structures in three-dimensional flows. Physica D 258:77–92. https://doi.org/10.1016/j.physd.2013.05.005
Uleysky MY, Budyansky MV, Prants SV (2010) Mechanism of destruction of transport barriers in geophysical jets with rossby waves. Phys Rev E 81:017,202. https://doi.org/10.1103/PhysRevE.81.017202
Zaslavsky GM (1998) Physics of chaos in Hamiltonian dynamics. Imperial College Press, London
Funding
The reported study was partially supported by the POI FEB RAS Program “Mathematical simulation and analysis of dynamical processes in the ocean” (117030110034-7) and by the Russian Foundation for Basic Research, project no. 17 − 05 − 00035. EAR was partially supported by the Ministry of Education and Science of Russian Federation, project no. MK − 172.2017.1. The work of KVK in obtaining the analytical relations for the positions of the critical points in the steady state was supported by the Russian Science Foundation, project no. 16 − 17 − 10025.
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Responsible Editor: Sergey Prants
This article is part of the Topical Collection on the International Conference “Vortices and coherent structures: from ocean to microfluids”, Vladivostok, Russia, 28–31 August 2017
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Ryzhov, E.A., Koshel, K.V. Advection of passive scalars induced by a bay-trapped nonstationary vortex. Ocean Dynamics 68, 411–422 (2018). https://doi.org/10.1007/s10236-018-1140-1
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DOI: https://doi.org/10.1007/s10236-018-1140-1