Abstract
Within the framework of the quasi-geostrophic approximation, the interactions of two identical initially circular vortex patches are studied using the contour dynamics/surgery method. The cases of barotropic vortices and of vortices in the upper layer of a two-layer fluid are considered. Diagrams showing the end states of vortex interactions and, in particular, the new regime of vortex triplet formation are constructed for a wide range of external parameters. This paper shows that, in the nonlinear evolution of two such (like-signed) vortices, the filaments and vorticity fragments surrounding the merged vortex often collapse into satellite vortices. Therefore, the conditions for the formation and the quasi-steady motions of a new type of triplet-shaped vortex structure are obtained.
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Assassi C, Morel Y, Vandermeirsch F, Chaigneau A, Pegliasco C, Morrow RM, Colas F, Fleury S, Carton X, Klein P, Cambra R (2016) An index to distinguish surface and subsurface intensified vortices from surface observations. J Phys Oceanogr 46(8):2529–2552
Bambrey RR, Reinaud JN, Dritschel DG (2007) Strong interactions between two co-rotating quasi-geostrophic vortices. J Fluid Mech 592:117–133
Bertrand C, Carton XJ (1993) Vortex merger on the beta-plane. C R Acad Sci Paris 316:1201–1206
Capéran P, Verron J (1988) Numerical simulation of a physical experiment on two-dimensional vortex merger. Fluid Dyn Res 3(1–4):87–92
Capuano TA, Speich S, Carton X, Blanke B (2018) Mesoscale and submesoscale processes in the Southeast Atlantic and their impact on the regional thermohaline structure. J Geophys Res Oceans 123:1961. https://doi.org/10.1002/2017JC013396 to appear (online 12 March 2018)
Carnevale GF, Cavazza P, Orlandi P, Purini R (1991) An explanation for anomalous vortex merger in rotating tank experiments. Phys Fluids A 3:1411–1415
Carton XJ (1992) On the merger of shielded vortices. Europhys Lett 18:697–703
Carton XJ (2001) Hydrodynamical modelling of oceanic vortices. Surv Geophys 22(3):179–263
Carton X, Maze G, Legras B (2002) A two-dimensional vortex merger in an external strain field. J Turbul 3:045
Carton X, Ciani D, Verron J, Reinaud J, Sokolovskiy M (2016) Vortex merger in surface quasi-geostrophy. Geophys Astrophys Fluid Dyn 110(1):1–22
Carton X, Morvan M, Reinaud JN, Sokolovskiy MA, L’Hegaret P, Vic C (2017) Vortex merger near topographic slope in a homogeneous rotating fluid. Regular Chaotic Dyn 22(5):455–478
Cerretelli C, Williamson CHK (2003) The physical mechanism for vortex merging. J Fluid Mech 475:41–77
Chelton DB, Schlax MG, Samelson RM (2011) Global observations of nonlinear mesoscale eddies. Prog Oceanogr 91(2):167–216
Christiansen JP, Zabusky NJ (1973) Instability, coalescence and fission of finite-area vortex structures. J Fluid Mech 61(part2):219–243
Ciani D, Carton X, Verron J (2016) On the merger of subsurface isolated vortices. Geophys Astrophys Fluid Dyn 110(1):23–49
Ciani D, Carton X, Barbosa Aguiar AC, Peliz A, Bashmachnikov I, Ienna F, Chapron B, Santolieri R (2017) Surface signature of Mediterranean water eddies in a long-term high-resolution numerical model. Deep Sea Res Part I Oceanogr Res Pap 130:12–29
Delbende I, Piton B, Rossi M (2015) Merging of two helical vortices. Eur J Mech B Fluid 49(Part B):363–372
Dritschel DG (1986) The nonlinear evolution of rotating configurations of uniform vorticity. J Fluid Mech 172:157–182
Dritschel DG, Waugh DW (1992) Quantification of the inelastic interaction of inequal vortices in two-dimensional vortex dynamics. Phys Fluids A 4(8):1737–1744
Dritschel DG, Zabusky NJ (1996) On the nature of vortex interactions and models in unforced nearly-inviscid two-dimensional turbulence. Phys Fluids 8(5):1252–1256
Filyushkin BN, Sokolovskiy MA (2011) Modeling the evolution of intrathermocline lenses in the Atlantic Ocean. J Mar Res 69(2–3):191–220
Freymuth P, Bank W, Palmer M (1984) First experimental evidence of vortex splitting. Phys Fluids 27(5):1045–1046
Freymuth P, Bank W, Palmer M (1985) Futher experimental evidence of vortex splitting. J Fluid Mech 152:289–299
Hopfinger EJ, van Heijst GJF (1993) Vortices in rotating fluids. Ann Rev Fluid Mech 25:241–289
Katsumata K (2016) Eddies observed by Argo floats. Part I. Eddy transport in the upper 1000 dbar. J Phys Oceanogr 46(11):3471–3486
Kirchhoff G (1876) Vorlesungen űber mathematische Physik: Mechanik. Taubner, Leipzig
Kozlov VF, Makarov VG (1984) Evolution modeling of unstable geostrophic eddies in a barotropic ocean. Oceanology 24(5):556–560
Lamb H (1932) Hydrodynamics, 6th edn. Cambridge University Press, Cambridge
Love AEH (1893) On the stability of certain vortex motion. Proc Lond Math Soc s1–25:18–43
Melander MV, Zabusky NJ, McWilliams JC (1988) Symmetric vortex merger in two dimensions: causes and conditions. J Fluid Mech 195:303–340
Meunier P, Leweke T (2001) Three-dimensional instability during vortex merging. Phys Fluids 13(10):2747–2751
Meunier P, Ehrenstein U, Leweke T, Rossi M (2002) A merger criterion for two-dimensional co-rotating vortices. Phys Fluids 14(8):2757–2766
Overman EA, Zabusky NJ (1982) Evolution and merger of isolated vortex structures. Phys Fluids 25:1297–1305
Polvani LM, Zabusky NJ, Flierl GR (1989) Two-layer geostrophic vortex dynamics: 1. Upper-layer V-states and merger. J Fluid Mech 205:215–242
Reinaud JN, Dritschel DG (2002) The merger of vertically offset quasi-geostrophic vortices. J Fluid Mech 469:287–315
Reinaud JN, Dritschel DG (2005) The critical merger distance between two co-rotating quasi-geostrophic vortices. J Fluid Mech 522:357–381
Roberts KV, Christiansen JP (1972) Topics in computational fluid mechanics. Comput Phys Commun 3(Suppl 1):14–32
Roshko A (1976) Structure of turbulent shear flows: a new look. AIAA Journal 14th Aerospace Sciences Meeting Paper No 76–78
Rousselet L, Doglioli A, Maes C, Blanke B, Petrenko A (2016) Impacts of mesoscale activity on the water masses and circulation in the Coral Sea. J Geophys Res Oceans 121:7277–7289
Saffman PG, Szeto R (1980) Equilibrium shape of a pair of equal vortices. Phys Fluids 23(12):2339–2342
Sokolovskiy MA, Verron J (2000) Finite-core hetons: stability and interactions. J Fluid Mech 423:127–154
Sokolovskiy MA, Verron J (2014) Dynamics of vortex structures in a stratified rotating fluid. Series atmospheric and oceanographic sciences library, vol 47. Springer, Switzerland
Valcke S, Verron J (1993) On interactions between two finite-core hetons. Phys Fluids A 5(8):2058–2060
Verron J, Hopfinger E (1991) The enigmatic merging conditions of two-layer baroclinic vortices. C R Acad Sci Paris Ser II 313(7):737–742
Verron J, Valcke S (1994) Scale-dependent merging of baroclinic vortices. J Fluid Mech 264:81–106
Verron J, Hopfinger E, McWilliams JC (1990) Sensitivity to initial conditions in the merging of two-layer baroclinic vortices. Phys Fluids A 2(6):886–889
von Hardenberg J, McWilliams JC, Provenzale A, Shchepetkin A, Weiss JB (2000) Vortex merging in quasi-geostrophic flows. J Fluid Mech 412:331–353
Waugh DW (1992) The efficiency of symmetric vortex merger. Phys Fluids A 4(8):1745–1758
Winant CD, Browand FK (1974) Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J Fluid Mech 63(2):237–255
Zabusky NJ, Hughes MH, Roberts KV (1979) Contour dynamics for the Euler equations in two dimensions. J Comput Phys 30(1):96–106
Zhang Z, Zhong Y, Tian J, Yang Q, Zhao W (2014) Estimation of eddy heat transport in the global ocean from Argo data. Acta Oceanol Sin 33(1):42–47
Funding
This is a contribution to PRC CNRS/RFBR 1069/16-55-150001. From the side of the MAS work was carried out within the framework of the state task no. 0149-2018-0001. This study receive support from Russian Foundation for Basic Research (Project No 16-05-00121), Russian Scientific Foundation (Project No. 14-50-00095), Ministry of Education and Science of Russian Federation (Project No. 14.W.03.31.0006).
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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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Sokolovskiy, M.A., Verron, J. & Carton, X.J. The formation of new quasi-stationary vortex patterns from the interaction of two identical vortices in a rotating fluid. Ocean Dynamics 68, 723–733 (2018). https://doi.org/10.1007/s10236-018-1163-7
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DOI: https://doi.org/10.1007/s10236-018-1163-7