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Dynamics of localized vortices on the beta plane. (English) Zbl 1204.76014

Summary: This paper studies the joint influence that rotation and the earth’s sphericity have on the dynamics of localized synoptic scale vortices within the quasi-geostrophic barotropic model in the beta-plane approximation. Rossby solitons (two-dimensional vortices exponentially localized in space which propagate without changing their form along the latitude circles) are considered in the first part of the article. The general properties of such solutions are discussed. The simplest examples are presented, and a brief review of the main results is given. The second part is dedicated to the theory of nonstationary monopoles. The physical mechanisms governing the evolution of such vortices are described; different stages of this evolution are determined for intense vortices. Analytical and numerical results are used to confirm the qualitative explanations.

MSC:

76F06 Transition to turbulence
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
86A05 Hydrology, hydrography, oceanography
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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[1] G. I. Marchuk, V. P. Dymnikov, and V. B. Zalesnyi, Mathematical Models in Geophysical Hydrodynamics and Numerical Methods of Their Realization (Gidrometeoizdat, Leningrad, 1987) [in Russian].
[2] J. Pedlosky, Geophysical Fluid Dynamics (Springer, Heidelberg, 1981; Mir, Moscow, 1984). · Zbl 0429.76001
[3] V. D. Larichev, ”Quasigeostrophic Nonlinear Waves and Turbulence in the Ocean,” Doctoral Dissertation in Physics and Mathematics (IO AN SSSR, Moscow, 1987) [in Russian].
[4] G. R. Flierl, V. D. Larichev, J. C. McWilliams, et al., ”The Dynamics of Baroclinic and Barotropic Solitary Eddies,” Dyn. Atmos. Oceans 5(1), 1–41 (1980).
[5] Z. Kizner, D. Berson, G. M. Reznik, et al., ”Baroclinic Topographic Modons on the beta-Plane,” Geoph. Astr. Fluid Dyn. 97(3), 175–211 (2003). · Zbl 1206.86014
[6] G. M. Reznik, ”Point Vortices on {\(\beta\)}-Plane and Rossby Solitons,” Okeanologiya 26(2), 165–173 (1986).
[7] V. M. Gryanik, ”Singular Geostrophic Vortices on {\(\beta\)}-Plane as a Model of Synoptic Vortices,” Okeanologiya 26(2), 174–179 (1986).
[8] G. M. Reznik, ”Singular Vortices on {\(\beta\)}-Plane,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 28(4), 398–405 (1992). · Zbl 0756.76015
[9] G. M. Reznik, ”Dynamics of Singular Vortices on a Beta-Plane,” J. Fluid Mech. 240, 405–432 (1992). · Zbl 0756.76015
[10] G. R. Flierl, ”Isolated Eddy Models in Geophysics,” Ann. Rev. Fluid Mech. 19, 493–530 (1987).
[11] V. D. Larichev and G. M. Reznik, ”Two-Dimensional Rossby Solitons,” Dokl. Akad. Nauk 231(5), 1077–1079 (1976).
[12] M. E. Stern, ”Minimal Properties of Planetary Eddies,” J. Mar. Res. 33(1), 1–13 (1975).
[13] G. M. Reznik, ”The Structure and Dynamics of Two-Dimensional Rossby Soliton,” Okeanologiya 27(5), 716–720 (1987).
[14] V. V. Meleshko and G. J. F. van Heijst, ”On Chaplygin’s Investigations of Two-Dimensional Vortex Structures in an Inviscid Fluid,” J. Fluid Mech. 272, 157–182 (1994). · Zbl 0819.76018
[15] W. T. M. Verkley, ”The Construction of Barotropic Modons on a Sphere,” J. Atmos. Sci. 41(16), 2492–2504 (1984).
[16] J. J. Tribbia, ”Modons in Spherical Geometry,” Geophys. Astrophys. Fluid Dyn. 30(1–2), 131–168 (1984).
[17] K. V. Klyatskin and G. M. Reznik, ”Point Vortices on a Rotating Sphere,” Okeanologiya 29(1), 21–27 (1989).
[18] G. M. Reznik, ”Exact Solution for Two-Dimensional Topographic Rossby Soliton,” Dokl. Akad. Nauk 282(4), 981–985 (1985).
[19] A. L. Berestov, ”Rossby Solitons,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 15(6), 648–654 (1979).
[20] Z. I. Kizner, ”Rossby Solitons with Axisymmetric Baroclinic Modons,” Dokl. Akad. Nauk 282, 1495–1498 (1984).
[21] G. M. Reznik and G. G. Sutyrin, ”Baroclinic Topographic Modons,” J. Fluid Mech. 431, 121–142 (2001). · Zbl 1054.76014
[22] Z. Kizner, D. Berson, and R. Khvoles, ”Non-Circular Baroclinic beta-Plane Modons: Constructing Stationary Solutions,” J. Fluid Mech. 489, 199–228 (2003). · Zbl 1055.76055
[23] R. Khvoles, D. Berson, and Z. Kizner, ”The Structure and Evolution of Barotropic Elliptical Modons,” J. Fluid Mech. 530, 1–30 (2005). · Zbl 1071.76010
[24] E. S. Benilov, ”Beta-Induced Translation of Strong Isolated Eddies,” J. Phys. Oceanogr. 26, 2223–2229 (1996).
[25] Z. Kizner, G. Reznik, B. Fridman, et al., ”Shallow-Water Modons on the F-Plane,” J. Fluid Mech. 603, 305–329 (2008). · Zbl 1151.76618
[26] S. V. Muzylev and G. M. Reznik, ”On Proofs of Stability of Drift Vortices in Magnetized Plasmas and Rotating Fluids,” Phys. Fluids 4(9), 2841–2843 (1992).
[27] J. Nycander, ”Refutation of Stability Proofs for Dipole Vortices,” Phys. Fluids A 4(3), 467–476 (1992). · Zbl 0748.76056
[28] G. M. Reznik and Z. Kizner, ”Two-Layer Quasigeostrophic Singular Vortices Embedded in Regular Flow. Pt I. Invariants of Motion and Stability of Vortex Pairs,” J. Fluid Mech. 584 185–202 (2007). · Zbl 1118.76070
[29] J. C. McWilliams, G. R. Flierl, V. D. Larichev, et al., ”Numerical Studies of Barotropic Modons,” Dyn. Atm. Oceans 5(4), 219–238 (1981).
[30] G. E. Swaters, ”Ekman Layer Dissipation in an Eastward-Traveling Modon,” J. Phys. Oceanogr. 15(9), 1212–1216 (1985).
[31] G. F. Carnevale, G. K. Vallis, R. Purini, et al., ”Propagation of Barotropic Modons over Topography,” Geophys. Astrophys. Fluid Dyn. 41(1–2), 45–101 (1988). · Zbl 0643.76025
[32] G. E. Swaters, ”Barotropic Modon Propagation over Slowly Varying Topography,” Geophys. Astrophys. Fluid Dyn. 361(2), 85–113 (1986). · Zbl 0597.76019
[33] M. Makino, T. Kamimura, and T. Taniuti, ”Dynamics of Two-Dimensional Solitary Vortices in a Low-{\(\beta\)} Plasma with Convective Motion,” J. Phys. Soc. Jpn. 50(3), 980–989 (1981).
[34] V. D. Larichev and G. M. Reznik, ”Numerical Experiments on Studying Collisions of Two-Dimensional Rossby Solitons,” Dokl. Akad. Nauk 264(1), 229–233 (1982).
[35] M. Swenson, ”Instability of Equivalent-Barotropic Riders,” J. Phys. Oceanogr. 17(4), 492–506 (1987).
[36] Z. Kizner, D. Berson, and R. Khvoles, ”Baroclinic Modon Equilibria on the Beta-Plane: Stability and Transitions,” J. Fluid Mech. 468, 239–260 (2002). · Zbl 1019.76020
[37] G. M. Reznik and S. Kravtsov, ”Dynamics of Barotropic Singular Monopole on the {\(\beta\)}-Plane,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 32(6), 762–769 (1996).
[38] G. G. Sutyrin, J. S. Hesthaven, J. P. Lynov, et al., ”Dynamical Properties of Vortical Structures on the beta-Plane,” J. Fluid Mech. 268, 103–131 (1994). · Zbl 0818.76007
[39] G. M. Reznik and W. Dewar, ”An Analytical Theory of Distributed Axisymmetric Barotropic Vortices on the Beta-Plane,” J. Fluid Mech. 269, 301–321 (1994). · Zbl 0809.76013
[40] G. M. Reznik, R. Grimshaw, and E. Benilov, ”On the Long-Term Evolution of an Intense Localized Divergent Vortex on the beta-Plane,” J. Fluid Mech. 422, 249–280 (2000). · Zbl 1003.76015
[41] W. Horton, ”Drift Wave Vortices and Anomalous Transport,” Phys. Fluids 1(3), 524–536 (1989).
[42] G. G. Sutyrin and G. R. Flierl, ”Intense Vortex Motion on the Beta Plane: Development of the beta Gyres,” J. Atmos. Sci. 51(5), 773–790 (1994).
[43] V. M. Kamenkovich, M. N. Koshlyakov, and A. S. Monin, Synoptic Vortices in the Ocean (Gidrometeoizdat, Leningrad, 1982) [in Russian].
[44] G. M. Reznik, R. Grimshaw, and K. Sriskandarajah, ”On Basic Mechanisms Governing Two-Layer Vortices on a Beta-Plane,” Geoph. Astrophys. Fluid. Dyn. 86(1–4), 1–42 (1997).
[45] J. C. McWilliams and G. R. Flierl, ”On the Evolution of Isolated, Nonlinear Vortices,” J. Phys. Oceanogr. 9(6), 1155–1167 (1979).
[46] R. P. Mied and G. J. Lindemann, ”The Birth and Evolution of Eastward-Propagating Modons,” J. Phys. Oceanogr. 12(3), 213–230 (1982).
[47] G. M. Reznik and Z. Kizner, ”Two-Layer Quasigeostrophic Singular Vortices Embedded in a Regular Flow. Pt II: Steady and Unsteady Drift of Individual Vortices on a Beta-Plane,” J. Fluid Mech. 584, 203–223 (2007). · Zbl 1118.76071
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