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Dynamics of singular vortices on a beta-plane

Published online by Cambridge University Press:  26 April 2006

G. M. Reznik
G. M. Reznik
Affiliation:
P. P. Shirshov Institute of Oceanology of the Academy of Sciences of Russia. Krasikova, 23, 117218, Moscow, Russia
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Abstract

A new singular-vortex theory is presented for geostrophic, beta-plane dynamics. The stream function of each vortex is proportional to the modified Bessel function Ko(pr), where p can be an arbitrary positive constant. If p−1 is equal to the Rossby deformation scale Rd, then the vortex is a point vortex; for p−1 ≠ Rd the relative vorticity of the vortex contains an additional logarithmic singularity. Owing to the β-effect, the redistribution of the background potential vorticity produced by the vortices generates a regular field in addition to the velocity field induced by the vortices themselves. Equations governing the joint evolution of singular vortices and the regular field are derived. A new invariant of the motion is found for this system. If the vortex amplitudes and coordinates are set in a particular way then the regular field is zero, and the vortices form a system moving along latitude circles at a constant speed lying outside the range of the phase velocity of linear Rossby waves. Each of the systems is a discrete two-dimensional Rossby soliton and, vice versa, any distributed Rossby soliton is a superposition of the singular vortices concentrated in the interior region of the soliton. An individual singular vortex is studied for times when Rossby wave radiation can be neglected. Such a vortex produces a complicated spiral-form regular flow which consists of two dipoles with mutually perpendicular axes. The dipoles push the vortex westward and along the meridian (cyclones move northward, and anticyclones move southward). The vortex velocity and trajectory are calculated and applications to oceanic and atmospheric eddies are given.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 240 , July 1992 , pp. 405 - 432
Copyright
© 1992 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. 1964 Handbook of Mathematical Functions. National Bureau of Standards.
Adem, J. 1956 A series solution for the barotropic vorticity equation and its application to the study of atmospheric vortices. Tellus 8, 364.Google Scholar
Bogomolov, V. A. 1977 The dynamics of vorticity on a sphere. Izv. Akad. Nauk SSSR, Mekh. Ghid. Gaza 13, 57.Google Scholar
Bogomolov, V. A. 1979 On the two-dimensional hydrodynamics on a sphere. Izv. Akad. Nauk SSSR, Phys. Atmos. Okeana 15, 29.Google Scholar
Bogomolov, V. A. 1985 On the vortex motion on a rotating sphere. Izv. Akad. Nauk SSSR, Phys. Atmos. Okeana 21, 391.Google Scholar
Firing, E. & Beardsley, R. 1976 The behavior of a barotropic eddy on a β-plane. J. Phys. Oceanogr. 6, 57.Google Scholar
Flierl, G. R. 1987 Isolated eddy models in geophysics. Ann. Rev. Fluid Mech. 19, 493.Google Scholar
Flierl, G. R., Larichev, V. D., McWilliams, J. C. & Reznik, G. M. 1980 The dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans 5, 1.Google Scholar
Flierl, G. R., Stern, M. E. & Whitehead, J. A. 1983 The physical significance of modons: Laboratory experiments and general integral constraints. Dyn. Atmos. Oceans 7, 233.Google Scholar
Gryanik, V. M. 1983a The dynamics of localized vortices: ‘vortex charges’ in baroclinic fluid. Izv. Akad. Nauk SSSR, Phys. Atmos. Okeana 19, 467.Google Scholar
Gryanik, V. M. 1983b The dynamics of singular geostrophic vortices in two-level model of the atmosphere (ocean). Izv. Akad. Nauk SSSR, Phys. Atmos. Okeana 19, 227.Google Scholar
Gryanik, V. M. 1986 Singular geostrophic vortices on a β-plane as a model of synoptic eddies. Okeanologiya 26, 174. (Transl. Oceanology 26, 126.)Google Scholar
Gryanik, V. M. 1988 Localized vortices: ‘vortex charges’ and ‘vortex filaments’ in baroclinic differentially rotating fluid. Izv. Akad. Nauk SSSR, Phys. Atmos. Okeana 24, 1251.Google Scholar
Gryanik, V. M. & Tevs, M. V. 1989 The dynamics of singular geostrophic vortices in an n-layer model of the atmosphere (ocean). Izv. Akad. Nauk SSSR, Phys. Atmos. Okeana 25, 243.Google Scholar
Horton, W. 1989 Drift wave vortices and anomalous transport.. Phys. Fluids B 1, 524.Google Scholar
Kamenkovich, V. M. 1989 On the evolution of Rossby waves under localized effects. Okeanologiya 29, 5. (Transl. Oceanology 29, 1.)Google Scholar
Klyatskin, K. V. & Reznik, G. M. 1989 On point vortices on rotating sphere. Okeanologiya 29, 21. (Transl. Oceanology 29, 12.)Google Scholar
Kono, M., Horton, W. & Matsuoka, C. 1989 Point vortex description of drift wave vortices: dynamics and transport. University of Texas. Inst. for Fusion Studies Tech. Rep. DOE/ET-53086–410.Google Scholar
Larichev, V. D. & Reznik, G. M. 1976 Two-dimensional Rossby soliton: an exact solution. Rep. USSR Acad. Sci. 231, 1077.Google Scholar
Mcwilliams, J. C. & Flierl, G. R. 1979 On the evolution of isolated, nonlinear vortices. J. Phys. Oceanogr. 9, 1155.Google Scholar
Morikawa, G. K. 1960 Geostrophic vortex motion. J. Met. 17, 148.Google Scholar
Obukhov, A. M. 1949 On the geostrophic wind. Izv. Acad. Nauk SSSR, Geograph. 8, 281.Google Scholar
Olver, F. W. J. 1974 Introduction to Asymptotic and Special Functions. Academic.
Pedlosky, J. 1982 Geophysical Fluid Dynamics. Springer.
Reznik, G. M. 1986 Point vortices on a β-plane and Rossby solitary waves. Okeanologiya 26, 165. (Transl. Oceanology 26, 119.)Google Scholar
Smith, D. C. & Reid, H. O. 1982 A numerical study of nonfrictional decay of mesoscale eddies. J. Phys. Oceanogr. 12, 244.Google Scholar
Stern, M. E. 1975 Minimal properties of planetary eddies. J. Mar. Res. 40, 57.Google Scholar
Zabusky, N. J. & McWilliams, J. C. 1982 A modulated point vortex for geostrophic beta-plane dynamics. Phys. Fluids 25, 2175.Google Scholar