Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2025-01-02T12:58:41.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Multipolar vortices in two-dimensional incompressible flows

Published online by Cambridge University Press:  26 April 2006

Yves G. Morel  and
Xavier J. Carton
Yves G. Morel
Affiliation:
SHOM, LEGI/IMG, BP53X, 38041 Grenoble Cedex, France
Xavier J. Carton
Affiliation:
SHOM, GRGS, Centre National d’Etudes Spatiales, 18 Ave. Edouard Belin, 31055 Toulouse Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

In a two-dimensional incompressible fluid, the barotropic instability of isolated circular vortices can lead to multipole formation. The multipoles we study here are composed of a core vortex surrounded by two or more identical satellite vortices, of opposite-sign vorticity to the core, and the total circulation is zero. First, we present the generation of multipoles from unstable piecewise-constant monopoles perturbed on a monochromatic azimuthal mode. The stationary multipoles formed by this nonlinear evolution retain the same energy, circulation and angular momentum as the original monopoles, but possess a lower enstrophy. These multipolar steady states are then compared to multipolar equilibria of the Euler equation, obtained either analytically by a perturbation expansion or numerically via a relaxation algorithm. Finally the stability of these equilibria is studied. Quadrupoles (one core vortex bound to three satellites) prove relatively robust, whether initially perturbed or not, and resist severe permanent deformations (mode-2 shears or strains of amplitude up to 0.1ζ(max). Amplification of the mode-3 deformation proves more destructive. More complex multipoles degenerate in less than a turnover period into end-products of a lesser complexity, via vortex splitting, pairing or merging. We use the conservation of integral properties to classify the large variety of instability mechanisms along physical guidelines. To conclude, we synthetize the connections between these various vortex forms.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 267 , 25 May 1994 , pp. 23 - 51
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acton, E. & Dhanak, M. R. 1993 The motion and stability of a vortex array above a pulsed surface. J. Fluid Mech. 247, 231–246.Google Scholar
Bell, G. I. 1990 The nonlinear evolution of a perturbed axisymmetric eddy. Woods Hole GFD Summer School Reports. Woods Hole Oceanography Institution.
Carnevale, G. F. & Kloosterziel, R. C. 1994 Emergence and evolution of triangular vortices. J. Fluid Mech. 259, 305–331.Google Scholar
Carton, X. J., Flierl, G. R. & Polvani, L. M. 1989 The generation of tripoles from unstable axisymmetric isolated vortex structures. Europhys. Lett. 9, 339–344.Google Scholar
Carton, X. J. & McWilliams, J. C. 1989 Barotropic and baroclinic instabilities of axisymmetric vortices in a quasi-geostrophic model. In Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence (ed. J. C. J. Nihoul & B. M. Jamart), Vol.50. Elsevier.
Carton, X. J. 1992 On the merger of shielded vortices. Europhys. Lett. 18, 697–703.Google Scholar
Carton, X. J. & Legras, B. 1994 The life-cycle of tripoles in two-dimensional incompressible flows. J. Fluid Mech. 267, 53–82.Google Scholar
Dritschel, D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134.Google Scholar
Dritschel, D. G. 1986 The nonlinear evolution of rotating configurations of uniform vorticity. J. Fluid Mech. 172, 157–182.Google Scholar
Dritschel, D. G. 1988 Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 10, 77.Google Scholar
Dritschel, D. G. 1989 The stabilisation of a two-dimensional vortex strip by an adverse shear. J. Fluid Mech. 206, 193–221.Google Scholar
Flierl, G. R. 1988 On the instability of geostrophic vortices. J. Fluid Mech. 197, 349–388.Google Scholar
Gent, P. R. & McWilliams, J. C. 1986 The instability of barotropic circular vortices. Geophys. Astrophys. Fluid Dyn. 35, 209–233.Google Scholar
Heijst, G. J. F. van & Flor, J. B. 1989 Laboratory experiments on dipole structures in a stratified fluid. In Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence (ed. J. C. J. Nihoul & B. M. Jamart), Vol. 50. Elsevier.
Heijst, G. J. F. van, Kloosterziel, R. C. & Williams, C. W. M. 1989 Laboratory experiments on the tripolar vortices in a rotating fluid. J. Fluid Mech. 225, 301–331.Google Scholar
Hesthaven, J. S., Lynov, J. P., Rasmussen, J. J. & Sutyrin, G. G. 1992 Generation of tripolar vortical structures on the beta-plane. Preprint.
Hopfinger, E. J. & Heijst, G. J. F. van 1993 Vortices in rotating fluids. Ann. Rev. Fluid Mech. 25, 241–289.Google Scholar
Kloosterziel, R. C. & Heijst, G. J. F. van 1989 On tripolar vortices. In Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence (ed. J. C. J. Nihoul & B. M. Jamart), Vol. 50. Elsevier.
Kozlov, V. F. & Makarov, V. G. 1985 Simulation of the instability of axisymmetric vortices using the contour dynamics method. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 1, 33–39. (English transl., 1985: Fluid Dyn. 1, 28–34.)Google Scholar
Legras, B. & Dritschel, D. G. 1993 A comparison of the contour surgery and pseudospectral methods. J. Comput. Phys. 104, 287–302.Google Scholar
Legras, B., Santangelo, P. & Benzi, R. 1988 High-resolution numerical experiments for forced two-dimensional turbulence. Europhys. Lett. 5, 37–42.Google Scholar
Leith, C. E. 1981 Minimum-enstrophy vortices. Phys. Fluids 7, 1388–1395.Google Scholar
Lin, S. J. 1992 Contour dynamics of tornado-like vortices. J. Atmos. Sci. 49, 1745–1756.Google Scholar
McWilliams, J. C. & Zabusky, N. J. 1982 Interactions of isolated vortices. I: Modons colliding with modons. Geophys. Astrophys. Fluid Dyn. 19, 207–227.Google Scholar
McWilliams, J. C. 1983 Interactions of isolated vortices. II: Modon generation by monopole collision. Geophys. Astrophys. Fluid Dyn. 24, 1–22.Google Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in the turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
McWilliams, J. C. 1990 The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361–385.Google Scholar
Melander, M. V., Zabusky, N.J. & McWilliams, J. C. 1988 Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340.Google Scholar
Mied, R. P., Kirwan, A. D. Jr & Lindemann, G. J. 1992 Rotating modons over isolated topographic features. J. Phys. Oceanogr. 22, 1569–1582.Google Scholar
Morel, Y. G. & Carton, X. J. 1991 Generation and stability of multipolar vortices - a report from the EGS 16th General Assembly, session NP3. SHOM Internal Reports, and Annales Geophys., Suppl. to vol.9, C560.
Morikawa, G. K. & Swenson, E. V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 1058–1073.Google Scholar
Orlandi, P. & Heijst, G. J. F. van 1992 Numerical simulation of tripolar vortices in 2D flow. Fluid Dyn. Res. 9, 179–206.Google Scholar
Pierrehumbert, R. T. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144.Google Scholar
Pingree, R. D. & LeCann, B. 1992 Three anticyclonic Slope Water Oceanic eddies (SWODDIES) in the Southern Bay of Biscay in 1990. Deep-Sea Res. in press.Google Scholar
Polvani, L. M. & Carton, X. J. 1990 The tripole: a new coherent vortex structure of incompressible two-dimensional flows. Geophys. Astrophys. Fluid Dyn. 51, 87–102.Google Scholar
Stern, M. E. 1974 Minimal properties of planetary eddies. J. Mar. Res. 33, 1–13.Google Scholar
Thomson, J. J. 1883 A Treatise on the Motion of Vortex Rings, pp. 94–108. Macmillan.
Wu, H. M., Overman II, E. A. & Zabusky, N. J. 1984 Steady-state solutions of the Euler equations in two dimensions: Rotating and translating V-States with limiting cases. I. Numerical algorithms and results. J. Comput. Phys. 53, 42–71.CrossRefGoogle Scholar