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  • Journal of Fluid Mechanics, Volume 614
  • November 2008, pp. 381-405

Thresholds for the formation of satellites in two-dimensional vortices

  • M. R. TURNER (a1) and A. D. GILBERT (a1)
  • DOI:
  • Published online: 10 November 2008

This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m = 2 added to it. If the perturbation is weak, then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero. However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it.

The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative diagnostics, the appearance of an inflection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier–Stokes equations using a family of profiles based on the tanh function.

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L. A. Barba & A. Leonard 2007 Emergence and evolution of tripole vortices from net-circulation initial conditions. Phys. Fluids 19 (1), 017101.

A. J. Bernoff & J. F. Lingevitch 1994 Rapid relaxation of an axisymmetric vortex. Phys. Fluids 6, 37173723.

R. J. Briggs , J. D. Daugherty & R. H. Levy 1970 Role of Landau damping in crossed-field electron beams and inviscid shear flow. Phys. Fluids 13, 421432.

B. Fornberg 1977 A numerical study of 2-D turbulence. J. Comput. Phys. 25, 131.

I. M. Hall , A. P. Bassom & A. D. Gilbert 2003 aThe effect of fine structure on the stability of planar vortices. Eur. J. Mech. B./Fluids 22, 179198.

I. M. Hall , A. P. Bassom & A. D. Gilbert 2003 bThe effect of viscosity on the stability of planar vortices with fine structure. Q. J. Mech. Appl. Maths 56 (4), 649657.

P. Koumoutsakos 1997 Inviscid axisymmetrization of an elliptical vortex. J. Comput. Phys. 138, 821857.

B. Legras & D. Dritschel 1993 Vortex stripping and the generation of high vorticity gradients in two-dimensional flows. Appl. Sci. Res. 51, 445455.

S. G. Llewellyn Smith 1995 The influence of circulation on the stability of vortices to mode-one disturbances. Proc. R. Soc. Lond. A 451, 747755.

C. Macaskill , A. P. Bassom & A. D. Gilbert 2002 Nonlinear wind-up in a strained planar vortex. Eur. J. Mech. B./Fluids 21, 293306.

J. D. Möller & M. T. Montgomery 1999 Vortex Rossby waves and hurricane intensification in a barotropic model. J. Atmos. Sci. 56, 16741687.

R. D. Pingree & B. Le Cann 1992 Anticyclonic eddy X91 in the southern bay of Biscay, May 1991 to February 1992. J. Geophys. Res. 97, 14 353–14 367.

L. F. Rossi , J. F. Lingevitch & A. J. Bernoff 1997 Quasi-steady monopole and tripole attractors for relaxing vortices. Phys. Fluids 9 (8), 23292338.

D. A. Schecter , D. H. E. Dubin , A. C. Cass , C. F. Driscoll , I. M. Lansky & T. M. O'Neil 2000 Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids 12 (10), 23972412.

J. Thuburn & V. Lagneau 1999 Eulerian mean, contour integral, and finite-amplitude wave activity diagnostics applied to a single-layer model of the winter Stratosphere. J. Atmos. Sci. 56, 689710.

M. R. Turner , A. D. Gilbert & A. P. Bassom 2008 Neutral modes of a two-dimensional vortex and their link to persistent cat's eyes. Phys. Fluids 20 (2), 027101-1027101-10.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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