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Stability and evolution of uniform-vorticity dipoles

Published online by Cambridge University Press:  11 February 2011

V. G. MAKAROV  and
Z. KIZNER
V. G. MAKAROV
Affiliation:
Interdisciplinary Center of Marine Sciences of the National Polytechnic Institute, La Paz, Baja California Sur 23096, Mexico
Z. KIZNER*
Affiliation:
Departments of Physics and Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
*
†Email address for correspondence: [email protected]
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Abstract

Using an iterative algorithm, a family of stationary two-dimensional vortical dipoles is constructed, including translational (symmetric and asymmetric about the translation axis) and orbital (i.e. moving in circles) dipoles. The patches of uniform vorticity comprising a dipole possess symmetry about the axis passing through their centroids and are, generally, unequal in area and absolute value of vorticity. The solutions are discriminated by three parameters, the ratio of the areas of individual vortices, the ratio of their vorticities and the separation between the centroids of the patches. The dipole stability and evolution of unstable states are studied numerically with a contour dynamics method, where the perturbations allowed are, generally, asymmetric. The diagrams of convergence of the iterative algorithm (without any symmetry constraints) are built in three cross-sections of the parameter space: at opposite vorticity of the individual vortices, at equal areas of the vortices and at zero net circulation of the vortex pairs (when inequality of areas of the individual vortices is offset by inequality of the absolute values of vorticity). The convergence bound is shown to be close to the stability bound in the parameter space, and the larger is the separation, the stronger are the perturbations needed to move the dipole out of equilibrium. Typical scenarios of the evolution of unstable symmetric translational dipoles and weakly stable dipoles of other kinds are described, including the transition of a dipole into an oscillating tripole – the scenario that has not been discussed so far.

JFM classification

Vortex Flows: Contour dynamics Vortex Flows: Vortex breakdown Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 307 - 325
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

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.CrossRefGoogle Scholar
Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: stationary ‘V-states’, interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859–862.CrossRefGoogle Scholar
Dritschel, D. G. 1988 Contour surgery: a topological reconnection scheme for extended contour integrations using contour dynamics. J. Comput. Phys. 77, 240–266.CrossRefGoogle Scholar
Dritschel, D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269–303.CrossRefGoogle Scholar
Gent, P. R. & McWilliams, J. C. 1986 The instability of barotropic circular vortices. Geophys. Astrophys. Fluid Dyn. 35, 209–233.CrossRefGoogle Scholar
van Heijst, G. J. F. & Kloosterziel, R. C. 1989 Tripolar vortices in a rotating fluid. Nature 338, 369–371.CrossRefGoogle Scholar
van Heijst, G. J. F., Kloosterziel, R. C. & Williams, C. V. M. 1991 Laboratory experiments on the tripolar vortex in a rotating fluid. J. Fluid Mech. 225, 301–331.CrossRefGoogle Scholar
Kizner, Z. 2006 Stability and transitions of hetonic quartets and baroclinic modons. Phys. Fluids 18, 056601(1–12).CrossRefGoogle Scholar
Kizner, Z. 2008 Hetonic quartet: exploring the transitions in baroclinic modons. In IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (ed. Borisov, A. V. et al. , vol. 6, pp. 125–133. Springer.CrossRefGoogle Scholar
Kizner, Z., Berson, D. & Khvoles, R. 2002 Baroclinic modon equilibria on the beta-plane: stability and transitions. J. Fluid Mech. 468, 239–270.CrossRefGoogle Scholar
Kizner, Z., Berson, D. & Khvoles, R. 2003 Noncircular baroclinic beta-plane modons: constructing stationary solutions. J. Fluid Mech. 489, 199–228.CrossRefGoogle Scholar
Kizner, Z. & Khvoles, R. 2004 The tripole vortex: experimental evidence and explicit solutions. Phys. Rev. E 70, 016307.CrossRefGoogle ScholarPubMed
Kozlov, V. F. 1983 The method of contour dynamics in model problems of the ocean topographic cyclogenesis. Izv. Atmos. Ocean. Phys. 19, 635–640.Google Scholar
Kozlov, V. F. & Makarov, V. G. 1998 Vortex patch dynamics in a coastal current. Oceanology 38, 456–462.Google Scholar
Makarov, V. G. 1991 Computational algorithm of the contour dynamics method with changeable topology of domains under study. Model. Mekh. 5, 83–95.Google Scholar
Makarov, V. G. 1996 Numerical simulation of the formation of tripolar vortices by the method of contour dynamics. Izv. Atmos. Ocean. Phys. 32, 40–49.Google Scholar
Makarov, V. G. & Bulgakov, S. N. 2008 Regimes of near-wall vortex dynamics in potential flow through gaps. Phys. Fluids 20, 086605(1–11).CrossRefGoogle Scholar
Morel, Y. J. & Carton, X. J. 1994 Multipolar vortices in two-dimensional incompressible flows. J. Fluid Mech. 267, 23–51.CrossRefGoogle Scholar
Orlandi, P. & van Heijst, G. J. F. 1992 Numerical simulation of tripolar vortices in 2D flows. Fluid Dyn. Res. 9, 179–206.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144.CrossRefGoogle Scholar
Sadovskii, V. S. 1971 Vortex regions in a potential stream with a jump of Bernoulli's constant at the boundary. J. Appl. Math. Mech. 35, 773–779.CrossRefGoogle Scholar
Saffman, P. G. & Tanveer, S. 1982 The touching pair of equal and opposite uniform vortices. Phys. Fluids 25, 1929–1930.CrossRefGoogle Scholar
Sokolovskiy, M. A. & Verron, J. 2000 Finite-core hetons: stability and interactions. J. Fluid Mech. 423, 127–154.CrossRefGoogle Scholar
Wu, H. M., Overman, E. A. II & Zabusky, N. J. 1984 Steady-state solutions of the Euler equations: rotating and translating V-states with limiting cases. Part I. Numerical algorithms and results. J. Comput. Phys. 53, 42–71.CrossRefGoogle Scholar
Zabusky, N. J., Hughes, M. N. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two-dimensions. J. Comput. Phys. 30, 96–106.CrossRefGoogle Scholar