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Steady vortex dipoles with general profile functions



Vortex dipoles in a two-dimensional, inviscid flow are obtained by prescribing the profile function relating the vorticity to the stream function. The profile functions used are smooth, and the solutions obtained have a smooth transition from the exterior flow to the interior of the vortex. The dipoles are nearly elliptical, and this relates this work to the ‘supersmooth’ dipoles obtained recently by Kizner & Khvoles (Regular Chaotic Dyn., vol. 9, 2004, pp. 509–518). The solutions found here are obtained by an iterative method for solving the nonlinear partial differential equation for the stream function. This iterative method is both robust and flexible. Solutions are also obtained in a β-plane, and they are shielded, as has also been found in previous work.


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Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177190.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Benjamin, T. B. 1976 The alliance of practical and analytical insights into the non-linear problems of fluid mechanics. In Applications of Methods of Functional Analysis to Problems in Mechanics (ed. Germain, P. & Neyroles, B.), Lecture Notes in Mathematics, vol. 503, pp. 829. Springer.
Boyd, J. & Ma, H. 1990 Numerical study of elliptical modons using spectral methods. J. Fluid Mech. 221, 597611.
Burton, G. R. 1989 Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex. Acta Math. 163, 291309.
Burton, G. R. 1996 Uniqueness for the circular vortex-pair in a uniform flow. Proc. R. Soc. Lond. A 452, 23432350.
Elcrat, A. R., Fornberg, B., Horn, M. & Miller, K. 2000 Some steady vortex flows past a circular cylinder. J. Fluid Mech. 409, 1327.
Elcrat, A. R., Fornberg, B. & Miller, K. 2001 Some steady axisymmetric vortex flows past a sphere. J. Fluid Mech. 433, 315328.
Elcrat, A. R., Fornberg, B. & Miller, K. G. 2005 Stability of vortices in equilibrium with a cylinder. J. Fluid Mech. 544, 5368.
Elcrat, A. R., Fornberg, B. & Miller, K. G. 2008 Steady axisymmetric vortex flows with swirl and shear. J. Fluid Mech. 613, 395410.
Elcrat, A. R. & Miller, K. 1995 Rearrangements in steady multiple vortex flows. Commun. Part. Differ. Equ. 20, 14811490.
Eydeland, A. & Turkington, B. 1988 A computational method of solving free-boundary problems in vortex dynamics. J Comput. Phys. 78, 194214.
Flierl, G. 1987 Isolated eddy models in geophysics. Annu. Rev. Fluid Mech. 19, 493530.
Flierl, G., Larichev, V., McWilliams, J. & Reznik, G. 1980 The dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans 5, 141.
Flierl, G., Stern, M. & Whitehead, J. Jr. 1983 The physical significance of modons: laboratory experiments and integral constraints. Dyn. Atmos. Oceans 7, 233263.
Hesthaven, J. S., Lynov, A. H., Nielsen, A. H., Rasmussen, J. J. & Schmidt, M. R. 1995 Dynamics of a nonlinear dipole vortex. Phys. Fluids 7, 22202229.
Khvoles, R., Berson, D. & Kizner, Z. 2005 The structure and evolution of elliptical barotropic modons. J. Fluid Mech. 530, 130.
Khvoles, R., Kizner, Z. & Kessler, D. 2010 Viscous selection of an elliptical dipole. J. Fluid Mech. 658, 492508.
Kizner, Z. & Khvoles, R. 2004 Two variations on the theme of Lamb–Chaplygin: supersmooth dipole and rotating multipoles. Regular Chaotic Dyn. 9, 509518.
Kizner, Z. & Reznik, G. 2010 Localized dipoles: from 2D to rotating shallow water. Theor. Comput. Fluid Dyn. 24, 101110.
Kloeden, P. 1987 On the uniqueness of solitary Rossby waves. J. Austral. Maths Soc. B 28, 476485.
Meleshko, V. & van Heijst, G. 1994 On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157182.
Pierrehumbert, R. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129144.
Trieling, R., Sandbergen, R., van Heijst, G. & Kizner, Z. 2010 Barotropic elliptical dipoles in a rotating fluid. Theor. Comput. Fluid Dyn. 24, 111115.
Turkington, B. 1983 On steady vortex flows in two dimensions. Part II. Commun. Part. Differ. Equ. 8, 10311071.
Verkley, W. T. M. 1993 A numerical method for finding form-preserving free solutions of the barotropic vorticity equations on a sphere. J. Atmos. Sci. 50, 14881503.
Wu, H. M., Overman, E. A. II & Zabusky, N. J. 1984 Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases. Part I. Numerical algorithms and results. J. Comput. Phys. 53, 4271.
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Steady vortex dipoles with general profile functions



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