Three-dimensional quasi-geostrophic staggered vortex arrays. (English) Zbl 1479.76042

Summary: We determine and characterise relative equilibria for arrays of point vortices in a three-dimensional quasi-geostrophic flow. The vortices are equally spaced along two horizontal rings whose centre lies on the same vertical axis. An additional vortex may be placed along this vertical axis. Depending on the parameters defining the array, the vortices on the two rings are of equal or opposite sign. We address the linear stability of the point vortex arrays. We find both stable equilibria and unstable equilibria, depending on the geometry of the array. For unstable arrays, the instability may lead to the quasi-regular or to the chaotic motion of the point vortices. The linear stability of the vortex arrays depends on the number of vortices in the array, on the radius ratio between the two rings, on the vertical offset between the rings and on the vertical offset between the rings and the central vortex, when the latter is present. In this case the linear stability also depends on the strength of the central vortex. The non-linear evolution of a selection of unstable cases is presented exhibiting examples of quasi-regular motion and of chaotic motion.


76E20 Stability and instability of geophysical and astrophysical flows
76B47 Vortex flows for incompressible inviscid fluids
76U60 Geophysical flows
Full Text: DOI


[1] von Helmholtz, H., Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math., 55, 25-55 (1858)
[2] Kirchhoff, G., Vorlesungen über mathematische Physik (1876), Leipzig: Teubner, Leipzig · JFM 08.0542.01
[3] Thomson, W., Vortex Statics, Proc. Roy. Soc. Edinburgh, 9, 59-73 (1878) · JFM 08.0613.01
[4] Thomson, W., Floating Magnets, Nature, 18, 13-14 (1878)
[5] Mayer, A. M., On the Morphological Laws of the Configurations Formed by Magnets Floating Vertically and Subjected to the Attraction of a Superposed Magnet; with Notes on Some of the Phenomena in Molecular Structure Which These Experiments May Serve to Explain and Illustrate, Am. J. Sci. Arts, Ser. 3, 16, 94, 247-256 (1878)
[6] Thomson, J. J., Treatise on the Motion of Vortex Rings (1883), London: Macmillan, London · JFM 15.0854.02
[7] Havelock, T. H., The Stability of Motion of Rectilinear Vortices in Ring Formation, Philos. Mag., 11, 70, 617-633 (1931) · Zbl 0001.08102
[8] Morton, W. V., Vortex Polygons, Proc. R. Irish Acad., Sect. A, 42, 21-29 (1935) · JFM 61.0912.03
[9] Khazin, L. G., Regular Polygons of Point Vortices and Resonance Instability of Steady States, Sov. Phys. Dokl., 21, 567-570 (1976)
[10] Mertz, G. T., Stability of Body-Centered Polygonal Configurations of Ideal Vortices, Phys. Fluids, 21, 7, 1092-1095 (1978) · Zbl 0379.76042
[11] Stewart, H. J., Periodic Properties of the Semi-Permanent Atmospheric Pressure Systems, Quart. Appl. Math., 1, 262-267 (1943) · Zbl 0063.07196
[12] Stewart, H. J., Hydrodynamic Problems Arising from the Investigation of the Transverse Circulation in the Atmosphere, Bull. Amer. Math. Soc., 51, 781-799 (1945) · Zbl 0063.07197
[13] Morikawa, G. K.; Swenson, E. V., Interacting Motion of Rectilinear Geostrophic Vortices, Phys. Fluids, 14, 6, 1058-1073 (1971)
[14] Aref, H., Stability of Reactive Equilibria of Three Vortices, Phys. Fluids, 21, 9 (2009) · Zbl 1183.76077
[15] Kizner, Z., Stability of Point-Vortex Multipoles Revisited, Phys. Fluids, 23, 6 (2001) · Zbl 1308.76051
[16] Kizner, Z., On the Stability of Two-Layer Geostrophic Point-Vortex Multipoles, Phys. Fluids, 26, 4 (2014) · Zbl 1321.76028
[17] Kurakin, L. G.; Yudovich, V. I., The Stability of Stationary Rotation of a Regular Vortex Polygon, Chaos, 12, 3, 574-595 (2002) · Zbl 1080.76520
[18] Kurakin, L. G.; Ostrovskaya, I. V., On Stability of the Thomson’s Vortex \(N\)-Gon in the Geostrophic Model of the Point Bessel Vortices, Regul. Chaotic Dyn., 22, 7, 865-879 (2017) · Zbl 1401.76037
[19] Kurakin, L. G.; Ostrovskaya, I. V.; Sokolovskiy, M. A., On the Stability of Discrete Tripole, Quadrupole, Thomson’ Vortex Triangle and Square in a Two-Layer/Homogeneous Rotating Fluid, Regul. Chaotic Dyn., 21, 3, 291-334 (2016) · Zbl 1346.76204
[20] Kurakin, L. G.; Melekhov, A. P.; Ostrovskaya, I. V., A Survey of the Stability Criteria of Thomson’s Vortex Polygons outside a Circular Domain, Bol. Soc. Mat. Mex., 22, 2, 733-744 (2016) · Zbl 1348.76041
[21] Kurakin, L. G.; Ostrovskaya, I. V.; Sokolovskiy, M. A., Stability of Discrete Vortex Multipoles in Homogeneous and Two-Layer Rotating Fluid, Dokl. Phys., 60, 5, 217-223 (2015)
[22] Kurakin, L. G., Influence of Annular Boundaries on Thomson’s Vortex Polygon Stability, Chaos, 14, 2 (2014) · Zbl 1345.70034
[23] Kurakin, L. G., The Stability of the Steady Rotation of a System of Three Equidistant Vortices outside a Circle, J. Appl. Math. Mech., 75, 2, 227-234 (2011) · Zbl 1272.76116
[24] Kurakin, L. G.; Ostrovskaya, I. V., Stability of the Thomson Vortex Polygon with Evenly Many Vortices outside a Circular Domain, Siberian Math. J., 51, 3, 463-474 (2010) · Zbl 1211.76027
[25] Kurakin, L. G.; Ostrovskaya, I. V., Nonlinear Stability Analysis of a Regular Vortex Pentagon Outside a Circle, Regul. Chaotic Dyn., 17, 5, 385-396 (2012) · Zbl 1252.76017
[26] Kurakin, L. G., On the Stability of Thomson’s Vortex Configurations inside a Circular Domain, Regul. Chaotic Dyn., 15, 1, 40-58 (2010) · Zbl 1229.37055
[27] Kurakin, L. G., On Stability of a Regular Vortex Polygon in the Circular Domain, J. Math. Fluid Mech., 7, 376-386 (2005) · Zbl 1097.76031
[28] Kurakin, L. G., On Nonlinear Stability of the Regular Vortex Systems on a Sphere, Chaos, 14, 3, 592-602 (2004) · Zbl 1080.76031
[29] Kurakin, L. G., Stability, Resonances, and Instability of the Regular Vortex Polygons in the Circular Domain, Dokl. Phys., 49, 11, 658-661 (2004)
[30] Kizner, Z.; Khvoles, R.; McWilliams, J. C., Rotating Multipoles on the \(f\)- and \(\gamma \)-planes, Phys. Fluids, 19, 1, 016603-016618 (2007) · Zbl 1146.76443
[31] Dritschel, D. G., The Stability and Energetics of Corotating Uniform Vortices, J. Fluid Mech., 157, 95-134 (1985) · Zbl 0574.76026
[32] Crowdy, D. G., Exact Solutions for Rotating Vortex Arrays with Finite-Area Cores, J. Fluid Mech., 469, 209-235 (2002) · Zbl 1019.76011
[33] Xue, B. B.; Johnson, E. R.; McDonald, N. R., New Families of Vortex Patch Equilibria for the Two-Dimensional Euler Equations, Phys. Fluids, 29, 12 (2017)
[34] Sokolovskiy, M. A.; Koshel, K. V.; Dritschel, D. G.; Reinaud, J. N., \(N\)-Symmetric Interaction of \(N\) Hetons: Part 1. Analysis of the Case \(N=2\), Phys. Fluids, 32, 9 (2020)
[35] Reinaud, J. N., Three-Dimensional Quasi-Geostrophic Vortex Equilibria with \(m\)-Fold Symmetry, J. Fluid Mech., 863, 32-59 (2019) · Zbl 1415.86017
[36] Reinaud, J. N.; Dritschel, D. G., The Stability and Nonlinear Evolution of Quasi-Geostrophic Toroidal Vortices, J. Fluid Mech., 863, 60-78 (2019) · Zbl 1415.86018
[37] Dritschel, D. G., Ring Configurations of Point Vortices in Polar Atmospheres, Regul. Chaotic Dyn., 26, 4, 467-481 (2021)
[38] Vallis, G. K., Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation (2017), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 1374.86002
[39] Dritschel, D. G.; Boatto, S., The Motion of Point Vortices on Closed Surfaces, Proc. A, 471, 2176 (2015) · Zbl 1371.76042
[40] Reinaud, J. N.; Carton, X., Existence, Stability and Formation of Baroclinic Tripoles in Quasi-Geostrophic Flows, J. Fluid Mech., 785, 1-30 (2015) · Zbl 1381.76050
[41] Adriani, A.; Mura, A.; Orton, G.; Hansen, C.; Altieri, F.; Moriconi, M. L.; Rogers, J.; Eichstdt, G.; Momary, T.; Ingersoll, A. P.; Filacchione, G.; Sindoni, G.; Tabataba-Vakili, F.; Dinelli, B. M.; Fabiano, F.; Bolton, S. J.; Connerney, J. E. P.; Atreya, S. K.; Lunine, J. I.; Tosi, F.; Migliorini, A.; Grassi, D.; Piccioni, G.; Noschese, R.; Cicchetti, A.; Plainaki, C.; Olivieri, A.; O’Neill, M. E.; Turrini, D.; Stefani, S.; Sordini, R.; Amoroso, M., Cluster of Cyclones Encircling Jupiter’s Poles, Nature, 555, 7695, 216-219 (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.