Skip to main content Accessibility help

Three-dimensional quasi-geostrophic vortex equilibria with $m$ -fold symmetry

  • Jean N. Reinaud (a1)


We investigate arrays of $m$ three-dimensional, unit-Burger-number, quasi-geostrophic vortices in mutual equilibrium whose centroids lie on a horizontal circular ring; or $m+1$ vortices where the additional vortex lies on the vertical ‘central’ axis passing through the centre of the array. We first analyse the linear stability of circular point vortex arrays. Three distinct categories of vortex arrays are considered. In the first category, the $m$ identical point vortices are equally spaced on a circular ring and no vortex is located on the vertical central axis. In the other two categories, a ‘central’ vortex is added. The latter two categories differ by the sign of the central vortex. We next turn our attention to finite-volume vortices for the same three categories. The vortices consist of finite volumes of uniform potential vorticity, and the equilibrium vortex arrays have an (imposed) $m$ -fold symmetry. For simplicity, all vortices have the same volume and the same potential vorticity, in absolute value. For such finite-volume vortex arrays, we determine families of equilibria which are spanned by the ratio of a distance separating the vortices and the array centre to the vortices’ mean radius. We determine numerically the shape of the equilibria for $m=2$ up to $m=7$ , for each three categories, and we address their linear stability. For the $m$ -vortex circular arrays, all configurations with $m\geqslant 6$ are unstable. Point vortex arrays are linearly stable for $m<6$ . Finite-volume vortices may, however, be sensitive to instabilities deforming the vortices for $m<6$ if the ratio of the distance separating the vortices to their mean radius is smaller than a threshold depending on  $m$ . Adding a vortex on the central axis modifies the overall stability properties of the vortex arrays. For $m=2$ , a central vortex tends to destabilise the vortex array unless the central vortex has opposite sign and is intense. For $m>2$ , the unstable regime can be obtained if the strength of the central vortex is larger in magnitude than a threshold depending on the number of vortices. This is true whether the central vortex has the same sign as or the opposite sign to the peripheral vortices. A moderate-strength like-signed central vortex tends, however, to stabilise the vortex array when located near the plane containing the array. On the contrary, most of the vortex arrays with an opposite-signed central vortex are unstable.


Corresponding author

Email address for correspondence:


Hide All
Adriani, A., Mura, A., Orton, G., Hansen, C., Altieri, F., Moriconi, M. L., Rogers, J., Eichstdt, G., Momary, T., Ingersoll, A. P. et al. 2018 Clusters of cyclones encircling Jupiter’s poles. Nature 555, 216219.10.1038/nature25491
Aref, H. 2009 Stability of relative equilibria of three vortices. Phys. Fluids 21, 094101.10.1063/1.3216063
Burbea, J. 1982 On patches of uniform vorticity in a plane of irrotational flow. Arch. Rat. Mech. Anal. 77, 349358.10.1007/BF00280642
Carnevale, G. F. & Kloosterziel, R. .C. 1994 Emergence and evolution of triangular vortices. J. Fluid Mech. 259, 305331.10.1017/S0022112094000157
Chelton, D. B., Schlax, M. G. & Samelson, R. M. 2011 Global observations of nonlinear mesoscale eddies. Prog. Oceanogr. 91, 161216.10.1016/j.pocean.2011.01.002
Crowdy, D. G. 2002 Exact solutions for rotating vortex arrays with finite-area cores. J. Fluid Mech. 469, 209235.10.1017/S0022112002001817
Crowdy, D. G. 2003 Polygonal n-vortex arrays: a Stuart model. Phys. Fluids 15 (12), 37103717.10.1063/1.1623766
Dijkstra, H. A. 2008 Dynamical Oceanography. Springer.
Dritschel, D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95134.10.1017/S0022112085002324
Dritschel, D. G. 1988 Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77, 240266.10.1016/0021-9991(88)90165-9
Dritschel, D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269303.10.1017/S0022112095001716
Dritschel, D. G. 2002 Vortex merger in rotating stratified flows. J. Fluid Mech. 455, 83101.10.1017/S0022112001007364
Dritschel, D. G. & Saravanan, R. 1994 Three-dimensional quasi-geostrophic contour dynamics, with an application to stratospheric vortex dynamics. Q. J. R. Meteorol. Soc. 120, 12671297.10.1002/qj.49712051908
Ebbesmeyer, C. C., Taft, B. A., McWilliams, J. C., Shen, C. Y., Riser, S. C., Rossby, H. T., Biscaye, P. E. & Östlund, H. G. 1986 Detection, structure and origin of extreme anomalies in a western Atlantic oceanographic section. J. Phys. Oceanogr. 16, 591612.10.1175/1520-0485(1986)016<0591:DSAOOE>2.0.CO;2
Gryanik, V. M. 1983 Dynamics of localized vortex perturbations on vortex charges in a baroclinic fluid. Izv. Atmos. Acean. Phys. 19, 347352.
Kizner, Z. 2011 Stability of point-vortex multipoles revisited. Phys. Fluids 23, 064104.10.1063/1.3596270
Kizner, Z. 2014 On the stability of two-layer geostrophic point-vortex multipoles. Phys. Fluids 26, 046602.10.1063/1.4870239
Kizner, Z. & Khvoles, R. 2004a The tripole vortex: experimental evidence and explicit solutions. Phys. Rev. E 70 (1), 016307.
Kizner, Z. & Khvoles, R. 2004b Two variations on the theme of Lamb-Chaplygin: supersmooth dipole and rotating multipoles. Regular Chaotic Dyn. 9, 509518.10.1070/RD2004v009n04ABEH000293
Kizner, Z., Khvoles, R. & McWilliams, J. C. 2007 Rotating multipoles on the f- and 𝛾-planes. Phys. Fluids 19 (1), 016603.10.1063/1.2432915
Kizner, Z., Shteinbuch-Fridman, B., Makarov, V. & Rabinovich, M. 2017 Cycloidal meandering of a mesoscale eddy. Phys. Fluids 29, 086601.10.1063/1.4996772
Kurakin, L. G. & Yudovich, V. I. 2002 The stability of stationary rotation of a regular vortex polygon. Chaos 12 (3), 574595.10.1063/1.1482175
Morikawa, G. K. & Swenson, E. V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14 (6), 10581073.10.1063/1.1693564
Peterson, M. .R., Williams, S. J., Maltrud, M. E., Hecht, M. W. & Hamann, B. 2013 A three-dimensional eddy census of a high-resolution global ocean simulation. J. Geophys. Res. Oceans 118, 17571774.
Pierrehumbert, R. T. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129144.10.1017/S0022112080000559
Reinaud, J. N. & Carton, X. 2015 Existence, stability and formation of baroclinic tripoles in quasi-geostrophic flows. J. Fluid Mech. 785, 130.10.1017/jfm.2015.614
Reinaud, J. N. & Carton, X. 2016 The interaction between two oppositely travelling, horizontally offset, antisymmetric quasi-geostrophic hetons. J. Fluid Mech. 794, 409443.10.1017/jfm.2016.171
Reinaud, J. N. & Dritschel, D. G. 2002 The merger of vertically offset quasi-geostrophic vortices. J. Fluid Mech. 469, 297315.10.1017/S0022112002001854
Reinaud, J. N. & Dritschel, D. G. 2005 The critical merger distance between two co-rotating quasi-geostrophic vortices. J. Fluid Mech. 522, 357381.10.1017/S0022112004002022
Reinaud, J. N. & Dritschel, D. G. 2009 Destructive interactions between two counter-rotating quasi-geostrophic vortices. J. Fluid Mech. 639, 195211.10.1017/S0022112009990954
Reinaud, J. N. & Dritschel, D. G. 2018 The merger of geophysical vortices at finite Rossby and Froude number. J. Fluid Mech. 848, 388410.10.1017/jfm.2018.367
Reinaud, J. N. & Dritschel, D. G. 2019 The stability and nonlinear evolution of quasi-geostrophic toroidal vortices. J. Fluid Mech. 863, 6078.10.1017/jfm.2018.1013
Reinaud, J. N., Dritschel, D. G. & Koudella, C. R. 2003 The shape of vortices in quasi-geostrophic turbulence. J. Fluid Mech. 474, 175192.10.1017/S0022112002002719
Reinaud, J. N., Sokolovskiy, M. A. & Carton, X. 2017 Geostrophic tripolar vortices in a two-layer fluid: linear stability and nonlinear evolution of equilibria. Phys. Fluids 29 (3), 036601.10.1063/1.4978806
Safman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Shteinbuch-Fridman, B., Makarov, V. & Kizner, Z. 2015 Two-layer geostrophic tripoles comprised by patches of uniform potential vorticity. Phys. Fluids 27, 036602.10.1063/1.4916283
Shteinbuch-Fridman, B., Makarov, V. & Kizner, Z. 2017 Transitions and oscillatory regimes in two-layer geostrophic hetons and tripoles. J. Fluid Mech. 810, 535553.10.1017/jfm.2016.738
Sokolovskiy, M. A. & Verron, J. 2008 On the motion of a + 1 vortices in a two-layer rotating fluid. IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, IUTAM Bookseries Vol. 6. p. 481. Springer.10.1007/978-1-4020-6744-0_43
Thomson, J. J. 1883 A Treatise of Vortex Rings. MacMillan.
Trieling, R. R., van Heijst, G. J. F. & Kizner, Z. 2010 Laboratory experiments on multipolar vortices in a rotating fluid. Phys. Fluids 22, 094104.10.1063/1.3481797
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press.10.1017/CBO9780511790447
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 (1), 4271.10.1016/0021-9991(84)90051-2
Xue, J. J., Johnson, E. R. & McDonald, N. R. 2017 New families of vortex patch equilibria for the two-dimensional Euler equations. Phys. Fluids 29 (12), 123602.10.1063/1.5009536
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for Euler equations in two dimensions. J. Comput. Phys. 30 (1), 96106.10.1016/0021-9991(79)90089-5
Zhang, Z., Wang, W. & Qiu, B. 2014 Oceanic mass transport by mesoscale eddies. Science 345, 322324.10.1126/science.1252418
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed