Three-dimensional quasi-geostrophic vortex equilibria with \(m\)-fold symmetry. (English) Zbl 1415.86017

Summary: 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.


86A05 Hydrology, hydrography, oceanography
76B47 Vortex flows for incompressible inviscid fluids
76U05 General theory of rotating fluids
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[1] Adriani, A.; Mura, A.; Orton, G.; Hansen, C.; Altieri, F.; Moriconi, M. L.; Rogers, J.; Eichstdt, G.; Momary, T.; Ingersoll, A. P., Clusters of cyclones encircling Jupiter’s poles, Nature, 555, 216-219, (2018)
[2] Aref, H., Stability of relative equilibria of three vortices, Phys. Fluids, 21, (2009) · Zbl 1183.76077
[3] Burbea, J., On patches of uniform vorticity in a plane of irrotational flow, Arch. Rat. Mech. Anal., 77, 349-358, (1982) · Zbl 0495.76032
[4] Carnevale, G. F.; Kloosterziel, R. . C., Emergence and evolution of triangular vortices, J. Fluid Mech., 259, 305-331, (1994)
[5] Chelton, D. B.; Schlax, M. G.; Samelson, R. M., Global observations of nonlinear mesoscale eddies, Prog. Oceanogr., 91, 161-216, (2011)
[6] Crowdy, D. G., Exact solutions for rotating vortex arrays with finite-area cores, J. Fluid Mech., 469, 209-235, (2002) · Zbl 1019.76011
[7] Crowdy, D. G., Polygonal n-vortex arrays: a Stuart model, Phys. Fluids, 15, 12, 3710-3717, (2003) · Zbl 1186.76116
[8] Dijkstra, H. A., Dynamical Oceanography, (2008), Springer
[9] Dritschel, D. G., The stability and energetics of corotating uniform vortices, J. Fluid Mech., 157, 95-134, (1985) · Zbl 0574.76026
[10] Dritschel, D. G., Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics, J. Comput. Phys., 77, 240-266, (1988) · Zbl 0642.76025
[11] Dritschel, D. G., A general theory for two-dimensional vortex interactions, J. Fluid Mech., 293, 269-303, (1995) · Zbl 0854.76018
[12] Dritschel, D. G., Vortex merger in rotating stratified flows, J. Fluid Mech., 455, 83-101, (2002) · Zbl 1028.76053
[13] Dritschel, D. G.; Saravanan, R., Three-dimensional quasi-geostrophic contour dynamics, with an application to stratospheric vortex dynamics, Q. J. R. Meteorol. Soc., 120, 1267-1297, (1994)
[14] Ebbesmeyer, C. C.; Taft, B. A.; Mcwilliams, J. C.; Shen, C. Y.; Riser, S. C.; Rossby, H. T.; Biscaye, P. E.; Östlund, H. G., Detection, structure and origin of extreme anomalies in a western Atlantic oceanographic section, J. Phys. Oceanogr., 16, 591-612, (1986)
[15] Gryanik, V. M., Dynamics of localized vortex perturbations on vortex charges in a baroclinic fluid, Izv. Atmos. Acean. Phys., 19, 347-352, (1983)
[16] Kizner, Z., Stability of point-vortex multipoles revisited, Phys. Fluids, 23, (2011) · Zbl 1308.76051
[17] Kizner, Z., On the stability of two-layer geostrophic point-vortex multipoles, Phys. Fluids, 26, (2014) · Zbl 1321.76028
[18] Kizner, Z.; Khvoles, R., The tripole vortex: experimental evidence and explicit solutions, Phys. Rev. E, 70, 1, (2004)
[19] Kizner, Z.; Khvoles, R., Two variations on the theme of Lamb-Chaplygin: supersmooth dipole and rotating multipoles, Regular Chaotic Dyn., 9, 509-518, (2004) · Zbl 1102.76008
[20] Kizner, Z.; Khvoles, R.; Mcwilliams, J. C., Rotating multipoles on the f- and 𝛾-planes, Phys. Fluids, 19, 1, (2007) · Zbl 1146.76443
[21] Kizner, Z.; Shteinbuch-Fridman, B.; Makarov, V.; Rabinovich, M., Cycloidal meandering of a mesoscale eddy, Phys. Fluids, 29, (2017)
[22] 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
[23] Morikawa, G. K.; Swenson, E. V., Interacting motion of rectilinear geostrophic vortices, Phys. Fluids, 14, 6, 1058-1073, (1971)
[24] Peterson, M. . R.; Williams, S. J.; Maltrud, M. E.; Hecht, M. W.; Hamann, B., A three-dimensional eddy census of a high-resolution global ocean simulation, J. Geophys. Res. Oceans, 118, 1757-1774, (2013)
[25] Pierrehumbert, R. T., A family of steady, translating vortex pairs with distributed vorticity, J. Fluid Mech., 99, 129-144, (1980) · Zbl 0473.76034
[26] 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
[27] Reinaud, J. N.; Carton, X., The interaction between two oppositely travelling, horizontally offset, antisymmetric quasi-geostrophic hetons, J. Fluid Mech., 794, 409-443, (2016)
[28] Reinaud, J. N.; Dritschel, D. G., The merger of vertically offset quasi-geostrophic vortices, J. Fluid Mech., 469, 297-315, (2002) · Zbl 1152.76347
[29] Reinaud, J. N.; Dritschel, D. G., The critical merger distance between two co-rotating quasi-geostrophic vortices, J. Fluid Mech., 522, 357-381, (2005) · Zbl 1065.76093
[30] Reinaud, J. N.; Dritschel, D. G., Destructive interactions between two counter-rotating quasi-geostrophic vortices, J. Fluid Mech., 639, 195-211, (2009) · Zbl 1183.76733
[31] Reinaud, J. N.; Dritschel, D. G., The merger of geophysical vortices at finite Rossby and Froude number, J. Fluid Mech., 848, 388-410, (2018) · Zbl 1404.76062
[32] Reinaud, J. N.; Dritschel, D. G., The stability and nonlinear evolution of quasi-geostrophic toroidal vortices, J. Fluid Mech., 863, 60-78, (2019)
[33] Reinaud, J. N.; Dritschel, D. G.; Koudella, C. R., The shape of vortices in quasi-geostrophic turbulence, J. Fluid Mech., 474, 175-192, (2003) · Zbl 1022.76025
[34] Reinaud, J. N.; Sokolovskiy, M. A.; Carton, X., Geostrophic tripolar vortices in a two-layer fluid: linear stability and nonlinear evolution of equilibria, Phys. Fluids, 29, 3, (2017)
[35] Safman, P. G., Vortex Dynamics, (1992), Cambridge University Press
[36] Shteinbuch-Fridman, B.; Makarov, V.; Kizner, Z., Two-layer geostrophic tripoles comprised by patches of uniform potential vorticity, Phys. Fluids, 27, (2015) · Zbl 1326.76034
[37] Shteinbuch-Fridman, B.; Makarov, V.; Kizner, Z., Transitions and oscillatory regimes in two-layer geostrophic hetons and tripoles, J. Fluid Mech., 810, 535-553, (2017) · Zbl 1384.86004
[38] Sokolovskiy, M. A.; Verron, J., 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, 481, (2008), Springer · Zbl 1207.76040
[39] Thomson, J. J., A Treatise of Vortex Rings, (1883), MacMillan · JFM 15.0854.02
[40] Trieling, R. R.; Van Heijst, G. J. F.; Kizner, Z., Laboratory experiments on multipolar vortices in a rotating fluid, Phys. Fluids, 22, (2010)
[41] Vallis, G. K., Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, (2006), Cambridge University Press · Zbl 1374.86002
[42] Wu, H. M.; Overman, E. A. Ii; Zabusky, N. J., 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, 42-71, (1984) · Zbl 0524.76029
[43] Xue, J. J.; Johnson, E. R.; Mcdonald, N. R., New families of vortex patch equilibria for the two-dimensional Euler equations, Phys. Fluids, 29, 12, (2017)
[44] Zabusky, N. J.; Hughes, M. H.; Roberts, K. V., Contour dynamics for Euler equations in two dimensions, J. Comput. Phys., 30, 1, 96-106, (1979) · Zbl 0405.76014
[45] Zhang, Z.; Wang, W.; Qiu, B., Oceanic mass transport by mesoscale eddies, Science, 345, 322-324, (2014)
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