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On nonlinear stability of the regular vortex systems on a sphere. (English) Zbl 1080.76031

Summary: We present necessary and sufficient conditions for stability and instability of the stationary rotation of a system of \(n\) identical point vortices located at the same latitude on a sphere at vertices of a regular \(n\)-gon. We also examine stability of the equilibrium configuration of identical point vortices situated at the vertices of a regular polyhedra. It is proved that vortex tetrahedron, octahedron, and icosahedron are stable, while vortex cube and dodecahedron are unstable.

MSC:

76E30 Nonlinear effects in hydrodynamic stability
76B47 Vortex flows for incompressible inviscid fluids
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[1] DOI: 10.1038/018013b0
[2] DOI: 10.1103/PhysRevLett.43.214
[3] DOI: 10.1007/BF00683912
[4] DOI: 10.1103/PhysRevLett.75.3277
[5] DOI: 10.1063/1.870307 · Zbl 1149.76365
[6] DOI: 10.1080/14786443109461714
[7] DOI: 10.1134/1.1493390
[8] DOI: 10.1134/1.1493390
[9] DOI: 10.1063/1.1482175 · Zbl 1080.76520
[10] DOI: 10.1016/S0065-2156(02)39001-X
[11] Bogomolov V. A., Izv., Acad. Sci., USSR, Atmos. Oceanic Phys. 15 pp 243– (1979)
[12] DOI: 10.1063/1.532602 · Zbl 0927.37013
[13] DOI: 10.1070/rd2000v005n02ABEH000141 · Zbl 0967.76023
[14] DOI: 10.1134/1.1560737
[15] Bogomolov V. A., Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 6 pp 57– (1977)
[16] DOI: 10.1017/S0022112093002381 · Zbl 0793.76022
[17] DOI: 10.1016/0393-0440(92)90015-S · Zbl 0761.58017
[18] DOI: 10.1016/S0167-2789(00)00167-6
[19] DOI: 10.1016/S0167-2789(97)00236-4 · Zbl 0962.76516
[20] DOI: 10.1070/rd1998v003n04ABEH000094 · Zbl 0958.76013
[21] DOI: 10.1088/0951-7715/15/1/307 · Zbl 0999.76030
[22] Boatto S., SIAM (Soc. Ind. Appl. Math) J. Appl. Math. 64 pp 216– (2004)
[23] F. Laurent-Polz, J. Montaldi, and M. Roberts, ”Stability of relative equilibria of point vortices on the sphere,” preprint INLN, 2004 (available at arXiv: math.DS/0402430). · Zbl 1248.70020
[24] DOI: 10.1070/RD2003v008n03ABEH000243 · Zbl 1150.76348
[25] DOI: 10.1063/1.1581451 · Zbl 1080.37603
[26] DOI: 10.1007/BF01090320 · Zbl 0442.76020
[27] Bogomolov V., Izv., Acad. Sci., USSR, Atmos. Oceanic Phys. 21 pp 298– (1985)
[28] DOI: 10.1017/S0022112076000591 · Zbl 0343.76041
[29] Klyatskin K., Oceanology (Engl. Transl.) 29 pp 12– (1989)
[30] DOI: 10.1017/S0022112097008100 · Zbl 0908.76096
[31] F. Laurent-Polz, ”Point vortices on a rotating sphere,” preprint INLN, 2003 (available at arXiv: math.DS/0301360). · Zbl 1120.76012
[32] DOI: 10.1063/1.524322 · Zbl 0446.76027
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