## On the stability of Thomson’s vortex $$N$$-gon and a vortex tripole/quadrupole in geostrophic models of Bessel vortices and in a two-layer rotating fluid: a review.(English)Zbl 1446.76114

Summary: In this paper the two-layer geostrophic model of the rotating fluid and the model of Bessel vortices are considered. Kirchhoff’s model of vortices in a homogeneous fluid is the limiting case of both of these models. Part of the study is performed for an arbitrary Hamiltonian depending on the distances between point vortices.
The review of the stability problem of stationary rotation of regular Thomson’s vortex $$N$$-gon of identical vortices is given for $${N\geqslant 2}$$. The stability problem of the vortex tripole/quadrupole is also considered. This axisymmetric vortex structure consists of a central vortex of an arbitrary intensity and two/three identical peripheral vortices. In the model of a two-layer fluid, peripheral vortices belong to one of the layers and the central vortex can belong to either another layer or the same.
The stability of the stationary rotation is interpreted as orbital stability (the stability of a one-parameter orbit of a stationary rotation of a vortex system). The instability of the stationary rotation is instability of equilibrium of the reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied.
The parameter space is divided into three parts: $$\mathbf{A}$$ is the domain of stability in an exact nonlinear setting, $$\mathbf{B}$$ is the linear stability domain, where the stability problem requires nonlinear analysis, and $$\mathbf{C}$$ is the instability domain.
In the stability problem of a vortex multipole, another definition of stability is used; it is the stability of an invariant three-parametric set of all trajectories of the families of stationary orbits. It is shown that in the case of non zero total intensity, the stability of the invariant set implies orbital stability.

### MSC:

 76E07 Rotation in hydrodynamic stability 76B47 Vortex flows for incompressible inviscid fluids 76E20 Stability and instability of geophysical and astrophysical flows 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
Full Text:

### References:

  Borisov, A. V. and Mamaev, I. S., Mathematical Methods in the Dynamics of Vortex Structures, R&C Dynamics, Institute of Computer Science, Izhevsk, 2005, 368 pp. (Russian) · Zbl 1119.76001  Cabral, H. E. and Schmidt, D. S., “Stability of Relative Equilibria in the Problem of $$N+1$$ Vortices”, SIAM J. Math. Anal., 31:2 (1999/2000), 231-250 · Zbl 0961.76014  Campbell, L. J., “Transverse Normal Modes of Finite Vortex Arrays”, Phys. Rev. A, 24:1 (1981), 514-534  Izv. Akad. Nauk SSSR. Fiz. Atmos. Okeana, 19:3 (1983), 227-240 (Russian)  Havelock, T. H., “The Stability of Motion of Rectilinear Vortices in Ring Formation”, Philos. Mag., 11:70 (1931), 617-633 · JFM 57.1109.03  Kizner, Z., “Stability of Point-Vortex Multipoles Revisited”, Phys. Fluids, 23:6 (2001), 064104, 11 pp. · Zbl 1308.76051  Kizner, Z., “On the Stability of Two-Layer Geostrophic Point-Vortex Multipoles”, Phys. Fluids, 26:4 (2014), 046602, 18 pp. · Zbl 1321.76028  Kurakin, L. G. and Yudovich, V. I., “The Stability of Stationary Rotation of a Regular Vortex Polygon”, Chaos, 12:3 (2002), 574-595 · Zbl 1080.76520  Dokl. Akad. Nauk, 462:2 (2015), 161-167 (Russian)  Kurakin, L. G., Ostrovskaya, I. V., and 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 (2016), 291-334 · Zbl 1346.76204  Kurakin, L. G. and 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 (2017), 865-879 · Zbl 1401.76037  Kurakin, L. G., Lysenko, I. A., Ostrovskaya, I. V., and Sokolovskiy, M. A., “On Stability of the Thomson”s Vortex $$N$$-Gon in the Geostrophic Model of the Point Vortices in Two-Layer Fluid”, J. Nonlinear Sci., 29:4 (2019), 1659-1700 · Zbl 1423.76076  Mertz, G., “Stability of Body-Centered Polygonal Configurations of Ideal Vortices”, Phys. Fluids, 21:7 (1978), 1092-1095 · Zbl 0379.76042  Morikawa, G. K. and Swenson, E. V., “Interacting Motion of Rectilinear Geostrophic Vortices”, Phys. Fluids, 14:6 (1971), 1058-1073  Thomson, W., “Floating Magnets (Illustrating Vortex Systems)”, Nature, 18 (1878), 13-14  Thomson, J. J., Treatise on the Motion of Vortex Rings, Macmillan, London, 1883, 156 pp. · JFM 15.0854.02  Sokolovskiy, M. A. and Verron, J., “Some Properties of Motion of $$A + 1$$ Vortices in a Two-Layer Rotating Fluid”, Nelin. Dinam., 2:1 (2006), 27-54 (Russian)  Sokolovskiy, M. A. and Verron, J., Dynamics of Vortex Structures in a Stratified Rotating Fluid, Atmos. Oceanogr. Sci. Libr., 47, Springer, Cham, 2014, XII, 382 pp. · Zbl 1384.86001  Stewart, H. J., “Periodic Properties of the Semi-Permanent Atmospheric Pressure Systems”, Quart. Appl. Math., 1 (1943), 262-267 · Zbl 0063.07196  Stewart, H. J., “Hydrodynamic Problems Arising from the Investigation of the Transverse Circulation in the Atmosphere”, Bull. Amer. Math. Soc., 51 (1945), 781-799 · Zbl 0063.07197
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.