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

    2019, Vol. 15, no. 4, pp.  533-542

    Author(s): Kurakin L. G., Ostrovskaya I. V.

    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: $\bf A$ is the domain of stability in an exact nonlinear setting, $\bf B$ is the linear stability domain, where the stability problem requires nonlinear analysis, and $\bf C$ 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.
    Keywords: $N$-vortex problem, Thomson's vortex $N$-gon, point vortices, two-layer fluid, stability, Hamiltonian equation
    Citation: Kurakin L. G., Ostrovskaya I. V., 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, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp.  533-542

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    [1] 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)  mathscinet
    [2] 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  crossref  mathscinet
    [3] Campbell, L. J., “Transverse Normal Modes of Finite Vortex Arrays”, Phys. Rev. A, 24:1 (1981), 514–534  crossref  adsnasa
    [4] Gryanik, V. M., “Dynamics of Singular Geostrophical Vortices in a $2$-Level Model of the Atmosphere (Ocean)”, Izv. Atmos. Ocean Phys., 19:3 (1983), 171–179  mathscinet; Izv. Akad. Nauk SSSR. Fiz. Atmos. Okeana, 19:3 (1983), 227–240 (Russian)  mathscinet  adsnasa
    [5] Havelock, T. H., “The Stability of Motion of Rectilinear Vortices in Ring Formation”, Philos. Mag., 11:70 (1931), 617–633  crossref
    [6] Kizner, Z., “Stability of Point-Vortex Multipoles Revisited”, Phys. Fluids, 23:6 (2001), 064104, 11 pp.  crossref  adsnasa
    [7] Kizner, Z., “On the Stability of Two-Layer Geostrophic Point-Vortex Multipoles”, Phys. Fluids, 26:4 (2014), 046602, 18 pp.  crossref  zmath  adsnasa  elib
    [8] Kurakin, L. G. and Yudovich, V. I., “The Stability of Stationary Rotation of a Regular Vortex Polygon”, Chaos, 12:3 (2002), 574–595  crossref  mathscinet  zmath  adsnasa
    [9] Kurakin, L. G., Ostrovskaya, I. V., and Sokolovskiy, M. A., “Stability of Discrete Vortex Multipoles in Homogeneous and Two-Layer Rotating Fluid”, Dokl. Phys., 60:5 (2015), 217–223  crossref  mathscinet  adsnasa  elib; Dokl. Akad. Nauk, 462:2 (2015), 161–167 (Russian)  mathscinet
    [10] 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  mathnet  crossref  mathscinet  zmath  adsnasa
    [11] 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  mathnet  crossref  mathscinet  zmath  adsnasa
    [12] 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  crossref  mathscinet  zmath  adsnasa
    [13] Mertz, G., “Stability of Body-Centered Polygonal Configurations of Ideal Vortices”, Phys. Fluids, 21:7 (1978), 1092–1095  crossref  zmath  adsnasa
    [14] Morikawa, G. K. and Swenson, E. V., “Interacting Motion of Rectilinear Geostrophic Vortices”, Phys. Fluids, 14:6 (1971), 1058–1073  crossref  adsnasa
    [15] Thomson, W., “Floating Magnets (Illustrating Vortex Systems)”, Nature, 18 (1878), 13–14  crossref  adsnasa; Kelvin, W. T., Mathematical and Physical Papers, v. 4, Cambridge Univ. Press, Cambridge, 1910, 162–164
    [16] Thomson, J. J., Treatise on the Motion of Vortex Rings, Macmillan, London, 1883, 156 pp.  zmath
    [17] 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)  mathnet  crossref  mathscinet
    [18] 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.  crossref  mathscinet  zmath
    [19] Stewart, H. J., “Periodic Properties of the Semi-Permanent Atmospheric Pressure Systems”, Quart. Appl. Math., 1 (1943), 262–267  crossref  mathscinet  zmath
    [20] Stewart, H. J., “Hydrodynamic Problems Arising from the Investigation of the Transverse Circulation in the Atmosphere”, Bull. Amer. Math. Soc., 51 (1945), 781–799  crossref  mathscinet  zmath

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