[AOS21-12] On the Stability of Discrete Vortex Structures in Geostrophic Models Two-Layer Rotating Fluid and Bessel Vortices
Keywords:Vortex dynamics, Stability, Point vortex
A two-layer quasigeostrophic model is considered. The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a circle of radius R in one of layers is presented. The vortices have identical intensity and length scale is 1/g>0. The problem has three parameters: N, gR and b, where b is the ratio of the fluid layers thicknesses. The stability of the stationary rotation is interpreted as orbital stability. The instability of the stationary rotation is instability of system reduced equilibrium.
The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied.
The parameters space (N,gR,b) is divided on three parts: A is the domain of stability in an exact nonlinear setting, B is the linear stability domain, where the stability problem requires the nonlinear analysis, and C is the instability domain. The case A takes place for N=2,3,4 for all possible values of parameters gR and b. In the case of N=5 we have two domains: A and B. In the case N=6 part B is curve, which divides the space of parameters (gR,b) into the domains: A and C. In the case of N=7 there are all three domains: A, B, and C. The instability domain C takes place always if N=2n>7. In the case of N=2l+1>7 there are two domains: B and C.
Similar research of the stability has been carried out for the vortex structure consisting of a central vortex of arbitrary intensity G and two/three identical peripheral vortexes. The identical vortexes, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer.
The stability of the Thomson vortex N-gon is also studied in the case of the model of the Bessel vortices for any N.
A number of statements about the stability is obtained for the systems of interacting particles with the general Hamiltonian depending only on distances between the particles.
The results of theoretical analysis are confirmed by numerical calculations of the vortex trajectories.
The main results are published in the papers [1-5].
I. Lysenko was supported by Ministry of Education and Science of the Russian Federation, Southern Federal University (Project No. 1.5169.2017/8.9), and the other three authors was supported by the Russian Foundation for Basic Research (Projects No. 20-55-10001).
Bibliography.
1. L.G. Kurakin, I.V. Ostrovskaya, and M.A. Sokolovskiy. Stability of discrete vortex multipoles in homogeneous and two-layer rotating fluid // Doklady Physics. 2015. Vol. 60, no 5. P. 217-223.
2. L.G. Kurakin, I.V. Ostrovskaya, and M.A. Sokolovskiy. On the stability of discrete tripole, quadrupole, Thomson' vortex triangle and square in a two-layer/homogeneous rotating fluid // Regul. Chaotic Dyn. 2016. Vol. 21, no. 3. P. 291-334.
3. L.G. Kurakin, and I.V. Ostrovskaya. On stability of the Thomson's vortex N-gon in the geostrophic model of the point Bessel vortices // Regul. Chaotic Dyn. 2017. Vol. 22, no. 7. P. 865-879.
4. L. Kurakin, I. Lysenko, I. Ostrovskaya, and M. Sokolovskiy. On stability of the Thomson's vortex N-gon in the geostrophic model of the point vortices in two-layer fluid // J. of Nonlinear Science. 2019. vol. 29, iss. 4. P. 1659–1700.
5. L.G. Kurakin, and I.V. Ostrovskaya. 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. P. 533-542.
The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied.
The parameters space (N,gR,b) is divided on three parts: A is the domain of stability in an exact nonlinear setting, B is the linear stability domain, where the stability problem requires the nonlinear analysis, and C is the instability domain. The case A takes place for N=2,3,4 for all possible values of parameters gR and b. In the case of N=5 we have two domains: A and B. In the case N=6 part B is curve, which divides the space of parameters (gR,b) into the domains: A and C. In the case of N=7 there are all three domains: A, B, and C. The instability domain C takes place always if N=2n>7. In the case of N=2l+1>7 there are two domains: B and C.
Similar research of the stability has been carried out for the vortex structure consisting of a central vortex of arbitrary intensity G and two/three identical peripheral vortexes. The identical vortexes, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer.
The stability of the Thomson vortex N-gon is also studied in the case of the model of the Bessel vortices for any N.
A number of statements about the stability is obtained for the systems of interacting particles with the general Hamiltonian depending only on distances between the particles.
The results of theoretical analysis are confirmed by numerical calculations of the vortex trajectories.
The main results are published in the papers [1-5].
I. Lysenko was supported by Ministry of Education and Science of the Russian Federation, Southern Federal University (Project No. 1.5169.2017/8.9), and the other three authors was supported by the Russian Foundation for Basic Research (Projects No. 20-55-10001).
Bibliography.
1. L.G. Kurakin, I.V. Ostrovskaya, and M.A. Sokolovskiy. Stability of discrete vortex multipoles in homogeneous and two-layer rotating fluid // Doklady Physics. 2015. Vol. 60, no 5. P. 217-223.
2. L.G. Kurakin, I.V. Ostrovskaya, and M.A. Sokolovskiy. On the stability of discrete tripole, quadrupole, Thomson' vortex triangle and square in a two-layer/homogeneous rotating fluid // Regul. Chaotic Dyn. 2016. Vol. 21, no. 3. P. 291-334.
3. L.G. Kurakin, and I.V. Ostrovskaya. On stability of the Thomson's vortex N-gon in the geostrophic model of the point Bessel vortices // Regul. Chaotic Dyn. 2017. Vol. 22, no. 7. P. 865-879.
4. L. Kurakin, I. Lysenko, I. Ostrovskaya, and M. Sokolovskiy. On stability of the Thomson's vortex N-gon in the geostrophic model of the point vortices in two-layer fluid // J. of Nonlinear Science. 2019. vol. 29, iss. 4. P. 1659–1700.
5. L.G. Kurakin, and I.V. Ostrovskaya. 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. P. 533-542.