Abstract
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 the layers is presented. The vortices have identical intensity and length scale is \(\gamma ^{-1}>0\). The problem has three parameters: N, \(\gamma R\) and \(\beta \), where \(\beta \) is the ratio of the fluid layer 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 parameter space \((N,\gamma R,\beta )\) is divided on three parts: \(\varvec{A}\) is the domain of stability in an exact nonlinear setting, \(\varvec{B}\) is the linear stability domain, where the stability problem requires the nonlinear analysis, and \(\varvec{C}\) is the instability domain. The case \(\varvec{A}\) takes place for \(N=2,3,4\) for all possible values of parameters \(\gamma R\) and \(\beta \). In the case of \(N=5\), we have two domains: \(\varvec{A}\) and \(\varvec{B}\). In the case \(N=6\), part \(\varvec{B}\) is curve, which divides the space of parameters \((\gamma R, \beta )\) into the domains: \(\varvec{A}\) and \(\varvec{C}\). In the case of \(N=7\), there are all three domains: \(\varvec{A}\), \(\varvec{B}\) and \(\varvec{C}\). The instability domain \(\varvec{C}\) takes place always if \(N=2n\geqslant 8\). In the case of \(N=2\ell +1\geqslant 9\), there are two domains: \(\varvec{B}\) and \(\varvec{C}\). The results of research are presented in two versions: for parameter \(\beta \) and parameter \(\alpha \), where \(\alpha \) is the difference between layer thicknesses. A number of statements about the stability of the Thomson N-gon 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.
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Acknowledgements
The work of the first three authors was supported by the Ministry of Education and Science of the Russian Federation, Southern Federal University (Project No. 1.5169.2017/8.9). MAS was supported by the Ministry of Education and Science of the Russian Federation (Project No. 14.W.03.31.0006, numerical simulation), Russian Science Foundation (Project No. 14-50-00095, application to ocean) and Russian Foundation for Basic Research (Projects Nos. 16-55-150001 and 16-05-00121, vortex dynamics). The authors are grateful to M. Yu. Zhukov for valuable discussions. We express our gratitude to the anonymous reviewer #2 for useful comments and recommendations.
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Appendices
Appendix A: The Properties of Polynomial \(P(N,1,\Lambda )\)
According to formulas (3.22) and (3.16), the equality
is valid.
Thus, the polynomial \(P(N,1,\Lambda )\) given by formulas (3.25), (3.26) has the form:
The coefficient \(p_0(N,1)\) is factorized as
If the function \(V=W\) is defined by formula (2.2), the values \(\lambda _{21}\), \(\lambda _{01}\) are given by equalities (4.2)–(4.4), and the functions \(T_N^\pm \) have the form:
Thus, inequality \(p_0(N,1)>0\) is valid. The inequalities \(\lambda _{21}>0\) and \(p_1(N,1)<0\) are valid. It follows from formulas (4.2)–(4.4). Therefore, roots of the polynomial\(P(N,1,\Lambda )\), which is given by formulas (3.25), (3.26), are positive for any\(N>2\)in the case, if the function\(V=W\)is defined by equality (2.2).
Appendix B: The Case \(N\geqslant 5\)
In this section, we present the results obtained for Bessel vortices in Kurakin and Ostrovskaya (2017), which are necessary to study of our problem for \(N\geqslant 5\).
1.1 Appendix B.1: The Properties of Functions \(\lambda _{1k}^*\)
The asymptotics of functions \({\lambda }_{1k}^*(N,R)\) given by formula (4.4)
is valid. To construct this formula, we used asymptotics of the modified Bessel function \(K_\nu (\xi )\) (Abramowitz and Stegun 1964):
Here we take into account that \(\min \limits _{1\le m \le \lfloor \frac{N}{2} \rfloor }B_m=B_1\). Here \(B_m =\sqrt{2-2C_m}\), \(C_m=\cos \frac{2\pi m}{N}\).
The case of even\(N=2n\geqslant 8 \). The value \(a_{1n}=0\) and the asymptotics
are valid.
Figure 19 shows the graphics of the functions \(\delta _{2n,1}(R)\lambda _{1n}^*(2n,R).\) The weight function
is positive for \(N\geqslant 6.\)
The inequality
is valid for \(N=2n\geqslant 8\).
The case of odd\(N=2\ell +1\geqslant 7 \). According to asymptotics (B.7), (B.8), for large enough R the following inequalities can be regarded as equivalent
We introduce the following notations. The value \(R_{*k}(N)\) is real root of the equation \(\lambda _{1k}^*(N,R)=0\) if it exists, \(R_{*}(N)=R_{*\ell }\), and \(R_{ak}(N)=-\dfrac{b_{1k}}{a_{1k}}\).
We note that
The results of calculations with the help of the approximation formulas:
are presented in Table 1 and in Fig. 20.
There is an asymptotics: \(R_*(N)\rightarrow R_a(N)\), \(N=2\ell +1\rightarrow \infty \).
The inequalities
are valid. Accordingly (B.13), the conditions
are equivalent to the condition
1.2 Appendix B.2: The Properties of the Functions \(p_0^*(N,k)\)
We introduce the denotation
where \(\lambda _{1k}^*\), \(\lambda _{2k}^*\) and \(\lambda _{0k}^*\) are given by (4.4).
The asymptotics
is valid.
Figure 21 shows the graphics functions \(\delta _{N,2}(R) p_0^*(N,\frac{N-1}{2}).\) The weight function
is positive for odd \(N=2\ell +1\geqslant 5.\) The functions shown in Fig. 21 are negative: \(p_0^*(N,\frac{N-1}{2})<0\) for the pentagon \((N=5)\) at \(R>R_{05}\), and in all other cases \(N=2\ell +1\ge 7\) for all \(R>0\). Therefore, for \(N=2\ell +1\ge 7\) the condition (4.15) of Proposition 4.1 is valid for all \(R>0\).
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Kurakin, L.G., Lysenko, I.A., Ostrovskaya, I.V. et al. 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, 1659–1700 (2019). https://doi.org/10.1007/s00332-018-9526-2
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DOI: https://doi.org/10.1007/s00332-018-9526-2