On Stability of the Thomson’s Vortex N-gon in the Geostrophic Model of the Point Vortices in Two-layer Fluid

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.

Appendices

Appendix A: The Properties of Polynomial $P(N,1,\Lambda )$

According to formulas (3.22) and (3.16), the equality

$$\begin{aligned} \lambda _{11} (N,R,\beta ) = \frac{1}{R^4} \lambda _{21}(N,R,\beta ) \end{aligned}$$

(A.1)

is valid.

Thus, the polynomial $P(N,1,\Lambda )$ given by formulas (3.25), (3.26) has the form:

$$\begin{aligned} P(N,1,\Lambda )= & {} \Lambda ^2 +p_1(N,1)\Lambda +p_0(N,1), \end{aligned}$$

(A.2)

$$\begin{aligned} p_1(N,1)= & {} -\left( 1+\dfrac{1}{R^4}\right) \lambda _{21},\quad p_0(N,1)=\dfrac{1}{R^4}\lambda _{21}^2-\lambda _{01}^2. \end{aligned}$$

(A.3)

The coefficient $p_0(N,1)$ is factorized as

$$\begin{aligned} p_0(N,1)=T_N^-\cdot T^+_N,\quad T_N^\pm =\frac{1}{R^2}\lambda _{21}\pm \lambda _{01}. \end{aligned}$$

(A.4)

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:

$$\begin{aligned} T_N^-&= \frac{N-1}{4R^2} +\frac{1}{4R} \cdot \beta \sum \limits _{m=1}^{N-1} B_m K_{1} (RB_m) >0, \end{aligned}$$

(A.5)

$$\begin{aligned} T_N^+&= \frac{N-1}{4R^2} + \frac{1}{2} \cdot \beta \sum \limits _{m=1}^{N-1} \left( S_m^2 K_0(R B_m) +\dfrac{1}{2R} B_m K_1(R B_m) \right) >0. \end{aligned}$$

(A.6)

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)

$$\begin{aligned} {\lambda }_{1k}^*= & {} \frac{K_1(R\,B_1)}{4R^2}\left( a_{1k}+\frac{b_{1k}}{R} + O\left( \frac{1}{R^2}\right) \right) ,\quad R\rightarrow \infty ,\quad k=1,\ldots , \left\lfloor \frac{N}{2} \right\rfloor , \end{aligned}$$

(B.7)

$$\begin{aligned} a_{1k}= & {} \left( 1+ \cos {\frac{2\pi k}{N}}\right) B_{1}^2,\quad b_{1k}=\frac{4}{B_{1}}\left( 1-2C_{1}+\cos {\frac{2\pi k}{N}}\right) -\frac{1}{2B_1}a_{1k} \end{aligned}$$

(B.8)

is valid. To construct this formula, we used asymptotics of the modified Bessel function $K_\nu (\xi )$ (Abramowitz and Stegun 1964):

$$\begin{aligned} K_\nu (\xi )&=\sqrt{\frac{\pi }{2\xi }}\mathrm{{e}}^{-\xi }\left( 1+\frac{4\nu ^2-1}{8\xi }+O\left( \frac{1}{\xi ^2}\right) \right) ,\quad \xi \rightarrow \infty , \\ \frac{K_0(\xi )}{K_1(\xi )}&=1-\frac{1}{2\xi }+O\left( \frac{1}{\xi ^2}\right) ,\quad \xi \rightarrow \infty . \end{aligned}$$

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

$$\begin{aligned} {\lambda }_{1n}^*(2n,R)= & {} \frac{K_1(R\,B_1)}{4R^2}\left( -\frac{8C_1}{B_1\,R} + O\left( \frac{1}{R^2}\right) \right) , \quad R\rightarrow +\infty \end{aligned}$$

(B.9)

are valid.

Figure 19 shows the graphics of the functions $\delta _{2n,1}(R)\lambda _{1n}^*(2n,R).$ The weight function

$$\begin{aligned} \delta _{2n,1}(R)=\frac{R^3B_1}{2C_1K_{1}(RB_1)} \end{aligned}$$

(B.10)

is positive for $N\geqslant 6.$

The inequality

$$\begin{aligned} \lambda _{1n}^*(N,R)<0 \end{aligned}$$

(B.11)

is valid for $N=2n\geqslant 8$.

Fig. 19

Case of even $N=2n\geqslant 6$. The graphs of the functions $\delta _{2n,1}(R)\lambda _{1n}^*(2n,R)$ for $N=6,8,10,40,42,44,200,400,600$ in the interval $0< R\leqslant 20$. Graphics are given from top to bottom in order of increasing N

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

$$\begin{aligned} {\lambda }_{1k}^*(N,R)>0 \quad \text {and}\quad a_{1k}+\frac{b_{1k}}{R}>0, \quad k=1,\dots ,\ell ,\quad \ell =\frac{N-1}{2}. \end{aligned}$$

(B.12)

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

$$\begin{aligned} \max _{1\leqslant k \leqslant \ell }R_{a k}(N)= -\frac{b_{1 \ell }}{a_{1\ell }}=R_a(N). \end{aligned}$$

(B.13)

The results of calculations with the help of the approximation formulas:

$$\begin{aligned} \begin{aligned}&R_*(N)\approx R_a(N), \\&R_a(N)= \frac{4}{B_1^3}\left[ \frac{2C_{1}}{1+ \cos {\frac{\pi (N-1)}{N}}}-1\right] +\frac{1}{2B_1} \end{aligned} \end{aligned}$$

(B.14)

are presented in Table 1 and in Fig. 20.

There is an asymptotics: $R_*(N)\rightarrow R_a(N)$, $N=2\ell +1\rightarrow \infty$.

Table 1 Case odd $N=2\ell +1\geqslant 7$

Fig. 20

Function $R_a(N)$ on the interval $7\leqslant N\leqslant 41$

The inequalities

$$\begin{aligned} \lambda _{1\ell }^*<0,\quad 0<R<R_*(N),\quad \text {and} \quad \lambda _{1\ell }^*>0,\quad R>R_*(N), \end{aligned}$$

(B.15)

are valid. Accordingly (B.13), the conditions

$$\begin{aligned} \lambda _{1k}^* \geqslant 0,\quad R\geqslant R_*(N), \quad k=1,\ldots ,\ell \end{aligned}$$

(B.16)

are equivalent to the condition

$$\begin{aligned} \lambda _{1\ell }^* \geqslant 0,\quad R\geqslant R_*(N). \end{aligned}$$

(B.17)

1.2 Appendix B.2: The Properties of the Functions $p_0^*(N,k)$

We introduce the denotation

$$\begin{aligned} p_0^*(N,k)= \lambda _{1k}^*\lambda _{2k}^*-{\lambda _{0k}^*}^2,\quad k=1,\ldots ,\left\lfloor \frac{N-1}{2} \right\rfloor , \end{aligned}$$

(B.18)

where $\lambda _{1k}^*$, $\lambda _{2k}^*$ and $\lambda _{0k}^*$ are given by (4.4).

The asymptotics

$$\begin{aligned} p_0^*\left( N,\frac{N-1}{2}\right)= & {} K_{1}^2(R\,B_1)\left( \frac{Q(N)}{R}+O\left( \frac{1}{R^2}\right) \right) ,\quad R\rightarrow \infty , \end{aligned}$$

(B.19)

$$\begin{aligned} Q(N)= & {} \frac{1}{4B_1}\left( 1-\cos {\frac{\pi (N-1) }{N}}\right) \left( 1+\cos {\frac{\pi (N-1)}{N}}-C_1-C_1^2\right) \end{aligned}$$

(B.20)

is valid.

Figure 21 shows the graphics functions $\delta _{N,2}(R) p_0^*(N,\frac{N-1}{2}).$ The weight function

$$\begin{aligned} \delta _{N,2}(R)=-\frac{1}{Q(N)K_{1}^2(RB_1)}>0 \end{aligned}$$

(B.21)

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 $R>0$.

Fig. 21

Case of odd $N=2\ell +1\geqslant 5$. The graphs of the functions $\delta _{N,2}\cdot p_0^*(N,\frac{N-1}{2})$ at $N=5,7,\dots , 21$ in the interval $0< R\leqslant 20$. The graphs are located from top to bottom in order of increasing N