Toroidal Geometry Stabilizing a Latitudinal Ring of Point Vortices on a Torus

Abstract

We carry out the linear stability analysis of a polygonal ring configuration of N point vortices, called an N-ring, along the line of latitude $\theta _0$ on a torus with the aspect ratio $\alpha$. Deriving a criterion for the stability depending on the parameters N, $\theta _0$ and $\alpha$, we reveal how the aspect ratio $\alpha$ contributes to the stability of the N-ring. While the N-ring necessarily becomes unstable when N is sufficiently large for fixed $\alpha$, the stability is closely associated with the geometric property of the torus for variable $\alpha$; for low aspect ratio $\alpha \sim 1$, $N=7$ is a critical number determining the stability of the N-ring when it is located along a certain range of latitudes, which is an analogous result to those in a plane and on a sphere. On the other hand, the stability is determined by the sign of curvature for high aspect ratio $\alpha \gg 1$. That is to say, the N-ring is neutrally stable if it is located on the inner side of the toroidal surface with a negative curvature, while the N-ring on its outer side with a positive curvature is unstable. Furthermore, based on the linear stability analysis, we describe nonlinear evolution of the N-ring when it becomes unstable. It is difficult to deal with this problem, since the evolution equation of the N point vortices is formulated as a Hamiltonian system with N degrees of freedom, which is in general non-integrable. Thus, we reduce the Hamiltonian system to a simple integrable system by introducing a cyclic symmetry. Owing to this reduction, we successfully find some periodic orbits in the reduced system, whose local bifurcations and global transitions for variable $\alpha$ are characterized in terms of the fundamental group of the torus.

Derivation of the Linearized Equation

Starting from

$$\begin{aligned} \theta _m(t) = \theta _0+\varepsilon \vartheta _m(t), \quad \phi _m(t) = \frac{2\pi m}{N}+tV_N(\theta _0)+\varepsilon \varphi _m(t), \end{aligned}$$

(37)

we derive the evolution equation of $(\vartheta _m(t),\varphi _m(t))$ by the asymptotic expansion of $O(\varepsilon )$. For $(\theta _m(t),\phi _m(t))$ and $(\theta _j(t),\phi _j(t))$ with $m\ne j$, $\zeta _m/\zeta _j$ is asymptotically expanded as follows:

$$\begin{aligned} \zeta _m/\zeta _j=\left( 1-\varepsilon \frac{\vartheta _m(t)-\vartheta _j(t)}{\alpha -\cos \theta _0}\right) \exp \left\{ \mathrm {i}\frac{2\pi (m-j)}{N}+\varepsilon \mathrm {i}(\varphi _m(t)-\varphi _j(t))\right\} +o(\varepsilon ). \end{aligned}$$

(38)

For an integer j, define $\omega _j$ by $\omega _j=\mathrm {e}^{2\pi \mathrm {i} j/N}.$ Since $K(\zeta )$ is a holomorphic function at $\omega _{m-j}=\exp (\mathrm {i}(2\pi /N)(m-j))\ne 1$, we obtain the following expansion of $K(\zeta _m/\zeta _j)$,

$$\begin{aligned} K(\zeta _m/\zeta _j) = K(\omega _{m-j})+K'(\omega _{m-j})(\zeta _m/\zeta _j-\omega _{m-j})+o(\vert \zeta _m/\zeta _j-\omega _{m-j}\vert ). \end{aligned}$$

Using (38), we have

$$\begin{aligned} \zeta _m/\zeta _j-\omega _{m-j} = -\varepsilon \left( \frac{\vartheta _m(t)-\vartheta _j(t)}{\alpha -\cos \theta _0}-\mathrm {i}(\varphi _m(t)-\varphi _j(t))\right) \omega _{m-j}+o(\varepsilon ), \end{aligned}$$

which yields

$$\begin{aligned}&K(\zeta _m/\zeta _j)+\overline{K(\zeta _m/\zeta _j)}\nonumber \\&\quad = K(\omega _{m-j})+\overline{K(\omega _{m-j})}-\varepsilon \left( \omega _{m-j}K'(\omega _{m-j})+\overline{\omega _{m-j}K'(\omega _{m-j})}\right) \frac{\vartheta _m(t)-\vartheta _j(t)}{\alpha -\cos \theta _0}\nonumber \\&\qquad + \mathrm {i}\varepsilon \left( \omega _{m-j}K'(\omega _{m-j})-\overline{\omega _{m-j}K'(\omega _{m-j})}\right) (\varphi _m(t)-\varphi _j(t))+o(\varepsilon ), \end{aligned}$$

(39)

$$\begin{aligned}&\mathrm {i}\left( K(\zeta _m/\zeta _j)-\overline{K(\zeta _m/\zeta _j)}\right) \nonumber \\&\quad = \mathrm {i}\left( K(\omega _{m-j})-\overline{K(\omega _{m-j})}\right) \nonumber \\&\qquad -\mathrm {i}\varepsilon \left( \omega _{m-j}K'(\omega _{m-j})-\overline{\omega _{m-j}K'(\omega _{m-j})}\right) \frac{\vartheta _m(t)-\vartheta _j(t)}{\alpha -\cos \theta _0}\nonumber \\&\qquad -\varepsilon \left( \omega _{m-j}K'(\omega _{m-j})+\overline{\omega _{m-j}K'(\omega _{m-j})}\right) (\varphi _m(t)-\varphi _j(t))+o(\varepsilon ). \end{aligned}$$

(40)

Since

$$\begin{aligned} \zeta ^{-1}K'(\zeta ^{-1}) = -\frac{\zeta ^{-1}}{(\zeta ^{-1}-1)^2}-\sum _{n\ge 1} \frac{\rho ^n\zeta ^{-1}}{(\rho ^n\zeta ^{-1}-1)^2}+\frac{\rho ^n\zeta ^{-1}}{(\zeta ^{-1}-\rho ^n)^2} = \zeta K'(\zeta ), \end{aligned}$$

we have

$$\begin{aligned} \overline{\omega _{m-j}K'(\omega _{m-j})}=\omega _{m-j}^{-1}K'(\omega _{m-j}^{-1})=\omega _{m-j}K'(\omega _{m-j}). \end{aligned}$$

It follows from $K(\zeta ^{-1})+K(\zeta )=1$ that Eqs. (39) and (40) are simplified as follows:

$$\begin{aligned} K(\zeta _m/\zeta _j)+\overline{K(\zeta _m/\zeta _j)}&= 1-2\varepsilon \frac{\omega _{m-j}K'(\omega _{m-j})}{\alpha -\cos \theta _0}\left( \vartheta _m(t)-\vartheta _j(t)\right) +o(\varepsilon ),\\ \mathrm {i}\left( K(\zeta _m/\zeta _j)-\overline{K(\zeta _m/\zeta _j)}\right)&= \mathrm {i}\left( K(\omega _{m-j})-K(\overline{\omega _{m-j}})\right) \\&\quad -2\varepsilon \omega _{m-j}K'(\omega _{m-j})(\varphi _m(t)-\varphi _j(t))+o(\varepsilon ). \end{aligned}$$

Summing them up from $j=1$ to N except $j=m$, we have

$$\begin{aligned} \sum _{j\ne m}^N K(\zeta _m/\zeta _j)+\overline{K(\zeta _m/\zeta _j)}&=\sum _{j\ne m}^N 1-\varepsilon \frac{ 2\omega _{m-j}K'(\omega _{m-j})}{\alpha -\cos \theta _0}\left( \vartheta _m(t)-\vartheta _j(t)\right) +o(\varepsilon )\nonumber \\&= N-1-\varepsilon \sum _{j=1}^{N-1} \frac{2\omega _{j}K'(\omega _{j})}{\alpha -\cos \theta _0}\vartheta _m(t)\nonumber \\&\quad +\varepsilon \sum _{j\ne m}^N \frac{2\omega _{m-j}K'(\omega _{m-j})}{\alpha -\cos \theta _0}\vartheta _j(t)+o(\varepsilon ),\\ \mathrm {i}\sum _{j\ne m}^N K(\zeta _m/\zeta _j)-\overline{K(\zeta _m/\zeta _j)}&= \mathrm {i}\sum _{j\ne m}^N \left( K(\omega _{m-j})-K(\overline{\omega _{m-j}})\right) \\&\quad -\varepsilon \sum _{j\ne m}^N 2\omega _{m-j}K'(\omega _{m-j})(\varphi _m(t)-\varphi _j(t))+o(\varepsilon )\nonumber \\&= \mathrm {i}\sum _{j=1}^{N-1} \left( K(\omega _{j})-K(\omega _{N-j})\right) -\varepsilon \sum _{j=1}^{N-1} 2\omega _{j}K'(\omega _{j})\varphi _m(t)\\&\quad +\varepsilon \sum _{j\ne m}^N 2\omega _{m-j}K'(\omega _{m-j})\varphi _j(t)+o(\varepsilon )\\&= -\varepsilon \sum _{j=1}^{N-1} 2\omega _{j}K'(\omega _{j})\varphi _m(t)\\&\quad +\varepsilon \sum _{j\ne m}^N 2\omega _{m-j}K'(\omega _{m-j})\varphi _j(t)+o(\varepsilon ). \end{aligned}$$

We can rewrite them as

$$\begin{aligned} \sum _{j\ne m}^N K(\zeta _m/\zeta _j)+\overline{K(\zeta _m/\zeta _j)}&= N-1+\varepsilon \sum _{j=1}^N \frac{k_{m-j}}{\alpha -\cos \theta _0}\vartheta _j(t)+o(\varepsilon ), \end{aligned}$$

(41)

$$\begin{aligned} \mathrm {i}\sum _{j\ne m}^N K(\zeta _m/\zeta _j)-\overline{K(\zeta _m/\zeta _j)}&= \varepsilon \sum _{j=1}^N k_{m-j}\varphi _j(t)+o(\varepsilon ). \end{aligned}$$

(42)

The remaining part of (3) is expanded as follows:

$$\begin{aligned} \frac{\alpha \theta _m - \sin \theta _m}{4\pi ^2 \alpha }+\frac{r_c(\theta _j)}{4\pi ^2\mathscr {A}} - \frac{1}{4\pi }&= \frac{\alpha \theta _0 - \sin \theta _0}{4\pi ^2 \alpha }+\frac{r_c(\theta _0)}{4\pi ^2\mathscr {A}} - \frac{1}{4\pi }\\&\quad +\frac{\alpha - \cos \theta _0}{4\pi ^2 \alpha }(\theta _m-\theta _0)+o(\vert \theta _m-\theta _0\vert )\\&\quad -\frac{1}{4\pi ^2{\mathscr {A}}(\alpha -\cos \theta _0)} (\theta _j-\theta _0)+o(\vert \theta _j-\theta _0\vert )\\&= \frac{\alpha \theta _0 - \sin \theta _0}{4\pi ^2 \alpha }+\frac{r_c(\theta _0)}{4\pi ^2\mathscr {A}} - \frac{1}{4\pi }\\&\quad +\varepsilon \left\{ \frac{\alpha - \cos \theta _0}{4\pi ^2 \alpha }\vartheta _m-\frac{1}{4\pi ^2{\mathscr {A}}(\alpha -\cos \theta _0)} \vartheta _j\right\} \\&\quad +o(\varepsilon ),\\ \frac{\alpha \theta _m - \sin \theta _m}{4\pi ^2\alpha }+\frac{r_c(\theta _m)}{4\pi ^2\mathscr {A}}+\frac{1}{4\pi } \sin \theta _m&= \frac{\alpha \theta _0 - \sin \theta _0}{4\pi ^2\alpha }+\frac{r_c(\theta _0)}{4\pi ^2\mathscr {A}}+\frac{1}{4\pi } \sin \theta _0\\&\quad +\varepsilon \left\{ \frac{\alpha - \cos \theta _0}{4\pi ^2\alpha }\vartheta _m-\frac{1}{4\pi ^2{\mathscr {A}}(\alpha -\cos \theta _0)}\vartheta _m\right. \\&\quad \left. +\frac{\cos \theta _0}{4\pi } \vartheta _m\right\} +o(\varepsilon ). \end{aligned}$$

Hence, we have

$$\begin{aligned}&r^2(\alpha -\cos \theta _m)^2 \frac{\mathrm {d}\phi _m}{\mathrm {d}t} \nonumber \\&\quad = \frac{\varGamma }{4\pi }\left( N-1+\varepsilon \sum _{j=1}^N \frac{k_{m-j}}{\alpha -\cos \theta _0}\vartheta _j(t)\right) \nonumber \\&\qquad +\,\sum _{j\ne m}^N \varGamma \left[ \frac{\alpha \theta _0 - \sin \theta _0}{4\pi ^2 \alpha }+\frac{r_c(\theta _0)}{4\pi ^2\mathscr {A}} - \frac{1}{4\pi }\right. \nonumber \\&\left. \qquad +\,\varepsilon \left\{ \frac{\alpha - \cos \theta _0}{4\pi ^2 \alpha }\vartheta _m-\frac{1}{4\pi ^2{\mathscr {A}}(\alpha -\cos \theta _0)} \vartheta _j\right\} \right] \nonumber \\&\qquad +\,\varGamma \left[ \frac{\alpha \theta _0 - \sin \theta _0}{4\pi ^2\alpha }+\frac{r_c(\theta _0)}{4\pi ^2\mathscr {A}}+\frac{1}{4\pi } \sin \theta _0\right. \nonumber \\&\qquad \left. +\,\varepsilon \left\{ \frac{\alpha - \cos \theta _0}{4\pi ^2\alpha }\vartheta _m-\frac{\vartheta _m}{4\pi ^2{\mathscr {A}}(\alpha -\cos \theta _0)}+\frac{\cos \theta _0}{4\pi } \vartheta _m\right\} \right] +o(\varepsilon )\nonumber \\&\quad =N\varGamma \left[ \frac{\alpha \theta _0 - \sin \theta _0}{4\pi ^2 \alpha }+\frac{r_c(\theta _0)}{4\pi ^2\mathscr {A}}\right] +\frac{\varGamma }{4\pi } \sin \theta _0\nonumber \\&\qquad +\, \varepsilon \left[ \frac{\varGamma }{4\pi }\sum _{j=1}^N \frac{k_{m-j}}{\alpha -\cos \theta _0}\vartheta _j(t)+ N\varGamma \left( \frac{\alpha - \cos \theta _0}{4\pi ^2\alpha }+\frac{\cos \theta _0}{4\pi N}\right) \vartheta _m\right. \nonumber \\&\qquad \left. -\,\varGamma \sum _{j=1}^N\frac{\vartheta _j}{4\pi ^2{\mathscr {A}}(\alpha -\cos \theta _0)}\right] +o(\varepsilon )\nonumber \\&\quad =(R-r\cos \theta _0)^2V_N(\theta _0)\nonumber \\&\qquad +\, \varepsilon \left[ \frac{\varGamma }{4\pi }\sum _{j=1}^N \frac{k_{m-j}}{\alpha -\cos \theta _0}\vartheta _j(t)+ N\varGamma \left( \frac{\alpha - \cos \theta _0}{4\pi ^2\alpha }+\frac{\cos \theta _0}{4\pi N}\right) \vartheta _m\right. \nonumber \\&\qquad \left. -\,\varGamma \sum _{j=1}^N\frac{\vartheta _j}{4\pi ^2{\mathscr {A}}(\alpha -\cos \theta _0)}\right] +o(\varepsilon ). \end{aligned}$$

(43)

Owing to

$$\begin{aligned} \alpha -\cos \theta _m= & {} \alpha -\cos \theta _0+\varepsilon \sin \theta _0\vartheta _m+o(\varepsilon ),\\ (\alpha -\cos \theta _m)^2= & {} (\alpha -\cos \theta _0)^2\left( 1+2\varepsilon \frac{\sin \theta _0\vartheta _m}{\alpha -\cos \theta _0}\right) +o(\varepsilon ), \end{aligned}$$

we obtain

$$\begin{aligned}&r^2(\alpha -\cos \theta _m)^2 \frac{\mathrm {d}\phi _m}{\mathrm {d}t} \nonumber \\&\quad = r^2(\alpha -\cos \theta _0)^2\left( 1+2\varepsilon \frac{\sin \theta _0\vartheta _m}{\alpha -\cos \theta _0}\right) \left( V_N(\theta _0)+\varepsilon \frac{\mathrm {d}\varphi _m}{\mathrm {d}t}\right) +o(\varepsilon )\nonumber \\&\quad = (R-r\cos \theta _0)^2V_N(\theta _0)+2\varepsilon r^2(\alpha -\cos \theta _0)\sin \theta _0V_N(\theta _0)\vartheta _m\nonumber \\&\qquad +\varepsilon (R-r\cos \theta _0)^2\frac{\mathrm {d}\varphi _m}{\mathrm {d}t}+o(\varepsilon ). \end{aligned}$$

(44)

Comparing (43) with (44), and taking the limit as $\varepsilon \rightarrow 0$, we finally obtain the linearized equation for $\varphi _m$,

$$\begin{aligned} \frac{\mathrm {d}\varphi _m}{\mathrm {d}t}&=\frac{\varGamma }{4\pi r^2}\sum _{j=1}^N \frac{k_{m-j}}{(\alpha -\cos \theta _0)^3}\vartheta _j(t)-\varGamma \sum _{j=1}^N\frac{\vartheta _j}{4\pi ^2 r^2{\mathscr {A}}(\alpha -\cos \theta _0)^3}\\&\quad +\frac{N\varGamma }{r^2(\alpha -\cos \theta _0)^2}\left( \frac{\alpha - \cos \theta _0}{4\pi ^2\alpha }+\frac{\cos \theta _0}{4\pi N}-\frac{2r^2V_N(\theta _0)\sin \theta _0(\alpha -\cos \theta _0)}{N\varGamma }\right) \vartheta _m. \end{aligned}$$

Similarly, we derive the linearized equation for $\vartheta _m$,

$$\begin{aligned} \frac{\mathrm {d}\vartheta _m}{\mathrm {d}t}=\frac{\varGamma }{4\pi r^2}\sum _{j=1}^N \frac{k_{m-j}}{\alpha -\cos \theta _0}\varphi _j(t){,} \end{aligned}$$

by comparing (42) with

$$\begin{aligned} r^2(\alpha -\cos \theta _m)\frac{\mathrm {d}\theta _m}{\mathrm {d}t}= & {} r^2(\alpha -\cos \theta _0+\varepsilon \sin \theta _0\vartheta _m)\varepsilon \frac{\mathrm {d}\vartheta _m}{\mathrm {d}t}+o(\varepsilon )\\= & {} \varepsilon r^2(\alpha -\cos \theta _0)\frac{\mathrm {d}\vartheta _m}{\mathrm {d}t}+o(\varepsilon ). \end{aligned}$$