## Abstract

The paper gives the equation of motion for *N* point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles, called the circular domain. The velocity field induced by the point vortices is described in terms of the Schottky–Klein prime function associated with the circular domain. The explicit representation of the equation enables us not only to solve the Euler equations through the point-vortex approximation numerically, but also to investigate the interactions between localized vortex structures in the circular domain. As an application of the equation, we consider the motion of two point vortices with unit strength and of opposite signs. When the multiply connected domain is symmetric with respect to the real axis, the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle, which we investigate in detail.

## 1. Introduction

The study of incompressible and inviscid flows in planar multiply connected domains is not only one of the fundamental subjects in the field of mathematical fluid dynamics, but it also contributes towards the understanding of geophysical flows with many islands and artificial obstacles such as lakes, inland seas and coastal regions. This is because these regions are mathematically represented by multiply connected domains.

The motion of incompressible and inviscid flow is described by two-dimensional Euler equations. According to Kelvin’s theorem, circulation is conserved along the path of a fluid particle and thus vorticity is neither generated nor disappears during its evolution. Hence, in order to solve the Euler equations, we have only to investigate the evolution of the non-zero vorticity domain at the initial moment. Based on this observation, we discretize the initial non-zero vorticity domain with a set of *N* points, called *point vortices*, whose strengths are determined by the circulations around these points. Then, we track the evolutions of the *N* point vortices. This discretization method for the Euler equations is known as the vortex method (Cottet & Koumoutsakos 2000), which reduces the Euler equations to a system of ordinary differential equations for the *N* point vortices. The point-vortex system is often used as a simple mathematical model to describe interactions between localized vortex structures where the vorticity is concentrated in small regions. Readers can find many results and references for this topic in the books of Newton (2001) and Saffman (1992).

The point-vortex system is formulated as a Hamiltonian dynamical system, whose Hamiltonian is conventionally called the *Kirchhoff–Routh function*. Suppose that the *N* point vortices with strengths {*Γ*_{λ}|*λ*=1,…,*N*} are in a domain . Then, the Kirchhoff–Routh function is represented by
1.1
where denotes the complex conjugation of *z*_{λ}. The function *G*(*z*;*w*) is the hydrodynamic Green function satisfying the following Poisson equation:
1.2
with a certain boundary condition imposed on ∂𝒟, in which the *δ* function represents a source at *w*∈𝒟. The function ℛ(*z*;*w*) is derived from the Green function by
which is called the *Robin function* (Flucher & Gustafsson 1997). Note that *H*_{G} and *H*_{R} represent the vortex–vortex interaction and the vortex–boundary interaction, respectively. Lin (1941*a*, showed how to derive the equation for the *N* point vortices from the Kirchhoff–Routh function, which is stated as follows (Newton 2001).

## Theorem 1.1.

*Let* *be the Kirchhoff–Routh function for N point vortices {z _{λ}|λ=1,…,N} with strengths {Γ_{λ}|λ=1,…,N} in a domain . Then, the equation of motion for the N point vortices is given by*
1.3

*where*

*denotes the complex unit*.

Kirchhoff–Routh functions for the unbounded plane, the surface of a sphere and some simple domains with boundaries have been obtained (Saffman 1992; Newton 2001). On the other hand, Crowdy & Marshall (2005*a*) recently considered a bounded multiply connected domain inside the unit circle with many circular boundaries, which is called the *circular domain*, for which they gave an analytic representation of the Kirchhoff–Routh function. It has been applied to describe the steady irrotational uniform flow past many cylindrical obstacles in a plane (Crowdy 2006*a*) and to compute the lift on cylindrical obstacles (Crowdy 2006*b*).

Crowdy & Marshall (2005*b*) investigated the motion of a single point vortex, which is equivalent to a contour line of the Kirchhoff–Routh function, since the Hamiltonian dynamical system (1.3) for *N*=1 is always integrable. On the other hand, the motion of more than one point vortex is no longer integrable owing to a lack of special symmetry of the circular domain, and it has not been investigated well. This is because of the lack of an explicit representation of the equation for the *N* point vortices. Thus, the primary purpose of the present paper is deriving the equation for the *N* point vortices in the circular domain from the Kirchhoff–Routh function given by Crowdy & Marshall (2005*a*) and Lin’s theorem. Then, we make use of the equation to investigate the motion of two point vortices with unit strength and of opposite signs in the circular domain.

The paper consists of four sections. In §2, we derive the equation of the *N* point vortices in the multiply connected circular domain from the Kirchhoff–Routh function given by Crowdy & Marshall (2005*a*). In §3, we investigate the motion of two point vortices in a circular domain that is symmetric with respect to the real axis. Owing to the symmetry, it is reduced to the motion of a single point vortex in the multiply connected upper semicircle. Section 4 gives a summary and concluding remarks.

## 2. Equation for the *N* point vortices in circular domains

We introduce a special multiply connected domain *D*_{ζ} in the complex *ζ*-plane, called the circular domain, inside the unit circle |*ζ*|≤1 with *M* circular obstacles. The circular domain is regarded as a canonical multiply connected domain since it is mathematically shown that, for any bounded domain with *M* holes, there exists a conformal mapping that maps the domain to a circular domain with *M* circular obstacles (Nehari 1952). Let *C*_{0} denote the unit circle and {*C*_{i}|*i*=1,…,*M*} represent the boundaries of the disjoint *M* circular obstacles inside the unit circle, whose centres and radii are denoted by *δ*_{i}∈*D*_{ζ} and . The conjugation map *ϕ*_{i}(*ζ*) with respect to the unit circle associated with the circle *C*_{i} is given by
with which we define the Möbius maps ϑ_{i}(*ζ*) as
Here, we use the notations (*ϕ*(*ζ*))*=*ϕ**(*ζ**) and (*ϕ*(*ζ**))*=*ϕ**(*ζ*) for the conjugation of the map.

The infinite free group of maps generated by the basic Möbius maps ϑ_{i}(*ζ*) and their inverses for *i*=1,…,*M* is called the *Schottky group*, which is denoted by *Θ*. Note that every element *θ*_{i}(*ζ*) in the Schottky group is represented by a composition of these 2*M* basic maps. Let represent a subset of the Schottky group *Θ* that excludes the identity map and all inverse mappings. Then, the Schottky–Klein prime function *ω*(*ζ*,*α*) is defined as follows (Baker 1995):
2.1
For the unit circle |*ζ*|≤1, the only element in the Schottky group is the identity map, since there is no obstacle inside. Then, the subset is the empty set from the definition. For the doubly connected concentric annulus {*q*<|*ζ*|<1}, there is only one Möbius map ϑ_{1}(*ζ*)=*q*^{2}*ζ* associated with the inner boundary |*ζ*|=*q*. Since the Schottky group consists of all compositions of ϑ_{1}(*ζ*)=*q*^{2}*ζ* and its inverse , any element in the subset is represented by *θ*_{k}(*ζ*)=*q*^{2k}*ζ* for *k*>1.

Crowdy & Marshall (2005*a*) gave the Green function *G*(*ζ*;*α*) and the Robin function ℛ(*α*;*α**) for the circular domain *D*_{ζ} as follows:
The Kirchhoff–Routh function (1.1), with the Green function and the Robin function, enables us to derive the equation for the *N* point vortices in the circular domain *D*_{ζ} using theorem 1.1. The following theorem is one of the main results of this paper.

## Theorem 2.1.

*Let {z _{λ}|λ=1,…,N} denote the positions of N point vortices with strengths {Γ_{λ}|λ=1,…,N} in the circular domain* .

*Then, the motion of the point vortices is described by*2.2

*in which*.

*ω*_{ζ}(*ζ*,*α*)=(∂/∂*ζ*)*ω*(*ζ*,*α*) andThe two terms in the summation represent the contributions from the point vortex at *z*_{α} and its conjugation point with respect to the unit circle at , while the last term is the velocity field induced by the point vortex at the conjugation point of *z*_{λ}. Since the function *ω*_{ζ}(*ζ*,*α*)/*ω*(*ζ*,*α*) plays a key role in equation (2.2), the following expression is useful.

## Proposition 2.2.

*The function ω_{ζ}(ζ,α)/ω(ζ,α) is given by*
2.3

*in which*.

## Proof.

The calculation is straightforward. Let us write , where Since we have the proof is completed by 2.4 ▪

Owing to this proposition, we can write down the explicit representations of *ω*_{ζ}(*ζ*,*α*)/*ω*(*ζ*,*α*) for the simply connected unit disc |*ζ*|≤1 and the doubly connected concentric annulus . For the unit circle, we have *ω*(*ζ*,*α*)=(*ζ*−*α*), since the subset *Θ*′′ is the empty set, which leads to
2.5
Regarding the doubly connected concentric annulus, any element in is represented by *θ*_{k}(*ζ*)=*q*^{2k}*ζ* for *k*>1. Hence, it follows from proposition 2.2 and
that we have
2.6

In the following, we give the proof of theorem 2.1. First, let us show some basic properties for the Schottky–Klein prime function *ω*(*ζ*,*α*).

## Lemma 2.3.

*The Schottky–Klein prime function ω(ζ,α) satisfies the following*:

*ω*(*ζ*,*α*)=−*ω*(*α*,*ζ*),*for z∈D*_{ζ},*and**for z∈D*._{ζ},*ω*_{ζ}(z^{*−1},z)=−*ω*_{α}(z,z^{*−1})

## Proof.

Since properties (i)–(iii) have already been confirmed in Crowdy & Marshall (2005*a*), we prove the last two properties. Differentiating with respect to *ζ* and *α*, as in the proof of proposition 2.2, we have
2.7
and
2.8
Substituting *ζ*=*α*=*z*∈*D*_{ζ}, we have owing to .

Regarding property (v), we first show for *z*∈*D*_{ζ}. It is easy to see from equations (2.7) and (2.8) that
Since owing to property (iii), we have . Then, it follows from
2.9
that we have
▪

Now, we calculate the right-hand side of equation (1.3). First, we deal with the contribution from the vortex–vortex interaction part *H*_{G}. Let us rewrite Green’s function *G*(*z*_{λ};*z*_{α}) as
2.10
The terms in *H*_{G} that contain the variable *z*_{λ} are given by
Since the derivative of *G*(*z*_{λ};*z*_{α}) with respect to becomes
we have
2.11

Next, we consider the contribution from the vortex–boundary interaction part *H*_{R}. The following lemma gives us a simpler expression of the Robin function.

## Lemma 2.4.

*The Robin function ℛ( α;α*) is expressed as*
2.12

## Proof.

It follows from property (i) of lemma 2.3 that we have Thus, we have . Then the Robin function ℛ is rewritten as ▪

Owing to this lemma, becomes Differentiating it with respect to , we have 2.13 Note that the third term after the first equality vanishes owing to property (iv) of lemma 2.3. To reduce it further, we need the following lemma.

## Lemma 2.5.

2.14

## Proof.

It follows from properties (i) and (ii) of lemma 2.3 that we obtain
Differentiating both sides of property (i) of lemma 2.3 with respect to *ζ*, we have
which leads to
Hence, owing to properties (i), (ii) and (v) of lemma 2.3, we obtain
▪

Substituting equation (2.14) into equation (2.13), we finally obtain 2.15 Consequently, equation (2.2) is derived from equations (2.11) and (2.15).

## 3. Motion of two point vortices in the circular domain

### (a) Reduction to a single point vortex in a semicircle

We consider the motion of two point vortices located at *z*_{1} and *z*_{2} in the circular domain *D*_{ζ} with strengths *Γ*_{1}=−*Γ*_{2}=1. Then, equation (2.2) for the two point vortices is given by
3.1
and
3.2
When the domain is the unit circle or the doubly connected concentric annulus {*ζ*|*q*<|*ζ*|<1}, their motion is integrable, since the system admits an additional invariant quantity *I*=|*z*_{1}|^{2}−|*z*_{2}|^{2}, which corresponds to the invariance of the system with respect to the rotation around the origin. This is directly confirmed from equations (3.1) and (3.2) as follows:
This vanishes if we have, for arbitrary *z*, ,
For the unit circle, these conditions hold, since equation (2.5) satisfies
We also have the invariance of *I* for the concentric annulus, since it follows from equation (2.6) that
and

Here, let us consider a circular domain that is symmetric with respect to the real axis. In other words, for arbitrary obstacle *C*_{i}, there exists an integer *j*∈{1,…,*M*} such that the obstacles *C*_{j} and *C*_{i} are symmetric with respect to the real axis, i.e. and *q*_{j}=*q*_{i}. Note that, if *j*=*i*, the obstacle *C*_{i} is symmetric with respect to the real axis. In terms of the maps in the Schottky group, it is equivalent to say that, for arbitrary map , there exists a map *θ*_{j}(*ζ*) such that , since for any generating functions of the Schottky group, we have
Then, for arbitrary *ζ* and *α*∈*D*_{ζ}, we obtain
3.3
Suppose that the initial location of the two point vortices is . Then, owing to equation (3.3), we have
Thus, we prove the following theorem.

### Theorem 3.1.

*Let the circular domain D _{ζ} be symmetric with respect to the real axis. Suppose also that the initial configuration of the two point vortices with Γ_{1}=−Γ_{2}=1 satisfies* .

*Then, we have*

*for t≥0*.

Owing to this theorem, the orbits of the two point vortices in the circular domain are symmetric with respect to the real axis, which are observed by plotting the contour lines of the Hamiltonian with , i.e. 3.4 Hence, it is possible to investigate the motion of the two point vortices by plotting the contour lines of the Hamiltonian (3.4).

Finally, let us mention how to compute the Schottky–Klein prime function *ω*(*ζ*,*α*) numerically. Since it is unable to calculate the infinite product in equation (2.1), we have to truncate it up to a finite number of the Möbius maps in . As in Crowdy & Marshall (2005*a*), we introduce the *level* of all possible compositions of the generating maps ϑ_{i}(*ζ*) and . The *level-zero map* is the identity map. The *level-one maps* contain all generating maps ϑ_{i}(*ζ*) and their inverse for *i*=1,…,*M*. All compositions of any two of the level-one maps that cannot be reduced to the identity map constitute the *level-two maps*. We can define recursively the *level-three maps* that consist of all possible combinations of three of the level-one maps and that are not reduced to a lower-level map. In the present paper, we have truncated the infinite product (2.1) up to the level-three maps. The truncated finite product approximates the Schottky–Klein prime function accurately if the obstacles *C*_{i} for *i*=1,…,*M* are well separated, as is confirmed in Crowdy & Marshall (2005*a*). As for the infinite summation (2.3) in proposition 2.2, we similarly truncate it up to the level-three maps.

### (b) Motion of a point vortex in multiply connected semicircles

#### (i) Two obstacles

Owing to the reflectional symmetry, the two obstacles *C*_{1} and *C*_{2} are symmetric with respect to the real axis. Namely, the centres and the radii of the two obstacles are given by
3.5
for *a*∈(−1,1) and *b*>0. The radius *r* satisfies , since the obstacle *C*_{1} must be contained in the upper semicircle.

Figure 1 shows the contour plots of the Hamiltonian (3.4) for *a*=0, *r*=0.05 and various *b* in equation (3.5). Saddle and centre points of the contour plot correspond to unstable and neutrally stable fixed configurations of the two point vortices, respectively. The contour plots are symmetric with respect to the real axis, which means that when the point vortex with positive unit strength goes along a contour line in the upper semicircle, the other point vortex with negative unit strength always moves along the contour line in the lower semicircle that is symmetric with respect to the real axis. Thus, it is sufficient for us to focus on the contour lines in the upper semicircle. When *b*=0.2 in figure 1*a*, there are two fixed configurations, a centre point above the obstacle *C*_{1} and a saddle point between the obstacle and the real axis. The saddle point is connected by homoclinic orbits. For *b*=0.4 in figure 1*b*, while the saddle point with the homoclinic connections below the obstacle remains, a new saddle point appears above the obstacle. The new saddle point is connected by homoclinic orbits that surround two centre points. At *b*=0.4885 in figure 1*c*, the homoclinic orbits with respect to the two saddle points coincide, and then two saddles with homoclinic connections appear again for *b*=0.6 in figure 1*d*. Finally, when the island approaches the outer boundary for *b*=0.8 in figure 1*e*, the saddle point below the obstacle *C*_{1} changes to a centre point.

The motion of the two point vortices changes largely in the neighborhood of the saddle points of the contour plot. For instance, in figure 1*a*, when we set the point vortex below the saddle point at the initial moment, it goes along the real axis and the boundary of the unit circle. On the other hand, it rotates around the obstacle *C*_{1} when it is initially set above the saddle. Thus, in order to characterize the motion of the single point vortex in the upper semicircle, we pay attention to the topological structure that consists of the saddle/centre points, their heteroclinic/homoclinic orbits in the contour lines of the Hamiltonian (3.4) and the obstacle *C*_{1}. In figure 2, we show the topological patterns corresponding to the contour plots from figure 1*a*–*e*. The topological patterns for *b*=0.2 and 0.8 and those for *b*=0.4 and 0.6 are the same topologically. Hence, the topological patterns are referred to as Ia for *b*=0.2, IIa for *b*=0.4, III for *b*=0.4885, IIb for *b*=0.6 and Ib for *b*=0.8. Figure 2 also illustrates how the topological patterns change with respect to *b*. The changes from pattern Ia to IIa and from pattern IIb to Ib are because of a pitchfork bifurcation from the neutrally stable fixed configuration to an unstable one with homoclinic connections, which gives rise to the two neutrally stable fixed configurations. The transition between patterns IIa and IIb occurs because of the reconnection of the homoclinic orbits through the degenerate pattern III. Table 1 gives the classification of the topological patterns of the contour lines for the other cases *a*=0.3 and 0.7, with various *b* and *r*, in which we typically observe patterns Ia, Ib, IIa and IIb. The degenerate pattern III also appears for the parameters between patterns IIa and IIb. We have examined the topological patterns for the other values of *a*, but we did not find any other patterns.

#### (ii) Three obstacles

Since the circular domain with three obstacles has reflectional symmetry, the centre of one obstacle is on the real axis. Thus, the centres and the radii of the three obstacles are given by
3.6
for *a*, *c*∈(−1,1) and positive *b*, *r*_{1} and *r*_{2}. Figure 3 shows contour plots of the Hamiltonian (3.4) in the circular domains with the three obstacles (3.6) for *a*=0, *b*=0.8, *r*_{1}=0.05 and *c*=0. We change the radius *r*_{2} of the obstacle *C*_{3}. For small *r*_{2}, the topological pattern is equivalent to pattern Ib in figure 2. As *r*_{2} gets larger, the topological pattern changes to IIb and IIa. Figure 4 gives other contour plots of the Hamiltonian (3.4) for *a*=0, *b*=0.6, *r*_{1}=0.1 and *c*=0.3, which clearly demonstrates the transition of the contour pattern from IIb to IIa through the degenerate pattern III at *r*_{2}=0.2969. We have checked other configurations of the three obstacles, but we are unable to find any other topological patterns except those in figure 2. Hence, we could say that the topological patterns observed here are the same as those in §3*b*(i).

#### (iii) Four obstacles

We need to consider many parameters in order to determine the locations of four islands. Hence, in this section, we deal with a special configuration of the four circular obstacles of the same radius. The centres of the obstacles are represented by 3.7 The radii of the four obstacles are fixed to 0.1.

First, we fix *b*=0.5 and change *a*. Figure 5 shows the contour plots of the Hamiltonian (3.4). When *a*=0.15 in figure 5*a*, in which the two obstacles *C*_{1} and *C*_{3} are close, there exists a saddle point between the two obstacles with homoclinic connections. In addition, we have two saddles and three centres below the obstacles and two saddles and one centre above the obstacles. Each saddle point is connected by homoclinic and heteroclinic orbits. For *a*=0.2 in figure 5*b* and *a*=0.25 in figure 5*c*, the number of saddle and centre points remains the same as in figure 5*a*, but their global topological patterns of the homoclinic and heteroclinic orbits are different. As the parameter *a* increases, the saddle point between the two obstacles changes to a centre point, as in figure 5*d* for *a*=0.3. The two saddles and the three centres below the obstacles collapse to a centre in figure 5*e* for *a*=0.5. We obtain five topological patterns, which are referred to as I for *a*=0.15, II for *a*=0.2, III for *a*=0.25, IV for *a*=0.3 and V for *a*=0.5, respectively.

The transitions between patterns I, II and III occur not because of the change of stability of the fixed configuration, but because of the reconnection of the homoclinic and heteroclinic orbits through the degenerate cases for *a*=0.1935 in figure 6*a* and for *a*=0.21269 in figure 6*b*. The subcritical pitchfork bifurcation from saddle to centre results in the change of the topological pattern from III to IV. The transition between patterns IV and V occurs because of a degenerate pinching of the contour lines of the Hamiltonian around the centre point below the obstacles, as shown in figure 7*a*.

Next, we change *b* with *a*=0.15. Figure 8 shows the contour plots of the Hamiltonian (3.4). When the two obstacles *C*_{1} and *C*_{3} are close to the real axis for *b*=0.15 in figure 8*a*, we have two saddle points with heteroclinic and homoclinic connections between the obstacles and the real axis and a centre point above them. The pattern is topologically equivalent to pattern V for figure 5*e*. Another degenerate pinching as in figure 7*b* generates a saddle point with homoclinic connections for *b*=0.2 in figure 8*b*, which gives rise to a new pattern VI. A global reconnection of the heteroclinic and homoclinic orbits occurs from figure 8*b* to figure 8*c* for *b*=0.3, which is identified as a new pattern VII. The degenerate topological pattern between the patterns VI and VII for *b*=0.2838 is given in figure 6*c*. The pinching bifurcation of figure 7*a* around the centre above the obstacles results in the transition from pattern VII to I for *b*=0.4 and 0.5 in figure 8*d*,*e*. Then, the global reconnection from pattern I to II is observed for *b*=0.72 in figure 8*f* through the degenerate pattern, as in figure 6*a*. The transition from pattern II to VI is because of the pinching bifurcation of figure 7*b* around the centre point below the two obstacles. These seven topological patterns I–VII are typically observed for the other configurations of the obstacles (3.7). Table 2 gives the classification of the topological patterns of the contour lines for *a*=0.2, 0.25 and 0.3 with various *b*, in which we observe patterns II–V. For *a*=0.2, a new transition from pattern II to IV is observed owing to the degenerate pinching of figure 7*b*. We summarize the transitions and the bifurcations of the topological patterns of the contour lines in the upper semicircle in figure 9. Let us note that we are unable to find any other generic patterns except the seven patterns, although we have examined patterns for other possible configurations of the four obstacles.

### (c) Comparison with the motion of a single point vortex in circles

As is shown in Crowdy & Marshall (2005*b*), since the motion of a single point vortex in any circular domain is integrable, its orbit corresponds to a contour line of the following Hamiltonian:
3.8
which is obtained by taking *N*=1 in equation (1.1). We investigate the topological pattern of the contour lines of equation (3.8) for circular domains with the same connectivity as the semicircles considered in §3*b*.

We first deal with the motion of a point vortex in a doubly connected circular domain with one obstacle, which we compare with the results in §3*b*(i) and 3*b*(ii). Because of the rotational symmetry of the unit circle, the centre of the obstacle is given by *δ*_{1}=*b*i for *b*>0 without loss of generality. Figure 10 shows the contour plot of *H*^{(s)} for *b*=0.5. The radius of the obstacle is 0.1. We have one saddle point with homoclinic connections and one centre point, whose corresponding topological pattern is equivalent to pattern Ib in figure 2. For the other value of *b* and the radius of the obstacle, the only topological pattern we observe is pattern Ib.

Next, we consider the motion of a point vortex in a triply connected circular domain. Since the configuration of the four obstacles (3.7) in §3*b*(iii) has special symmetry, for the sake of comparison, we assume that the centres of two obstacles in the circular domain are located at
3.9
and the radii of the two obstacles are both 0.1. Figure 11 shows the contour plots of the Hamiltonian (3.8) with *b*=0 and various *a*. The topological patterns for *a*=0.2, 0.3 and 0.5 are equivalent to patterns VII, VI and V in figure 9, respectively. The global reconnection of the homoclinic orbits occurs between patterns VII and VI through the degenerate pattern, as in figure 6*c*. The saddle point between the obstacles in figure 11*b* becomes the centre point in figure 11*c* because of the pitchfork bifurcation. The transition and bifurcation between the three patterns are observed for the other parameters *a* and *b*, as shown in table 3.

These results indicate that the possible topological patterns of the contour lines for the motion of the single point vortex in the circular domain are given as a part of those observed in the semicircle with the same connectivity. Hence, we can observe a wide variety of topological patterns of the contour lines and the transitions and bifurcations between them in the motion of the single point vortex in the multiply connected semicircle compared with those in the multiply connected circle.

### (d) Non-integrable motion of the two point vortices without symmetry

When either the circular domain or the initial configuration of the two point vortices is not symmetric with respect to the real axis, the system is no longer integrable. Here, we give some numerical samples of the motion of the two point vortices without reflectional symmetry. Equations (3.1) and (3.2) are integrated using the fourth-order Runge–Kutta method with time-step size Δ*t*=0.001. We consider the same symmetric circular domain with the two obstacles (3.5) for *a*=0, *b*=0.4 and *r*=0.05, as in figure 1*b*. On the other hand, we deal with an initial configuration without reflectional symmetry. Let *ζ*_{1} and *ζ*_{2} denote the two saddle fixed points obtained in figure 1*b*. Then, the initial configuration is given by
3.10
for *ε*_{1}, . For given *ε*_{1}, the parameter *ε*_{2} is determined numerically so that the value of the Hamiltonian for equation (3.10) is equivalent to that of the Hamiltonian (3.4) with reflectional symmetry at *z*_{1}=*ζ*_{k} up to 12 digits.

Figure 12 shows the trajectories of the two point vortices for the initial data (*a*) ** z**(0)=

*z*_{1}and (

*b*)

**(0)=**

*z*

*z*_{2}with

*ε*

_{1}=1.0×10

^{−4}. Although the initial disturbance

*ε*

_{1}is quite small, they are far from being integrable. Figure 13

*a*shows the evolutions of for

**(0)=**

*z*

*z*_{1}and

*z*_{2}with

*ε*

_{1}=1.0×10

^{−4}, which indicate that the reflectional symmetry between the two point vortices is breaking, since

*d*(

*t*)≡0 for all time if the system is integrable. In addition, in order to see the dependence on the initial disturbance, we observe the centre of the two point vortices, i.e.

*c*(

*t*)=1/2(

*z*

_{1}(

*t*)+

*z*

_{2}(

*t*)), which evolves in the real axis if the motion is integrable. Figure 13 also shows the trajectories of

*c*(

*t*) for (

*b*)

**(0)=**

*z*

*z*_{1}and (

*c*)

**(0)=**

*z*

*z*_{2}with

*ε*

_{1}=1.0×10

^{−4}and 3.0×10

^{−4}, which indicate that the motion of the two point vortices is sensitive to the initial disturbance, since the trajectories of

*c*(

*t*) for

*ε*

_{1}=1.0×10

^{−4}and 3.0×10

^{−4}are different.

The motion of the two point vortices is reduced to a three-dimensional dynamical system since equations (3.1) and (3.2) define the Hamiltonian dynamical system in the four-dimensional phase space (Re *z*_{1},Im *z*_{1},Re *z*_{2},Im *z*_{2}), and the orbit evolves in the same energy surface of the Hamiltonian. Thus, in order to observe the motion of the two point vortices from another point of view, we show the Poincaré section {Re *z*_{1}=0} in figure 12*c*,*d* corresponding to the trajectories in figure 12*a*,*b*. For ** z**(0)=

*z*_{1}, the orbits spread in a region near the homoclinic orbits to

*ζ*

_{1}. On the other hand, they spread inside the whole unit circle for

**(0)=**

*z*

*z*_{2}. We infer from them that the evolution of the two point vortices without the reflectional symmetry could be chaotic.

## 4. Summary and concluding remarks

We have derived the equation of motion for the *N* point vortices in a multiply connected domain, called the circular domain, inside the unit circle that has *M* circular obstacles. The interaction between the point vortices is expressed by the function *ω*_{ζ}(*ζ*,*α*)/*ω*(*ζ*,*α*), in which *ω*(*ζ*,*α*) is the Schottky–Klein prime function associated with the circular domain. Owing to the explicit representation, it is possible to approximate the solution of the Euler equations in the circular domain numerically with the vortex method (Cottet & Koumoutsakos 2000). Moreover, for given *N* point vortices {*z*_{λ}|*λ*=1,…,*N*} with strengths {*Γ*_{λ}|*λ*=1,…,*N*}, we have the following expression for the instantaneous velocity field at any point *z* in the circular domain:
which is available to compute the motion of passive scalars advected by the *N* point vortices.

Let us mention the equation for the *N* point vortices for general multiply connected domains. Suppose that the conformal mapping *ζ*=*f*(*w*) from a given multiply connected domain 𝒟_{w} in the complex *w*-plane to a circular domain 𝒟_{ζ} in the complex *ζ*-plane is constructed. Then, the motion of the *N* vortex points {*w*_{λ}|*λ*=1,…,*N*} in 𝒟_{w} is described by a Hamiltonian dynamical system, whose Hamiltonian *H*^{(w)} is represented by
in which *H*^{(ζ)} denotes the Kirchhoff–Routh function for the *N* point vortices {*ζ*_{λ}|*λ*=1,…,*N*} in the circular domain 𝒟_{ζ} with *ζ*_{λ}=*f*(*w*_{λ}).

We have applied the equation to investigate the motion of two point vortices with unit strength and of opposite signs. When the domain is either the simply connected circle or the doubly connected concentric annuls, their motion is integrable. On the other hand, for the circular domains with more than one obstacle, it is not integrable in general. However, when the circular domain is symmetric with respect to the real axis, the motion of the two point vortices is reduced to that of a single point vortex in the upper semicircle, if their initial configuration has the same symmetry. We have described the motion of the single point vortex when the circular domain contains two, three and four obstacles by plotting the contour lines of the Hamiltonian. We focused on the topological pattern in the semicircle that consists of fixed configurations, homoclinic/heteroclinic orbits and the obstacles. Then, we described the transitions and bifurcations between the topological patterns, which occured because of the pitchfork bifurcation, the degenerate pinching bifurcations of the contour line around the centre point and the reconnection of the heteroclinic/homoclinic orbits. Comparing the results with the motion of the single point vortex in the circular domains with the same connectivity, we found that more topological patterns of the contour lines and more complicated transitions and bifurcations between them are possible for semicircles than for circles. Moreover, we have given several numerical examples of the motion of two point vortices when the saddle points in the integrable system are slightly perturbed. Although the initial perturbation is quite small, the evolution of the two point vortices without the reflectional symmetry becomes complicated, which is sensitive to the initial amplitude of disturbance. A Poincaré section for trajectories illustrates that the motion of the two point vortices becomes chaotic.

## Acknowledgements

This study is partially supported by JSPS grant no. 19654014 and by JST PRESTO. I would like to show my gratitude to the School of Applied Mathematics at University of Sheffield for providing me with a nice research environment during my stay as a visiting scholar.

## Footnotes

- Received February 9, 2009.
- Accepted May 7, 2009.

- © 2009 The Royal Society