2.1. The invariant function

For unbounded ideal fluid, planar motion of an ensemble of N point vortices with strengths $\gamma _n$ and coordinates $(x_n,y_n)$ occurs with conservation of the Hamiltonian H (i.e. interaction energy), combined invariant M and the total vortex momentum L

(2.1)\begin{gather}H = -\displaystyle{1 \over {4{{\rm \pi} }}}\sum\limits_{n = 1}^{N-1} {\sum\limits_{k = n + 1}^N {\gamma _n\gamma _k\log (r_{nk}^2 )} ,} \end{gather}

(2.2)\begin{gather}M = \Gamma J-I_x^2 -I_y^2 = \sum\limits_{n = 1}^{N-1} {\sum\limits_{k = n + 1}^N {\gamma _n\gamma _kr_{nk}^2 } } ,\end{gather}

(2.3)\begin{gather}L = 2{{\rm \pi} }\sum\limits_{n = 1}^N {\gamma _n(x_n\,\hbox{d}y_n/\hbox{d}t-y_n\,\hbox{d}\kern0.06em x_n/\hbox{d}t)} = \sum\limits_{n = 1}^{N-1} {\sum\limits_{k = n + 1}^N {\gamma _n\gamma _k} } ,\end{gather}

where $r_{nk}$ are intervortex separations. Here, $\Gamma = \sum\nolimits_{n = 1}^N {\gamma _n} $ is the total circulation, $J = \sum\nolimits_{n = 1}^N {\gamma _n(x_n^2 + y_n^2 )} $ is the angular momentum, $I_x = \sum\nolimits_{n = 1}^N {\gamma _nx_n} $, $I_y = \sum\nolimits_{n = 1}^N {\gamma _ny_n} $ are the components of the linear momentum; all are conserved quantities. Note that $M = \Gamma J$ when the origin of the coordinates system lies in the centre of vorticity $(I_x/\Gamma ,I_y/\Gamma )$. The system of 2N equations $\gamma _n\,\hbox{d}\kern0.06em x_n/\hbox{d}t = \partial H/\partial y_n$, $\gamma _n\,\hbox{d}y_n/\hbox{d}t = -\partial H/\partial x_n$ describes the absolute motion of N vortices.

Considering relative motion as the evolution of the geometrical form of an ensemble of vortices, we can use polar coordinates with origin at the position of any vortex (the first for definiteness) and a reference polar axis that contains the segment $r_{12}$ (again for definiteness). Choosing then the magnitude of $r_{12}$ as a variable space scale, one can introduce 2(N−2) independent variables: polar angles $\varphi _n$ and scaled radii $\rho _n = r_{1n}/r_{12},\;n \ge 3$, which uniquely define geometrical form (but not its size, position and orientation in space) of the vortex configuration at any time.

For $r_{nk} = r_{12}\tilde{r}_{nk}$, $n = 1, \ldots ,N,\;k \gt n$, where $\tilde{r}_{nk}$ are the normalized intervortex separations, the expressions (2.1) and (2.2) become $H = \tilde{H}-L\log (r_{12}^2 )/(4{{\rm \pi} })$ and $M = r_{12}^2 \tilde{M}$, where $\tilde{H} = H(\tilde{r}_{nk})$, $\tilde{M} = M(\tilde{r}_{nk})$. Thus, the invariants of motion (2.1)–(2.3) can be combining into one invariant function of relative motion which depends on the above-defined normalized phase variables only

(2.4)\begin{align} F& = -4{{\rm \pi} }H-L\log |M| = -4{{\rm \pi} }\tilde{H}-L\log |\tilde{M}|\nonumber\\ & = \gamma _1\sum\limits_{n = 3}^N {\gamma _n\log (\rho _n^2 )} + \sum\limits_{n = 2}^{N-1} {\gamma _n} \sum\limits_{k = n + 1}^N {\gamma _k\log [\rho _n^2 + \rho _k^2 -2\rho _n\rho _k\cos (\varphi _n-\varphi _k)]} -L\log |R|,\end{align}

where a scaled function

(2.5)\begin{equation}R = \tilde{M} = M/r_{12}^2 = \sum\limits_{n = 2}^N {\gamma _n(\Gamma -\gamma _n)\rho _n^2 -2} \sum\limits_{n = 2}^{N-1} {\gamma _n\rho _n} \sum\limits_{k = n + 1}^N {\gamma _k\rho _k\cos (\varphi _n-\varphi _k)} ,\end{equation}

is introduced and $\varphi _2 = 0,\;\rho _2 = 1$. The cosine rule is used here to express the intervortex separations $r_{nk}$ at $n \gt 1,\;k \gt n$.

The invariant function F, in fact, represents the Hamiltonian H restricted to a level surface of the invariant M. An isoline of this function represents the evolution of geometrical shapes of a vortex ensemble, but not of its sizes (because of the normalization on r ₁₂). More rigorously, from this point on, we are discussing the behaviour of a new dynamical system with the phase space $(\varphi _3,\rho _3, \ldots ,\varphi _N,\rho _N)$, in which the absolute motion and orientation of a vortex ensemble are ignored and geometrically similar configurations of vortices are indistinguishable. Only in this sense shall we talk below of phase trajectories, stationary points (or equilibria) and their stability. For example, now not only conventional relative equilibria, but also a self-similarly expanding or collapsing vortex ensemble becomes represented in this phase space by a stationary point, whose stability/instability may be interpreted as stability/instability of the non-stationary self-similar motion. A solution of a set of equations $\partial F/\partial \varphi _n = 0,\;\partial F/\partial \rho _n = 0$ gives stationary points in the phase space, i.e. equilibrium configurations. The type of these points (maximum/minimum or saddle) defines the nonlinear stability of the corresponding stationary state. To do this, it is enough to calculate the eigenvalues of the Hessian, a square matrix of second-order partial derivatives of the invariant function.

2.2. Necessary condition for group scattering

In addition to stationary points in the phase space $(\varphi _3,\rho _3, \ldots ,\varphi _N,\rho _N)$, there are also singular points induced by the logarithmic terms in the invariant function. The set of these points determines the topology of the phase space and (qualitatively) the evolution of a given vortex configuration depending on its initial location in this space. If the phase trajectory touches some singular point (or curve), the initially N-vortex system asymptotically splits into certain independent groups of vortices. When not all of the circulations of the individual vortices have the same sign, some intervortical distances may increase to infinity. For example, an N-vortex system during the well-known self-similar motion can either break up to N infinitely distant vortices (within infinite time), or collapse in one point within finite time (with the velocity growing to infinity). In the case of three vortices (see e.g. Newton Reference Newton2001) the necessary and sufficient condition for self-similar complete scattering/collapse is simultaneous vanishing of the invariants L and M (or R). Nevertheless, at any N, the invariant function remains finite, as directly follows from its definition (2.4).

A necessary condition for complete group scattering can be derived as follows. Let N vortices cluster during the evolution into K groups ($1 \lt K \le N$) with the circulations $\Gamma _k$ (the sum of strengths of all vortices belonging to kth group). Clustering and the hypothetical scattering of the groups mean that the intervortex separations remain finite within each group only. Therefore, the invariants of motion H and M will remain finite only if the summarized contribution of the terms in (2.1) and (2.2) tending to infinity is zero. After regrouping the terms, the sought-for necessary condition for group scattering takes the form

(2.6)\begin{equation}L_G = \sum\limits_{k = 1}^{K-1} {\sum\limits_{m = k + 1}^K {\Gamma _k\Gamma _m} } = 0.\end{equation}

Here, $L_G$ is the group analogue of the total vortex momentum L. Condition (2.6) is also valid in the case of emission of inseparable groups with zero group circulation.

3.1. Invariant function and singular points

Consider now three vortices with strengths $\gamma _1 = \gamma ,\;\gamma _2 = \alpha ,\;\gamma _3 = \beta $ (see figure 1). Setting $\varphi _3 = \varphi ,\;\rho _3 = b/a = q$ in formulae (2.4) and (2.5), we will obtain the following expression for the invariant function

(3.1)\begin{equation}F(\varphi ,q) = -L\log |R(\varphi ,q)| + \beta \gamma \log q^2 + \alpha \beta \log P(\varphi ,q),\end{equation}

where

(3.2)\begin{gather}P(\varphi ,q)\equiv c^2/a^2 = q^2-2q\cos \varphi + 1,\end{gather}

(3.3)\begin{gather}R(\varphi ,q) = \beta (\alpha + \gamma )q^2-2\alpha \beta q\cos \varphi + \alpha (\beta + \gamma ),\end{gather}

(3.4)\begin{gather}L = \alpha \gamma + \beta \gamma + \alpha \beta .\end{gather}

Figure 1.

Parameters of three vortices in the plane; a, b and c are dimensional, 1 and q, normalized sizes.

The invariant function is 2${\rm \pi}$-periodic and symmetric relative to the vertical axis $\varphi = {{\rm \pi} }$. Except for the cases where emission or complete scattering can occur, all phase trajectories are either 2${\rm \pi}$-periodic or closed (see examples in figure 2). Periodicity of phase trajectories, however, does not mean strict periodicity of the absolute motion on the $(x,y)$ plane.

Figure 2.

Phase diagrams of the relative motion of three vortices at (a) $\alpha = \beta = \gamma = 1$ $(\Gamma = 3,\;L = 3)$; (b) $\alpha = 0.5,\;\beta = -1.5,\;\gamma = 1$ $(\Gamma = 0,\;L = -1.75)$; (c) $\alpha = 1,\;\beta = -0.75,\;\gamma = 1$ $(\Gamma = 1.25,\;L = -0.5)$. Black solid circles mark the stationary points representing collinear equilibria at $\varphi = 0,{{\rm \pi} }$ and the equilateral triangle at $\varphi = {\rm \pi}/3,\;q = 1$; dashed bold lines, the separatrices passing through the saddle points. Grey circles mark the singular points S2; bold solid line in (c), singular curve S1. Arrows indicate the direction of motion in the phase trajectories.

For any set of strengths, the phase diagram necessarily includes a singular point with coordinates (0, 1) for which $P(\varphi ,q) = 0$. In this point two of the three vortices ‘merge’ into one ‘vortex’ and the initial problem reduces to N = 2. So, hereinafter we will refer to this type of singularity as S2. The same reduction occurs also at $q = 0$ and $q = \infty $. In the phase diagrams presented in figure 2 points S2 appear as centres.

Singular points of another type, S1 (when the initial problem may reduce to N = 1, see below), fill the curve described by the equation $R(\varphi ,q) = 0$. The condition for the existence of the singular curve S1 is easily derived using (3.3):

(3.5)\begin{equation}s = \gamma \Gamma /(\alpha \beta ) \lt 0.\end{equation}

If the strengths of all vortices have the same sign (figure 2a), curve S1 does not exist. If $\Gamma = 0$ (figure 2b), S1 consists of one point which is the stationary point $(0,-\alpha /\beta )$ at $\alpha /\beta \lt 0$ or $({{\rm \pi} },\alpha /\beta )$ at $\alpha /\beta \gt 0$. In the phase diagram, this point is represented by the centre. Generally (see figure 2c), the curve S1 consists of a set of points, in each of which the local infinite extremum of the invariant function is reached, while the Hamiltonian of the system is finite.

At $L = 0$ the curve S1 (where $R = 0$) presents a set of initial vortex configurations, for which the motion becomes self-similar, i.e. either total scattering or collapse occurs. At $L\ne 0$ curve S1 in the phase diagram represents a special line in which the absolute motion is periodic. We will discuss some features of such a motion in § 4.

3.2. Nonlinear stability of equilibria

Stationary states, or equilibria, of three vortices and their linear stability were investigated earlier by Tavantzis & Ting (Reference Tavantzis and Ting1988) and Aref (Reference Aref2009). Here we search for equilibria and analyse their nonlinear stability using the invariant function F. Its partial derivatives are

(3.6a,b)\begin{gather}F_\varphi = Aq\sin \varphi ,\quad F_q = A(q-\cos \varphi ) + Cq,\end{gather}

(3.7a)\begin{gather}F_{\varphi \varphi } = Aq\cos \varphi -Bq^2\sin ^2\varphi ,\end{gather}

(3.7b)\begin{gather}F_{qq} = A + C-B(q-\cos \varphi )^2-Dq^2 + 2Eq(q-\cos \varphi ),\end{gather}

(3.7c)\begin{gather}F_{\varphi q} = F_{q\varphi } = [A-Bq(q-\cos \varphi ) + Eq^2]\sin \varphi ,\end{gather}

where

(3.8)\begin{equation}\left. \begin{gathered}A = 2\alpha \beta \left( {\displaystyle{1 \over P}-\displaystyle{L \over R}} \right),\quad B = 4\alpha \beta \left( {\displaystyle{1 \over {P^2}}-\displaystyle{{\alpha \beta L} \over {R^2}}} \right),\quad C = 2\beta \gamma \left( {\displaystyle{1 \over {q^2}}-\displaystyle{L \over R}} \right),\\ D = 4\beta \gamma \left( {\displaystyle{1 \over {q^4}}-\displaystyle{{\beta \gamma L} \over {R^2}}} \right),\quad E = \displaystyle{{4\alpha \beta^2\gamma L} \over {R^2}}.\end{gathered}\right\}\end{equation}

As seen from (3.6) and (3.8), the condition of vanishing $F_\varphi $ and $F_q$ determining the stationary points is

(3.9)\begin{equation}\left. \begin{gathered} q(R-LP)\sin \varphi = 0,\\ \alpha q(q-\cos \varphi )(R-LP) + \gamma P(R-Lq^2) = 0. \end{gathered}\right\}\end{equation}

When a stationary point is found, its type and hence the nonlinear stability of the corresponding equilibrium is defined by the sign of the Hessian determinant

(3.10)\begin{equation}G = F_{\varphi \varphi }F_{qq}-F_{\varphi q}F_{q\varphi }.\end{equation}

At positive (negative) G, a stationary point in the phase plane is a centre (saddle), and the corresponding equilibrium is stable (unstable).

Two cases follow from the first equation in (3.9): either $R-LP = 0$ or $\sin \varphi = 0$. In the first case, (3.9) take a simple form

(3.11)\begin{equation}\left. \begin{gathered} R-LP = 0,\\ R-Lq^2 = 0. \end{gathered} \right\}\end{equation}

At $L\ne 0$ the system has a single solution $\varphi = {{\rm \pi} }/3,\;q = 1$; that is an equilateral triangle which is the relative equilibrium existing at any set of strengths (see figure 2). For this solution P = 1 and R = L, coefficients A and C in (3.8) are equal to zero, and the Hessian determinant (3.10) is easily calculated, $G = 0.75(BD-E^2) = 12(\alpha \beta \gamma )^2/L$. Thus, equilateral triangles are unstable at L < 0 and are nonlinearly stable at L > 0 (in particular, equilateral triangles with zero total circulation are all unstable). The same result was obtained using trilinear coordinates by Synge (Reference Synge1949). Also, the nonlinear stability of the equilateral triangle with equal strengths of vortices was shown by Kizner (Reference Kizner2014). At $L = 0$ the equilateral triangle is stable according to Aref (Reference Aref2009). Some details of this case will be considered in § 5.

In the second case, when $\sin \varphi = 0$, two sets of the stationary points (at $\varphi = 0$ and $\varphi = {{\rm \pi} }$) satisfy (3.9), both corresponding to collinear vortex equilibria. Let

(3.12)\begin{equation}\sigma = \cos \varphi = \begin{cases} + 1, & {\varphi = 0},\\ {-1,}& {\varphi = {{\rm \pi} }}.\end{cases}\end{equation}

Then, with the exception of singular solutions, the cubic equation

(3.13)\begin{equation}(\alpha + \gamma )q^3-\sigma (2\alpha + \gamma )q^2-(2\beta + \gamma )q + \sigma (\beta + \gamma ) = 0,\end{equation}

can be derived for determination of the collinear stationary states. Positive real roots of (3.13) define equilibria at given strengths. A similar equation (obtained in a different way) already appeared in Tavantzis & Ting (Reference Tavantzis and Ting1988) for the case $\sigma = 1$ and in Aref (Reference Aref2009) for the case $\sigma = -1$ (see also Borisov & Lebedev Reference Borisov and Lebedev1998). Note that for collinear configurations the second term in the Hessian determinant (3.10) vanishes due to (3.7c).

In some special cases, relationship (3.13) can be simplified. For example, at $\alpha = \beta = \gamma $ one obtains three stationary saddle points (0, 0.5), (0, 2) and $({{\rm \pi} },1)$ in the phase diagram (see figure 2a). Direct calculation shows that the Hessian determinant is negative in these points. In fact, these three stationary points represent the same symmetric collinear configuration. Topologically, however, all three points are important since the separatrices divide the phase space into areas with different behaviours of phase trajectories.

When two vortices have equal (e.g. unit) strengths, there are two symmetric collinear equilibria in the points (0, 0.5) and $({{\rm \pi} },1)$, for which the central vortex can be of arbitrary strength $\lambda $. The Hessian determinant (3.10) for both stationary points is proportional to $-(4\lambda + 5)[\lambda /(\lambda + 2)]^2$. From here the necessary and sufficient condition $\lambda \lt -1.25$ for nonlinear stability is obtained. The same result was obtained earlier by Kizner (Reference Kizner2014), who used another nonlinear approach, and by Aref (Reference Aref2009) and Kurakin et al. (Reference Kurakin, Ostrovskaya and Sokolovskiy2016) within the linear stability analysis. Note that in the interval $-1.25 \lt \lambda \lt -0.5$ the equilateral triangle and one of the collinear configurations can both be unstable (see figure 2c).

Equation (3.13) can be written in an alternative form

(3.14)\begin{equation}R^{(\sigma )}(q)\left( {q-\sigma \displaystyle{\gamma \over {\Gamma -\beta }}} \right)-L\displaystyle{{\Gamma + \beta } \over {\Gamma -\beta }}\left( {q-\sigma \displaystyle{{\Gamma -\alpha } \over {\Gamma + \beta }}} \right) = 0,\end{equation}

where $R^{(\sigma )}(q) = \beta (\Gamma -\beta )q^2-2\alpha \beta \sigma q + \alpha (\Gamma -\alpha )$ is derived from (3.3) using (3.12). At $L = 0$ and $R^{(\sigma )}\ne 0$ from (3.14) follows the solution $q = -\sigma \beta /\alpha $ representing the known fixed (i.e. motionless) collinear equilibrium (see e.g. Newton Reference Newton2001). Direct substitution of this solution in (3.7) gives $F_{\varphi \varphi } = -2\gamma ^2 \lt 0$, $F_{qq} = 2(\alpha \gamma /\beta )^2 \gt 0$, therefore, all fixed collinear equilibria are unstable. This statement was earlier proved in the linear approximation by Aref (Reference Aref2009).

At zero total circulation, $\Gamma = 0$, the single solution following from (3.14) is the singular solution S1 (see figure 2b). For this solution, $R = 0$, therefore, it is necessary to use asymptotic analysis for estimation of the sign of the Hessian determinant. Since the coefficient at the term $L/R^2$ in the expression for $F_{qq}$ vanishes, the leading terms in $F_{\varphi \varphi }$ and $F_{qq}$ which define the sign of G are $2\alpha ^2L/R$ and $2\beta ^2L/R$, respectively. The product of these two infinite quantities is always positive, therefore, any collinear equilibrium with zero total circulation is nonlinearly stable. Earlier, Aref (Reference Aref2009) proved the linear stability of such configurations, and Kizner (Reference Kizner2011, Reference Kizner2014) proved their nonlinear stability at $\alpha = \beta $.

5.1. Complete scattering

Consider now a case where functions R and L both vanish. In this case, (3.9) turn into identities, thus all points of the singular curve S1 become stationary points where the geometrical form of three vortices remains unchanged with time. On the other hand, the solutions at $R = 0$ considered in § 4 cease being periodic in the limit $L\to 0$ and only expansion or convergence of vortices are possible for each point on S1 except for points corresponding to the relative equilibria. Such kind of motion is known as self-similar motion (Gröbli Reference Gröbli1877; Synge Reference Synge1949; Newton Reference Newton2001; Aref Reference Aref2010).

At $L = 0$, condition (3.5) for the existence of the singular curve S1 is automatically satisfied, and equation for determining the configurations that expand or collapse self-similarly takes the form

(5.1)\begin{equation}R = q^2-2(\mu + 1)q\cos \varphi + \mu = 0,\quad \mu = \beta /\alpha .\end{equation}

At different points of the line S1, function F now takes different finite values, and each isoline that crosses S1 is divided in the crossing point into parts with opposite directions of passing through (compare figures 3a and 5). The crossing points in which the two parts of an isoline converge (diverge) represent a vortex configuration that expands (collapses) self-similarly. The points on S1 which correspond to relative equilibria (dark solid circles in figure 5) divide S1 into segments within which only collapse or expansion occurs. The relative equilibria on S1 are the equilateral triangle in points $( \pm {{\rm \pi} }/3,1)$, and two collinear states at $\varphi = 0$ and/or $\varphi = {{\rm \pi} }$ with positive $q = \sigma (\mu + 1) \pm \sqrt {\mu ^2 + \mu + 1} $.

Figure 5.

Phase diagrams of relative motion of three vortices at $L = 0$ and (a) $\alpha = -0.5,\;\beta = \gamma = 1$; (b) $\alpha = -3,\;\beta = -1.5,\;\gamma = 1$. Bold white line, singular curve S1; grey solid circle, singular point S2. Dark solid circles, stationary points. The scattering domain is shaded by grey. Bold dark lines pass through stable relative equilibria; bold dashed lines are saddle point separatrices. Dotted lines show phase trajectories obtained in numerical simulations presented in figure 6; cross and hollow triangle mark the starting and the ending points.

At $L\to 0$ and $R\to 0$ one can derive from (3.7), (3.8) that $F_{\varphi \varphi }\to k_{\varphi \varphi }^2 L/R^2$, $F_{qq}\to k_{qq}^2 L/R^2$, $F_{\varphi q}\to k_{\varphi \varphi }k_{qq}L/R^2$, where $k_{\varphi \varphi } = 2\alpha \beta q\sin \varphi $ and $k_{qq} = -2\alpha (\gamma q + \beta \cos \varphi )$. Hence, within these limits, the Hessian determinant (3.10) vanishes. But, since in points on S1 that corresponds to self-similar expanding (or collapsing) states the phase trajectories converge (or diverge), these states are stable (or unstable). Relative equilibria are stable. The same results were earlier obtained using linear approximation by Tavantzis & Ting (Reference Tavantzis and Ting1988), Kimura (Reference Kimura1990) and Aref (Reference Aref2010). At $L = 0$, in the phase diagrams (figure 5) there are also a singular point S2 (presented by centre) and a saddle point $(0,-\beta /\alpha )$ or $({{\rm \pi} },\beta /\alpha )$; the latter corresponds to an unstable fixed collinear equilibrium (Aref Reference Aref2009; see also § 3.2).

Curve S1 and all phase trajectories that cross it lie completely in the domain (shaded by grey in figure 5) which is bounded by the isolines of F passing through the points representing the equilateral triangle and one of the collinear states. Starting the motion from any interior point (with $R\ne 0$) of this domain we will asymptotically approach some point Z on S1 (note that the phase diagram is symmetric about the straight line $\varphi = {{\rm \pi} }$). In the absolute motion, this appears as asymptotic scattering: all intervortex separations increase to infinity, but the changing geometrical form of the configuration tends to the triangle at the point Z. It is clear also that $R\to 0$ at $t\to \infty $. Thus, the necessary condition for asymptotic scattering is $L = 0$, and the sufficient condition is belonging of the initial state to the above-mentioned domain which can be called the scattering domain. The collapse is more delicate; it can occur for self-similar motion only when both R and L vanish.

It is known (e.g. Newton Reference Newton2001; Aref Reference Aref2010) that the trajectories of the vortices in a self-similar motion are similar logarithmic spirals, and the intervortex separations increase or decrease in proportion to $(1 + \tau t)^\kappa $ with $\kappa = 1/2$, where parameter $\tau $ depends on the initial vortex configuration. Our numerical simulations show that, in general, this is correct with good accuracy for the asymptotically self-similar scattering too.

Two examples of asymptotic scattering are presented in figure 6(a,b) for the combinations of strengths as in figure 5(a,b) respectively. In the first case, the initially collinear (but not stationary) state asymptotically transforms into an equilateral triangle (dotted curve in figure 5a). The initial point $(0,q_0)$ in the phase plane (the cross in figure 5a) is located at the lower boundary of the scattering domain, where $q_0\approx 0.618$ is the solution of the equation $F(0,q_0) = F({{\rm \pi} }/3,1) = 0$. In the second example, the initial point $(\varphi _0,1)$, where $\varphi _0\approx -1.402$ is the solution of the equation $F(\varphi _0,1) = F(0,q_s)$, is placed at the phase trajectory passing through the point $(0,q_s)$ which corresponds to the collinear equilibrium with $q_s = 0.5(3 + \sqrt 7 )$. In this case, where the initial state is an isosceles triangle, the configuration asymptotically transforms into a collinear state (dotted curve in figure 5b).

Figure 6.

Asymptotic scattering at $L = 0$ (but $R\ne 0$) for combinations of strengths and initial vortex configurations as in figures 5(a) and 5(b) respectively. Trajectories of the absolute motion of three vortices (dotted line, the vortex with the strength $\alpha $) are shown. Black solid circles and arrows show the initial and the final (at $t = 3000$) positions.

Some features of motion when both L and R are small but not equal to zero are discussed in Leoncini et al. (Reference Leoncini, Kuznetsov and Zaslavsky2000) for the case of two equal strengths.

5.2. Emission of a translating dipole

The necessary condition for emission of a dipole translating to infinity is vanishing of its total circulation. Under this condition L becomes negative, hence the stationary point $({{\rm \pi} }/3,1)$, which corresponds to equilateral triangle on the phase plane, is a saddle point. The stationary points corresponding to collinear equilibria are defined at $\alpha + \beta = 0$ by the equation $(q-\sigma )[(\eta + 1)q^2-\sigma \eta q + (\eta -1)] = 0$, where $\eta = \alpha /\gamma $. In addition to the singular solution S2, this equation describes two branches of collinear equilibria (at $\sigma = \pm 1$) which exist at $|\eta |\le 2/\sqrt 3 $ (see figure 7a). At $\sigma = -1$ (when $\varphi = {{\rm \pi} }$), the Hessian determinant G is always positive, hence, the corresponding equilibria are stable. At $\sigma = 1$ (when $\varphi = 0$) G takes negative values in the interval $2-\sqrt 3 \lt q \lt 2 + \sqrt 3 $. In the points where $\eta = \pm 2/\sqrt 3 $, $F_{qq} = 0$, and the plot of the invariant function F calculated over stationary points changes its behaviour here (figure 7b). Since $q(-\eta ) = 1/q(\eta )$ and $F(-\eta ) = F(\eta )$ the plots in figure 7 are presented for positive $\eta $ only.

Figure 7.

(a) Collinear relative equilibria at $\alpha + \beta = 0$ on the $(\eta = \alpha /\gamma ,q)$ plane; solid (dotted) lines represent solutions at $\varphi = 0$ ($\varphi = {{\rm \pi} }$). Vertical dashed lines A, B and C pass through $\eta = 0.75$, 1 and 1.1, respectively. (b) Invariant function F calculated over collinear equilibria with $\varphi = 0$. Solutions on bold solid lines are unstable; blank circle is the point in which stability changes to instability.

In the case of emission of a dipole the singular curve S1 always exists, and at $\alpha + \beta = 0$ its form is described by the equation $R = (\eta + 1)q^2-2\eta q\cos \varphi + (\eta -1) = 0$, which also has the singular solution of S2 type in the point (0, 1). In the special case, $\eta = 1$ or $\eta = -1$, when there exist two pairs of vortices with zero group circulation, the equation $R = 0$ reduces to equalities $q = \cos \varphi $ or $q\cos \varphi = 1$, which describe a set of rectangular triangles. In these cases curve S1 either goes to infinity, asymptotically approaching the straight line $\varphi = \pm {{\rm \pi} }/2$, or it tends to approach the point $( \pm {{\rm \pi} }/2,0)$. The points at $q = \infty $ or $q = 0$ are the singular points of S2 type by its definition in § 3.1 Thus, in all cases, S1 asymptotically approaches the singular point S2, in which the dynamics of three vortices actually turns into the dynamics of two vortices. In terms of absolute motion, this directly implies the emission of a dipole. In the phase plane not only singular curve S1 is attracted by the point S2; there also exists a manifold of trajectories that are attracted by S2 and are entirely located inside some domain, which will be termed the emission domain. Outside this domain, the phase trajectories are closed or periodic.

Examples of phase diagrams are presented in figure 8. At $|\eta |\lt 1$ (figure 8a) the boundaries of the emission domain are $2{{\rm \pi} }$-periodic. At $1 \lt |\eta |\lt 2/\sqrt 3 $, the finite emission domain is multiply connected (figure 8c) because of the existence of two branches in the family of collinear equilibria at $\varphi = 0$ (see figure 7a). At $|\eta |\gt 2/\sqrt 3 $ there is only one equilibrium, the unstable equilateral triangle, and the emission domain is bounded by a closed branch of the separatrix described by the equation $F(\varphi ,q) = \alpha ^2\ln \alpha ^2$. The aforementioned special case $\eta = 1$ (i.e. $\alpha = -\beta = \gamma $) is presented in figure 8b. Note that if the starting point is set on S1, the absolute motion of the individual vortices is executed along parallel straight lines while the vortex triangle remains rectangular. An example of such an evolution (but only its initial stage without mentioning of emission) was considered by Gröbli (Reference Gröbli1877); see also Aref et al. (Reference Aref, Rott and Thomann1992).

Figure 8.

Phase diagrams of relative motion of three vortices at $\alpha + \beta = 0$, $\gamma = 1$. (a) $\alpha = 0.75$ (line A in figure 7a); (b) $\alpha = 1$ (line B); and (c) $\alpha = 1.1$ (line C). Shaded is the emission domain. Bold dark (dashed) lines pass through the saddle points corresponding to the equilateral triangle (collinear equilibrium). Other notations as in figures 2 and 5.

In a dipole, which translates to infinity, the intervortical distance tends to $d = a_0q_0^{1/\eta } \sqrt {q_0^2 -2q_0\cos \varphi _0 + 1} $. This can be derived by equating the Hamiltonians of the limiting dipole and the initial configuration of three vortices.

6.1. Relative motion and stability of five vortices

Consider now a centrally symmetric configuration in which four vortices are located in the vertices of a parallelogram, and the fifth is located in the geometrical centre. Vortices at the opposite ends of diagonals are assumed to have the same strengths (see figure 9). In general, the invariant function F depends on six phase variables (see § 2.1). However, often, in numerical simulations and laboratory experiments, unstable vortex ensembles, initially possessing central symmetry, practically retain it in the course of their evolution, even when no special restrictions are imposed (e.g. Trieling, van Heijst & Kizner Reference Trieling, van Heijst and Kizner2010). The assumption of keeping the central symmetry makes it possible to reduce the dimension of the phase space to two. Kizner (Reference Kizner2011, Reference Kizner2014) proposed such a consideration for the case where four or three peripheral vortices have equal strength. In our case, one can derive the expression for the invariant function F which depends on two variables only. Using the designations $q = b/a,\;p = c/a,\;h = d/a$ (by analogy with the case N = 3) we obtain

(6.1)\begin{equation}F(\varphi ,q) = -L\log |R(q)| + \beta (\beta + 2\gamma )\log q^2 + 2\alpha \beta \log P(\varphi ,q),\end{equation}

where now

(6.2)\begin{gather}P(\varphi ,q)\equiv p^2h^2 = q^4-2q^2\cos 2\varphi + 1,\end{gather}

(6.3)\begin{gather}R(q) = \alpha + \beta q^2,\end{gather}

(6.4)\begin{gather}L = 2(\alpha \gamma + \beta \gamma + \alpha \beta ) + (\alpha + \beta )^2.\end{gather}

In expression (6.1) the term $C_F = -L\log |\Gamma | + [2(\alpha ^2 + \beta ^2)-L]\log 2$ with $\Gamma = \gamma + 2(\alpha + \beta )$, is omitted as a constant. Due to the symmetry, M is presented by the angular momentum only, and the function R is independent of $\varphi $. The invariant function F is ${{\rm \pi} }$-periodic and symmetric with respect to $\varphi = {{\rm \pi} }/2$.

Figure 9.

Scheme of the relative arrangement of five vortices with central symmetry; a, b, c and d are dimensional, 1 and q, normalized sizes.

On the phase plane $(\varphi ,q)$ there are singular points with coordinates $(0,1)$, $(\varphi ,0)$ or $(\varphi ,\infty )$. These points (of S3 type) represent the limiting symmetric tripole resulting either from pairwise ‘merger’ (collapse) of the peripheral vortices (of the original configuration), which creates the two satellites of the tripole, or from the ‘merger’ of two peripheral vortices and the central vortex (of the original configuration), which creates the central vortex of the tripole. The singular curve S1 (where $R = 0$) has the same sense, as in the case of three vortices, representing either limiting transition to five independent vortices, or collapse to one vortex, both at $L = 0$. According to (6.3), S1 is a straight line $q = \sqrt {-\alpha /\beta } $ and exists when $\alpha $ and $\beta $ have opposite signs. At $L\ne 0$ line S1, as for three vortices, represents a locus of initial vortex configurations which are periodic in absolute motion (when the central symmetry is kept).

Since $F_\varphi = 8\alpha \beta q^2\sin 2\varphi /P$, equilibria with central symmetry can be only collinear configurations, at $\varphi = 0$, or rhombuses with a central vortex, at $\varphi = {{\rm \pi} }/2$. Designating

(6.5)\begin{equation}\sigma = \cos 2\varphi = \begin{cases} { + 1,} & {\varphi = 0}, \\ {-1,} & {\varphi = {{\rm \pi} }/2},\end{cases}\end{equation}

from the equation $F_q = 0$ we derive the relation for both families of stationary points

(6.6)\begin{equation}R[(\alpha + 2\gamma )q^2-\sigma (2\Gamma -3\alpha )]-L(q^2-\sigma ) = 0.\end{equation}

Since $F_{\varphi q} = F_{q\varphi } = 0$ at the stationary points, a function

(6.7)\begin{equation}g = -4R^2-\sigma L(q^2-\sigma )^2, \end{equation}

which completely defines the sign of the Hessian determinant, can be derived. From consideration of (6.7) it is seen that rhombuses at $L \le 0$ and collinear states at $L \ge 0$ are unstable. In particular, at $\alpha = \beta $ for a square (when $\varphi = {{\rm \pi} }/2,\;q = 1$) direct substitution to (6.7) gives the instability condition $\alpha (\alpha + 2\gamma ) \lt 0$ obtained earlier (in a different way) by Kizner (Reference Kizner2011, Reference Kizner2014). In case of zero total circulation (when L is always negative) all rhombuses are unstable, but collinear equilibria can be stable relative to perturbations which obey the central symmetry; from here, however, stability to unrestricted perturbations does not follow.

6.2. Symmetrical scattering of five vortices

Consider first the situation when two dipoles formed by peripheral vortices move away from the motionless central vortex in opposite directions. A necessary condition for such a development is $\alpha + \beta = 0$. Because this condition entails $L = -2\alpha ^2$ and $R = \alpha (1-q^2)$, all equilibria (rhombuses at $q^2 = (3\eta \pm 2\sqrt {2\eta ^2 + 1} )/(2 + \eta )$ and collinear states at $q^2 = (2-\eta )/(2 + \eta )$ where $\eta = \alpha /\gamma $) are unstable due to (6.7), i.e. all stationary points are saddles. The singular point S3 at $\beta = -\alpha $ formally represents an unstable collinear equilibrium and lies on the singular line S1. Topologically (see figure 10a) this point attracts all phase trajectories entirely located in the emission domain (shaded in the plot), bounded by separatrices passing through stationary saddle points. Thus, the dipole emission is only possible for the vortex configurations represented by the points lying inside this domain.

Figure 10.

Symmetrical scattering of five vortices. Phase diagrams for (a) emission of two dipoles at $\alpha = 1,\;\beta = -\alpha ,\;\gamma = 1$; and (b) complete scattering at $L = 0$ and $\alpha = 1,\;\beta = -3 + \sqrt 6 ,\;\gamma = 1$. Notations as in figures 2, 5 and 8.

For the complete scattering (where $L = 0$), from (6.7) it follows that all equilibria described by relation (6.6) are unstable, except for the case $R = 0$. When both L and R are zero, the singular line S1 is a set of isolated points for which self-similar motion with keeping the position of the central vortex takes place. As for the case of three vortices, on the phase plane (figure 10b) there exists the scattering domain where, for any interior point, asymptotic scattering (without keeping form) occurs.

6.3. Relative motion of four vortices

This case follows from the previous one under the assumption that $\gamma = 0$ in formulae (6.1)–(6.7). Two families of equilibria (collinear states at $\sigma = 1$ and rhombuses at $\sigma = -1$) are defined by the relation $q^4-[(4 + \sigma ) + (1 + 4\sigma )\mu ]q^2 + \sigma \mu = 0$ where $\mu = \beta /\alpha $. From (6.7) it follows that the equilibria of both types are unstable at $q^2 \lt q_1^2 = 2-\sqrt 3 $ and $q^2 \gt q_2^2 = 2 + \sqrt 3 $, and are stable (with respect to symmetrical perturbations) in the interval $(q_1^2 ,q_2^2 )$. Similar results were reported by Roberts (Reference Roberts2013) who constructed and investigated for stability stationary solutions for rhombuses, thought in a different way. Collinear equilibria of four vortices were considered by Hampton et al. (Reference Hampton, Roberts and Santoprete2014). As follows from our general approach with the use of four phase variables in this case (not presented here), rhombuses in the interval $(q_1^2 ,q_2^2 )$ remains stable also with respect to general perturbations, but collinear equilibria do not.

The change from stability to instability (or vice versa) occurs at $\mu = \mu _{1,2} = -2\mp \sqrt 3 $ (when $L = 0$) in the stationary points $(0,q_{1,2})$ and $({{\rm \pi} }/2,q_{1,2})$. In these points $\mu q^2 + 1 = 0$ (i.e. $R = 0$), thus they lie on the line S1. All other points from S1 represent the configurations that undergo self-similar scattering/collapse of four vortices. The same criterion of self-similarity ($q^2 = -1/\mu = 2 \pm \sqrt 3 $) was derived earlier by Novikov & Sedov (Reference Novikov and Sedov1979). The domain on the phase plane, in which all four-vortex configurations undergo asymptotic scattering can be found by analogy with the case of N = 5.

For four centrally symmetric vortices, the necessary condition of dipole emission is zero total circulation. Since the linear momentum is also equal to zero, the system becomes integrable and can be reduced to the relative motion of three vortices (Eckhardt Reference Eckhardt1988; Rott Reference Rott1990; Aref & Stremler Reference Aref and Stremler1999). The initial symmetry is kept in this case. At $\Gamma = 0$ there are only unstable rhombus equilibria with $q^2 = 3 \pm 2\sqrt 2 $; separatrices passing through corresponding saddle points bound the emission domain on the phase plane where any initial vortex configuration eventually breaks up into a pair of dipoles moving in opposite directions.

The invariant function depending only on two variables can be constructed also under the condition that, for all times, the configuration can be obtained by mirror symmetry reflection of a pair of vortices relative to a straight line, the physical interpretation of this case being the motion of two vortices near an impenetrable solid wall. Clearly, the four vortices form an isosceles trapezoid in this case. Relative equilibria of two pairs of vortices with equal strengths considered by Roberts (Reference Roberts2013) and Hampton et al. (Reference Hampton, Roberts and Santoprete2014) also have this shape.