2.1. Point vortices

Point vortices have a vorticity distribution of the form (Newton Reference Newton2001)

(2.5)\begin{equation} \zeta ={-}\nabla^2\psi = \sum_{j=1}^M\varGamma_j\,\delta(z-z_j), \end{equation}

where $z_j=x_j+\mathrm {i}y_j$ are the time-dependent locations of the point vortices and the constants $\varGamma _j$ are the circulations or strengths of the point vortices. The corresponding stream function is

(2.6)\begin{equation} \psi(z,\bar z) ={-}\frac{1}{2{\rm \pi}}\sum_{j=1}^M\varGamma_j\log|z-z_j|, \end{equation}

which is the imaginary part of the complex potential

(2.7)\begin{equation} f(z) = \frac{1}{2{\rm \pi}\mathrm{i}}\sum_{j=1}^M\varGamma_j\log(z-z_j). \end{equation}

The velocity field arising from the point vortices is the derivative of the complex potential

(2.8)\begin{equation} u-\mathrm{i}v = f'(z) = \frac{1}{2{\rm \pi}\mathrm{i}}\sum_{j=1}^M\frac{\varGamma_j}{z-z_j}. \end{equation}

The complex potential (2.7) and the complex velocity (2.8) are complex-analytic functions of $z$. The velocity of a point vortex is obtained from (2.8) after subtracting off the singular term and evaluating at the point vortex location

(2.9)\begin{equation} \overline{\frac{\mathrm{d}z_k}{\mathrm{d}t}} = \left. \left[(u-\mathrm{i}v)-\frac{\varGamma_k}{2{\rm \pi}\mathrm{i}}\frac{1}{z-z_k}\right]\right|_{z=z_k} = \frac{1}{2{\rm \pi}\mathrm{i}}\sum_{\genfrac{}{}{0pt}{}{j=1}{j\neq k}}^M\frac{\varGamma_j}{z_k-z_j}, \end{equation}

for $k=1,2,\ldots ,M$ (Saffman Reference Saffman1992; Newton Reference Newton2001; Llewellyn Smith Reference Llewellyn Smith2011).

In this paper, we study stationary configurations of point vortices in which all vortex velocities are zero. Then (2.9) reduces to the $M$ algebraic conditions on the $M$ unknown vortex positions $z_1,\ldots ,z_M$ as

(2.10)\begin{equation} \sum_{\substack{j=1\\ j\neq k}}^M\frac{\varGamma_j}{z_k-z_j}=0\quad\text{for } k=1,2,\ldots,M, \end{equation}

for given values of the circulations $\varGamma _1,\ldots ,\varGamma _M$. For further discussion of point vortex equilibria, see Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020) and the review article Aref et al. (Reference Aref, Newton, Stremler, Tokieda and Vainchtein2003).

2.2. Stuart vortices

Stuart (Reference Stuart1967) considered steady solutions by choosing $V(\psi ) = \exp (-2\psi )$ in (2.4). In this case, we get the Liouville equation (1.1) with $a=1$ and $b=-2$, the general solution of which is given by (1.2). Then, in our present notation, Stuart's original solution is obtained by substituting

(2.11)\begin{equation} h(z) = A\tan\left(\frac{z}{2}\right) \quad\implies\quad h'(z) = \frac{A}{2}\sec^2\left(\frac{z}{2}\right) \end{equation}

in (1.2), where $A$ is a real constant. Subtracting an unimportant constant leads to the stream function

(2.12)\begin{equation} \psi(z,\bar{z};A) = \log\left[\frac{1}{A}\left|\cos^2\left(\frac{z}{2}\right)\right| + A\left|\sin^2\left(\frac{z}{2}\right)\right|\right], \end{equation}

which is smooth for all finite values of $A$ owing to the fact that $h'(z)$ does not vanish anywhere in the complex plane.

By taking appropriate limits as $A\to 0,\infty$ in (2.12), we find

(2.13a)\begin{gather} \lim_{A\to 0}\left[\psi+\log A\right] = 2\log\left|\cos\left(\frac{z}{2}\right)\right|, \end{gather}

(2.13b)\begin{gather}\lim_{A\to\infty}\left[\psi-\log A\right]= 2\log\left|\sin\left(\frac{z}{2}\right)\right|. \end{gather}

These limiting stream functions are the imaginary parts of the complex potentials

(2.14a,b)\begin{equation} G(z) = 2\mathrm{i}\log\cos\left(\frac{z}{2}\right) \quad\text{and}\quad F(z) = 2\mathrm{i}\log\sin\left(\frac{z}{2}\right). \end{equation}

Both the complex potentials $G(z)$ and $F(z)$ correspond to an infinite row of point vortices with circulations $-4{\rm \pi}$ each and consecutive vortices separated by a distance $2{\rm \pi}$ (Saffman Reference Saffman1992). Figure 2 shows streamline plots for the Stuart vortices and their limiting cases.

Figure 2. Streamlines and vorticity for Stuart vortices (Stuart Reference Stuart1967) given by the stream function (2.12). Panels with a white background show streamline patterns for point vortices with negative ($-$, red) circulation in an otherwise irrotational flow. Panels (b–c) show the everywhere rotational and smooth flow for finite $A\neq 0$. In the limiting cases $A=0,\infty$, the smooth vorticity concentrates into a periodic row of point vortices with complex potentials (2.14a,b), surrounded by irrotational flow.

2.3. Polygonal $N$-vortex equilibria

Stuart (Reference Stuart1967) made the choice (2.11) because he was interested in obtaining everywhere smooth solutions, i.e. without point vortex singularities. Crowdy (Reference Crowdy2003) instead considered the function (rewritten in our notation)

(2.15)\begin{equation} h(z) = A(z^N+C) \quad\implies\quad h'(z) = ANz^{N-1}, \end{equation}

for integers $N\geq 2$, and where $A$ is a real constant and $C$ is a complex constant. Substituting (2.15) into (1.2), again with $a=1$ and $b=-2$, we get the stream function

(2.16)\begin{equation} \psi(z,\bar{z};A,C) = \log\left[\frac{1}{A}\frac{1}{|z|^{N-1}}+A\frac{|z^N+C|^2}{|z|^{N-1}}\right], \end{equation}

where we have dropped an unimportant constant $\log N$. The hybrid stream function (2.16) shows a point vortex singularity at $z_1=0$ with strength $\varGamma _1=2{\rm \pi} (N-1)/2$. The point vortex at the origin is surrounded by $N$ smooth vortices arranged on a regular polygon, as shown in figure 3 for the case $N=4$. The symmetry of the solution guarantees that the point vortex at the origin in the hybrid solution (2.16) remains stationary according to (1.4).

Figure 3. Panels with a white background show streamline patterns for point vortices with positive ($+$, blue) and negative ($-$, red) circulation in an otherwise irrotational flow. Panels (b) and (c) show point vortices embedded in a sea of smooth Liouville-type background vorticity, which is negative and shaded in red. Streamlines and vorticity are shown for the $N$-polygonal equilibria (Crowdy Reference Crowdy2003) given by the stream function (2.16). The limit $A\to 0$ is an isolated point vortex, whereas the limit $A\to \infty$ is a centred polygon of stationary point vortices. Both of these limiting equilibria contain point vortices in an otherwise irrotational flow.

Just as was done for (2.12), it is useful to look at the two limits ${A\to 0}$ and ${A\to \infty }$ of (2.16). We find

(2.17a)\begin{gather} \lim_{A\to 0}[\psi+\log A] ={-}\log\left|z\right|^{N-1}, \end{gather}

(2.17b)\begin{gather}\lim_{A\to\infty}[\psi-\log A] ={-}\log\frac{|z|^{N-1}}{|z^N+C|^2}. \end{gather}

We emphasize that while the stream function (2.16) is not harmonic for $0\lt A \lt \infty$, both the limits in (2.17) are harmonic except at a finite set of point vortices. Indeed, the limits are the imaginary parts of the respective complex potentials

(2.18a,b)\begin{equation} G(z) = \frac{\varGamma_1}{2{\rm \pi}\mathrm{i}}\log z \quad\text{and}\quad F(z;C) = \frac{\varGamma_1}{2{\rm \pi}\mathrm{i}}\log{z}+2\mathrm{i}\,\log(z^N+C). \end{equation}

In the $A\to 0$ limit we thus recover a single pure point vortex flow (strength $\varGamma _1$), but in the $A\to \infty$ limit we recover the centred-polygon pure point vortex equilibria studied by Morikawa & Swenson (Reference Morikawa and Swenson1971). The stationary polygon consists of a central point vortex with strength $\varGamma _1$ surrounded by $N$ satellite point vortices with strengths $-4{\rm \pi}$. It is interesting that, in contrast to the identical point vortex limits (up to a translation) of the Stuart vortex solutions (2.12) as ${A\to 0,\infty }$, the polygonal solutions (2.16) have distinct pure point vortex limits.

4.1. Input seed and the first Liouville link

We begin by making a simple choice for the input equilibrium function (3.4),

(4.1)\begin{equation} g'(z) = z \quad\implies\quad g(z) = \frac{z^2}{2}, \end{equation}

which corresponds to a single point vortex ($M=1$) at $z_1=0$ with circulation $\varGamma _1=1/2$. This formula for $g'(z)$ is arrived at by considering the complex potential $f(z)$ given by (2.7), for the single point vortex above, and taking $g'(z)=\exp (4{\rm \pi} \mathrm {i} f(z))$. Substituting (4.1) into (3.2), we obtain the hybrid stream function

(4.2)\begin{equation} \psi(z,\bar{z};A,C) ={-}\frac{1}{4{\rm \pi}}\log\left[\frac{4A|z|}{4+A^2|z^2+2C|^2}\right], \end{equation}

where the hybrid parameter $A$ varies in the range $0\lt A\lt \infty$. We have not added in an explicit integration constant $C$ to $g(z)$ in (4.1), this is added separately in (4.2). It is clear that the only singularity in (4.2) is a stationary point vortex of strength $+1/2$ at $z=0$, which is seen to be stationary owing to a two-fold rotational symmetry of the solution about the origin. For all other values of $z$, (4.2) solves (1.1). The stream function and the velocity field are both smooth elsewhere, i.e. the point vortex at $z=0$ is surrounded by everywhere smooth and non-zero Liouville-type vorticity. The hybrid stream function (4.2) is therefore a solution of (1.3) with $\tilde {M}=1$, $\tilde {z}_1=0$ and $\tilde {\varGamma }_1=1/2$.

Rewriting (4.2) as

(4.3)\begin{equation} \psi(z,\bar{z};A,C) ={+}\frac{1}{4{\rm \pi}}\log\left[\frac{1}{Az}+A\frac{|z^2+2C|^2}{4z}\right], \end{equation}

and taking careful limits as $A\to 0,\infty$ leads us to the stream functions

(4.4a)\begin{align} \lim_{A\to 0}\left[\psi+\frac{1}{4{\rm \pi}}\log A\right] & ={-}\frac{1}{4{\rm \pi}}\log\left|z\right|, \end{align}

(4.4b)\begin{align} \lim_{A\to\infty}\left[\psi-\frac{1}{4{\rm \pi}}\log A\right] & ={-}\frac{1}{4{\rm \pi}}\log\frac{4|z|}{|z^2+2C|^2}. \end{align}

These stream functions are clearly the imaginary parts of the complex potentials

(4.5a,b)\begin{equation} G(z) = \frac{1}{4{\rm \pi}\mathrm{i}}\log z \quad\text{and}\quad F(z;C) = \frac{1}{4{\rm \pi}\mathrm{i}}\log\frac{4z}{(z^2+2C)^2}. \end{equation}

The complex potential $G(z)$ corresponds to a single point vortex at $z_1$ with strength $\varGamma _1$; we have thus recovered the input point vortex equilibrium in the limit $A\to 0$. The other limit $A\to \infty$ yields $F(z;C)$, which corresponds to a colinear three point vortex equilibrium comprising a central point vortex at the origin of circulation $1/2$ and two satellite vortices at $\pm \sqrt {-2C}$ of circulation $-1$ each. It is a simple matter to check that these three point vortices satisfy (2.10) and are therefore in stationary equilibrium. Note that (4.1) corresponds to the choice for $g(z)$ made by Crowdy (Reference Crowdy2003) with $N=2$. The family of hybrid equilibria and its point vortex limits are shown as the first Liouville link in figure 4.

Figure 4. An example of the schematic Liouville chain shown in figure 1. The mathematical details for this example are provided in § 4. We start with a simple input seed equilibrium, an isolated point vortex in an otherwise irrotational flow, with the corresponding $g_0'(z)$ defined by (4.1). For finite $A_0\neq 0$, the stream function (4.2) is a solution of (1.3) and we refer to this set of hybrid solutions as a Liouville link. In the limiting case $A_0=0$ we recover the input seed equilibrium, but for $A_0=\infty$ we obtain a new pure point vortex equilibrium. After scaling the circulations of the point vortices in this $A_0=\infty$ equilibrium (we call this a twist operation), we can obtain a new input equilibrium function $g_1'(z)$, which allows us to create a second link in the Liouville chain. The stream function for this second link is given by (4.11). We can keep adding links to the chain indefinitely, which creates a new equilibrium solution with a larger number of vortices at each stage. Every pair of point vortex limits is connected by the transformation (6.7) discussed in Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020). Here $C_0=-1/2$, $C_1=0$ and $C_2=6$.

4.2. The necessity of the twist parameter

A natural question now arises. The above construction started with a known point vortex equilibrium and produced another one. Can the same process be re-initiated with a new input equilibrium function in (3.2), which corresponds to the new point vortex equilibrium $F(z;C)$ given by (4.5b)? Additionally, will an analogous $A \to \infty$ limit of this hybrid stream function produce yet another distinct point vortex equilibrium?

Let us relabel the input equilibrium function in (4.1) as $g_0'$, whose primitive is now called $g_0$. The corresponding family of hybrid equilibria (4.2) gets relabelled as $\psi _0$ with hybrid parameter $A_0$ and integration constant $C_0$. Finally, we introduce the notation $G_0$ and $F_0$ for the complex potentials (4.5a,b).

Instead of the input equilibrium function (4.1), we now reinterpret the equilibrium $F_0$ as an input, i.e. we choose

(4.6)\begin{equation} g_1'(z) = \frac{z}{(z^2/2 +C_0)^2} \quad\implies\quad g_1(z) ={-} \frac{1 }{(z^2/2 +C_0)}, \end{equation}

which is obtained from (4.5a,b) via the formula $g_1' = \exp (4{\rm \pi} \mathrm {i}F_0)$. The hybrid stream function follows from substituting (4.6) into (3.2). After some algebra, we get

(4.7)\begin{equation} \psi_1(z,\bar{z};A_1,C_0,C_1) ={-}\frac{1}{4{\rm \pi}}\log\left[ \frac{4A_1|z|}{|z^2+2C_0|^2+A_1^2|C_1|^2|z^2+2(C_0-1/C_1)|^2}\right], \end{equation}

where we have called the new integration constant $C_1$ and the hybrid parameter $A_1$. Note that (4.7) depends on both integration constants $C_0$ and $C_1$. Clearly, the only singularity in (4.7) is a point vortex at $z_1=0$ with strength $\varGamma _1=1/2$, and it is stationary owing to symmetry. The hybrid stream function (4.7) is therefore a solution of (1.3) with $\tilde {M}=1$, $\tilde {z}_1=0$ and $\tilde {\varGamma }_1=1/2$.

Taking limits as $A_1\to 0,\infty$ of (4.7), in a similar manner as (4.4), we obtain the complex potentials

(4.8a)\begin{align} \tilde{G}_1(z;C_0) &= \frac{1}{4{\rm \pi}\mathrm{i}}\log\left[\frac{4z}{(z^2 +2C_0)^2}\right], \end{align}

(4.8b)\begin{align} \tilde{F}_1(z;C_0,C_1) &= \frac{1}{4{\rm \pi}\mathrm{i}}\log\left[\frac{1 }{C_1^2}\frac{4z } {(z^2+2(C_0-1/C_1))^2}\right]. \end{align}

It is seen from (4.5b) and (4.8a) that $\tilde{G}_1=F_0$. Also, $\tilde{F}_1$ is just a rescaling of $\tilde{G}_1$ in (4.8). The point vortex limits of the hybrid stream function (4.7) are therefore not distinct. Unfortunately, the choice (4.6) gives nothing new; no new point vortex equilibrium is produced to continue the iteration.

4.3. Second Liouville link: from three to ten point vortices

In spite of this apparent setback, progress can still be made, and this is where the idea of a twist, using a twist parameter $\alpha$, comes in. A trivial but crucial observation is: if $f(z)$ is the complex potential for a stationary point vortex equilibrium then so too is $\alpha f(z)$, for any real $\alpha$. Suppose we take $\alpha _0 = -2$ and rescale $F_0$ in (4.5b), i.e. define

(4.9)\begin{equation} G_1(z;C_0) = \alpha_0 F_0(z;C_0) = \frac{1}{4{\rm \pi}\mathrm{i}}\log\frac{(z^2+2C_0)^4}{z^2}, \end{equation}

where, for convenience, we have dropped a constant $-(\log 4)/4{\rm \pi} \mathrm {i}$ from the complex potential. The complex potential $G_1$ is the previous $F_0$ but now multiplied (we say ‘twisted’) by $\alpha _0 =-2$. All we have done with this twist parameter $\alpha _0$ is to rescale the point vortex circulations without changing the state of hydrodynamic equilibrium.

We can now reinitiate the construction, not with the input equilibrium function (4.6), but with the choice of $g_1'=\exp (4{\rm \pi} \mathrm {i} G_1)$:

(4.10a)\begin{align} g_1'(z) &= \frac{(z^2 + 2C_0)^4 }{z^2}, \end{align}

(4.10b)\begin{align} \implies\quad g_1(z) &= \frac{z^7 }{7} + \frac{8}{5}C_0 z^5 + 8 C_0^2 z^3+ 32 C_0^3 {z} - \frac{16 C_0^4 }{z}. \end{align}

Dropping the constant in (4.9) has meant that both the numerator and denominator polynomials in $g_1'(z)$ are monic. The hybrid stream function (3.2) takes the form

(4.11)\begin{equation} \psi_1(z,\bar{z};A_1,C_0,C_1) ={-}\frac{1}{4{\rm \pi}}\log\left[\frac{A_1|z^2+2C_0|^4} {|z|^2+A_1^2|z(g_1(z)+C_1)|^2}\right]. \end{equation}

For $A_1>0$, (4.11) is now a solution of (1.3) in the case of $\tilde {M}=2$ with $\tilde {z}_1= - \tilde {z}_2 = \sqrt {-2C_0}$ and $\tilde {\varGamma }_1=\tilde {\varGamma }_2=+2$. There are now two point vortices embedded in the smooth background sea of Liouville-type vorticity. It is necessary that these point vortices are stationary according to (1.4) to obtain steady solutions. This was directly shown to be true for the stream function (4.11) in Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2019) with $C_0=-1/2$ (of course, this also follows from the general theory in § 5).

Turning now to the limits $A_1\to 0,\infty$ of (4.11), we obtain the complex potentials

(4.12a)\begin{gather} G_1(z;C_0) = \frac{1}{4{\rm \pi}\mathrm{i}}\log\frac{(z^2+2C_0)^4}{z^2}, \end{gather}

(4.12b)\begin{gather}F_1(z;C_0,C_1)= \frac{1}{4{\rm \pi}\mathrm{i}}\log\frac{(z^2+2C_0)^4}{(z(g_1(z)+C_1))^2}. \end{gather}

We have obtained the input equilibrium $G_1$ in the $A_1\to 0$ limit, but $F_1$, obtained in the $A_1\to \infty$ limit, is a new pure point vortex equilibrium. Comparing (4.8) and (4.12), we see that the twist operation has resulted in a new point vortex equilibrium given by $F_1$ in (4.12b). This emergent equilibrium is found to comprise two point vortices located at $\pm \sqrt {-2C_0}$ and of circulations $+2$ each, along with eight point vortices located at the roots of the degree-eight polynomial $z(g_1(z)+C_1)$ and of circulations $-1$ each. With $C_0=-1/2$, the functions in (4.10) are essentially those given in (3.8b) and (3.9b) of Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2019), who explore in detail this set of hybrid equilibria for $0\lt A_1\lt \infty$, which includes the highly non-trivial point vortex equilibrium that emerges in the $A_1 \to \infty$ limit. This family of hybrid equilibria and its point vortex limits are shown as the second Liouville link in figure 4.

From this explicit example, it should be clear how a function $g'(z)$ associated with a known point vortex equilibrium gives, on substitution into (3.2), a family of hybrid equilibria for any $0 < A < \infty$, called a Liouville link. These hybrid equilibria are bracketed by two point vortex equilibria corresponding to $A=0$ and $A=\infty$. After a suitable twist operation, a second Liouville link can be added to the first, and the procedure can be iterated to produce a Liouville chain. Figure 4 shows the third Liouville link in this chain. We refer the reader back to figure 1 where this process is depicted schematically. The example discussed in this section can be continued indefinitely to form an infinite Liouville chain, see § 9.2.

Not all Liouville chains can be continued indefinitely. We give examples of single-link Liouville chains in § 7, Liouville chains of finite length in § 8 and two further examples of infinite Liouville chains in § 9.

5.1. Proof that $h(z)$ is rational

We begin by showing that restricting the point vortex strengths according to (3.3) leads to a function $h(z)$ that is free of logarithms and hence to a single-valued velocity field (5.2). This proof can also be found in the context of the transformation described in Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020), but we include it here for completeness. This condition is equivalent to $h'(z)$ having zero residue at each of its poles, which are clearly at point vortex locations $z_k$ with negative circulations $\varGamma _k = -1$. Near any such $z_k$, we rewrite

(5.6)\begin{equation} h'(z) = A\frac{H_k(z)}{(z-z_k)^{2}}, \end{equation}

where

(5.7)\begin{equation} H_k(z) = \prod_{\substack{j=1\\ j\neq k}}^M (z-z_j)^{2\varGamma_j} \quad\text{for }k=1,2,\ldots,M. \end{equation}

Because the vortex positions are non-overlapping, $H_k(z_k)$ is finite and non-zero. The series representation for $h'(z)$ near $z_k$ is

(5.8)\begin{equation} h'(z) = A \left(\frac{H_k(z_k)}{(z-z_k)^{2}} + \frac{H_k^{\,\prime}(z_k)}{(z-z_k)} + \frac{H_k''(z_k)}{2} + \cdots\right). \end{equation}

Hence $h'(z)$ will have zero residue at $z_k$ if, and only if, the coefficient $H_k^{\,\prime }(z_k)$ vanishes. Combining (5.7) and (2.10) yields

(5.9)\begin{equation} \frac{H_k^{\,\prime}(z_k)}{H_k(z_k)} = \left.\left(\log H_k(z)\right)'\right|_{z=z_k} = 2\sum_{\substack{j=1\\ j\neq k}}^M \frac{\varGamma_j}{z_k-z_j}=0, \end{equation}

and hence $H_k^{\,\prime }(z_k)=0$ as desired. Similar arguments show that allowing for $\varGamma _k = -1/2$ in (3.3) would always lead to non-rational $h(z)$. Allowing for larger negative circulations, say $\varGamma _k = -3/2$, would require the corresponding coefficient $H_k''(z_k)$ to vanish, which is not true in general. On the other hand, this can happen in specific examples, for instance the trivial example of a single point vortex.

5.2. Proof that singularities are stationary point vortices

With the choice of (5.5) for $h'(z)$, the stream function is smooth away from the zeros and poles of $h'(z)$ and $h(z)$ and therefore satisfies the modified Liouville equation (1.3). Because $h'(z)$ and $h(z)$ are both rational functions, singularities $\tilde {z}_k$ of the stream function (5.1) can only appear at their roots and poles. It remains to show that at each of these singularities, the velocity field is of the form (1.4) and hence that the stream function satisfies (1.3). We have the following three cases to consider: (a) zeros of $h'(z)$, (b) poles of $h'(z)$ and (c) zeros of $h(z)$. The poles of $h(z)$ and $h''(z)$ coincide with the poles of $h'(z)$, and so do not need to be checked separately.

Using (5.5), we see that the term

(5.10)\begin{equation} \frac{h''(z)}{h'(z)} = (\log h'(z))^\prime = 2\sum_{j=1}^M\frac{\varGamma_j}{z-z_j} \end{equation}

is proportional to the pure point vortex velocity field (2.8). Because the point vortices are stationary, from (2.10) we have

(5.11)\begin{equation} \frac{h''(z)}{h'(z)} = \frac{2\varGamma_k}{z-z_k} + O(z-z_k)\quad\text{as }z\to z_k \end{equation}

near any zero or pole $z_k$ of $h'(z)$.

First consider (a) a zero $z_k$ of $h'(z)$. The second term in (5.2) vanishes at $z_k$ and we have from (5.11)

(5.12)\begin{equation} u-\mathrm{i}v = \frac{1}{2{\rm \pi}\mathrm{i}}\frac{\varGamma_k}{z-z_k} + O(z-z_k)\quad\text{as }z\to z_k, \end{equation}

which corresponds to a stationary point vortex at $z_k$ with circulation $\tilde {\varGamma }_k=\varGamma _k$. Note that the zeros of $h'(z)$ correspond to point vortices with positive circulations $\varGamma _k$. Next, considering (b), it was already noted by Crowdy (Reference Crowdy2003) that a simple-pole term in $h(z)$ leads to a smooth velocity field at the location of the pole. Because from (3.3) the only poles of $h'(z)$ are second-order poles, the only poles of $h(z)$ will be simple poles. Going back to the stream function (5.1), we see that at a simple pole $z_k$ of $h(z)$, the argument of the logarithm has a constant non-zero leading term, and hence the stream function is regular at $z_k$ owing to the structure of the Liouville solution. The same conclusion may also be reached by expanding (5.2) near $z_k$. It remains to consider (c), of which there are two types, simple and multiple zeros. At a simple zero $\hat {z}_k$ (say) of $h(z)$, the second term in (5.2) vanishes while the first term is regular because $h'(z)$ is regular at $\hat {z}_k$. The velocity field is therefore regular at a simple zero of $h(z)$. If $\hat {z}_j$ is a multiple zero of $h(z)$, then it must also be a zero of $h'(z)$ and so $\hat {z}_j=z_k$ for some $k=1,\ldots ,M$. This case has already been covered in (a) and corresponds to a stationary point vortex at $z_k$ of strength $\tilde {\varGamma }_k=\varGamma _k$.

To summarise: under the restriction (3.3), a rational $h'(z)$, defined by (5.5), leads to a rational $h(z)$. The roots of $h'(z)$, which correspond to the positive circulation point vortices in the input equilibrium function (5.5), are preserved as point vortices at the same locations (renamed as $\tilde {z}_k$) with their strengths remaining the same. The poles of $h'(z)$, which correspond to point vortices with circulations $-1$, become the background sea of smooth Liouville-type vorticity.

If the value of the constant $b$ is chosen to be different from $-8{\rm \pi}$, all elements of the proofs above remain the same except for the strengths $\tilde {\varGamma }_k$, which now scale as $\tilde {\varGamma }_k=(-8{\rm \pi} /b)\varGamma _k$. Appendix B contains more details.

6.1. Point vortex limits of a Liouville link and a transformation between them

The stream function (3.2) for the Liouville link contains a real parameter $A$ and a complex parameter $C$. In this section, we are interested in the behaviour of the hybrid stream function in the limits $A\to 0,\infty$ and show that, in these limits, the rotational solutions approach distinct stationary point vortex equilibria.

Taking $A>0$ and rearranging the argument of the logarithm in the hybrid stream function (3.2), we obtain

(6.1a)\begin{align} \psi(z,\bar{z};A,C) &={+}\frac{1}{4{\rm \pi}}\log\left[\frac{1+A^2|g(z)+C|^2}{A|g'(z)|}\right] \end{align}

(6.1b)\begin{align} &={+}\frac{1}{4{\rm \pi}}\log\left[\frac{1}{A|g'(z)|} + A\frac{|g(z)+C|^2}{|g'(z)|}\right]. \end{align}

Note that we can take $A>0$ without loss of generality because if $A$ were negative or indeed complex, then $|A|>0$ would appear in (6.1b) in place of $A$.

Consider now the limiting case $A\to 0$. In this case, the second term in the argument of the logarithm drops out and the stream function (6.1b) can be written, after renormalising for the infinite term $-(1/4{\rm \pi} )\log A$, as

(6.2a)\begin{equation} \lim_{A\to 0}\left[\psi(z,\bar{z};A,C)+\frac{1}{4{\rm \pi}}\log A\right] ={-}\frac{1}{4{\rm \pi}} \log |g'(z)|. \end{equation}

Similarly, in the limiting case $A\to \infty$, we see that the first term in the argument of the logarithm drops out and the stream function (6.1b) can be written, after renormalising for the infinite term $(1/4{\rm \pi} )\log A$, as

(6.2b)\begin{equation} \lim_{A\to\infty}\left[\psi(z,\bar{z};A,C)-\frac{1}{4{\rm \pi}}\log A\right] ={-}\frac{1}{4{\rm \pi}}\log \frac{|g'(z)|}{|g(z)+C|^2}. \end{equation}

The stream functions (6.2a) and (6.2b) are respectively the imaginary parts of the complex potentials

(6.3a,b)\begin{equation} G(z)=\frac{1}{4{\rm \pi}\mathrm{i}}\log g'(z) \quad\text{and}\quad F(z;C)=\frac{1}{4{\rm \pi}\mathrm{i}}\log\left[\frac{g'(z)}{(g(z)+C)^2}\right], \end{equation}

determined by the input equilibrium $g'(z)$. Comparing (6.3a,b) with (5.4), we see that $G(z)=f(z)$ is the complex potential of the stationary point vortex equilibrium with which we started. The complex potential $F(z;C)$ also corresponds to a stationary point vortex equilibrium, distinct from $G(z)$, as explained in the following.

To relate the two different complex potentials $G(z)$ and $F(z)$ defined in (6.3a,b), we first define a new function $\hat {g}'(z)$ via the transformation

(6.4)\begin{equation} g'(z)\mapsto\hat{g}'(z) = \hat{E}\left[\frac{g'(z)}{(g(z)+C)^2}\right]^\alpha, \end{equation}

where $\alpha$ is the twist parameter and $\hat {E}$ is some constant. Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020) showed that the transformation (6.4) takes a given stationary point vortex equilibrium $g'(z)$ into a new equilibrium $\hat {g}'(z)$ if the point vortex circulations in $g'(z)$ belong to the set (3.3). The primitive of $g'(z)$ is then also a rational function and $\hat {g}'(z)$ viz. (6.4) takes the same mathematical form as $g'(z)$ viz. (3.4), but with some of the vortex positions and circulations changed. The proof for this assertion is very similar to the proof presented in §§ 5.1 and 5.2, and is detailed in Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020).

The stationary point vortex equilibria connected by the transformation (6.4) are closely related to the complex potentials $G(z)$ and $F(z;C)$ in (6.3a,b) of stationary point vortex equilibria that exist at the end points of a Liouville link. Indeed, let us define $\hat {G}(z;C)$ after scaling the circulations in $F(z;C)$ by $\alpha$, and adding a constant in terms of $\hat {E}$:

(6.5)\begin{equation} \hat{G}(z;C)=\alpha F(z;C) + \frac{1}{4{\rm \pi}\mathrm{i}}\log\hat{E}. \end{equation}

We can rewrite (6.3a,b) in terms of $\hat {G}(z;C)$ as

(6.6a,b)\begin{equation} G(z)=\frac{1}{4{\rm \pi}\mathrm{i}}\log g'(z) \quad\text{and}\quad \hat{G}(z;C)=\frac{1}{4{\rm \pi}\mathrm{i}}\log\hat{g}'(z). \end{equation}

Thus $G(z)=f(z)$ is the stationary point vortex equilibrium with which we started, defined by $g'(z)$, whereas $\hat {G}(z;C)$ (or $F(z;C)$) is a new stationary point vortex equilibrium defined by $\hat {g}'(z)$, which is given by the transformation (6.4). We emphasize that $\hat {G}(z;C)$ is obtained after scaling all the circulations in the $A\to \infty$ limit (6.3a,b) of the Liouville link by the twist parameter $\alpha$. Finally, we note that, by definition, $G(z)$ is independent of $C$.

6.2. Twist operations and Liouville chains

A brief summary of Liouville links and their limits, which we have discussed so far in §§ 5 and 6.1, is as follows. The input equilibrium function $g'(z)$ in the hybrid stream function (3.2) of the Liouville link can be chosen to be of the form (3.4), which consists of stationary point vortices whose circulations belong to the set (3.3). Then, for any finite value of the parameter $A$, the Liouville link solution exists. The transformation (6.4) allows us to jump directly between the end points of the Liouville link, which are pure point vortex equilibria. These latter equilibria are recovered as distinct limits of the rotational hybrid solutions, as $A\to 0,\infty$.

The transformation (6.4) between stationary pure point vortex equilibria can sometimes be iterated to produce hierarchies of pure point vortex equilibria consisting of increasing numbers of point vortices with each iteration (Krishnamurthy et al. Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020). It is also possible to have the iteration continue indefinitely. At a given $n$th stage of the iteration, if $g_n'(z)$ is a stationary point vortex equilibrium of the form (3.4) with the circulations belonging to the set (3.3), then $g_n(z)$ is a rational function and $g_{n+1}'(z)$, defined by the iterated transformation (for some constants $E_n$)

(6.7)\begin{equation} g_{n+1}'(z) = E_{n+1}\left[\frac{g_n'(z)}{(g_n(z)+C_n)^2}\right]^{\alpha_n}, \end{equation}

is a stationary point vortex equilibrium. A new twist parameter $\alpha _n$ is defined at every stage of the iteration (6.7). If $\alpha _n$ can be chosen so that the circulations in $g_{n+1}'(z)$ also belong to the set (3.3), then the iteration can be continued. Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020) showed that if we choose at any stage the special value $\alpha _n=1$, then the resulting equilibrium will simply be a space-shifted version of the equilibrium at the previous stage. They also discussed various classes of stationary point vortex equilibria that can be generated from a ‘seed equilibrium function’ of the form

(6.8)\begin{equation} g_0'(z)=z^{2\varGamma} \end{equation}

with different choices of $\varGamma$ and $\alpha _n$. Here, $\varGamma$ is the circulation of the seed point vortex. In this manner, the Adler–Moser polynomials found by Adler & Moser (Reference Adler and Moser1978) and the two polynomial hierarchies discussed by Loutsenko (Reference Loutsenko2004) are all produced from the same seed (6.8) through the iterated transformation (6.7). The point vortices in equilibrium are at the roots of successive polynomials in these hierarchies.

A Liouville chain is a sequence of Liouville links joined together by applying a twist operation at an end point of each Liouville link. Every link in the Liouville chain is a family of hybrid stream functions, defined by (3.6), and continuously parametrised by $0\lt A_n\lt \infty$. The end points of every Liouville link are pure point vortex equilibria obtained in the limits $A_n\to 0,\infty$ and have the complex potentials

(6.9a,b) \begin{equation} G_n(z) = \frac{1}{4{\rm \pi}\mathrm{i}}\log g_n'(z) \quad\text{and}\quad F_n(z;C_n) = \frac{1}{4{\rm \pi}\mathrm{i}}\log\left[\frac{g_n'(z)}{(g_n(z)+C_n)^2}\right]. \end{equation}

The twist operation is encoded by the twist parameter $\alpha _n$, which scales the circulations of all the point vortices at the $A_n\to \infty$ limit of the Liouville link. The point vortex equilibrium, obtained after scaling the circulations and adding a constant, is used to build the next link in the chain:

(6.10)\begin{equation} G_{n+1}(z;C_n) = \alpha_n F_n(z;C_n) + \frac{1}{4{\rm \pi}\mathrm{i}}\log E_{n+1}. \end{equation}

The stream function at the $n$th stage will contain $n+2$ parameters: the real parameter $A_n$ and $n+1$ complex parameters $C_0,\ldots,C_n$. Every function $g_n'(z)$ in the chain is of the form (3.4) and is obtained via the iterated transformation (6.7). As long as the sequence of rational functions $g_n'(z)$ corresponding to stationary pure point vortex equilibria exists, a corresponding sequence of stream functions given by (3.6) also exists.

7.1. Liouville link between equilibria with three and eleven point vortices

Consider a stationary equilibrium (O'Neil Reference O'Neil2006; Krishnamurthy et al. Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020) consisting of three point vortices of strengths $3$, $3/2$, $-1$ located at $-2$, $1$, $0$. The input equilibrium function $g'(z)$ corresponding to this point vortex equilibrium is written from (3.4) as

(7.1)\begin{equation} g'(z) = \frac{(z+2)^6(z-1)^3}{z^2}. \end{equation}

Because the point vortex circulations are restricted to the set (3.3), it follows that the primitive of $g'(z)$ must be rational. Indeed, we get after an explicit calculation that

(7.2)\begin{equation} g(z) = \frac{z^8}{8}+\frac{9 z^7}{7}+\frac{9 z^6}{2}+3 z^5-18 z^4-36 z^3+24 z^2+144 z+\frac{64}{z}. \end{equation}

Substituting in (3.2) and taking $C$ as an integration constant, we get the hybrid stream function

(7.3)\begin{equation} \psi(z,\bar{z};A,C) ={-}\frac{1}{4{\rm \pi}}\log\left[ \frac{A|z+2|^6|z-1|^3}{|z|^2 + A^2|z(g(z)+C)|^2}\right], \end{equation}

which defines a Liouville link solution for the range of parameter values $0\lt A\lt \infty$. It is clear from (7.3) that the point vortex at $z=0$ in (7.1), with circulation $-1$, has been smoothed out. The two point vortices at $z=-2$ and $z=1$ in (7.1) remain embedded in the flow retaining the values of their circulations, are stationary, and are surrounded by an everywhere smooth rotational flow of Liouville-type.

The transformation (6.4) defines the function $\hat {g}'(z)$. Setting $\alpha =1$, $\hat {E}=1/64$ in (6.4) and using (7.1), (7.2), we get

(7.4)\begin{equation} \hat{g}'(z) = \frac{(z+2)^6(z-1)^3}{(8z(g(z)+C))^2}, \end{equation}

where $8z(g(z)+C)$ is a degree-9 monic polynomial, as seen from (7.2). Comparing the form of $\hat {g}'(z)$ with (3.4), we see that it consists of 11 point vortices: the two positive point vortices at $z=-2$ and $z=1$ together with nine negative point vortices of strength $-1$ located at the roots of the polynomial $8z(g(z)+C)$.

In the limits $A\to 0$ and $A\to \infty$, the stream function (7.3) becomes the imaginary part of the complex potentials (6.6a,b) with $g'(z)$ and $\hat {g}'(z)$ given by (7.1) and (7.4). To see this, notice that the stream function (7.3) can be re-written as

(7.5)\begin{equation} \psi(z,\bar{z};A,C) ={+}\frac{1}{4{\rm \pi}}\log\left[ \frac{1}{A}\frac{|z|^2}{|z+2|^6|z-1|^3}+A\frac{|z(g(z)+C)|^2}{|z+2|^6|z-1|^3}\right], \end{equation}

which in the limits $A\to 0,\infty$ becomes

(7.6a)\begin{align} \lim_{A\to 0}\left[\psi+\frac{1}{4{\rm \pi}}\log A\right] &={-}\frac{1}{2{\rm \pi}}\log\left|\frac{(z+2)^3(z-1)^{3/2}}{z}\right|, \end{align}

(7.6b)\begin{align}\lim_{A\to\infty}\left[\psi-\frac{1}{4{\rm \pi}}\log A\right] &={-}\frac{1}{2{\rm \pi}}\log\left|\frac{(z+2)^3(z-1)^{3/2}}{z(g(z)+C)}\right|. \end{align}

The stream function (7.3) is thus a Liouville link between the three point vortex equilibrium represented by (7.1) and the 11 point vortex equilibrium represented by (7.4). The streamline patterns for this Liouville link are shown in figure 5 for various values of parameters $A$ and $C$; the figure also shows the limiting point vortex equilibria.

Figure 5. Streamline patterns for the stream function (3.2) with the input equilibrium function (7.1). The Liouville link exists for the range of parameter values $0\lt A\lt \infty$. The formation of the limiting point vortex equilibria (6.6a,b), with $g'(z)$ and $\hat {g}'(z)$ given by (7.1) and (7.4), can be seen as $A$ becomes small and large. This process is shown for two values of $C$. When $C=0$, the hybrid solutions are symmetric with respect to the $x$-axis but when $C=100+180\mathrm {i}$ (complex valued), this symmetry is lost. The $A= 0$ limit is independent of $C$ by definition.

7.2. Liouville link between equilibria with four and ten point vortices

Figure 6 shows the streamline patterns for the stream function (3.2) with the input equilibrium function (3.4) formed from a four point vortex equilibrium (O'Neil Reference O'Neil2006; Krishnamurthy et al. Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020)

(7.7)\begin{equation} g'(z) = \frac{(z+\sqrt{3}+3\mathrm{i})^4(z-2\mathrm{i})^2(z-\sqrt{3})}{z^2}. \end{equation}

The point vortex strengths in (7.7) satisfy the constraints (3.3) and the rational function $g(z)$ is

(7.8) \begin{align} g(z) &= \frac{z^6}{6} + \frac{1}{5} (3\sqrt{3}+8\mathrm{i}) z^5 - (1-3\sqrt{3}\mathrm{i}) z^4 + 4(2\sqrt{3}+3\mathrm{i}) z^3\nonumber\\ &\quad - 12(1-3\sqrt{3}\mathrm{i}) z^2 + 24(\sqrt{3}-9\mathrm{i}) z + \frac{288(\sqrt{3}+3\mathrm{i})}{z}. \end{align}

Setting the twist parameter $\alpha =1$ along with $\hat {E}=1/36$, the transformed point vortex equilibrium (6.4) is given by

(7.9a)\begin{equation} \hat{g}'(z) = \frac{(z+\sqrt{3}+3\mathrm{i})^4(z-2\mathrm{i})^2(z-\sqrt{3})}{q^2(z)}, \end{equation}

where the polynomial $q(z)$ is

(7.9b) \begin{align} q(z) &= z^7 + \frac{6}{5} (3\sqrt{3}+8\mathrm{i}) z^6 - 6(1-3\sqrt{3}\mathrm{i}) z^5 + 24(2\sqrt{3}+3\mathrm{i}) z^4\nonumber\\ &\quad - 72(1-3\sqrt{3}\mathrm{i}) z^3 + 144(\sqrt{3}-9\mathrm{i}) z^2 + 6Cz + 1728(\sqrt{3}+3\mathrm{i}). \end{align}

Figure 6. Streamline patterns for the hybrid stream function (3.2) with input equilibrium (7.7). In the limit as $A\to 0,\infty$, the hybrid solution goes over into the stationary point vortex equilibria given by (6.6a,b), with $g'(z)$ and $\hat {g}'(z)$ as in (7.7) and (7.9). Varying the integration constant $C$ alters the locations of the centres and saddles in the flow.

The Liouville link shown in figure 6 consists of the three positive point vortices in (7.7) embedded in a background smooth Liouville-type vorticity for the range of parameter values $0\lt A\lt \infty$. The $A\to 0,\infty$ limits of the Liouville link are pure point vortex equilibria with four and ten point vortices. Their complex potentials are given by (6.6a,b), where $g'(z)$ and $\hat g'(z)$ are given by (7.7) and (7.9).

9.1. Liouville chain in terms of Adler–Moser polynomials

Successive Adler–Moser polynomials are produced by (6.7) with the seed equilibrium function and twist parameters (Krishnamurthy et al. Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020)

(9.1a,b)\begin{equation} g_0^{\,\prime}(z)= z^{2} \quad\text{and}\quad \alpha_n={-}1 \quad\text{for}\quad n\geq 0. \end{equation}

Thus $\varGamma =1$ in (6.8). The first few rational functions $g_n'(z)$ are

(9.2a)\begin{align} g_1'(z) &= \frac{(z^{3} + 3 C_0)^{2}}{z^{2}}, \end{align}

(9.2b)\begin{align}g_2'(z) &= \frac{(z^{6} + 15 C_0 z^{3} + 5 C_1 z- 45 C_0^{2})^{2}}{(z^{3} + 3 C_0)^{2}}, \end{align}

(9.2c)\begin{align}g_3'(z) &= \frac{q^2(z)}{(z^{6} + 15 C_0 z^{3} + 5 C_1 z - 45 C_0^{2})^{2}}, \end{align}

where the polynomial $q(z)$ is

(9.2d)\begin{gather} q(z) = z^{10} + 45 C_0 z^{7} + 35 C_1 z^{5} + 7 C_2 z^{3} - 525 C_0 C_1 z^{2}\nonumber\\ + 4725 C_0^{3} z + 21 C_0 C_2 - \frac{175}{3}C_1^{2}. \end{gather}

The Adler–Moser polynomials $p_n(z)$ are then read off from (9.2) via the formula

(9.3)\begin{equation} g_n'(z)=\frac{p_{n+1}^2(z)}{p_n^2(z)}\quad\text{for } n\geq~0. \end{equation}

To compare $p_n(z)$ with the polynomials given in Adler & Moser (Reference Adler and Moser1978), one must redefine their parameters $\tau _2,\tau _3,\ldots$ as $\tau _2=3C_0$, $\tau _3=5C_1$, $\tau _4=7C_2$ and so on.

The hybrid stream functions follow from substituting (9.2) into (3.6) and the corresponding streamline patterns are shown in figure 8. The limiting cases $A_0=0,\infty$ are stationary pure point vortex equilibria given by $g_0'(z)$ and $g_1'(z)$; the limiting cases $A_1=0,\infty$ are stationary pure point vortex equilibria given by $g_1'(z)$ and $g_2'(z)$, and so on; see the schematic in figure 1. The point vortex locations in these limiting patterns are, of course, given by the roots of successive Adler–Moser polynomials and are shown in figure 3 of Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020). The value of the twist parameter $\alpha _n$ in (9.1a,b) is chosen so that, at each step, the iteration can be continued.

Figure 8. Streamline patterns for a Liouville chain formed from the hierarchy of Adler–Moser polynomials. The panels corresponding to $(s_n)$ and $(a_n)$ show the hybrid stream functions $\psi _n(z,\bar {z})$ given by (3.6) for $n=1,2,3$; the corresponding rational functions $g_n'(z)$ are given by (9.2). Panels (a–c) corresponding to $(s_n)$ are symmetric configurations whereas (d–f) corresponding to $(a_n)$ are asymmetric configurations – they differ in the choice of constants $C_0,\ldots ,C_3$ in (9.2). The values of these constants are given in table 1. The point vortex limits of ($s_n$) and ($a_n$) are shown in panels (S1)–(S4) and (A1)–(A4) of figure 3 in Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020).

9.2. Liouville chains in terms of the Loutsenko polynomials

There are two hierarchies of polynomials described in Loutsenko (Reference Loutsenko2004), both of which are produced by the iterated transformation (6.7) (Krishnamurthy et al. Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020). The first hierarchy results from the choice of seed equilibrium function and twist parameters

(9.4a,b)\begin{equation} g_0^{\,\prime}(z) = z, \quad\text{and}\quad \alpha_n = \begin{cases} -2 & \text{for }n\text{ even}, \\ -1/2 & \text{for }n\text{ odd}. \end{cases} \end{equation}

This corresponds to setting $\varGamma =1/2$ in (6.8). We see that $g_0'(z)$ in (9.4a,b) is actually the input equilibrium function (4.1) with which we started the construction in § 4; the twist parameters used there are also the same as those used in (9.4a,b). In fact, the Liouville chain constructed in § 4 is given in terms of the first hierarchy of Loutsenko polynomials. The first few rational functions in this hierarchy, produced by (6.7), are

(9.5a)\begin{align} g_1'(z) &= \frac{(z^{2} + 2 C_0)^{4}}{z^{2}}, \end{align}

(9.5b)\begin{align}g_2'(z) &= \frac{z^{8} + \frac{56}{5}C_0 z^{6} + 56 C_0^{2} z^{4} + 224 C_0^{3} z^{2} + 7 C_1 z - 112 C_0^{4}}{(z^{2} + 2 C_0)^{2}}, \end{align}

(9.5c)\begin{align}g_3'(z) &= \frac{(z^{7} + 14 C_0 z^{5} + 140 C_0^{2} z^{3} + 5 C_2 z^{2} - 280 C_0^{3} z + 10 C_0 C_2 - \frac{35}{2} C_1)^{4}} {(z^{8} + \frac{56}{5} C_0 z^{6} + 56 C_0^{2} z^{4} + 224 C_0^{3} z^{2} + 7 C_1 z - 112 C_0^{4})^{2}}. \end{align}

The hybrid equilibria follow from substitution of (9.5) into (3.6). Streamline patterns for the Liouville links $n=0,1,2$ in this chain are shown in figure 4.

Polynomials $p_n(z)$ are read off from the rational function $g_n'(z)$ using the formula

(9.6)\begin{gather} g_n^{\,\prime}(z) = \begin{cases} \dfrac{p_{n+1}(z)}{p_n^2(z)} & \text{for }n\text{ even}, \\ \dfrac{p_{n+1}^4(z)}{p_n^2(z)} & \text{for }n\text{ odd}. \end{cases} \end{gather}

To compare $p_n(z)$ with the polynomials described in Loutsenko (Reference Loutsenko2004), the branch $i \le 0$ in his notation, one needs to first rename the polynomials $p_n(z)$ according to $p_n\to p_{-(n+1)/2}$ for $n$ odd and $p_n\to q_{-n/2}$ for $n\geq 2$ and even. We can make the identification after redefining his parameters $\tau _{-1},t_{-2},\tau _{-2},\ldots$ as $\tau _{-1}=2C_0$, $t_{-2}=7C_1$, $\tau _{-2}=5C_2$ and so on. The second hierarchy of Loutsenko is produced using (6.7) with the seed equilibrium function and twist parameters taken to be

(9.7a,b)\begin{equation} g_0^{\,\prime}(z) = z^4, \quad\text{and}\quad \alpha_n = \begin{cases} -1/2 & \text{for }n\text{ even}, \\ -2 & \text{for }n\text{ odd}. \end{cases} \end{equation}

The first few rational functions are

(9.8a) \begin{align} g_1'(z) &= \frac{z^{5} + 5 C_0}{z^{2}},\quad g_2'(z) = \frac{(z^{5} + 4 C_1 z - 20 C_0)^{4}}{(z^{5} + 5 C_0)^{2}},\quad g_3'(z) = \frac{q(z)}{(z^{5} + 4 C_1 z - 20 C_0)^{2}}, \end{align}

where the numerator polynomial $q(z)$ in $g_3'(z)$ is

(9.8b) \begin{align} q(z) &= z^{16} + \frac{176}{7}C_1 z^{12} - 160 C_0 z^{11} + 352 C_1^{2} z^{8} - \frac{42240}{7} C_0 C_1 z^{7}\nonumber\\ &\quad+ 35200 C_0^{2} z^{6} + 11 C_2 z^{5} - 2816 C_1^{3} z^{4} + 28160 C_0 C_1^{2} z^{3} \nonumber\\ &\quad- 140800 C_0^{2} C_1 z^{2} + 352000 C_0^{3} z - \frac{2816}{5} C_1^{4} + 55 C_0 C_2. \end{align}

We can read off polynomials $p_n(z)$ from (9.8) using the formula

(9.9)\begin{equation} g_n^{\,\prime}(z) = \begin{cases} \dfrac{p_{n+1}^4(z)}{p_n^2(z)} & \text{for }n\text{ even}, \\ \dfrac{p_{n+1}(z)}{p_n^2(z)} & \text{for }n\text{ odd}. \end{cases} \end{equation}

To compare $p_n(z)$ with the polynomials from the branch $i \geq 0$ in Loutsenko (Reference Loutsenko2004), one must rename $p_n(z)$ according to $p_n\to p_{n/2}$ for $n\geq 0$ even and $p_n\to q_{(n+1)/2}$ for $n$ odd; and redefine his parameters as $t_{1}=5C_0$, $\tau _2=4C_1$ and so on.

Figure 9 shows the streamline patterns for the hybrid equilibria obtained from (9.8), for the cases $n=1,2,3$. The corresponding limiting point vortex patterns of these Liouville links are shown in figure 5 of Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020). It is seen in figure 9 that the inter-streamline distance alternately increases and decreases as we move up the hierarchy. The total circulation of the hybrid equilibria is calculated in Appendix A. From (A5), we see that for any finite $A_n$, the total circulation of the hybrid solution is simply given in terms of the net circulation of the underlying point vortex equilibrium, $\varGamma _{hyb}=-(\varGamma _{pv}+1)$ because $\varGamma _{pv}>0$. It is easily calculated from (9.4a,b) and (9.7a,b) that $\varGamma _{pv}$ oscillates up and down as we move up the hierarchy, which is the reason for the alternating streamline patterns. Although this is true also for figure 4, the pattern is not as easily visible there.

Figure 9. Symmetric $(s_n;\ a - c)$ and asymmetric $(a_n; d - f)$ streamline patterns of the hybrid stream functions (3.6) for $n=1,2,3$ given in terms of the second Loutsenko hierarchy (9.8). The values of the constants used here are given in table 1. The corresponding point vortex limits are shown in figure 5 of Krishnamurthy et al. (Reference Krishnamurthy, Wheeler, Crowdy and Constantin2020).