A transformation between stationary point vortex equilibria

Abstract

A new transformation between stationary point vortex equilibria in the unbounded plane is presented. Given a point vortex equilibrium involving only vortices with negative circulation normalized to −1 and vortices with positive circulations that are either integers or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant. When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations, each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler–Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko (Loutsenko 2004 J. Phys. A: Math. Gen. 37 , 1309–1321 (doi:10.1088/0305-4470/37/4/017)). For the latter polynomials, the existence of such a transformation appears to be new. The new transformation, therefore, unifies a wide range of disparate results in the literature on point vortex equilibria.

Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

“…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. (2020).…”

“…The corresponding limiting point vortex patterns of these Liouville links are shown in figure 5 of Krishnamurthy et al. (2020). It is seen in figure 9 that the inter-streamline distance alternately increases and decreases as we move up the hierarchy.…”

“…For further discussion of point vortex equilibria, see Krishnamurthy et al. (2020) and the review article Aref et al. (2003).…”

“…Every pair of point vortex limits is connected by the transformation (6.7) discussed in Krishnamurthy et al. (2020). Here , and .…”

“…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 2006; Krishnamurthy et al. 2020) The point vortex strengths in (7.7) satisfy the constraints (3.3) and the rational function is Setting the twist parameter along with , the transformed point vortex equilibrium (6.4) is given by where the polynomial is

Figure 6.Streamline patterns for the hybrid stream function (3.2) with input equilibrium (7.7). In the limit as , the hybrid solution goes over into the stationary point vortex equilibria given by (6.6 a , b ), with and as in (7.7) and (7.9).

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