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In this paper a family of sequences of polynomials is studied that generalize the Adler–Moser and Loutsenko polynomial sequences related to point vortex equilibria. The polynomials come in pairs with roots that correspond to point vortex configurations in equilibrium, and they have continuous parameters so that the equilibria form positive-dimensional sets in configuration space. The polynomial pairs satisfy a bilinear differential equation and consequently have special properties. The differential recursion relations used to generate the sequences are found by exploiting a symmetry of this equation. Raising and lowering operators are constructed which map between the different levels of the lattice of solutions. In this way the new polynomials are shown to have some of the same special algebraic properties as the Adler–Moser polynomials, such as the Wronskian structure and Stieltjes root sums. • New point vortex equilibria are described using the polynomial method. • Polynomial sequences generalize the Adler–Moser and Loutsenko sequences. • Special algebraic properties are revealed through a set of lattice operators. • Wronskian structure and Stieltjes sums are proven. [ABSTRACT FROM AUTHOR]

A new exact method is presented for obtaining uniformly rotating distributions of vorticity in a two dimensional ideal fluid. The vorticity is confined to the union of a straight line and a finite collection of points; i.e., the distribution is a collection of point vortices together with a number of vortex sheets lying on the common line. The vorticity density of the vortex sheets and the velocity field of the fluid are expressed in terms of a rational function in which the point vortex positions and strengths appear as parameters. For many values of these parameters, the vortex sheet portion of the distribution is not unique, and there is a continuous family of vortex sheets which combine with the point vortices to form relative equilibria. Several examples are worked out in detail. [ABSTRACT FROM AUTHOR]

The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way. [ABSTRACT FROM AUTHOR]

Equilibrium configurations of point vortices with circulations of two discrete values are associated with the zeros of a sequence of polynomials having many continuous parameters: the Adler-Moser polynomials in the case of circulation ratio −1, and the Loutsenko polynomials in the case of ratio −2. In this paper a new set of polynomial sequences generalizing the vortex system to three values of circulations is constructed. These polynomials are extensions of the previously known polynomials in the sense that they are special cases of the latter when the third circulation is zero. The polynomials are naturally connected with rational functions that satisfy a second-order differential equation. [ABSTRACT FROM AUTHOR]

Point vortex equilibria in which the vortices are arranged in clusters are examined. The vortex velocities in these configurations are all equal. Necessary conditions for their existence are established that relate the circulations within the clusters to the cluster radius. A method for generating these configurations by singular continuation is proved to be valid for the generic case. Finally, a partial analysis of exceptional cases is given and their connection to the existence of parametrized families of equilibria is described. [ABSTRACT FROM AUTHOR]

Abstract: Numerical evidence is presented for the existence of relative equilibrium configurations of vortex sheets (one-dimensional singular distributions of vorticity) in a two-dimensional ideal fluid. Point vortices are used to approximate the vortex sheets. [Copyright &y& Elsevier]

Abstract: Point vortices in a two-dimensional fluid may be arranged so that all resulting vortex velocities are identical; these are equivalent to force-free arrangements of 2D Coulomb charges in a homogeneous field. For generic choice of vortex circulations, the number of stationary configurations is finite, but a few infinite solution families (of arbitrary positive dimension) are known to exist. This paper extends the classes of point vortex systems known to have a finite or infinite number of stationary configurations respectively. The main tool is a new minimal system of polynomial equations characterizing these configurations. The polynomial system is extended to include symmetric vortex configurations such as vortex streets and rings, as well as circulation symmetries (vortices with identical circulations). By expressing the systems as a multilinear differential equation, one-dimensional solution sets can be constructed for a wide range of systems, including vortex streets and rings. [Copyright &y& Elsevier]

Stationary configurations of identical point vortices on the sphere are investigated using a simple numerical scheme. Configurations in which the vortices are arrayed along curves on the sphere are exhibited, which approximate equilibrium configurations of vortex sheets on the sphere. Other configurations (found after starting from random initial conditions) exhibit net-like distributions of vorticity, dividing the sphere into many cells that contain no vorticity or diffuse vorticity and forming a stationary ‘vortex foam’ on the sphere. They may be viewed as intermediate-energy elements in the set of all identical point vortex equilibria on the sphere. In the continuum limit, these foam states may correspond to stationary states of multiple intersecting vortex sheets. Stationary configurations of point vortices are not found to have this character when vortices of opposite circulations are included. [ABSTRACT FROM AUTHOR]