Continuous family of point vortex lattice equilibria in doubly periodic rectangular domains with a Liouville-type background vorticity.

The stationary equilibria of N point vortices in a rectangular domain with doubly periodic boundary conditions offer a foundational framework for understanding the stationary lattices of coherent vortex structures. In this paper, we obtain new continuous families of these lattices. These solutions are constructed using methods from complex analysis including conformal mapping and special function theory. First, the doubly periodic rectangular domain is mapped to an annulus by a conformal mapping. Next, a loxodromic function is used to construct a solution consisting of stationary point vortices embedded in a smooth background vorticity field which is exponentially related to the stream function (Liouville-type background vorticity). This solution contains three continuous parameters, and by considering the limits of one of these parameters, a stationary point vortex solution without a background vorticity field is constructed. Until now, stationary point vortex solutions under doubly periodic boundary conditions have been mainly symmetric. Varying the other two parameters noted above, we generate families of asymmetric stationary point vortex lattices.