The N-vortex Problem in Doubly-periodic Domains with Background Vorticity

We study the N-vortex problem in a doubly periodic rectangular domain in the presence of a background vorticity field. We first consider a constant background field and derive an explicit formula for the hydrodynamic Green's function using a conformal mapping approach. We show that the point vortices form a Hamiltonian system and that the two-vortex problem is integrable. Several fixed lattice configurations are obtained for general N, some of which consist of vortices with inhomogeneous strengths and lattice defects. We then consider a smooth background field given by the Liouville PDE and show example solutions in which point vortices exist in stationary equilibrium with the background.