Multipolar vortices and algebraic curves

This paper demonstrates that analytical solutions to the steady two-dimensional Euler equations possessing all the qualitative properties of multipolar vortices observed experimentally and numerically can be constructed using the theory of algebraic curves and quadrature domains. The solutions consist of a finite set of line vortices superposed on finite-area patches of uniform vorticity. By way of example, new solutions are presented in which the support of the vorticity is a quintuply connected vortex patch with five vorticity maxima modelling a symmetric pentapolar vortex consisting of a central core vortex, four satellite vortices and four enclosed zones of irrotational fluid between the core and the satellite vortices. The method of construction is amenable to the derivation of exact solutions of the two-dimensional Euler equations having vorticity distributions of even greater geometrical complexity.