The construction of exact multipolar equilibria of the two-dimensional Euler equations

Using ideas involving the Schwarz function of analytic curves, a new class of exact multipolar equilibria of the two-dimensional Euler equations characterized by an annular region of vorticity enclosing a region of irrotational fluid is constructed. The results generalize a recently derived class of exact solutions for multipolar vortex equilibria [Crowdy, Phys. Fluids 11, 2556 (1999)]. The solutions have many qualitative similarities to the multiple-vortex nonlinear saturation states of an unstable annulus of uniform vorticity. More generally, the results suggest the possibility of constructing multipolar equilibria of the steady Euler equations having distributed vortical regions of more or less arbitrary geometrical complexity.