Growing vortex patches

Abstract

This paper demonstrates that two well-known equilibrium solutions of the Euler equations—the corotating point vortex pair and the Rankine vortex—are connected by a continuous branch of exact solutions. The central idea is to “grow” new vortex patches at two stagnation points that exist in the frame of reference of the corotating point vortex pair. This is done by generalizing a mathematical technique for constructing vortex equilibria first presented by Crowdy [D. G. Crowdy, “A class of exact multipolar vortices,” Phys. Fluids 11, 2556 (1999)]. The solutions exhibit several interesting features, including the merging of two separate vortex patches via the development of touching cusps. Numerical contour dynamics methods are used to verify the mathematical solutions and reveal them to be robust structures. The general issue of how simple vortex equilibria can be continued continuously to more complicated ones with very different vortical topologies is discussed. The solutions are examples of exact solutions of the Euler equations involving multiple interacting vortex patches.