Stability and transport properties of multiple-patch quasi-equilibria

A novel subclass of exact solutions to the Euler equations in two dimensions has been put forward recently [D. Crowdy, "A class of exact multipolar vortices," Phys. Fluids 11, 2556 (1999)]. The solutions show vortical equilibria which can be described by only two parameters. The first one designates the multipolar aspect of these equilibria, i.e., the number of point vortices involved, while the other parameter signatures the shape of the finite area of uniform vorticity in which the point vortices are embedded. The main aspect of these equilibria is that the vortical configuration is static, meaning that the velocity induced at the patch edge, outside the vortical area, and also at the locations of the point vortices is zero. We show with numerical experiments that quite remarkably the linearly stable equilibria of Crowdy seem to mix very efficiently in contrast to the unstable vortex solutions. In the second part of this paper we report on the dynamics, stability, and mixing properties of similar vortex systems where the point vortices are regularized to vortex patches (with uniform vorticity). Several of these multiple-patch vortices turn out to be remarkably stable, although the regularization itself should be considered as a (symmetric) perturbation of Crowdy's multipolar solutions.