New families of vortex patch equilibria for the two-dimensional Euler equations

Various modified forms of contour dynamics are used to compute multipolar vortex equilibria, i.e., configurations of constant vorticity patches which are invariant in a steady rotating frame. There are two distinct solution families for “N + 1” point vortex-vortex patch equilibria in which a finite-area central patch is surrounded by N identical point vortices: one with the central patch having opposite-signed vorticity and the other having same-signed vorticity to the satellite vortices. Each solution family exhibits limiting states beyond which no equilibria can be found. At the limiting state, the central patch of a same-signed equilibrium acquires N corners on its boundary. The limiting states of the opposite-signed equilibria have cusp-like behaviour on the boundary of the central patch. Linear stability analysis reveals that the central patch is most linearly unstable as it approaches the limiting states. For equilibria comprising a central patch surrounded by N identical finite-area satellite patch...