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

Abstract

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 patches, again two distinct families of solutions exist: one with the central patch and satellite patches having the same-signed vorticity and the other in which they are opposite-signed. In each family, there are two limiting behaviours in which either the central patch or the satellite patches develop corners or cusps. Streamline plots and time-dependent simulations indicate that opposite-signed multipolar equilibria are robust structures and same-signed equilibria are generally less stable. Streamlines also reveal stable and unstable (saddle point) stagnation points, indicating the existence of new equilibria in which additional patches of vorticity are “grown” at the stagnation points. Examples of such equilibria are computed, and a general numerical routine is briefly described for finding even more complex finite-area equilibria. Finally, new nested polygonal vortex equilibria consisting of two sets of polygonally arranged vortex patches (named “N + N” equilibria here) are computed for two distinct cases: one with the corners of the polygons aligned with each other and the other when they are staggered. Various limiting states are computed for these equilibria. Time-dependent simulations reveal that the aligned equilibria are susceptible to instability, while the staggered equilibria survive a relatively long time. In some parameter regimes, following instability, these structures evolve into known structures such as “N + 1” multipolar vortex equilibria and N-polygon co-rotating equilibria.

Multipole vortex patch equilibria for active scalar equations

“…This remains equally true for the body-centered polygonal configurations (γ 2 = 0), treated in [49], as well as for the nested polygons without a central patch (γ 0 = 0). The latter solutions were first observed numerically in [50].…”

“…Very recently, Godard-Cadillac, Gravejat and Smets [24] extended Turkington's result for the gSQG equations, while Ao, Dávila, Del Pino, Musso and Wei [1] have obtained related families of smooth solutions via gluing techniques. See [39,40,41,46,50] for additional references on multiply connected patches.…”

“…Remark 1.10. The parameter ε > 0 can be chosen uniformly as any of the parameters b 0 , b 1 and b 2 tend to 0, and therefore we may recover the point vortex-vortex patch configurations discussed in [13,50]; see Remark 2.4.…”

Three-dimensional Quasi-geostrophic Staggered Vortex Arrays

“…Following these early studies, a vast body of literature has extended the results on the stability of point vortex arrays [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] as well as finite core vortex arrays e.g. [30][31][32][33]. Additionally, Sokolovskiy et al (2020) [34] have recently studied the evolution of non-equilibrium four-vortex arrays in a two-layer quasi-geostrophic system.…”

Multipole Vortex Patch Equilibria for Active Scalar Equations

“…Very recently, Godard-Cadillac, Gravejat and Smets [25] extended Turkington's result to the gSQG equations, while Ao, Dávila, Del Pino, Musso and Wei [1] have obtained related families of smooth solutions via gluing techniques. See [41,42,44,49,53] for additional references on multiply connected patches.…”