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The repeated filamentation of two-dimensional vorticity interfaces

Undular disturbances of arbitrarily small steepness to the boundary of a circular patch of uniform vorticity are found to give way to a qualitatively different type of behaviour after a time inversely proportional to the initial wave steepness squared. Repeatedly, thin filaments are drawn out at a frequency of nearly half the vorticity jump across the interface (the intrinsic frequency of linear waves on the interface), and the interaction between the filaments and the undulating boundary generate new disturbances from which still greater numbers of filaments of increasingly complicated shape are drawn out. The vortex boundary thereby experiences an extraordinary, continual and apparently irreversible growth in complexity. Essentially the same phenomenon occurs on all vortex-patch equilibria, e.g. the Kirchhoff elliptical vortex, with even greater complexity. The steepening of a disturbance on a non-circular vortex proceeds faster than that on a circular vortex, because of the varying mean strain and shear seen by the disturbance as it travels around the vortex boundary. Generalizations to more than one vorticity interface, to flows on the surface of a sphere, and to sharp but not infinitely sharp vorticity gradients are also discussed. The results support the view that almost any two-dimensional, inviscid, incompressible flow with large vorticity gradients will exhibit repeated filamentation.