Self-similar collapse of three vortices in the generalised Euler and quasi-geostrophic equations

Abstract

We revisit the classical problem of the self-similar, finite-time collapse of three vortices. We extend the study to the generalised two-dimensional Euler equations as well as the generalised three-dimensional quasi-geostrophic equations. In both these situations, the flow is fully controlled by a materially-conserved field of a generalised vorticity (or potential vorticity) related to the streamfunction by a modified Poisson’s equation, in which the standard Laplacian is replaced by a fractional Laplacian. We first determine the conditions for the self-similar collapse of three point vortices, as well as the collapse time in a broad parameter space. We then consider the evolution of finite-core, two-dimensional and three-dimensional vortices under initial conditions corresponding to the collapse of equivalent point vortices. We show that the interaction precipitates the merger of the two like-signed vortices in the vortex triad.