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Point vortex models are presented for the generalized Euler equations, whichare characterized by a fractional Laplacian relation between the active scalarand the streamfunction. Special focus is given to the case of the surfacequasi-geostrophic (SQG) equations, for which the existence of finite-timesingularities is still a matter of debate. Point vortex trajectories areexpressed using Nambu dynamics. The formulation is based on a noncanonicalbracket and allows for a geometrical interpretation of trajectories asintersections of level sets of the Hamiltonian and Casimir. Within thissetting, we focus on the collapse of solutions for the three point vortexmodel. In particular, we show that for SQG the collapse can be eitherself-similar or non-self-similar. Self-similarity occurs only when theHamiltonian is zero, while non-self-similarity appears for non-zero values ofthe same. For both cases, collapse is allowed for any choice of circulationswithin a permitted interval. These results differ strikingly from the classicalpoint vortex model, where collapse is self-similar for any value of theHamiltonian, but the vortex circulations must satisfy a strict relationship.Results may also shed a light on the formation of singularities in the SQGpartial differential equations, where the singularity is thought to be reachedonly in a self-similar way.

Collapse of generalized Euler and surface quasi-geostrophic point-vortices