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#### Collapse of generalized Euler and surface quasigeostrophic point vortices

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##### Citation

Badin, G., & Barry, A. M. (2018). Collapse of generalized Euler and surface quasigeostrophic
point vortices.* Physical Review E,* *98*: 023110. doi:10.1103/PhysRevE.98.023110.

Cite as: http://hdl.handle.net/21.11116/0000-0002-1484-2

##### Abstract

Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.