• DOI:10.1103/PhysRevE.98.023110
• Corpus ID: 51688376

Collapse of generalized Euler and surface quasigeostrophic point vortices.

@article{Badin2018CollapseOG,
  title={Collapse of generalized Euler and surface quasigeostrophic point vortices.},
  author={Gualtiero Badin and Anna M. Barry},
  journal={Physical review. E},
  year={2018},
  volume={98 2-1},
  pages={
          023110
        }
}

• G. Badin, Anna M. Barry
• Published 23 May 2018
• Physics
• Physical review. E

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.