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

TL;DR

This paper investigates the collapse behavior of point vortices in generalized Euler and SQG equations, revealing conditions for self-similar and non-self-similar singularity formation, which differ from classical models.

Contribution

It introduces a Nambu dynamics-based formulation for point vortex trajectories in generalized Euler and SQG models, analyzing collapse scenarios and their implications for singularity formation.

Findings

Collapse can be self-similar or non-self-similar in SQG vortices.

Self-similar collapse occurs only when the Hamiltonian is zero.

Collapse is possible for any circulation within a certain interval.

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 streamfunction. Special focus is given to the case of the surface quasi-geostrophic (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 non-zero 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.