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

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.

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Cited by in corpus (7)

• Regularization by noise for the point vortex model of mSQG equations
• Velocity statistics for point vortices in the local α-models of turbulence
• Hölder regularity for collapses of point vortices
• Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts
• Long time localization of modified surface quasi-geostrophic equations
• The Motion of a Point Vortex in Multiply Connected Polygonal Domains
• On the Dynamics of Point Vortices with Positive Intensities collapsing with the boundary