Collapse of generalized Euler and surface quasigeostrophic point vortices

Area of Science:

• Fluid Dynamics and Mathematical Physics
• Analysis of Partial Differential Equations

Background:

• Generalized Euler equations involve fractional Laplacian relations.
• Surface quasigeostrophic (SQG) equations are a key focus, with debated finite-time singularity existence.
• Point-vortex models offer a framework for studying these dynamics.

Purpose of the Study:

• To analyze point-vortex models for generalized Euler equations, particularly SQG.
• To investigate the nature of solution collapse in three-point-vortex systems.
• To explore the conditions leading to self-similar versus non-self-similar collapse in SQG.

Main Methods:

• Formulation of point-vortex dynamics using Nambu mechanics and a noncanonical bracket.
• Geometrical interpretation of trajectories via intersections of Hamiltonian and Casimir level sets.
• Analysis of the three-point-vortex model to study solution collapse.

Main Results:

• SQG solutions exhibit both self-similar and non-self-similar collapse.
• Self-similar collapse occurs when the Hamiltonian is zero; non-self-similar collapse occurs for non-zero Hamiltonians.
• Collapse is permitted for a wide range of vortex circulations, unlike classical models.

Conclusions:

• The findings challenge classical point-vortex models regarding collapse conditions and self-similarity.
• Results provide potential insights into the formation mechanisms of singularities in SQG partial differential equations.
• The study highlights the diverse collapse behaviors possible within the SQG framework.