Collapse and Burst of generalized Surface Quasi-Geostrophic point Vortices

TL;DR

This work studies burst and collapse phenomena in generalized Surface Quasi-Geostrophic point-vortex dynamics. It extends a known Euler-based framework to $\alpha\neq 2$ by isolating a self-similar 3-vortex core and analyzing its stability under external forcing, then uses a fixed-point approach to lift stability to an $N$-vortex configuration, yielding bursts for small times. The main theoretical contribution is the verification of Hypothesis $\text{A}$ for $\alpha=1$ (SQG) and supporting numerical evidence that the hypothesis holds for a range $(\alpha_-,\alpha_+)$ around $[1,2]$, enabling explicit constructions of bursts and collapses. This provides a mechanism to generate and control singular vortex configurations in gSQG dynamics and connects stability properties of self-similar solutions to the occurrence of finite-time singular events.

Abstract

We consider the generalized Surface Quasi-Geostrophic point vortices dynamics, and identify a sufficient condition implying existence of bursts out of (and collapses into) any given initial configuration of vortices. The condition is related to the stability of the linearized dynamics around three vortices evolving in a self-similar fashion.

Figures (1)

• Figure 1: Values of $x_+(\alpha)$ (red) and $x_-(\alpha)$ (blue) between which the condition \ref{['eq:otherconditionmu']} implying the stability required by \ref{['hyp:A']} is satisfied. The red inset of the first plot is enlarged in the second plot.

Theorems & Definitions (12)

• Definition 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Lemma 2.2
• Proposition 2.3
• proof : Proof of \ref{['prop:existenceexternal']}
• Remark 2.4
• Proposition 2.5
• proof : Proof of \ref{['thm:Nvortices']}