K\'arm\'an vortex street for the generalized surface quasi-geostrophic equation

TL;DR

This paper investigates the existence and properties of periodic traveling-wave solutions, known as von Kármán vortex streets, for the generalized surface quasi-geostrophic equation, revealing their structure and relationships between vortex parameters.

Contribution

It introduces a variational approach to construct $C^1$ vortex street solutions for the gSQG equation, including the Euler case, and explores vortex size and street structure relationships.

Findings

Existence of $C^1$ periodic vortex street solutions.

Relationship between vortex size, speed, and street structure.

Regularization of Kármán point vortex street.

Abstract

We are concerned with the existence of periodic travelling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation(including incompressible Euler equation), known as von K\'arm\'an vortex street. These solutions are of $C^1$ type, and are obtained by studying a semilinear problem on an infinite strip whose width equals to the period. By a variational characterization of solutions, we also show the relationship between vortex size, travelling speed and street structure. In particular, the vortices with positive and negative intensity have equal or unequal scaling size in our construction, which constitutes the regularization for K\'arm\'an point vortex street.