Geometrical classification of self-similar motion of two-dimensional three point vortex system by deviation curvature on Jacobi field
type: Item
instance of: scholarly article
title: Geometrical classification of self-similar motion of two-dimensional three point vortex system by deviation curvature on Jacobi field
zbMATH Open document ID: 1483.37070
DOI: 10.1155/2021/9979529
author: Yuma Hirakui
author: Takahiro Yajima
published in: Advances in Mathematical Physics
publication date: 2021-11-25T00:00:00Z
review text: Summary: In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.
Mathematics Subject Classification ID: 37J25
Mathematics Subject Classification ID: 37J39
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 70G45
zbMATH DE Number: 7433526
zbMATH Keywords: point vortex system
zbMATH Keywords: deviation curvatures
zbMATH Keywords: Jacobi field
Wikidata QID: Q114069814
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2021/9979529
OpenAlex ID: W3208931435
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description: scientific article; zbMATH DE number 7433526