Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

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 s...

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Autores principales:	Yuma Hirakui, Takahiro Yajima
Formato:	article
Lenguaje:	EN
Publicado:	Hindawi Limited 2021
Materias:	Physics QC1-999
Acceso en línea:	https://doaj.org/article/1e1931bbdbb64990840194a822be3ace
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spelling	oai:doaj.org-article:1e1931bbdbb64990840194a822be3ace2021-11-08T02:35:41ZGeometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field1687-913910.1155/2021/9979529https://doaj.org/article/1e1931bbdbb64990840194a822be3ace2021-01-01T00:00:00Zhttp://dx.doi.org/10.1155/2021/9979529https://doaj.org/toc/1687-9139In 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.Yuma HirakuiTakahiro YajimaHindawi LimitedarticlePhysicsQC1-999ENAdvances in Mathematical Physics, Vol 2021 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Physics QC1-999
spellingShingle	Physics QC1-999 Yuma Hirakui Takahiro Yajima Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field
description	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.
format	article
author	Yuma Hirakui Takahiro Yajima
author_facet	Yuma Hirakui Takahiro Yajima
author_sort	Yuma Hirakui
title	Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field
title_short	Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field
title_full	Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field
title_fullStr	Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field
title_full_unstemmed	Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field
title_sort	geometrical classification of self-similar motion of two-dimensional three point vortex system by deviation curvature on jacobi field
publisher	Hindawi Limited
publishDate	2021
url	https://doaj.org/article/1e1931bbdbb64990840194a822be3ace
work_keys_str_mv	AT yumahirakui geometricalclassificationofselfsimilarmotionoftwodimensionalthreepointvortexsystembydeviationcurvatureonjacobifield AT takahiroyajima geometricalclassificationofselfsimilarmotionoftwodimensionalthreepointvortexsystembydeviationcurvatureonjacobifield
_version_	1718443164346351616

Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

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