Stability of relative equilibria of three vortices

被引:30
作者
Aref, Hassan [1 ,2 ]
机构
[1] Virginia Tech, Dept Engn Sci & Mech, Blacksburg, VA 24061 USA
[2] Tech Univ Denmark, Ctr Fluid Dynam, DK-2800 Lyngby, Denmark
基金
新加坡国家研究基金会;
关键词
eigenvalues and eigenfunctions; flow instability; pattern formation; vortices; 3-VORTEX MOTION; VORTEX;
D O I
10.1063/1.3216063
中图分类号
O3 [力学];
学科分类号
08 ; 0801 ;
摘要
Three point vortices on the unbounded plane have relative equilibria wherein the vortices either form an equilateral triangle or are collinear. While the stability analysis of the equilateral triangle configurations is straightforward, that of the collinear relative equilibria is considerably more involved. The only comprehensive analysis available in the literature, by Tavantzis and Ting [Phys. Fluids 31, 1392 (1988)], is not easy to follow nor is it very physically intuitive. The symmetry between the three vortices is lost in this analysis. A different analysis is given based on explicit formulas for the three eigenvalues determining the stability, including a new formula for the angular velocity of rotation of a collinear relative equilibrium. A graphical representation of the space of vortex circulations is introduced, and the resultants between various polynomials that enter the problem are used. This approach adds considerable transparency to the solution of the stability problem and provides more physical understanding. The main results are summarized in a diagram that gives both the stability and instability of the various collinear relative equilibria and their sense of rotation.
引用
收藏
页数:22
相关论文
共 50 条
[31]   Shallow-water vortex equilibria and their stability [J].
Plotka, H. ;
Dritschel, D. G. .
13TH EUROPEAN TURBULENCE CONFERENCE (ETC13): VORTICITY DYNAMICS, 2011, 318
[32]   Singular continuation of point vortex relative equilibria on the plane and sphere [J].
O'Neil, Kevin A. .
NONLINEARITY, 2013, 26 (03) :777-804
[33]   The effect of ageostrophy on the stability of vortices in a two-layer ocean [J].
Benilov, E. S. ;
Flanagan, J. D. .
OCEAN MODELLING, 2008, 23 (1-2) :49-58
[34]   Selective mode excitation in three-chained magnetic vortices [J].
Hasegawa, Norinobu ;
Sugimoto, Satoshi ;
Fujimori, Hiroaki ;
Kondou, Kouta ;
Niimi, Yasuhiro ;
Otani, YoshiChika .
APPLIED PHYSICS EXPRESS, 2015, 8 (06)
[35]   New Types of Three-Dimensional Vortices in the Heisenberg Model [J].
Borisov, A. B. ;
Dolgikh, D. V. .
PHYSICS OF METALS AND METALLOGRAPHY, 2021, 122 (05) :423-427
[36]   Something Old, Something New: Three Point Vortices on the Plane [J].
Stremler, Mark A. .
REGULAR & CHAOTIC DYNAMICS, 2021, 26 (05) :482-504
[37]   Line vortices in a three-dimensional ordered Josephson medium [J].
M. A. Zelikman .
Technical Physics, 2005, 50 :36-43
[38]   New Types of Three-Dimensional Vortices in the Heisenberg Model [J].
A. B. Borisov ;
D. V. Dolgikh .
Physics of Metals and Metallography, 2021, 122 :423-427
[39]   Morse Theory and Relative Equilibria in the Planar n-Vortex Problem [J].
Roberts, Gareth E. .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2018, 228 (01) :209-236
[40]   Interaction and pinning of plane vortices in a three-dimensional Josephson medium and possible distances between two isolated vortices [J].
M. A. Zelikman .
Technical Physics, 2001, 46 :831-839