The stability and energetics of corotating uniform vortices. (English) Zbl 0574.76026

Equilibrium shapes of two-dimensional rotating configurations of uniform vortices are numerically calculated for two to eight corotating vortices. Additionally, a perturbation series is developed which approximately describes the vortex shapes. The equilibrium configurations are subjected to a linear stability analysis. This analysis both confirms existing results regarding point vortices and shows that finite vortices may destabilize via a new form of instability derived from boundary deformations. Finally, we examine the energetics of the equilibrium configurations. We introduce a new energy quantity called ’excess energy’, which is particularly useful in understanding the constraints on the evolution of unstable near-equilibrium configurations. This theory offers a first glance at nonlinear stability. As an example, the theory explains some features of the merger of two vortices.


76B47 Vortex flows for incompressible inviscid fluids
76M99 Basic methods in fluid mechanics
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[1] Dritschel, J. Fluid Mech. 61 pp 219– (1985)
[2] DOI: 10.1017/S0022112073000686 · Zbl 0266.76039
[3] Arnol’D, Dokl. Akad. Nauk SSSR 162 pp 975– (1965)
[4] Wu, Univ. Pittsburgh Tech. Rep. 17 pp 497– (1982)
[5] Ting, J. Fluid Mech. 17 pp 497– (1983)
[6] DOI: 10.1175/1520-0469(1978)035 2.0.CO;2
[7] DOI: 10.1063/1.862935
[8] Saffman, J. Fluid Mech. 117 pp 171– (1982)
[9] DOI: 10.1017/S0022112081002656__S0022112081002656
[10] DOI: 10.1063/1.863907 · Zbl 0489.76033
[11] DOI: 10.1063/1.1693564
[12] DOI: 10.1017/S0022112080000559 · Zbl 0473.76034
[13] Love, Proc. Lond. Math. Soc. 35 pp 18– (1893)
[14] Havelock, Phil. Mag. 11 pp 617– (1931)
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