Linear and nonlinear instability of a vortex ring

Yasuhide Fukumoto, Yuji Hattori

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


A new linear instability mechanism of curvature origin is established for a vortex ring. The curvature effect reduces O(2) × SO(2) symmetry of a circularcylindrical tube to O(2), and fuels a pair of Kelvin waves whose azimuthal wavenumbers on the core are separated by one. For Kelvin's vortex ring, the growth rate and eigenfunctions are written out in closed form. In the inviscid case, the curvature effect dominates over the elliptically straining effect, but the former suffers from enhanced viscous damping. There are numerous excitable modes. As a first step toward an understanding of the route to a matured stage, we derive equations for weakly nonlinear evolution of amplitudes of the curvature instability. Our direct numerical simulation successfully captures the elliptical instability.

Original languageEnglish
Title of host publicationIUTAM Symposium on Elementary Vortices and Coherent Structures
Subtitle of host publicationSignificance in Turbulence Dynamicsa
PublisherKluwer Academic Publishers
Number of pages12
ISBN (Print)9781402041808
Publication statusPublished - 2006

Publication series

NameFluid Mechanics and its Applications
ISSN (Print)0926-5112


  • Curvature instability
  • Hamiltonian spectra
  • Normal form
  • Vortex ring
  • Weakly nonlinear stability


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