Elliptical instability of a vortex tube and drift current induced by it

Research output: Contribution to journalConference article

8 Citations (Scopus)

Abstract

We revisit, with some extension to the nonlinear regime, the Moore-Saffman-Tsai-Widnall instability, which is the three-dimensional instability of a straight vortex tube elliptically strained by a pure shear flow. A circular cylindrical vortex tube supports neutrally stable three-dimensional waves called Kelvin waves. A pure shear breaks the circular symmetry, causing a parametric resonance between two Kelvin waves whose azimuthal wavenumbers are separated by two. The eigenvalues and functions are written out in full in terms of the Bessel and the modified Bessel functions, and so is the growth rate. That this instability has a link with the elliptical instability in the short-wave limit is established in this study. This result is studied in the light of Krein's theory for Hamiltonian spectra. We develop a Lagrangian (particle) formulation that makes feasible an efficient calculation of iso-vortical disturbances up to second order in amplitude and thereby facilitates calculation of the energy of Kelvin waves. It is found that a Kelvin wave induces a drift current along the vortex tube.

Original languageEnglish
Article number014041
JournalPhysica Scripta T
VolumeT132
DOIs
Publication statusPublished - Dec 1 2008
EventInternational Conference 'Turbulent Mixing and Beyond' - Trieste, Italy
Duration: Aug 18 2007Aug 26 2007

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vortex tubes
Kelvin waves
Kelvin wave
vortex
Vortex
Tube
Kelvin
Bessel functions
shear flow
eigenvalue
symmetry
Modified Bessel Functions
Parametric Resonance
Three-dimensional
eigenvalues
disturbances
Friedrich Wilhelm Bessel
Shear Flow
shear
formulations

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics
  • Mathematical Physics
  • Condensed Matter Physics

Cite this

Elliptical instability of a vortex tube and drift current induced by it. / Fukumoto, Yasuhide; Hirota, M.

In: Physica Scripta T, Vol. T132, 014041, 01.12.2008.

Research output: Contribution to journalConference article

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AU - Hirota, M.

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N2 - We revisit, with some extension to the nonlinear regime, the Moore-Saffman-Tsai-Widnall instability, which is the three-dimensional instability of a straight vortex tube elliptically strained by a pure shear flow. A circular cylindrical vortex tube supports neutrally stable three-dimensional waves called Kelvin waves. A pure shear breaks the circular symmetry, causing a parametric resonance between two Kelvin waves whose azimuthal wavenumbers are separated by two. The eigenvalues and functions are written out in full in terms of the Bessel and the modified Bessel functions, and so is the growth rate. That this instability has a link with the elliptical instability in the short-wave limit is established in this study. This result is studied in the light of Krein's theory for Hamiltonian spectra. We develop a Lagrangian (particle) formulation that makes feasible an efficient calculation of iso-vortical disturbances up to second order in amplitude and thereby facilitates calculation of the energy of Kelvin waves. It is found that a Kelvin wave induces a drift current along the vortex tube.

AB - We revisit, with some extension to the nonlinear regime, the Moore-Saffman-Tsai-Widnall instability, which is the three-dimensional instability of a straight vortex tube elliptically strained by a pure shear flow. A circular cylindrical vortex tube supports neutrally stable three-dimensional waves called Kelvin waves. A pure shear breaks the circular symmetry, causing a parametric resonance between two Kelvin waves whose azimuthal wavenumbers are separated by two. The eigenvalues and functions are written out in full in terms of the Bessel and the modified Bessel functions, and so is the growth rate. That this instability has a link with the elliptical instability in the short-wave limit is established in this study. This result is studied in the light of Krein's theory for Hamiltonian spectra. We develop a Lagrangian (particle) formulation that makes feasible an efficient calculation of iso-vortical disturbances up to second order in amplitude and thereby facilitates calculation of the energy of Kelvin waves. It is found that a Kelvin wave induces a drift current along the vortex tube.

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