TY - JOUR

T1 - Elliptical instability of a vortex tube and drift current induced by it

AU - Fukumoto, Yasuhide

AU - Hirota, M.

PY - 2008/12/1

Y1 - 2008/12/1

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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U2 - 10.1088/0031-8949/2008/T132/014041

DO - 10.1088/0031-8949/2008/T132/014041

M3 - Conference article

AN - SCOPUS:70350633426

VL - T132

JO - Physica Scripta

JF - Physica Scripta

SN - 0031-8949

M1 - 014041

T2 - International Conference 'Turbulent Mixing and Beyond'

Y2 - 18 August 2007 through 26 August 2007

ER -