### Abstract

We revisit the Moore-Saffman-Tsai-Widnall instability, a parametric resonance between left- and right-handed bending waves of infinitesimal amplitude, on the Rankine vortex strained by a weak pure shear flow. The results of Tsai & Widnall (1976) and Eloy & Le Dizès (2001), as generalized to all pairs of Kelvin waves whose azimuthal wavenumbers m are separated by 2, are simplified by providing an explicit solution of the linearized Euler equations for the disturbance flow field. Given the wavenumber k_{0} and the frequency ω_{0} of an intersection point of dispersion curves, the growth rate is expressible solely in terms of the modified Bessel functions, and so is the unstable wavenumber range. Every intersection leads to instability. Most of the intersections correspond to weak instability that vanishes in the short-wave limit, and dominant modes are restricted to particular intersections. For helical waves m = ±1, the growth rate of non-rotating waves is far larger than that of rotating waves. The wavenumber width of stationary instability bands broadens linearly in k_{0}, while that of rotating instability bands is bounded. The growth rate of the stationary instability takes, in the long-wavelength limit, the value of ε/2 for the two-dimensional displacement instability, and, in the short-wavelength limit, the value Of 9ε/16 for the elliptical instability, being larger at large but finite values of k_{0}. Here ε is the strength of shear near the core centre. For resonance between higher azimuthal wavenumbers m and m + 2, the same limiting value is approached as k_{0} → ∞, along sequences of specific crossing points whose frequency rapidly converges to m + 1, in two ways, from above for a fixed m and from below for m → ∞. The energy of the Kelvin waves is calculated by invoking Cairns' formula. The instability result is compatible with Krein's theory for Hamiltonian spectra.

Original language | English |
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Pages (from-to) | 287-318 |

Number of pages | 32 |

Journal | Journal of Fluid Mechanics |

Issue number | 493 |

DOIs | |

Publication status | Published - Oct 25 2003 |

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### All Science Journal Classification (ASJC) codes

- Mechanics of Materials
- Computational Mechanics
- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

**The three-dimensional instability of a strained vortex tube revisited.** / Fukumoto, Yasuhide.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - The three-dimensional instability of a strained vortex tube revisited

AU - Fukumoto, Yasuhide

PY - 2003/10/25

Y1 - 2003/10/25

N2 - We revisit the Moore-Saffman-Tsai-Widnall instability, a parametric resonance between left- and right-handed bending waves of infinitesimal amplitude, on the Rankine vortex strained by a weak pure shear flow. The results of Tsai & Widnall (1976) and Eloy & Le Dizès (2001), as generalized to all pairs of Kelvin waves whose azimuthal wavenumbers m are separated by 2, are simplified by providing an explicit solution of the linearized Euler equations for the disturbance flow field. Given the wavenumber k0 and the frequency ω0 of an intersection point of dispersion curves, the growth rate is expressible solely in terms of the modified Bessel functions, and so is the unstable wavenumber range. Every intersection leads to instability. Most of the intersections correspond to weak instability that vanishes in the short-wave limit, and dominant modes are restricted to particular intersections. For helical waves m = ±1, the growth rate of non-rotating waves is far larger than that of rotating waves. The wavenumber width of stationary instability bands broadens linearly in k0, while that of rotating instability bands is bounded. The growth rate of the stationary instability takes, in the long-wavelength limit, the value of ε/2 for the two-dimensional displacement instability, and, in the short-wavelength limit, the value Of 9ε/16 for the elliptical instability, being larger at large but finite values of k0. Here ε is the strength of shear near the core centre. For resonance between higher azimuthal wavenumbers m and m + 2, the same limiting value is approached as k0 → ∞, along sequences of specific crossing points whose frequency rapidly converges to m + 1, in two ways, from above for a fixed m and from below for m → ∞. The energy of the Kelvin waves is calculated by invoking Cairns' formula. The instability result is compatible with Krein's theory for Hamiltonian spectra.

AB - We revisit the Moore-Saffman-Tsai-Widnall instability, a parametric resonance between left- and right-handed bending waves of infinitesimal amplitude, on the Rankine vortex strained by a weak pure shear flow. The results of Tsai & Widnall (1976) and Eloy & Le Dizès (2001), as generalized to all pairs of Kelvin waves whose azimuthal wavenumbers m are separated by 2, are simplified by providing an explicit solution of the linearized Euler equations for the disturbance flow field. Given the wavenumber k0 and the frequency ω0 of an intersection point of dispersion curves, the growth rate is expressible solely in terms of the modified Bessel functions, and so is the unstable wavenumber range. Every intersection leads to instability. Most of the intersections correspond to weak instability that vanishes in the short-wave limit, and dominant modes are restricted to particular intersections. For helical waves m = ±1, the growth rate of non-rotating waves is far larger than that of rotating waves. The wavenumber width of stationary instability bands broadens linearly in k0, while that of rotating instability bands is bounded. The growth rate of the stationary instability takes, in the long-wavelength limit, the value of ε/2 for the two-dimensional displacement instability, and, in the short-wavelength limit, the value Of 9ε/16 for the elliptical instability, being larger at large but finite values of k0. Here ε is the strength of shear near the core centre. For resonance between higher azimuthal wavenumbers m and m + 2, the same limiting value is approached as k0 → ∞, along sequences of specific crossing points whose frequency rapidly converges to m + 1, in two ways, from above for a fixed m and from below for m → ∞. The energy of the Kelvin waves is calculated by invoking Cairns' formula. The instability result is compatible with Krein's theory for Hamiltonian spectra.

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U2 - 10.1017/S0022112003006025

DO - 10.1017/S0022112003006025

M3 - Article

SP - 287

EP - 318

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

IS - 493

ER -