1. Elliptical instability of a vortex tube and drift current induced by it Turbulent Mixing and Beyond International Conference International Conference August 18-26, 2007 (Aug. 24) August 18-26, 2007 (Aug. 24) The Abdus Salam International Centre for Theoretical Physics (ICTP) Trieste, Italy Trieste, Italy Yasuhide Fukumoto and Makoto Hirota Graduate School of Mathematics, Kyushu University, Fukuoka, Japan

2. Aircraft trailing vortices (Higuchi 1993) Cessna Citation IV from B25

3. Instability of trailing vortices (Crow 1970) from Van Dyke: An Album of Fluid Motion B-47

4. Short-wave Instability of trailing vortices (Leweke & Williamson ’98)

5. Axial flow in a vortex ring Naitoh, Fukuda, Gotoh, Yamada & Nakajima (’02) cf. Maxworthy (’77)

6. Contents 1. Introduction 2. Influence of a pure shear on Kelvin waves “A global stability of the Rankine vortex to three-dimensional disturbances" Moore & Saffman ('75), Tsai & Widnall ('76) Eloy & Le Dizès ('01), Y. F. ('03) elliptical instability (local stability) cf. vortex ring: Hattori & Y. F. ('03), Y. F. & Hattori ('05) 3. Energy of Kelvin waves Cairn’s formula (’79), Y. F. ('03), Hirota & Y. F. ('07) for continuous spectra 4. Weakly nonlinear corrections to Kelvin waves kinematically accessible variations (= isovortical perturbations) → drift current

7. Elliptically strained vortex

8. Expand infinitesimal disturbance in Suppose that the core boundary is disturbed to the linearized Euler equations

9. Example of a Kelvin wave m=4

10. Dispersion relation of Kelvin waves m=-1 (solid lines) and m=1 (dashed lines)

11. Equations for disturbance of

12. Solution of disturbance of For the m wave, we find, from the Euler equations, and (radial wave numbers) Disturbance field is explicitely written out.

13. Growth rate of helical waves (m=±1) Instability occurs at every intersection points of dispersion curves of (m, m+2) waves (m, m+2) waves ???

14. Krein’s theory of Hamiltonian spectra Spectra of a finte-dimensional Hamilton system

15. Energy of a Kelvin wave (averaged) Excess energy for generating a Kelvin wave base flow disturbance Kelvin wave stationary component ??? (no strain)

16. Carins’ formula (Carins ‘79)

17. Energy of a helical wave (m=1)

18. Energy signature of helical waves (m=±1) m=-1 (solid lines) and m=1 (dashed lines)

19. Difficulty in Eulerian treatment Excess energy base flow disturbance Complicated calculation would be required for

20. Steady Euler flows iso-vortical sheets Kinematically accessible variation (= preservation of circulation) (= preservation of circulation) Theorem (Kelvin, Arnold ’66) A steady Euler flow is a coditional extremum of energy H on an iso-vortical sheet coditional extremum of energy H on an iso-vortical sheet (= w.r.t. kinematically accessible variations). (= w.r.t. kinematically accessible variations).

21. Variational principle for stationary vortical region ☆ Volume preserving displacement of fluid particles: ☆ Iso-vorticity: Then. using

22. First and second variations The first variation Given which satisfies Then is a solution of The secobd variation Further, given which satisfies Then is a solution of ( : projection operator )

23. Wave energy in terms of iso-vortical disturbance Excess energy by Arnold’s theorem It is proved that and that and that is the wave-energy does not contribute to are linear disturbances!!

24. Drift current Take the average over a long time For the Rankine vortex Substitute the Kelvin wave There is no contribution from There is no contribution from For 2D wave, For 2D wave, genuinly 3D effect !!

25. Drift current caused by Kelvin waves Displacement vector of m wave Flow-flux, of m wave, in the axial direction

26. Axial flow-flux of buldge wave (m=0), elliptic wave (m=2) m=0 (dashed lines) and m=2 (solid lines) Dispersion relation For the principal mode, 1.242, -1.242 1.242, -1.242 3.370, -0.2443 3.370, -0.2443 7.058, -0.09046 7.058, -0.09046 8.882, -0.06828 8.882, -0.06828 12.521, -0.04564 12.521, -0.04564

27. Axial flow-flux of a helical wave (m=1) For the principal mode （＝ stationary) 2.505 2.505 4.349 4.349

28. Axial current of staionary helical modes For stationary modes time average is not necessary : Given,

29. Summary 1.Tsai & Widnall ('76) is simplified; Disturbance field and growth rate are written out in terms of the Bessel and modified Bessel functions. 2.Energetics: Energy of the Kelvin waves is calculated by adapting Cairns’ formula (= black box) consistent with Krein’s theory Linear stability of an elliptic vortex, a straight vortex tube subject to a pure shear, to three-dimensional disturbances is calculated. This is a parametric resonance instability between two Kelvin waves caused by a perturbation breaking S-symmetry of the circular core. Modification of mean field at 2 nd order ： 3. Lagrangian approach: Energy of the Kelvin waves is calculated by restricting disturbance to kinematically accessible field linear perturbation is sufficient to calcilate energy, quadratic in amplitude! linear perturbation is sufficient to calcilate energy, quadratic in amplitude! 4. Axial current: For the Rankine vortex, 2 nd-order drift current includes not only azimuthal but also axial component