Three-dimensional nonlinear stability theory of vortices from the view point of Hamiltonian dynamicalsysytems
| Project/Area Number |
16540345
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
FUKUMOTO Yasuhide Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30192727)
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| Co-Investigator(Kenkyū-buntansha) |
HATTORI Yuji Kyushu Institute of Technolog, Faculty of Engineering, Associate Professor, 工学部, 助教授 (70261469)
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| Project Period (FY) |
2004 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | Kelvin's vortex ring / curvature instability / Krein theory / weakly nonlinear instability / three-wave resonance / laser-matter interactions / helical vortex tube / energy balance of magnetic eddies / Dysonの方法 / ノーマルフォーム / 振幅方程式 / らせん渦 / ハミルトン的スペクトル理論 / 理想NHD流の渦界面ダイナミックス / 回転乱流のシェルモデル |
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| Research Abstract |
A new instability mechanism "curvature instability", originating from the curvature of vortex lines, is found for Kelvin's vortex ring. The eigenfunction is explicitly writing out and thereby asymptotic form of the growth rate at large wavelengths is derived. We have elucidated structure of the spectrum from the viewpoint of the Krein theory for Hamiltonian systems. By repeating irradiation of pulses of excimer laser on a Co-coated substrate, unstable vortex rings are created. By analyzing the micrographs of their frozen pictures. we identified four types of instability modes. We made a weakly nonlinear stability analysis of Kelvin's vortex ring. It is found from numerical analysis of the derived amplitude equations that the system exhibits a chaotic behavior. We also conducted a direct numerical simulation of a vortex ring, and clarified a detailed structure of the Widnall instability.
We exploited the Hamiltonian normal form, associated with the SO(2) x O(2)-symmetry, to select possib
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le form of the weakly nonlinear evolution equations of amplitude of the bending modes (azimuthal wavenumber m=1,-1) of Kelvin waves on an elliptically strained vortex tube. The coefficient of the amplitude equations are evaluated for non-rotating modes. The nonlinear terms make the elliptical instability saturate. However, it is pointed out that three-wave resonance of modes of m=3,4 and a bending mode derives a secondary instability.
By extending Dyson's technique to three dimensions, we developed a method of asymptotic expansions of the Biot-Savart integral, which takes account of the influence of finite core thickness on a vortex tube. This method is applied to calculation of velocity field around a helical vortex tube. In the neighborhood of the core, we cannot ignore the dipoles arranged on the tube center line which reflects the distribution of vorticity in the core.
A formulation of contour dynamics is given to the magnetohydrodynamic evolution of axisymmetric magnetic eddies. A family of exact solutions is found. Their significance in energy balance is clarified by both numerical implementation of the contour dynamics and a direct numerical simulation. Less
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Report
(3 results)
Research Products
(27 results)
