Three-dimensional nonlinear stability theory of vortices from the view point of Hamiltonian dynamicalsysytems

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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 … More 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