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

在开尔文涡旋环中发现了一种新的不稳定机制“曲率不稳定”，它起源于涡旋线的曲率。显式地写出了特征函数，从而导出了大波长下增长率的渐近形式。本文从Krein理论的角度对哈密顿系统的谱结构进行了阐述。准分子激光脉冲在共涂基板上反复照射，形成不稳定的涡环。通过分析他们冰冻照片的显微照片。我们确定了四种不稳定模式。对开尔文涡旋环进行了弱非线性稳定性分析。通过对所导出的振幅方程的数值分析发现，系统表现为混沌行为。我们还进行了旋涡环的直接数值模拟，明确了Widnall不稳定性的详细结构。我们利用与SO(2) x O(2)-对称相关的哈密顿标准形式，选择了椭圆应变涡管上开尔文波弯曲模（方位角波数m=1,-1）振幅的弱非线性演化方程的可能形式。对非旋转模态下振幅方程的系数进行了计算。非线性项使椭圆不稳定性趋于饱和。然而，指出m=3,4模态和弯曲模态的三波共振导致二次失稳。通过将Dyson技术推广到三维空间，我们提出了一种考虑涡流管有限芯厚影响的Biot-Savart积分的渐近展开方法。将该方法应用于螺旋涡管的速度场计算。在核心附近，我们不能忽略排列在管中心线上的偶极子，它们反映了核心内涡度的分布。给出了轴对称磁涡流磁流体动力学演化的轮廓动力学公式。得到了一组精确解。通过轮廓动力学的数值实现和直接数值模拟，阐明了它们在能量平衡中的意义。少

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