Symmetry breaking and Hamiltonian bifurcation theory for three-dimensional instability of a vortex tube

1.从哈密顿谱理论的角度研究了椭圆涡旋和涡旋环的三维不稳定性。(1)纯剪切破坏了朗肯涡旋的SO(2)xO(2)对称性，使圆核变形为椭圆。我们已经成功地用贝塞尔函数明确地写出了无穷小的扰动。阐明了其与椭圆不稳定性的关系。(2)对于涡旋环，对称破缺摄动为曲率效应。开尔文涡环上的微小扰动以封闭形式表示出来。我们发现了方位角波数为2的开尔文波对之间存在参数共振的可能性。利用WKB方法进行了局部稳定性分析，表明高斯核上不稳定模的结构与开尔文涡环上不稳定模的结构有很大的不同。(3)利用与Z2xO(2)对称相关的等变矢量场，导出了纯剪切作用下涡管上几种不稳定模态振幅的弱非线性演化方程。当几十纳秒的准分子激光脉冲照射在共涂层基板上时，旋涡细丝立即产生，淬火和冻结。通过局部感应近似的稳定性分析估计时间尺度，我们从冻结图像的显微照片中提取了动态图像。(2)通过多次重复脉冲辐照，得到不稳定涡环。对于平面波的边带不稳定性，证明了在线性临界点附近，基于包络方程、基于振幅方程和基于二次不稳定性分析的方法都是等价的。对于Rayleigh-Benard对流，采用中心流形约简导出了五阶振幅方程，并在密度作为展开参数的温度二次依赖的简并点附近对其分岔进行了分析，明确了直角六边形、直角三角形和拼布被子模式作为一般正规形式方程的初级分岔解是不稳定的。少

1.Three-dimensional instability of an elliptic vortex and a vortex ring was investigated from the viewpoint of the Hamiltonian spectral theory.(1)A pure shear breaks the SO(2)xO(2)-symmetry of the Rankine vortex and deforms the circular core into an ellipse.We have succeeded in explicitly writing down the infinitesimal disturbances in terms of the Bessel functions.The relation with the elliptical instability is clarified.(2)For a vortex ring, symmetry-bresking perturbation is the curvature effect. The infinitesimal disturbances on Kelvin's vortex ring are written out in a closed form.We have found a possibility of parametric resonance between pairs of Kelvin waves whose azimuthal wavenumbers are separated by two.A local stability analysis is also made, using the WKB method It is clarified that the structure of unstable modes on a Gaussian core is very different from that on Kelvin's vortex ring.(3)We have derived weakly nonlinear evolution equations of amplitudes of several unstable mo … More des on a vortex tube subjected to a pure shear, with the aid of equivariant vector fields associated with Z2xO(2) symmetry.2.When a pulse of excimer laser of a few ten nanoseconds is irradiated on a Co-coated substrate, vortex filaments are created, quenched and frozen at once.By estimating the time scales from a stability analysis in the localized induction approximation, we have extracted a dynamical picture from the micrographs of the frozen pictures.(2)By repeating the irradiation, of pulses several times, unstable vortex rings are obtained.3.For the side-band instability of a plane wave, it is proven that, near the linear critical point, the methods based on envelope equations, on amplitude equations and on a secondary-instability analysis are all equivalent.4.For the Rayleigh-Benard convection, the fifth-order amplitude equations are derived, using the center-manifold reduction, and their bifurcation analysis is made near the degeneracy point with quadratic dependence on temperature of the density being an unfolding parameter It is clarified that right-hexagonal, right-triangle and patch-work quilt patterns are unstable as a primary bifurcation solution of the generic normal form equations. Less