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

1.从Hamilton谱理论的角度研究了椭圆涡和涡环的三维不稳定性。(1)A纯剪切破坏了Rankine涡旋的SO（2）xO（2）对称性，使圆核变形为椭圆，我们成功地用Bessel函数明确地写出了微扰，阐明了微扰与椭圆不稳定性的关系。（2）对于涡环，曲率扰动是涡环的曲率效应。本文将Kelvin涡环上的无穷小扰动写成封闭形式，发现了方位波数为2的Kelvin波对之间发生参激共振的可能性，并用WKB方法作了局部稳定性分析，阐明了高斯核上的不稳定模的结构与Kelvin涡环上的不稳定模的结构有很大的不同。(3)We导出了几种不稳定波振幅的弱非线性演化方程 ...更多信息 利用Z_2xO（2）的等变矢量场，对纯剪切作用下的涡流管进行了数值模拟. 2.当几十纳秒的准分子激光脉冲照射在镀钴基片上时，涡流丝立即产生、猝灭和冻结.通过局域诱导近似下的稳定性分析，估计了时间尺度，我们已经从冻结图像的显微图像中提取出了动态图像。(2)By 3.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、振幅方程和二次不稳定性分析的方法是等价的; 4.对于Rayleigh-Benard对流，利用中心流形约化，导出了五阶振幅方程，并以密度对温度的二次依赖性为展开参数，在简并点附近进行了分支分析。阐明了直角六边形、直角三角形和拼缝绗缝图案作为一般规范形方程的初始分支解是不稳定的。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