Development of Lagrangian hydrodynamics for three-dimensional nonlinear instability of vortices

嵌入在无粘不可压缩流体中的圆柱形涡是中性稳定的，但可以被破坏平衡的扰动所破坏。欧拉方法在计算波浪非线性诱导的平均流时存在困难。我们已经开发了一种方法，利用拉格朗日变量计算明确的平均流量，从而所有的振幅方程的系数被确定为三阶振幅。通过对因变量进行归一化处理，将这些方程转化为正则的哈密顿方程。此外，我们利用WKB方法研究了短波长稳定性，以补充简正模分析。

A circular-cylindrical vortex embedded in an inviscid incompressible fluid is neutrally stable, but can be destabilized by symmetry-breaking perturbations. The Eulerian treatment has difficulty in calculating the mean flow nonlinearly induced by waves. We have developed a method of exploiting the Lagrangian variables for calculating unambiguously the mean flow, whereby all the coefficients of the amplitude equations were determined to third order in amplitude. These equations were converted into canonical Hamltonian equations by normalizing dependent variables. Besides, we investigated the short-wavelength stability by use of the WKB method, to complement the normal-mode analyses.