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筒形涡旋是中性稳定的，但可以通过对称破缺扰动而不稳定。用欧拉法计算非线性波浪引起的平均流有困难。我们开发了一种利用拉格朗日变量来明确计算平均流量的方法，其中振幅方程的所有系数都确定为振幅的三阶。通过对因变量进行归一化，将这些方程转化为标准哈密顿方程。此外，我们还利用WKB方法对其短波稳定性进行了研究，以补充正态分析。

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.