Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations.
Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations.
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分层和旋转流体中涡对的锯齿状不稳定性。
DOI：
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发表时间：
2010
影响因子：
3.7
通讯作者：
P. Billant
中科院分区：
文献类型：
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作者：
P. Billant
在分层旋转流体中，成对的柱状垂直涡受到三维弯曲不稳定性的影响，这种不稳定性在准地转极限中被称为Z字形不稳定性或高柱不稳定性。本文提出了一个一般的渐近理论，这些不稳定性。本文导出了在长垂直波长和两涡分离时，即涡核半径R小于涡分离距离B时，分层旋转流体中各涡柱应变与慢弯曲波相互作用的方程。这些方程与均匀流体中涡丝的方程具有相同的形式，只是互感函数和自感函数的表达式不同。一个关键的区别是，当流体强烈分层时，自感函数的符号与均匀流体相反：|Max| < N（其中N是Brunt-Väisälä频率，max是涡旋的最大角速度），适用于任何涡旋轮廓和行星自转的大小。物理上，这意味着当流体与均匀流体相比分层旋转时，涡流的缓慢弯曲波以与涡流内部的流动相同的方向旋转。当分层较弱时，即|Max|> N时，自感函数是复杂的，因为在涡的角速度等于Brunt-Väisälä频率的径向位置处，弯曲波被粘性临界层阻尼。以前的理论只适用于强分层的非旋转流体，而现在的理论则适用于任何行星的旋转速率，并且当应变小于Brunt-Väisälä频率时：Γ/（2πb2）<$N，其中Γ是涡旋环流。由于应变很小，这个条件在很宽的分层范围内都能满足：从弱分层到强分层的流体。该理论进一步推广到任何基本流的分层和旋转流体中的任意数量的涡。当无临界层时，粘性和扩散效应也被考虑在雷诺数的领先顺序。在第2部分（Billant等人，流体力学杂志，2010，doi：10.1017/S 002211201000282 X），将使用本理论研究涡对的稳定性，并且将显示预测与直接数值稳定性分析的结果非常一致。与均匀流体相比，分层和旋转流体中涡旋对的锯齿形不稳定性和独特的稳定性特性的存在将被证明源于自感函数的符号反转。
In stratified and rotating fluids, pairs of columnar vertical vortices are subjected to three-dimensional bending instabilities known as the zigzag instability or as the tall-column instability in the quasi-geostrophic limit. This paper presents a general asymptotic theory for these instabilities. The equations governing the interactions between the strain and the slow bending waves of each vortex column in stratified and rotating fluids are derived for long vertical wavelength and when the two vortices are well separated, i.e. when the radii R of the vortex cores are small compared to the vortex separation distance b. These equations have the same form as those obtained for vortex filaments in homogeneous fluids except that the expressions of the mutual-induction and self-induction functions are different. A key difference is that the sign of the self-induction function is reversed compared to homogeneous fluids when the fluid is strongly stratified: |max| < N (where N is the Brunt–Väisälä frequency and max the maximum angular velocity of the vortex) for any vortex profile and magnitude of the planetary rotation. Physically, this means that slow bending waves of a vortex rotate in the same direction as the flow inside the vortex when the fluid is stratified-rotating in contrast to homogeneous fluids. When the stratification is weaker, i.e. |max| > N, the self-induction function is complex because the bending waves are damped by a viscous critical layer at the radial location where the angular velocity of the vortex is equal to the Brunt–Väisälä frequency. In contrast to previous theories, which apply only to strongly stratified non-rotating fluids, the present theory is valid for any planetary rotation rate and when the strain is smaller than the Brunt–Väisälä frequency: Γ/(2πb2) ≪ N, where Γ is the vortex circulation. Since the strain is small, this condition is met across a wide range of stratification: from weakly to strongly stratified fluids. The theory is further generalized formally to any basic flow made of an arbitrary number of vortices in stratified and rotating fluids. Viscous and diffusive effects are also taken into account at leading order in Reynolds number when there is no critical layer. In Part 2 (Billant et al., J. Fluid Mech., 2010, doi:10.1017/S002211201000282X), the stability of vortex pairs will be investigated using the present theory and the predictions will be shown to be in very good agreement with the results of direct numerical stability analyses. The existence of the zigzag instability and the distinctive stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids will be demonstrated to originate from the sign reversal of the self-induction function.