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Barotropic instability of shear flows
Studies in Applied Mathematics ( IF 2.7 ) Pub Date : 2020-02-17 , DOI: 10.1111/sapm.12297
Zhiwu Lin 1 , Jincheng Yang 2 , Hao Zhu 3, 4
Affiliation  

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and find sharp stability boundary in the whole parameter space, which correct previous results in the fluid literature. The addition of the Coriolis force is found to bring some fundamental changes to the stability of shear flows. Moreover, we study the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping near the shear flows. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian PDEs. The last one is to study the singular and non-resonant neutral modes by using hypergeometric functions and singular Sturm-Liouville theory.

中文翻译:

剪切流的正压不稳定性

我们考虑具有科里奥利效应的不可压缩流体的剪切流的正压不稳定性。对于一类剪切流,我们开发了一种新方法来寻找急剧稳定条件。我们详细研究了具有 Sinus 剖面的流动,并在整个参数空间中找到了尖锐的稳定性边界,这纠正了流体文献中先前的结果。发现科里奥利力的加入给剪切流的稳定性带来了一些根本性的变化。此外,我们研究了非平凡行波解的分岔和剪切流附近的线性无粘阻尼。我们证明的第一个要素是对中性模式的仔细分类。第二种方法是以哈密顿量形式写出线性化流体方程,然后对一般哈密顿偏微分方程使用不稳定性指数理论。
更新日期:2020-02-17
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