Normal Mode Decomposition and Dispersive and Nonlinear Mixing in Stratified Fluids
Normal Mode Decomposition and Dispersive and Nonlinear Mixing in Stratified Fluids
复制标题
分层流体中的简正模态分解以及色散和非线性混合
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
J. Saut
中科院分区：
文献类型：
--
作者：
B. Desjardins;D. Lannes;J. Saut
在分析层化海洋中内波传播的基础上，本文考虑了平面条带中密度变的不可压缩欧拉方程，并研究了密度垂直稳定层结对应的静力平衡扰动的演化。我们证明了方程在这种构型下的局部适定性，并提供了其线性逼近的详细研究。根据与背景层结相关的Sturm-Liouville问题进行模式分解，我们证明了线性近似可以用一系列线性波动方程的色散扰动来描述。当所谓的Brunt-vaisäLä频率不是常数时，我们证明了这些方程是耦合的，从而呈现出色散混合现象。然后我们考虑了更具体的浅水构型(当水平尺度远大于深度时)；在Boussinesq近似下(即忽略动量方程中的密度变化)，我们提供了一个适定性定理，对于该定理，我们能够根据相关的物理尺度来控制存在时间。然后，我们可以将模式分解推广到非线性情况，并显示出与上述色散混合不同性质的非线性混合。最后，我们讨论了一些观点，如预期收敛到双流体系统的尖锐层结极限。
Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical stratification of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm–Liouville problem associated with the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so-called Brunt–Vaisälä frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approximation (i.e., neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a nonlinear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluid systems.