论文标题
建模浅水波
Modeling shallow water waves
论文作者
论文摘要
我们在这里回顾了许多最重要的模型，这些模型出现在文献（主要是沿海海洋学）中，以描述浅水中的波浪。我们表明，可以使用“湍流”和非静态术语的各种渐近膨胀来获得这些模型，这些渐方术语出现在方程式中，这是由自由表面Euler方程的垂直整合产生的。其中包括众所周知的非线性浅水（NSW），Boussinesq和Serre-Green-Naghdi（SGN）方程，我们为此我们审查了几个未决的开放问题。在统一的形式主义下，还对诸如多层新南威尔士州多层或SGN系统以及Isobe-kakinuma方程等最新模型进行了审查。我们还评论各种浅水系统的标量版本，可用于描述水平尺寸中的单向波$ d = 1 $;其中包括KDV，BBM，Camassa-Holm和Whitham方程。最后，我们展示了如何在浅水建模中考虑涡度效应，具体侧重于湍流项的行为。作为超出当前数学依据范围的挑战的例子，我们回顾了使用具有涡流的浅水模型来描述波浪破裂的最新作品，并为在强电流上传播浅水波的传播模型。
We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the "turbulent" and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre-Green-Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe-Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension $d=1$; among them are the KdV, BBM, Camassa-Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.
