Bidirectional Whitham type equations for internal waves with variable topography
Bidirectional Whitham type equations for internal waves with variable topography
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DOI：
10.1016/j.oceaneng.2022.111600
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发表时间：
2022-08
影响因子：
5
通讯作者：
C. Yuan;Zhan Wang
中科院分区：
文献类型：
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作者：
C. Yuan;Zhan Wang
在海洋重力内波的背景下，在检查波动动力学和解释观测数据方面最广泛使用的理论之一是Korteweg-de弗里斯（KdV）方程。然而，单向传播和无界相速度和群速度的特性限制了它在一些一般情况下的应用（Benjamin等人，1972年）。因此，使用Dirichlet-Neumann算子与刚性盖近似，我们推导出双向和单向Whitham型方程的哈密顿框架中，它保留了完整的线性色散关系的欧拉方程。地形的影响也被纳入建模，由于其实际意义，虽然调用的缩放合理地排除了一个显着的底部变化的住宿。新提出的方程虽然形式简洁，但没有明确给出内孤立波的解析解。因此，一个修改的Petviashvili迭代方法来获得数值解，以规避这一困难。利用这些方法，我们对KdV方程、Whitham型方程和原始方程进行了数值实验，并对实验结果进行了比较。各种模型之间的差异和相似之处共同表明全色散和双向传播的优势，因此，Whitham型方程的有效性。
In the context of oceanic internal gravity waves, one of the most widely-used theories on examining wave dynamics and interpreting observational data is the Korteweg–de Vries (KdV) equation. Nonetheless, the characters of unidirectional propagation and unbounded phase and group velocities restrict its application to some general cases (Benjamin et al., 1972). Thus, using the Dirichlet–Neumann operator with the rigid-lid approximation, we derive both bidirectional and unidirectional Whitham type equations in the Hamiltonian framework, which retain the full linear dispersion relation of the Euler equations. The effect of topography is also incorporated in modeling due to its practical relevance, although the invoked scaling plausibly excludes the accommodation of a significant bottom variation. There are no analytic solutions of internal solitary waves explicitly given in the newly proposed equations, even though these equations possess a concise form. Therefore, a modified Petviashvili iteration method is implemented to obtain the numerical solutions to circumvent this difficulty. Facilitated by these techniques, several numerical experiments are investigated and compared among different models: the KdV equation, the Whitham type equations, and the primitive equations. The discrepancies and similarities between the various models jointly indicate the advantage of full dispersion and bidirectional propagation and, thus, the effectiveness of the Whitham type equations.