On the Connection Between Step Approximations and Depth-Averaged Models for Wave Scattering by Variable Bathymetry
On the Connection Between Step Approximations and Depth-Averaged Models for Wave Scattering by Variable Bathymetry
复制标题
变测深波散射步长近似与深度平均模型之间的联系
DOI：
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发表时间：
2020
影响因子：
0.9
通讯作者：
R. Porter
中科院分区：
文献类型：
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作者：
R. Porter
两种流行且计算成本低廉的方法来近似面波在二维变水深测量中的传播，它们是“步长近似”和“深度平均模型”。在前者中，水深测量被离散成由垂直台阶连接的恒定深度的短区段。通过$2×2$传递矩阵的乘积计算整个水深测量的散射值，这些传递矩阵的条目编码了独立于所有其他步骤所采取的每个垂直步长的散射特性。在后者中，假设下垫面速度场具有可分离的深度相关性，并进行垂直平均处理，从而得到一个二阶常微分方程组。在这篇文章中，步长近似被重新讨论，并被证明是等价于描述零步长限制下的深度平均模型的颂歌。常微分方程取决于正则垂直阶跃问题的解是如何近似的。如果使用浅水近似，则得到众所周知的线性浅水方程。如果采用平面波变分近似，则可以恢复缓坡方程的一个新的变种。
Two popular and computationally inexpensive class of methods for approximating the propagation of surface waves over two-dimensional variable bathymetry are ‘step approximations’ and ‘depth-averaged models’. In the former, the bathymetry is discretised into short sections of constant depth connected by vertical steps. Scattering across the bathymetry is calculated from the product of $2 imes 2$ transfer matrices whose entries encode scattering properties at each vertical step taken in isolation from all others. In the latter, a separable depth dependence is assumed in the underlying velocity field and a vertical averaging process is implemented leading to a second-order ordinary differential equation (ODE). In this article, the step approximation is revisited and shown to be equivalent to an ODE describing a depth-averaged model in the limit of zero-step length. The ODE depends on how the solution to the canonical vertical step problem is approximated. If a shallow water approximation is used, then the well-known linear shallow water equation results. If a plane-wave variational approximation is used, then a new variant of the mild-slope equations is recovered.