On shoaling of solitary waves
On shoaling of solitary waves
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论孤立波的浅滩
DOI：
10.1017/jfm.2018.395
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发表时间：
2018
影响因子：
3.7
通讯作者：
H. Yeh
中科院分区：
文献类型：
--
作者：
J. Knowles;H. Yeh
绿色定律是浅水波放大的经典分析预测之一-波幅与$h^{-1/4}$成比例增长，其中$h$是当地水深。绿色定律适用于线性浅水波在逐渐变化的水深中单向传播。另一方面，机械能守恒表明，如果波形保持其孤波特性，则孤波的浅化波振幅增长为$a\propto h^{-1}$。尽管如此，最近的一些实验室和现场测量表明，增长的长波在浅滩是慢于预测的绿色的法律。绿色定律中明显缺失的因素是非线性和频散效应以及来自海滩的波反射，而绝热变浅过程不识别有限斜率和长度的海滩上的波形的变换。在这里，我们首先研究这个问题的分析的基础上变系数扰动Korteweg-de弗里斯方程。得到了三种不同极限的解析解：（1）线性和非色散极限的绿色定律，（2）线性和色散极限的较慢幅度增长率，以及（3）非线性和非色散极限。然后，为了表征各种入射波和海滩条件下的浅水行为，我们实现了一个五阶伪谱数值模型的全水波欧拉理论。我们发现，绿色的法律是不规范的，但仅限于小入射波振幅时，波长仍然是小的相比，海滩长度尺度。一般情况下，浅水过程中的波浪放大率不遵循幂律。当入射波为有限长时，当波长与海滩长度之比较小时，浅水放大比格林定律的浅水放大快，而当波长与海滩长度之比增大时，浅水放大比格林定律的浅水放大慢。我们还发现，入射波开始放大之前，其波峰到达海滩脚趾由于波浪反射。孤立波浅化的其他突出特点和行为进行了讨论。
One of the classic analytical predictions of shoaling-wave amplification is Green’s law – the wave amplitude grows proportional to $h^{-1/4}$ , where $h$ is the local water depth. Green’s law is valid for linear shallow-water waves unidirectionally propagating in a gradually varying water depth. On the other hand, conservation of mechanical energy shows that the shoaling-wave amplitude of a solitary wave grows like $a\propto h^{-1}$ , if the waveform maintains its solitary-wave identity. Nonetheless, some recent laboratory and field measurements indicate that growth of long waves during shoaling is slower than what is predicted by Green’s law. Obvious missing factors in Green’s law are the nonlinearity and frequency-dispersion effects as well as wave reflection from the beach, whereas the adiabatic shoaling process does not recognize the transformation of the waveform on a beach of finite slope and length. Here we first examine this problem analytically based on the variable-coefficient perturbed Korteweg–de Vries equation. Three analytical solutions for different limits are obtained: (1) Green’s law for the linear and non-dispersive limit, (2) the slower amplitude growth rate for the linear and dispersive limit, as well as (3) nonlinear and non-dispersive limit. Then, in order to characterize the shoaling behaviours for a variety of incident wave and beach conditions, we implement a fifth-order pseudo-spectral numerical model for the full water-wave Euler theory. We found that Green’s law is not the norm but is limited to small incident-wave amplitudes when the wavelength is still small in comparison to the beach length scale. In general, the wave amplification rate during shoaling does not follow a power law. When the incident wave is finite, the shoaling amplification becomes faster than that of Green’s law when the ratio of the wavelength to the beach length is small, but becomes slower when the length ratio increases. We also found that the incident wave starts to amplify prior to its crest arriving at the beach toe due to the wave reflection. Other prominent characteristics and behaviours of solitary-wave shoaling are discussed.