双曲余弦海脊上海啸俘获波的解析与数值研究

Analytical and numerical investigation of tsunami trapped waves over a hyperbolic-cosine squared ocean ridge

海啸能被大洋海脊引导以俘获波的形式沿其传播上万千米,且因其特殊的运动方式,携带巨大能量影响远场地区的港口,严重威胁海岸安全。本文首先基于线性浅水方程,推导了双曲余弦平方海脊上俘获波的波面解,其为μ阶ν次的连带勒让德函数的第一类解和第二类解的组合。进一步推导出其对应的频散关系,其中对于确定的频率ω,存在无穷多个波数ky与之对应。采用MIKE21-BW模型,模拟了产生于海脊脊顶处的海啸在理想双曲余弦平方海脊上的传播变形过程。结果表明,小部分能量以自由先驱波进行传播,海啸波的波能大部分被海脊俘获。海脊俘获波沿着海脊方向为行进波,随着海啸波传播时间的增加,波浪在沿着海脊方向的延展范围也逐渐增大,波高逐渐减小、波的个数逐渐增加。俘获波能量主要由不同频率以相同速度传播的具有孤立波特性的波浪成分和能量主要集中在特定频率范围内的波浪成分组成。

Tsunami can be guided by oceanic ridges propagation as trapped waves transferring energy to thousands of kilometers away,which threatens the safety of harbor and coastal.Based on the linear shallow water approximation,analytical solutions of trapped waves over a hyperbolic-cosine squared ocean ridge are derived,which could be described by combining the associated Legendre functions of the first and second kinds.The corresponding dispersion relation is also obtained.For various fixed frequencyω,there are infinite wavenumber kysatisfied the dispersion relation.MIKE21-BW was used to simulate tsunami generated on the top of the ridge and propagation along it.The results show that only small part of the energy transfers as free waves and most of it is rapped over the ridge.The trapped wave propagates as traveling wave along the ocean ridge.The envelope of trapped waves is prolonged while the maximum amplitude decreases and the number of the trapped waves increases.The energy of trapped wave is mainly composed of two parts：solitary wave characteristic component with different frequencies travelling with the same speed and dispersive wave characteristic component with fixed frequency range.