海洋内波作为分层流体重要的现象之一，对人们的生产生活、海洋工程、海 洋军事以及水下通信等方面产生重要影响。此外，内波还具有强大的输运能力， 在海洋生态系统物质输运方面承担着重要的作用。关于海洋内波的研究已经持 续了几十年，尤其是在最近的十年，多模态内波成为了物理海洋学研究热点之 一。为了深入理解海洋内波，本文采用理论分析和数值模拟相结合的方式，建立 了海洋内波的模型，并重点研究了其动力学特性。

首先，本文对两层流体中海洋内波的相关工作进行了完善，并考虑了内孤 立波和内周期波这两种情形。对于内孤立波，通过对 Ablowitz-Fokas-Musslimani (AFM)全局关系公式渐近分析分别得到了强非线性模型方程和弱非线性模型方 程;通过边界积分法数值计算了完全非线性欧拉方程。使用三种方式对比计算 了内孤立波的波形以及全局分岔曲线。此外为了理解内孤立波的输运能力，本 文对其流场结构和粒子轨迹进行了详细的刻画。对于内周期波，同样也使用边 界积分法数值模拟欧拉方程且使用经典的 Stokes 展开法来计算行波的三阶近似 解。在此基础上，本文也对内周期波的流场结构和粒子轨迹进行了数值模拟，并重点研究了周期波中的 Stokes 漂移现象。

其次，为了研究多模态内波，本文将两层流体推广至三层流体并对三层流 体中的内波建模。假设海洋由三个密度不同的均匀流体层组成且形成了稳定的 分层结构。根据特征波长与水深的关系，基于无旋流的 AFM 公式，为“浅水-浅 水-深水”和“深水-浅水-深水”海洋情景开发了强非线性和弱非线性模型。用数 值迭代方法计算了内孤立波，并用数值延拓方法得到了它们的全局分岔图，并对 不同模型进行了比较。对于“浅水-浅水-深水”情况，流体中存在一模态和二模 态的内孤立波，并且在二模态分支中观察到脉冲展宽现象。而在“深水-浅水-深 水”情况下，只能获得二模态孤立波。通过求解基于 MITgcm 模型的原始方程， 证实了二模态孤立波的存在性和稳定性。

此外，本文还重点关注了二维深水流体中强非线性内波的动力学特性。基于 AFM 全局关系公式的推广，针对包含地形的三层深水流体系统，发展了一种强 非线性模型方程，即类 Miyata-Choi-Camassa(MCC)系统。使用 Petviashvili 迭 代技术在类 MCC 方程中数值求解了一模态内孤立波，并在基于边界积分方法计 算的欧拉方程中找到了这种波。通过结果比较，验证了类 MCC 系统的广泛适用 性。利用类 MCC 方程中的多模态孤立波解，本文应用保形映射技术计算了流场 结构和粒子轨迹。对于非定常模拟，本文提出了一个伪谱算法来处理多界面时间 耦合的方程，并研究了由于刚性壁上的局部障碍物的移动效应而产生的多模态 内孤立波的生成演化机制，以及在顶壁平坦时它们的相互作用过程。对于刚性壁 上障碍物的流动，流速可以分为跨临界和超临界。多模态内孤立波的脱落仅发生 在流速的跨临界区域。在流速的超临界区域，生成的多模态内波以与障碍物相同速度一起向前移动。本文实现了具有拐点的分岔曲线的刻画，以实现区域的划 分，并从物理角度解释了生成机制。通过多模态内波的相互作用模拟，我们发现 脉冲展宽的二模态内孤立波是稳定的。

最后，本文考虑了“浅水-浅水-深水”海洋情景中二模态内孤立波对海洋工 程结构的影响，研究了柔性立管在其作用下的水动力学响应;基于类 MCC 模型 方程，数值计算由二模态内波引起的流体力。结合 MCK 结构控制方程，讨论了 在不同振幅二模态内波作用下柔性立管的涡激振动效应以及挠度特征等。通过 数值模拟发现，二模态内孤立波可以激发多摸和多频的顺流向及横流向的涡激 振动效应。此外，随着内孤立波振幅的增加，顺流向和横流向的最大涡激振动振 幅以及横流向的偏移值会增大。

Oceanic internal waves, as one of the significant phenomena in stratified fluids, have crucial impacts on various aspects including people’s production and living, ma- rine engineering, marine military operations, underwater communication, and so on. Additionally, internal waves possess strong transportation capabilities, playing a cru- cial role in material transport within the entire marine ecosystem. Research on oceanic internal waves has been ongoing for decades. Especially in recent years, multi-mode internal waves have become one of the hot topics in physical oceanography. In order to deepen the understanding of oceanic internal waves, this thesis uses a combination of theoretical analysis and numerical simulation to complete the modeling of oceanic internal waves and focus on studying their dynamic characteristics.

Firstly, this thesis complements the relevant work on oceanic internal waves in a two-layer fluid and considers two scenarios: internal solitary waves and internal pe- riodic waves. For internal solitary waves, asymptotic analyses were conducted to ob- tain strongly nonlinear model equations and weakly nonlinear model equations based on the Ablowitz-Fokas-Musslimani (AFM) global relation formula, respectively. The fully nonlinear Euler equations are numerically calculated using the boundary integral method. The wave profiles and global bifurcation curves of internal solitary waves are computed and compared using three methods. Furthermore, to understand the trans- port capability of internal solitary waves, this thesis provides detailed descriptions of their flow field structure and particle trajectories. For internal periodic waves, the Euler equations are numerically calculated using the boundary integral method, and the third- order approximate solutions of traveling waves are computed using the classical Stokes expansion method. We also numerically simulated the flow field structure and particle trajectories of internal periodic waves, and focused on the Stokes drift phenomenon.

Secondly, to investigate multi-mode internal waves, we generalize the two-layer fluids to the three-layer fluids and model the internal waves in three-layer fluids. The ocean is assumed to be composed of three homogeneous fluid layers of different den- sities in a stable stratified configuration. According to the relationship between char- acteristic wavelength and water depth, strongly nonlinear and weakly nonlinear models were developed for the “shallow-shallow-deep”and “deep-shallow-deep”oceanic scenarios based on the AFM formula of irrotational flow. The internal solitary waves were calculated using the numerical iteration method, and their global bifurcation di- agrams were obtained using the numerical continuation method, and different models were compared. For the“shallow-shallow-deep”scenario, mode-1 and mode-2 internal solitary waves are observed and the pulse broadening phenomenon is observed in the mode-2 branch. In the“deep-shallow-deep”scenario, only mode-2 solitary waves are obtained. The existence and stability of mode-2 internal solitary waves are confirmed by solving the original equations based on the MITgcm model.

Furthermore, special attention is given to the dynamic characteristics of strongly nonlinear internal waves in two-dimensional fluids of great depth. A fully nonlinear model for a three-layer fluid of great depth containing topography, a Miyata-Choi- Ca- massa (MCC) type system, is developed based on the generalization of the AFM global relation formulation for free-surface water waves. Mode-1 internal solitary waves are numerically found in the MCC- type equation using the Petviashvili iteration technique and in the full Euler equations based on the boundary integral equation method. The comparison between the results validates the broad applicability of the MCC-type sys- tem. Using the multi-mode internal solitary-wave solutions in the MCC-type equations, we apply the conformal mapping technique to calculate streamlines and particle trajec- tories. For unsteady simulations, we propose a pseudo-spectral algorithm to handle the time-coupled equations and investigate the generation mechanism of multi-mode inter- nal solitary waves due to the resonance effect of local protrusion on the rigid wall, as well as their collisions when the wall is flat. For flows past protrusion on the rigid wall, the flow speed can be divided into transcritical, and supercritical. The shedding of multi- mode internal solitary waves can occur only in the transcritical region of flow speed. In the supercritical region of flow velocity, the generated multi-mode internal waves move forward at the same speed as the obstacle. We implement the carving of bifurcation curves with inflection to achieve the regional delimitation and the interpretation of the generation mechanism from a physical point of view. Through the interaction simula- tion of multi-mode internal waves, we found that the pulse broadened mode-2 internal solitary waves are stable.

Finally, this thesis considers the influence of mode-2 internal solitary waves on the marine engineering structure in the“shallow-shallow-deep”oceanic scenario, and studies the hydrodynamic response of the flexible riser under its action. Based on the MCC-type model equations, the fluid force caused by the mode-2 internal waves is nu- merically calculated. Combined with the MCK structural control equation, the vortex- induced vibration effect and deflection characteristics of the flexible riser under the ac- tion of mode-2 internal waves of different amplitudes are discussed. Through numerical simulations, it has been discovered that mode-2 internal solitary waves can induce multi- mode and multi-frequency vortex-induced vibration effects in both inline and crossflow directions. Additionally, with the increase in the amplitude of internal solitary waves, the maximum vortex-induced vibration amplitude in both inline and crossflow direc- tions as well as the crossflow displacement value increase.