Nonlinear Resonant Wave Interactions in Density-Stratified Flows

在温度和盐度随深度变化的密度分层海洋中，经常观测到非线性内波及其与表面波的相互作用。特别是，非线性共振相互作用在表面波谱和内波波谱的长期演化中起着至关重要的作用。此外，最近的遥感观测表明，内波可以在很大程度上改变海洋表面的表现。因此，更好地了解非线性表面波和内波的动力学及其相互作用在许多学科中都是至关重要的，包括物理海洋学、环境科学以及海洋和海岸工程。归根结底，这些信息将有助于研究气候变化的研究人员，因为跨越海空边界和海洋内部的能量和动量转移对他们的工作至关重要。该项目将通过建模、数值模拟和NJIT新开发的波浪研究实验室的实验相结合的方式，解决内波和面波的动力学相互作用的几个方面。该项目还将为本科生和研究生提供参与研究的机会。该项目将对最近开发的两层流中多方向波的显式哈密顿系统进行数值研究，并将推导出窄带二维表面波与大振幅长内波之间近共振相互作用的简化模型。考虑到实际的海洋条件，该项目将研究波向、带宽和高阶非线性对内波诱导的表面波调制以及随后两种波模之间的能量传递的影响。重点是满足所谓的群共振条件的短表面波包的演化，该条件要求包的群速度与长内波的相速度大致匹配。在现实的海洋条件下，这一条件很容易满足。然后，将用通过非定常保角变换技术获得的欧拉方程的完全非线性数值解和先前的实验室观测来验证数学模型对于一维波的有效性。将两层流的理论方法推广到三层流，并研究了前两个内波模式发生共振相互作用的条件及其动力学。除了时间周期的共振相互作用外，还将研究一种没有能量交换的特殊共振相互作用，以找到行波和稳态解决方案。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Nonlinear internal waves and their interaction with surface waves have been frequently observed in density-stratified oceans, where the temperature and salinity vary with depth. In particular, nonlinear resonant interactions have been known to play a crucial role in the long-term evolution of surface and internal wave spectra. In addition, recent observations from remote sensing have shown that internal waves can substantially change the surface expression of the ocean. Therefore, a better understanding of the dynamics of nonlinear surface and internal waves and their interactions is crucial in many disciplines, including physical oceanography, environmental science, and ocean and coastal engineering. Ultimately this information would help researchers working on climate change as the transfer of energy and momentum across the air-sea boundary and in the interior of the ocean is fundamental to their work. This project will address several aspects of the dynamical interaction of internal and surface waves through a combination of modeling, numerical simulations, and experiments at the newly developed wave research laboratory at NJIT. The project will also provide opportunities for the involvement of undergraduate and graduate students in the research.This project will study numerically the recently developed explicit Hamiltonian system for multi-directional waves in two-layered flows and will derive reduced models for near-resonant interactions between narrow-banded 2D surface waves and a large amplitude long internal wave. With keeping realistic oceanic conditions in mind, the project will study the effects of wave direction, bandwidth, and high-order nonlinearity on the modulation of surface waves induced by internal waves and the subsequent energy transfer between the two wave modes. The focus is on the evolution of a packet of short surface waves satisfying the so-called group resonance condition, which requires the group velocity of the packet to match approximately the phase speed of a long internal wave. This condition can be easily met under realistic oceanic conditions. The mathematical models will be then validated for one dimensional waves with fully nonlinear numerical solutions of the Euler equations obtained via an unsteady conformal mapping technique and previous laboratory observations. The theoretical approach for two-layered flows will be extended to three-layered flows, and the conditions for resonant interactions between the first two internal wave modes and their dynamics will be studied. In addition to time-periodic resonant interactions, a special resonant interaction with no energy exchange will be examined to find traveling wave and steady-state solutions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.