Mathematical Sciences: Nonlinear Coupling of Long Internal Waves With Small-Scale Disturbances

9404673 AKYLAS内部重力波具有相当大的地球物理意义。大振幅内波扰动是深海和沿海地区以及低层大气中海洋温跃层的共同特征。这项工作的重点是了解长振幅内波与短尺度扰动的相互作用。这是一个与各种海洋和大气现象有关的完全非线性机制。例如，孤立内波与短尺度表面波的耦合表现为表面粗糙度的变化，即所谓的“表面撕裂”，通常伴随着现场的内波，并且在海面的雷达图像中是可见的。主要目的是从物理和数学的角度来理解这种非线性耦合机制。重点计算了长内波引起的短波尾的振幅。为此，针对弱非线性扰动，将发展波数域的奇异摄动技术，并对大振幅波进行充分的数值计算。

9404673 AKYLAS Internal gravity waves are of considerable geophysical interest. Large-amplitude internal wave disturbances are common features in the ocean thermocline both in deep water and in coastal regions, as well as in the lower atmosphere. This work focuses on understanding the interaction of long large-amplitude internal waves with short-scale disturbances. This is a fully nonlinear mechanism that is relevant to various oceanic and atmospheric phenomena. For example, the coupling of solitary internal waves with short-scale surface waves manifests as surface roughness variations---the so-called 'surface rips'---that often accompany internal waves in the field, and are visible in radar images of the sea surface. The main objective is to obtain an understanding of this nonlinear coupling mechanism both from a physical and a mathematical point of view. Particular emphasis is placed on calculating the amplitude of short-wave tails induced by long internal waves. For this purpose, a singular perturbation technique in the wavenumber domain will be developed for weakly nonlinear disturbances, and fully numerical computations will be carried out for large-amplitude waves.