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

孤立内波是海洋中常见的特征，由于存在温度和盐度的变化，在能量和动量的传输中起着重要作用。我们目前对内部孤立波动力学的理论认识在很大程度上是建立在弱非线性长波理论的基础上的，这种理论是建立在只有长波存在的假设上的。另一方面，从实验室和实地观察得到的证据表明，一般来说，大振幅孤立内波并不像这些近似理论所建议的那样保持局部受限，相反，它们发展出短波长振荡尾，导致辐射阻尼。这种辐射反过来又提供了一种非线性机制，将大振幅孤立波扰动与不同模态的短尺度振荡波耦合在一起。例如，孤立的内波有可能将能量传递给与内波保持锁相的短尺度表面波，这种情况使人联想到在野外经常伴随着内波的表面撕裂，并且在海面的雷达图像中是可见的。为了填补目前在理论认识上的空白，本研究计划利用奇异摄动和数值方法研究孤立内波振荡尾的产生和相关的辐射衰减机制。特别强调在现实条件下计算短波尾的振幅，包括深部流体和存在自由表面的可能性。为此，将开发一种用于弱非线性扰动的非线性WKB技术，并对大振幅孤立波进行充分的数值计算。此外，从孤立波到不同模态的小尺度扰动（短尺度表面波或低模态内波）的能量转移将与实际中发生的其他物理过程（如粘性耗散）进行比较。

Solitary internal waves are common features in the ocean, due to existing temperature and salinity variations, and play an important part in the transport of energy and momentum. Our current theoretical understanding of internal solitary-wave dynamics is based, to a large extent, on weakly nonlinear long- wave theories which are derived on the assumption that only long waves are present. On the other hand, there is evidence from laboratory and field observations that, in general, large- amplitude solitary internal waves do not remain locally confined as suggested by these approximate theories, but, rather, they develop short-wavelength oscillatory tails that cause radiation damping. This radiation, in turn, provides a nonlinear mechanism for coupling large-amplitude solitary-wave disturbances with short-scale oscillatory waves of different modes. For example, it is possible for solitary internal waves to transfer energy to short-scale surface waves that remain phase-locked with the internal waves P a situation reminiscent of the surface rips that often accompany internal waves in the field, and are visible in radar images of the sea surface. In order to fill the gap currently existing in our theoretical understanding, this research program investigates the generation of oscillatory tails and the related radiation- dampling mechanism of solitary internal waves, using singular- perturbation and numerical methods. Particular emphasis is placed on calculating the amplitude of short-wave tails under realistic conditions, including the possibility of deep fluids and the presence of a free surface. For this purpose, a nonlinear WKB technique will be developed for weakly nonlinear disturbances, and fully numerical computations will be carried out for large-amplitude solitary waves. In addition, the transfer of energy from solitary waves to small-scale disturbances of different modes (short-scale surface or lower- mode internal waves) will be examined in comparison with other physical processes, like viscous dissipation, that occur in practice.