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

由于温度和盐度的变化，孤立内波是海洋中的常见特征，在能量和动量的传输中发挥着重要作用。我们目前对内孤立波动力学的理论理解在很大程度上是建立在弱非线性长波理论的基础上的，这些理论是建立在只存在长波的假设基础上的。另一方面，实验室和现场观测的证据表明，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，而是发展出引起辐射衰减的短波长振荡尾巴。这种辐射反过来提供了一种非线性机制，用于将大幅度孤立波扰动与不同模式的短尺度振荡波耦合。例如，孤立的内波有可能将能量转移到与内波P保持锁相的短尺度表面波，这种情况使人想起在野外经常伴随着内波的表面撕裂，并在海面的雷达图像中可见。为了填补目前理论认识上的空白，本研究项目利用奇异摄动和数值方法研究了孤立内波振荡尾部的产生和相关的辐射衰减机制。特别强调了在现实条件下计算短波尾部的振幅，包括深部流体的可能性和自由表面的存在。为此，将发展一种针对弱非线性扰动的非线性WKB技术，并将对大振幅孤立波进行充分的数值计算。此外，从孤立波到不同模式的小尺度扰动(短尺度表面波或低模内波)的能量转移将与实践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.