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

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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.