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Theoretical studies of the generation of surface waves and the propagation of internal waves in the sea Manton, Michael John


The theory of three distinct problems arising in geophysical fluid dynamics is considered. Part I concerns the generation of sea waves. The sea state and the air flow above the sea during active wave generation are discussed; in particular, it is shown that a significant fraction of the momentum flux from the air is transferred to the sea in the form of wave drag. Different mechanisms for transferring energy and momentum into a wave field are considered. Phillips' resonance model of the initial generation of waves is modified to include the dispersive effects of the wave field. It is shown that the irrotational flow caused by surface waves interacts with the turbulent velocity fluctuations in the air to produce an insignificant energy flux to the sea. Miles' theory, which involves the interaction of the wave-induced fluctuation in the air with the mean velocity field, is discussed and the streamline configuration near the critical height is considered. We show that a closed streamline "cat's-eye" may lie either over the wave crest or over the wave trough, depending upon the behaviour of the mean wind profile. The most unsatisfactory simplification in Miles' theory would seem to be the neglect of air turbulence. By considering the full set of energy equations and closing them with the aid of some simple physical assumptions, it is found that the effects of turbulence are essentially restricted to the neighbourhood of the critical height. The turbulence acts to diffuse momentum across this critical layer in a manner analogous to the action of molecular viscosity. When the critical height is small compared with the wavelength of the surface wave, the critical layer extends to the sea surface, and the streamline cat's-eye lies over the wave trough. The propagation of two-dimensional waves in a rotating Boussinesq fluid with constant Brunt-Väisälä frequency and variable depth is considered in Part II. A method is developed for the investigation of the problem, which involves first reducing it to find the solution of a certain functional equation. For a linear depth profile, an exact solution to this equation is obtained, which agrees with the previously known solution. For a more general profile, an analytic solution is derived in a form involving infinite series that converge provided that the depth variations are strictly transmissive. For a slowly varying depth, the solution is also obtained by means of a two-scale perturbation expansion. Finally, exact solutions of the functional equation which correspond to marginally transmissive depth profiles are obtained. In Part III the diffraction of internal waves by a semi-infinite vertical barrier in a uniformly rotating Boussinesq fluid with constant Brunt- Väisälä frequency, N , and constant depth is discussed. For the frequency passband f < σ < N , where f and σ are respectively the inertial and wave frequencies, the presence of rotation gives rise to internal Kelvin waves which propagate without attenuation away from the barrier and which have amplitudes that decay exponentially in the direction along the barrier.

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