A mixing length treatment of the effect of turbulence on the wind generation of water waves

Abstract

A theoretical study is made of the shearing flow over a sinusoidal boundary representing the interface between the ocean and the atmosphere, with the main purpose being to investigate the pressure distribution at the interface and to calculate the energy transfer between the two media. The theory is developed on a model of turbulent flow making use of Prandtl's mixing length theory to represent the shear stress terms in the basic Navier-Stokers equations. A curvilinear coordinate system which follows the wave train is used in order to simplify the equations, and to allow for a linearized solution which requires only that the wave amplitude be small in comparison to the wavelength. All parameters are non-dimensionalized and the analysis is made without restriction as to the type of velocity profile. Various velocity profiles are investigated and, in general, the phase relations between the presence at the interface and wave elevation imply a situation which could allow an energy input from the air flow to the wave.