An Improved Parameterization for Energy Exchange from Wind to Stokes Waves

Abstract

Previous analysis for the generation of non-linear surface waves by shear flow (Croft & Sajjadi 1993, Sajjadi et al. 1997 and Sajjadi 1998) is extended by: (i) presenting results on Stokes wave; (ii) imposing the boundary condition at the surface wave, rather than at the mean surface; and (iii) including the dominant viscous term in the complete Orr-Sommerfeld equation. The inclusion (i) yields an energy transfer that is larger than those predicted for monochromatic waves, while (ii) has no real effect and (iii) only a small effect for gravity waves. The present analysis is mainly confined to the second-order Stokes wave but its extension to higher-order Stokes waves is straightforward and suggested. Results are also generalized for fully non-linear Stokes waves (nth order) which confirms the findings of previous studies by Sajjadi and coworkers. The energy transfer parameters are obtained via numerical integration of the full Orr-Sommerfeld equation, and it is found the results agree well with that of Conte & Miles (1959) and confirms that the extra energy associated with the growth of non-linear waves is due to higher harmonics of Stokes wave. An expression is derived for parameterized form of this energy exchange from wind to Stokes waves which is could be used in spectral wave models. The present parameterization is compared with other models formulations and its is shown to agree well with the numerical simulation of full Reynolds stress equations, Sajjadi (2002b).