Wind-wave amplification mechanisms: possible models for steep wave events in finite depth

P Montalvo,M A Manna,C Kharif,R Kraenkel

doi:10.5194/nhess-13-2805-2013

Abstract

Abstract. We extend the Miles mechanism of wind-wave generation to finite depth. A β-Miles linear growth rate depending on the depth and wind velocity is derived and allows the study of linear growth rates of surface waves from weak to moderate winds in finite depth h. The evolution of β is plotted, for several values of the dispersion parameter kh with k the wave number. For constant depths we find that no matter what the values of wind velocities are, at small enough wave age the β-Miles linear growth rates are in the known deep-water limit. However winds of moderate intensities prevent the waves from growing beyond a critical wave age, which is also constrained by the water depth and is less than the wave age limit of deep water. Depending on wave age and wind velocity, the Jeffreys and Miles mechanisms are compared to determine which of them dominates. A wind-forced nonlinear Schrödinger equation is derived and the Akhmediev, Peregrine and Kuznetsov–Ma breather solutions for weak wind inputs in finite depth h are obtained.

Highlights

The pioneer theories to describe surface wind-wave growth in deep water began with the works of Jeffreys (1925), Phillips (1957) and Miles (1957, 1997), and the modern investigations take nonlinearity and turbulence effects into account. Janssen (2004) has provided a thorough review of the topic.1.1 Miles’ and Jeffreys’ mechanisms of wind-wave growthMiles’ and Jeffreys’ theories consider both air and water to be incompressible and disregard viscosity effects
A β-Miles linear growth rate depending on the depth and wind velocity is derived and allows the study of linear growth rates of surface waves from weak to moderate winds in finite depth h
For constant depths we find that no matter what the values of wind velocities are, at small enough wave age the β-Miles linear growth rates are in the known deep-water limit

Summary

Introduction

The pioneer theories to describe surface wind-wave growth in deep water began with the works of Jeffreys (1925), Phillips (1957) and Miles (1957, 1997), and the modern investigations take nonlinearity and turbulence effects into account. Miles’ and Jeffreys’ theories consider both air and water to be incompressible and disregard viscosity effects. Both theories are linear and Miles’ mechanism is limited to the deepwater domain. They give the linear wave growth γMiles and γJeffreys (respectively noted γM and γJ) of wind-generated normal Fourier modes of wave number k. The resonant mechanism happens at a critical height where the airflow speed matches the phase velocity of the surface wave

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