Miles' mechanism for generating surface water waves by wind, in finite water depth and subject to constant vorticity flow

TL;DR

This paper extends Miles' theory of wind-generated surface water waves to include finite water depth and constant vorticity, deriving new analytical growth rates and revealing how vorticity influences wave amplification.

Contribution

It introduces a generalized wave growth model incorporating vorticity and finite depth, providing analytical expressions and insights into their combined effects on wave amplification.

Findings

Vorticity shifts the maximum wave age for wave growth.

Growth coefficients are affected by shear gradient in water flow.

Analytical growth rates depend on water depth and vorticity.

Abstract

The Miles theory of wave amplification by wind is extended to the case of finite depth h and a shear flow with (constant) vorticity {\Omega}. Vorticity is characterised through the non-dimensional parameter {\nu} = {\Omega} U_1 /g, where g the gravitational acceleration, U_1 a characteristic wind velocity and k the wavenumber. The notion of 'wave age' is generalised to account for the effect of vorticity. Several widely used growth rates are derived analytically from the dispersion relation of the wind/water interface, and their dependence on both water depth and vorticity is derived and discussed. Vorticity is seen to shift the maximum wave age, similar to what was previously known to be the effect of water depth. At the same time, a novel effect arises and the growth coefficients, at identical wave age and depth, are shown to experience a net increase or decrease according to the shear gradient in the water flow.