Asymptotic interpretation of the Miles mechanism of wind-wave instability

Abstract

When wind blows over water, ripples are generated on the water surface. These ripples can be regarded as perturbations of the wind field, which is modelled as a parallel inviscid flow. For a given wavenumber, the perturbed streamfunction of the wind field and the complex phase speed are the eigenfunction and the eigenvalue of the so-called Rayleigh equation in a semi-infinite domain. Because of the small air-water density ratio, ρa/ρw ≡ ϵ ≪ 1, the wind and the ripples are weakly coupled, and the eigenvalue problem can be solved perturbatively. At the leading order, the eigenvalue is equal to the phase speed C0 of surface waves. At order ϵ, the eigenvalue has a finite imaginary part, which implies growth. Miles (J. Fluid Mech., vol. 3, 1957, pp. 185-204) showed that the growth rate is proportional to the square modulus of the leading-order eigenfunction evaluated at the so-called critical level Z = ZC, where the wind speed is equal to and the waves extract energy from the wind. Here, we construct uniform asymptotic approximations of the leading-order eigenfunction for long waves, which we use to calculate the growth rate as a function of. In the strong wind limit, we find that the fastest growing wave is such that the aerodynamic pressure is in phase with the wave slope. The results are confirmed numerically.