Asymptotic interpretation of the Miles mechanism of wind-wave instability. (10th August 2022) - Bonfils, A.F.; Mitra, Dhrubaditya; Moon, W.; Wettlaufer, J.S.
Abstract: 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 k, 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, ρ _a/ρ _wε 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 c_0 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=z_c, where the wind speed is equal to c_0 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 k . 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.