Asymptotic analysis of surface waves interacting with a wind and a current

Abstract: Following Miles (1957), surface waves are regarded as perturbations of the wind, modeled as an inviscid parallel shear flow; a water current can be included in the model. The linear stability analysis of the shear flow leads to an eigenvalue problem. The real part of the eigenvalue is the phase speed of the waves while the imaginary part times the wavenumber is the growth rate of the wave amplitude. The streamfunction of the perturbed flow, or eigenfunction, obeys the Rayleigh equation with coupled boundary conditions at the air-water interface. First, I will show how Miles simplified this problem using the small air-water density ratio. Next, for waves whose wavelength is much larger than the characteristic length scale of the shear, I will solve the Rayleigh equation asymptotically and infer the complex eigenvalue. Finally I will show that, in the strong wind limit, the fastest growing waves are those for which the aerodynamic pressure is in phase with the wave slope.

References:
[1] J. W. Miles, J. Fluid Mech., 3:185–204, 1957.
[2] A. F. Bonfils, D. Mitra, W. Moon, and J. S. Wettlaufer, J. Fluid Mech., 944:A8, 2022.
[3] A. F. Bonfils, D. Mitra, W. Moon, and J. S. Wettlaufer, arXiv:2211.02942.