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Asymptotic interpretation of the Miles mechanism of wind-wave instability
- Source :
- Journal of Fluid Mechanics,944,A8(2021)
- Publication Year :
- 2021
-
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, $\rho_{\rm{a}}/\rho_{\rm{w}}\equiv\epsilon<<1$, 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 $\epsilon$, the eigenvalue has a finite imaginary part, which implies growth. Miles (1957) 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.
- Subjects :
- Physics - Fluid Dynamics
Subjects
Details
- Database :
- arXiv
- Journal :
- Journal of Fluid Mechanics,944,A8(2021)
- Publication Type :
- Report
- Accession number :
- edsarx.2107.06844
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1017/jfm.2022.441