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Asymptotic multi-layer (AML) analyses and computation of solutions for turbulent flows over steady and unsteady monochromatic surface waves are reviewed in the limits of low-turbulence stresses and small wave amplitude. The structure of the flow is defined in terms of asymptotically matched thin layers, namely the surface layer and a critical layer (CL), whether it is ‘elevated’ or ‘immersed’, corresponding to its location above or within the surface layer. The results particularly demonstrate the physical importance of the singular flow features and physical implications of the elevated CL in the limit of the unsteadiness tending to zero. These agree with the variational mathematical solution of Miles (J Fluid Mech, 3:185–204, ) for a small but finite growth rate, but they are not consistent physically or mathematically with his analysis in the limit of a growth rate tending to zero. As this and other studies conclude, in the limit of zero growth rate, the effect of the elevated CL is eliminated by finite turbulent diffusivity, so that the perturbed flow and the drag force are determined by the asymmetric or sheltering flow in the surface shear layer and its matched interaction with the upper region. But for groups of waves, in which the individual waves grow and decay, there is a net contribution of the elevated CL to the wave growth. CLs, whether elevated or immersed, affect this asymmetric sheltering mechanism, but in quite a different way from their effect on growing waves. These AML methods lead to physical insights and suggest approximate methods for analysing higher-amplitude and more complex flows, such as flow over wave groups.

Asymptotic multi-layer analysis of wind over unsteady monochromatic surface waves