Wave growth by non-separated sheltering
\textit{H. Jeffreys} (1925) put forward the sheltering mechanism of wind-induced growth of surface water waves, whereby the air flow was assumed to separate in the lee of the wave crest, leading to a pressure difference across the wave crests, and thus to wave growth. Jeffreys found the wave growth rate in terms of an undetermined parameter, which he called sheltering coefficient. Most ocean waves have gentle slopes without separation, and recent studies show how for these gentle waves turbulent stresses in air flow lead to a reduction in wind speed in the lee, so that streamlines reach their maximal vertical displacement slightly downwind of the wave crest. There is an associated pressure difference across the wave, and hence an associated wave growth. In other words, for waves of small slope and without separation, there is a non-separated sheltering that leads to the growth of waves. This mechanism, which is associated with turbulence in the wind, complements the celebrated critical-layer mechanism of wave growth first analyzed by \textit{J. W. Miles} [J. Fluid Mech. 30, 163-175 (1967; Zbl 0153.30303)], which is an inviscid shear-flow instability. There have been studies of how the non-separated sheltering and the critical-layer mechanisms act together, but no definitive conclusions have been reached. The detailed and rigorous analyses of non-separated sheltering over slow waves and fast waves are mathematically difficult. The purpose of this paper is to develop a more heuristic method, with no intention of rigor, but which sheds light on the physical processes responsible for non-separated sheltering. The method is based on analyzing the displacement of streamlines over the wave, and is based on the elegant formalism introduced by \textit{J. W. Miles} [J. Fluid Mech. 256, 427-441 (1993)]. The present study can be considered as a step towards analyzing together the effects of non-separated sheltering and the critical layer. The solutions obtained here using physically-based heuristic arguments are in full agreement with those calculated using formal asymptotic methods by the first author and \textit{J. C. R. Hung} [J. Fluid Mech. 251, 109-148 (1993; Zbl 0787.76032)]. The understanding gained from the new method suggests a nonlinear correction to the formula for wave growth that tends to reduce the wave growth rate for steeper waves, in agreement with computations. Finally, a general advantage of more heuristic methods is that they distinguish the key physical assumptions from the mathematical assumptions, and so demonstrate how the results can be extended to more general situations. In particular, the present analysis suggests how the results can be generalized for waves of moderate slope, when nonlinear effects are significant.
wind-induced growth of surface waver waves
scientific article; zbMATH DE number 1358007