How wind feeds waves

Wind gets faster with height, and somewhere above the water it’s moving exactly as fast as the waves. That height is the critical layer. Ride along with a wave and the air there goes nowhere: it turns over in a row of slow loops, one over each trough, called cat’s eyes. Below them the air slides backwards past the wave, following its shape. The eddies are small and quick near the water and bigger and lazier higher up.

None of that would grow the waves if the air pressure on the water were in step with them. It isn’t quite: it’s a little higher on the back of each wave than on its front, so it pushes down where the water is falling and pulls up where it’s rising, and that work goes into the wave. In a gale the critical layer sinks into the thin layer right next to the water, and the push is strongest. When the waves outrun the wind there’s no height where the two match, and they stop gaining.

My work is deriving that push from first principles, with no constants fitted to wave data. The hard part is that thin layer next to the water: how the turbulence there responds to each passing wave sets how hard the wind pushes, and the standard turbulence models disagree about it.

Schematic, and not my model. The airflow is the textbook linear picture: a logarithmic wind, streamlines that follow the surface near the water, and cat’s eyes where the wind matches the wave speed, after Miles (1957). The inner layer, thicker behind each crest, and the pressure leaning onto the back of each wave follow the non-separated sheltering of Belcher and Hunt (1993). The two belong to different regimes: sheltering does most of the work for slow waves under strong wind, where the real critical layer is extremely thin and sits inside the inner layer, and Miles’s cat’s eyes matter more for faster waves. The waves are far steeper than real ones and grow in seconds instead of over hundreds of wave periods. The critical layer is drawn much higher above the water than it really sits, the cat’s eyes are sized by hand, the inner layer’s thickening and the pressure’s lean are exaggerated, and the eddies are fewer and larger than real turbulence.