A relationship between wind stress and wave slope

TLDR

Recent wind‑wave tank and ocean data show that the growth rate β of energy and momentum transfer from wind to surface waves follows β = [(0.04±0.02) u* 2 ω cos θ]/c² across a wide frequency range. By applying the β relation and enforcing that wind‑to‑wave momentum flux does not exceed wind stress, the authors derive a theoretical upper bound on mean‑square wave slope between g/(2πU₁₀) and 20 Hz, which measurements in tanks and the ocean confirm when the air‑water interface is well defined. The derived slope limit predicts that mean‑slope spectral densities decrease as the peak frequency of a wind wave spectrum decreases, explaining why mean‑square dominant wave slopes are typically smaller on the ocean than in wavetanks.

Abstract

Recently published data taken in wind‐wave tanks and on the ocean are shown to yield a growth rate, β, describing the transfer of energy and momentum directly from wind to surface waves, that is well described by the relation β = [(0.04±0.02) u * 2 ω cos θ]/ c 2 over a wide range of frequencies. Here u * is air friction velocity, ω is radian wave frequency, c is phase speed, and θ is the angle between wind and waves. Using this form and the requirement that the momentum flux from wind to waves not exceed the wind stress, we show that the total mean‐square, upwind/downwind wave slope between the frequencies g /2π U 10 and 20 Hz must be less than ρ a /[ρ ω (0.04±0.02)], where ρ a and ρ ω are air and water densities, g is gravitational acceleration, and U 10 is wind speed at 10 m. Measurements of mean square slope both in wavetanks and on the ocean seem to show agreement with this limitation as long as the air‐water interface is well defined. One implication of such a slope limitation is that mean slope spectral densities are limited to values which decrease as the peak frequency of a wind wave spectrum decreases. This may provide an explanation for the observation that mean‐square, dominant wave slopes are typically smaller on the ocean than in wavetanks.