Wave growth by non-separated sheltering

Abstract

Atmospheric boundary layer flow over surface water waves of small slope is analysed with a new heuristic method that clearly shows the underlying physical mechanisms. In this method we consider how the wave displaces mean streamlines in the air flow. Turbulence in the air flow is found to affect the flow over the wave only in a thin inner region that lies close to the interface. The streamline displacement at the top of this inner region has three contributions: displacement over the undulating wave surface; a Bernoulli contribution associated with pressure variations over the wave (which is associated with higher wind speeds at the waves crests and lower wind speeds in the troughs); and a displacement caused by the turbulent stresses in the air flow. The displacement caused by turbulent stresses is a factor (u ∗ U i ) 2 smaller than the other two contributions ( u ∗ is the friction velocity and U i is the wind speed at the top of the inner region), but is important because it leads to the winds being slightly faster on the upwind side of the wave crest compared to in the lee and to the streamline-displacement pattern being shifted slightly downwind of the wave crest. This then leads to a small surface pressure difference across the wave crest and thence wave growth. This is the non-separated sheltering mechanism. The solutions obtained here using physically-based heuristic arguments are in full agreement with those calculated using formal asymptotic methods by Belcher & Hunt (1993) and Cohen & Belcher (1999). 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.