Coupled dynamics of short wind waves and the air flow over long surface waves

A study is presented of modulations induced by a dominant long surface wave
(LW) in the coupled system\ of the air flow - short wind waves. The
modulation of short waves in the gravity range results from their
interaction with the LW orbital velocity and their interaction with the wind
stress modulated by the LW. The modulation of waves in the capillary range
is mainly due to the modulation of the rate of generation of parasitic
capillaries by steep short gravity waves. The variation of the wind surface
stress is described by the dynamics of the turbulent air flow over the LW
with the surface roughness varying along its profile. The variation of the
surface roughness is caused by the variation of the form drag supported by
modulated short waves. In turn the LW-induced variation of the surface
stress affects the short wave modulation. This provides a feed-back in the
coupled air flow-short waves system in the presence of the LW. The model
results show that the amplitude of the surface roughness modulation can be
large. In terms of the modulation transfer function it can reach values of
10-20. The modulation of the form drag, which causes the modulation of the
surface roughness, comes mainly from short breaking waves strongly modulated
by dominant waves. The modulation of the surface roughness considerably
affects the dynamics of the air flow over the LW and thus the LW wind growth
rate. Models of the air flow above waves assuming a constant roughness
parameter underestimate the growth rate parameter approximately two-three
times as compared to the measured values. The present study show that when
the variation in the surface roughness is accounted for, the growth rate
parameter increases roughly twice, which to a large extent reduces the
discrepancy with measurements.