On the Interaction Between Long and Short Surface Waves

Abstract

Short, dissipative, surface waves superposed on longer waves cause a growth of the long wave momentum Ml at a ratewhere kl, al are the amplitude and wavenumber of the long waves, so that klal is their steepness; Sa is the radiation stress of the short waves and τs, the rate of transfer of momentum to the short waves by the wind; and the angle braces denote an average over the long-wave phase ? = klx??lt. The first term in the above equation is the radiation stress interaction (Phillips, 1963; Hasselmann, 1971) and is generally negligible compared with the second term, neglected by Hasselmann (1971), which shows that long waves can grow if short wave generation (rather than dissipation) is correlated with the long wave orbital velocity. Even if the modulation of τs is only O(klal) times ?τs?, this mechanism can contribute a significant fraction of long wave momentum. However, even a substantially greater modulation of τs, perhaps due to varying exposure of short waves to the wind, is unlikely to account for all the alleged momentum input to long waves, due to the upper bound klal on the efficiency of the process.