An Interaction Mechanism between 

 Large and Small Scales for 

 Wind-Generated Water Waves 



Marten Landahl, 

 Sheila Widnall, 



and 

 Lennart Hultgran 



Massachusetts Institute of Technology 

 Cambridge, Massachusetts 



ABSTRACT 



By aid of a non-linear two-scale analysis it is 

 shown that large-scale water waves can experience 

 growth due to spatial non-uniformities in the 

 growth rate of the small-scale waves in the non- 

 uniform wind field associated with the large-scale 

 waves. The growth rate is shown to be proportional 

 to the mean-square slope of the small-scale waves 

 and their growth rates, but inversely proportional 

 to the difference between the phase velocity of the 

 large-scale wave and the group velocity of the small- 

 scale waves. It is suggested that this mechanism 

 can transfer wind energy to short gravity waves at 

 a higher rate than the direct linear transfer 

 mechanism of Miles (1962) . The analysis also 

 predicts that a large-scale wave moving against the 

 wind will be damped by the action of the small-scale 

 waves. 



1. INTRODUCTION 



The mechanism whereby wind generates water waves 

 has long proven a difficult and challenging problem 

 in theoretical fluid mechanics which has not yet 

 been satisfactorily resolved. The simple linear 

 mechanism of forcing by pressure fluctuations 

 [Phillips (1957)] and by instability induced by 

 the mean wind field [Miles (1957), 1962)] have 

 been found inadequate to account for the high values 

 of energy transfer from wind to waves observed for 

 longer waves, both in the laboratory and in the 

 open sea. For short waves in the capillary regime, 

 laboratory experiments [Larson and Wright (1975)] 

 have given good agreement between observed growth 

 rates and Miles' instability theory, particularly 

 when the surface drift velocity in the water is 

 taken into account [Valenzuela (1976)]. For waves 

 in the short gravity range, however, recent experi- 

 ments by Plant and Wright (1977) give growth rates 

 much in excess of that predicted by the instability 



theory with the discrepancy beginning at a wave 

 length of about 10 cm and increasing with wave 

 length. Open-sea measurements have also produced 

 energy transfer rates for gravity waves which are 

 much in excess of the values according to Miles. 

 [See, for example, the recent review of Barnett 

 and Kenyon (1975)] . 



In view of the failure of linear theory one is 

 forced to look for nonlinear mechanisms for energy 

 transfer. Nonlinear interaction between waves in 

 the gravity range [Phillips (1966)] is a compara- 

 tively weak process (of third order in amplitude) 

 which causes redistribution of the energy from 

 waves of intermediate wave numbers to waves of lower 

 and higher wave numbers. This could be effective 

 for the eventual saturation of the spectrum but is 

 unlikely to be strong enough to make a large change 

 in the initial growth. A more tenable proposition 

 is that the modification of the turbulence in the 

 air by the wave induced velocity field could change 

 the phase shift between surface elevation and the 

 pressure so as to alter the energy transfer rate. 

 This effect has been investigated by many authors 

 [Manton (1972), Davies (1972), and Townsend (1972), 

 among others] employing different turbulence models. 

 These investigations point to the possibility that 

 the modulation of the turbulence by the wind could 

 have an important effect, but it is difficult to 

 assess the adequacy of the postulated turbulence 

 models employed. 



An interesting possibility for transfer of energy 

 to gravity waves is through nonlinear interaction 

 with capillary waves which can draw energy from 

 wind at a much higher rate than the longer waves. 

 The interaction between short and long surface 

 waves has been subject to a great deal of discussion 

 in the literature. A train of short waves riding 

 on a long wave becomes modulated by the orbital 

 velocity field of the long wave so as to make their 

 wave length smaller - and hence their amplitude 

 greater - in the region near the crest of the long 

 wave. Longuet-Higgins (1969) argued that the 



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