Hasselmann 



study of Snyder and Cox (3), which indicates that the combined Miles- Phillips 

 mechanisms fail to account for the observed wave growth by almost an order of 

 magnitude. 



We shall attempt here to outline a complete theory based on a systematic 

 expansion of the coupled fields. Besides the Miles and Phillips processes, the 

 theory yields three further processes at lowest order: a nonlinear interaction 

 with the mean boundary- layer flow and two forms of wave -turbulence interaction. 



The problem may be divided into two parts: the analysis of the coupling 

 between the wave field and the turbulent boundary- layer flow, and the determi- 

 nation of the energy transfer due to the coupling. The first part concerns the 

 details of the interaction expansion. The second part may be regarded as a 

 particular application of a general transfer theory for random wave fields in 

 weakly coupled systems. 



WAVE-ATMOSPHERE INTERACTIONS 



We present here only the general structure of the interaction analysis; a 

 detailed derivation is given in Ref . 4. Let l, be the surface displacement, 

 u = U+ u' be the turbulent velocity field in the atmosphere, consisting of a mean 

 flow U and a superimposed fluctuating component u' of zero mean, and Su be 

 the wave- induced perturbation of the turbulent velocity field. We assume that 

 all fields are statistically homogeneous, so that they may be represented as a 

 superposition of mutually statistically orthogonal Fourier components of ampli- 

 tude i^, u^, Su,^, where k is the two-dimensional, horizontal wavenumber 

 vector. 



The equations of motion of the coupled wave -atmosphere system may then 

 be expressed in the form 



L(Su^) = Q(u^, Su,^) (z>0) , (1) 



Su^ = R[u^, i^] (z=0) , (2) 



Cfc + ct2^^ = S[^^, u^, SuJ (z=0) , (3) 



where cr = (gk tanh kH)*''^, g is the gravitational acceleration, H is the water 

 depth, 2 is the vertical coordinate, measured positive upward, L represents a 

 linear (essentially the Orr-Sommerfeld) operator, and Q, R, and s are nonlinear 

 functionals of the coupled fields. 



The wave -atmosphere interactions are proportional to the air-to-water 

 density ratio and are therefore weak. This suggests a solution by iteration. To 

 first order, the forcing function s in the harmonic-oscillator equation (Eq. (3)) 

 can be neglected, yielding a stationary wave field of free, sinusoidal waves. The 

 free-wave field can then be substituted in the boundary condition (Eq. (2)), which 

 together with Eq. (1) determines the wave-induced velocity field Su. Substitution 

 of the solution hu in the forcing function S then determines a second-order solu- 

 tion for the wave field, and so forth. 



586 



