Generation of Waves by Turbulent Wind 



Miles and Phillips introduce additional simplifications. Phillips ignores the 

 dependence of s on the fields l,^^ and Su,j, so that Eq. (3) reduces to a purely ex- 

 ternal excitation by the turbulence field u ' (physically, s corresponds in this 

 case to the unmodified turbulent surface pressure). Miles ignores the nonlinear 

 term Q in Eq. (1) (i.e., the wave-induced perturbation of the turbulent Reynolds 

 stress) and the u' -dependence of R in Eq. (2). This reduces Eqs. (1) and (2) to 

 linear, constant -coefficient equations, and the boundary- value problem of deter- 

 mining Su becomes tractable. 



If the Miles approximation is regarded as an acceptable first-order solu- 

 tion, the general case can be approached by introducing a second iteration loop 

 in which Q and the u' dependence of R are treated as further perturbations. 

 The nth iteration is obtained by solving Eqs. (1) and (2) with the (n - l )th itera- 

 tion substituted in the right-hand sides. In this manner, the wave-induced 

 velocity field is obtained as a power series in the components ^^ and u'^, and 

 the forced-harmonic-oscillator equation (Eq. (3)) takes the form 



^"k + ^'^k = Pk +B,^, + L q , Ck u' + ••• . (4) 



kj tkj :k 1 2 1 2 



The first two terms on the right correspond to the Phillips and Miles approxi- 

 mations, respectively; P^ denotes the external forcing term due to random 

 turbulent pressure fluctuations; and B^, C^jk^, • • are coupling coefficients, 

 which are determined by solving the Orr-Sommerfeld equation, Eq. (1), under 

 boundary condition (2). In general, this is possible only by numerical methods 

 or by restriction to simple boundary -layer models. (It is known that a simple 

 constant-velocity or constant- slope profile is inadequate in the Miles approxi- 

 mation, in which the energy transfer is determined by the local profile curvature 

 at the critical layer. However, the detailed properties of the velocity profile are 

 probably less important for the higher order processes.) 



Correlation measurements of wave height and surface pressure by Longuet- 

 Higgins et al. (5) indicate that the Miles approximation does indeed yield a rea- 

 sonable first-order description of the wave-induced fluctuations in the atmos- 

 phere. It should be noted, however, that this does not necessarily apply to the 

 Miles transfer expression. In Miles' approximation, the wave- induced pressure 

 fluctuations are almost 90 degrees out of phase with the wave height over most 

 of the wave spectrum, so that only a small fraction of the pressure field is effec- 

 tive in generating waves. It is therefore conceivable that the higher order pres- 

 sure fluctuations, although smaller in absolute magnitude, lead to a larger 

 energy transfer. 



THE ENERGY TRANSFER 



After determining the coefficients of the interaction expansion, the problem 

 remains of evaluating the energy transfer resulting from the coupled equations 

 (Eqs. (4)). The analysis is basically straightforward but involved algebraically. 

 We summarize here only the results, referring to Refs. 4 and 6 for details. The 

 problem may be regarded as a generalization of the theory of wave-wave inter- 

 actions, first considered by Peierls (7) in his classic study on the heat conduction 



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