A Simplified Quasi-Linear Model for Wave Generation and Air–Sea Momentum Flux

Abstract

A simplified model is described for wave generation and air?sea momentum flux. The model is based upon the quasilinear theory employed by Fabrikant and Janssen, in which the mean flow is approximated to second order in the wave amplitude and fluctuating quantities are approximated to first order. The wave generation rate is computed using Miles' wave generation theory and the numerical method of Conte and Miles. The computationally expensive iterative procedure used in Janssen's quasi-linear model is avoided by specifying in advance the vertical wind velocity profile as a combination of a logarithmic profile above height z = zp and a square root profile below that height, zp being given by U(zp) = c(kp), where U is the wind velocity, kp is the wavenumber at the peak of the wave spectrum, and c(kp) is the associated phase speed. In contrast to Janssen's model and his subsequent simple surface-layer model, no explicit surface roughness parameter is specified: the whole of the applied wind stress is assumed to be taken up by waves with the simple one-dimensional energy spectrum E(k) = (1/2)αPk?3 for wavenumbers k between kp and ∞, E(k) = 0 for k < kp. The square root part of the velocity profile is derived by assuming that the form of the sea surface is stochastically self-similar, and that the velocity profile is also self-similar. A self-similar velocity profile requires kzc(k) to be independent of k, where U[zc(k)] = c(k). The model gives a rather crude approximation to the wave generation rate. The quantity σ?1E?in(k)/E(k), where σ is the wave angular frequency and E?in(k) is the rate of wave energy input, is independent of k for a spectrum of the form E(k) ? k?3. It is proportional to (U*/cp)2 if the Phillips parameter αP is constant and to (U*/cp)1/2 if αP ? (cp/U*)?3/2 where U* is the friction velocity and cp = c(kp). Air?sea momentum flux, as described by the drag coefficient CD(10 m) = [U*/U(10 m)]2, is represented rather better. For wave age cp/U* above 5?10 it decreases with increasing wave age if U* is kept constant. The predicted drag coefficient, however, tends to decrease more rapidly with wave age for the older wind seas than field measurements indicate [if we assume αP ? (cp/U*)?3/2]. For very young seas with cp/U* < 5, the drag coefficient increases with wave age. The model predicts a significant variation of drag coefficient with wave age even when αP is constant. For sea states so ?old? that zp > 10 m, the wind speed depends just on cp/U*, and CD(10 m) increases again with wave age. This last, counterintuitive situation will be modified if we allow more of the air?sea momentum flux to be supported directly by turbulence.