On the Doppler distortion of the sea-wave spectra

Discussions on a form of a frequency spectrum of wind-driven sea waves just above the spectral maximum have continued for the last three decades. In 1958 Phillips made a conjecture that wave breaking is the main mechanism responsible for the spectrum formation [O.M. Phillips, J. Fluid Mech. 4 (1958) 426]. That leads to the spectrum decay ∼ω−5∼ω−5, where ωω is the frequency of waves. There is a contradiction between the numerous experimental data and this spectrum. Experiments frequently show decay ∼ω−4∼ω−4 [Y. Toba, J. Oceanogr. Soc. Japan 29 (1973) 209; M.A. Donelan, J. Hamilton, W.H. Hui, Phil. Trans. R. Soc. London A315 (1985) 509; P.A. Hwang, et al., J. Phys. Oceanogr. 30 (1999) 2753]. There are several ways of the explanation of this phenomenon. One of them (proposed by Banner [M.L. Banner, J. Phys. Oceanogr. 20 (1990) 966]) takes into account the Doppler effect due to surface circular currents generated by underlying waves in the Phillips model.In this article the influence of the Doppler effect on an arbitrary averaged spectrum is considered using both analytic and numerical approaches. Although we mostly concentrated on the very important case of Phillips model, the developed technique and general formula can be used for the analysis of other spectra.For the particular case of Phillips spectra we got analytic asymptotics in the vicinity of spectral maximum and for high frequencies. Results were obtained for two most important angular dependences of the spectra: isotropic and strongly anisotropic. Together with the analytic investigation we performed numerical calculations in a wide range of frequencies. Both high and low frequency asymptotics are in very good agreement with the numerical results.It was shown that at least at low frequencies, the correction to the spectrum due to the Doppler shift is negligible. At high frequencies there is an asymptotic with tail ∼ω−3∼ω−3.