Frequency downshift in narrowbanded surface waves under the influence of wind

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Abstract
It is well known that the spectral peak of wind-induced gravity waves on the sea surface tends to shift to lower frequencies as the fetch increases. In past theories the nonlinear dynamics subsequent to Benjamin–Feir instability has been found to initiate the downshift in narrow-banded waves in the absence of wind. However, these weakly nonlinear theories all predict the downshift to be only the first phase of an almost cyclic process. Limited by the length of a wave tank, existing experiments are usually made with relatively steep waves which often break. Although there is a theory on how breaking adds dissipation to stop the reversal of the initial trend of downshift, the details of breaking must be crudely characterized by semi-empirical hypotheses. Since the direct role of wind itself must be relevant to the entire development of wind-wave spectrum, we examine here the effect of wind on the nonlinear evolution of unstable sidebands in narrow-banded waves. We assume that the waves do not break and consider the case where the nonlinear effects that initiate the downshift, energy input by wind and damping by internal dissipation all occur on the same timescale. This means that not only must the waves be mild but the wind stress intensity must also lie within a certain narrow range. With these limitations we couple the air flow above the waves with Dysthe's extension of the cubic Schrödinger equation, and examine the initial as well as the long-time evolution of a mechanically generated wavetrain. For a variety of wind intensities, downshift is indeed found to be enhanced and rendered long lasting.

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