Frequency Diffusion in Wave-Mean Flow Interaction

Several recent studies have shown that dispersive wave system, in presence of a weak (and stationary) random mean flow (of much smaller magnitude than the group velocity), may show a scattering behaviour. Starting from a set of PDEs describing dispersive waves in presence of a mean flow, we study the possibility of having a frequency shift producing statistical scattering of the waves. We identify two relevant parameters for our study : a first parameter controlling the power law of the dispersion relation; and a second parameter indicating the initial ratio between the average value of the stationary mean flow and the initial group velocity. We then investigate the corresponding phase diagram thanks to a 2D ray tracing scheme in order to identify the different asymptotic regimes. This scheme is derived from the wave-mean flow Hamiltonian system. We demonstrate that, as shown in prior studies, there is a range of dispersive systems for which the frequency stays constant and diffusion is only observed in wave number angle. Moreover, we show that, for lower values of the power law controlling the dispersion relation, the assumption that the group velocity is larger than the mean flow can break, yielding to a diffusion to larger values in frequency, as well as in wave number.