Mean flow in high-order wave evolution equations at arbitrary depth

Fluid motion under water waves includes an Eulerian return flow in the direction opposite to wave propagation that is of importance for accurately modelling the transport of tracers in the ocean like sediments, plastic pollution, oil, etc. The return flow is related to the mean flow, that is to the derivative in space of the zero-harmonic component of the velocity potential φ0. It turns out that this component is not consistently taken into account in some derivations of the high-order nonlinear Schrödinger equation (HONLS) using the multiscale development at arbitrary depth, which therefore do not correctly reproduce experimental results in the deep-water limit. We show how to formulate a Neumann problem for φ0 that can be solved at fourth order in steepness for arbitrary depth. The derivative of such term is thus included in the HONLS in both cases of propagation in time and in space. We compare the results of the simulations obtained using our model to those obtained with previously published fourth-order models without the mean-flow term, to observations and to accurate simulations performed with the high-order spectral method. While our model is equivalent to alternative formulations in intermediate water, it is more accurate in deep water. Our model therefore provides the first fourth-order NLS relevant for depths ranging from intermediate to deep water.