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The evaluation and consideration of the mean flow in wave evolution equationsare necessary for the accurate prediction of fluid particle trajectories underwave groups, with relevant implications in several domains, from the transportof pollutants in the ocean, to the estimation of energy and momentum exchangesbetween the waves at small scales and the ocean circulation at large scale. Wederive an expression of the mean flow at finite water depth which, in contrastto other approximations in the literature, accurately accords with thedeep-water limit at third order in steepness, and is equivalent to second-orderformulations in intermediate water. We also provide envelope evolutionequations at fourth order in steepness for the propagation of unidirectionalwave groups either in time or space that include the respective mean flow term.The latter, in particular, is required for accurately modelling experiments inwater wave flumes in arbitrary depths.

Mean flow modelling in high-order nonlinear Schr\"odinger equations