Steady Streaming Generated by a Monochromatic Surface Wave Close to a Rough Bottom

The velocity field within the bottom boundary layer generated by a propagating surface wave is quite important because it allows to determine the bottom shear stress, which is relevant from a practical point of view since, for example, it allows to verify whether the sediments, which make up the bottom, are set into motion and transported by the flow. If the surface wave is of small amplitude and the sea bottom is assumed to be smooth, the flow in the bottom boundary layer was determined by Stokes (1851) and Longuet-Higgins (1953), who showed that weak nonlinear effects generate a steady streaming. Laboratory experiments by Sleath (1988) showed that a laminar flow over a rough bottom made up of medium sand can be observed up to values of the Reynolds number equal to about 250. Therefore, real situations exist such that the flow regime is laminar and the bottom can be considered hydrodynamically rough. Recently, Blondeaux and Vittori (2023) proposed a simplified model for the oscillating boundary layer close to a rough bottom and derived the solution of the linearized problem. In the present contribution their model is used to solve the problem assuming weak non-linear effects, thus extending the solution by Longuet-Higgins (1953) to the more realistic case of rough bottom.