Steady streaming under a surface wave propagating over a rough bottom: A model of the bottom boundary layer

The steady streaming generated by nonlinear effects at the bottom of a propagating surface wave is determined when the bottom is characterized by a roughness, the size of which scales with the boundary layer thickness. Therefore, the cornerstone contribution by Longuet-Higgins, who considered a smooth bottom, is extended to sea waves and sandy bottoms characterized by a grain size that ranges from fine silt to fine gravel. For values of the grain size d* up to 0.05 delta* , delta* being the thickness of the viscous bottom boundary layer, the velocity profile is practically coincident with that predicted by Longuet-Higgins. If the grain size is further increased, the steady velocity component becomes larger and reaches a maximum value that is approximately 70% larger than that predicted by Longuet-Higgins. The maximum of the steady velocity component is attained for d* = 0.6 delta*. A further increase in d* leads to a decrease in the steady velocity component that, however, keeps always larger than that predicted for a smooth bottom. As the roughness size increases up to the values typical of medium sand, the steady velocity component increases. Then, a further increase in the roughness size leads to a decrease in the steady streaming even though, in the range of the roughness size presently investigated, the steady velocity component is always larger than that predicted for a smooth bottom.