Wave-induced setup of the mean surface over a sloping beach

A new theoretical approach for the wave-induced setup over a sloping beach is presented that takes into consideration the explicit variations of the surface waves due to bottom slope and viscosity. In this way, the wave forcing of the mean Lagrangian volume fluxes is calculated without assuming that the local depth is constant. The analysis is valid in the region outside the surf zone and is based on the shallow-water assumption. A novel approach for separating the viscous damping of the waves from the frictional damping of the mean flow is introduced, where the mean Eulerian velocity is applied in the bottom stress for the mean fluxes. In the case where the onshore Lagrangian mean transport is zero, a new formula is derived for the Eulerian mean free surface slope, in which the effects of bottom slope, viscous wave damping and frictional bottom drag on the mean flow are clearly identified. The analysis suggests that viscous damping of the waves and frictional dissipation of the Eulerian near-bed return flow could lead to setup outside the surf zone.