Integral flow properties of the swash zone and averaging

Abstract

The swash zone is that part of a beach over which the instantaneous shoreline moves back and forth as waves meet the shore. This zone is discussed using the nonlinear shallow water equations which are appropriate for gently sloping beaches. A weakly three-dimensional extension of the two-dimensional solution by Carrier & Greenspan (1958) of the shallow water equations for a wave reflecting on an inclined plane beach is developed and used to illustrate the ideas. Thereafter attention is given to integrated and averaged quantities. The mean shoreline might be defined in several ways, but for modelling purposes we find the lower boundary of the swash zone to be more useful. A set of equations obtained by integrating across the swash zone is investigated as a model for use as an alternative boundary condition for wave-resolving studies. Comparison with sample numerical computations illustrates that they are effective in modelling the dynamics of the swash zone and that a reasonable representation of swash zone flows may be obtained from the integrated variables. The longshore flow of water in the swash zone is in many ways similar to the Stokes’ drift of propagating water waves. Further averaging is made over short waves to obtain results suitable as boundary conditions for longer period motions including the effect of incident short waves. In order to clearly present the work a few simplifications are made. The main result is that in addition to the kinematic type of boundary condition that occurs on a simple, e.g. rigid, boundary two further conditions are found in order that both the changing position of the swash zone boundary and the longshore flow in the swash zone may be determined. Models of the short waves both outside and inside the swash zone are needed to complete a full wave-averaged model; only brief indication is given of such modelling.