III. THE NUMERICAL MODEL 



1. Description . 



There are several methods of modeling bathymetric changes due to the 

 presence of a littoral barrier. An attempt can be made to either model the 

 complete hydrodynamics and the resulting sediment transport or model using a 

 combination of analytical and empirical sediment transport equations. The 

 second method was chosen due to past relative success. 



At least two methods of employing sediment transport equations exist: a 

 fixed longshore and cross-shore grid system where the depth is allowed to 

 vary or a fixed longshore and depth system where the cross-shore distance is 

 allowed to change. Although it may seem somewhat awkward, the latter system 

 was chosen for the model. This method allows the modeler to think of 

 bathymetric changes due to a littoral barrier in terms of the effect on the 

 contours; i.e., the contour realinement due to the structure's presence is 

 observed. One limitation of this, approach, at least as it was applied here, 

 is that each depth contour must be single-valued; it is not possible to 

 represent bars. 



The next step in formulating the model was choosing the specific 

 representation of the bathymetry. The model is an n-line representation of 

 the surf zone in which the longshore direction x is divided into equal 

 segments each ax in length. The bathymetry is represented by n-contour 

 lines, each a specified depth, which change in offshore location according to 

 the equation of continuity. There are two components of sediment transport 

 at each of the contour lines, a longshore component, Qx, and an offshore 

 component, Qy. Figure 1 is a definition sketch showing the beach profile 

 representation in a series of steps and the planform profile representation 

 and notations used. 



Implementation of the sediment transport equations requires knowledge of 

 the wave field and the equilibrium offshore profile. A discussion of the 

 refraction and diffraction schemes follows. The equilibrium profile is 

 introduced when it is convenient. As an introduction to the logic used in 

 the numerical model, a flow chart is presented in Figure 2. 



2. Refraction . 



A refraction scheme compatible with variable Ay's was required because 

 of the variable distance to fixed depth contours (as opposed to the more 

 usual fixed grid system where a grid center has a longshore and offshore 

 coordinate with a variable depth). One of the benefits of the n-line model 

 is the ease with which the response of the contours to a particular wave and 

 structure condition can be visualized. A fixed grid system and an 

 interpolation scheme could have been used to obtain the wave field; however, 

 this would have reduced accuracy and increased computation time. The scheme 

 developed also saves computation time because it does not use differential 

 products terms. 



