The second input required by the continuity equation to predict the 

 bathymetric changes is the cross-shore sediment transport. The governing 

 equation for onshore-offshore transport (after Bakker, 1968) is 



\.r"X,oh-i-^^-.j^"^^J 



where Cqcc is an activity factor (inside-the surf zone = 10 feet per second 

 for the prototype simulation herein, 10" feet per second for the physical 

 model simulation) (see App. A. for a discussion) and WE;Q(i,j) is the 

 positive equilibrium profile distance between y(i,j) and y(i,j-l), determined 

 from the equilibrium profile used in the numerical model h = Ay^/^ (Dean, 

 1977). See Appendix A for discussion of the value of A. The physical 

 interpretation of equation (28) is that as this profile steepens (flattens), 

 sediment is transported offshore (onshore). 



b. Methods of Solution . Three separate finite-difference techniques 

 were used to solve the equations: 



(1) Explicit longshore-continuity and explicit cross-shore 

 continuity; 



(2) Implicit longshore-continuity and explicit cross-shore 

 continuity for half a time-step then vice versa; and 



(3) Implicit longshore-cross-shore continuity. 



An explicit formulation was first developed which used the refraction 

 scheme, the distribution of longshore sediment transport across the surf 

 zone, and the onshore-offshore sediment transport equation. Problems in 

 addition to the usual ones which are encountered with explicit methods (e.g., 

 computation time and cost) were immediately realized. In the explicit 

 method, both transport computations are based on the former values of the 

 contour locations and are completely uncoupled. Stability of an explicit 

 scheme requires a small time-step. In addition, the noncoupled nature of the 

 equations, in some cases, resulted in crossing of the contours due to the 

 transport computed. 



It is logical to assume that an implicit formulation of the longshore 

 transport equation used as input to the continuity equation along with the 

 explicit onshore-offshore transport component would help the numerical 

 stability (on the other half time-step, the longshore component would be 

 computed explicitly and the onshore-offshore transport equation would be 

 solved implicitly with the continuity equation). Although this scheme would 

 be superior to the explicit procedure, it still would be susceptible to 

 crossing contours. It should be noted that the magnitude of the coefficient 

 used in the onshore-offshore equation is very important to the extent that 

 the simulation models natural phenomena. If the coefficient is \/ery small or 

 vanishes, sediment will not move offshore and contours will cross because of 

 the variation in the distribution of longshore sediment transport across the 

 surf zone. If the coefficient is too large, the onshore-offshore transport, 

 may become large enough that on a particular time step, an offshore contour 



22 



