since the location of the break point and any point where wave reformation 

 occurs are not known a priori. 



427. The slope -dependent term in Equation 33 improves numerical stabil- 

 ity of the model. Some simulations performed omitting this term experienced 

 numerical oscillation at the shoreward bar face as the slope grew steep and 

 the trough became more pronounced. The slope -term tends to flatten the 

 trough, since the transport rate at the shoreward bar face is reduced. 



428. The numerical scheme proved to be very stable under a wide range 

 of conditions in spite of the irregular bathymetry that occurs if bars are 

 formed. Typical length and time -steps used in the model are Ax = 0.5-5.0 m 

 and At = 5-20 min. The length step has to be chosen so as to resolve the 

 main morphologic features. A shorter length step requires a correspondingly 

 shorter time -step to maintain numerical stability. An effort was made to 

 derive an explicit stability criterion but was not successful. Therefore, at 

 the present time, trial and error must be used to determine appropriate values 

 of Ax and At for the particular application. 



429. For a beach profile exposed to constant wave and water level 

 conditions, the profile shape predicted by the model approaches a steady- 

 state, resulting in an equilibrium profile. The approach to equilibrium is 

 controlled by the rate at which energy dissipation in the surf zone attains 

 the equilibrium value D . A bar, if formed, causes the break point to 

 translate in the seaward direction as it grows, making the offshore boundary 

 of the surf zone move accordingly. At equilibrium, the break point is 

 stationary and the energy dissipation per unit water volume is constant 

 throughout the surf zone, being approximately equal to D^^ , corrected by the 

 slope-dependent term in Equation 33. 



Calibration and Verification 



430. The numerical model was applied to simulate beach profile evolu- 

 tion for a number of erosional cases from the LWT experiments. As an objec- 

 tive criterion for judging agreement between the simulated and measured beach 

 profile, the sxom of squares of the difference of measured and calculated 

 depths was formed according to 



175 



