Because of a singularity (described below), the equilibrium principle cannot 

 be applied at the shoreline; the equilibrium concept-based portion of the 

 model must be terminated some distance seaward of the water's edge. Ad hoc 

 geometrical-based assumptions are employed to simulate profile change in the 

 swash zone and on the beach face and dune. Despite these apparent severe 

 limitations, the Kriebel model has been demonstrated to yield qualitatively 

 and quantitatively reasonable results (Kriebel 1982, 1986). 

 Governing equations 



76. The following discussion is based on the work of Dean (1977), 

 Kriebel (1982), Moore (1982), and Kriebel and Dean (1985a, b). The two 

 reports by Kriebel (1984a, b) are recommended for a more complete discussion. 

 The fundamental assumption is that the beach profile is well approximated by 

 the following relationship: 



h(x) = Ax b (9) 



which is the general power law form of Equation 1. As discussed in Part II, 

 the most appropriate value of the scale coefficient b is 2/3, which can be 

 explained by uniform energy dissipation (Dean 1977). However, use of a 

 variable exponent, although highly empirical, results in better fits to some 

 profiles (Balsillie 1982). The best fit scale coefficients determined in 

 this manner do not deviate greatly from the value of 2/3. The shape 

 parameter A was empirically related to the mean grain size (Figure 8) by 

 Moore (1982). Moore recommended that A be represented by a smooth line 

 drawn through somewhat scattered results to give a monotonically increasing 

 function of grain size. This relationship expresses the well-known result 

 that coarse-grained beaches tend to be steeper than fine-grained beaches. 



77. The basic equation for the cross-shore transport rate Q c per 

 unit beach width is assumed to be given by 



Q c = k(D - D eq ) (10) 



where 



k = empirical coefficient 



D = energy dissipation per unit volume (assuming spilling waves) 



D e q = energy dissipation corresponding to the equilibrium 

 profile for a beach of given grain size 



39 



