would move too far shoreward, thereby crossing an inshore contour or vice 

 versa. Once the contours cross, not only does the bathymetry become 

 unrealistic, but mathematically, the equation which computes the longshore 

 distribution across the surf zone changes signs at some locations and the 

 entire model becomes physically unrealistic. 



To circumvent these problems, an implicit scheme that simultaneously 

 solves the three governing equations, was developed. Utilizing equation (26) 

 and the one-dimensional Jacobian (Ay/Ah) to convert to Qx(h), the total 

 longshore transport equation (27), the following equation is obtained, 



, ^„ ,3/2^ u «3/2 3, , ,^ ,3/2^ , „3/2 3, 



ir / / 1 >j-i 



i,j 



r \\ — ^^^^^^ — ) ;■ "' \\ '-" ^ ) j 



K) 



X sin (2e - 2a^) (29) 



Qx^TsJ) represents the sediment transport between depths h(i,j) and h(i,j-l) 

 (see Fig. 1). The term in brackets represents the normalized distribution of 

 longshore transport between h(i,j) and h(i,j-l); e is the averaged wave angle 

 at the location of Qx(i>j) and uq is the local contour orientation angle. 

 Defining everything except sin (2e - 2ac) as v(i,j) and using a superscript 

 to denote a time step, this equation can be written 



Qf^ = V. . sin (2e - 2a"''M (30) 



^i,j ^'-^ ^ 



The assumption has been made that the wave field (H and e) do not vary during 

 the bathymetric changes over the time-step. Using the following trigonometric 

 identities, 



sin (2a - 2b) = sin 2a cos 2b - cos 2a sin 2b (31a) 



cos 2a =2 cos2 a - 1 (31b) 



sin 2a = 2 sin a cos a (31c) 



and recognizing that the following expression is an approximation 



1 , n+1 n+1 ^ n " i 

 sin uTh. . ~- ^ ''^>^ "^i-l-J ^^J ''^-^^y (32) 



c 'i,j 



K^'^,-.j-,--i.j'^r 



23 



