elevation. Such a transformation leads to (1) the same order of 

 numerical accuracy in the vertical direction at all horizontal 

 locations, and (2) a smooth representation of the bottom topography. 

 Although additional terms are introduced by this transformation, the 

 advantages warrant its application. Models using regular rectangular 

 grid in the vertical direction cannot accurately resolve the shallow 

 coastal area unless a large number of grid points is used in the deeper 

 offshore area. In addition, if the bottom is approximated by a series 

 of rectangular steps, estimate on bottom stress may be distorted and 

 hence is not suitable for studying sediment transport problems. 



To better resolve the complex shoreline geometries and bottom 

 features, a non-uniform grid is often required in the x and y 

 directions (Butler and Sheng, 1982). To allow ease in numerical 

 analysis and as shown in Figure 2b, this non-uniform grid (x,y,z) is 

 further mapped into a uniform grid (a,Y,o): 



X = a 



X ■" ^" 



y = a^ + byT ^ 



(11) 



The transformed three-dimensional equations of motion in a,Y,o 

 grid system are rather complex. Detailed equations and boundary 

 conditions in non-dimensional form can be found elsewhere (Sheng, 

 1981). Staggered numerical grid is used in both the horizontal and 

 vertical directions. 



External Mode 



struc 

 separ 

 (exte 

 signi 

 three 

 (Shen 

 three 

 compu 

 model 



In the present study, numerical computation of the vertical flow 

 tures (internal mode), which are governed by slower dynamics, are 

 ated from the computation of the vertically-integrated variables 

 rnal mode). This so-called "mode splitting" technique resulted in 

 ficant improvement of the numerical efficiency of a 

 -dimensional hydrodynamic model for Lake Erie 



g et al., 1978). It allows for computation of the 



-dimensional flow structures with minimal additional cost over 

 tation of the two-dimensional flow with a vertically-integrated 



The external mode is described by the water level (;) and the 

 vertically-integrated mass fluxes (U,V) = J^ (u,v) Hda. Performing 

 vertical integration of the transformed three-dimensional equations of 

 motion, and rewriting (ci,y) as (x,y) for simplicity, we obtain: 



3t 



_i_ 



8U 

 3x 





3V 

 ay " 









8U _ 



at " 



- 





9 A 

 9X \, 







VH 







1 



PoUv 



-/ 







ghj 



lid 

 ax 



+ 



(12) 



£^=.^^H_U-L^^U H(u<.) 



f.{( 



- fV 



pda+op 



5!ii£ + Jl(, 



ax 



Hdo + (H.D.). 



sx" bx 



(13) 



271 



Sheng 



