smoothing procedures were needed every 10th time step to dampen the time- 

 splitting instability inherent with three level schemes. Considerable aver- 

 aging of all dependent variables was incorporated within the difference for- 

 mulations which were essentially centered for all space derivatives. The 

 lateral mixing stress gradients were evaluated at the lowest time level 

 (n-1) for stability reasons. 



The numerical model results for longshore current profile under oblique 

 wave attack on a plane beach with and without lateral mixing are shown in 

 Figure 35(a) (Ebersole and Dalrymple, 1980). The analytic model results for 

 the same example are given in Figure 35(b) using the Longuet-Higgins (1970) 

 model. A significant difference is apparent. With no lateral mixing, the 

 large difference between the analytic and numerical results is completely due 

 to numerical "viscosity," i.e., truncation error terms stemming mainly from 

 the local and convective accelerations that give nimierical dispersion. The 

 dotted line in Figure 35(a) is from the 1976 model which omitted the convective 

 acceleration term. The difference can be attributed to additional numerical 

 dispersion associated with the truncation error from finite-differencing the 

 convective term. Decreasing the grid scales will help but it is apparent 

 that the physical mixing desired will generally be dominated by the undesir- 

 able numerical mixing inherent in the scheme employed. Use of the model to 

 quantify longshore current profiles, nearshore circulations, and rip currents 

 is questionable for this reason. Circulation cells will be generated and rip 

 currents foirm but how certain can the magnitudes of the results be when the 

 numerical information losses dominate the physical processes? 



Ebersole and Dalrymple (1979, 1980) recognize the importance of including 

 the convective acceleration and lateral mixing terms for accuracy in the 

 physical processes being modeled. They also noted the numerical mixing pre- 

 sent in Figure 35 (a and b) . However, the time step was selected "...to be 

 significantly lower than the two-dimensional Courant stability criterion..." 

 and no discussions of numerical accuracy are presented. 



d. Others in United States . Allender, et al. (1978) used the numerical 

 model developed by Birkemeier and Dalrymple (1975, 1976) to compare field 

 observations in Lake Michigan (see Ch. 4). Hudspeth (1979)^^ describes ex- 

 tensions to the MIT model (Liu and Mei, 1975) to include tide- induced currents. 

 The finite-element method is being implemented. 



Two other U.S. computer-based models are included in Table 6 for complete- 

 ness. These were developed by Fox and Davis (1971, 1976). In the 1971 simu- 

 lation model for eastern Lake Michigan, longshore current is computed as the 

 first derivative of the local barometric pressure variation with time. For 

 small water bodies where local storms generate the wind waves at the coast, 

 this approach is physically defensible. The scaling parameters are found ex- 

 perimentally and are not generally transferable to other locations. The 1976 

 Coastal Storm Model (Fox and Davis, 1976) forecasts or hindcasts longshore 

 current conditions for a given storm size, shape, intensity, and path. Stan- 

 dard wave height forecasting procedures are used to compute period, breaker 



^''huDSPETH, R. , "Effects of Jetties in Steady Currents," Research in Ocean 

 'Engineering^ Massachusetts Institute of Technology, Cambridge, Mass., 

 Vol. 1, No. 3, 1979 (not in bibliography). 



127 



