(2) Weak Currents at Large Angles . As observed in the laboratory 

 and field (see Ch. 2), it is the occurrence of relatively large wave incidence 

 angles that drives the longshore current (see Fig. 12) . Liu and Dalrymple 

 (19 78) concluded from studies of available field data (Inman and Quinn, 1951; 

 Balsillie, 1975; Komar, 1976a) that a weak current (U/iig «1) large wave angle 

 model for bottom shear stress was consistent with field observations. Using 

 this assumption they derived the expression 



pC. _ ^ _ ^ 



T^ = — - u^ [(vsin2a)i + 2v(l + sin2a)j] (81) 



By 77 Bm 



with 1, j the unit vectors in the x and y directions, respectively. For small 

 angles a, the original model formulation of Longuet-Higgins (1970), equation 

 (51) is recovered, as expected. 



Of major interest was the theoretical longshore current velocity profile 

 that resulted in comparison with that derived by Longuet-Higgins (1970) , ne- 

 glecting lateral mixing stresses and corrected for wave setup due to normal 

 wave incidence, i.e., equation (71), Liu and Dalrymple (1978) derived the fol- 

 lowing complicated expression for the longshore current without lateral mixing 



V* = ^.JL ghtanB(^)[5-6(^)V][l-0'gh]-^. 



{(1+3y2/8) + (l-5Y2/16)(^)'gh - ^(^) g2h2}"' (82) 



Equation (79) uses Snell's Law to include wave setup from refraction. Conse- 

 quently, the large-angle velocity profile that results from equation (82) is 

 different from the small-angle modified form (eq. 71) for two reasons; namely, 

 the bottom shear-stress formulation and the inclusion of wave refraction in 

 the wave setup. Comparison of the results of the two theories must be viewed 

 with this in mind. 



The results of two comparisons made by Liu and Dalrymple (1978) for their 

 weak current large-angle theory are shown in Figure 26. Here, X is the di- 

 mensionless surf zone distance such that X = 1 at the breaker. The symbol 

 V* means the modified theory of Longuet-Higgins given by equations (71) in 

 general and equation (72) at the breaker. Figure 26 (a and b) demonstrates a 

 significant deviation between the two theories as incident wave angle in- 

 creases. Even at a = 10°, approximately a 20-percent difference is observed 

 at the breaker line ^Fig. 26, a) and across the surf zone (Fig. 26, b). It was 

 also shown that the v*/il ratio depended upon two factors, (tan3/C^) and 

 wave angle a. When the ratio tang/C^ was small (mild slopes or large bottom 

 friction as usually found in the field), the v*u^ ratio was also relatively 

 low across the surf zone for < a <45°. This gives further credence to the 

 weak current large-angle theory for dissipative-type beaches . 



Liu and Dalrymple also developed a validity diagram (see Fig. 5 in Liu 

 and Dalrymple, 1978) for longshore current theories that depended upon the 

 ratio v*/vL ), at the breaker line to separate the weak and strong current 



95 



