All other models for lateral mixing shear stress, tl, use some different 

 combinations of characteristic velocity and length scales to approximate the 

 eddy viscosity. They are summarized in Table 4 along with the resulting ex- 

 pression for Vl- Three distinct categories of thought have emerged for both 

 the velocity u, and length £ scales employed. 



(1) Reference Velocity, u . Thornton (1969, 1970a) and Jonsson, 

 Skovgaard, and Jacobsen (1974) used time averaged (over one wave period) values 

 of the maximum orbital velocity near the bottom Ug^^^ as the reference velocity 

 u. To reduce its influence outside the breakers where turbulent mixing is 

 smaller, Jonsson, Skovgaard, and Jacobsen (1974) used the mean overdepth, u , 

 in place of the bottom value. 



Madsen, Ostendorf, and Reynolds (1978) used the maximum orbital velocity 

 predicted by linear long wave theory. For shallow water this reduces to 

 equation (53) and is identical to the original model theory used by Longuet- 

 Higgins (1970). It should be noted that 



and thus u-^^ is related to the surf zone celerity. Longuet-Higgins (1970, 

 1972a) never stated that the reference velocity taken was the wave celerity 

 as reported by Thornton (1976, Table 3). Equation (90) also relates the 

 time-averaged and maximum values of these two approaches. Kraus and Sasaki 

 (1979) took slightly different models inside and outside the surf zone and 

 followed the original model of Longuet-Higgins (1970). 



A completely different approach was taken by Battjes (1975). He felt 

 surf zone turbulence was generated by wave breaking so that the velocity scale 

 chosen should reflect this fact. Since kinetic energy transport is propor- 

 tional to velocity cubed, Battjes took the one-third powe^r of the wave energy 

 dissipation per unit area and per unit mass, i.e., (D/p) ^ as the reference 

 velocity. Here, D, the rate of energy dissipation per unit area, was found 

 from equation (50). Skovgaard, Jonsson and Olsen (1978) took the same result 

 inside the surf zone but a simple proportion of v, (at the breakers) to give 

 less mixing outside the breakers. 



Inman, Tait, and Nordstrom (1970) related the characteristic velocity to 

 the breaker height and number of waves in the surf zone. The end result was 

 simply H^/T. 



(2) Characteristic Length Scale, £ . .Thornton (1969, 1970a) and 

 Jonsson, Skovgaard, and Jacobsen (1974) used the excursion amplitude to get 

 results for v^ that were twice those derived by Thornton. The discrepancy is 

 discussed by Jonsson, Skovgaard, and Jacobsen (1974), Nielsen (1977), and Gourlay 

 (1978) but without resolution. Fortunately, such differences in v^ produce 

 relatively small (10 percent) changes in v magnitude and little shape change as 

 indicated by Jonsson, Skovgaard, and Jacobsen (1974). Inman, Tait, and Nordstrom 

 (1970) used solitary wave theory to calculate the horizontal excursion distance 

 employed in their model. Further discussion can be found in Longuet-Higgins 

 (1972a). 



100 



