c. Lateral Turbulent Mixing Stress, dT /dx. For the time-averaged 

 lateral mixing force over the water depth, T Longuet-Higgins (1970) derived 



\ - h^L = K i = h(Np-x(gh)^) f (55) 



where 



tl = the lateral turbulent eddy stress due to waves 

 y-j- = the lateral turbulent eddy viscosity due to waves 

 N = a dimensionless, turbulent closure coefficient for lateral wave 



mixing proposed by Longuet-Higgins (1970) to be of the order NlO.OlC 



The term Tg is reserved for a true, turbulent Reynolds stress due to random 

 turbulent velocity interactions at scales far less than the wave orbital velo- 

 city scale. 



If wave setup is again neglected taking h = d = xtang gives 



dT 



dx dx 



so that the lateral turbulent mixing stress becomes 



^=Npg^(tanB)'/2 4 [it'^ % (57) 



dx dx dx 



d. Dimensional Longshore Currents . In summary, Longuet-Higgins (1970) 

 derived the momentum balance equation (42) 



dS dT^ 



dx B dx 



with equations (49), (54), and (57) 



^=Npg^(tanB)'/2 x'/2 ^ (56) 



dS 



^ 5 % 2/^ D\% .sina,-^/2 -% 



!: = IT Pg Y (tan3) ^ ( — — )x -^ = rx ^ 



dx 16 



r-, = - pC^g\(tan6) ^x^v = qxS^ 



'^^L ., %, .3/2 d r-% dv, d .-\ dv, 



dT" = NPs (^^^^) d^[^ df ] = P d^ f^ dS^ 



where we define three new constants 



r = -^PgV(tan3)'/2(^) (58a) 



85 



