23
and a spilling breaker assumption. Longuet-Higgins wrote
/ dU dV\ ..
n = p'23)
in which ex and ey Eire mixing length coefficients.
Longuet-Higgins reasoned that the mixing length parameter ex must tend to zero as
the shoreline was approached. He assumed that ex is proportional to the distance from the
shoreline, X, multiplied by a velocity scale which he chose as the shallow water celerity
y/gh. Thus ex is written as
ex = NX \/gh (2-24)
where N is a dimensionless constant for which Longuet-Higgins chose the limits as
0 < N < 0.016
(2.25)
Obviously, for physical reasons there needs to be some limit on the scale of these eddies.
Ebersole and Dalrymple arbitrarily chose the value of ex near the breaker line to be the
maximum value and for ex to be constant offshore from the breaker line. The same limit
upon ex is used in this study. The coefficient ev is assumed to be a constant and it is
arbitrarily assigned the maximum value of ex.
In the present study the further assumption is made in computing that the cross
derivatives are negligible (a common assumption in fluid mechanics). Therefore, the lateral
mixing terms become
(£)
(2.26)
/ av\
\£xdx)
(2.27)
in the x direction and
in the y direction. This means that the lateral mixing term is represented by a purely lateral
diffusion of momentum. That the lateral mixing term is a lateral diffusion term has some
significance in that the diffusion term is expressed explicitly and thus there results a time
step limitation to an implicit model. This will be discussed in section 4.1 dealing with the
finite differencing of the equations governing the circulation model.