44
will generate currents off-shore of the breaker line in the circulation model. Although
such currents in general will be of a small magnitude, they can cause some numerical
computational problems. The problem though is easily corrected by using as the dispersion
relationship
u) = a + k cos 9 U (3.76)
and the substitution
_jA(x,y) j(f kcottdx)
(3.77)
Rather than using a splitting method, the procedure will be to go directly from Eq. (3.51)
using Eq. (3.77) and writing
u>2 a1 = 2uik cos 9U (k cos 6 U)"1
(3.78)
in place of Eq. (3.55). Dropping the (j)zx term and all terms containing products of the
current components that cannot be reduced as explained above results in the following
parabolic equation.
Cg cos 6 + U'
(Cg cos 6 + U)AZ +
At
A + V Av +
§a
-cos20)A- i CCg{^)yJ +^A = 0
Now adding a phase shift
t cotdz-j icoafdz)
so that the free surface can be recovered by
rj(x)V) = A'ei(flcose'dz)
(3.79)
(3.80)
(3.81)
where 6 is the wave angle obtained by Snells law in the absence of any obstructions or y
variation of the depth or current. The following governing equation is thus obtained.
CgCosO + U'
(Cg cos 6 + U)A'X -|- i(k cos 9 k cos 9)(Ct cos 9 + U)A' +
At
A'
+VA'v + 2 (7) A> ~ 2kCi^ ~ Cs2 ~ 2
CC~)v
+ ^A' = 0 (3.82)