51
Using Eq. (4.15) to eliminate V* in Eq. (4.19) and (4.18) to eliminate U* in Eq. (4.14) the
following are obtained. For the x direction
AT AT
r + daxSiU = DayS,V'
AT
Un+1 + g8tfÂˇ* = Un
and for the y direction
AT AT
r+1 + v'"+1 = +
AT
yn+1 + 9^Svfjn+1 = vn
(4.20)
(4.21)
(4.22)
(4.23)
The above equations are numerically stable for any combination of AT, AX, and AY.
However as a circulation model they are incomplete in that they do not contain any of the
forcing terms that drive the circulation. In particular in this model wave forcing represented
by gradients in the radiation stresses and bottom friction are of concern. Bottom friction is
especially important because at steady state conditions gradients in the radiation stresses
must be balanced by either gradients in the water surface elevation or by bottom friction
forces. In the absence of friction forces the model could yield continuously accelerating
currents. The method of adding the forcing terms is somewhat arbitrary in that the rational
derivation of the finite differencing scheme of Sheng and Butler cannot be expanded to
include these essentially non- linear terms. The addition of these terms to the model also
limits the latitude of the ratio of jn a manner that is not analyzable. That both the
friction and the radiation stresses are nonlinear should be recognized. The non-linearity
of the velocity friction term is quite evident since it is represented by the quadratic term
pF\U\U. The radiation stress terms, which represent excess momentum flux induced by
waves, are functions of the wave height squared, based on linear wave theory. In shallow
water the wave height H is a function of the total water depth, h0 + fj so that H2 is a
function of fj2.
All the forcing terms and the nonlinear acceleration terms are added at the present or
time level with the exception of the bottom friction term. The quadratic friction term