53
The following notation is used to describe the finite differencing of Eq. (4.26): Fj
represents the value of F(i,j) where F is any term and i and j designate the grid row
and grid column. Because of the staggered grid system used for the circulation model as
previously explained, in which the velocity components are specified at the grid edges, the
velocity terms U and V are averaged across the grid so as to produce a grid centered term
for use in the wave model. Thus 17) in the wave model is actually \(Uj + ^)+1) and similarly
for V and other grid designations.
A Crank-Nicholson finite differencing scheme is used for Eq. (4.26). All terms not
involving an x derivative are an average of the value of the term on the grid row and on
the i + l grid row. Terms having y derivatives are centered at, j on the row and +1, j
(X'+1X'.)
on the + 1 grid row. The one term with an x derivative is written as 1 1 Thus all
terms are centered at + |,j. The finite differencing of Eq. (4.26) is written as
1
+ 2
4+1-4
AX
+ '- [(jfc+1 cos $<+1 Jfc)+1 cos 0,+1)A;+1 + (jfc* cos 0* k) cos 0*)Aj]
1 f [(C7g cos g + C/)/o-]->-1 ~ [(Cg cos g +- f7)/cx]-
h 2AX \ \(Ca cos 0 + U)/(r]fl + \(Ca cos 0 + U)/o
Va
(Ce cos 6 + U)
r
- 4I
2AY
l}(4++4)
( Vo V |a;+i a;.,]
^(<7,0)8 0 +t/)^. 2AY
1
+ 2
2(Cycos0 + U)
(Cg cos0 + C7)
<+' r(V/);i\ (V/a))t\
l
+
2(C¡ cos 0 + 17)
)!(
2AY
w*)Ui PH-1
2AY
4
t
4
kCg{l -cosJ0)l*+1 A<+1 +
(Cg cos 0 + U)
kC¡( 1 cos2 0)
(#),
(Cg cos 0 + U)
i+l
'P
'+!
Vj
(),
+
A>+1
V)
(Cg cos 0 + i/);+1 (Cg cos 0 + U))
/
1 w*+1 A,+1 w*- A\ )
+4 {(Cycos0 + u)*1 + (Cgcos0 + u)) j 0 ^4'27)