A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

172 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
n – 1 ∂
( HDIFFx) ---- A
i + 1⁄ 2,
jk , ∂x
Hh u ∂ ⎛ ----- ⎞ ∂
---- A
⎝ ∂x⎠k∂y
Hh u ∂ n – 1
= ⎛ + ⎛ ----- ⎞ ⎞ =
⎝ ⎝ ∂y⎠k⎠
+ ⁄ , j
i 1 2
1
( Δx)
2
n – 1 n – 1 n – 1 n – 1
+ ------------- ( hi + 1,
j, kAH
( u 3
i + 1,
j, k i + ⁄ 2 , j, k–
ui + 1⁄ 2,
j, k)
n – 1 n – 1 n – 1 n – 1
– hi, j, kAH
( u
i, j, k i + 1⁄ 2,
j, k–
ui – 1⁄ 2,
j, k))
1
8( Δy)
2
n – 1
n – 1 n – 1
n – 1
+ ---------------- ( ( ( hi + 1,
j + 1,
k + hi, j+ 1,
k)
⋅ ( AH + A
i + 1,
j + 1,
k H )
i, j + 1,
k
n – 1 n – 1 n – 1
n – 1
+ ( hi + 1,
jk , + hi, j, k)
⋅ ( AH + A
i + 1,
j, k H ))
i, j, k
n – 1
n – 1
× ( + , j + 1,
k – + ⁄ , jk , )
ui 1⁄ 2
u i 1 2
n – 1 n – 1 n – 1
n – 1
– ( ( hi + 1,
j, k+
hi, j, k)
⋅ ( AH + A
i + 1,
jk , H )
i, j, k
n – 1
n – 1 n – 1
n – 1
+ ( hi + 1,
j – 1,
k + hi, j– 1,
k)
⋅ ( AH + A
i + 1,
j – 1,
k H ))
i, j– 1,
k
n – 1 n – 1
× ( + , j, k–
+ ⁄ , j – 1,
k)
) .
ui 1⁄ 2
u i 1 2
For the y-momentum equation (eq. 4.24), the finite-difference expressions are
n ∂(
Uv)
k ∂(
Vv)
k k – 1 n
⁄ 2
( ADVy) = ⎛---------------- + ---------------- + ( vw)
⎞ =
i, j+ 1⁄ 2 , k ⎝ ∂x
∂y
k + 1⁄ 2⎠
, + ⁄
i j 1 2
1 n
n
n
n
+ --------- ( ( U
4Δx i + 1⁄ 2,
j + 1, k + Ui + 1⁄ 2 , jk , ) ⋅ ( vi + 1,
j + 1⁄ 2,
k + vij , + 1⁄ 2 , k)
–
n
( Ui – 1⁄ 2 , j + 1,
k + U 1 i – ⁄ 2 , j, k)
⋅ vij 1⁄ 2
1
+ (
4Δy
–
n
n
--------- ( V 3 ij , + ⁄ 2 , k + Vi, j + 1⁄ 2,
k)
⋅ v 3 ij ⁄ 2
n
n
( Vi, j + 1⁄ 2 , k + Vi, j – 1⁄ 2 , k)
⋅ vi j 1⁄ 2
n
n
n
( , + , k + vi 1,
+ ⁄ , k))
– j 1 2
n
n
( , + , k + , + ⁄ , k)
v ij 1 2
n
n
( , + , k + , – ⁄ , k))
v ij 1 2
1 n
n
n
n
+ -- ( ( w
4 i, j+ 1,
k – 1⁄ + w
2 ijk , , – 1⁄ ) ⋅ ( v
2 ij , + 1⁄ 2 , k – 1 + vij , + 1⁄ 2 , k)
(D.4)
n
n
n
n
– wi, j 1,
+ ⁄ + , , + ⁄ ⋅ ( , + ⁄ , k + , + ⁄ , k + 1))
, (D.5)
n
n
( CORy) = ( fU
ij , + 1⁄ 2 , k k)
=
i, j+ 1⁄ 2 , k
( + k 1 w
2 i j k 1 ) v
2 ij 1
2
v ij 1 2
f n
n
n
n
+ -- ( U
4 i + 1⁄ 2 , j + 1, k + Ui – 1⁄ 2 , j + 1, k + Ui – 1⁄ 2 , j, k+
Ui + 1⁄ 2 , j, k)
, (D.6)