Geoscience Reference
In-Depth Information
U
x
∂
U
x
∂
U
y
∂
U
x
∂
g
∂
z
s
1
ρ
∂
T
xx
∂
1
ρ
∂
T
xy
∂
−
τ
bx
ρ
+
=−
x
+
+
(6.41)
x
y
∂
x
y
h
−
τ
by
ρ
U
x
∂
U
y
∂
U
y
∂
U
y
∂
g
∂
z
s
1
ρ
∂
T
yx
∂
1
ρ
∂
T
yy
∂
+
=−
+
+
(6.42)
x
y
∂
y
x
y
h
Corresponding to Eq. (6.40), the stream function
ψ
in the depth-averaged 2-Dmodel
is defined by
1
h
∂ψ
1
h
∂ψ
U
x
=
y
,
U
y
=−
(6.43)
∂
∂
x
The vorticity is still defined as
=
∂
U
y
∂
−
∂
U
x
∂
(6.44)
x
y
Therefore, the following stream-function equation is obtained by inserting Eq. (6.43)
into Eq. (6.44):
2
2
∂
ψ
+
∂
ψ
1
h
∂
h
x
∂ψ
1
h
∂
h
y
∂ψ
−
x
−
=−
h
(6.45)
∂
x
2
∂
y
2
∂
∂
∂
∂
y
Cross-differentiating Eqs. (6.41) and (6.42) with respect to
y
and
x
and subtracting
them yields the vorticity equation:
∂(
U
x
)
+
∂(
U
y
)
=
∂
∂
)
∂
∂
+
∂
∂
)
∂
∂
(ν
+
ν
(ν
+
ν
+
S
(6.46)
t
t
∂
x
∂
y
x
x
y
y
where
∂
∂
2
2
2
y
+
∂
U
y
∂
U
y
∂
∂
ν
−
∂
ν
U
x
∂
2
∂
ν
y
−
∂
U
x
∂
t
t
t
S
=
+
x
2
y
2
∂
∂
∂
∂
x
x
y
x
2
∂
2
U
y
∂
2
U
y
∂
2
U
y
∂
2
∂
2
U
x
2
U
x
∂
2
U
x
∂
+
∂
+
∂ν
+
∂
−
∂ν
+
∂
t
t
+
∂
x
2
y
2
∂
∂
∂
x
2
y
2
∂
∂
x
x
y
y
x
y
τ
by
ρ
τ
bx
ρ
−
∂
∂
+
∂
∂
x
h
y
h
Similarly, differentiating Eq. (6.41) with respect to
x
and Eq. (6.42) with respect
to
y
and adding them together leads to the following Poisson equation for the