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
+
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
 
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