Geoscience Reference
In-Depth Information
be integrated from a depth
z
to the water surface
η
η
η
−=
∂
∂
p
x
∫
∫
ppz
()
dx
=−
g
ρ
dx
=−
g
ρ η
(
−
z
)
0
3
0
3
0
3
z
z
Assuming the atmospheric pressure
p
0
to be constant, we can write the horizontal
pressure gradient as
1
∂
∂
p
x
∂
∂
η
1
∂
∂
p
x
∂
∂
η
=
g
=
g
ρ
x
ρ
x
0
1
1
0
2
2
In this way the pressure can be completely eliminated from the equations and is
substituted by the water level
.
One more term that has to be dealt with is the turbulent friction term. If this
term is integrated over the whole water column we obtain
η
s
s
b
2
−
∂
∂
u
x
∂
∂
u
x
∂
∂
u
x
∫
v
1
dx
=
v
1
v
1
M
2
3
M
M
3
3
3
b
where the indices
s
and
b
stand for surface and bottom. These two terms are exactly
the stress per density in the
x
1
direction that is exerted over the two interfaces of
the surface
τ
s
and the bottom
τ
b
:
s
2
∂
∂
u
x
1
ρ
(
)
∫
s
b
v
1
dx
=
ττ
−
.
M
2
3
1
1
3
0
b
These values are boundary values that have to be imposed on the moving fluid or,
as is the case with the bottom friction, can be computed from the actual flow field.
Their exact formulation will be given in Section 3.5 on boundary processes.
If the vertical integrated equations are divided by the total depth
H
, the 2D
shallow water equations result:
2
2
∂
∂
u
t
∂
∂
u
x
∂
∂
u
x
∂
∂
u
x
+
∂
∂
u
x
−=−
∂
∂
η
1
(
)
+
s
b
H
1
+
u
1
+
u
1
fu
g
+
ττ
−
v
1
1
(3.25a)
1
2
2
1
1
M
2
2
x
ρ
H
1
2
1
0
1
2
and
2
2
∂
∂
u
t
∂
∂
u
x
∂
∂
u
x
∂
∂
u
x
+
∂
∂
u
x
+=−
∂
∂
η
1
(
)
+
s
b
H
2
+
u
2
+
u
2
fu
g
+
ττ
−
v
2
2
(3.25b)
1
2
1
2
2
M
2
2
x
ρ
H
1
2
2
0
1
2
Search WWH ::
Custom Search