Geoscience Reference
In-Depth Information
by Eq. (2.76). Note that
∂
h
/∂
t
in Eq. (2.79) may be replaced by
∂
z
s
/∂
t
, because the
bed change is omitted.
Integrating the
x
-momentum equation (2.66) over the flow depth yields
z
s
z
s
z
s
z
s
u
x
)
∂
∂
u
x
∂
∂(
∂(
u
y
u
x
)
∂(
u
z
u
x
)
t
dz
+
dz
+
dz
+
dz
x
∂
y
∂
z
z
b
z
b
z
b
z
b
g
z
s
z
b
z
s
z
s
z
s
∂
∂τ
xx
∂
∂τ
∂τ
xz
∂
z
s
1
ρ
1
ρ
1
ρ
xy
=−
x
dz
+
x
dz
+
y
dz
+
dz
(2.80)
∂
∂
z
z
b
z
b
z
b
and then applying the Leibniz rule to this equation yields
z
s
u
x
dz
z
s
u
x
dz
∂
∂
u
hx
∂
z
s
∂
u
bx
∂
z
b
∂
+
∂
∂
u
hx
∂
z
s
u
bx
∂
z
b
−
+
−
x
+
t
t
t
x
∂
∂
x
z
b
z
b
z
s
u
y
u
x
dz
+
∂
∂
u
hy
u
hx
∂
z
s
u
by
u
bx
∂
z
b
−
+
+
u
hz
u
hx
−
u
bz
u
bx
y
∂
y
∂
y
z
b
z
s
z
b
τ
xx
dz
gh
∂
z
s
1
ρ
∂
∂
1
ρ
τ
xx
,
s
∂
z
s
1
ρ
τ
xx
,
b
∂
z
b
=−
x
+
−
x
+
∂
x
∂
∂
x
z
s
z
b
τ
xy
dz
1
ρ
∂
∂
1
ρ
τ
xy
,
s
∂
z
s
1
ρ
τ
xy
,
b
∂
z
b
1
ρ
(τ
+
−
+
+
−
τ
)
(2.81)
xz
,
s
xz
,
b
∂
∂
y
y
y
Substituting boundary conditions (2.68) and (2.71) into Eq. (2.81) results in the
depth-integrated
x
-momentum equation:
hU
x
)
∂
+
∂(
hU
y
U
x
)
∂(
hU
x
)
+
∂(
gh
∂
z
s
1
ρ
∂
[
h
(
T
xx
+
D
xx
)
]
=−
x
+
∂
∂
∂
∂
t
x
y
x
1
ρ
∂
[
h
(
T
xy
+
D
xy
)
]
1
ρ
(τ
+
+
−
τ
bx
)
sx
∂
y
(2.82)
where
T
xx
and
T
xy
are the depth-averaged normal and shear stresses;
D
xx
and
D
xy
account for the dispersion momentum transports due to the vertical non-uniformity of
velocity, defined as
D
xx
=−
h
z
s
=−
h
z
s
2
dz
and
D
xy
z
b
(
u
x
−
U
x
)
z
b
(
u
x
−
U
x
)(
u
y
−
U
y
)
dz
;
sx
is the
x
-component of shear force per unit horizontal area, usually due to
wind driving at the water surface, defined as
τ
τ
=
τ
−
τ
∂
z
s
/∂
x
−
τ
∂
z
s
/∂
y
;
sx
xz
,
s
xx
,
s
xy
,
s
τ
bx
is the
x
-component of bed shear force per unit horizontal area, defined as
τ
bx
=
τ
xz
,
b
−
τ
xx
,
b
∂
and
z
b
/∂
x
−
τ
xy
,
b
∂
z
b
/∂
y
. Note that
τ
bx
may be written as
τ
bx
=
m
b
τ
bx
,
in which
τ
bx
is the
x
-component of bed shear force per unit bed surface area, and
m
b
is the bed slope coefficient defined as
m
b
=[
2
2
1
/
2
.
1
+
(∂
z
b
/∂
x
)
+
(∂
z
b
/∂
y
)
]
