Geoscience Reference
In-Depth Information
Substituting Eqs. (5.141), (5.184), and (5.185) into the discretized sediment
transport equation (5.137) yields
Q
n
+
1
tk
,
i
Q
n
+
1
tk
,
i
Q
tk
,
i
+
1
Q
tk
,
i
F
3
k
=
ψ
−
ψ
1
+
1
1
−
+
−
n
tk
,
i
1
U
i
+
1
n
tk
,
i
U
i
t
n
+
1
1
U
n
+
1
β
t
tk
,
i
U
n
+
1
n
+
1
β
β
β
+
i
+
i
tk
,
i
+
⎨
⎩
+
θ
1
−
θ
Q
tk
,
i
)
+
θψ
Q
n
+
1
tk
,
i
Q
n
+
1
tk
,
i
Q
tk
,
i
+
1
−
Q
n
+
1
tk
,
i
x
(
−
)
+
x
(
+
1
L
n
+
1
t
,
i
+
1
+
1
⎡
⎛
⎞
⎤
⎬
N
⎣
e
k
,
i
+
1
Q
n
+
1
tk
,
i
⎝
r
k
,
i
+
1
Q
n
+
1
tk
,
i
⎠
+
⎦
Q
∗
n
+
1
tk
,
i
−
+
f
k
,
i
+
1
+
s
i
+
1
g
k
,
i
+
1
+
1
+
1
+
1
⎭
k
=
1
⎧
⎨
⎡
⎣
e
k
,
i
Q
n
+
1
tk
,
i
⎛
⎝
⎞
⎠
+
⎤
⎦
Q
∗
n
+
1
tk
,
i
⎫
⎬
N
+
θ(
−
ψ)
L
n
+
1
t
,
i
1
Q
n
+
1
tk
,
i
r
k
,
i
Q
n
+
1
tk
,
i
−
+
+
f
k
,
i
s
i
g
k
,
i
⎩
⎭
=
k
1
Q
tk
,
i
+
1
−
Q
t
∗
k
,
i
+
1
L
t
,
i
+
1
Q
tk
,
i
−
Q
t
∗
k
,
i
L
t
,
i
+
(
−
θ)
ψ
+
(
−
ψ)
1
1
q
n
+
1
tlk
,
i
q
n
+
1
tlk
,
i
q
tlk
,
i
+
1
+
(
q
tlk
,
i
]=
−
θ
[
ψ
+
(
1
−
ψ)
]−
(
1
−
θ)
[
ψ
1
−
ψ)
0
+
1
(
k
=
1, 2,
...
,
N
)
(5.188)
For the channel with
I
−
1 reaches shown in Fig. 5.7, the system of equations
(5.186)-(5.188) has
(
2
+
N
)(
I
−
1
)
equations, which are used to determine
(
2
+
N
)
I
unknowns:
A
,
Q
, and
Q
tk
(
k
N
boundary conditions. For simplicity, the Manning roughness coefficient and flow
density are treated as intermediate variables. An alternative treatment for flow density
may be to remove it from the left-hand sides of Eqs. (5.177) and (5.178), as described
in Section 9.2.
Eqs. (5.186)-(5.188) can be solved by many methods. The following Newton-
Raphson solution procedure is given as an example, which is almost the same as
that used by Holly and Rahuel (1990).
The Newton-Raphson correction equations for each reach are written in the
following matrix form:
=
1, 2,
...
,
N
). The system is closed by imposing 2
+
[
L
i
]{
δ
W
i
}+[
R
i
]{
δ
W
i
+
1
}+{
S
i
}=
0
(5.189)
where
{
δ
W
i
}
is the vector of unknown corrections to the 2
+
N
primary variables:
δ
A
i
,
δ
is the known vector of functions
F
1
,
F
2
, and
F
3
k
defined in Eqs. (5.186)-(5.188); and
Q
i
,
δ
Q
t
1,
i
,
δ
Q
t
2,
i
,
...
, and
δ
Q
tN
,
i
;
{
S
i
}
[
L
i
]
[
R
i
]
and
are the matrices of Jacobian
derivatives with
(
2
+
N
)
×
(
2
+
N
)
elements, e.g.,
⎡
⎣
⎤
⎦
∂
F
1
/∂
A
i
∂
F
1
/∂
Q
i
∂
F
1
/∂
Q
t
1,
i
···
∂
F
1
/∂
Q
tN
,
i
∂
F
2
/∂
A
i
∂
F
2
/∂
Q
i
∂
F
2
/∂
Q
t
1,
i
···
∂
F
2
/∂
Q
tN
,
i
∂
F
31
/∂
A
i
∂
F
31
/∂
Q
i
∂
F
31
/∂
Q
t
1,
i
···
∂
F
31
/∂
Q
tN
,
i
[
L
i
]=
.
.
.
.
···
∂
F
3
N
/∂
A
i
∂
F
3
N
/∂
Q
i
∂
F
3
N
/∂
Q
t
1,
i
···
∂
F
3
N
/∂
Q
tN
,
i
(5.190)
