Geoscience Reference
In-Depth Information
This coupled solution procedure can eliminate those non-physical phenomena that
exist in the decoupled procedure. However, the set of discretized sediment equations
should be solved simultaneously. An iteration method is normally needed, and then the
computational effort will be significantly increased. To avoid this, the direct solution
method proposed by Wu (1991) can be used (Wu
et al
., 2004a), as described below.
For convenience, Eq. (5.138) is written as
Q
n
+
1
tk
,
i
e
k
Q
n
+
1
t
=
+
e
0
k
(5.142)
+
1
∗
k
,
i
+
1
c
1
L
n
+
1
c
2
Q
n
+
1
tk
,
i
c
3
Q
tk
,
i
+
1
+
c
4
Q
tk
,
i
+
c
0
k
)/
c
1
, and
c
0
k
is
where
e
k
=
θψ/(
)
,
e
0
k
=
(
+
t
,
i
+
1
Q
n
+
1
t
L
n
+
1
t
,
i
c
0
k
without
θψ
1
/
1
.
∗
k
,
i
+
+
Eq. (5.139) is written as
f
1
Q
n
+
1
tk
,
i
f
2
Q
n
+
1
t
A
bk
,
i
+
1
=
1
−
1
+
f
0
k
(5.143)
+
∗
k
,
i
+
L
n
+
1
t
,
i
p
m
)
Q
tk
,
i
+
1
−
Q
t
∗
k
,
i
+
1
)/
where
f
1
=
f
2
=
θ
t
/
[
(
1
−
]
, and
f
0
k
=
(
1
−
θ)
t
(
+
1
p
m
)
L
t
,
i
+
1
]
[
(
1
.
Inserting Eqs. (5.141) and (5.142) into Eq. (5.143) yields
−
p
n
+
1
bk
,
i
1
Q
∗
n
+
1
A
bk
,
i
+
1
=
(
f
1
e
k
−
f
2
)
1
+
(
f
1
e
0
k
+
f
0
k
)
(5.144)
+
tk
,
i
+
and then substituting Eq. (5.135) into Eq. (5.144) leads to
Q
∗
n
+
1
tk
,
i
1
p
∗
bk
,
i
+
1
A
n
+
1
m
,
i
A
b
,
i
+
1
(
f
2
−
f
1
e
k
)
(
f
1
e
0
k
+
f
0
k
)
+
+
1
A
bk
,
i
+
1
=
1
+
A
n
+
1
m
,
i
Q
∗
n
+
1
tk
,
i
A
n
+
1
m
,
i
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
f
1
e
k
)
+
(
f
2
−
f
1
e
k
)
+
1
+
1
+
+
1
Q
∗
n
+
1
tk
,
i
A
n
+
1
m
,
i
p
bk
,
i
+
1
A
m
,
i
+
1
+
p
∗
bk
,
i
+
1
(
A
m
,
i
+
1
)
]
(
f
2
−
f
1
e
k
)
1
[
−
+
+
1
−
A
n
+
1
m
,
i
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
f
1
e
k
)
+
1
+
1
(5.145)
Summing Eq. (5.145) over all size classes and using Eq. (5.133) yields the following
equation for the total change in bed area:
⎨
⎩
−
Q
∗
n
+
1
tk
,
i
A
n
+
1
m
,
i
p
bk
,
i
+
1
A
m
,
i
+
1
+
p
∗
bk
,
i
+
1
(
A
m
,
i
+
1
)
]
N
(
f
2
−
f
1
e
k
)
1
[
−
+
1
+
A
b
,
i
+
1
=
A
n
+
1
m
,
i
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
f
1
e
k
)
+
1
+
1
k
=
1
⎬
A
n
+
1
m
,
i
N
(
f
1
e
0
k
+
f
0
k
)
+
1
+
A
n
+
1
m
,
i
Q
∗
n
+
1
tk
,
i
⎭
+
(
f
2
−
f
1
e
k
)
+
1
+
1
k
=
1
⎡
⎣
1
⎤
⎦
Q
∗
n
+
1
tk
,
i
N
1
p
∗
bk
,
i
+
1
(
f
2
−
f
1
e
k
)
+
−
(5.146)
A
n
+
1
m
,
i
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
f
1
e
k
)
+
1
k
=
1
+
1
