Geoscience Reference
In-Depth Information
5.3.3 1-D unsteady non-equilibrium sediment transport
model
5.3.3.1 Discretization of unsteady sediment transport
equations
For simplicity, the bed-material load transport model introduced in Section 5.1.2.1 is
adopted here. Applying the Preissmann scheme to discretize the total-load transport
equation (5.34) yields (Wu
et al
., 2004a)
Q
n
+
1
tk
,
i
Q
n
+
1
tk
,
i
Q
tk
,
i
+
1
Q
tk
,
i
ψ
1
−
ψ
+
1
1
−
+
−
n
tk
,
i
1
U
i
+
1
tk
,
i
U
i
n
n
+
1
1
U
n
+
1
n
tk
,
i
U
n
+
1
+
1
t
β
t
β
β
β
+
tk
,
i
+
i
+
i
+
θ
1
−
θ
Q
n
+
1
tk
,
i
Q
n
+
1
tk
,
i
Q
tk
,
i
+
1
−
Q
tk
,
i
)
x
(
−
)
+
x
(
+
1
Q
n
+
1
tk
,
i
Q
n
+
1
t
Q
n
+
1
tk
,
i
Q
n
+
1
t
−
−
+
1
∗
k
,
i
+
1
∗
k
,
i
+
θ
ψ
+
(
1
−
ψ)
L
n
+
1
t
,
i
L
n
+
1
t
,
i
+
1
Q
tk
,
i
+
1
−
Q
t
∗
k
,
i
+
1
L
t
,
i
+
1
Q
tk
,
i
−
Q
t
∗
k
,
i
L
t
,
i
+
(
1
−
θ)
ψ
+
(
1
−
ψ)
q
n
+
1
tlk
,
i
q
n
+
1
q
tlk
,
i
+
1
+
(
q
tlk
,
i
]
=
θ
[
ψ
1
+
(
1
−
ψ)
tlk
,
i
]+
(
1
−
θ)
[
ψ
1
−
ψ)
+
(5.137)
which can be written as
c
1
Q
n
+
1
tk
,
i
c
2
Q
n
+
1
tk
,
i
c
3
Q
tk
,
i
+
1
+
c
4
Q
tk
,
i
+
=
+
c
0
k
(5.138)
+
1
where
ψ
t
+
θ
x
+
θψ
c
1
=
n
+
1
1
U
n
+
1
L
n
+
1
t
,
i
β
tk
,
i
+
i
+
1
+
1
1
−
ψ
t
+
θ
x
−
θ(
1
−
ψ)
L
n
+
1
t
,
i
c
2
=−
tk
,
i
U
n
+
1
n
+
1
β
i
ψ
1
−
θ
−
(
1
−
θ)ψ
L
t
,
i
+
1
c
3
=
t
−
n
tk
,
i
1
U
i
+
1
β
x
+
1
−
ψ
1
−
θ
x
−
(
1
−
θ)(
1
−
ψ)
c
4
=
t
+
tk
,
i
U
i
n
L
t
,
i
β
Q
n
+
1
t
Q
n
+
1
t
Q
t
∗
k
,
i
+
1
L
t
,
i
+
1
Q
t
∗
k
,
i
L
t
,
i
∗
k
,
i
+
1
k
,
i
L
n
+
1
t
,
i
∗
c
0
k
=
θψ
+
θ(
1
−
ψ)
+
(
1
−
θ)ψ
+
(
1
−
θ)(
1
−
ψ)
L
n
+
1
t
,
i
+
1
q
n
+
1
tlk
,
i
q
n
+
1
tlk
,
i
q
tlk
,
i
+
1
+
(
q
tlk
,
i
+
θψ
1
+
θ(
1
−
ψ)
+
(
1
−
θ)ψ
1
−
θ)(
1
−
ψ)
+
In order to satisfy sediment continuity, the sediment exchange terms in Eqs. (5.34)
and (5.36) should be discretized using the same scheme. Thus, the bed change
