Geoscience Reference
In-Depth Information
c
1
=
0.480 and
c
2
=
0.502; with Equation (7.45) the corresponding Muskingum param-
eters are
K
0.24. All these values are close to the corresponding ones
obtained in Example 7.2 using the standard trial-and-error procedure; in fact, a plot of
the resulting hydrograph
Q
e
would be difficult to distinguish from the curve shown in
Figure 7.10 for
K
=
1.97 h and
X
=
=
1.99 h and
X
=
0.3.
7.3.2
From physical characteristics of the channel
Diffusive behavior of the Muskingum wave
Among the properties of the linear kinematic wave is the fact, explained in Section 5.4.3,
that it propagates along the channel with a celerity
c
k0
, but without any change in shape.
Therefore, it was puzzling to practitioners for some time in the past that this is apparently
not the case for the Muskingum wave, even though it is based on the same approximation
as outlined in Section 5.4.4. Indeed, as illustrated in Figure 7.10, a Muskingum flood
wave undergoes not only translation in time but also a change in shape.
It is now realized that the change in shape of the calculated wave is not the result of
the underlying physics, but rather the result of numerical diffusion caused by the approx-
imation of derivatives by ratios of finite differences. Any lumped kinematic approach is
based on the storage equation (7.11); that equation is a discretized form of the continuity
equation, in which the spatial derivative is approximated by a difference over a distance
x
. As pointed out by Cunge (1969), it is this approximation that causes the spreading
of the calculated wave.
The diffusion introduced by finite difference approximations can be determined by
trying to recover the partial differential equation (5.111), without lateral inflow, from
(7.37). Recall that the subscripts 1 and 2 refer to the beginning and end of the time
interval
t
; similarly, the subscripts i and e refer to the inflow and exit end of the spatial
reach
x
. For the present purpose, the four
Q
terms can be expressed in terms of the rate
of flow
Q
(
x
,
t
) by a Taylor expansion as follows
Q
i1
=
Q
2
Q
∂
+
∂
Q
∂
1
2
∂
t
)
2
Q
i2
=
Q
t
t
+
(
+···
t
2
+
∂
2
∂
2
Q
∂
Q
∂
1
x
)
2
Q
e1
=
x
+
+···
Q
x
(
(7.46)
x
2
2
Q
∂
+
∂
Q
∂
1
2
∂
t
)
2
Q
e2
=
Q
t
t
+
(
+···
t
2
Q
∂
∂
+
∂
1
2
∂
2
Q
∂
x
)
2
···+
t
t
+···
x
+
(
Q
···
) (
+···
∂
x
2
x
where the terms of order higher than 2 have been neglected. Substitution of (7.46)
into (7.37) and division by [(1
−
c
0
)
t
] produces the following partial differential
