Geoscience Reference
In-Depth Information
disappears; this is not the case for the celerity in (7.51), because the time of travel
K
then
also goes to zero.
Physically based estimation of the parameters: the MCD method
The MCD acronym stands for the Muskingum-Cunge-Dooge method and it refers to
the names of the two investigators who contributed independently to its development.
As proposed by Cunge (1969), the numerical diffusion in the Muskingum formulation
can be put to good use in the estimation of the parameters
X
and
K
; this is done simply
by requiring this numerical diffusion to be equal to the physical diffusion resulting from
the hydraulic characteristics of the flow. Accordingly, equating (7.51) and (7.52) with
(5.94) and (5.93), respectively, one obtains
x
K
=
(7.53)
dQ
0
/
dA
c
0
which for wide channels is equal to
x
/
[(
a
+
1)
V
0
], and
1
2
−
bQ
0
c
k0
B
c
S
0
X
=
(7.54)
x
The symbol
Q
0
is a typical reference flow rate in the channel,
b
the parameter of (5.39)
which is normally taken as 1
2 for turbulent flow (see Table 5.2),
c
k0
is the kinematic
wave celerity,
B
c
is the channel width,
S
0
is the bed slope, and
/
x
is the length of the
channel reach. With the more accurate expression (5.98) for the diffusivity, this is
x
1
a
2
Fr
0
1
2
−
bQ
0
c
k0
B
c
S
0
X
=
−
(7.55)
In cases when the channel is sufficiently wide, the Kleitz-Seddon principle (5.108) can
be used to express (7.55) in even simpler terms, namely
x
1
a
2
Fr
0
1
2
−
bh
0
X
=
−
(7.56)
(
a
+
1)
S
0
All these expressions for
X
indicate that it is normally smaller than 0.5.
In a different development, Dooge (1973) determined the parameters
K
and
X
of the
Muskingum formulation by equating its first two moments (7.31) and (7.34) with the first
two moments obtainable from the unit response (5.72); recall that this response function
is obtained by the exact solution of the linearized complete shallow water equation
(5.67). The details of this derivation are beyond the present scope, but it is easy to show
that the resulting expressions are the same as (7.53) and (7.56). This indicates that the
application of the Muskingum formulation with (7.53) and (7.56) will ensure that it
produces a wave motion whose average speed of propagation and dispersion are the
same as those obtainable with the exact solution. It also means that the expressions for
the Muskingum parameters in Equations (7.53)-(7.56) are even better than would be
suggested by a cursory review of Cunge's (1969) derivation. These expressions conform
not only with the diffusion approximation, but with the exact solution of the linearized
complete shallow water equation (5.67), as well.
