Geoscience Reference
In-Depth Information
Solution of the linearized equation
In most practical applications of this approach, a linearized form of the equation has been
used. This is readily obtained from (5.87) by proceeding as before in Section 5.4.1; thus,
decompose the variables into a uniform steady part and a perturbation, or
q
=
q
0
+
q
p
and
h
=
h
0
+
h
p
, and retain only the first-order terms, to obtain
dq
0
dh
0
∂
bq
0
S
0
∂
2
q
p
∂
∂
q
p
∂
q
p
∂
t
+
x
=
(5.88)
x
2
This is a linear diffusion equation, in which the constant diffusivity is
bq
0
S
0
D
0
=
(5.89)
with
b
=
1
/
2 for turbulent flow, and the constant advectivity is
dq
0
dh
0
c
d0
=
(5.90)
In a linear diffusion channel, this advectivity is clearly the same as the celerity (
a
1)
V
0
of the main body of the wave of the complete linear solution, given by Equation (5.77).
This will become clear in Section 5.4.3, but it can already be verified by determining
(
dq
0
/
+
dh
0
) from (5.43) for uniform steady flow. This means that, as will be shown below,
the advectivity of the diffusion equation is also the kinematic wave celerity
c
k0
,or
c
d0
=
c
k0
(5.91)
The linear diffusion formulation in (5.88) was derived for a very wide channel; for a
channel cross section
A
c
=
A
c0
+
A
cp
with a flow rate
Q
=
Q
0
+
Q
p
, the basic equation
can be written as
∂
2
Q
p
∂
Q
p
∂
c
k0
∂
Q
p
∂
D
0
∂
+
=
(5.92)
t
x
x
2
which is the linearized form of (5.87). The diffusivity is now
bQ
0
B
c
S
0
D
0
=
(5.93)
and the advectivity is, in light of (5.91),
dQ
0
dA
c0
c
k0
=
(5.94)
This can again be approximated by
c
k0
=
1)
V
0
, if the channel is wide enough.
The diffusion approximation of free surface flow has been the subject of numerous
investigations (see Sch onfeld, 1948; Hayami, 1951; Appleby, 1954; Daubert, 1964; Van
de Nes and Hendriks, 1971; Dooge, 1973). The general case, with an inflow
q
u
(
t
)atthe
upstream boundary of the channel at
x
(
a
+
t
), has
been presented by Brutsaert (1973) as a special case of (5.69) with (5.70). To allow a
comparison with the solution of the complete shallow water equations, consider again the
same example as before with zero lateral inflow and with a known value of the upstream
inflow
q
u
(
t
).
=
0 and a nonzero lateral inflow
i
=
i
(
x
,
