Geoscience Reference
In-Depth Information
axis is made to shrink and the vertical axis is made to stretch, as
x
+
increases. As men-
tioned, the first part
u
1
becomes negligible very quickly. For the
c
ase shown in Figure 5.8,
Equation (5.79) indicates that at
x
+
=
(0
,
1
,
5, 20), the relative volume of
u
1
decreases as
exp(
−
e
1
x
+
)
=
(1
.
0
,
0
.
411
,
0
.
0117
,
1
.
9
×
10
−
8
).
The main point of this simple example of a linear analysis, leading to the solution (5.72)
with (5.73) and (5.74) or with (5.79) and (5.80), is to show how in the propagation of free
surface disturbances there are two main types of wave, namely dynamic and kinematic
waves. As shown in Equations (5.60) and (5.64), the former are the result of the first three
terms in (5.22) (or in the second of (5.44)); as shown in (5.77), the latter arise when also the
two slope terms are included in the analysis. Except in unusual cases, such as supercritical
flow (when Fr
0
>
1) or dynamic shock (see Chapter 7), the former are normally faster than
the latter, but they tend to decay relatively quickly; comparing (5.64) with (5.77), one sees
that the kinematic wave is faster than the dynamic wave, only when (
aV
0
)
>
(
gh
0
)
1
/
2
,or
Fr
0
<
a
); as noted, two paragraphs earlier, this is also the criterion for bore formation or
dynamic shock. Moreover, when 1
<
Fr
0
<
(1
/
a
), both
e
4
and
e
6
become imaginary; this
changes the modified Bessel function I
1
( ) to a regular Bessel function J
1
( ), which exhibits
oscillatory behavior. In the linear analysis the two types of waves appear separately, and the
total disturbance is the result of their simple superposition, as shown in (5.72). Because the
momentum shallow water equation (5.22) is quite nonlinear, in real world situations one
can expect these two special types of propagation to interact with each other. Nevertheless,
the linear analysis has clearly illustrated some of their most important features.
As an aside, it is of interest to point out that the ratio of the celerity of the dynamic
waves relative to the mean velocity
V
, and the relative celerity of the kinematic wave (
aV
)
is also referred to as the Vedernikov number, namely Ve
=
(
gh
)
1
/
2
(1
/
/
(
aV
) (Vedernikov, 1946;
Chow, 1959). As shown above for the linear case, Ve
>
1 is the criterion for bore formation.
5.4.2
The diffusion analogy: a first approximation
In many situations encountered in nature, the flow velocities change relatively slowly, so
that the acceleration (inertia or dynamic) terms (
x
) often are rather
small compared to the other terms in the governing equations. For example, it was noted
by Iwasaki (1967), that in the upper Kitakami, a river some 195 km long draining an
area of 7860 km
2
in northern Honshu, these inertia terms were observed to be at most
1.5%, and usually smaller than 1% of the stage gradient
g
[(
∂
V
/∂
t
) and
V
(
∂
V
/∂
∂
/∂
−
S
0
]. Similarly,
the following values were presented in the
Flood Studies Report
(Natural Environment
Research Council, 1975) as being typical for British rivers.
h
x
)
S
f
S
0
∼
∂
h
/∂
x
10
−
2
0
.
9
∼
2
×
S
0
(5.81)
∂
V
/∂
t
V
∂
V
/∂
x
10
−
3
∼
∼
1
.
7
×
gS
0
gS
0
Recall that the governing equations of free surface flow describe the conservation of
mass and the conservation of momentum. In this section the consequences are considered
of neglecting these inertia terms in the momentum equations (5.22) or (5.25). However,
the continuity equations (5.13) or (5.24) are left intact.
