Geoscience Reference
In-Depth Information
where
ε
is an effective storage coefficient (here assumed constant). Substituting for
h
gives the
same equation as for surface flow on a variable width slope:
W
x
∂q
c
∂W
x
q
∂x
∂t
=−
+
cW
x
r
(B5.7.10)
exp
(
fh
)
f
K
s
sin
˛
ε
K
s
sin
˛
ε
but here
c
=
for the constant conductivity case and
c
=
for the exponentially
declining conductivity case.
For saturated subsurface runoff on a hillslope, Beven (1981) showed that the kinematic wave
description was a good approximation to a more complete Dupuit-Forchheimer equation
description, if the value of a non-dimensional parameter, defined by
4
K
s
sin
ˇ
i
=
(B5.7.11)
where
i
the effective rate of storm recharge to the slope, was greater than about 0.75. When
this condition is met, any drawdown of the water table at the lower end of the slope due to an
incised channel, is unlikely to have a great effect on the predicted discharges.
Given the mass conservation of Equation (B5.7.1), there is only one primary assumption
underlying the derivation of the kinematic wave equation:
A1 A functional relationship between storage and discharge can be specified for the particular
flow process being studied.
Several examples of such relationships for both surface and subsurface flow have been
demonstrated above. The limitations of the kinematic wave approach must be appreciated,
but a major advantage is that it is not restrictive in its assumptions about the nature of the
flow processes, only that discharge should be a function of storage. Analytical solutions of the
kinematic wave equation require that this functional relationship should be univalued (and
generally of simple form). The simplicity of the kinematic wave assumptions have allowed
the exploration of analytical solutions for different shapes of hillslope (e.g. Troch
et al.
, 2002;
Norbiato and Borga, 2008). Numerical solutions do not have such a restriction and it is pos-
sible to envisage a kinematic wave solution that would have a hysteretic storage-discharge
relationship that would more closely mimic the solution of the full surface or subsurface flow
equations (in the same way that hysteretic soil moisture characteristics are sometimes used in
unsaturated zone models Jaynes, 1990). It seems that no-one has tried to implement such a
model in hydrology, although there has been some analysis of the storage-discharge hystere-
sis that arises at the hillslope and catchment scale (e.g. Ewen and Birkinshaw, 2006; Beven,
2006b; Norbiato and Borga, 2008; Martina
et al.
, 2011).
