Geology Reference
In-Depth Information
argument. In inter-rill (unconcentrated) flows, it
has been suggested that the entrainment rate is a
function of the square of rainfall energy (e.g.
Meyer, 1981) modulated by a function of depth of
overland flow once generated. Torri
et al
. (1987)
suggested that this function is a negative expo-
nential of flow depth to a power. The median
travel distance of particles in shallow flows can
be considered to be a non-linear function of rain-
fall energy and flow energy as well as particle
mass (Parsons & Stromberg, 1998). On a planar,
uniform slope with a uniform infiltration rate
and uniform roughness, flow hydraulics can
be considered to be simple functions of distance
downslope (discharge increases linearly, so that
depth increases with
x
2/3
, flow velocity with
x
1/3
and flow energy with
x
5/3
). Under these assump-
tions, Equation (6.2) can be rewritten as:
1996). Under these assumptions, Equation (6.2)
now becomes:
+
+=
xLx
()
d
ò
j
(
xL
)
Kudu
(6.4)
d
3
x
where
K
3
is a parameter reflecting slope and soil
(and thus flow) conditions. Equation (6.4) is non-
linear because
L
d
is proportional to
x
2.18
, but it is
analytically solvable, producing flux as a function
of slope length as shown in Fig. 6.1(b). Thus, until
sediment exhaustion becomes an issue, erosion
rates in rills and gullies increase with scale, albeit
at a decreasing rate with distance.
On homogeneous slopes such as the ones con-
sidered thus far, the occurrence of multiple process
domains will produce scale-erosion relationships
with different forms depending on the location
where concentrated flows start, so will be a func-
tion of initial conditions and storm characteris-
tics,
ceteris paribus
. The difference in the order of
magnitude of the fluxes under the different proc-
ess regimes may make it empirically difficult to
differentiate the curves where concentrated flows
are initiated close to the divide.
Wainwright
et al
. (2001) relaxed the assump-
tion that all particles travel the same distance
under the same flow conditions. It has long been
recognized that particles travel a range of differ-
ent distances as a result of flow variability (espe-
cially turbulent bursts where transitional or
turbulent flows occur), microtopographic varia-
tions and other factors (Wainwright & Thornes,
1991; Hassan
et al
., 1992; Parsons & Stromberg,
1998). Travel distance of particles can thus be
considered to follow a distribution function, with
individual particles moving different distances
with a given probabilistic form. If
F
L
d
is the cumu-
lative probability distribution function of travel
distances of particles of size
d
, then the flux at
any point on the slope is:
xLx
+
()
d
4/9
ò
-
2
u
j
(
xL
+=
)
Ke
du
(6.3)
d
d
1
x
(see Parsons
et al
., 2004, for full details of the
steps in producing Equation (6.3) ) where
K
1
is a
parameter relating to rainfall intensity and energy
and flow roughness, and
u
is an integration vari-
able [L]. Equation (6.3) must be solved numeri-
cally, and examples of its form for different
particle sizes are shown in Fig. 6.1(a). Typically,
at short distances, flux increases with distance
downslope, until reaching a maximum several
metres from the divide, after which the flux
decreases. Parsons
et al
. (2006a), Rejman
et al
.
(1999) and Wilcox
et al
. (1997) have demonstrated
that this pattern is observed in field settings
where there are measurements of flux at different
distances from the divide.
In concentrated flows in rills and gullies, dif-
ferent relationships have been found for travel
distance of particles, entrainment rates and flow
hydraulics. Travel distance is approximated as a
function of excess stream power and particle
size (Hassan
et al
., 1992), entrainment can be
considered a function of flow shear stress to the
1.5 power (Yalin, 1977), flow discharge will
increase with catchment area, or as a function
of
x
1.67
(Hack, 1957), and flow depth is equal to
discharge to the power 0.4 (Abrahams
et al
.,
x
0
é
ù
j
()
xExl
=
(
-
) 1
-
Fl
()
l
,
(6.5)
ë
û
d
d
L
d
or in other words, the total number of particles
entrained from a given distance upslope (
l
) that
travel that threshold distance or further. Although
