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where
Rasmussen, Sørensen and Willetts, 1985; Sherman, 1992;
Sherman and Farrell, 2008).
The Owen (1964) equation has been used as the basis
for constructing self-regulatory models of saltation (An-
derson and Haff, 1988, 1991; Werner, 1990; McEwan and
Willetts, 1991, 1993). These steady-state models work on
the premise that as
u
∗
(and hence saltation) increases, so
too does the effective aerodynamic roughness (
z
0
). This
increased aerodynamic roughness exerts an extra drag on
the airflow and this effect propagates upwards through the
velocity profile as an internal boundary layer. This results
in near-surface winds being reduced, eventually to reach
a steady-state value where as many grains are leaving the
surface as are falling onto it (Anderson, Sørensen and
Willetts, 1991). An equilibrium is therefore established
between
u
∗
,
z
0
and saltation load (Sherman, 1992).
The difficulties in determining a meaningful
z
0
, as de-
scribed above, are a function of the complexity of the
parameter and its response to changing surface charac-
teristics in time and space (King, Nickling and Gillies,
2006, 2008). Blumberg and Greeley (1993) regard one of
the most difficult hurdles as being able to take account of
the effects of a surface roughness change into the develop-
ment of a velocity profile, for as there is a transition from a
smooth to rough surface (or rough to smooth), a boundary
layer grows downwind in response to that transition (Fig-
ure 18.6). Hence, different parts of the velocity profile are
likely to be responding to different surface roughnesses
with many subsidiary boundary layers responding to
roughnesses provided at different scales by such elements
as individual grains, surface ripples or isolated vegetation.
In the example shown in Figure 18.6, the wind passes
from a smooth surface (perhaps a flat sand sheet) to a
n
=
intercept statistic of the regression equation
m
=
gradient statistic of the regression equation
Such an approach is common practice and works well for
flat and noncomplex surfaces (Bauer
et al.
, 2009; Gillies,
Nickling and King, 2007; Gillies
et al.
, 2000; King, Nick-
ling and Gillies, 2005; Sherman
et al.
, 1998; Wiggs
et al.
,
1994; Weaver, 2008). However, defining a clear intercept
on complex or patterned (e.g. ploughed) surfaces is prob-
lematic where small variations in wind direction during
measurement can have a dramatic effect on calculated
values of
z
0
as the wind profile responds to the chang-
ing effective configuration of surface properties that pro-
vide the surface drag (Gillette, Herrick and Herbert, 2006;
Wiggs and Holmes, 2010). Furthermore, on sloping sur-
faces the regression equation approach to calculating
z
0
suffers from the same problems as when calculating
u
∗
,
the disruption of the log-linear velocity profile due to flow
acceleration.
An additional issue arises when a sand bed responds
to increasing wind shear by allowing grain entrainment
and saltation to take place. In such cases the saltating
sand extracts momentum from the wind, carried in the
form of a grain-borne shear stress, and this acts as an
additional drag on the airflow and so increases the value of
z
0
(McKenna-Neuman and Nickling, 1994; Sherman and
Farrell, 2008; Bauer, Houser and Nickling, 2004, 2009).
In order to satisfy the concept of an increasing
z
0
with
increasing
u
∗
when saltation is occurring, several workers
(Rasmussen, Sørensen and Willetts 1985; Anderson and
Haff, 1991; McEwan, 1993) have turned their attention to
the relationship applied by Owen (1964):
z
0
=
C
0
u
2
∗
/
2
g
(18.6)
u
*
0
where
Velocity
x
z
0
=
aerodynamic roughness height during saltation
x
u
*
0
Internal
boundary
layer
C
0
=
≈
constant (
0.02)
y
In this case, the depth of saltation (associated with the
aerodynamic roughness height,
z
0
) is related to the lift-
off velocity of the individual sand grains, which in turn is
governed by the shear velocity (
u
∗
). While this relation-
ship has been widely used to determine
u
∗
in wind tunnel
studies, it has yet to be fully utilised in field situations
Smooth surface
Rough surface
Distance
Figure 18.6
The growth of an internal boundary in response
to a changing surface roughness (after Greeley and Iversen,
