Soil Science Society of America Journal 64:852-858 (2000)
© 2000 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Flattened Residue Effects on Wind Speed and Sediment Transport
Geert Sterk
Erosion and Soil & Water Conservation Group, Dep. of Environmental Sciences, Wageningen Univ., Nieuwe Kanaal 11, 6709 PA Wageningen, The Netherlands
geert.sterk{at}users.tct.wag-ur.nl
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ABSTRACT
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Mulching with flattened crop residues is widely used to protect soils from wind erosion. Several wind tunnel and field experiments have shown decreased protection of some soil covers with increasing wind speed. In some studies, sediment transport was enhanced with flattened residue as compared with the bare soil condition. The purpose of this article was to determine the behavior of wind speed and sediment transport when the soil surface is covered with randomly applied, flattened crop residues. A literature review was conducted to evaluate recent insights in turbulent flow properties and related sediment transport. A conceptual model was then developed to explain decreasing soil protection of a certain residue quantity when the free stream wind speed increases. The main reason is the change in turbulent flow properties of the near-surface wind when no-erodible roughness elements are added to an otherwise smooth surface. The average wind speed is reduced by the roughness but the probability distribution of instantaneous wind speed becomes wider and positively skewed. If free stream wind speed increases, at a certain moment the changed turbulence will cause more wind gusts that exceed the threshold wind speed for soil particle movement than would occur over a bare surface. But it is likely that this only happens when soil cover is less than 10%. For higher soil covers, increasing wind speed will also cause a decrease in soil protection, but natural wind speeds are normally not sufficient to cause enhanced sediment transport, because average wind speed is sufficiently reduced.
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INTRODUCTION
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MULCHING WITH ORGANIC MATERIALSc is widely promoted as a means to protect agricultural land from degradation in wind erosion prone areas. Applied mulch consists of non-edible roughness elements which decrease the wind stress on a surface by absorbing a significant fraction of the downward momentum flux from the airflow above (Raupach et al., 1993). Post-harvest crop residues are usually the most readily available mulch material. But in many semiarid areas where agriculture is characterized by few inputs, biomass production is often low and a strong competition may exist for crop residues. For instance, in the Sahelian zone of Africa, residues of pearl millet [Pennisetum glaucum (L.) R. Br.], which is the main staple crop, are needed for fuel, fodder, and construction material (Lamers and Feil, 1993). With an average millet stover production of 2200 kg ha-1 (Manu et al., 1991), the quantities available for soil protection are usually well below 2000 kg ha-1. Still, crop residues are by far the main source of mulch material.
It is well understood that the protective properties of residue are better when the material is left standing rather than flattened (e.g., Siddoway et al., 1965). However, standing residue is sometimes not agronomically acceptable, such as in the Sahel, where millet stalks may harbor crop-damaging stem borers (Acigona ignefusalis Hampson). When the stalks are cut down after harvest, the larvae of stem borers are killed by heat at the soil surface, which reduces the risk of infestation during the next cropping season (Ndoye and Gahukar, 1987).
Results of wind tunnel experiments with flattened residue show exponential decrease of soil particle transport with increasing soil cover (Siddoway et al., 1965; Fryrear, 1985; Bilbro and Fryrear, 1994). It is therefore often concluded that any quantity, even very small ones, reduces sediment transport as compared with an unprotected surface. This does not agree, however, with data from several wind tunnel experiments (Sterk and Enninga, 1998; Funk, 1995; Horning et al., 1998) and a field experiment (Sterk and Spaan, 1997). These studies showed that residue quantities covering <10% of the soil surface may not always reduce soil particle transport. In some cases soil particle transport was even enhanced due to the presence of non-erodible roughness elements. It is therefore questionable whether the observed exponential relationships between soil cover and wind-blown particle transport are valid for small flattened residue quantities. Moreover, it is uncertain if the wind tunnel derived relationships are also valid for field conditions, where wind is more turbulent than under controlled conditions in a wind tunnel.
The objective of this article was to develop a conceptual model of wind speed and sediment transport when the soil surface is covered with flattened residue, based on a literature review.
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Previous Experimental Results
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The traditional method to quantify erosion suppression by roughness elements is through the "soil flux ratio", which is the ratio of the streamwise soil flux1
QR (kg m-1 s-1) in the presence of roughness elements to the soil flux Qs over a bare soil without roughness elements, exposed to the same wind conditions:
 | (1) |
In previous work, RQ has often been called the "soil loss ratio" (e.g., Bilbro and Fryrear, 1994), but this terminology is slightly inaccurate because soil loss is actually the streamwise derivative of soil flux (Raupach et al., 1993).
The results fo wind tunnel experiments with different flattened residues by Siddoway et al. (1965), Fryrear (1985), and Bilbro and Fryrear (1994) all indicated that RQ decreases exponentially with increasing soil cover. Bilbro and Fryrear (1994) derived the following equation by combining several data sets from different sources:
 | (2) |
where SC (%) is the fraction of the soil covered by non-erodible roughness elements. Figure 1 shows a plot of the fitted equation through the experimental domain, which ranged from 8 to 95% SC.
Sterk and Enninga (1998) conducted a wind tunnel experiment to determine the soil protection created by four quantities of dry and flattened maize (Zea mays L.) stalks (500, 1000, 1500, and 2000 kg ha-1) for two different free-stream wind speeds (9.5 and 11.0 m s-1). Unfortunately, SC was not measured and thus the results are not easily compared with Fig. 1. It is likely though that SC ranged from only 2% for 500 kg ha-1 to about 10% for 2000 kg ha-1. This is based on the similarity between maize stalks and pearl millet stalks, of which data were collected in the Sahelian zone of Niger (Sterk, 1995, unpublished data). The observed RQ indicated a better soil protection with increasing amounts of flattened residue. But, at the same time, it was observed that for a certain SC, RQ was lower with a wind speed of 9.5 m s-1 than with 11.0 m s-1. Thus, the soil protection created by a certain quantity of flattened residue decreased with increasing wind speed. During several runs, even more particle transport was observed with flattened residue than without (RQ > 1). This was explained by enhanced turbulence around maize stalks.
Funk (1995) used a portable wind tunnel to determine soil flux ratios for field plots with seedlings in northeastern Germany. The effect of sugar beet (Beta vulgaris L.) and maize plants on soil particle transport was tested. During a run, free-stream wind speed was continuously increased from 0 to 18 m s-1. Values of RQ were significantly larger than one for vertically-projected SC of less than 5%. Hence, there was more soil particle transport with plants on the soil surface than without. This enhanced transport was explained by contraction of the flow between rows of plants, and increased turbulence near the soil surface.
Horning et al. (1998) tested the combined effects of flat cereal residue surface cover and random roughness on sediment transport at two field locations on the Columbia Plateau, Washington State. Three free-stream wind speeds (12, 15, and 18 m s-1) were applied with a portable wind tunnel on bare and mulched plots. The RQ values for the lowest roughness densities were greater than one. This was not only true for the lowest soil covers
, but also for the lowest random roughness values. The results were unexpected by the authors, and were attributed to measurement errors.
Sterk and Spaan (1997) conducted a field experiment in the Sahelian zone of Niger. Dry, flat and randomly distributed pearl millet stalks were applied at quantities of 1000 and 1500 kg ha-1. The corresponding SC's were 4.7 and 7.1%, respectively. The average measured RQ was 0.58 for the 1000 kg ha-1 soil cover and 0.36 for the 1500 kg ha-1 soil cover. A clear relationship between RQ and the average wind speed of a storm was observed (Fig. 2)
. It was concluded that the protective properties of the flattened residue decrease with increasing average wind speed of a storm. Soil particle transport was even enhanced by the lowest soil cover compared with bare soil during a strong storm with an average wind speed of 11.3 m s-1 at 2 m height. This was explained by enhanced turbulence created by the flattened residue.
The relationship between average wind speed and RQ also can be observed from the results of Siddoway et al. (1965) and Bilbro and Fryrear (1994), but was not discussed by them. Sterk and Enninga (1998), Funk (1995), and Sterk and Spaan (1997) are probably the first studies that paid attention to the phenomenon of low soil covers causing increased sediment transport. They all give the same explanation, i.e., enhanced turbulence due to the presence of roughness elements, but only limited explanations were provided in all three studies. Therefore, a more theoretically based description of the impact of non-erodible roughness elements on turbulent flow properties and sediment transport needs to be developed.
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Theoretical considerations
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Flow properties
The flow properties of air moving over a solid boundary are affected by the roughness of the surface. Two surface roughness conditions are distinguished here—smooth and rough. A smooth surface means a flat, sandy surface with no other roughness elements than those formed by the sand grains themselves. Hence, the smooth surface condition as defined is not an aerodynamically smooth surface. The term smooth is used to distinguish it from a rough surface, which is formed by a flat, sandy surface on which non-erodible roughness elements like maize or millet stalks are placed.
The air flow in a wind tunnel or the atmosphere is nearly always, and certainly during wind erosion events, of a turbulent nature. The turbulent boundary layer with thickness 
consists of two distinct parts, an inner layer and an outer layer. In the outer layer, which extends approximately from 0.15 to , the drag caused by the surface is only felt in an indirect way (Hinze, 1975). Because of the distance between the lower limit of the outer layer and the soil surface, this layer is only of minor relevance for wind erosion processes, and is therefore not considered here. In the inner layer, the flow experiences a direct dynamical influence of the surface. It can be subdivided into the inertial sublayer, in which height above the effective surface provides the appropriate length scale, and a sublayer adjoining the surface itself, in which the flow depends explicitly on surface defined length scales. This region is called the roughness sublayer (Raupach et al., 1980).
Before continuing with the description of average wind speed, shear stress and turbulent properties in the inner layer, some definitions are given first. The vector of instantaneous wind speed at a certain position in the flow is defined as (u,v,w), where u is the streamwise component in Horizontal Direction x, v is the lateral component in Horizontal Direction y, and w is the normal component in Vertical Direction z. Each instantaneous velocity component can be considered the sum of a temporal average (
, 
, 
) and a fluctuating or turbulent part (u',v',w'). The Reynolds stress or average stress is defined as -
(N m-2), which is equal to the downward flux density of streamwise momentum. Dividing the shear stress by density () results in the kinematic shear stress - (m2 s-2), which is equal to the square of the shear velocity u* (m s-1). The average kinematic shear stress at a single point is made up of the sum of four different events, which can be identified by the four quadrants i in the (u', w') plane. These events are outward interactions
, ejections
, inward interactions
, and sweeps
(Wallace et al., 1972). Ejections and sweeps result in a downward momentum transport and therefore contribute positively to the shear stress (-u'w' > 0). Inward and outward interactions result in upward momentum transport and thus contribute negatively to the shear stress (-u'w' < 0). For natural flow conditions always a positive average shear stress exists. So, the absolute magnitude of the negative contributions from inward and outward interactions has to be lower than the positive contributions from sweeps and ejections.
The vertical profile of average horizontal wind speed 
in the inertial sublayer can be described with the well-known semi-logarithmic wind profile (Panofsky and Dutton, 1984):
 | (3) |
where 
is the Von Karman constant, d is the zero-plane displacement length, and z0 is the roughness length. According to Raupach et al. (1980) the wind profile can be extended downward to a height of
 | (4) |
where zr is the height of the roughness sublayer, h is the average height of roughness elements, and Ds is the average separation distance between nearest neighbor roughness elements. Hence, the lower the density of non-erodible roughness elements the greater the vertical extent of the roughness sublayer. But when the surface is smooth, which means a surface without non-erodible roughness elements, h and Ds are approximately zero and the semi-logarithmic wind profile describes the flow in the entire inner layer, except for the very thin viscous sublayer adjoining the surface itself.
The mean wind speed profile in the roughness sublayer cannot be described by Eq. [3] (Mulhearn and Finnigan, 1978; Raupach et al., 1980; Jacobs and van Boxel, 1991). In general, roughness slows down the average wind speed, but at wide element spacings, wakes generated by the roughness elements complicate the flow. The form of the wind profile at a certain position is related to several roughness element length scales including height, width, shape, spacing and arrangement (Wolfe and Nickling, 1993). The wind tunnel experiments of Raupach et al. (1980) showed that the flow close to the roughness elements is characterized by spatial variation in horizontal wind speed, which was named horizontal inhomogeneity. Mean wind speed profiles measured at different locations were characterized by increasing scatter as the surface was approached. Defining a mean wind speed profile in the roughness sublayer requires spatial averaging of temporal average profiles measured at several locations. The experiment of Jacobs and Van Boxel (1991) showed that at a certain height the horizontal wind speed within a maize row canopy can vary by 20% from its spatial average value.
There exists also an important difference between the inertial sublayer and the roughness sublayer for the kinematic shear stress (-u'w'). The average stress in the inertial sublayer is constant with height and equal to u2*. This implies that the flow in this layer experiences the surface as uniformly rough and its detailed roughness structure is unimportant. In the roughness sublayer, however, the shear stress rapidly decreases when the surface is approached. Much of the stress is absorbed by the roughness elements. The total shear stress exerted to the flat surface between the roughness elements is lower than the total stress that is felt in the inertial sublayer. Moreover, the surface shear stress is characterized by spatial variation (Mulhearn and Finnigan, 1978).
Turbulent flow properties are influenced by roughness elements as well. In the inertial sublayer, the standard deviation 
u of the streamwise velocity component u is more or less constant with height (Raupach, 1981). Roughness, however, affects this standard deviation significantly. The value of u at a certain height is much larger for a rough surface than for a smooth surface, when the free stream velocity is the same (Lyles et al., 1971). This means that the turbulence intensity at a certain height, defined as
 | (5) |
is higher for a rough surface than for a smooth surface. This not only results from the higher u, but also from the lower average wind speed () over the rough surface. Tu increases with decreasing height in the inertial sublayer, because u is about constant while decreases. In addition, Tu for a given height and surface is independent of average wind speed (Lyles et al., 1971). This means that u increases proportionately with . Below the inertial sublayer, u decreases and shows spatial variation when the surface is approached (Mulhearn and Finnigan, 1978; Raupach, 1981; Krogstad et al., 1992). As the average wind speed also decreases and is characterized by spatial variability, turbulence intensities within the roughness sublayer will show a large spatial variation. Resulting values of Tu are generally very high, typically between 0.5 and 5.0 (Raupach et al., 1991).
The experiments of Raupach (1981) and Krogstad et al. (1992) showed an additional effect of roughness on turbulent flow properties. The skewness values of u and w are close to zero for a smooth surface throughout the inertial sublayer. Over a rough surface, the skewness of u becomes strongly positive with decreasing height, whereas the skewness of w shows a sharp decrease and becomes negative near the roughness elements. This means that over a rough surface, sweeps (u' > 0, w' < 0) are dominant and by far the main contributors to the shear stress both within and just above the roughness elements. The stress contibutions by ejections (u' < 0, w' > 0) are only of minor importance. Over a smooth surface, it was found that the stress contibutions of sweeps and ejections are equally important.
Summarizing, roughness elements slow down the average wind speed in the roughness sublayer, but at the same time they cause spatial variation in horizontal wind speeds. Defining a mean wind speed profile requires therefore spatial averaging of temporal average profiles. Average shear stress, which is constant with height in the inertial sublayer, decreases rapidly with height in the roughness sublayer. Much of the stress is absorbed by the roughness elements, and less stress is exerted on the surface between the elements. Turbulent flow properties are affected by the roughness as well. The turbulence intensity Tu is generally high and characterized by spatial variation in the roughness sublayer. The intermittent stress contributions are mainly caused by sweeps over a rough surface because of the large positive skew in the distribution of u and negative skew in the distribution of w.
Sediment Transport
Wind-blown sediment transport can occur in three different modes: creep, saltation, and suspension (Bagnold, 1973). Because the main mass of wind-blown sediment on sandy soils is normally transported by saltation, only this transport mode is considered in the continuation. In most predictive transport equations, the saltation mass flux is described as a function of shear velocity u*, which is proportional to the square root of average shear stess at the surface. However, in three recent publications (Bauer et al., 1988; Butterfield, 1998; Sterk et al., 1998, it was concluded that the use of u* or shear stress is inappropriate when considering saltation transport at time scales in the order of one second.
Bauer et al. (1988) studied the character of unsteady wind and related sand transport at a beach in the Guadeloupe-Nipomo Dunes Preserve, California. Hot-film anemometry and continuous-weighing sand traps were used to determine the relation between wind characteristics and saltation transport. The results showed a good correspondence between instantaneous streamwise wind speed and saltation flux, but the correspondence between shear stress fluctuations and saltation flux was poor.
Butterfield (1998) conducted detailed wind tunnel measurements of saltation transport in unsteady winds. Wind speed was measured with hot-wire anemometers, while the saltation mass flux was quantified with a calibrated laser system. It was concluded that the saltation flux correlates more strongly with velocities measured in the outer layer than with shear velocity u*.
Sterk et al. (1998) studied the effect of instantaneous wind speed and shear stress fluctuations on saltation transport during a field experiment on a bare, sandy alfisol in southwest Niger. Wind speed and shear stress fluctuations were measured with a fast response propeller anemometer at 3-m height. The saltation flux was quantified with a saltiphone, which is an acoustic saltation sensor that counts particle impacts with a microphone. A good correlation between horizontal, streamwise velocity fluctuations and instantaneous saltation flux was obtained (Fig. 3)
. High saltation fluxes were exclusively associated with sweeps and outward interactions. Both structures have a positive u' and create opposite contributions to the shear stress. Hence, no correlation between shear stress fluctuations and saltation flux was obtained.
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Fig. 3 Instantaneous horizontal wind speed (u) and saltation flux (S) vs. time for a typical wind erosion event on a sandy alfisol in southwest Niger on 27 June 1994
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These three experimental studies lead to the conclusion that the saltation flux is mainly driven by wind speed fluctuations in the inertial sublayer. Fluctuations in shear stress measured in the inertial sublayer seem to be irrelevant. In particular sweeps, but also outward interactions are the dominant turbulent structures that initiate and sustain soil particle transport.
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Conceptual model
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A conceptual model of the effect of non-erodible roughness elements on turbulent flow and sediment transport is now developed. The model is based on the above described information, and the concept of describing the initiation of soil particle movement as a stochastic process (Grass, 1970). Consider a dry, loose sandy soil with particles of similar density and shape, but with a range of particle sizes. For each particle size, a threshold wind speed ut at a certain height zt can be defined. The height zt is taken as the distance between the sandy surface and a height just above the roughness sublayer when the surface is partially covered with non-erodible elements. As the soil consists of a range of particle sizes, there exists also a range of threshold wind speeds. Hence, a probability density function (p.d.f.) of ut can be determined. It is assumed here that the ut values are Gaussian distributed. The exact shape of the p.d.f. is not relevant for the conceptual model.
A soil particle will be moved by the wind if the instantaneous wind speed u at zt exceeds the ut value of that particle. Since u is also a stochastic parameter, again a p.d.f. can be determined. The shape of this p.d.f. depends on the surface condition. Over a smooth surface, the distribution of u is close to Gaussian, whereas over a rough surface the distribution is positively skewed.
The conceptual model of the effect of roughness elements on saltation transport is based on the p.d.f.'s of u and ut. Both p.d.f.'s are drawn on the same axis in Fig. 4
, where two free-stream wind speed conditions (U1 and U2) are considered. In case of a smooth sandy surface without roughness elements, both p.d.f.'s are approximately Gaussian. If there is no overlap in the two p.d.f.'s (Fig. 4A), then all soil particles are stable and there is no sediment transport possible. So, the maximum instantaneous value of u is still below the minimum threshold wind speed for the soil. If the wind becomes stronger and the free stream wind speed increases from U1 to U2, the p.d.f. of u will move towards the p.d.f. of ut. Also, the distribution of u will become wider, as u will increase with increasing average wind speed. At a certain moment the two p.d.f.'s will partially overlap (Fig. 4B). This means that the wind is now strong enough to cause soil particle movement. The gusts with the highest instantaneous u values, i.e., sweeps and outward interactions, cause movement of the particles with lowest ut values. When the wind speed becomes still stronger, more overlap occurs and the number of particles that are lifted and transported will increase.
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Fig. 4 Probability density functions of horizontal wind speed u and threshold wind speed ut for a smooth surface condition (A and B), and a rough surface condition (C and D). Symbols U1 and U2 indicate two free-stream wind speeds, dashed lines indicate average wind speeds
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When the sandy soil is partially covered with non-erodible roughness elements, the situation will be different. The p.d.f. of ut does not change, bu the p.d.f. of u is wider because of the higher standard deviation u. Also, the distribution is positively skewed, which means that sweeps and outward interactions are more frequent than ejections and inward interactions. For the free stream wind speed U1, the distance between the averages of the p.d.f.'s will be wider for the rough surface (Fig. 4C), because the wind speed is slowed down by the roughness elements. The positive skew, however, causes disproportionately less distance between the two tails of the p.d.f.'s. With increasing wind speed, the p.d.f. of u will move towards the p.d.f. of ut. In addition, the p.d.f. of u will become wider (large u) and possibly more skewed. With wind speed increasing even more, at a certain moment there is more overlap between the p.d.f.'s than in the case of the smooth surface (Fig. 4D). Hence, the roughness elements cause higher instantaneous streamwise wind speeds, even though the free stream wind speed is the same. This potentially results in more sediment transport as compared with a smooth surface condition.
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Discussion and conclusions
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The conceptual model explains why application of flattened residue can enhance wind-blown sediment transport. When a certain quantity of non-erodible roughness elements is added to an otherwise smooth surface, the average wind speed will be slowed down, but the gustiness of the wind will be enhanced. If the free stream wind speed increases, at a certain moment, the reduction in average wind speed is nullified by the enhanced gustiness, and the reduction in sediment transport becomes zero
. The level of soil cover will determine how quickly this point is reached. In general, a low SC will only cause a minor decrease in average wind speed, but will enhance gustiness, and therefore the critical wind speed will be relatively low. The higher SC the more the average wind speed is slowed down and the higher the critical wind speed at which
. This is indicated in Fig. 2 for two different quantities (1000 and 1500 kg ha-1) of flattened residue. The lower SC (4.7%) had a critical wind speed of 11.1 m s-1 (at 2 m), while the higher SC (7.1%) had a critical wind speed of 16.0 m s-1. On the basis of the limited experimental data described here, it is assumed that under natural wind speed conditions only enhancement of sediment transport can be expected when SC < 10%. For higher SC's the critical wind speed is probably higher than what can be expected to occur under natural conditions.
The conceptual model is based on the characteristics of instantaneous streamwise wind speed just above the roughness sublayer. Within the roughness sublayer the spatial variation in horizontal wind speed (horizontal inhomogeneity) causes an even wider range of instantaneous wind speeds. Each individual obstacle will have its own wake zones and zones where the wind speed is accelerated. Complicated erosional–depositional patterns are created, which are related to the dimensions, number and distribution of the obstacles (Logie, 1982; Iversen et al., 1991).
At least one deficiency exists in the conceptual model. The process of particle entrainment during a storm cannot be explained by a single, static p.d.f. of ut. Once saltation transport has started, the moving sand cloud will cause ejections of particles whose ut values are not reached by the wind alone. But the sum of forces caused by wind drag and momentum of falling grains is sufficient to initiate movement of particles with higher ut values (Bagnold, 1973). This aspect, however, is believed to be similar for the two surface conditions, and is therefore less important when the effect of roughness on sediment transport is considered.
Other aspects that are not yet fully understood is how roughness affects the average and skewness of the u distribution when the free-stream wind speed increases. In Fig. 4, it was assumed that the roughness causes similar decreases in average wind speed for the two free-stream wind speeds. It was also assumed that the skewness becomes larger when the free-stream wind speed increases. No evidence was found in the literature for both assumptions.
In principle, the conceptual model is valid for wind tunnel and atmospheric conditions. There exists, however, an important difference in the flow properties for the two conditions. Turbulence intensities are generally lower in wind tunnels than in the atmosphere. Thus, in a wind tunnel, the p.d.f. of u will be narrower than in the field, which results in a higher critical free-stream wind speed for obtaining RQ values larger than one. It is therefore expected that the risk of enhancing sediment transport by small quantities of flattened residue is bigger under atmospheric conditions than what is expected from wind tunnel experiments.
Finally, it is concluded that a simple exponential model like Eq. [2] is not sufficient for describing the effect of soil cover on sediment transport. It does not allow enhanced sediment transport as compared with the smooth situation, and the model is independent of wind speed. In addition, the wind tunnel derived relationships are likely to overestimate the reduction in sediment transport by flattened residue under atmospheric conditions. The conceptual model as described here could assist in the development of a process-based model describing the effect of flat mulch material on wind-blown sediment transport. However, more experimental work on the behavior of instantaneous velocity fluctuations and sediment transport as affected by non-erodible roughness elements remains to be done.
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ACKNOWLEDGMENTS
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I am grateful to Adrie Jacobs and Ed Skidmore for critical reading of and commenting on the draft version of this paper, and to Emiel van Loon for helping draw one graph. The constructive comments given by three anonymous reviewers are appreciated.
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NOTES
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1 Soil flux is similar to the often used term "mass transport rate." 
Received for publication July 6, 1999.
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