A new model of wind erosion in the presence of vegetation

Authors


Abstract

[1] Vegetation is known to impact strongly the erosion of soil by the wind. Lateral cover is the primary parameter used to represent the amount of vegetation in aeolian research and, in particular, shear stress partitioning research. Although lateral cover provides a simple means for representing how much vegetation is in an area, it is not capable of characterizing how vegetation is distributed. A new, nonequilibrium model for the representation of nonerodible roughness elements is presented that uses the size distribution of erodible gaps between plants to characterize the ratio of the maximum shear stress to the average shear stress at the surface. The model shows very good agreement with measured shear stress ratios from the laboratory and field experiments. The model also satisfactorily explains relatively high horizontal aeolian sediment flux at high lateral cover. The relationship between this model and another shear stress partitioning model is explored, and the new model is found to be superior to the existing model because it (1) utilizes parameters with physical meaning that are measurable in the field or laboratory, (2) explains observations of horizontal flux at high cover, (3) overcomes difficulties inherent in the use of lateral cover to characterize vegetation on the surface, (4) is scale-explicit, and (5) can be used at multiple scales from an individual unvegetated gap to an entire landscape.

1. Introduction

[2] Nonerodible roughness elements, principally wind-immobile clasts and vegetation, modulate the erosion of soil by the wind. Lateral cover (λ) is an index of the amount of roughness encountered by the wind as it flows over a surface. It is most simply related to plant cover by λ = NAp, where N is the number density of plants and Ap is the area of each of these plants projected onto a plane perpendicular to the surface (i.e., the area of the plant when viewed in profile) [Okin, 2005]. For more than 3 decades, lateral cover has been used as the main parameter to represent nonerodible elements in aeolian research (i.e., since Marshall [1971]). For more than a decade, lateral cover has been the primary variable representing the amount of vegetation in shear stress partitioning models [e.g., Raupach, 1992] that have come to dominate models of wind erosion and dust flux from vegetated surfaces [Marticorena and Bergametti, 1995; Mahowald et al., 2002; Zender et al., 2003].

[3] Unfortunately, when plotted against lateral cover, field observations of horizontal aeolian flux do not match what is predicted from the prevailing shear stress partitioning model [Raupach et al., 1993]. Gillies et al. [2007] demonstrated a good relationship between a roughness density and sediment transport for regularly arrayed solid elements. Roughness porosity and distribution of the roughness (i.e., departure from regular array) causes the relationship to weaken. Field observations, for instance, show that there is a quasi-logarithmic decrease in the horizontal flux with increasing lateral cover, and that there can be measurable horizontal flux even at relatively high lateral cover (Figure 1). In contrast, when the shear-stress partitioning model of Raupach et al. [1993] is combined with a model of horizontal flux like that of Shao and Raupach [1993], horizontal flux decreases much more rapidly with increasing lateral cover (Figure 2). In this model, horizontal flux reaches zero at relatively low lateral cover and above this threshold aeolian transport does not occur. The exact form of the relationship between total horizontal flux and the threshold shear velocity in the Shao and Raupach [1993] model is not responsible for this behavior. Instead, it is the fact that the Raupach et al. [1993] model overestimates the threshold shear velocity for the vegetated surface and requires that the threshold shear velocity be the same everywhere. Okin [2005] for instance, showed that variability of vegetation cover has a major impact on total horizontal flux, particularly at high lateral cover.

Figure 1.

Horizontal flux versus lateral cover (λ) for two field experiments. (top) Data from individual storms from Owens Dry Lake [Lancaster and Baas, 1998]. (bottom) Data from two seasons in the Chihuahuan Desert [Li et al., 2007].

Figure 2.

Estimates of total horizontal flux using the shear-stress partitioning model of Raupach et al. [1993] and the flux equation of Shao and Raupach [1993]. Estimates are plotted for two values of m, 1.0 and 0.5. Light lines denote horizontal flux estimates at constant shear velocity of 100 cm/s. Heavy lines denote horizontal flux estimates for the wind speed record of the Jornada Experimental Range (JER) in New Mexico from 1997 to 2001.

[4] To partially compensate for the overestimation of the threshold shear velocity for vegetated surfaces, Raupach et al. [1993] introduced a parameter, m, that accounts for the difference between the average shear stress and the maximum shear stress on that surface. Here m was defined such that τs″(λ) = τs′() where τs″ is the maximum shear stress experienced anywhere on the surface and τs′ is the average shear stress for the surface. In effect, the m parameter, which must vary on the interval [0,1], serves to reduce the effective lateral cover thereby decreasing the threshold shear velocity. Reported values for m vary between the value of ∼1.0 suggested by Raupach et al. [1993] to 0.16 reported by Wyatt and Nickling [1997]. Crawley and Nickling [2003] reported intermediate values of m = 0.53–0.58. The m parameter is essentially a nonphysical tuning factor that has been used to account for the spatial variability of the shear stress acting on the surface.

[5] The inhomogeneity of the surface shear stress field suggests a weakness in the lateral cover parameter itself because the lateral cover quantifies how much roughness is on a surface but not how that roughness is distributed. This weakness has been highlighted as the “telephone pole problem” (W. G. Nickling, personal communication) (Figure 3). The telephone pole problem, simply stated, is that a telephone pole in a field may provide adequate lateral cover to effectively increase the threshold shear velocity, at least in shear stress partitioning models using λ. In contrast, field experience and physical intuition suggests that the same lateral cover divided into small blocks spread across the field would be more effective in increasing the threshold shear velocity and thus decreasing wind erosion. Although the average shear stress on the surface in both cases is the same, the telephone pole has little impact on the maximum shear stress experienced by the surface and therefore little impact on the threshold shear velocity. The distributed blocks have a significant impact on the maximum shear stress experienced at the surface and therefore greater impact on the effective threshold shear velocity.

Figure 3.

The Telephone Pole problem. Both of these depicted surfaces have the same lateral cover. In the left image, lateral cover is partitioned into sixteen objects, whereas in the right image, lateral cover is partitioned into only four objects that are four times the height of objects in the image on the left. Under the Raupach et al. [1993] shear stress partitioning model, both of these surfaces would experience the same horizontal flux for a given wind condition. In the model presented here, the surface depicted on the right will experience greater horizontal flux for a given wind condition than the surface depicted on the left.

[6] The purpose of this report is to present a new model for shear stress partitioning on vegetated surfaces that is not subject to the weaknesses of lateral cover as an index of the sheltering effect of vegetation. This model utilizes unvegetated gap size to characterize the surface and thus explicitly accounts for spatial variability in the shear stress experienced by the soil. Not only does this model better match the patterns of horizontal flux observed in the field (e.g., Figure 1), but the model also fits measured shear stress ratios, and explains the diversity of values for m reported in the literature.

2. Model Description

[7] The new shear stress partitioning model for vegetated surfaces begins with the physically reasonable assumption that plants are porous objects. The porosity of vegetation has either been measured in the field or imposed in wind tunnel studies by several investigators [Grant and Nicking, 1998; Gillies et al., 2000, 2002; Minvielle et al., 2003]. Because wind flows through porous vegetation, reverse flow in the lee of plants can be negligible and windward surface shear velocities in the wake of the plant can be greater than zero. Some studies [e.g., Bowker et al., 2006] have shown that there can be reverse flow on the lee side of plants. The model used by Bowker et al. [2006], the Quick Urban and Industrial Complex (QUIC) model, does not allow porous objects, which partly explains the existence of reverse flow in their study. This may be appropriate in the case of mesquite coppice dunes studied by these authors, however, because these features are composed primarily of sand and as a result probably behave as bluff bodies. At any rate, proceeding from the assumption that return flow in the lee of porous plants is negligible, it will be shown that the new model accounts equally well for shear stress partitioning of bluff and porous bodies.

[8] Studies of shear velocity downwind of porous fences indicates that the surface shear stress increases from a small value immediately downwind of a plant and asymptotically approaches the surface shear stress further downwind in the absence of vegetation [Bradley and Mulhearn, 1983] (Figure 4). This creates an area of reduced, but not zero, shear stress downwind of plants that attenuates as the distance from the plant increases. As a result, the ability of the wind to erode in the immediate lee of a plant is much less than in an unsheltered gap between plants.

Figure 4.

The ratio of shear velocity behind a porous windbreak (u*s) to the shear velocity on an identical surface without a windbreak (u*) adapted from Bradley and Mulhearn [1983]. These data have been scaled so that the ratio asymptotically approaches one. The ratio (u*s/u*) is a function of the downwind distance from the windbreak expressed in units of windbreak height (h) and is fit by equation (4). For these data, the best-fit coefficients are (u*s/u*)x=0 = 0.32 and c1 = 4.8.

[9] Wind erosion occurs in the model at all points where the wind shear velocity exceeds the threshold shear velocity of the soil and increases with increasing excess shear velocity. Because the model seeks to model explicitly the distribution of shear stress on the surface, the model is formulated probabilistically, utilizing the probability distribution function of the distance to the nearest upwind plant. Specifically, the total horizontal flux, QTot, is calculated as:

equation image

where Pd(x/h) is the probability that any point in the landscape is a certain distance from the nearest upwind plant expressed in units of height of that plant, x/h, qx/h is the horizontal flux for a point x/h away from the nearest upwind plant. The formulation of QTot in equation (1) is simply the one-dimensional version of the relationship given by Raupach and Lu [2004], who suggested that for a specific area, the total flux is given by the sum of different sources weighted by the fraction of the area that they occupy. At small distances from a plant, qx/h is significantly less than the total horizontal flux in the absence of vegetation, qmax, due to the depression of surface shear velocity in the wake of the plants. As x/h increases, qx/h asymptotically approaches qmax as the sheltering effect of the vegetation dissipates.

[10] In principle, the model does not assume any specific distribution of vegetation because the windward plant spacing is specified explicitly and probabilistically. The probability distribution of the size of unvegetated gaps between plants, x/h, is specified as Pg(x/h). Pg(x/h) is used in the model to determine the probability that any point in the landscape is a distance (expressed in units of plant height) from the nearest upwind plant, Pd(x/h). These two probability distributions are related to one another by:

equation image

Because they are probability distributions, the integral from zero to infinity of Pg(x/h) and Pd(x/h) must be equal to one, and this constraint sets the coefficient of proportionality in this relationship.

[11] In practice, McGlynn and Okin [2006] have shown that Pd(x) can be represented as an exponential function Pd(x) = eequation image where equation image is a scaling parameter. Thus, by assuming constant plant height and using equation (2), the gap size distribution is given by:

equation image

which is a gamma distribution, and the average gap size is given by equation image. In units of plant height, the average gap size is given by equation image/h. Other formulations for Pg(x/h) and Pd(x/h) can be used. In particular, it is prudent to truncate Pd(x/h) so that there that there is no probability that erodible ground exists within a small fraction of equation image from the nearest upwind plant. Within this small distance from a plant, the boundary layer of the object itself predominates and wind erosion cannot occur. The cutoff of approximately 0.05% equation image gives a good fit with experimental data. This variation from the pure gamma distribution and any other changes to Pg(x/h) or Pd(x/h) will impact calculated values of QTot, SSR, and meff (discussed below).

[12] In the model, plants are assumed to be distributed on the land surface with each plant associated with a reduced shear stress wake zone (Figure 5). In this sheltered zone, the ratio of the surface shear velocity, u*s, to the surface shear velocity in the absence of plants, u*, is less than one. In contrast, the model of Raupach et al. [1993] assumes a zone of zero surface shear stress in the lee of plants (i.e., u*s/us = 0). To parameterize the zone of reduced shear stress, we use an exponential curve:

equation image
Figure 5.

The distribution of reduced shear stress on the surface during wind erosion. (left top) The Raupach et al. [1993] shear stress partition model envisions a triangle downwind of a plant that experiences no shear stress at the surface (black area). In the model presented here (left bottom), shear stress increases from the downwind edge of the plant and eventually approaches the shear stress for an equivalent unvegetated surface (dark areas have lower shear stress). The landscape is envisioned (right) as being comprised of a collection of plants with their reduced shear stress areas. The model does not assume any specific spatial distribution of plants.

[13] For the data of Bradley and Mulhearn [1983] (Figure 4), the best-fit coefficients are (u*s/u*)x=0 = 0.32 and c1 = 4.8. The e-folding distance for recovery of the shear stress is approximately 4.8 times the plant height, h. With this curve, the shear stress recovers to ∼90% of its value in the absence of vegetation after a distance of approximately 10 times the plant height. This result is consistent with that of Minvielle et al. [2003], who determined that the shear stress downwind of porous vegetation in the wind tunnel recovered to nearly the upwind value in about the same distance. We do not consider the fit through the data of Bradley and Mulhearn [1983] to be definitive for all atmospheric conditions and plant porosities. Further field and wind tunnel research is required to clarify how porous vegetation impacts downwind surface shear stress profiles. The experimental results of Alfaro and Gomes [1995], for instance, suggest that there may be a relatively flat zone of (u*s/u*) at low x/h, though it is not clear whether those results are strictly applicable here because roughness elements were solid and shear stress was measured downwind of arrays of roughness elements rather than a single roughness element. For the purposes of this report, the exponential fit through the data of Bradley and Mulhearn [1983] was used, except where noted.

[14] The shear stress ratio (SSR) is the ratio of the average shear stress on the soil surface in the presence of vegetation to the shear stress on the soil surface in the absence of vegetation. SSR can be calculated directly in the model by SSR = equation image(u*s/u*)Pd(x/h)d(x/h), where (u*s/u*) is from equation (4).

[15] We use the formulation of Shao and Raupach [1993] to calculate the total horizontal mass flux for a point x/h distance from the nearest upwind plant:

equation image

where A is a constant assumed to be equal to one, ρ is the density of air, g is the acceleration due to gravity, and u*t is the threshold shear velocity of the unvegetated soil. Conceptually, the model is not dependent on the form of the equation for qx/h as long as qx/h is expressed as a function of u*t and u*s (which substitutes for u*). In practice, qx/h must be evaluated for all possible values of x/h at a specific u* where u*s is determined from equation (4). If qx/h is desires for a time period where the probability distribution of wind velocity is known or assumed, it can be calculated as:

equation image

where qx/h,var is the average horizontal flux at a point x/h from the nearest upwind plant for the desired wind record and Pu* is the probability of shear velocity u* in the wind record.

[16] Finally, to make direct comparisons between the model presented here and the model of Raupach et al. [1993], as well as to compare with existing experimental measurements, we derive a relationship between the unvegetated gap size and lateral cover, λ. Consider a line transect across vegetation arranged in circular clumps. The average length of a gap and the adjacent (upwind) plant, equation image, will be given by equation image, where equation image is the average length of a plant along the transect and equation image is the average length of a gap. For circular plants, equation image is equal to π/4 of the plant diameter. The fractional cover of plants, C, is the fraction of the transect that is covered by plants, and is given by C = equation image. Okin [2005] has shown that relationship exists between C, and lateral cover λ, λ = (AP/AB)C, where AP is the average profile area of plants and AB is the average basal area of plants. Combining the relationship between λ and C with the relationship between C and equation image gives:

equation image

equation (7) can be used to calculate λ for all values of equation image, which allows model results to be plotted against either λ or average gap size/canopy height (equation image/h).

3. Results

[17] There is very good agreement between the model-calculated SSR and experimental data from King et al. [2005] when (u*s/u*)x=0 = ∼0.0–0.3 (Figure 6). The same model results plotted as SSR versus average gap size/canopy height (equation image/h) are shown in Figure 7. Several features of the experimental data are explained by the model results. First, both the experimental measurement and model predictions show the same general decrease in SSR with increasing lateral cover. Second, the experimental data shows little scatter at low lateral cover but increasing scatter with increasing lateral cover. Similarly, at low lateral cover the model predictions of SSR converge to a relatively narrow range, whereas at high lateral cover, model predictions of SSR cover a relatively wide range depending on the value of (u*s/u*)x=0. Third, experimental SSR measurements for porous objects (Figure 6, open symbols), both in laboratory and field tests, have generally higher SSR than solid objects at the same lateral cover.

Figure 6.

Shear stress ratio (SSR) versus lateral cover (λ) calculated from the model presented here for four different values of (u*s/u*)x=0. For all cases, c1 = 4.8 (equation (4)). Also plotted are values from field and laboratory results. Data plotted as solid symbols correspond to solid roughness elements. Data plotted as open symbols correspond to porous roughness elements. Experimental data from King et al. [2005].

Figure 7.

Shear stress ratio (SSR) plotted against the ratio of average gap width to canopy height for four different values of (u*s/u*)x=0. For all cases, c1 = 4.8 (equation (4)).

[18] The model-calculated SSR at lateral cover of 1 (SSRλ=1), which also corresponds to the SSR at average gap size approaching zero, for 101 model runs with (u*s/u*)x=0 varying from 0.0 to 1.0 shows a linear relationship between these two variables (Figure 8). The intercept of this relationship is nearly zero (0.0074) and the slope is nearly 1 (0.9926) with R2 = 1.0. When the intercept is forced to be zero, the slope of the relationship is 1.0038 with R2 = 0.9998. This result makes sense. If the size of unvegetated gaps were equal to zero, then, according to the model, the shear stress at the surface of the (infinitesimally small) gaps between plants would be equal to value of (u*s/u*) immediately downwind of the vegetation, that is, (u*s/u*)x=0. Thus, it becomes clear that the shear stress ratio at very high lateral cover must be equal to (u*s/u*)x=0 and that SSRλ=∞ = (u*s/u*)x=0. By this reasoning, the model allows a physical interpretation of SSR at high lateral cover or low gap size. King et al. [2005] showed that SSRλ=1 was sensitive to β, the ratio of the drag coefficient of an individual roughness element to the drag coefficient of the unvegetated soil. King et al.'s [2005] results for the Raupach et al. [1993] shear stress partitioning model indicate that SSRλ=1 increases with increasing β. The new model is not sensitive to β because this parameter is not used. In light of the new model, we can infer that (u*s/u*)x=0 increases with increasing β, or more to the point, increasing drag coefficient of an individual roughness element. Further experimental work is required to clarify this relationship.

Figure 8.

Plot of the modeled shear stress ratio at λ = 1, SSRλ=1, versus the ratio of the shear velocity immediately downwind of a roughness element to the shear velocity in the absence of the roughness element (u*s/u*)x=0.

[19] Furthermore, although (u*s/u*)x=0 behaves in the model as somewhat of a tuning parameter, it differs from the m parameter of Raupach et al. [1993] in that it is both physically meaningful and measurable. Measurements in the laboratory or the field of (u*s/u*)x=0, as well as the e-folding distance of recovery of shear stress in the lee of roughness elements, c1, are required to further constrain the model, but data in Figure 6 suggests that for porous objects, (u*s/u*)x=0 ∼ 0.2 and for solid objects (u*s/u*)x=0 ∼ 0.1. Direct measurements of (u*s/u*)x=0 might be made in wind tunnels and in the field immediately downwind of roughness elements using an Irwin sensor [Irwin, 1980; Crawley and Nickling, 2003; Gillies et al., 2007]. In the field, near-surface anemometer measurements in the immediate lee of objects might also be used and scaled to the upwind shear stress. Carrying out these measurements for elements of different porosities would provide a means to quantify the porosity effect on shear stress reduction in the lee of the vegetation.

[20] The new model has been used to calculate QTot for several cases (Figure 9). In all cases, a threshold shear velocity of the unvegetated surface u*t of 24 cm s-1 was used. The fit of (u*s/u*) from the data of Bradley and Mulhearn [1983] was also used. QTot was calculated for the case where u* = 100 cm s−1 as well as the case where the wind velocity is variable (equation (6)) using an experimental wind record from the Jornada Experimental Range (JER) in south central New Mexico. QTot was calculated for a range of values of equation image, the average unvegetated gap size, but constant canopy height, h = 100 cm. New model results are plotted as solid lines against both equation image/h (Figure 9, left) and λ (Figure 9, right). Results show significant horizontal flux rising from relatively low (but nonzero) values at small values of equation image/h to values approaching the expected value of QTot in the absence of vegetation. The model never reaches this maximum value of QTot because the presence of even a small amount of vegetation in the model means that there are areas that experience reduced surface shear stress. When plotted against λ, QTot decreases almost logarithmically (e.g., appears nearly linear in the log-linear plot), which is consistent with field observations (Figure 1). Also consistent with field observation, the model predicts significant aeolian flux at high lateral covers.

Figure 9.

Estimates of total horizontal flux for the Raupach et al. [1993] model (dashed) and the model presented here (solid). (a and b) Model estimates for the case of uniform unvegetated gap size. (c and d) Model estimates for the case where unvegetated gap size is given by a gamma distribution. Light lines denote estimates at constant shear velocity of 100 cm/s. Heavy lines denote estimates for wind speeds from the Jornada Experimental Range (JER) in New Mexico from 1997 to 2001. Estimates are plotted against lateral cover (λ) (right) and average unvegetated gap size in units of canopy height (left). In contrast to the Raupach et al. [1993] shear stress partitioning model, horizontal flux rates are never zero, even at high lateral cover or landscapes with small average gap size.

[21] Equation (7) allows calculation of the threshold shear velocity for the vegetated surface at any equation image/h using the model of Raupach et al. [1993], which in turn allows us to calculate total horizontal flux for this model (Figure 9, dashed lines). Horizontal flux drops to zero at relatively low lateral covers (Figure 9, right) and at relatively small plant spacing (Figure 9, left). At λ = 0, horizontal flux calculated with the model of Raupach et al. [1993] is equal to expected flux in the absence of vegetation. This value is slightly higher than the maximum value of QTot calculated with the new model as discussed above. In all, the new model presented here predicts slightly lower flux at low λ (high equation image/h), but higher flux at high λ (low equation image/h).

[22] To investigate the relationship between the new model presented here with that of Raupach et al. [1993], the horizontal flux predicted by the Raupach et al. [1993] shear stress partitioning scheme in combination with the flux model of Shao and Raupach [1993], QTot,Raupach, was derived by replacing the threshold shear velocity term in the Shao and Raupach [1993] flux model with the expression for the threshold shear velocity on a vegetated surface from Raupach et al. [1993], thusly:

equation image

where σ = equation image, and u* is the shear velocity. QTot,Raupach was set equal to the flux calculated by the model presented here, QTot,new:

equation image

and the resulting equation was rearranged to solve for m, yielding a straightforward but long solution to the quadratic equation that is shown here. The m parameter derived in this way will be referred hereafter to the effective m parameter, meff, or the value of the m parameter required to give the same total horizontal flux as the new model.

[23] For cases when all gaps are the same size (Figure 10, solid line with symbols), meff is less than one for all values of λ, decreases with increasing λ, and spans a range from ∼1 to ∼0.1. For all values of λ, QTot,Raupach < QTot,new (Figure 9, top right) when all gaps are the same size, and at low values of λ, QTot,RaupachQTot,new, resulting in meff ∼ 1. These results thus suggest that the m parameter in the shear stress partitioning model of Raupach et al. [1993] is not a constant, as hypothesized by those authors, but in fact, varies with λ. This interpretation is consonant with Raupach et al.'s definition of the m parameter. m was defined such that τs″(λ) = τs′() where τs″ is the maximum shear stress experienced anywhere on the surface and τs′ is the average shear stress for the surface. In effect, the statement that the m parameter depends on λ implies that the relationship between τs″ and τs′ also depends on these factors. This is physically intuitive; the maximum shear stress on the surface depends on the size of unvegetated gaps.

Figure 10.

Effective m parameter (meff) versus lateral cover (λ). The solid line with symbols is meff for a constant shear velocity of 100 cm/s and uniform unvegetated gap size. The light solid line is meff for a gamma distribution of unvegetated gap size and constant wind shear velocity of 100 cm/s. The heavy solid line is meff for a gamma distribution of unvegetated gap size and wind speeds from the Jornada Experimental Range (JER) in New Mexico from 1997 to 2001.

[24] For cases when the unvegetated gap size is represented as a gamma distribution (Figure 10, lines without symbols), QTot,Raupach > QTot,new for low lateral cover and QTot,Raupach < QTot,new for high lateral cover (Figure 9, bottom right). Likewise, meff > 1 at low lateral cover and meff < 1 at high lateral cover. At the point where QTot,Raupach = QTot,new, meff is equal to 1. These results highlight a significant difference between the model presented here and that of Raupach et al. [1993], particularly with respect to the calculation of horizontal aeolian sediment flux. Specifically, the model presented here predicts higher flux than the model of Raupach et al. at high lateral cover, but lower flux at low lateral cover. For cases when the new model predicts higher flux than that of Raupach et al., the assertion that m ≥ 1 holds, though m still varies as a function of λ. However, for cases when the new model predicts lower flux than that of Raupach et al., meff is found to have values greater than 1, outside the range originally proposed by those authors. In addition, because meff depends on the magnitude of the threshold shear velocity, we conclude that, in all likelihood, the m parameter does as well, though further experimental testing would be required to verify this result.

4. Discussion and Conclusions

[25] Although lateral cover provides a simple means for representing the amount of vegetation encountered by wind moving over a surface, it is not capable of characterizing how that vegetation is distributed. Recent results from field observations suggest that horizontal flux can be strongly dependent on vegetation distribution [e.g., Okin and Gillette, 2001; Gillette et al., 2006]. The model presented here provides a new way of looking at shear stress partitioning and aeolian sediment flux. In contrast to the lateral cover-based approaches to estimating shear stress partitioning and horizontal flux, the approach suggested here utilizes the size distribution of unvegetated gaps on the landscape to characterize the land surface. This approach yields predictions of SSR that are at least as good as those of the model of Raupach et al. [1993] without requiring a model parameter that is impossible to measure and difficult to justify physically, the m parameter. Indeed, although the model was derived assuming porous vegetation so that the impact of return flow behind solid objects could be ignored, the model does a remarkable job representing the SSR of solid objects (Figure 6), suggesting that its utility may extend to some arrays of solid objects. In addition, the model provides nonzero estimates of horizontal flux at relatively high vegetation cover that are consistent with field observations (e.g., Figure 1), in contrast to the shear stress partitioning model of Raupach et al. [1993], which suggests that above some relatively low threshold lateral cover, no horizontal aeolian flux should be possible (Figure 2).

[26] An additional advantage of the model presented here is that the primary input variables related to vegetation cover, the gap size distribution (Pg) and vegetation height (h), can be obtained by the standard line-intercept vegetation survey technique with height measurements of each intersected plant [Okin et al., 2006], or an image-based technique [McGlynn and Okin, 2006] supplemented by knowledge of plant height. Lateral cover, in contrast, is relatively difficult parameter to measure in the field, and no standard techniques exist to estimate it.

[27] Despite its superior fit with experimental shear stress partitioning data, there is still a problem for any shear stress partitioning model that is linked with sediment transport: the scale of the roughness, especially its height, has a profound effect on transport efficiency. Gillies et al. [2007] showed that when roughness is short (i.e., much less than the saltation cloud height), transport efficiency is much higher than when roughness is tall, even at the same roughness density. The physical interaction of the saltating grains with the roughness is very important. So, although the shear stress partitioning models effectively model aerodynamics over a wide range of roughness density or gap size, effective models of sediment flux may require an additional correction made for roughness height-dependent transport efficiencies not included here.

[28] Despite the absence of this correction, major advantages of the approach presented here over lateral cover-based approaches have to do with how the model represents variability in the landscape as well as the explicit treatment of scaling in the new approach. By using a linear parameter to characterize vegetation (windward gap size) as opposed to an areal parameter (lateral cover), the new model is able to incorporate directional variability into estimates of shear stress partitioning and wind erosion or dust flux. Anisotropic vegetation patterns are common in both natural and anthropogenic landscapes. Types of anisotropic natural vegetation include the well-known striped vegetation on gentle slopes in deserts [e.g., Klausmeier, 1999] or anisotropic coppice dune distributions often found in wind-dominated landscapes (e.g., “streets” in the work of Okin and Gillette [2001] and McGlynn and Okin [2006]). Anisotropic vegetation patterns in human-dominated landscapes might occur as furrows, roads, or windbreaks aligned along a particular direction. It is, for instance, a common sight to see dust emitted from dirt roads parallel to a strong wind. Okin et al. [2006, Figure 6] have shown that farmers in southwestern South Africa leave E-W oriented strips of natural vegetation to protect against strong south winds in the region. These vegetated strips cannot be expected to provide equal protection from east or west winds.

[29] Okin [2004] has highlighted the influence of areal spatial variability on aeolian flux by using a stochastic version of the Raupach et al. [1993] shear stress partitioning model. Because it is a nonlinear, threshold-controlled process, Okin [2004] found that small areas of relatively low cover dominated flux estimates. The problem with the approach of Okin [2004], however, is that the scale of the variability was not specified. That is to say, the model did not specify whether relatively low-cover areas needed to be at least a few meters across, a few tens of meters across, or a few hundred meters across. This deficiency arises from the unspecified scale of the model of Raupach et al. [1993] itself, and a more intrinsic problem of characterizing lateral cover in the field. Lateral cover, in short, is not a scale-independent variable: at fine scales (i.e., several meters), lateral cover depends on the exact arrangement of plants and the location where lateral cover is measured. At coarse scales (i.e., hundreds of meters), lateral cover depends on the arrangement of vegetation patches with relatively homogeneous vegetation. This deficiency of lateral cover is minimized in large homogenous areas of randomly or regularly spaced vegetation of the same height or the conditions usually represented in wind tunnels. Natural environments, in contrast, are often structurally diverse (i.e., have plants of many different heights and growth forms), and also often spatially heterogeneous, with contiguous patches composed of different species, growth forms, heights, and densities [Peters et al., 2006].

[30] In contrast, the model proposed here explicitly treats its scale of operation. In the model, variability in surface shear stress, which leads to variability in horizontal sediment flux, arises at the scale of individual unvegetated gaps. A landscape is composed of a collection of unvegetated gaps, each scaled by the height of the upwind sheltering plant, with a specific probability density. Although this probability density is not scale independent, it can be coupled with models or observations of plant distribution [e.g., Caylor et al., 2004] and demography [e.g., Peters, 2002] to investigate scaling dependencies. By treating the landscape as a collection of unvegetated gaps that behave somewhat independently, the model also provides a way to scale from the flux at a single point to the landscape-scale flux. Furthermore, the model allows estimation of flux at the downwind edge of a single unvegetated gap. It is at this gap-plant interface that the geomorphic process of wind erosion influences plant biology and soil texture, depth, and biogeochemistry [Okin et al., 2006].

[31] Many global models of dust emission and transportation exist, and some of these use shear-stress partitioning to represent the effect of vegetation on wind erosion and dust emission [e.g., Marticorena et al., 1997; Mahowald et al., 1999; Zender et al., 2003]. The model and data presented here suggests that wind erosion, and hence dust emission [Gillette, 1977], can occur at relatively high lateral cover, in contrast to existing shear stress partitioning schemes. It would be fruitful to implement the present wind erosion scheme in these global models to investigate the impact that this method would have on total atmospheric dust loading as well as the distribution of dust emission hotspots worldwide.

[32] Furthermore, there is considerable controversy about whether human activities in arid and semiarid regions impact atmospheric dust loading. For instance, Prospero et al. [2002] argue that major dust sources occur only in areas with no vegetation and that these areas are not impacted to a significant degree by human activities at the local scale. They thus conclude that global atmospheric dust loading is not strongly impacted by local land use. In contrast, Tegen and Fung [1995] and Mahowald et al. [2002] argue that humans may significantly impact global dust loading. Model results and data presented here show that there can be significant aeolian flux at relatively high cover. Relatively high cover landscapes are more likely to be used by humans for activities such as pastoralism, agriculture, and habitation than areas with little or no cover and are thus more susceptible to vegetation change resulting from land cover change. Furthermore, relatively high cover landscapes are likely to experience greater fluctuations of cover under climate change scenarios [Intergovernmental Panel on Climate Change, 2001] than landscapes with no or low cover. Thus the results presented here suggest that further investigation is required of shear stress partitioning and dust flux in relatively high-cover environments. The model presented here provides an initial step in this investigation.

Acknowledgments

[33] This research was supported by NSF-Ecosystems studies grant 0316320. Many thanks are offered to D. A. Gillette, who provided Jornada wind data as well as many stimulating discussions that have led to this model. The author would like to acknowledge the contribution of James King who provided shear stress partitioning data. Ian McGlynn contributions in the development of the model are also sincerely appreciated. The author would also like to thank the associate editor and two reviewers for their helpful comments.

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