JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, D11208, doi:10.1029/2004JD005288, 2005
Dependence of wind erosion and dust emission on surface heterogeneity: Stochastic modeling Gregory S. Okin Department of Environmental Sciences, University of Virginia, Charlottesville, Virginia, USA Received 27 July 2004; revised 17 February 2005; accepted 28 February 2005; published 8 June 2005.
[1] A method for incorporating surface heterogeneity into models of wind erosion and
dust emission is presented. Parameters that control threshold shear velocity (bare-soil threshold shear velocity, vegetation cover, and vegetation shape) are parameterized as modified-normal distributions with defined means and coefficients of variation. A parameterized bootstrap (Monte Carlo) approach is used to estimate histograms of threshold shear velocities for vegetated surfaces using the model of Raupach et al. (1993). Horizontal mass flux is calculated from threshold shear velocity histograms and a wind record. Vertical mass flux (dust emission) can be inferred from horizontal mass flux. The results from these simulations indicate that wind erosion and dust emission are dominated by the few instances with unusually high surface erodibility (e.g., low threshold shear velocity). This finding is supported by observations of dust emission ‘‘hot spots’’ in landscapes undergoing dust emission. As a result, wind erosion and dust emission flux rates are much higher in simulations with variability than they are in nonvariable cases using only parameter means. The magnitude of highly nonlinear, thresholdcontrolled processes such as wind erosion and dust flux is essentially controlled by variability in the landscape. This result argues for inclusion of surface heterogeneity in global and regional dust emission models. Citation: Okin, G. S. (2005), Dependence of wind erosion and dust emission on surface heterogeneity: Stochastic modeling, J. Geophys. Res., 110, D11208, doi:10.1029/2004JD005288.
1. Introduction [2] Atmospheric mineral dust, produced in deserts by wind erosion, is an integral component of the entire Earth system. It can affect the global energy balance by changing the radiative forcing of the atmosphere through its ability to scatter and absorb light [Tegen and Lacis, 1996; Sokolik and Toon, 1999]. Atmospheric mineral aerosols may also provide surfaces for reactions, change the concentration of other aerosols in the atmosphere, and affect cloud nucleation and optical properties [Dentener et al., 1996; Levin et al., 1996; Dickerson et al., 1997; Wurzler et al., 2000]. The iron in dust is thought to play a major role in ocean fertilization and oceanic CO2 uptake thereby affecting the global carbon budget [Duce and Tindale, 1991; Coale et al., 1996; Piketh et al., 2000]. Dust transported to downwind terrestrial ecosystems can play a major role in soil formation and nutrient cycling [Chadwick et al., 1999; Reynolds et al., 2001; Okin et al., 2004] and present serious health concerns [Griffin et al., 2001]. [3] Modeling efforts by many investigators on regional [e.g., Gillette and Hanson, 1989; Shao and Leslie, 1997; Lu and Shao, 2001] and global [e.g., Ginoux et al., 2001; Zender et al., 2003] scales have been undertaken to provide an understanding of dust source distributions and the Copyright 2005 by the American Geophysical Union. 0148-0227/05/2004JD005288
characteristics of dust-emitting landscapes. Modeling studies provide the further promise of understanding the impacts of climate change and land cover conversion on dust emission and distribution [e.g., Tegen and Fung, 1995; Mahowald and Luo, 2003]. [4] Surface characterization is a major constraint on accurate modeling of desert dust because the threshold wind speed above which dust emission occurs depends on both soil and vegetation characteristics [Raupach et al., 1993]. Because the soil and vegetation parameters that control wind erosion and dust emission vary in both space and time, consideration of this heterogeneity is vital in understanding both fine- and coarse-scale dust emissions. Observationally, dust emission often occurs in ‘‘hot spots’’ arising from surface heterogeneity, and Gillette [1999] has pointed out that observations of dust storms, particularly in vegetated terrains, often display distinct plumes emanating from relatively small areas which, despite their size, account for the majority of airborne dust. This ‘‘hot spot’’ phenomenon has been cited by Okin and Gillette [2004] as a major source of error in model estimates of dust emissions. [5] Wind erosion and dust emission occur when the erosivity of the wind, typically quantified as the wind shear velocity, u* (cm s1) exceeds the erodibility of the surface, typically quantified as a threshold shear velocity. As the shear velocity of the wind exceeds the threshold shear velocity of a surface, aerodynamic forces detach saltationsized particles from the surface and carry them downwind
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[Bagnold, 1941]. The shear velocity of the wind depends on free-stream wind velocity and the roughness of the surface it is passing over. The threshold shear velocity of the surface depends on soil particle size, particle cohesion, and the presence of physical or biological crusts [Marticorena and Bergametti, 1995; Marticorena et al., 1997b; Lu and Shao, 1999, 2001] and is modulated by the presence and distribution of nonerodible elements [Raupach et al., 1993; Okin and Gillette, 2001]. Saltation flux accounts for the bulk of the mass and kinetic energy flux of aeolian sediment transport. Dust emission (vertical flux) is a byproduct of saltation because saltating particles sandblast the surface and eject fine particles that cannot be detached by purely aerodynamic forces [Gillette, 1977; Alfaro et al., 1998]. Thus improvement in the estimation of horizontal flux will improve estimation of dust emission. [6] The purpose of this study is to evaluate the role of spatial heterogeneity in determining the erodibility of vegetated surfaces in a modeling context. This study provides a quantitative framework to estimate the importance of heterogeneity, and presents a general method for the incorporation in horizontal aeolian flux and dust emission model estimates. We conclude that the inclusion of spatial heterogeneity is vital in providing realistic estimates of horizontal sediment flux and dust emission.
2. Modeling 2.1. Relations Controlling Wind Erosion and Dust Emission [7] Saltation mass flux has been related by a large number of authors using different approaches. Bagnold [1941] approached the problem first and provided a physically reasonable solution that was later confirmed by the analysis of Owen [1964]. Equations that predict saltation have been suggested by many authors and verified by wind tunnels, field experiments, and alternate theoretical derivations. One such equation is [Shao et al., 1993] q¼A
rX 2 u u u2 ; *tv g u * * *
ð1Þ
where q is the instantaneous horizontal (saltation) mass flux (g cm1 s1), A is a unitless parameter (usually assumed to be equal to 1) related to supply limitation [Gillette and Chen, 2001], r is the density of air (g cm3), g is the acceleration of gravity (cm s2), u* is the wind shear velocity (cm s1), and u*tv is the threshold shear velocity for a vegetated surface (cm s1). In equation (1), u* represents the drag felt by the total surface (i.e., both the vegetation and soil). For consistency, u*tv also represents the shear velocity felt by the total surface, but can be thought of as the shear velocity at the soil surface at threshold conditions adjusted for the presence of vegetation (see equation (4)). Because of sandblasting by saltation-sized particles, vertical mass flux (dust flux) Fa (g cm2 s1) is linearly related to q by a constant K (cm1) that is typically on order of 105 – 106 but can vary by several orders of magnitude [Gillette et al., 1997]. The value of K is strongly dependent on soil texture, crusting, and moisture.
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[8] The shear velocity is related to wind speed at height z (cm), U(z) (cm s1), under neutral conditions by a modified Law of the Wall, u zD ; U ð zÞ ¼ * ln k zo
ð2Þ
where k is von Karmann’s (unitless) constant, zo is the roughness height (cm), and D is the displacement height (cm). If D > 0, then turbulent flow in wakes predominates and no wind erosion can occur; thus modeling approaches to wind erosion must assume that D = 0 at least part of the time, so that wind erosion is allowed on vegetated surfaces, and roughness height, zo, must be assumed to be the sole determinant of the relationship between U(z) and u*. [9] Several authors have proposed relations to predict the impact of surface roughness on aerodynamic roughness. A typical parameter used to describe surface roughness is ‘‘lateral cover,’’ the total frontal area of vegetation intercepted by the wind per unit area. Lateral cover is the vertical analog to fractional cover, C = NAB, where AB is the average basal area (cm2) of the plants (i.e., the average area of the silhouette of the plants projected onto a plane parallel to the ground). Raupach et al. [1993] defined lateral cover for cylindrical plants as l = ndh/S, where n is the total number of plants in area S, and d and h are the average basal width (i.e., diameter) and height of the plants in area S. Generalizing, lateral cover can be defined as l = NAP where N is the number density of plants (N = n/S with units of cm2) and AP is the average profile or frontal area (cm2) of the plants (i.e., the average area of the silhouette of the plants projected onto a plane perpendicular to the direction of the wind). For cylindrical plants, AP = dh, and the formulation for lateral cover defined as NAP is identical to that given by Raupach et al. [1993]. [10] Lateral cover is thus a measure of the area of canopy intercepted by the wind in a particular area and a good proxy for the amount of drag due to the presence of surface roughness elements. Lateral cover has been related to zo, aerodynamic roughness length by Lettau [1969] and Wooding et al. [1973] using zo = h2c l, where hc, is equal to canopy height (cm) and l is lateral cover. This relation holds for 0 < l < 0.1. For l > 0.1, zo/hc remains approximately constant. Marticorena et al. [1997a] have suggested a relation that incorporates the leveling-off of the linear relationship with increasing l,
zo ¼
8 < ð0:479l 0:01Þhc
for
l 0:11
:
for
l > 0:11
0:05hc
:
ð3Þ
Equation (3) differs slightly from the formulation given by Marticorena et al. [1997a] because of a small typographical error in the formulation in that paper. [11] Surface roughness impacts the erosivity of the wind, by controlling the roughness length, but also controls the erodibility of the surface. In an analysis of drag partitioning over vegetated surface, Raupach et al. [1993] considered the impact of roughness elements on the threshold shear velocity. For conditions of partial, randomly distributed vegetation cover, they determined that u*tv, the threshold
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Table 1. Input Parameters for Sensitivity Tests Parameter
Mean
CV
u*ts C R(= h/d) b zo
29 cm/s 20% 1.0 100 8.1 cm
1 – 50% 1 – 50% 1 – 50% 1 – 50% 0%
shear velocity for a vegetated surface, is related to the threshold shear velocity for an unvegetated surface u*ts by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u*tv ¼ u*ts ð1 slÞð1 þ blÞ;
ð4Þ
where s is equal to the ratio of the average basal area to average profile area (i.e., s = AB/AP) of the vegetation and b is the ratio of the drag coefficient of an isolated plant to the drag coefficient of the ground surface in the absence of the plant. Substituting the definition of the terms in equation (4), this equation may be re-expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi AP : u*tv ¼ u*ts ð1 C Þ 1 þ bC AB
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‘‘oversheltered’’ so that addition of vegetation has no further effect on the shear velocity within the canopy. Using the literature and empirical values published by Raupach [1992] (i.e., u*/U(h) = 0.3, the drag coefficient on the vegetation, CR = 0.3, the drag coefficient on the bare surface, CS = 0.0003, and an empirical constant, c = 0.37), lmax is approximately equal to 0.81. By their respective definitions, lateral cover is related to fractional cover by l = CAAPB , so for the case where AP = AB (the case considered below; see Table 1), the value of C where the surface becomes oversheltered is Cmax = 0.81. In other words, equations (4) and (5) are valid for values of C less than about 80%. [14] For a hemi-elliptical plant, the basal area (AB) is given by the area of a circle with radius d/2 where d is the plant radius (AB = p4 d2) and the profile area (AP) is given by one half of the area of an ellipse with axes d/2 and h, where h is the plant height (AP = p4 dh). Thus, for a hemi-elliptical plant, the ratio AP/AB = h/d and equation (5) becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h u*tv ¼ u*ts ð1 C Þ 1 þ bC : d
ð6Þ
ð5Þ
The first terms under the radical in equations (4) and (5) (i.e., [1 ls] and [1 C]) account for the fact that vegetation covers part of the surface, which leads to an amplification of the stress on the unvegetated portion of the ground. Although both forms of the first term under the radical are equivalent (that is, [1 ls] = [1 C]), the [1C] form in equation (5) provides the most intuitive expression of this amplification effect. Specifically, it arises from normalization of the drag force on the erodible portion of the surface by the fraction of the surface area that is exposed (i.e., [1 C]). [12] The second terms under the radical in equations (4) and (5) (i.e., [1 + bl] and [1 + bCAP/AB]) account for drag that the surface exerts on the wind in the presence of vegetation, reducing the ability of the wind to detach and transport particles. Although again, both of these forms of the second term under the radical are equivalent (that is, [1 + bl] = [1 + bCAP/AB]), the [1 bl] form of the term is the most easily understood. Lateral cover, l, is a quantification of the total vertical plant area intercepted by the wind per unit of ground area. Here b is the ratio of the drag coefficient of an individual plant to the drag coefficient of the surface. Therefore bl is the ratio of stress exerted on the roughness elements to the stress exerted on the ground, and is a measure of the amount of vegetation ‘‘seen’’ by the wind, in terms of the actual height, width, and density of vegetation (l), but also in terms of the aerodynamic barrier the vegetation presents to the wind flow (b). Thus the term [1 + bl] is the total stress exerted by the wind on the surface in the presence of vegetation. [13] What are the limits of equations (4) and (5)? As C ! 1 in equation (5), the threshold shear velocity in the presence of vegetation approaches zero (u*tv ! 0). This is a counterintuitive result and is not supported by field or wind tunnel data. Raupach’s [1992] theory of drag and drag partitioning on vegetated surfaces suggests that as lateral cover increases to some value (lmax), the surface becomes
[15] For a cylindrical plant, the basal area is equal to that of a hemi-elliptical plant, and the profile area is given by dh. Thus the ratio for these plants of AP/AB is given by (p/4)(h/ d) and equation (5) becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ph : u*tv ¼ u*ts ð1 C Þ 1 þ bC 4d
ð7Þ
2.2. Stochastic Modeling [16] A parametric bootstrap (i.e., Monte Carlo) estimation method [Efron, 1982] for the threshold shear velocity of vegetated surfaces, u*tv, was implemented in the Interactive Data Language (Research Systems, Inc., Boulder, Colorado). In each iteration of the model, random values of u*ts, C, b, and the ratio of vegetation height to vegetation diameter, R(= h/d), are drawn from a modified normal distribution and are used to calculate a value of u*tv for that iteration using equation (6). The distributions of u*ts, C, b, and R are defined in the model by their means and coefficients of variation (CV). In order to eliminate values very close to zero for any of the input parameters (i.e., u*ts, C, R, and b), modification of the normal distributions is implemented in the model. This modification is typically only required when the CV is large (typically >40%). Randomly generated values for any input parameter that are less than 0.1 of the mean of that parameter are discarded. Values greater than mean +(0.9 mean) are also discarded so that estimate of the mean for the distribution mean remains approximately equal to the input mean. Discarded values are replaced by uniformly distributed values in the range [0.1 mean, mean + (0.9 mean)]. Modifying input parameter distributions in this way reduces the CV relative to input CV. Expansion of the input parameter distribution CV is implemented by successive replacement of 10% of the modified-normal distribution by uniformly distributed values in the range [0.1 mean, mean + (0.9 mean)] until the distribution CV is greater than or equal to
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a strong theoretical or empirical approach to estimate the fine-scale variability in zo, this parameter is treated as constant. Allowing zo to vary in the model would also add significant parameter and computational complexity to the model. First, it would require specification of the coefficient of variability of canopy height in addition to mean canopy height, adding a dimension to the model’s parameter space. Second, if zo were to be recalculated for every iteration, this would require recalculation of the histogram of u* and total saltation flux, QTot, for each iteration. Currently, estimates of QTot are produced by maxðu Þ
QTot ¼
1 TTot
X*
min u
Figure 1. Histogram of wind shear velocity, u*, constructed from 14.9-m wind speed data at the Jornada Basin LTER site from 28 March to 10 July 2002. Wind speeds were not recorded below 125 cm s1. Wind speed data are courtesy of D. Gillette. the input CV. This procedure results in distributions that are increasingly uniformly distributed as the input CV increases. Distributions for variables with CV less than approximately 40% are nearly indistinguishable from normal distributions. [17] Histograms of u*tv are constructed by successive random draws from the distribution of input parameters and evaluation of equation (6). Stabilization of the mean estimate of u*tv typically occurs in a few thousand computational iterations, but histograms presented here are based on 500,000 iterations. Final histograms are normalized by the total number of iterations, resulting in frequency distributions of u*tv with unity integrals. [18] A 105-day wind record (28 March 2000 to 10 July 2000) from the Jornada Experimental Range Geomet site (approximately 32350N, 106490W) near Las Cruces, New Mexico, was used to construct a histogram of wind speed at 1490 cm. Wind data for nonerosive winds (U(1490 cm) < 125 cm/s) were not recorded. Alternatively, the method of Gillette and Hanson [1989] could be used to create a simulated wind record. From the wind record, a histogram of shear velocity was created using equation (2) (Figure 1). Roughness height, zo, is determined by evaluating equation (3) using values of l = 0.2 and hc = 160 cm to give a value of zo = 8.1 cm. In eroding landscapes, the saltation layer itself acts like a roughness element, and thus influences the momentum transfer from the air to the soil. This feedback is not taken into account in our study. [19] The zo almost certainly varies over a heterogeneous landscape, as vegetation height and lateral cover vary. However, zo also varies on a fine scale within the canopy and is a therefore very likely to be dependent on the local distribution of vegetation and the wind direction (D. A. Gillette et al., Wind characteristics of mesquite streets in the northern Chihuahuan Desert, New Mexico, USA, submitted to Environmental Fluid Mechanics, 2005), factors that are beyond the spatial resolution of the present model. The same holds true for displacement height, D, which is assumed to be zero everywhere. Thus, in the absence of
Xi qi DT ;
ð8Þ
*tv
where Xi is the normalized frequency of u*tv, and qi is the horizontal mass flux calculated from equation (1) for a value of u*tv. DT is the time period for which individual measurements of U(z) are made (e.g., 5 min or 300 s) and Ttot is the length of the total wind record (e.g., 105 days 9.1 106 s). The summation in equation (8) is evaluated for every shear velocity between the minimum threshold shear velocity for the surface, min(u*tv) and the maximum wind shear velocity, max(u*).
3. Results and Discussion 3.1. Distribution of u*tv Resulting From Varying C, R, and B, and u*ts [20] The sensitivity of u*tv to variability in u*ts, C, R, and b was evaluated using the parameters in Table 1 (Figures 2 and 3). Using equation (6), the mean values of u*ts, C, R, and b in Table 1 give a value u*tv = 118.9 cm s1. However, the effect of the varying the coefficient of variation of each input parameter can clearly be seen on histograms of u*tv resulting from Monte Carlo simulation (Figure 2). [21] Increasing the variability of any of the parameters that control the threshold shear velocity for a vegetated surface in the Raupach et al. [1993] theory (i.e., u*ts, C, R, and b) increases the variability of u*tv. In practical terms, this means that the threshold shear velocity of a vegetated surface is represented by a range instead of a single value. This has important consequences for both the amount and distribution of wind erosion and dust flux from natural surfaces, which will be discussed below. [22] The distribution of threshold shear velocities for vegetated surfaces depends on the exact form of the relationship between u*tv and u*ts, C, R, and b in equations (4)– (7). The near-normal distributions of u*tv with varying u*ts are due to the linear relationship between these terms, and broadening of the histogram of u*tv is a direct result of broadening of the histogram of u*ts with increasing CV. Thus the mean estimate for u*tv is invariant with the coefficient of variation of u*ts if all other parameters do not vary. In other words, the square-root term in equation (6) can be expressed as a constant, c, and the mean (expectation value) of u*tv is given by: E(u*tv) = E(u*tsc) = c E(u*ts). [23] In contrast, the histogram of u*tv becomes increasingly skewed with increasing coefficient of variation of C, R, and b, a result of the square-root relationships between
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Figure 2. Modeled histograms of u*tv. In each panel, the CV of one parameter (i.e. u*ts, C, b, and R) is varied (5%, 10%, 15%, 20%, 30%, 40%, and 50%) and the CV of all others is set to 1%. The plot of q in g cm1 day1versus u*tv for the Jornada wind record is shown in each panel.
u*tv and these terms. Because b and R each occur once in equations (4)– (7) and both in the same term, the sensitivity of model estimates of u*tv to changes in the coefficient of variation of either b or R is identical. Physically, variability in b tends to increase variability u*tv because b represents the drag exerted on vegetation cover relative to unvegetated soils. Variability in R tends to increase u*tv because R(= h/d) represents the degree to which vegetation penetrates into the airstream at a given fractional cover, thus increasing the drag between the wind and the vegetation. [24] Although C appears twice under the radical in equations (4) –(7), the distributions of u*tv with varying C is somewhat narrower than the distribution of u*tv with either b or R. This is due to the fact that vegetation cover acts to simultaneously decrease u*tv by amplification of surface stress (i.e., in the first term under the radical in equations (4) and (5): [1 + bl] or [1 + bCAP/AB]) and to increase u*tv by increasing the drag of the surface (i.e., in the second term under the radical in equations (4) and (5): [1 ls] and [1 C]). Thus the fact that variability in cover is somewhat less effective than b or R at creating variability in u*tv is a result of the fact that the effect of cover in removing momentum from the air is offset by the effect of cover in amplifying surface stress. [25] The mean of u*tv is dependent on the CV of C, R, and b, but not on the CV of u*ts (Figure 3). This is because of the
linear relationship between u*tv and u*ts; when u*ts varies alone, the resulting distribution is symmetrical around the mean value of u*tv of 118.9 cm s1. When C, R, and b vary, however, the resulting distribution becomes skewed, and the median deviates from the mean. Specifically, the increasingly skewed nature of the modeled distribution of u*tv causes the median estimate of u*tv to decrease with increasing coefficient of variation of C, R, and b. [26] Two important physical insights about natural vegetated environments arise from these results, namely that in an environment where both soil are varying (i.e., u*ts varies) and vegetation cover varies (i.e., C, b, and R vary). (1) The mean threshold shear velocity will be less than that predicted if only the mean soil and vegetation characteristics are considered, and (2) wind erosion will and dust emission will occur in some places in the environment at a given freestream wind speed and not others except in the most extreme cases when the wind is strong enough to initiate aeolian processes across the entire landscape. This latter insight is consistent with the appearance of hot spots in dust-emitting landscapes as discussed by Gillette [1999]. 3.2. Total Horizontal Flux, QTot, Calculated Using Derived Distributions of u*tv [27] Broadening of the modeled distribution of u*tv results in both (1) increasing the number of instances when u*
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Figure 3. Mean, median, and coefficient of variation of modeled estimates of u*tv for varying coefficients of variation of beta (b), cover (C), and threshold shear velocity for bare soil (u*ts). exceeds u*tv and 2) increasing the magnitude of q for those instances in which u* exceeds u*tv (Figure 2). As a result, QTot increases monotonically with increases in the coefficients of variation of C, R, b, and u*ts (Figure 4). This relationship tends to be nonlinear because of (1) the frequency distribution of u*, (2) the frequency distribution of u*tv, and (3) the nonlinear relationship between these parameters and q (equation (1)). Furthermore, the extension of a tail into low values of u*tv in the frequency distributions for increasing coefficients of variation of C, R, and b contributes to the increase in estimates of QTot. Because q is a strong function of decreasing u*tv, the few instances of low values in the left tail come to dominate the estimates of
QTot. This result is consistent with the findings of Gillette [1999] and Okin and Gillette [2004] that dust emission ‘‘hot spots’’ dominate both local and regional dust emission budgets. 3.3. Sensitivity Study: Relationships Between Effective Mean Values and Varying Values [28] Effective percent cover (CE) and effective u*ts (u*ts,E) were calculated to evaluate the sensitivity of Qtot to variation of the parameters in the Raupach et al. [1993] model relative to changes in the mean values of these parameters under the condition of zero variation (CV for all parameters = 0). CE was estimated for all cases in Figure 4 by determining the
Figure 4. Horizontal mass flux (QTot) in g cm1 day1 as a function of the coefficient of variation of C, b, and u*ts. 6 of 10
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value of C in equation (6) that provides an estimate of total horizontal flux, QTot, equal to the estimate of QTot in Figure 4. The values of R, b, and u*ts were set to those given in Table 1. Calculation of u*ts,E for all cases in Figure 4 was accomplished by inverting equation (6) for u*ts and determining the value of u*ts that provides estimate of QTot in Figure 4. [29] Effective cover (CE) and effective soil threshold shear velocity (u*ts,E), the value of C and u*ts that yield the same mass flux in models with no variability as obtained here in the varying model, indicate that increasing the coefficient of variation of u*ts up to at least 50% effectively reduces the soil threshold for wind erosion by 30% and reduces the effect of vegetation, measured by CE, by 60%. Increasing the coefficient of variation of C up to at least 50% effectively reduces the soil threshold for wind erosion by approximately 20% and reduces CE by approximately 40%. [30] Marticorena and Bergametti [1995, equations (6) and (7)] simplified the relationships of Iversen and White for the threshold velocity on the Earth’s surface of soils with narrow (in some cases uniform) particle size distributions. Using these equations, the decrease of u*ts,E from approximately 30 cm s1 to approximately 20 cm s1 seen in this study by increasing the CV of u*ts to 50% is equivalent to a decrease of particle diameter from approximately 330 mm to about 75 mm if soils are assumed to have uniform particle sizes, or about a quartering in effective particle diameter. This dramatic change in effective particle diameter illustrates the importance of the consideration of variance in wind erosion models, and not simply means values. In this case, the change in the effective particle diameter is due to a diversity of soil threshold shear velocities in the model, and the fact that the lowest values dominate the horizontal flux. Of course, real soils are comprised of many different particles sizes, and this may be one of the causes for variations in u*ts in soils. However, fine-scale variations in the presence of nonerodible clasts, crusts (physical or biological), soil moisture, and other factors will also contribute to variability of u*ts in real soils. [31] Analysis of CE and u*ts,E indicates that the wind erodibility of the surface increases with increasing coefficient of variation and that u*ts exerts the strongest control on erodibility, measured by u*tv (Figure 5). In other words, increasing variability of u*ts, C, R, and b in the landscape results in effective reduction in cover and soil threshold velocity because those cases with low values of u*tv dominate the total horizontal flux estimate. The impact of changing parameter variance is nearly the same magnitude as the impact of changing the parameter means, and the impact of changing both parameter variance and means is additive. A small degree of variability (low values of the coefficient of variation of u*ts, C, R, and b), however, actually increases CE and u*ts,E relative to mean values of C and u*ts (Table 1), respectively. This effect arises from the nonlinearity of the relationship between QTot and u*tv. Representing u*tv as a distribution instead of a single value increases the instances of reduced erosion (u* < u*tv) but also increases the instances of enhanced erosion (u* > u*tv). When u*tv just exceeds u*, however, q is nearly zero and is a concave-up function of u*tv. Thus, for cases when the distribution of u*tv barely intersects the distribution of u*,
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instances of reduced erosion outweigh the instances of enhanced erosion and result in a landscape that experiences slightly less flux than would be experienced in the no variability case.
4. Conclusions and Implications [32] This study supports the conclusions of Liu and Westphal [2001] that wind erosion and dust emission are scale-dependent, nonlinear processes, although Liu and Westphal’s [2001] study focused primarily on scale-dependent biases in modeled wind fields. Accurate and physically realistic model estimates of wind erosion and dust emission depend on incorporation of scale dependence in the surface and meteorological fields for models. Stochastic modeling provides a framework for the inclusion of spatial and temporal variability not previously considered in models of wind erosion and dust flux at scales from the landscape to the globe. [33] In modeling dust emissions at a landscape scale at the Jornada Long-Term Ecological Research Site near Las Cruces, New Mexico, Okin and Gillette [2004] found that horizontal flux was vastly underestimated for vegetated surfaces. The reason for this failing is now clear: Variability in the parameters that control the threshold shear velocity for these surfaces, u*ts, C, R, and b in the Raupach et al. [1993] model, provides a stronger control over threshold estimates than do the mean values of these parameters. That is to say, for a threshold-dominated and nonlinear process such as wind erosion, mean parameter values are meaningless in determining the aggregate behavior at the landscape scale. This is particularly true when the shear velocity of the wind never exceeds the value of threshold shear velocity for the surface calculated using mean values. In this case, soil and vegetation heterogeneity are the sole controls on surface erodibility and erosion will occur in localized areas but not over the entire landscape. Because there is no way to know, a priori, if erosion is to occur in localized areas or over the entire landscape simply by considering parameter means, estimates of soil and vegetation variability are required for accurate assessment of aeolian sediment flux. Thus the corrections to flux estimates provided by the inclusion of variability in the present model are pivotal in understanding the horizontal and vertical aeolian flux from a vegetated landscape. [34] Gillette [1999] examined the geophysical origins behind the often-observed phenomenon of dust emission ‘‘hot spots,’’ locations in the landscape that have consistently stronger dust emissions than the surrounding area. Dust emitted from these hot spots represents the majority of dust emission from the landscape. The sources of dust hot spots were broken down into a prioritized list of the most importance contributors to dust emission (Table 2). Hot spots arise from the few instances in which the erodibility of the wind is high and/or the threshold shear velocity of the surface is low. The model presented here builds on Gillette’s [1999] reasoning by expressing the idea of a hot spot in a model context. Hot spots arise from the few instances when u* exceeds u*tv, and are a simple consequence of natural variability within the landscape. The model described here provides a method for representing hot spots at multiple scales: This model can be run at multiple locations in the
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Figure 5. Equivalent percent cover and equivalent threshold shear velocity of bare soil (u*ts) as a function of the coefficient of variation of u*ts, cover (C), beta (b), and R. Equivalent percent cover is the cover in a nonvarying model (coefficients of variation all set to zero) that will give the same horizontal flux as the varying model (coefficients of variation in figure). Equivalent u*ts is the threshold shear velocity for bare soil in a nonvarying model that will give the same horizontal flux as the varying model.
landscape where each location is defined by a different set of parameters. This was the approach used by Okin and Gillette [2004] using 30 m 30 m grid cells for modeling dust emission and wind erosion across the entire Jornada LTER site. They were able to show that different combinations of vegetation cover and soils resulted in differing amounts of dust emission at the JER. In particular, those few areas with very low vegetation and highly erodible soils were highlighted as significant sources of dust in the basin. However, their model underestimated dust emission in large portions of the basin because they did not include the intragrid cell heterogeneity that is included in this stochastic model. Thus, for modeling dust emission on the landscape scale, combining the spatially distributed modeling approach of Okin and Gillette [2004] and intra-grid cell variability (this study) will provide multiple-scale representation of hot spots and will improve estimation of dust emissions from heterogeneous landscapes. Some method for incorporating landscape variability is also required for global dust emission models such as that of Ginoux et al. [2001] and Zender et al. [2003].
[35] Remote sensing may provide some possibility for including measured variability into flux models at regional to global scales. At regional scales, the use of high-resolution imagery may provide a means to estimate the coefficient of variation of vegetation cover at different scales. Phinn et al. [1996], Okin and Gillette [2001], Laliberte and Rango [2004], and I. O. McGlynn and G. S. Okin (Highspatial resolution determination of fetch and anisotropy in a mixed desert shrubland, submitted to Journal of Arid Environments, 2005) have used high-resolution imagery to quantify the fractional cover, spacing, variability, and spatial patterning of vegetation in desert shrublands. At regional to global scales, high-resolution imagery may be used to provide estimates of canopy size and variance in small areas nested within coarser images where fractional cover is determined by alternate means (e.g., multiple endmember spectral mixture analysis [Roberts et al., 1998]). Alternatively, inter-pixel variability may be used as a proxy for intra-pixel variability where multispectral remote sensing techniques are used to determine soil and canopy mean characteristics on a pixel-by-pixel basis. For instance,
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Table 2. Prioritized Lists of Mechanisms of Enhanced Particle Flux From Gillette [1999] and Their Representation in the Model Presented in This Study Mechanism
Representation in This Model
High winds Lack of momentum partitioning, especially to vegetation, cobbles, boulders, or gravel Presence of particles 80 mm < diameter < 20 mm High particle availability Owen effect (i.e., long fetch distances) Lack of trapping Lack of cohesion by soil moisture Low binding energies of suspended particles in the soil matrix
geometric-optical canopy reflectance models like those of Franklin and Strahler [1988] and Li and Strahler [1992] may be used to derive information on vegetation cover, plant density, and canopy size, and the method of Okin and Painter [2004] may be used to estimate soil surface grain size. No matter what techniques are used, the importance of landscape heterogeneity in dust production indicates a need for new methods to quantify heterogeneity at multiple scales in sparsely vegetated landscapes.
Notation A AB AP B C C CE Cmax CR CS d D F g h hc k K l lmax n N q QTot R
supply-limitation coefficient, 0 A 1, unitless. average basal area of plants in an area, S, cm2. average profile area of plants in an area, S, cm2. ratio of drag coefficient of vegetated surface to drag coefficient of unvegetated surface, unitless. an empirical constant (= 0.37) [Raupach, 1992], unitless. fractional cover, 0 C 1, C = NAB, unitless. equivalent fractional cover, the unvarying value of C that would give the same QTot as that calculated using varying parameters, unitless. fractional cover at which u*/U(h) is maximum, unitless. drag coefficient of the wind on an isolated roughness element (plant), unitless. drag coefficient of the wind on unvegetated soil, unitless. average plant diameter, cm. displacement height, cm. instantaneous vertical sediment (dust) flux for specified set of conditions, g cm2 s1. acceleration due to gravity (981 cm/s2), cm s2. average plant height, cm. canopy height, cm. Von Karmann’s constant (0.4), unitless Ratio of vertical flux, F, to horizontal flux, q, cm1. lateral cover, l nbh/s = Nbh = NAP [Raupach et al., 1993], unitless. lateral cover at which u*/U(h) is maximum, unitless. number of plants in an area, S, unitless. plant number density, N n/S, cm2. instantaneous horizontal sediment flux for specified set of conditions, g cm1 s1. average horizontal sediment flux for a specified time period and set of conditions, g cm1 s1. ratio of plant height to plant diameter (R h/d), unitless.
mean and CV of u* mean and CV of C, b, R mean and CV of u*ts mean and CV of A mean and CV of C not explicitly included mean and CV of u*ts mean and CV of u*ts
r particle density, g cm3. s ratio of the average plant basal area to average profile area, s AB/AP, unitless. S area of an individual model grid cell, cm2. DT time between wind speed records, s. Ttot total time of wind speed record, s. u* wind shear velocity, cm s1. u*ts threshold shear velocity for an unvegetated surface, cm s1. u*ts,E equivalent threshold shear velocity for an unvegetated surface, the unvarying value of u*ts that w3ould give the same QTot as that calculated using varying parameters, cm s1. u*tv threshold shear velocity for an vegetated surface, cm s1. U(z) wind speed at height, z, cm s1. X probability of u*tv, unitless. z height above the surface, cm. zo roughness length, cm.
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G. S. Okin, Department of Environmental Sciences, University of Virginia, 291 McCormick Road, Charlottesville, VA 22904-4123, USA. (
[email protected])
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