1 Estimating aerodynamic resistance of rough surfaces using angular reflectance. 1 2 Adrian ...

35 Pages·2010·0.5 MB·English

1 Estimating aerodynamic resistance of rough surfaces using angular reflectance.
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3 Adrian Chappell1, Scott Van Pelt2, Ted Zobeck3 and Zhibao Dong4
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5 1CSIRO Land and Water, GPO Box 1666, Canberra, ACT 2601, Australia ([email protected])
6 2Agricultural Research Service, United States Department of Agriculture, Big Spring, TX, 79720 USA.
7 3Agricultural Research Service, United States Department of Agriculture, Lubbock, TX, 79720 USA.
8 2Key Laboratory of Desert and Desertification, Chinese Academy of Sciences, Lanzhou, China.
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10 Abstract
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12 Current wind erosion and dust emission models neglect the heterogeneous nature of surface roughness
13 and its geometric anisotropic effect on aerodynamic resistance, and over-estimate the erodible area by
14 assuming it is not covered by roughness elements. We address these shortfalls with a new model which
15 estimates aerodynamic roughness length (z ) using angular reflectance of a rough surface. The new model
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16 is proportional to the frontal area index, directional, and represents the geometric anisotropy of z . The
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17 model explained most of the variation in two sets of wind tunnel measurements of aerodynamic
18 roughness lengths (z ). Field estimates of z for varying wind directions were similar to predictions made
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19 by the new model. The model was used to estimate the erodible area exposed to abrasion by saltating
20 particles. Vertically integrated horizontal flux (F ) was calculated using the area not covered by non-
h
21 erodible hemispheres; the approach embodied in dust emission models. Under the same model conditions,
22 F estimated using the new model was up to 85% smaller than that using the conventional area not
h
23 covered. These F simulations imply that wind erosion and dust emission models without geometric
h
24 anisotropic sheltering of the surface, may considerably overestimate F and hence the amount of dust
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25 emission. The new model provides a straightforward method to estimate aerodynamic resistance with the
26 potential to improve the accuracy of wind erosion and dust emission models, a measure that can be
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27 retrieved using bi-directional reflectance models from angular satellite sensors, and an alternative to
28 notoriously unreliable field estimates of z and their extrapolations across landform scales.
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30 Keywords: Dust emission model; Wind erosion; Sheltering; Erodible; Flow separation; Drag; Wake;
31 Aerodynamic Resistance; Aerodynamic Roughness length; Shadow; Illumination; Ray-
32 casting; Digital elevation model; Roughness density; Frontal area index; Angular
33 reflectance; bi-directional reflectance.
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35 1. Introduction
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37 Soil-derived mineral dust contributes significantly to the global aerosol load. The direct and indirect
38 climatic effects of dust are potentially large. A prerequisite for estimating the various effects and
39 interactions of dust and climate is the quantification of global atmospheric dust loads (Tegen, 2003).
40 Recent developments in global dust emission models explicitly simulate areas of largely unvegetated dry
41 lake beds as sources of preferential dust emission (Tegen et al., 2002; 2006; Mahowald et al., 2003). In
42 the case of the Earth’s largest source of dust (Bodélé Depression; Warren et al., 2007) there are some
43 significant discrepancies between ground measurements of dust emission processes and model
44 assumptions (Chappell et al., 2008). Dust emission is produced by two related processes called saltation
45 and sandblasting. Saltation is the net horizontal motion of large particles or aggregates of particles
46 moving in a turbulent near-surface layer. Sandblasting is the release of dust and larger material caused by
47 saltators as they impact the surface (Alfaro and Gomez, 1995; Shao, 2001). Naturally rough (unvegetated)
48 surfaces usually comprise a heterogeneous mixture (size and spacing) of non-erodible roughness elements
49 that reduce the area of exposed and hence erodible substrate. When such rough surfaces are exposed to
50 the wind, wakes or areas of flow separation (Arya, 1975) are created downwind of all obstacles. These
51 sheltered areas reduce the area of exposed substrate still further and protect some of the roughness
52 elements from the wind (depending on their size and spacing). This heuristic formed the basis for the
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53 dimensional analysis of the Raupach (1992) model where dynamic turbulence was replaced by a concept
54 of effective shelter area and was portrayed as a wedge-shaped sheltered area in the lee of the element. The
55 size and shape of the sheltered area is influenced by the wind velocity (speed and direction) and the
56 heterogeneous nature of the surface (Figure 1). Consequently, the erodible area and the non-erodible
57 roughness elements that are exposed to, and protected from, drag are an anisotropic function of the
58 heterogeneous surface and wind speed.
59 [Figure 1]
60 Central to wind erosion and dust emission models is the turbulent transfer of momentum from the fluid to
61 the bed. The key assumption made by dust emission models (e.g., Marticorena and Bergametti, 1995; p.
62 16,418) is that the momentum extracted by roughness elements is controlled primarily by their roughness
63 density (λ; Marshall, 1971) and consequently the erodible area is that which is not covered by roughness
64 elements. The λ (also known as lateral cover or the frontal area index) is expressed as λ = n b h / S where
65 n is the number of roughness elements inside an area (or pixel) S and b and h are the breadth and height,
66 respectively of the roughness elements. This assumption forms one of the foundations for the dust
67 production model (Marticorena and Bergametti, 1995) and dust emission scheme (Marticorena et al.,
68 1997) upon which many dust emission models are based (e.g., Tegen et al., 2006). The approach assumes
69 that the roughness elements cover part of the surface, protect it from erosion and that they consume part
70 of the momentum available to initiate and sustain particle motion by the wind. The assumption manifests
71 itself in dust emission models (e.g., Marticorena and Bergametti, 1995; p. 16,422; Eq. 34):
r
72 G = EC aU3∫ (1+R)(1- R2)dS (D )dD (1)
tot g * D rel p p
p
73 as the ratio of the erodible area to the total surface area (E) and is set to 1 in the absence of information
74 about non-erodible roughness elements and of vegetation and snow (Tegen et al., 2006). The parameter C
75 is a constant of proportionality (2.61), ρ is the air density, g is a gravitational constant, U 3 is the cubic
a *
76 shear stress of the Prandtl-von Karman equation where U =u(z)(k/ln(z/z )) where u is the wind speed at a
* 0
77 reference height z, k is von Karman’s constant (0.4) and z is the aerodynamic roughness length. The
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78 threshold friction velocity defines R=U (D , z , z )/U where the threshold shear stress U is a function
*t p 0 0s * *t
79 of particle diameter D , z and the aerodynamic roughness length of the same surface without obstacles
p 0
80 (z ). The dS is a continuous relative distribution of basal surfaces formed by dividing the mass size
0s rel
81 distribution by the total basal surface and dDp is the particle diameter distribution. This approach includes
82 neither a sheltering effect nor any interaction between the momentum extraction of the roughness
83 elements and the downwind substrate area that they protect (wake). Furthermore, R implicitly assumes
84 homogeneous surface roughness and it does not account for the anisotropy of heterogeneous surface
85 roughness created by changing wind directions i.e., anisotropic z .
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86 Wind erosion and dust emission models should reach a compromise between the realistic
87 representation of the erosion / abrasion processes and the availability of data to parameterize or drive the
88 model (Raupach and Lu, 2004). The requirement here is to reduce the complexity of aerodynamic
89 resistance from an understanding of wake and shelter but capture the essence of the process to make
90 reasonable estimates, particularly across scales of variation. For example, Shao et al. (1996) provided one
91 of the first physically based wind erosion models to operate across spatial scales from the field to the
92 continent (Australia). One of the main reasons for its success was its approximation of λ using NDVI
93 (Normalised Difference Vegetation Index) data. To improve this approximation Marticorena et al. (2004)
94 argued that a proportional relationship existed between the protrusion coefficient (PC) derived from a
95 semi-empirical bidirectional reflectance (BRF) model (Roujean et al. 1992) and geometric roughness.
96 Although Roujean et al. (1992) stated the model’s limitation for unvegetated situations and Marticorena et
97 al. (2004) recognised this limitation, they developed a relationship between geometric roughness and z .
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98 They retrieved the PC from surface products of the space-borne POLDER (POLarization and
99 Directionality of the Earth’s Reflectances) instrument and compared it to geomorphic estimates of z
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100 (Marticorena et al. 1997; Callot et al. 2000). The authors concluded that z could be derived reliably from
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101 the PC in arid areas.
102 The main justification for the simplifying assumption of λ in wind erosion and dust emission
103 models appears to be the hypothesis that the configuration and shape of non-erodible (unvegetated)
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104 surface roughness elements are unimportant for explaining the drag partition. The concept of drag or
105 shear stress partitioning (Schlichting, 1936) is that the total force on a rough surface F can be partitioned
t
106 into two parts: F acting on the non-erodible roughness elements and F acting on the intervening
r s
107 substrate surface F = F + F . There is a growing body of evidence that supports this approach. For
t r s
108 example, Marshall (1971) studied drag partition experimentally in a wind tunnel and showed no
109 difference between cylinders placed on a regular grid, on a diagonal or at random across the wind tunnel
110 (λ = 0.0002 to 0.2). Raupach et al. (1993) reached a similar conclusion after inspecting Marshall’s data
111 and believed that there was only a weak experimental dependence of stress partition on roughness
112 element shape and the arrangement of elements on the surface. Drag balance instrumentation used by
113 Brown et al. (2008) in a wind tunnel, independently and simultaneously measured the drag on arrays of
114 cylinders and the intervening surface, separately. Results were interpreted as confirmation that an increase
115 in surface roughness enhanced the sheltering of the surface, regardless of roughness configuration i.e.,
116 irregular arrays of cylinders were analogous to staggered configurations in terms of drag partitioning.
117 The role of flow separation and much-reduced drag in sheltered regions, particularly downwind of
118 roughness elements, is significant for drag partitioning. We posit that the sheltered area is required to
119 account for anisotropic variation in aerodynamic resistance for realistic wind erosion and dust emission
120 models. Furthermore, we posit that current estimates of the erodible area using the area not covered by
121 protruding objects is a poor representation of the erodible substrate exposed to abrasion from mobile
122 material. The aim of the paper is to describe and evaluate the basis for using angular reflectance data to
123 quantify the geometric anisotropy of aerodynamic resistance, account for heterogeneity and estimate the
124 area exposed to abrasion.
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126 2. Estimating aerodynamic roughness length (z ) and erodible area using shadow
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128 2.1 Relationship between reflectance and frontal area index (λ)
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130 A new approach is presented here which is based on Chappell and Heritage (2007). The approach is
131 inspired by the dimensional analysis of the Raupach (1992; p. 377-378) model (effective shelter area) and
132 its replacement of dynamic turbulence with the scales controlling an element wake and how the wakes
133 interact (Shao and Yang, 2005) and by the heuristic model of Arya (1975) and hence its similarity with
134 the scheme of Marticorena and Bergametti (1995). In common with Marticorena et al. (2004), we show
135 the relationship between reflectance and aerodynamic resistance estimated by wind tunnel studies of
136 aerodynamic roughness length (z ) and explain the relationship between reflectance and the frontal area
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137 index (λ).
138 The λ is the projection of an obstacle’s frontal area onto a pre-defined area or pixel with a flat
139 surface. The projection is defined for a 45° illumination zenith angle i such that Tan i is used as a
140 multiplication factor which in this case is restricted to 1 and thus the entire frontal area of the object is
141 projected. If 0° < i < 45° then Tan i reduces the projected frontal area and when 45° < i < 90° then Tan i
142 increases the projected frontal area. If that pixel is viewed at nadir and illuminated for different i, the
143 shadow cast by the object is the same as the projection of the object onto the pixel for the given i. The
144 light reflected from the rough surface is reduced by the proportion of the area that is in shadow and
145 visible (Figure 2). In the case of many homogeneous objects the shadow area may be reduced if the
146 spacing between the objects is insufficient to allow the shadow cast to reach the underlying surface
147 (Figure 2). In other words the shadow is projected onto adjacent objects (mutual shadowing).
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149 [Figure 2]
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151 However, if objects with vertical sides (e.g., cylinders) are homogeneous their shadow is truncated
152 because although it is projected on to the adjacent objects it is not visible at nadir. Illumination of natural
153 surfaces demonstrates that a portion of the surface that may otherwise cast shadow may also be under
154 shadow and this effect is dependent on illumination azimuth relative to an arbitrary origin (Figure 3).
155
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156 [Figure 3]
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158 Since we believe a priori that the geometry of a rough surface influences aerodynamic resistance and that
159 roughness is also one of the main controls on the proportion of illumination there should be a relationship
160 between the aerodynamic resistance and illumination proportion (viewed at nadir).
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162 2.2 Relationship between shadow and erodible area
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164 The convention within current dust emission models is to approximate the erodible area as simply the
165 intervening surface not covered by non-erodible elements. However, saltating soil particles usually strike
166 the soil surface at an elevation angle of approximately 12°–15° (Sorensen, 1985). The point of impact is
167 influenced by the particle jump length, angle of descent and surface roughness (Potter et al., 1990). As the
168 particles bounce, they may jump over obstructions on the surface. If the obstruction is sufficiently tall that
169 the particle cannot jump over it and it is non-erodible, it will shelter a portion of the soil surface from
170 abrasion. Thus, upwind obstructions determine the angle of trajectory shelter angle a particle must
171 achieve in order to strike a given point within the horizontal bounce distance of the saltating particle. The
172 fraction of the soil surface impacted by saltating grains varies with the fraction of the surface sheltered by
173 non-erodible roughness elements. Potter et al. (1990) developed the cumulative shelter angle distribution
174 and it was included in the Wind Erosion Prediction System to make daily estimates of wind erosion
175 (Hagen, 1990). Here we propose to use the approach by Potter et al. (1990) and developed by Zobeck and
176 Popham (1998) to approximate the erodible area. The curves formed by field measurements by Zobeck
177 and Popham (1998; Figure 2 and 3) are a function of the surface roughness, the shelter angle is equivalent
178 to the illumination zenith angle described above and the proportion of the surface illuminated and viewed
179 at nadir is equivalent to the surface fraction. Consequently, we propose to approximate the erodible area
180 by predicting the proportion of surface in shadow with an illumination zenith angle of 75° (equivalent to
181 an elevation angle 15°) and viewed at nadir.
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183 2.3 Evaluation methodology
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185 A digital elevation model is used here to reconstruct the surface roughness configuration of previous
186 laboratory wind tunnel and field studies for which shadow/illumination measurements were not available.
187 A ray-casting approach demonstrates the means by which shadow can be estimated remotely. It makes
188 use of a fine resolution of elevation sufficient to discretise the surface obstacles. Such an approach is able
189 to handle heterogeneous bed situations where the object heights and spacings are different across a
190 surface. Variable illumination orientation, to represent wind direction, was accounted for with the same
191 computational procedure and an illumination azimuth angle f relative to a fixed arbitrary origin.
192 Although a number of models exist to estimate the proportion of reflectance from a rough surface
193 (Cierniewski, 1987; Hapke, 1993; Li and Strahler, 1992; Liang and Townsend, 1996) an empirical
194 function is used here for simplicity and to avoid the modeling becoming a distraction from the retrieval of
195 aerodynamic resistance information. The most suitable model for describing reflectance of surface
196 roughness across scales is the subject of ongoing research by the authors. As an alternative to these
197 models, the proportion of illuminated surface viewed at-nadir for given illumination zenith and azimuth
198 angles was fitted with a Gaussian model with an additional parameter. The function has the desirable
199 quality of resembling a positive exponential or a Gaussian model. Its isotropic form and model
200 parameters are:
  ia 
201 S(i)=cexp-  (2)
 a 
  r 
202 where S is the proportion of illuminated surface for a given illumination zenith angle (i). The function
203 approaches its sill (c) asymptotically and does not have a finite range. Instead, the distance parameter r
204 defines the extent of the model. For practical purposes it can be regarded as having an effective range of
205 approximately (3r)-α, where it reaches 95% of its sill (Webster and Oliver, 2001). The additional
206 parameter α describes the intensity of variation and the curvature. If α = 1 then the function is a positive
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207 exponential and if α = 2 the function is a Gaussian. As i→90°, c enables S to be greater than 0 to
208 represent the illumination of objects above a rough background or substrate. The proportion of
209 illuminated surface associated with the illumination zenith angle 75° is used to approximate the erodible
210 area (section 2.2). The goodness of fit was assessed using the RMSE which is here defined as the square
211 root of the mean squared difference divided by the number of degrees of freedom (df). The df is the
212 number of data minus the number of model parameters used.
213 The brightness of the rough surface was assumed to be the same as that of the background
214 (substrate) and in the case of the reconstructed wind tunnel studies to scatter according to the Lambertian
215 distribution. In this case, the Gaussian function was integrated over all illumination zenith (i) and azimuth
216 (f) angles and made relative to the incoming radiation (I) to form a single scattering albedo (SSA):
f=2p i=p /2   ia 
217 SSA= I/ ∫ ∫ cexp- . (3)
 a 
f=0 i=0   r 
218 In addition to the SSA a calculation is made for a specified direction over all i but for only a single f
219 viewed at nadir and made relative to the incoming radiation. This statistic is here defined as the relative
220 directional reflectance (RDR) for use as a measure of the geometric anisotropy of the aerodynamic
221 resistance.
222 Although the ray-casting approach described here is evidently capable of handling the anisotropic
223 nature of heterogeneous surface roughness, it is not intended to provide an operational method for the
224 retrieval of shadow. Surface illumination / shadow may be readily retrieved using angular sensors on
225 airborne and space-borne platforms. A photometric model can be used to characterize the surface
226 reflectance for given illumination and viewing conditions (cf. Hapke, 1993). The estimation of soil /
227 surface roughness using shadow and geometric models is well established (Cierniewski, 1987) and
228 approximations to radiative transfer theory by Hapke (1993) have provided parameterized models for soil
229 bi-directional reflectance measurements (Pinty et al., 1989; Jacquemoud et al., 1992; Chappell et al.,
230 2006; 2007; Wu et al., 2008). The description above amounts to a hypothesis that the proportion of
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231 illumination (viewed at nadir) can approximate the aerodynamic roughness length and the erodible area.
232 That hypothesis is evaluated against existing measurements of aerodynamic resistance.
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234 3. Evaluation data
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236 3.1 Isotropic wind tunnel roughness
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238 A wind tunnel study by Marshall (1971) investigated cylinder and hemisphere roughness elements made
239 from solid ‘varnished’ wood. The elements had a uniform height of 2.54 cm with a range of diameters
240 (1.27, 2.54, 5.08, 7.62 and 12.7 cm). The elements were arranged in the working section of a wind tunnel
241 on square and diagonal grids and also using randomly arranged patterns with spacing between elements
242 (2.54 – 125.73 cm) that produced various coverage (ratio of cylinder base area to the specific cover:
243 approx. 0.01 – 44.18 %). Each surface configuration was subjected to a single freestream velocity at 20.3
244 m s-1 and the total drag force of the surface roughness and that of the roughness elements separately was
245 measured to deduce the surface drag force in between elements.
246 Aerodynamic roughness lengths (z ) of Marshall’s surface configurations were derived using the
0
247 same approach as Raupach et al., (2006; p. 214):
z d   U 
248 0 = expkB- d  , (4)
h h   u 
*
249 where d » 3.3+15.0(lf )0.43(cm), k=0.4, B=2.5 and U =20.3 m s-1 from table 4 of Marshall (1971).
d
250 These data are plotted against λ in Figure 4. Their characteristics are summarized in Table 1.
251
252 [Figure 4]
253
254 An investigation of gravel surface aerodynamic resistance was conducted in a wind tunnel by Dong et al.
255 (2002). Six types of artificial gravel were fabricated in cement to form “parabolic-shaped” elements with
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