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Estimating aerodynamic resistance of rough surfaces using angular reflectance.
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Adrian Chappell1, Scott Van Pelt2, Ted Zobeck3 and Zhibao Dong4
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CSIRO Land and Water, GPO Box 1666, Canberra, ACT 2601, Australia ([email protected])
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Agricultural Research Service, United States Department of Agriculture, Big Spring, TX, 79720 USA.
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Agricultural Research Service, United States Department of Agriculture, Lubbock, TX, 79720 USA.
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Key Laboratory of Desert and Desertification, Chinese Academy of Sciences, Lanzhou, China.
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Abstract
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Current wind erosion and dust emission models neglect the heterogeneous nature of surface roughness
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and its geometric anisotropic effect on aerodynamic resistance, and over-estimate the erodible area by
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assuming it is not covered by roughness elements. We address these shortfalls with a new model which
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estimates aerodynamic roughness length (z0) using angular reflectance of a rough surface. The new model
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is proportional to the frontal area index, directional, and represents the geometric anisotropy of z0. The
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model explained most of the variation in two sets of wind tunnel measurements of aerodynamic
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roughness lengths (z0). Field estimates of z0 for varying wind directions were similar to predictions made
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by the new model. The model was used to estimate the erodible area exposed to abrasion by saltating
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particles. Vertically integrated horizontal flux (Fh) was calculated using the area not covered by non-
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erodible hemispheres; the approach embodied in dust emission models. Under the same model conditions,
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Fh estimated using the new model was up to 85% smaller than that using the conventional area not
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covered. These Fh simulations imply that wind erosion and dust emission models without geometric
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anisotropic sheltering of the surface, may considerably overestimate Fh and hence the amount of dust
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emission. The new model provides a straightforward method to estimate aerodynamic resistance with the
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potential to improve the accuracy of wind erosion and dust emission models, a measure that can be
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retrieved using bi-directional reflectance models from angular satellite sensors, and an alternative to
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notoriously unreliable field estimates of z0 and their extrapolations across landform scales.
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Keywords:
Dust emission model; Wind erosion; Sheltering; Erodible; Flow separation; Drag; Wake;
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Aerodynamic Resistance; Aerodynamic Roughness length; Shadow; Illumination; Ray-
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casting; Digital elevation model; Roughness density; Frontal area index; Angular
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reflectance; bi-directional reflectance.
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1. Introduction
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Soil-derived mineral dust contributes significantly to the global aerosol load. The direct and indirect
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climatic effects of dust are potentially large. A prerequisite for estimating the various effects and
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interactions of dust and climate is the quantification of global atmospheric dust loads (Tegen, 2003).
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Recent developments in global dust emission models explicitly simulate areas of largely unvegetated dry
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lake beds as sources of preferential dust emission (Tegen et al., 2002; 2006; Mahowald et al., 2003). In
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the case of the Earth’s largest source of dust (Bodélé Depression; Warren et al., 2007) there are some
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significant discrepancies between ground measurements of dust emission processes and model
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assumptions (Chappell et al., 2008). Dust emission is produced by two related processes called saltation
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and sandblasting. Saltation is the net horizontal motion of large particles or aggregates of particles
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moving in a turbulent near-surface layer. Sandblasting is the release of dust and larger material caused by
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saltators as they impact the surface (Alfaro and Gomez, 1995; Shao, 2001). Naturally rough (unvegetated)
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surfaces usually comprise a heterogeneous mixture (size and spacing) of non-erodible roughness elements
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that reduce the area of exposed and hence erodible substrate. When such rough surfaces are exposed to
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the wind, wakes or areas of flow separation (Arya, 1975) are created downwind of all obstacles. These
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sheltered areas reduce the area of exposed substrate still further and protect some of the roughness
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elements from the wind (depending on their size and spacing). This heuristic formed the basis for the
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dimensional analysis of the Raupach (1992) model where dynamic turbulence was replaced by a concept
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of effective shelter area and was portrayed as a wedge-shaped sheltered area in the lee of the element. The
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size and shape of the sheltered area is influenced by the wind velocity (speed and direction) and the
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heterogeneous nature of the surface (Figure 1). Consequently, the erodible area and the non-erodible
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roughness elements that are exposed to, and protected from, drag are an anisotropic function of the
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heterogeneous surface and wind speed.
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[Figure 1]
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Central to wind erosion and dust emission models is the turbulent transfer of momentum from the fluid to
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the bed. The key assumption made by dust emission models (e.g., Marticorena and Bergametti, 1995; p.
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16,418) is that the momentum extracted by roughness elements is controlled primarily by their roughness
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density (λ; Marshall, 1971) and consequently the erodible area is that which is not covered by roughness
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elements. The λ (also known as lateral cover or the frontal area index) is expressed as λ = n b h / S where
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n is the number of roughness elements inside an area (or pixel) S and b and h are the breadth and height,
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respectively of the roughness elements. This assumption forms one of the foundations for the dust
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production model (Marticorena and Bergametti, 1995) and dust emission scheme (Marticorena et al.,
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1997) upon which many dust emission models are based (e.g., Tegen et al., 2006). The approach assumes
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that the roughness elements cover part of the surface, protect it from erosion and that they consume part
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of the momentum available to initiate and sustain particle motion by the wind. The assumption manifests
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itself in dust emission models (e.g., Marticorena and Bergametti, 1995; p. 16,422; Eq. 34):
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Gtot = EC
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as the ratio of the erodible area to the total surface area (E) and is set to 1 in the absence of information
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about non-erodible roughness elements and of vegetation and snow (Tegen et al., 2006). The parameter C
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is a constant of proportionality (2.61), ρa is the air density, g is a gravitational constant, U*3 is the cubic
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shear stress of the Prandtl-von Karman equation where U*=u(z)(k/ln(z/z0)) where u is the wind speed at a
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reference height z, k is von Karman’s constant (0.4) and z0 is the aerodynamic roughness length. The
ρa
g
U *3 ∫ (1 + R )(1 − R 2 )dS rel ( D p )dD p
Dp
(1)
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threshold friction velocity defines R=U*t(Dp, z0, z0s)/U* where the threshold shear stress U*t is a function
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of particle diameter Dp, z0 and the aerodynamic roughness length of the same surface without obstacles
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(z0s). The dSrel is a continuous relative distribution of basal surfaces formed by dividing the mass size
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distribution by the total basal surface and dDp is the particle diameter distribution. This approach includes
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neither a sheltering effect nor any interaction between the momentum extraction of the roughness
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elements and the downwind substrate area that they protect (wake). Furthermore, R implicitly assumes
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homogeneous surface roughness and it does not account for the anisotropy of heterogeneous surface
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roughness created by changing wind directions i.e., anisotropic z0.
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Wind erosion and dust emission models should reach a compromise between the realistic
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representation of the erosion / abrasion processes and the availability of data to parameterize or drive the
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model (Raupach and Lu, 2004). The requirement here is to reduce the complexity of aerodynamic
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resistance from an understanding of wake and shelter but capture the essence of the process to make
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reasonable estimates, particularly across scales of variation. For example, Shao et al. (1996) provided one
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of the first physically based wind erosion models to operate across spatial scales from the field to the
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continent (Australia). One of the main reasons for its success was its approximation of λ using NDVI
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(Normalised Difference Vegetation Index) data. To improve this approximation Marticorena et al. (2004)
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argued that a proportional relationship existed between the protrusion coefficient (PC) derived from a
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semi-empirical bidirectional reflectance (BRF) model (Roujean et al. 1992) and geometric roughness.
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Although Roujean et al. (1992) stated the model’s limitation for unvegetated situations and Marticorena et
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al. (2004) recognised this limitation, they developed a relationship between geometric roughness and z0.
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They retrieved the PC from surface products of the space-borne POLDER (POLarization and
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Directionality of the Earth’s Reflectances) instrument and compared it to geomorphic estimates of z0
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(Marticorena et al. 1997; Callot et al. 2000). The authors concluded that z0 could be derived reliably from
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the PC in arid areas.
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The main justification for the simplifying assumption of λ in wind erosion and dust emission
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models appears to be the hypothesis that the configuration and shape of non-erodible (unvegetated)
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surface roughness elements are unimportant for explaining the drag partition. The concept of drag or
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shear stress partitioning (Schlichting, 1936) is that the total force on a rough surface Ft can be partitioned
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into two parts: Fr acting on the non-erodible roughness elements and Fs acting on the intervening
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substrate surface Ft = Fr + Fs. There is a growing body of evidence that supports this approach. For
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example, Marshall (1971) studied drag partition experimentally in a wind tunnel and showed no
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difference between cylinders placed on a regular grid, on a diagonal or at random across the wind tunnel
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(λ = 0.0002 to 0.2). Raupach et al. (1993) reached a similar conclusion after inspecting Marshall’s data
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and believed that there was only a weak experimental dependence of stress partition on roughness
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element shape and the arrangement of elements on the surface. Drag balance instrumentation used by
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Brown et al. (2008) in a wind tunnel, independently and simultaneously measured the drag on arrays of
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cylinders and the intervening surface, separately. Results were interpreted as confirmation that an increase
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in surface roughness enhanced the sheltering of the surface, regardless of roughness configuration i.e.,
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irregular arrays of cylinders were analogous to staggered configurations in terms of drag partitioning.
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The role of flow separation and much-reduced drag in sheltered regions, particularly downwind of
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roughness elements, is significant for drag partitioning. We posit that the sheltered area is required to
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account for anisotropic variation in aerodynamic resistance for realistic wind erosion and dust emission
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models. Furthermore, we posit that current estimates of the erodible area using the area not covered by
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protruding objects is a poor representation of the erodible substrate exposed to abrasion from mobile
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material. The aim of the paper is to describe and evaluate the basis for using angular reflectance data to
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quantify the geometric anisotropy of aerodynamic resistance, account for heterogeneity and estimate the
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area exposed to abrasion.
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2. Estimating aerodynamic roughness length (z0) and erodible area using shadow
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2.1 Relationship between reflectance and frontal area index (λ)
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A new approach is presented here which is based on Chappell and Heritage (2007). The approach is
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inspired by the dimensional analysis of the Raupach (1992; p. 377-378) model (effective shelter area) and
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its replacement of dynamic turbulence with the scales controlling an element wake and how the wakes
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interact (Shao and Yang, 2005) and by the heuristic model of Arya (1975) and hence its similarity with
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the scheme of Marticorena and Bergametti (1995). In common with Marticorena et al. (2004), we show
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the relationship between reflectance and aerodynamic resistance estimated by wind tunnel studies of
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aerodynamic roughness length (z0) and explain the relationship between reflectance and the frontal area
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index (λ).
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The λ is the projection of an obstacle’s frontal area onto a pre-defined area or pixel with a flat
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surface. The projection is defined for a 45° illumination zenith angle i such that Tan i is used as a
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multiplication factor which in this case is restricted to 1 and thus the entire frontal area of the object is
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projected. If 0° < i < 45° then Tan i reduces the projected frontal area and when 45° < i < 90° then Tan i
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increases the projected frontal area. If that pixel is viewed at nadir and illuminated for different i, the
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shadow cast by the object is the same as the projection of the object onto the pixel for the given i. The
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light reflected from the rough surface is reduced by the proportion of the area that is in shadow and
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visible (Figure 2). In the case of many homogeneous objects the shadow area may be reduced if the
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spacing between the objects is insufficient to allow the shadow cast to reach the underlying surface
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(Figure 2). In other words the shadow is projected onto adjacent objects (mutual shadowing).
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[Figure 2]
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However, if objects with vertical sides (e.g., cylinders) are homogeneous their shadow is truncated
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because although it is projected on to the adjacent objects it is not visible at nadir. Illumination of natural
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surfaces demonstrates that a portion of the surface that may otherwise cast shadow may also be under
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shadow and this effect is dependent on illumination azimuth relative to an arbitrary origin (Figure 3).
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[Figure 3]
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Since we believe a priori that the geometry of a rough surface influences aerodynamic resistance and that
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roughness is also one of the main controls on the proportion of illumination there should be a relationship
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between the aerodynamic resistance and illumination proportion (viewed at nadir).
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2.2 Relationship between shadow and erodible area
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The convention within current dust emission models is to approximate the erodible area as simply the
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intervening surface not covered by non-erodible elements. However, saltating soil particles usually strike
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the soil surface at an elevation angle of approximately 12°–15° (Sorensen, 1985). The point of impact is
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influenced by the particle jump length, angle of descent and surface roughness (Potter et al., 1990). As the
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particles bounce, they may jump over obstructions on the surface. If the obstruction is sufficiently tall that
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the particle cannot jump over it and it is non-erodible, it will shelter a portion of the soil surface from
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abrasion. Thus, upwind obstructions determine the angle of trajectory shelter angle a particle must
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achieve in order to strike a given point within the horizontal bounce distance of the saltating particle. The
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fraction of the soil surface impacted by saltating grains varies with the fraction of the surface sheltered by
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non-erodible roughness elements. Potter et al. (1990) developed the cumulative shelter angle distribution
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and it was included in the Wind Erosion Prediction System to make daily estimates of wind erosion
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(Hagen, 1990). Here we propose to use the approach by Potter et al. (1990) and developed by Zobeck and
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Popham (1998) to approximate the erodible area. The curves formed by field measurements by Zobeck
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and Popham (1998; Figure 2 and 3) are a function of the surface roughness, the shelter angle is equivalent
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to the illumination zenith angle described above and the proportion of the surface illuminated and viewed
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at nadir is equivalent to the surface fraction. Consequently, we propose to approximate the erodible area
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by predicting the proportion of surface in shadow with an illumination zenith angle of 75° (equivalent to
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an elevation angle 15°) and viewed at nadir.
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2.3 Evaluation methodology
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A digital elevation model is used here to reconstruct the surface roughness configuration of previous
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laboratory wind tunnel and field studies for which shadow/illumination measurements were not available.
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A ray-casting approach demonstrates the means by which shadow can be estimated remotely. It makes
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use of a fine resolution of elevation sufficient to discretise the surface obstacles. Such an approach is able
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to handle heterogeneous bed situations where the object heights and spacings are different across a
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surface. Variable illumination orientation, to represent wind direction, was accounted for with the same
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computational procedure and an illumination azimuth angle φ relative to a fixed arbitrary origin.
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Although a number of models exist to estimate the proportion of reflectance from a rough surface
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(Cierniewski, 1987; Hapke, 1993; Li and Strahler, 1992; Liang and Townsend, 1996) an empirical
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function is used here for simplicity and to avoid the modeling becoming a distraction from the retrieval of
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aerodynamic resistance information. The most suitable model for describing reflectance of surface
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roughness across scales is the subject of ongoing research by the authors. As an alternative to these
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models, the proportion of illuminated surface viewed at-nadir for given illumination zenith and azimuth
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angles was fitted with a Gaussian model with an additional parameter. The function has the desirable
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quality of resembling a positive exponential or a Gaussian model. Its isotropic form and model
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parameters are:
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  i α
S (i ) = c exp − α
  r
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where S is the proportion of illuminated surface for a given illumination zenith angle (i). The function
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approaches its sill (c) asymptotically and does not have a finite range. Instead, the distance parameter r
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defines the extent of the model. For practical purposes it can be regarded as having an effective range of
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approximately (3r)-α, where it reaches 95% of its sill (Webster and Oliver, 2001). The additional
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parameter α describes the intensity of variation and the curvature. If α = 1 then the function is a positive




(2)
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exponential and if α = 2 the function is a Gaussian. As i→90°, c enables S to be greater than 0 to
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represent the illumination of objects above a rough background or substrate. The proportion of
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illuminated surface associated with the illumination zenith angle 75° is used to approximate the erodible
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area (section 2.2). The goodness of fit was assessed using the RMSE which is here defined as the square
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root of the mean squared difference divided by the number of degrees of freedom (df). The df is the
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number of data minus the number of model parameters used.
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The brightness of the rough surface was assumed to be the same as that of the background
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(substrate) and in the case of the reconstructed wind tunnel studies to scatter according to the Lambertian
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distribution. In this case, the Gaussian function was integrated over all illumination zenith (i) and azimuth
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(φ) angles and made relative to the incoming radiation (I) to form a single scattering albedo (SSA):
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SSA = I /
φ = 2π i =π / 2
∫ ∫
φ =0
i =0
  i α
c exp − α
  r

 .


(3)
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In addition to the SSA a calculation is made for a specified direction over all i but for only a single φ
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viewed at nadir and made relative to the incoming radiation. This statistic is here defined as the relative
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directional reflectance (RDR) for use as a measure of the geometric anisotropy of the aerodynamic
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resistance.
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Although the ray-casting approach described here is evidently capable of handling the anisotropic
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nature of heterogeneous surface roughness, it is not intended to provide an operational method for the
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retrieval of shadow. Surface illumination / shadow may be readily retrieved using angular sensors on
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airborne and space-borne platforms. A photometric model can be used to characterize the surface
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reflectance for given illumination and viewing conditions (cf. Hapke, 1993). The estimation of soil /
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surface roughness using shadow and geometric models is well established (Cierniewski, 1987) and
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approximations to radiative transfer theory by Hapke (1993) have provided parameterized models for soil
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bi-directional reflectance measurements (Pinty et al., 1989; Jacquemoud et al., 1992; Chappell et al.,
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2006; 2007; Wu et al., 2008). The description above amounts to a hypothesis that the proportion of
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illumination (viewed at nadir) can approximate the aerodynamic roughness length and the erodible area.
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That hypothesis is evaluated against existing measurements of aerodynamic resistance.
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3. Evaluation data
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3.1 Isotropic wind tunnel roughness
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A wind tunnel study by Marshall (1971) investigated cylinder and hemisphere roughness elements made
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from solid ‘varnished’ wood. The elements had a uniform height of 2.54 cm with a range of diameters
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(1.27, 2.54, 5.08, 7.62 and 12.7 cm). The elements were arranged in the working section of a wind tunnel
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on square and diagonal grids and also using randomly arranged patterns with spacing between elements
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(2.54 – 125.73 cm) that produced various coverage (ratio of cylinder base area to the specific cover:
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approx. 0.01 – 44.18 %). Each surface configuration was subjected to a single freestream velocity at 20.3
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m s-1 and the total drag force of the surface roughness and that of the roughness elements separately was
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measured to deduce the surface drag force in between elements.
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Aerodynamic roughness lengths (z0) of Marshall’s surface configurations were derived using the
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same approach as Raupach et al., (2006; p. 214):
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 
z0 δ
U
= exp k  B − δ
h h
u*
 
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where δ ≈ 3.3 + 15.0(λφ ) 0.43 (cm), k=0.4, B=2.5 and U δ =20.3 m s-1 from table 4 of Marshall (1971).
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These data are plotted against λ in Figure 4. Their characteristics are summarized in Table 1.

 ,

(4)
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[Figure 4]
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An investigation of gravel surface aerodynamic resistance was conducted in a wind tunnel by Dong et al.
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(2002). Six types of artificial gravel were fabricated in cement to form “parabolic-shaped” elements with
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circular bases and diameters (19, 29, 38, 47, 57, 65 mm) and heights (12, 19, 24, 31, 37, 43 mm) with a
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diameter : height of approximately 1.5. The gravels were arranged in the working section of a wind tunnel
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in a diamond (staggered) pattern so that a range of coverage (1 – 92%) was provided. When the spacings
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of the objects with these coverages were evaluated in our model, coverage greater than 85% had a spacing
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of less than the model resolution (1 mm). Consequently, surface coverages greater than 85% were
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excluded in all subsequent analyses. Dong et al (2002) used ten free-stream velocities (4, 6, 8, 10, 12, 14,
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16, 18, 20, 22 m s-1) for each type of gravel and coverage. Wind profiles (the distribution of wind speed
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with height) were measured using an array of 10 Pitot-static probes mounted at ten heights (3, 6, 10, 15,
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30, 60, 120, 200, 350 and 500 mm above the surface). Aerodynamic roughness lengths (z0) were derived
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from measured wind profiles using least squares curve fitting (Dong et al., 2002). Results were eliminated
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by Dong et al. (2002) if logarithmic curve fitting gave a value of R2 < 0.98 at the 5% significance level.
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Only the data for 20 m s-1 freestream velocity was used here to ensure comparison with the results of
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Marshall (1971). These data are plotted against roughness density in Figure 4 and the characteristics are
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summarized in Table 1.
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3.3 Geometric anisotropic natural roughness
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Wind velocity measurements on a circular field (100 m radius; 86,180 m2) at the USDA, Agricultural
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Research Service experimental farm in Lubbock, Texas were made between March – May 2001 to
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provide a validation dataset for a larger unpublished study. At the centre of the field a 2 m high mast was
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erected and equipped with calibrated cup anemometers at heights of 0.5, 1.0, and 2.0 m and with a hot
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wire anemometer at 0.01 m. Data were logged on a CSI 21X data logger programmed to record the
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average wind speed at 1 min integrated intervals until the 2 m wind speed exceeded 3.5 m s-1 for 5
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minutes at which time the integrated sampling interval was reduced to 1 sec. Five minute average wind
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data were available and wind directions were merged into the data files. The parameter values (u* and z0)
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of the Prandtl equation were optimized against the 5 min average wind velocity using a non-linear
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weighted least squares routine (PROC NLIN in SAS version 8.2). The characteristics of these data are
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summarized in Table 2.
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4. Digital elevation model reconstructions of wind tunnel and field surface conditions
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Surface roughness configurations were illuminated using ray-casting coded in Matlab for rapid
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visualization and written to (re)create shapes that could easily be described using geometry. For example,
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the wind tunnel surface roughness configuration objects constructed by Dong et al. (2002) were recreated
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by approximating them using oblate spheroids with the equation x2/a2+y2/b2+z2/b2=1 where x, y and z are
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the dimensions of a cartesian co-ordinate system and a and b are the semi-major (height) and semi-minor
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(diameter) axes of an ellipsoid. The shapes were buried in the plane up to their semi-major axis length and
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their centres were placed with an equidistant spacing that matched the coverages used by Dong et al.
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(2002). To reconstruct the surface roughness configurations of Marshall’s (1971) wind tunnel study we
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created digital elevation surfaces of portions which were well represented using truncated hemispheroids
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in the digital reconstruction.
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4.1 Isotropic wind tunnel roughness
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For each simulated surface the number of objects was kept constant and 8 whole shapes were placed on
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the plane surface. Since spacing, and hence coverage of the objects varied between simulated surfaces,
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the simulation area also varied (Figure 2). Each surface was illuminated for a single φ and with many i.
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An example of one half of the configuration surface of hemispheroids for the same i is provided in Figure
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2a & b. Only the downwind half of the DEM surface was used in the calculation of the shadow area to
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avoid the edge effects of absent upstream objects that would contribute shadow to the surface (cf.,
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Chappell and Heritage, 2007).
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4.2 Geometric anisotropic natural roughness
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There is a dearth of available data that combines digital elevation models (DEMs) of natural surfaces with
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wind speed data and aerodynamic roughness length (z0) observations. In their absence we used a DEM
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taken from a circular field from which wind velocity measurements were available at its centre (section
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3.3). The field was prepared consistently with machinery to produce a minimal roughness (approx. 1 cm)
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to satisfy a working hypothesis of that study: there was no discernible preferential orientation of surface
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roughness to influence aerodynamic resistance. Prior to making wind velocity measurements at the site,
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the DEM was produced using a laser on a horizontal frame that measured the elevation relative to an
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arbitrary datum over an area of 1 m2 with a horizontal resolution of 5 mm. The natural surface elevation
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data were filtered to remove a few spurious measurements. The DEM of the surface is shown in Figure 3
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using examples overlayed of the proportion of illuminated surface (viewed at nadir) that represents four
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azimuth directions (0°, 45°, 90° and 135°). The shadow overlay is described below (section 8). Note that
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an illumination zenith angle of 75° was used and therefore figure 3 also represents the variability of
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erodible area with azimuth angle.
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5. Evaluation of the relationship between shadow and aerodynamic roughness length (z0)
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Perhaps the most fundamental assumption made in the current understanding of the relationship between
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roughness configuration and aerodynamic resistance is that frontal area index (roughness density; λ)
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adequately captures the characteristics. This is evident in Figure 4 which shows the relationship between
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λ and aerodynamic roughness length (z0) standardized by object height (h) using the data from Marshall
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(1971) and Dong et al. (2002). Using the same surface configurations for both studies, a range of
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illumination conditions were simulated using ray-casting and these data were fitted with equation (2) and
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then integrated over all illumination zenith angles using equation (3). All model fits achieved an RMSE
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value of less than 0.0045 of shadow proportion and consequently the model parameters were accepted.
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For brevity, the results of the model fitting and the integrated values are not shown. The values of the
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variance model parameter (c) were divided by the values of the exponent (α) and are plotted against λ to
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demonstrate the relationship between λ and model parameters representing illumination / shadow (Figure
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5).
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[Figure 5]
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Since there is a linear relationship between λ and the illumination function parameters, the relationship
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between the same parameter values and the aerodynamic roughness length (z0) standardized by object
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height (h) (Figure 6) is very similar to the relationship between λ and z0/h (Figure 4). The results show a
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small separation between the reconstructed wind tunnel studies of Dong et al. (2002) and Marshall (1971)
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due to the nature of the objects (discussed below).
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[Figure 6]
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This relationship is sufficient to enable predictions of z0/h for other surface roughness configurations.
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However, the relationship is restricted to this particular model and its parameter values. To investigate a
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more general approach involving other appropriate models (kernel) a relationship is sought between the
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single scattering albedo (SSA) and z0/h (Figure 7).
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[Figure 7]
355
356
There are strong separate relationships between the SSA and z0/h for the reconstructed surface roughness
357
of the two wind tunnel studies. The separation between the studies is similar to that evident in the model
358
parameter values (Figure 6) and is due to the bluff wall nature of the cylinders used in Marshall’s study as
359
opposed to the hemispheroids used by Dong et al. (2002). The former does not enable mutual shadowing
14
360
whilst the latter does. Consequently, when the spacing between objects is smaller than the projected area
361
of the objects (cast shadow) the mutual shadowing of the hemispheroids reduces the SSA but the vertical
362
wall of the cylinders does not allow it and SSA reduces at a slow rate. When the surface roughness
363
configuration is widely spaced the values of SSA for both studies are very similar and their different
364
geometry makes little difference.
365
Natural particles, aggregates of particles and non-erodible roughness elements (e.g., reg deflation
366
surface) tend towards hemispheroids because of the inherent strength of the shape and because of the
367
weathering / abrasion processes. The hemispheroids represent a more realistic behaviour in mutual
368
shadowing than the cylinders. Hemispheroids have also been found by other workers to adequately
369
represent the light scattering behaviour of a range of unvegetated (Cierniewski, 1987) and vegetated
370
surfaces (Li and Strahler, 1992). This provides a compelling basis to adopt Dong et al. (2002) results
371
using hemispheroids to establish a general relationship between SSA and z0/h (Figure 8).
372
373
[Figure 8]
374
375
A positive variant of the Gaussian model (Eq. 2) is shown in Figure 8 (c=0.2116, α=10.9940, r=0.8892)
376
and has a RMSE=0.0026 for z0/h. It is this model which is used in subsequent predictions described
377
below.
378
379
6. Geometric anisotropic aerodynamic resistance (z0) of a heterogeneous surface
380
381
The final stage of testing the shadow model is to make predictions of z0 over the heterogeneous surface
382
with changing wind directions. Recall that a digital elevation model (DEM) was measured over a 1 m2
383
area of a 100 m radius circular field prepared with minimal roughness. The DEM has already been
384
presented (Figure 3) but the shadow overlay has not been described. The pattern of shadow is created by
385
an illumination azimuth angle from an arbitrary origin (0°) that represents the wind direction and an
15
386
illumination zenith angle of 75° to represent the abrasion angle. The most noticeable aspect of the shadow
387
overlays (Figure 3) is the large proportion of the DEM that they cover. More than 70% of the surface is
388
sheltered with a wind direction at 0°. This finding has important implications for the estimation of
389
erodible area in dust emission models (see section 8). Secondly, differences amongst the shadow overlays
390
include the changing patterns of shadow and changing coverage of shadow with different azimuth angle.
391
These changes are responses to the variability in the surface elevation and the systematic decrease in
392
elevation towards the origin of the DEM surface.
393
The patterns evident in Figure 3 can be generalized as a shadow function using illumination zenith
394
angles and illumination azimuth angles (Figure 9). In this case, the shadow function is the proportion of
395
shadow visible at nadir; when the shadow function is close to 1 most of the surface is in shadow and
396
when close to 0 most of the surface is illuminated. Each curve shows an increase in the illumination
397
function with increasing illumination zenith angle. In general, the curves show only small differences in
398
their shape as a consequence of the prepared surface roughness. Nevertheless, the angular anisotropy of
399
the roughness remains evident in some curves. For a given azimuth angle (e.g., 0°) and its opposing angle
400
(180°) the rate of change is different and is readily evident as a contrast in the magnitude of the shadow at
401
the largest illumination angles. These patterns may reverse as is the case with the remaining azimuth
402
angles. Such complicated angular anisotropic responses are characterised readily by shadow with
403
changing illumination zenith angle offering the potential to model the aerodynamic resistance using
404
sensors which measure reflectance.
405
406
[Figure 9]
407
408
Using the previously established isotropic relationship between SSA and z0 (Figure 8) we
409
predicted the z0 of the heterogeneous surface for a range of wind directions (Table 2). Since the prediction
410
requires a value for object height (h) this was estimated when the average predicted z0 was the same as the
16
411
median measured z0 (0.056 mm) for all observations and resulted in a value for h=2.59 mm. This h value
412
is consistent with the small surface roughness of the prepared surface.
413
414
[Table 2]
415
416
Predicted z0 changed only slightly with wind direction. This pattern is consistent with the prepared nature
417
of the soil surface. However, it is clear that some anisotropy remains since the differences in predicted z0
418
exceed the uncertainty of the model estimate. Only those measured z0 were used with approximately the
419
same wind direction (± 5°) as those used to make predictions. The observed z0 values were highly skewed
420
(not shown) and consequently parametric statistics were avoided. Instead, the median z0 and half the z0
421
interquartile range are presented here to provide uncertainty about the estimate (Table 2). The measured
422
z0 values are highly variable and some measurement ranges do not contain the values of the predicted z0.
423
The interpretation of these data is difficult because field measurements of z0 are notoriously unreliable
424
(Lancaster et al., 1991; Bauer et al., 1992). Nevertheless, there is good general agreement between the
425
measurements and predictions.
426
427
7. Erodible area used in dust emission models
428
429
The proportion of a surface sheltered from abrasion by saltating material is not included in regional and
430
global dust emission estimates despite for example, a measure (the cumulative shelter angle distribution)
431
being included in the Wind Erosion Prediction System. Dust emission models assume that the relative
432
contribution to the total flux of each size range (e.g., Marticorena and Bergametti, 1995; p. 16,422) is “…
433
proportional to the relative [area] it occupies on the total surface.” There are several weaknesses with this
434
assumption:
435
436
1) non-erodible roughness elements protect
a. the underlying substrate from the wind and abrasion / sandblasting,
17
437
438
b. other non-erodible roughness elements from the wind and hence reduce the momentum
extracted by the roughness elements (i.e., z0).
439
2) the area protected will be a geometric anisotropic function of wind velocity (speed and
440
direction) and configuration of the roughness elements or surface roughness heterogeneity
441
3) the spatial organisation of the surface will play a significant rôle in the sheltering of substrate
442
and non-erodible roughness elements (e.g., ripples)
443
The implications for dust emission models of the geometric anisotropic sheltering of non-erodible surface
444
roughness are illustrated using the model of Marticorean and Bergametti (1995). The vertically integrated
445
horizontal flux (Fh) was calculated to consider the way in which the exposed erodible surface contributes
446
dust emission. Two configurations of the same hemispheroid (38 mm high and 24 mm diameter) used in
447
Dong’s et al (2002) wind tunnel study were included with different spacings (Table 3). The obstacle-free
448
surface was estimated following the approach of most dust emission models and the alternative erodible
449
area was calculated using the zenith abrasion angle (75°) to represent exposure to sandblasting. The Fh
450
used a single lognormal mode at 100 µm and σ=2, measured z0 and a smooth roughness z0s≈0.026 mm,
451
following the wind tunnel measurements of Dong et al. (2002).
452
453
[Table 3]
454
455
Table 3 shows that there are large differences in z0 and U* for each roughness configuration caused by the
456
spacing of the hemispheroids. In the case of widely-spaced hemispheroids the area not covered was 95%
457
of the surface area but using the abrasion angle only approximately 79% was erodible. Consequently,
458
estimates of Fh for the area not covered were 17% larger than the erodible area. In the case of closely-
459
spaced hemispheroids the area not covered was 25% but the erodible area was approximately 4% of the
460
surface area. Using the same z0 and U* values the Fh for the erodible area was 85% smaller than that of
461
the area not covered (Table 3). These results show that substantial over-estimates of Fh and hence dust
462
emission can arise from the use of area not covered to estimate the erodible area. The zenith abrasion
18
463
angle and sheltered area offers an alternative rapid and consistent estimate of the erodible area for use in
464
dust emission models.
465
466
8. Conclusions
467
468
Predictions of aerodynamic resistance were made using the relationship between single scattering albedo
469
(SSA) and aerodynamic roughness length (z0) provided by wind velocity profile measurements in a wind
470
tunnel study. This new model provided the ability to account for the anisotropy of aerodynamic resistance
471
caused by changing wind direction, which is particularly important over heterogeneous surfaces. These
472
achievements were possible because the model used illumination and shadow to represent the geometric
473
projection of frontal area index in any orientation. The estimation of shadow cast at a 75° illumination
474
zenith angle was equivalent to the sheltering from abrasion by saltating particles. The large area sheltered
475
from abrasion was in contrast to the area not covered used in current dust emission models. Consequently,
476
the findings above suggested strongly that wind erosion and dust emission models that do not account for
477
geometric anisotropic sheltering of the surface may considerably overestimate the horizontal flux and
478
hence dust emission.
479
The estimation of z0 using measured wind velocity profiles is notoriously unreliable particularly
480
over large areas or multiple dunes. That unreliability would perhaps be more evident if more studies
481
included the uncertainty associated with their estimates. Notwithstanding that omission, much of the
482
uncertainty in z0 field estimates arise from the difficulty in accounting for the inevitable geometric
483
anisotropy in z0 as a consequence of varying wind directions over heterogeneous surfaces. Consequently,
484
field z0 is likely to be neither precise nor accurate. The results of this study suggest that the new model
485
provides an alternative measure that is suitable to investigate geometric anisotropic aerodynamic
486
resistance across scales of variation (e.g., grain, ripple, dune etc.).
487
488
489
Acknowledgements
19
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The first author is grateful to M. Ekström and N. Chappell for their enduring support and critical
observations. He is also grateful to B. Marticorena and G. Bergametti for making the dust emission code
available. The authors would like to thank P. Hairsine for his incisive comments on an early version of
this manuscript and the two anonymous referees who provided constructive comments on the manuscript.
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21
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22
595
596
597
598
Tables
Table 1. Characteristics of the surfaces used to test the shadow model.
Marshall (1971)
Cylinders
Dong et al. (2002)
Hemispheroids
599
600
Layout
Lateral
cover
(λ)
Object
height
(mm)
Object
diameter
(mm)
Average
height
(mm)
Street
0.0002 –
0.22
25.4
12.7 –
127
0.002 –
11
Diagonal
0.01 –
0.61
19 – 65
Van Pelt unpublished
Natural surface
N/A
N/A
N/A
N/A = Not applicable because the surface is natural.
12 – 43
N/A
Wind
velocity
(m s-1)
at 0.6 m
20.3
Wind
direction
(degrees)
Fixed
0.07 – 19
at 0.6 m
4 – 22
Fixed
0.0008
at 2 m
0.7 – 16
1-358
23
601
602
603
Table 2. Aerodynamic roughness lengths (z0) measured in a field for which 1 m2 was used to predict
geometric anisotropic z0 using the new model.
Wind
direction
(azimuth
angle)
(± 5°)
0
45
90
135
180
225
270
315
All data
z0 median
z0 half
interquartile
range
Predicted z0
z0 RMSE
16
129
152
182
285
183
46
28
(mm)
0.058
3.641
0.529
0.033
0.053
0.254
0.043
0.012
(mm)
0.500
1.981
0.566
0.032
0.311
0.938
0.038
0.004
(mm)
0.540
0.516
0.522
0.477
0.530
0.529
0.542
0.534
(mm)
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
4786
0.079
0.459
0.530
0.002
Number
of data
604
24
605
606
607
608
Table 3. Vertically integrated horizontal flux (Fh) for two configurations of hemispheres (38 mm height
and 24 mm diameter) exposed to simulated sandblasting using the proportion of the wind tunnel not
covered and the erodible proportion using the zenith abrasion angle (see text for details).
Lateral
cover
Λ
0.04
0.60
Measured
z0
mm
0.65
0.21
U*
mm s-1
884.0
707.3
Not covered
Area
Fh
%
95.0
25.0
g mm-1 s-1
4.0
0.9
Erodible proportion
Area
Fh
%
78.8
3.7
g mm-1 s-1
3.4
0.1
Difference
Absolute
Fh
%
17
85
609
25
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
Figure captions
Figure 1.
Cylinders used to represent non-erodible roughness elements in wind tunnel studies and
parameterizations for wind erosion and dust emission models protect a portion of the substrate
surface that may include all or part of other roughness elements in a heterogeneous surface
(a). A change in wind direction redefines the area of the substrate protected from the wind and
may expose previously protected roughness elements (b).
Figure 2.
Digital elevation models of 38 mm diameter and 24 mm high hemispheroids on a diagonal
lattice with a coverage of 5% (a) and 75% (b) using an illumination zenith angle of 75° to cast
the shadow (darkest tone) and represent the area sheltered from abrasion (see text for details).
Figure 3.
Digital elevation model of a 1 m2 natural soil surface with an overlay of shadow (dark tones)
representing illumination azimuth angles of (a) 0°, (b) 45°, (c) 90° and (d) 135° for an
illumination zenith angle of 75° representing the abrasion angle (see text for details). The
proportion of shadow viewed at nadir covering the surface is 74%, 65%, 63% and 56% for
each azimuth direction, respectively.
Figure 4.
The results of wind tunnel measurements by Marshall (1971) and Dong et al., (2002) showing
aerodynamic roughness length (z0) standardised by object height (h) against frontal area index
(λ; roughness density).
Figure 5.
The sill variance (c) divided by the exponent (α) parameter values of the Gaussian model
fitted to the illuminated surface roughness of wind tunnel studies by Dong et al. (2002) and
Marshall (1971) plotted against the frontal area index (λ; roughness density).
Figure 6.
The sill variance (c) divided by the exponent (α) parameter values of the Gaussian model
fitted to the illuminated surface roughness of wind tunnel studies by Dong et al. (2002) and
Marshall (1971) plotted against the aerodynamic roughness length (z0) standardized by object
height (h).
Figure 7.
Single scattering albedo (SSA) of the reconstructed wind tunnel studies hemispheroid surface
roughness and its relationship with the measured aerodynamic roughness length (z0)
standardized by object height (h).
Figure 8.
Gaussian model fitted to the single scattering albedo (SSA) of Dong et al. (2002)
reconstructed wind tunnel surface roughness and its relationship with the measured
aerodynamic roughness length (z0) standardized by object height (h).
Figure 9.
Shadow function for several illumination zenith (i) and azimuth (φ) angles representing the
wind directions over the natural surface digital elevation model shown in Figure 3.
26
651
652
Figures
(a)
(b)
wind
wake
653
654
655
656
657
658
659
Figure 1.
Cylinders used to represent non-erodible roughness elements in wind tunnel studies and
parameterizations for wind erosion and dust emission models protect a portion of the substrate
surface that may include all or part of other roughness elements in a heterogeneous surface
(a). A change in wind direction redefines the area of the substrate protected from the wind and
may expose previously protected roughness elements (b).
27
660
661
662
663
664
665
666
667
(a)
Figure 2.
(b)
Digital elevation models of 38 mm diameter and 24 mm high hemispheroids on a diagonal
lattice with a coverage of 5% (a) and 75% (b) using an illumination zenith angle of 75° to
cast the shadow (darkest tone) and represent the area sheltered from abrasion (see text for
details). Note the differences in size of axes.
28
668
(a)
(b)
669
(c)
(d)
670
671
672
673
Figure 3.
Digital elevation model of a 1 m2 natural soil surface with an overlay of shadow (dark tones) representing illumination azimuth angles
of (a) 0°, (b) 45°, (c) 90° and (d) 135° for an illumination zenith angle of 75° representing the abrasion angle (see text for details). The
proportion of shadow viewed at nadir covering the surface is 74%, 65%, 63% and 56% for each azimuth direction, respectively.
29
674
675
676
677
Figure 4.
The results of wind tunnel measurements by Marshall (1971) and Dong et al., (2002) showing
aerodynamic roughness length (z0) standardised by object height (h) against frontal area index
(λ; roughness density).
30
678
679
680
681
Figure 5.
The sill variance (c) divided by the exponent (α) parameter values of the Gaussian model
fitted to the illuminated surface roughness of wind tunnel studies (Dong et al., 2002;
Marshall, 1971) and plotted against the frontal area index (λ; roughness density).
31
682
683
684
685
686
Figure 6.
The sill variance (c) divided by the exponent (α) parameter values of the Gaussian model
fitted to the illuminated surface roughness of wind tunnel studies (Dong et al., 2002;
Marshall, 1971) and plotted against the aerodynamic roughness length (z0) standardized by
object height (h).
32
687
688
689
690
Figure 7.
Single scattering albedo (SSA) of the reconstructed wind tunnel studies surface roughness and
its relationship with the measured aerodynamic roughness length (z0) standardized by object
height (h).
33
691
692
693
694
Figure 8.
Gaussian model fitted to the single scattering albedo (SSA) of Dong et al. (2002) reconstructed
wind tunnel surface roughness and its relationship with the measured aerodynamic roughness
length (z0) standardized by object height (h).
34
695
696
697
698
Figure 9.
Shadow function for several illumination zenith (i) and azimuth (φ) angles representing the
wind directions over the natural surface digital elevation model shown in Figure 3.
35