Modeling entrainment of sedimentary particles by wind and water: A generalized approach

Authors


Abstract

[1] For long-standing theoretical reasons, it is often asserted that the threshold shear stress for entrainment of sedimentary particles (τ*t = ρfu*t2, made dimensionless as A = ρfu*t2/((ρp − ρf)gd)) has a universal relationship with the particle Reynolds number (Re*t = u*td/ν), where u*t is the threshold friction velocity, ρf is the fluid density, ρp is the density of the particles, d is the particle diameter, g is the gravitational acceleration and ν is the kinematic viscosity of the fluid. However, experimental plots of A(Re*t) for sediment entrainment in air and water show two major differences: (1) For large Re*t, the values of A in water are, in general, a few times larger than those in air, and (2) when Re*t <1, A increases more rapidly in air than in water as Re*t decreases. This paper derives a new, general theory for A, which incorporates the effects of fluid turbulence, particle cohesion and probabilistic aspects of grain entrainment. It is found that difference (1) is explained by differences in the probability distribution of streamwise velocity fluctuations for typical situations in air and water, which follow from basic scaling laws for velocity variances in turbulent flow. Difference (2) is explained by the different behaviors of interparticle cohesion forces in air and water. The resulting expression is shown to compare well with experimental data.

1. Introduction

[2] The prediction of incipient motion of a particle lying in a bed of similar particles (i.e., the entrainment of sedimentary particles from the surface into the fluid when the fluid speed exceeds a certain threshold), is fundamental for understanding sediment transport by water and wind [Graf, 1971; Qian and Wan, 1983; Nickling, 1988; Raupach and Lu, 2004]. Shields [1936] pioneered research into incipient motion in open channels by introducing the dimensionless threshold shear stress A = τ*t/(ρp − ρf)gd, a measure of the ratio of the threshold hydrodynamic force on a surface particle to its weight (see Table 1 for definitions of symbols). Shields argued that A should be a unique function of the particle Reynolds number Re*t = u*td/ν, as

equation image

For particle entrainment by airflow, Bagnold [1941] derived a similar theory considering the balance between the aerodynamic drag and the gravitational force. He found that at large Re*t, A is nearly a constant and u*td1/2, which is essentially consistent with Shields' theory obtained in open channels. Across all particle sizes, Bagnold also assumed that A is a unique function of Re*t, again consistent with the proposals of Shields.

Table 1. Main Symbols (Omitting Symbols Used Only Once)
SymbolDefinitionFirst Use
Adimensionless threshold shear stress, A = ρfu*t2/((ρp − ρf)gd)Equation (1)
A1reference value of AEquation (2)
Bempirical constant in logarithmic law over a smooth wall (B = 5.0)Equation (12)
cicohesion coefficientEquation (6)
CD, CLdrag, lift coefficients (computations assume CD = 0.5, CL = 0.3)Equations (4) and (5)
dparticle diameterEquation (1)
FD, FL, FG, FCforces on particle from drag, lift, specific weight and cohesionEquations (3)(5)
F(Re*t)dimensionless function quantifying dependence of A on gravitational and aerodynamic forcesEquation (2)
fpcomponent of F associated with packing geometry (computations assume fp = 1.5)Equation (31)
F1component of F associated with first-order flow effectsEquation (32)
F2component of F associated with second-order flow effectsEquation (33)
G(d)dimensionless function quantifying dependence of A on cohesion forcesEquation (2)
ggravitational accelerationEquation (1)
ksNikuradse sand-grain roughnessEquation (14)
K1dimensional parameter for cohesion forceEquation (36)
LD, LL, LG, LCmoment arms for forces FD, FL, FG, FC (LD = dcosθ, LL = dsinθ, LG = dsinθ and LC = dsinθ)Equation (8)
nexponent specifying cohesion forceEquation (36)
pprobability of entrainmentEquation (23)
Reτflow Reynolds number, Reτ = u* δ/νSection 2.3
Re*tparticle Reynolds number at threshold, Re*t = u*tdEquation (1)
Shorizontal area of particle exposed to the flow (computations assume S = 0.2πd2)Equations (4) and (5)
ustreamwise velocity (dimensionless form: u+ = u/u*)Section 2
uΔreference velocity (at height yΔ) for defining FD and FLEquations (4) and (5)
utthreshold velocity (at height yΔ)Equation (10)
u*tthreshold friction velocityEquation (1)
υΔ, υtυΔ = ln uΔ, υt = ln utEquation (22)
yheight above level of effective drag (y = 0)Section 2
y+dimensionless height, y+ = u*ySection 2
yΔheight at which velocity uΔ is definedEquations (4) and (5)
yΔ+dimensionless reference height at which velocity uΔ is definedSection 2
z0roughness lengthEquation (16)
z1smallest separation between two spherical particles in equation (6)Equation (6)
βconstant relating reference height yΔ to dSection 2.1
δboundary layer depthSection 2.3
Δequation image+velocity-increment form of roughness functionEquation (13)
κvon Karman constant (κ = 0.41)Equation (12)
νkinematic viscosity of the fluidEquation (1)
ξinterpolation function for roughness in transition regimeEquation (15)
ρf, ρpfluid density, particle densityEquation (1)
θpivoting angle (see Figure 2; computations assume θ = 30°)Equation (8)
σstandard deviation of streamwise velocity (dimensionless form: σ+ = σ/u*)Equation (20)
τ*tthreshold shear stress, τ*t = ρfu*t2Equation (1)

[3] Over the last 70 years, experimental research into incipient motion has in general supported the theories of Shields and Bagnold [White, 1970; Graf, 1971; Yalin, 1972; Mantz, 1977; Buffington and Montgomery, 1997; Zingg, 1953; Greeley and Iversen, 1985; Nickling, 1988]. Figure 1 shows data obtained in experiments in air [Cleaver and Yates, 1973; Fletcher, 1976a, 1976b; Iversen et al., 1976; Iversen and White, 1982] and in water [Graf, 1971; Yalin, 1972; White, 1970], plotted on a typical Shields' A ∼ Re*t diagram. Three distinct regions can be identified: (1) Re*t ≥ 10, a region where the flow is fully turbulent, and where A attains a constant value of approximately 0.04 to 0.06 at Re*t ≥ 500 in water, and 0.01 to 0.03 in air (note that there are few data with Re*t ≥ 200 for airflows); (2) Re*t < 1, a region where particle entrainment is mainly due to viscous laminar flow, where A increases while Re*t decreases, at a rate which is steeper in air than in water; and (3) a transitional region, 1 ≤ Re*t ≤ 10, where the laminar sublayer partially covers the particles, but the outer turbulent flow is partially affected by the roughness of the grain bed. Within this region, A reaches a minimum value of approximately 0.03 to 0.06 for water [Yalin and Karahan, 1979], but such minima are not obvious for air entrainment [Greeley and Iversen, 1985].

Figure 1.

Dimensionless threshold shear stress A as a function of particle Reynolds number Re*t. The diagram shows that data obtained in water flow follow a different trend compared with those obtained in air stream. Data were extracted from the literature indicated in the legend.

[4] Figure 1 shows that data obtained from airflow and open channel flow do not collapse to a single A ∼ Re*t relationship. The upturn of the Shields curve at Re*t< 1 in airflow is much sharper than for water flow. For Re*t ≥ 5, A is about 3 to 4 times smaller in air than in water.

[5] Several researchers have noted the different behavior of A in air and in water, and that the expressions for A derived in one fluid cannot be directly applied to another [Bagnold, 1941; Graf, 1971; Iversen et al., 1987]. Bagnold [1941] suggested that the differences might be due to a difference in surface texture or bed packing conditions, or to observation errors. Iversen et al. [1987] attributed the difference to density ratio differences. By adding the impacting force of saltating particles to the force balance equation, they related A empirically to the particle-to-fluid density ratio. Although not directly targeting the different behavior of A in air and in water, a recent study of incipient motion in air has also incorporated the effect of the particle-to-fluid density ratio [Cornelis and Gabriels, 2004]. Interestingly, although Cornelis and Gabriels [2004] and Iversen et al. [1987] both related A to the density ratio, and both utilized the same data of Iversen and White [1982], the expression of Iversen et al. [1987] suggests that A decreases with increasing particle-to-fluid density ratio while the expression of Cornelis and Gabriels [2004] suggests A increases with increasing density ratio. Making the situation more confusing, a comprehensive review of incipient motion in open channel flow concludes that there is no obvious dependence of A on particle density, therefore on density ratio [Buffington and Montgomery, 1997]. These contradictions suggest that the quantitative relationships between A and flow conditions, fluid properties and particle properties remain unclear. It is also not clear why, for a given fluid and under similar experimental conditions, randomly varying values of A have often been obtained.

[6] Much research into incipient motion has been based on a force balance at the instant of particle entrainment [Bagnold, 1941; Phillips, 1980; Iversen and White, 1982], and/or analysis of dimensionless groups [Shields, 1936; Fletcher, 1976a, 1976b]. With certain assumptions, the functional forms for the parameter A are predetermined and then fitted to experimental data, and in some cases, excessive parameter fitting is involved [Greeley and Iversen, 1985; Cornelis and Gabriels, 2004]. These studies have provided a means of estimating threshold shear stress under idealized situations. Nevertheless, such a deterministic approach often results in partial explanations of the several processes involved in incipient motion. Apart from the inability to explain differences of A ∼ Re*t relationships in air and water, there is a lack of agreement about the values of A and the physical causes of why, for Re*t ≤ 1, the estimates of A vary by an order of magnitude or more for a given Re*t. Emphasizing the effects of flow, Bagnold [1956] argued that A will not exceed 0.4, but values of A greater than 10 have been found for d ≤ 10 μm [White, 1970; Cleaver and Yates, 1973; Fletcher, 1976a, 1976b]. By assuming laminar sublayer flow, Yalin [1972] derived A = 0.1/Re*t. With similar assumptions for the flow regime and force balance, Ling [1995] suggested that A is inversely proportional to Re*t for rolling grains, and inversely proportional to Re*t2 for a threshold lift condition. Both studies implied that the increase in A with decreasing Re*t is solely due to the transition in flow regime from turbulent to laminar flow.

[7] For particle entrainment in airflow, Iversen and White [1982] attributed the sharp upturn of A for Re*t < 1 to the effects of interparticle cohesion forces. To account for the effect of cohesion, Greeley and Iversen [1985] suggested that A should be of the form

equation image

where the dimensionless functions F(Re*t) and G(d) respectively represent the effects of aerodynamic and cohesion forces. Equation (2) essentially implies that the upturn of A for Re*t < 1 is partially due to the flow condition (through the term F(Re*t)) and partially due to interparticle cohesion (through the term G(d)). Though equation (2) overcomes the shortcomings of the early expression of Bagnold [1941] and is effective in describing the behavior of u*t for the entire particle size range, the expressions proposed by Greeley and Iversen [1985] involve the two empirical functions, F(Re*t) and G(d), and are difficult to relate to physical processes.

[8] While earlier theoretical analysis often focused on the effects of flow conditions by relating A to the particle Reynolds number Re*t, more recent empirical or semiempirical studies have to some degree sought to remove the flow term F(Re*t) in equation (2), or to replace it by a particle property. For instance, Marticorena and Bergametti [1995] simplified the expressions of Greeley and Iversen [1985] by expressing Re*t as a function of particle size only. Shao and Lu [2000] reanalyzed the wind tunnel data of Iversen and White [1982] and suggested that A = AN G(d), where AN is a constant of 0.013 and G(d) represents the effects of interparticle cohesion. More recently, Cornelis and Gabriels [2004] expressed A as a function of the particle-to-fluid density ratio, particle diameter and interparticle cohesion. Their expression involves five empirical constants that were determined by fitting to the data of Iversen and White [1982] using nonlinear regression. Such emphases on sedimentological controls are also abundant in the hydraulic literature. For example, in open channels with mixed bed material particle sizes, studies have commonly expressed A as an inverse empirical function of the ratio of particle size d to the underlying median bed particle size d50 [Komar and Li, 1988; Richards, 1990; Buffington and Montgomery, 1997].

[9] A clear advantage of using these semiempirical expressions of A is their simplicity. If A is not expressed as a function of Re*t, iteration is avoided, since the friction velocity u*t no longer appears on both sides of the expression for u*t. This is particularly appealing for large-scale spatially distributed applications [Marticorena and Bergametti, 1995]. However, the drawback of these expressions is their failure to account fully for the physical processes involved. As noted by Qian and Wan [1983], a variety of models could fit the experimental data well as long as the chosen expression has the right shape and contains at least two free parameters to describe the upturn of A for both large and small Re*t ranges. There is little basis for judging the superiority of an expression solely on goodness of statistical fit if the physical meaning of the fitted parameters is not clearly understood. A more general, physically based approach to the threshold of entrainment is needed.

[10] Realizing that deterministic approaches can only estimate the average values of threshold shear stress but are unable to predict the observed variations, some researchers have proposed that particle incipient motion is a stochastic process; that is, there exists a range of threshold shear stresses for a given particle size [Grass, 1970; Gessler, 1971]. Field and laboratory observations in open channel flow confirm a variability of threshold shear stress attributable to a number of random factors including temporal fluctuation of near bed turbulence, the bed packing condition, and heterogeneities in grain size and shape, etc [Einstein, 1950; Grass, 1970]. In the hydraulic literature, recent research into incipient motion has focused on finding appropriate probability distributions for instantaneous flow velocity or shear stress [Cheng and Chiew, 1998; Wu and Lin, 2002], the effects of random bed roughness and picking conditions [Papanicolaou et al., 2001, 2002], or the combined probabilistic nature of turbulent fluctuation and bed grain geometry [Wu and Chou, 2003]. Nevertheless, incorporating statistical/stochastic concepts remains rare in the modeling of particle incipient motion during wind erosion.

[11] The objective of this paper is to derive a general expression for A, to explain the differences in A ∼ Re*t plots between air and water. We incorporate the effect of fluid turbulence by utilizing recent advances in understanding of turbulence over rough walls, especially the scaling of velocity fluctuations with bulk Reynolds number (using boundary layer height δ as the length scale rather than particle diameter d). We also consider the probabilistic aspects of grain entrainment by adopting a lognormal distribution for the near-bed instantaneous flow velocity. The new expression also incorporates the influences of interparticle cohesion forces to account for the upturning of A for small values of Re*t. We use the new expression to analyze the effects of both the mean flow and turbulent fluctuations, and the physical causes of the different regions in the Shields diagram, which are shown to depend on changes in the relative magnitude of the appropriate forces. Though no direct curve-fitting is used, our results compare well with available experimental data.

2. Methods and the New Theory

[12] This paper considers a flat surface covered by uniformly sized erodible particles. The x axis lies in the plane of the bed surface, its direction parallel to the direction of flow, while the y axis is perpendicular to the bed and directed upward. The height origin (y = 0) is taken to be the aerodynamic height origin, defined as the level of effective drag or zero-plane displacement (Figure 2). An overbar denotes a temporal average. A superscript + with a velocity variable (either its mean or fluctuation) denotes normalization by the friction velocity u*, as in u+ = u/u*; a similar superscript with a length variable denotes normalization by the viscous length scale ν/u*, as in y+ = u*y/ν. Also, subscripts a and w denote the air and water cases, respectively.

Figure 2.

Schematic of an erodible particle resting on other similar particles. Forces acting on the particle sitting on the top include the aero/hydrodynamic drag, FD, the aero/hydrodynamic lift FL, the gravitational force, FG, and the net cohesive force, FC. The vertical distribution of stream-wise mean velocity equation image is also shown.

2.1. Forces Involved in Particle Entrainment

[13] Analysis of the particle threshold condition begins by defining a force balance on a static grain sitting on the bed surface. A particle resting on the bed (Figure 2) is subjected to forces of drag FD, lift FL, specific weight FG and a net cohesion force FC. We assume that FC is approximately equal in magnitude to the cohesion force between two adjacent individual particles, but follows the direction of FG. The forces other than FC can be expressed as

equation image
equation image
equation image

where uΔ = equation image + u′ is a streamwise reference instantaneous velocity at the threshold of grain motion, and other symbols are defined in Table 1. The frontal area S is approximately equal to 0.2πd2 for the packing geometry shown in Figure 2.

[14] Choosing the reference height yΔ where uΔ is defined can be difficult. Wu and Lin [2002] and Wu and Chou [2003] effectively calculated yΔ by integrating both equation image and u′ over the frontal area of the about-to-move particle exposed to the flow. Such an approach is difficult to justify in fluid mechanical terms, for several reasons. First, one cannot define the instantaneous velocity at a solid surface as zero, because of the no slip condition. Second, at any height within the roughness elements, the velocity field is complex and three-dimensional because fluid has to find its way around the roughness elements. Third, the drag on a surface-mounted roughness element does not satisfy a simple momentum integral constraint equivalent to the relationship for a wake of an obstacle in a free stream [Batchelor, 1967; Raupach, 1992], because of the absence of a free-stream velocity. Finally, an individual roughness element is exposed to a turbulent flow including not only a mean shear but also strong turbulence with contributions both from the large-scale boundary layer and also the wakes of other roughness elements.

[15] To handle these complexities, it is usual to define uΔ as the velocity at a reference height yΔ, horizontally averaged over an area large enough to smooth out spatial fluctuations caused by the bed roughness (in practice a horizontal distance of order 10d). This quantity is analogous to the free stream velocity for the drag coefficient of a body in a uniform flow. In the case of incipient motion, the logical choice for yΔ is the mean height of the most exposed roughness elements, i.e., yΔ = βd, where β is a constant. For instantaneous velocity, the likely choice for β is between 0.5 and 1.0, since the most exposed elements rest some way above the mean position of all bed elements. The level of effective drag or zero-plane displacement (our height origin y = 0) is approximately 0.5d to 0.7d above the bed substrate level [Jackson, 1981; Bridge and Bennett, 1992]. Cheng and Chiew [1998] suggested β = 0.6 for the most stable bed packing situation where particles of identical size rest in an interstice formed by closely packed bed surface particles. In section 3.1, the effects of β on the values of A will be discussed.

[16] The effects of cohesion forces on incipient motion and the dependence of cohesion forces on particle size are complex and poorly understood [Zimon, 1982; Shao and Lu, 2000; Cornelis and Gabriels, 2004]. Cohesion forces arise from both mutual attraction between solid particles, and interaction between the solid particles and those of the ambient fluid medium. The cohesion force is affected by the combination of molecular forces (including the van der Vaal's forces), Coulomb forces, electrostatic forces, capillary forces, and chemical bonding forces. Although these forces are functions of particle size d, their dependences on d are different, with molecular forces and electrostatic forces proportional to d, Coulomb forces proportional to 1/d2, the capillary force proportional to d(1 − dx−1), where x is the width of a microscopic water bridge in the contact zone of two particles [Zimon, 1982]. For instance, the cohesion forces between two identical spheres can be expressed as

equation image

where ci is the cohesion coefficient and zi is the smallest separation between two spherical particles [Theodoor and Overbeek, 1985]. Equation (6) indicates two possible relationships between FC and d: if zi is proportional to d, equation (6) gives FC proportional to 1/d (assuming no dependence of ci on d), while if zi is a constant, FC is proportional to d.

[17] Observations in air have resulted in various relationships between FC and d: direct, inverse and exponential dependence of the cohesion force on d, or even complete independence over a certain range of d [Zimon, 1982]. In water, the dependence of cohesion forces on particle size is simpler and most experiments suggest that FC is proportional to d [Corn, 1961; Zimon, 1982]. This is because in water, capillary and electrostatic forces do not apply as any charges on the particles will leak away and no capillary bridge develops within the particle contact zone [Theodoor and Overbeek, 1985]. This results in the cohesion force in a liquid medium being governed by molecular forces, which implies direct proportionality to d [Fuks, 1955]. Nevertheless, in the case of threshold conditions over a loosely packed bed with a small removal probability of grains (approximately 2%), direct proportionality between FC and d may hold in both water and air media as

equation image

where CC is a proportionality constant, according to Fuks [1955] and Zimon [1982].

[18] At the instant of particle motion, the combined retarding force moments must just balance the combined hydrodynamic driving force moment. Such a condition is expressed as

equation image

where LD = dcosθ, LL = LG = LC = dsinθ are the moment arms (about the pivot point P) of FD, FL, FG and FC (as shown in Figure 2). Following Wu and Lin [2002], the pivoting angle θ ranges from 30° when the grain is at its most exposed, to 90° when it is embedded within the surface grain layer.

[19] Combining equations (3)(5) and (7) with equation (8), the threshold condition can be written as

equation image

where

equation image

is the threshold velocity. Equation (9) simply states that the instantaneous reference velocity uΔ must exceed ut for particle detachment to occur. If we assume uΔ obeys a certain probability distribution, ut is therefore defined by a point on the velocity probability distribution of uΔ at the reference height y from the bed where uΔ is defined, and the probability of particle detachment from the bed is p = Prob(uΔut), as shown in Figure 3. The derivation of A based on a lognormal distribution of uΔ for a given probability removal p is given in section 2.4.

Figure 3.

Schematic diagram of probability of entrainment p equal to probability of exceedance Prob(uΔut), where uΔ and ut are the instantaneous reference velocity and the threshold velocity, respectively.

[20] We turn now to the values of CD and CL. Compared with the well-studied situation of an isolated object in a free stream, where the drag coefficient is a function of Reynolds number (Re* = u*d/ν) and is proportional to 1/Re* for small Re*, the drag coefficient for a particle resting on a surface is more complex and less well understood [Fischer et al., 2002; Jiménez, 2004]. It has been found that the drag coefficient CD ≈ 0.15 − 0.30 for a sphere resting on a surface [Tillman, 1944], but may be up to 1.25 for two-dimensional span-wise obstacles [Jiménez, 2004]. Chepil [1958] provided one of the most comprehensive data sets on the lift coefficient CL in air, and showed that the average ratio of lift to drag is nearly constant at 0.85 for a boundary layer friction Reynolds number of Reτ ≤ 5000, where Reτ = u* δ/ν and δ is the height of the boundary layer. Mainly for channel flows, James [1990] proposed

equation image

Equation (11) suggests that CL could be negative at Re* < 15. Marsh et al. [2004] derived a similar expression to equation (11) but with CL/CD ≈ 0.2 for Re* > 100. The data of Patnaik et al. [1994] for gravel bed rivers showed CL/CD ≈ 1, and an apparent decreasing trend of the lift-to-drag ratio with increasing Re* in the range 4,000 to 60,000. However, this has limited relevance to this study, where we focus on Re* in the range from 1 to 10,000. This is because aeolian transport involves a narrower range of particle sizes than in rivers, and in air, there is little reliable data on threshold velocities for Re*t > 100. Furthermore, the exact values for temporally averaged drag and lift coefficients are unknown at present, as they both depend upon flow conditions. Though the dependence of CD and CL on Re* is likely to have some effects on A, especially for small Re*, for simplicity, values of CD = 0.5 and CL = 0.3 are used in this study. According to Coleman [1967], for a 3D sphere and θ = 30°, LD = dequation image/2, and LL = LG = LC = d/2. These values are used here.

2.2. Mean Velocity Profile Over Rough Surfaces

[21] The classic theory of near-wall turbulent flow defines the universal logarithmic velocity profile for a smooth wall,

equation image

where equation image is the mean flow velocity at height y, κ is von Karman's constant (0.41), and B is an empirical constant with a value of 5.0 [Clauser, 1956; Panton, 2005]. The effect of roughness on the logarithmic velocity profile is a downward shift of equation (12), corresponding to the increase in skin friction [Raupach et al., 1991]. The logarithmic law for a rough surface can then be written as

equation image

There are three regimes in turbulent flow: smooth, transitionally rough and fully rough flow [Raupach et al., 1991]. Ligrani and Moffat [1986] suggested that these three flow regimes can be classified by ks+ < 2.25 when the flow is smooth, and ks+ > 90 when it is fully rough, where ks+ = u*ks/ν is the roughness Reynolds number, and ks is the equivalent sand grain roughness.

[22] The roughness function Δequation image+ in equation (13) has been measured experimentally by Nikuradse [1933] using sand roughness of different grain sizes, and in many other experiments over other kinds of rough surface; see Raupach et al. [1991] for a review. For a uniform sand bed, ks is equal to the diameter of the particles. Ligrani and Moffat [1986] used Nikuradse's data to obtain

equation image

in which the interpolation function

equation image

increases from 0 to 1 through the transitionally rough regime, 2.25 ≤ ks+ < 90. Equation (14) recovers the smooth wall log law in the smooth regime, and suggests that, in the transitionally rough regime, the roughness function Δequation image+ is near zero for ks+ below approximately 5.

[23] We note in passing that the rough wall law, equation (13), is commonly represented as

equation image

where z0 is the zero-velocity level of the logarithmic profile. The dimensionless quantities z0/ks, ξ and Δequation image+ carry equivalent information [Raupach et al., 1991]. Substituting equations (13) and (14) into equation (16) and solving for z0, we have

equation image

For fully rough flows, ξ = 1 (equation (15)). Substituting this into equations (13) and (17) gives

equation image

and

equation image

respectively. Equations (18) and (19) are consistent with common assumptions in the hydraulics literature and the “z0d/30 rule” that is commonly known to wind tunnel experimentalists (assuming d = ks for well sorted sand). In terms of threshold entrainment, all three flow regimes are relevant, depending on particle size and flow conditions. In this study, we shall compare the dimensionless shear stress A resulting from equations (13) and (14) with those derived from the fully rough limit, equations (16) and (19).

[24] Some researchers argue that equation image+ depends not only on y+ but also on the flow Reynolds number [Barenblatt and Chorin, 1998]. However, in comparison to that of roughness, such effects are relatively minor in the inner part of the mean velocity profile [Panton, 2005] and are not fully understood when roughness elements are present [Bergstrom et al., 2001].

2.3. Reynolds Number Dependence of Near-Bed Turbulent Velocity

[25] Like the mean velocity profile, the turbulence in air and in water is governed by universal dimensionless scaling laws. However, a major distinction between turbulent airflow and turbulent river flow lies in the flow Reynolds number Reτ. Under the conditions of incipient motion, the difference is mainly due to the intrinsic length scale δ, as the difference in viscous length scale u*/ν of the two media is relatively small. At the threshold condition, the boundary layer depth in a typical wind tunnel is of the order of 1 meter or more [Greeley and Iversen, 1985], whereas it is of the order of a few centimeters in laboratory flumes [Graf, 1971; Yalin, 1972]. The Reynolds number in the atmospheric boundary layer is about 100–1000 times higher than that observed in natural rivers, whereas the sublayer thickness remains nearly identical [Metzger et al., 2001]. We shall show that these differences are one of the reasons for the systematically smaller values of the dimensionless shear stress A observed in air relative to water (Figure 1).

[26] The turbulent velocity (here considered to be the standard deviation of the streamwise velocity, σ = equation image) is a strong function of the Reynolds number [DeGraaff and Eaton, 2000; Metzger et al., 2001; Marusic and Kunkel, 2003]. As shown in Figure 4, there is a clear dependence of σ+ = σ/u* on the flow Reynolds number Reτ across all values of the inner variable y+. Marusic and Kunkel [2003] proposed a scaling formulation to account for the full range of turbulence intensity σ+ in relation to y+ and Reτ, based upon the attached eddy hypothesis and the idea that the attached eddy motions in the log region and beyond impose a forcing on the viscous buffer zone and sublayer:

equation image

where α = equation image, b1 = 0.008, b2 = 0.115, b3 = 1.6, B1 = 2.39, B2 = 1.03, and B3 = 5.58. For 30 ≤ y+ ≤150, interpolation is needed. For simplicity, linear interpolation is used in this study. In essence, equation (20) describes the behavior of σ+ shown in Figure 4, i.e., σ+ increases with Reτ and peaks at y+ ≈ 15 for a given value of Reτ. This maximum value of σ+ can also be expressed as a function of Reτ

equation image

The σmax+ ∼ Reτ relationship described by equation (21) is plotted in Figure 5. Typical value ranges of Reτ for flumes, natural rivers, wind tunnels and atmospheric boundary layers are also shown. This shows that σmax+ ≈ 2.5 for open channel flow (assuming d = 1 mm, δ = 0.015 m and u*t = 0.03 m s−1), and σmax+ ≈ 3.5 for airflow in a wind tunnel (assuming d = 1 mm, δ = 1.2 m and u*t = 0.5 m s−1). The former value of σmax+ is consistent with those commonly used in the hydraulic literature [Cheng and Chiew, 1998; Wu and Lin, 2002].

Figure 4.

Normalized stream-wise turbulence intensities σ + as a function of normalized vertical height y+ and boundary layer friction Reynolds number Reτ. The lines are calculated using equation (20).

Figure 5.

Peak values of σ+ from Figure 4 as a function of Reτ. Typical ranges of Reτ for flumes, natural rivers, wind tunnels and atmospheric boundary layers are also shown.

[27] Though equations (20) and (21) were developed for smooth surfaces, recent studies show that roughness enhances the turbulence and the Reynolds shear stress over most of the boundary layer, and promotes isotropy as a result of mixing caused by the wakes generated by the roughness elements [Raupach et al., 1996; Krogstad and Antonia, 1999; Tachie et al., 2004]. However, for a sand surface, the increment of turbulence intensity is quite small for the streamwise component [Tachie at al., 2004], in contrast with surfaces of higher roughness such as vegetation. Therefore equations (20) and (21) are used in our analysis.

2.4. Entrainment Probability in a Lognormal Distribution of Instantaneous Velocities

[28] The near-bed turbulence is intense and dominated by gust-like eddy motions with length scales determined by the characteristic length scale of the roughness, ks or d [Raupach et al., 1991, 1996]. These gusts cause the streamwise velocity to show significant departure from a normal velocity distribution [Morrison et al., 2004], with strong positive skewness near the bed. It is reasonable to describe the highly positively skewed distribution for the instantaneous velocity uΔ as lognormal [Wu and Lin, 2002] (Figure 3). Thus we assume that if υΔ denotes the logarithm of uΔ(i.e., υΔ = ln uΔ), the probability density function fΔ) obeys a normal distribution:

equation image

where equation image and συ are the mean and standard deviation of υΔ, respectively.

[29] The probability of entrainment (p) can be expressed as

equation image

where υt = ln ut and ut is the threshold velocity determined by equation (10). Physically, p can be interpreted as the fraction of the time over which the instantaneous reference velocity uΔ must exceed the threshold velocity ut for detachment to occur. As will be shown later, p emerges as a critical parameter in estimating A. Using an approximation for the error function [Cheng and Chiew, 1998], we have

equation image

so that

equation image

where 0 < p < 1. The mean equation image and variance συ2 of υΔ can be estimated by first-order approximation using a Taylor series expansion, and then related to the mean and variance of uΔ, giving

equation image
equation image

Combining equations (24), (26) and (27), we have

equation image

Therefore substituting equation (10) into equation (28) and solving for the threshold friction velocity u*t by setting u* = u*t, we have

equation image

Substituting equation (29) into A = ρfu*t2/(ρp − ρf)gd, we have

equation image

where

equation image
equation image
equation image

and

equation image

All flow properties in equations (28)(34) are evaluated at the dimensionless reference height (yΔ+). In particular, from equations (13) and (14), equation image+ is a function of yΔ+ and roughness Reynolds number ks+, and from equation (20), σ+ is a function of yΔ+ and the flow Reynolds number Reτ. However, at the threshold of grain motion, both yΔ+ and ks+ are directly proportional to the threshold particle Reynolds number Re*t. Hence equation (30) replaces equations (1) and (2) with the more general formulation

equation image

Equation (35) states that the dimensionless threshold shear stress A is governed by four types of effect: (1) particle packing geometry, represented by the parameter fp; (2) the mean velocity profile (first-order flow effects), represented by F1(Re*t); (3) interaction between the mean velocity profile, turbulent fluctuations and particle removal probability p (second-order flow effects), represented by F2(Re*t, Reτ); and (4) interparticle cohesion, represented by G(d).

[30] Ignoring the turbulent fluctuations (i.e., σ+ = 0 and therefore F2 = 1) and interparticle cohesion (i.e., CC → 0 and therefore G(d) → 1), we have AfpF1. In this case, F1 can be interpreted as a first-order effect of the flow that depends solely on the mean velocity profile equation image+, which, in turn, depends only on Re*t. For similar reasons, F2 can be viewed as a second-order effect of the flow as it depends on the turbulent fluctuation σ+. If we assume A1 = fp (see equation (2)), the main contribution of our new expression (35) is to introduce F2, and further specify F1 and G(d). Note that the particle packing parameter fp depends on the drag and lift coefficients CD and CL, which both depend on the choice of reference height y and Re*t. Such dependences could be important in determining A and its variations. Certain aspects of the statistical effects of particle packing geometry have been discussed by Wu and Chou [2003] and Papanicolaou et al. [2002] and further research is needed to understand the effects of CD and CL in relation to Re*t.

[31] In this paper we focus on the effects of turbulent fluctuations on A. The statistical effect of the packing condition will not be considered, and fp is treated as a constant, assuming a fixed packing geometry (Figure 2). Using parameter values defined in section 2.1, we have fp ≈ 1 to 2. For simplicity, an average value of 1.5 is used throughout. In reality, F1 and F2 may also depend on particle packing conditions, but such effects are also excluded from this paper.

[32] Several expressions have been proposed for the effect of interparticle cohesion in the past. In general, they share a form similar to

equation image

where K1 and n are parameters which need to be calibrated against data. With n = 1 and K1 = equation image, equation (36) reduces to equation (34) and stands for the assumption of direct proportionality between the cohesion force FC and particle size d. Values of the parameters n and K1 from previous researchers are listed in Table 2. The resulting expressions for G(d) are plotted in Figure 6. This shows that interparticle cohesion becomes negligible when d > 100 μm (in air). However, cohesion is important in determining the upward trend in A with decreasing Re*t, at Re*t < 1 (Figure 1).

Figure 6.

Previously published relationships of G(d) as a function of particle size d.

Table 2. Parameter Values for and Implied Values of the Cohesion Parameter CC
K1CCNResearchers
6 × 10−7 N m−0.53.1 × 10−7 N m−0.51.5Greeley and Iversen [1985]
0.055a0.0291Iversen et al. [1987]
1.65 × 10−4 to 5.0 × 10−4 N m−18.6 × 10−5 to 2.6 × 10−4 N m−11Shao and Lu [2000]
1.69 × 10−4 to 1.77 × 10−4 N m−18.8 × 10−5 to 9.3 × 10−5 N m−11Cornelis and Gabriels [2004]
7 × 10−6 N m−1.33.7 × 10−6 N m−1.31.3Cornelis and Gabriels [2004]

[33] From Table 2 and Figure 6, it can be inferred that it is reasonable (given earlier work) to assume FC proportional to d, and that the most likely range of values for the cohesion coefficient CC is between 10−6 and 10−4 N m−1. Though the parameter values listed in Table 2 were all calibrated using the wind tunnel data of Iversen and White [1982], they are consistent with values of CC found in adhesion studies. For instance, CC was found to be of the order of 10−4 ∼10−2 N m−1 in air [Corn, 1961] and 10−6 ∼10−4 N m−1 in water [Zimon, 1982]. Accordingly, the estimates provided in Table 2 may represent the lower bounds of such effects in air.

[34] Zimon [1982] attributed the different relationships between FC and particle size d, resulting in a range of values of FC for a given d, to the probabilistic nature of particle removal. The variance of FC is larger for smaller particles, and may also be larger in air relative to water. For instance, for particle sizes in the range 10 to 20 μm in air, FC at a particle removal probability of 50% can be 102 to 103 times larger than at a small removal probability (2% to 5%), while for particles smaller than 10 μm, the difference can be as large as 105 [see Zimon, 1982, Figure 1.2]. Therefore, for small particles, the effect of the cohesion force on the entrainment threshold is not clear, as a large uncertainty and high variance in A are expected in both experimental data and theoretical estimates for particle sizes smaller than 10 μm.

3. Results and Discussion

[35] In this section we focus on the first- and second-order flow-related effects, F1 and F2, mainly considering the range Re*t > 10 where G(d) → 1 (i.e., medium to larger particles). In section 3.3, we compare our results with experimental data under a variety of different removal probabilities (p) and interparticle cohesion parameter values (CC). For all calculations, kinematic viscosity ν is set to 1.47 × 10–5 m2 s–1 in air and 1.0 × 10–6 m2 s–1 in water, particle density ρp to 2650 kg m–3, and fluid density ρf to 1 kg m–3 in air and 1000 kg m–3 in water.

3.1. First-Order Effects: Effects of the Mean Velocity Profile

[36] Figure 7 shows contour plots of F1 in ks+yΔ+ space (dashed lines), with equations accounting for all three flow regimes (equations (13), (14) and (15)) in Figure 7a, and for the fully rough regime only (equations (16) and (19)) in Figure 7b. In general, these diagrams show that contour lines of F1 are parallel to lines ks+ = ζyΔ+, where ζ is a constant. F1 reaches a maximum value of round 0.2 at ζ = 30. When ζ < 30, F1 decreases with y+ but increases with the roughness Reynolds number ks+, whereas when ζ > 30, F1 increases with y+ but decreases with ks+. In the region where ζ < 30, the opposite effects of ks+ and yΔ+ on F1 (and therefore on the dimensionless shear stress A) are the main reason that values of A are confined to a rather narrow range in both air and water. The region where ζ > 30 may have little to do with threshold entrainment. These points should become increasingly clear later in this section.

Figure 7.

Contour plots of F1 in y+ks+ space. (a) Equations (13)(15) were used for the calculation; (b) equation (16) with z0/ks = 0.031 was used for the calculation. Note that y+ is replaced by yΔ+ for the threshold condition of grain motion. See text for details.

[37] The differences in F1 between Figures 7a (using all three flow regimes) and Figure 7b (using only the fully rough regime) are most evident when ks+ ≤ 70, where the flow is either smooth or transitionally rough. This is consistent with the findings of Ligrani and Moffat [1986] and Raupach et al. [1991]. In the smooth flow regime, values of F1 are larger if the flow is treated as smooth rather than as fully rough. However, in the transitionally rough regime, values of F1 are smaller if the flow is treated as transitionally rough rather as fully rough. Therefore A can be underestimated for small particles but overestimated for medium particles if fully rough flow is assumed for all particle sizes. This implies that underestimated/overestimated values of A can arise as a result of the assumption of fully rough flow, and explains why the experimental Shields' curve (Figure 1) dips in the vicinity of Re*t ∼ 10 to 30. This is most clearly seen in the data for water. Iversen et al. [1987] stated there is no indication of such a “dip” in A with Re*t for their own and others' data in air. This may be because fully rough flow is assumed for all atmospheric boundary layer studies [Jiménez, 2004], including the wind tunnel studies of Iversen and his colleagues.

[38] In terms of threshold entrainment, the ratio between the equivalent sand roughness ks and the particle diameter d, namely α = ks/d, has been discussed by Ling [1995], who indicated that α may be interpreted as a measure of the packing density of roughness elements. For threshold conditions, α = 0.3 to 4 [Ling, 1995; Cheng and Chiew, 1998], implying that ks+ = αRe*t. Commonly used values of α are 1 for uniformly sized particles and 2 for mixed particle sizes [Wu and Lin, 2002; Wu and Chou, 2003]. On the other hand, according to section 2.1, y = βd, implying yΔ+ = βRe*t, where β lies between 0.5 and 1. We then have ks+ = ζyΔ+, where ζ = α/β varies approximately between 0.15 and 4. Larger values of ζ are possible for other types of packing geometry, as the packing geometry shown in Figure 2 represents the most exposed position for a sphere resting on a surface. Nevertheless, the contour lines outside of ζ = 0.1 and ζ = 10 may have little relevance for threshold entrainment.

[39] Some typical cases of threshold entrainment are shown in Figure 7. They are ζ = 0.5, 1, 2 and 4, respectively (solid lines). The upper bound of particle size common in gravelly fluvial systems is a grain size of about 100 mm [Graf, 1971; Buffington and Montgomery, 1997], whereas it is two orders of magnitude smaller in aeolian systems [Bagnold, 1941; Greeley and Iversen, 1985]. Thus the approximate upper bounds of ks+ and yΔ+ are of the order of 104 for fluvial systems (the lightly shaded area), and approximately 102 for aeolian systems (dark shading), respectively. This shows that, in air, threshold particle motion is likely to occur within the transitionally rough and smooth regimes, rather than in the fully rough regime. It also shows that, for ζ = 0.5 ∼ 4, the values of F1 are mainly limited to 0.01 ∼ 0.05. Multiplying these values by fp = 1.5 gives values of A as expected for fluvial entrainment [Graf, 1971; Qian and Wan, 1983; Buffington and Montgomery, 1997], but slightly larger than those obtained in wind tunnels [Bagnold, 1941; Iversen and White, 1982]. The physical cause of such differences will be investigated further in section 3.2.

[40] In comparison to uniformly sized beds, α tends to become larger and β tends to become smaller for mixed particle beds because of the hiding effect. As has been assumed previously by other researchers, for a given average bed particle size, ζ = 0.5 ∼ 1 represents more uniformly sized beds, and a larger ζ represents mixed-sized surfaces [Wu and Lin, 2002; Wu and Chou, 2003]. Figure 7 provides a theoretical explanation for the observation that A for mixed particle sizes is often larger than that for uniformly sized particles [Shields, 1936; Buffington and Montgomery, 1997]. In addition, as particle sorting processes cause both α and β to vary spatially and temporally, they inevitably increase the variations of A [Church, 1978; Andrews, 1983]. This explains why the “dip” in A disappears in data obtained from mixed-sized river beds.

3.2. Second-Order Effects: Effects of Boundary Layer Friction Reynolds Number Reτ

[41] The mean streamwise velocity is not the only factor affecting threshold entrainment. The turbulent velocity is also responsible for the initial dislodgement of sediment. Equation (33) provides us the means to investigate the second-order effects on A due to turbulent fluctuations, through the effects of bulk Reynolds number and its relation to particle removal probability. In order to understand the general effect of turbulent fluctuations on A, and for simplicity, σ+ = σmax+ is assumed at the height of the mean threshold velocity equation image and is applied to all particle sizes. Equation (21) is used to relate the maximum turbulent fluctuation σmax+ to Reτ.

[42] Figure 8 shows contour plots of F2 in σ+p space with F1 set to 0.01, 0.03 and 0.06 (Figure 8, left, middle, and right, respectively). For all three cases, F2 increases with removal probability p. This shows that, for any given Reτ, F2 has values smaller than 1 for 0 < p < 0.3 and larger than 1 for 0.3 < p < 1. Thus F2 acts to increase the variation of A. If we assume that A = fpF1 (as analyzed in section 3.1) represents the mean, multiplying this quantity by F2 reduces the actual value of A when 0 < p < 0.3 but enlarges it when 0.3 < p < 1.

Figure 8.

Contour plot of F2 in Reτp space. (left) F1 = 0.01, (middle) F1 = 0.03, and (right) F1 = 0.06.

[43] Figure 8 also shows that F2 increases gradually with Reτ when 0 < p < 0.3, and that this increase is more noticeable for smaller values of F1 than for larger ones. Physically, we would expect this to happen as the larger the turbulent fluctuations, the easier the initial dislodgement of sediment and therefore the greater the amount of reduction in A in comparison to its mean. We argue that this represents another reason for smaller values of A in air than those obtained in water (Figure 1). According to Figure 5, at the threshold, we may assume that Reτ,w = 500 in flumes and Reτ,a = 50000 in wind tunnels respectively. Figure 9 shows the estimated values of F2,w, F2,a and their ratio η = F2,w/F2,a in relation to p, with F1,w = F1,a = 0.01 and 0.04, respectively. The slope of the F2p relationship is larger in air than that is in water. The two F2p relationships intersect at approximately p = 0.2 ∼ 0.3, and F2 in air is about half of its value in water for p = 0.01 ∼ 0.05. The phenomenon that particle entrainment in air is easier for a smaller removal probability, but progressively harder for a larger removal probability for the same particle size entrained by water has also been found experimentally by Zimon [1982]. However, the explanation provided by Zimon [1982] was based upon the probabilistic nature of particle cohesion rather than statistical properties of near-wall turbulence, as argued here.

Figure 9.

(left) F2 as a function of p for the cases of air and water for (top) F1 = 0.04 and (bottom) F1 = 0.01. (right) Ratio of F2 in water and air for the same F1 values.

[44] Figure 1 suggests that the difference in A between air and water is actually larger than that predicted in Figure 9. This can be attributed to the following reasons. First, it is because of different definitions of particle removal probability at the threshold. Particle removal probabilities at the threshold are never clearly defined under experimental conditions. For instance, those values of A lower than 0.04 were obtained by visual observations, in comparison to those obtained using bed load transport rates [Buffington and Montgomery, 1997]. Second, it is shown in section 3.1 that the average F1 may be larger in water than in air. In fact, data reported in Figure 1 are mostly averaged from different experiments with similar conditions. If we assume F1,w = 2F1,a, fp,w = fp,w and Ga(d) = Gw(d) = 1 (applicable for medium to large particles), then Aw would be approximately 3 to 4 times larger than Aa at p = 0.02. Such a ratio is close to what is seen in Figure 1.

[45] Previous analyses in hydraulic literature treat the instantaneous pickup velocity ut and its fluctuation σ as linearly proportionality to the friction velocity u*, i.e., ut = Ctu* and σ = Cσu*, where Ct and Cσ are constants with values of approximately 5.5 and 2.0, respectively [Kironoto and Graf, 1994; Cheng and Chiew, 1998; Wu and Lin, 2002]. Also, fully rough turbulent flow is assumed with no allowance for variations in σ. A constant value of σ fails to predict the systematically lower values of A at large Re*t observed in air, in comparison to those observed in open channel flow. In this study, σ is allowed to vary with bulk Reynolds number as expressed by equations (20) and (21). Therefore linking incipient motion to the scaling of turbulent fluctuation σ is the key contribution of this study as this provides us with a fundamental basis to explain the systematically lower values of A in air than in water.

[46] The smaller values of A for smaller removal probabilities in air (and larger values for larger probabilities) suggest that the variance of threshold velocity is larger in air than it is in water. This may have profound implications for understanding wind erosion and aeolian transport. For instance, wind tunnel studies of Shao et al. [1993] found that dust emission is mainly due to saltation bombardment, and that dust emission arising from direct aerodynamic entrainment is generally small. On the other hand, direct field observations of Loosmore and Hunt [2000] found dust suspension can be initialized by turbulent eddies in surface winds well before the sandblasting mechanism. Similar observations were made by Roney and White [2004] at Owens Lake. Both Loosmore and Hunt [2000] and Roney and White [2004] found that the threshold velocity is smaller than the values estimated by the conventional “saltation thresholds.” From Figures 8 and 9, we suggest that the physical causes of the discrepancy are likely to be due to the different definitions of the threshold removal probability. While Shao et al. [1993] looked at sustained dust emission rates at an average p of approximately 30% to 50%, the latter two studies focused on initial uplifting at p = 1 ∼ 10%.

[47] Figures 7 and 8 also show that F1 may also affect F2. At smaller p, the effect of Reτ on A becomes less evident when F1 becomes larger. Also, for a given p, F2 increases with F1. Physically, this means that when A is larger, there is less reduction in its value by the second-order effect (i.e., when multiplying it by F2). Further experimental studies are needed to investigate these findings. In reality, the effects of Reτ and Re*t are more complex than those presented in Figures 7 and 8. This is because A, Reτ and Re*t are all related to threshold velocity u*t, which increases with particle size. In addition, boundary layer height δ may change with particle size.

[48] Direct field measurements by Roney and White [2004] at Owens Lake found that the dust suspension threshold varied between 50 and 75% of the values estimated from expressions derived from wind tunnel experiments. The general model proposed in this study provides the physical explanation for such an observation. Figure 10 shows three possible A ∼ Re*t relationships at boundary layer heights of δ = 0.01 m, 1 m, and 1000 m, respectively, all for p = 0.05. These curves were obtained, for an in-air situation, by substituting equations (13) and (20) in equations (32) and (33) with yΔ+ = 0.6Re*t, ks+ = Re*t and CC = 4 × 10–5 N m–1. The values of A were calculated through iteration over u*t for a given particle size d. This shows that A decreases with δ and the decrement increases when Re*t increases, suggesting that, for the same values of Re*t, direct field measurement would result in smaller A in comparison to that in wind tunnel conditions. The physical implication is that, for a given particle size, the average threshold velocity for dislodging particles could be smaller in the atmospheric boundary layer compared with that observed in wind tunnels. Alternatively, a particle that remains still in a wind tunnel may move if it is in an atmospheric boundary layer at the same average wind velocity.

Figure 10.

Computed A ∼ Re*t relationship in air for three different values of boundary layer height δ.

3.3. Comparison With Experimental Data

[49] In this section, using an approach similar to that employed to derive Figure 10, we shall use equations (13) and (20) to estimate F1 and F2. A is then calculated through iteration over u*t for a given particle size d. For all calculations below, we assume yΔ+ = 0.6Re*t. When comparing predictions with data from flume experiments, we assume ks+ = 3Re*t and δ = 0.02 m, while ks+ = Re*t and δ = 1.2 m for comparing with data from wind tunnels.

[50] Figure 11 shows predicted A ∼ Re*t relationships for four values of p in air and in water and three values of the cohesion coefficient CC, respectively. For a given fluid, the variation of A in relation to Re*t can be explained by differences in the removal probability p when Re*t > 10, by interparticle cohesion when Re*t < 1, and by both of them when Re*t is in between. Figure 11 shows that, with a mean value of CC = 10–5 ∼ 10–4 N m–1, the new expression agrees very well with experimental data for both air and water. The CC values used in this study are smaller than those suggested previously [Shao and Lu, 2000; Cornelis and Gabriels, 2004], but this is because flow-related effects play an increasingly important role in the new expression we propose. As discussed in section 3.1 and shown in Figure 7, at smaller ks+ and yΔ+ (therefore smaller Re*t), F1 (therefore A) increases with Re*t.

Figure 11.

Comparison between computed A ∼ Re*t relationships in air and water. The same experimental data shown in Figure 1 are used for comparison.

[51] Previous researchers have suggested that the particle-to-fluid density ratio Rρ = (ρp- ρf)/ρf is an important parameter in determining A [Iversen et al., 1987; Cornelis and Gabriels, 2004]. Our new expression suggests that A also depends on both Re*t and Reτ, not just on Rρ. Therefore, in terms of the effects of flow on A, not only the density but also the flow viscosity is important. For a set of fixed values of viscosity, particle size, and boundary layer height, our new expression suggests that A increases when Rρ decreases, as was shown by the wind tunnel data of Iversen and White [1982]. However, this would no longer be true if these parameters also vary, according to our generalized model. The experimental simulations of McKenna Neuman [2003, 2004] confirm that A and sediment transport rates depend not only on fluid density (or Rρ) but also on fluid viscosity. She showed both threshold velocity and mass transport rates of sedimentary particles vary with temperature [Selby et al., 1974] and humidity. For instance, it was found that the aerodynamic drag required to entrain sand size particles can be 30% lower under cold conditions in high-latitude regions, as compared to hot deserts and mass transport rates increase while temperature decreases. McKenna Neuman [2004] suggested that temperature-dependent changes in air density and viscosity, and turbulence, are the major affecting factors. This example shows that caution is needed in applying previous analytical and semiempirical models, and in assuming fixed parameter values, as both models and parameters may only be applicable to certain conditions.

[52] Both theoretical analysis and observation show a certain degree of variation in A. We therefore argue that threshold entrainment should be viewed statistically rather than deterministically, as most governing processes are probabilistic in nature, including near-surface turbulent flow, packing geometry [Kirchner et al., 1990; Papanicolaou et al., 2002] and interparticle cohesion. For instance, the most important process – rough wall turbulent motion – is fundamentally stochastic. The structure of near-bed turbulent flow is characterized by the spatial and temporal organization of coherent structures [Best, 1992], the links between outer and inner scales, the nature of intermittency and the role of anisotropy. All these turbulence characteristics depend on and interact with the bed condition in a complex manner. In addition, for small values of Re*t, the interparticle cohesion forces are also statistical in nature, and the relationships between cohesion forces and particle size are very complex and not well understood (see section 2.1). For all these reasons, a deterministic resolution of incipient motion may not be possible. This message is particularly relevant to wind erosion studies where a deterministic view of incipient motion remains common, and data on incipient motion are too sparse to separate the variety of physical processes involved.

4. Conclusions

[53] In this study, we have proposed a new, general expression for the dimensionless threshold shear stress A which is applicable to both aeolian and fluvial particle entrainment. It allows us to study the effects of mean flow, turbulent fluctuation, and interparticle cohesion on incipient motion separately. It suggests that the dimensionless threshold shear stress A is indeed related to Re*t, as earlier researchers proposed, and should also depend on flow Reynolds number Reτ, which is proposed for the first time. Therefore A depends on the flow condition and it should be a function of the viscosity (a flow property) as well as being a function of the density ratio and particle size.

[54] We have shown that the differences in the probability distribution of streamwise velocity fluctuations for typical situations in air and water are the main reason for the larger values of A in water than in air (Figure 1). This is because the turbulent fluctuations are not only related to the near-bed flow structure (i.e., Re*t,), as expected, but also depend on the bulk flow characteristics (i.e., Reτ). As velocity variance increases with the bulk flow Reynolds number Reτ, and typical values of Reτ in air are several orders larger than those in water under the conditions of incipient motion, so the values of A are smaller in air than in water. The upturn of A for small Re*t is partly due to the flow conditions and partly due to interparticle cohesion. Therefore incorporating descriptions of the flow condition into threshold entrainment is indispensable for all particle sizes.

[55] At large Re*t, the narrow range of values of A in relation to Re*t results from the opposite effects of the roughness Reynolds number ks+ and the dimensionless reference height yΔ+, both of which are proportional to the particle Reynolds number Re*t. Although the mean values of A may be within a narrow range, certain systematic variations exist. These variations are due to differences in particle packing density, which is incorporated in the first-order effect F1; and to the different removal probability (p) used in different experiments, and the bulk Reynolds number, which are incorporated in the second-order effect F2. In reality, these effects are likely to act interactively on A and are reflected in a combined but rather random manner in experimental studies.

Acknowledgments

[56] H.L. is grateful to Herbert Huppert at the Institute of Theoretical Physics, University of Cambridge, for his kind support. We would like to thank two anonymous reviewers for their thoughtful comments, which greatly improved this paper.

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