## 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}/(ρ_{p} − ρ_{f})*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*_{*t}*d*/ν, as

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*_{*t} ∝ *d*^{1/2}, 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.

Symbol | Definition | First Use |
---|---|---|

A | dimensionless threshold shear stress, A = ρ_{f}u_{*t}^{2}/((ρ_{p} − ρ_{f})gd) | Equation (1) |

A_{1} | reference value of A | Equation (2) |

B | empirical constant in logarithmic law over a smooth wall (B = 5.0) | Equation (12) |

c_{i} | cohesion coefficient | Equation (6) |

C_{D}, C_{L} | drag, lift coefficients (computations assume C_{D} = 0.5, C_{L} = 0.3) | Equations (4) and (5) |

d | particle diameter | Equation (1) |

F_{D}, F_{L}, F_{G}, F_{C} | forces on particle from drag, lift, specific weight and cohesion | Equations (3)–(5) |

F(Re_{*t}) | dimensionless function quantifying dependence of A on gravitational and aerodynamic forces | Equation (2) |

f_{p} | component of F associated with packing geometry (computations assume f_{p} = 1.5) | Equation (31) |

F_{1} | component of F associated with first-order flow effects | Equation (32) |

F_{2} | component of F associated with second-order flow effects | Equation (33) |

G(d) | dimensionless function quantifying dependence of A on cohesion forces | Equation (2) |

g | gravitational acceleration | Equation (1) |

k_{s} | Nikuradse sand-grain roughness | Equation (14) |

K_{1} | dimensional parameter for cohesion force | Equation (36) |

L_{D}, L_{L}, L_{G}, L_{C} | moment arms for forces F_{D}, F_{L}, F_{G}, F_{C} (L_{D} = dcosθ, L_{L} = dsinθ, L_{G} = dsinθ and L_{C} = dsinθ) | Equation (8) |

n | exponent specifying cohesion force | Equation (36) |

p | probability of entrainment | Equation (23) |

Re_{τ} | flow Reynolds number, Re_{τ} = u_{*} δ/ν | Section 2.3 |

Re_{*t} | particle Reynolds number at threshold, Re_{*t} = u_{*t}d/ν | Equation (1) |

S | horizontal area of particle exposed to the flow (computations assume S = 0.2πd^{2}) | Equations (4) and (5) |

u | streamwise velocity (dimensionless form: u^{+} = u/u_{*}) | Section 2 |

u_{Δ} | reference velocity (at height y_{Δ}) for defining F_{D} and F_{L} | Equations (4) and (5) |

u_{t} | threshold velocity (at height y_{Δ}) | Equation (10) |

u_{*t} | threshold friction velocity | Equation (1) |

υ_{Δ}, υ_{t} | υ_{Δ} = ln u_{Δ}, υ_{t} = ln u_{t} | Equation (22) |

y | height above level of effective drag (y = 0) | Section 2 |

y^{+} | dimensionless height, y^{+} = u_{*}y/ν | Section 2 |

y_{Δ} | height at which velocity u_{Δ} is defined | Equations (4) and (5) |

y_{Δ}^{+} | dimensionless reference height at which velocity u_{Δ} is defined | Section 2 |

z_{0} | roughness length | Equation (16) |

z_{1} | smallest separation between two spherical particles in equation (6) | Equation (6) |

β | constant relating reference height yΔ to d | Section 2.1 |

δ | boundary layer depth | Section 2.3 |

Δ^{+} | velocity-increment form of roughness function | Equation (13) |

κ | von Karman constant (κ = 0.41) | Equation (12) |

ν | kinematic viscosity of the fluid | Equation (1) |

ξ | interpolation function for roughness in transition regime | Equation (15) |

ρ_{f}, ρ_{p} | fluid density, particle density | Equation (1) |

θ | pivoting angle (see Figure 2; computations assume θ = 30°) | Equation (8) |

σ | standard deviation of streamwise velocity (dimensionless form: σ^{+} = σ/u_{*}) | Equation (20) |

τ_{*t} | threshold shear stress, τ_{*t} = ρ_{f}u_{*t}^{2} | Equation (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].

[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_{*t}^{2} 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

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* = *A*_{N }*G*(*d*), where *A*_{N} 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 *d*_{50} [*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.