Theoretical prediction of electric fields in wind-blown sand

Authors

  • TianLi Bo,

    1. Key Laboratory of Mechanics on Western Disaster and Environment, Lanzhou University, Lanzhou, China
    2. Department of Mechanics, Lanzhou University, Lanzhou, China
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  • Huan Zhang,

    1. Key Laboratory of Mechanics on Western Disaster and Environment, Lanzhou University, Lanzhou, China
    2. Department of Mechanics, Lanzhou University, Lanzhou, China
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  • Wei Zhu,

    1. Key Laboratory of Mechanics on Western Disaster and Environment, Lanzhou University, Lanzhou, China
    2. Department of Mechanics, Lanzhou University, Lanzhou, China
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  • XiaoJing Zheng

    Corresponding author
    1. Key Laboratory of Mechanics on Western Disaster and Environment, Lanzhou University, Lanzhou, China
    2. Department of Mechanics, Lanzhou University, Lanzhou, China
    • Corresponding author: X. Zheng, Department of Mechanics, Lanzhou University, Lanzhou, 730000, China. (xjzheng@lzu.edu.cn)

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Abstract

[1] Existing experimental and theoretical studies have identified the existence of an electric field in wind-blown sand, whose strength is hundreds to thousands of times larger than that of the fair-weather electric field (around 120v m− 1) and is related to the intensity of the wind-blown sand transport. The direction of electric field is often upward pointing and opposite to that of the fair-weather electric field. In this study, we performed theoretical predictions of electric fields in wind-blown sand transport by assuming a constant charge-to-mass ratio (i.e., − 60µc kg− 1) of the saltating particles and considering the streamwise spatial variation of particle concentration in the evolution of wind-blown sand. Our results show that there exist both vertical and horizontal electric fields in wind-blown sand transport. The numerical results of vertical electric fields and mass flux are in good agreement with the experimental data measured by Schmidt et al. [1998] and Shao and Raupach [1992], respectively, which suggests that our model is valid. The horizontal electric field demonstrates a vertical stratification feature and is about one order of magnitude bigger than the fair-weather electric field. Finally, the effects of the wind speed and the sand grain diameter on electric fields are discussed.

1 Introduction

[2] It has been generally accepted that the fair-weather atmospheric electric field has a value of around 120v m− 1 and directs vertically downward [Mather and Harrison, 2006]. However, during extreme weather like strong storms, dust devils, volcanic eruptions, and blizzards, the atmospheric electric field displays a severe disturbance caused by charged particles in air. In wind-blown sand transport, the electric field formed by moving charged particles is different from the fair-weather atmospheric electric field and is generally called the wind-blown sand electric field [Kok and Renno, 2008; Zheng et al., 2004]. The existence of electrostatic forces between charged particles and the electric field to a large extent influences the flow characteristics and transportation of sand particles [Murtomaa et al., 2004; Kok and Renno, 2008; Zheng et al., 2004, 2006]. For example, the electrical effects associated with dust storms and blizzards considerably affect the mass flux, the dust emission, and the trajectories of saltating particles [Schmidt et al., 1998, 1999; Zheng et al., 2003; Kok and Renno, 2008]. In addition, it also significantly affects the transmission of electromagnetic microwaves so as to influence the accuracy of experimental apparatus in the desert regions [Greeley and Leach, 1978]. For volcanic ash, the electrostatic force enhances the aggregation of dust particles which then affects the sedimentation processes of volcanic ash [Miura et al., 2002].

[3] Most existing studies have focused their attention on the vertical electric field of dusty phenomena. For example, Schmidt et al. [1998] measured the electric field and charge-to-mass ratio of saltating particles in wind-blown sand transport on a sand dune. In their measurements, the electric field was measured by a direct current electric field which is able to provide accurate measurement in the saltation layer. Their results showed that the vertical electric field was upward-pointing and as large as 166k vm− 1 at 1.7 cm height for a mean wind speed of 12m s− 1 at 1.5 m height. In addition, the vertical electric field decreased with height in a negative exponential way and approached 0 at about 20 cm. Above that, the wind-blown sand electric field changed to downward-pointing and approached −180v m− 1 at 2 m height. Because at the height above 2 m, the electric field induced by the creeping and saltating sand particles is nearly zero; therefore, the resultant electric field only contains the fair-weather electric field. A fitting formula was given for the variation of the vertical electric field with height. For the average charge-to-mass ratio, it was determined by measuring both the charge and mass of saltating sand particles collected in a Faraday cage with a copper tube connecting with the inner cylinder (so as to avoid the influence of additional charges produced by collisions between sand particles and copper tube). Using a Faraday cage mounted at 5 cm height, they measured an average charge-to-mass ratio of 60µc kg− 1 with 150 µm average diameters of saltating particles. Zheng et al. [2003] measured the electrification generated by wind-blown sands through wind tunnel experiments. Their results confirmed the conclusion that the direction of the field is upward. Moreover, they pointed out that the electric fields were primarily produced by the motion of charged particles and whose value increased with wind velocity and height. Subsequently, based on the point charge theory, and by assuming that the number density of sand particles N is a function only related to height z, Zheng et al. [2004] developed a general theoretical model for calculating the electric field and discussed the effect of wind velocity and wind-blown sand flux on the electric field strength.

[4] It should be noted that Kok and Renno [2008] made a comprehensive study of electrical effects in a wind-blown sand system. They first investigated the influence of electric field on the lifting of soil particles. By imposing a vertical electric field on the surface particles, they found that when the electric field exceeded 80kv m− 1, the threshold friction velocity was considerably reduced; for an electric field exceeding 150kv m− 1, sand particles can lift directly from the surface [Kok and Renno, 2006]. Subsequently, they presented a numerical model by considering the collisional charge transfer of saltating particles and found that the existence of electrostatic force enhanced the concentration of saltating particles, and the downward electrostatic force acted an effect of lower saltating particles' trajectories. In their study, the electric field was also assumed to be perpendicular to the surface [Kok and Renno, 2008]. In addition to the electric field in wind-blown sand, measurements in blizzards and volcanic ash also indicated that the electric field was perpendicular to the surface [Schmidt et al., 1998, 1999; Latham and Montagne, 1970; Hatakeyama and Uchikawa, 1952; Anderson et al., 1965; Miura et al., 2002].

[5] Until now, there has been little information about the horizontal electric field in wind-blown sand or dusty phenomena. The only exception refers to the field measurement conducted by Jackson and Farrell [2006] on the horizontal electric field in dust devils. They measured the horizontal electric field with electrometers mounted on the top of a car when it passed through a dust devil. Their results suggested that the maximum horizontal electric field values reached 120kv m− 1. Afterward, they modeled the dust devil in a cylindrical domain and the charges distributed on the cylindrical surface. Their simulation results are in agreement with their measurements. However, the cylinder charge model is difficult to be applied directly in wind-blown sand or other dusty phenomena because the spatial charge distribution in a dust devil is significantly different from the others.

[6] In this study, we present a numerical model that contains both horizontal and vertical electric field in the evolution of wind-blown sand transport. Section 2 gives the evolution of particle concentration in the wind-blown sand. In section 3, the charge density is calculated, and the electric fields are determined by Coulomb's law. The results of electric fields and the influence of wind speed and sand grain diameter on electric fields are given in section 4. Finally, section 5 is a summary of the main conclusions obtained in this study.

2 The Particle Concentration in Wind-Blown Sand Transport

[7] It has been well known that when wind speed exceeds a threshold value, sand particles are lifted by wind and jump along the sand surface. Both experiments and theories confirm that, even if the wind is steady, a considerable distance downwind of the leading edge of an erodible surface is required for the wind-blown sand transport to reach equilibrium. This distance is known as critical fetch distance [Delgado-Fernandez, 2010; Gillette et al., 1996; Dong et al., 2004]. The critical fetch distance is about a few meters in the wind tunnel while possibly over hundreds of meters in the natural environment. For example, Bagold [1941] suggested that the sand transport attains equilibrium at about 7 m. A measurement of streamwise mass flux in wind tunnel by Shao and Raupach [1992] found that the minimum critical fetch distance is approximately 15 m. However, field experiments by Gillette et al. [1996] and Fryrear and Saleh [1996] found that critical fetch distance is up to 150 m. It is possible that the limited vertical extent of wind tunnels restricts the development of internal boundary layer so that the response of the boundary layer flow due to momentum transfer by sand particles is much faster than that of flow in the natural environment [Bauer et al., 2004; Shao, 2000]. In addition, the effects of supply-limiting factors such as moisture, particle size, and surface crusts play an important role in increasing critical fetch distance in the natural environment [Delgado-Fernandez, 2010].

[8] As discussed above, the mass flux or number density of sand particles in wind-blown sand transport asymptotically reaches equilibrium value with the increase of streamwise distance. In this study, we model the evolution of saltation by three main processes [Anderson and Haff, 1991]: (i) the motion of sand particles, (ii) the coupling interaction between the sand particles and the wind flow, and (iii) the collisions between the saltating sand particles and bed surface, namely, as a saltating particle impacts on the surface, it may rebound itself and eject more particles into air.

[9] We use the large eddy simulation (LES) approach to simulate wind flows with saltation. In LES, the flow variables are decomposed as the sum of resolved (large-scale) and unresolved (small-scale) variables by a spatial filtering operation. The LES filtered continuity equation and LES filtered momentum equation with saltation [Ma and Zheng, 2011; Shao and Li, 1999] are shown as follows:

display math(1)
display math(2)

where i = 1 and 2 correspond to the streamwise and wall-normal directions (i.e., x1 = x, x2 = z, u1 = u, u2 = w) respectively; inline image and inline image represent the filtered wind speed and pressure; υ is the kinematic viscosity; ρ = 1.225kg ⋅ m− 3 is the air density; inline image is the porosity of the air flow, in which n is the number of sand particles in a unit volume and D is the sand grain diameter; δij = 1 if i = j, otherwise δij = 0; g is the gravitational acceleration; Fi is the reactive force per unit volume of sand exerted on wind; inline image are the subgrid scale (SGS) stresses, and they are modeled by means of a SGS turbulence model [Smagorinsky, 1963]

display math(3)

[10] The SGS viscosity is μSGS calculated as follows

display math(4)

where inline image and CSGS = 0.19 - 0.24 which require case-by-case adjustment [Versteeg and Malalasekera, 2007].

[11] As shown in Figure 1, the computational domain, which is meshed by 3000 × 300 grids, is 22 m long and 2 m high (denoted by L × H) in our simulation. In this study, the calculation of wind velocity field is based on the finite volume method. In particular, we use the QUICK differencing scheme by Hayase et al. [1992], the fully time implicit scheme, and the transient PISO algorithm [Versteeg and Malalasekera, 2007]. Since the typical time of sand transport saturation and sand particle saltation are approximately 2 and 0.05 s, respectively [Ma and Zheng, 2011], the integration time step for wind flow Δt and sand particles trajectories Δtp are 0.005 and 0.0001 s, respectively.

Figure 1.

Schematic diagram of the wind-blown sand transport simulation.

[12] The initial conditions of LES governing equations are

display math(5)
display math(6)

[13] Here z0 = D/30 is the aerodynamic surface roughness [Kok and Renno, 2009], and u* is the friction velocity of inflow.

[14] The boundary conditions of LES governing equations are

display math(7)
display math(8)
display math(9)
display math(10)

[15] As the previous studies [Kok and Renno, 2009; Ma and Zheng, 2011], we model the sand movement in two dimensions. The effect of mid-air collisions on particle trajectories is neglected since their effect remains poorly understood [Dong et al., 2005; Huang et al., 2007; Kok and Renno, 2008; Sorensen and McEwan, 1996]. With very few exceptions [Carneiro et al., 2011; Ren and Huang, 2010;], previous analytical and numerical studies have also neglected mid-air collisions [Andreotti, 2004; Creyssels et al., 2009; Kok and Renno, 2009; Pahtz et al., 2012]. The trajectory is mainly influenced by gravity, the aerodynamic drag and electrostatic force since they are much greater than the Magnus force and the Saffman force [Murphy and Hooshiari, 1982]. The motion of saltating particles can be described by

display math(11)
display math(12)

where q, xp,i, and up,i are the particle's charge, position, and velocity, respectively; g is the gravitational acceleration; Cd = [(32/Rep)2/3 + 1]3/2 is the drag coefficient [Cheng, 1997]; and Rep = VfρpD|UR|/μ is the particle Reynolds number [Kok and Renno, 2009], where ρp = 2650kg ⋅ m− 3 is the air density; inline image is the volume fraction of wind flow that is the sum of sand volume within one grid to the volume of flow within the same grid; ΔV is the control volume size; μ = 1.8 × 10− 5 is the air kinetic viscosity coefficient; inline image is the relative velocity of the particle with respect to air and inline image; and Ex and Ez are horizontal and vertical electric field, respectively, and are given in part 3.

[16] Detailed descriptions of our calculation procedure are as follows:

  1. [17] According to the initial and boundary conditions, the wind velocity field inline image at first time step Δt is calculated to convergence.

  2. [18] Using the expression inline image (k = 1, 2, …, 200) [Shao and Li, 1999], the number of particles entrained directly by aerodynamic shear stress at each discrete sand during Δt can be calculated, where u*,k is friction wind velocity at the k-th discrete sand bed, ηa,k is the number of aerodynamically entrained particles per unit area per unit time, and u* t is the threshold wind friction velocity. For time t = Δt, there are no collisions of sand particles with the surface, so the number of ejected particles from the surface is equal to the number of aerodynamically entrained particles inline image. After Δttp integration time step, through solving the equations of motion of saltating particles, the sand particles' location xp,k(t) and velocity field up,k(t) can be obtained. Finally, we can obtain the reactive force of sand with respect to wind inline image and the number density Nt(x,z) [Shao, 2005].

    display math(13)
    display math(14)

    where Δup,i is the velocity difference of sand particles within the time step Δt; ΔV is the control volume size; if the particle lifted with va and θa is located in the range of (x − 0.5Δx, x + 0.5Δx) ∪ (z − 0.5Δz, z + 0.5Δz), then δ(va,θa) = 1, else δ(va,θa) = 0, where Δx and Δz are the horizontal and vertical interval between two adjacent grids, respectively; inline image is the probability density function (PDF) of vertical lift-off velocities of the aerodynamically entrained particles, inline image is the mean lift-off velocity; inline image is the lift-off angle PDF of the aerodynamically entrained particles.

  3. [19] The new wind velocity field inline image is recalculated by substituting the reactive force inline image into the LES filtered momentum equations. According to the new wind velocity field, we can obtain the new friction velocity u*,k and ηa,k.

  4. [20] In terms of particles' locations calculated using equations ((11)) and ((12)), we can determine whether the sand particle collides with the surface. If the particles do not collide with the surface, then its location and velocity at this time step serve as the initial condition of next time step; if the particle collides with the surface, then the number of ejected particles is calculated by inline image, where vimp is the speed of particle impacting the surface; a is a dimensionless constant which is in the range of 0.01–0.05 [Kok and Renno, 2009]. Summing all of the ejection cases, we can obtain the total number of ejected particles. The number of rebounding particles is determined by the rebound PDF preb = 0.95[1 − exp(−γvimp)] [Anderson and Haff, 1991], namely, Nreb = Nimppreb, where γ ≈ 2m− 1s is a constant, Nimp is the number of particles impacting the surface, and then, the number of particles moving in the air inline image can be determined by the trajectories of particles.

  5. [21] According to the particles' locations and velocities calculated using equations ((1))–((4)), we can also calculate the reactive force inline image and the number density Nt + Δt(x,z) in the next time step through the following expressions:

    display math(15)
    display math(16)
    display math(17)
    display math(18)
    display math(19)
    display math(20)
    display math(21)
    display math(22)
    display math(23)
    display math(24)

    Similarly, if the particle is located in the range of

    display math

    Then, δa(va,θa) = δej(vej,θej)= δreb(vreb,θreb) = δair(uair,vair) = 1, else δa(va,θa) = δej(vej,θej) =δreb(vreb,θreb) = δair(uair,vair) = 0, where inline image is the total number of moving sand particles in the air; p(uair,vair) denotes the PDF of the particles' velocities in the air. The number of ejection particles with certain velocity and angle can be determined by the PDF of ejection velocities inline image and the PDF of ejection angles inline image. Analogously, the PDF of rebound velocities inline image and that of rebound angles inline image are employed to determine the number of rebounding particles inline image.

  6. [22] Repeating steps 2–5 until the wind velocity field satisfies the convergence condition inline image which means the wind-blown sand transport reaches eventual equilibrium state.

3 Theoretical Prediction Model of Electric Field

[23] It is well known that the electrification of wind-blown sand and dusty phenomena (such as dust storms and dust devils) is due to the charge transfer during dust/sand particle collisions. Several models, such as asymmetric contact and external electric fields, have been proposed to explain the charging of identical materials. However, the physical mechanism governing this charge transfer remains poorly understood [Kok and Renno, 2008; Lacks and Sankaran, 2011]. For charging of granular materials, smaller particles are generally negatively charged while larger particles are positively charged [Forward et al., 2009; Freier, 1960; Gill, 1948; Lacks and Sankaran, 2011; Latham, 1964; Schmidt et al., 1998; Zheng et al., 2003]. This charge segregation by particle size is consistent with measurements of vertical electric fields in wind-blown sand, dust storms, and dust devils. Different from the dusty phenomena, the water plays a key role in volcanic ash charge separation because of the substantial water contents in volcanic eruption clouds [Williams and McNutt, 2005]. In the wind-blown transport of sand particles, particles' motions can be divided into three types: (1) suspension (with diameter < 70µm), dust particles entrained into the atmosphere and often suspended in air; (2) saltation (with diameter 70 − 500µm), small sand particles that are driven by wind and hop along the surface; and (3) creep (with diameter > 500µm), large sand particles that are too heavy to be lifted by wind so that they roll or slide along the surface [Shao, 2000]. Therefore, in wind-blown sand, the saltating particles charge negatively relative to the creeping particles at the surface [Kok and Renno, 2008; Zheng et al., 2004].

[24] In this section, we establish a theoretical model to determine the electric fields in wind-blown sand. The saltating and creeping sand particles are usually negatively and positively charged, respectively [Kok and Renno, 2008; Zheng et al., 2004]. As the treatments of Zheng et al. [2004] and Yue and Zheng [2006], in our model, the average charge-to-mass ratio of saltating sand particles is presumed as − 60µc kg− 1, which was also obtained by the field measurements of Schmidt et al. [1998]. Here, we only account for the effect of charged particles in the rectangular domain 0 ≤ x ≤ + , −  < y < + , z0 < z < + , where the x axis is aligned with the wind velocity, the z axis is perpendicular to the surface, and the y axis parallels to the spanwise direction. Since the volumetric concentration of the suspended particles is only about 10− 8 − 10− 6 [Anderson and Hallet, 1986], which is much less than that of the saltating particles. Hence, in our model, the effect of suspended sand particles is neglected. Considering the charged particles of rolling or sliding along the surface and the saltating ones hopping in the air, the surface charge density and space charge density are written as ψ(x,y) and ζ(x,y,z), respectively. According to the law of charge conservation, the sum of creeping and saltating particles' charges in the whole region should be zero. Thus,

display math(25)

[25] As the treatment in previous studies, we assume that the average charge-to-mass ratio c of saltating particles is a constant [Zheng et al., 2004; Yue and Zheng, 2006]. If the sand particles are treated as identical spheres, then the space charge density ζ(x,y,z) can be expressed as

display math(26)

Here c is the average charge-to-mass ratio of saltating particles. Combining equations ((25)) and ((26)), the surface charge density ψ(x,y) can be expressed as

display math(27)

[26] For a small space element dxdydz′ and a small surface element dxdy″, which are located in space point P′(x′,y′,z′) and surface point P″(x″,y″,0), respectively. We denote the electric field at point P(x,0,z) due to the space element dxdydz′ and surface element dxdy″ as dE. According to the Coulomb's law, the electric field dE can be expressed as

display math(28)

where inline image is the vector pointing from P′(x′,y′,z′) toward P(x,0,z), inline image is the unit vector; inline image is the vector pointing from P″(x″,y″,0) to P(x,0,z), inline image is the unit vector; ε0 = 8.85 × 10− 12C2N− 1M− 2 is the electrical permittivity of air. In substituting equations ((25)) and ((26)) into equation ((27)) and integrating it over the computational domain, the horizontal and vertical electric filed can be expressed respectively as

display math(29)

where Ez0 = − 0.12kv m− 1 is the fair-weather electric field, the negative sign denotes the downward direction; inline image and inline image denote the horizontal electric filed due to the saltating and creep particles, respectively; inline image and inline image denote the vertical electric filed due to the saltating and creep particles, respectively; They can be respectively written as

display math(30)
display math(31)
display math(32)
display math(33)

[27] By substituting the number density N(x,z) into the equations ((30))–((33)), we can obtain the horizontal and vertical electric fields. It is worth noting that we only model the wind-blown sand transport in the limited extent of L × H, while the number density N(x,z) for x > L is equal to the eventual equilibrium value Ne(z).

4 Results and Discussions

[28] Figure 2a shows the comparison between the calculated vertical electric field with the measurement data of Schmidt et al. [1998]. In this case, the sand grain diameter is taken as D = 0.25 mm, wind speed and average charge-to-mass ratio are consistent with the value of Schmidt et al. [1998], namely, the former is 12 m s−1 at 1.5 m, and the latter is − 60µc kg− 1. As shown in Figure 2a, the calculated vertical electric field decreases rapidly with the increase of height, which is in agreement with the measurements. Furthermore, the simulated evolution of streamwise mass flux was compared with the observed results of Shao and Raupach [1992] in Figure 2b. As shown in the figure, the mass flux first increases to a maximum at position x ≈ 7m and then decreases to the equilibrium value at x ≈ 20m. Both vertical electric field and streamwise mass flux simulated results are in agreement with measurements. It indicates that our model and calculation procedure are valid. The minor difference between our result and measurements of vertical electric field at 14 cm might be due to the constant sand grain diameter and charge-to-mass ratio employed in our model [Zheng et al., 2004].

Figure 2.

Comparison of our simulated results with the experimental data: (a) the vertical electric field measured by Schmidt et al. [1998], and (b) the streamwsie mass flux measured by Shao and Raupach [1992]. Solid line denotes our result; squares denote the measurements.

[29] Figure 3a shows the height profiles of vertical electric field for several different streamwise distances. It can be seen that the variation of the vertical electric field with height is basically the same as that with streamwise distances. Namely, as the height increases, the upward-pointing vertical electric field first decreases to zero; then, it increases from zero to several kilovolts per meter in the opposite direction; and finally, it decreases with the height until it is equal to the fair-weather electric field. This law is consistent with the previous results of field measurements and theoretical predictions [Schmidt et al., 1998; Zheng et al., 2004]. However, the vertical electric fields at various streamwise distances reduce to zero at various elevations. For example, when streamwise distance x = 1.01m, the vertical electric field reduces to zero at 0.13 m, but when x = 10.94m, it reduces to zero at around 0.26 m. At the same height but various streamwise distances, the values of the vertical electric fields are significantly different. The evolution of vertical electric field acts as a function of streamwise distance at heights of 1.5, 5.8, and 10.2 cm, respectively (Figure 3b). It can be seen from Figure 3b, with the increase of streamwise distance, the vertical electric field gradually increases and eventually reaches an equilibrium value at about streamwise distance x = 25m.

Figure 3.

The vertical component of wind-blown sand electric field. (a) Height profile of the vertical electric field, where inline image is the eventual equilibrium values of the vertical electric field. (b) The vertical electric field varying with streamwise distance. In this situation, the friction velocity u* = 0.5 m s− 1 and sand grain diameter D = 0.25 mm.

[30] Figure 4a shows the height profiles of the horizontal electric field at different streamwise distances predicted by our model. It can be seen from Figure 4a that, in the developing stage of wind-blown sand, the maximum value of horizontal electric fields is up to several thousand volts per meter. It indicates that the spatial variation of particle concentration plays an important role in the generation of horizontal electric field. Different from the vertical electric field, which is generally upward-pointing near the surface and opposite to the fair-weather electric field, the direction of horizontal electric field varies with height. In the initial developing stage of wind-blown sand (i.e., the growth stage of mass flux), such as x = 1.01m and x = 5.42m, the horizontal electric field is pointing opposite to the streamwise direction in the lower region but along with the streamwise direction in the upper region. On the contrary, the horizontal electric field in the decline stage of mass flux pointing to the streamwise in the lower region. In addition, the height profile of horizontal electric field and its variation with the streamwise distance are significantly different from that of vertical electric field. On the one hand, the horizontal electric field is identified to be composed of three distinct regions. That is, near the sand surface, the intensity of horizontal electric field gradually reduces with height and an approach zero at 1–2 cm; then, it reverses direction and increases to a maximum value; and it eventually decreases to zero at around 20 m. On the other hand, significant discrepancies exist between the evolutions of vertical and horizontal electric fields. The evolution of horizontal electric field as a function of streamwise distance is shown in Figure 4b. As shown in Figure 4b, the horizontal electric field points oppositely in the growth and decline stage. As the streamwise distance increases, the horizontal electric field first increases, decreases to zero, then increases with an opposite sign, and finally decreases to the equilibrium value zero.

Figure 4.

The horizontal component of wind-blown sand electric field. (a) Height profile of the horizontal electric field. (b) The horizontal electric field varying with streamwise distance, where the friction velocity is u* = 0.5 m s− 1 and sand grain diameter is D = 0.25 mm.

[31] As discussed in section 2, in the developing stage of wind-blown sand, particle concentration or mass flux gradually increase with streamwise distance until reaching an eventual equilibrium value. In our model, sand particles are treated as point charges, so the spatial charge distribution is identical to sand particles' concentration distribution. Therefore, the vertical and horizontal electric fields in wind-blown sand can be interpreted by a simple electric dipole model. Figure 5a shows an electric dipole, the charges + Q and − Q are located at (0,0) and (0,h), respectively. The vertical and horizontal electric fields at location (x ′, z ′) produced by this electric dipole are given as follows:

display math(34)
display math(35)
Figure 5.

(a) Schematic of the electric dipole model. (b) The electric fields produced by an electric dipole; Ex and Ez denote horizontal and vertical electric fields, respectively. The electric fields are normalized by their maximum value.

[32] The height profiles of vertical and horizontal electric fields determined by equations ((34)) and ((35)) are shown in Figure 5b. In the case of x = 0.3m and h = 0.02m, the height profiles of electric fields are normalized by their maximum value. Comparing the height profile of vertical and horizontal electric fields produced by an electric dipole with the electric fields in wind-blown sand transport, we find that both characteristics of vertical and horizontal electric fields can be explained well by the electric dipole model.

[33] To reveal the effect of wind speed and sand grain diameter on the electric fields, we investigated the profile and distribution of horizontal and vertical electric fields at x = 10.94m for various sand grain diameters and wind speeds. Results are shown in Figures 6 and 7. Figure 6 show that at the same height, both horizontal and vertical electric fields increase with the wind speed. As the wind speed increases, moreover, the horizontal and vertical electric fields reduce to zero at higher altitude. It might be caused by the reason that for a large wind speed, more charged particles are lifted from surface, and particles fly higher. Compared with the effects of wind speed, the sand grain diameter affects horizontal and vertical electric fields oppositely. The reason for this is probably that the trajectories of larger sand particles are lower than the small ones. It is shown (Figure 7) that both horizontal and vertical electric fields decrease with the sand grain diameter.

Figure 6.

The effect of wind speed on the vertical and horizontal electric fields for streamwise distance x = 10.94 m and sand grain diameter D = 0.25 mm. (a) The profile of vertical electric fields. (c) The profile of horizontal electric fields.

Figure 7.

The effect of sand grain diameter on the vertical and horizontal electric fields for the streamwise distance x = 10.94 m and friction velocity u* = 0.5 m s− 1. (a) The height profile of vertical electric fields. (b) The profiles of horizontal electric fields for various sand grain diameters.

5 Conclusions

[34] By considering the streamwise spatial variation of particle concentration in the evolution of wind-blown sand transport, we model the horizontal and vertical electric fields quantitatively. The results show that the height profile of vertical electric field in the developing stage is similar to the steady case which is upward-pointing and reaches 100 kilovolts per meter in magnitude. In addition, we present a theoretical model to predict the horizontal electric field in wind-blown sand. It is suggested that the horizontal electric field is as large as several kilovolts per meter and is composed of three distinct regions: near the sand surface, the intensity of horizontal electric field gradually reduces with height to zero at 1–2 cm height; then it reverses direction and increases to a maximum value; and eventually, it decreases to zero at around 20 m.

[35] The horizontal and vertical electric fields are considerably affected by the streamwise distance, wind speed, and sand grain diameter. In the developing stage, the vertical electric field gradually increases with the streamwise distance until reaching equilibrium; however, the horizontal electric field increases to its maximum first and then decreases to zero. Both horizontal and vertical electric fields increase with wind speed and decrease with sand grain diameter.

[36] The previous studies suggest that the vertical electric field has significant influence on wind-blown sand transport. For example, Kok and Renno [2008] revealed that consideration of vertical electric field in saltation model makes simulated results closer to experimental measurements than that of nonvertical electric field consideration. By taking the horizontal electric field into account, this study found an enhancement of sand transport rate due to the horizontal electric field. We will perform further studies on the influence of the horizontal electric field on the wind-blown sand transport.

Acknowledgments

[37] This research was supported by grants from the National Natural Science Foundation of China (Nos. 11072097, 11232006, 11121202 and 11202088), the Science Foundation of Ministry of Education of China (No.308022), Fundamental Research Funds for the Central Universities (No. lzujbky-2012-3,lzujbky-2009-k01), National Key Technology R&D Program (2013BAC07B01), and the Project of the Ministry of Science and Technology of China (No.2009CB421304).The authors express their sincere appreciation for this support.