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**Atmospheric Chemistry and Physics**
An interactive open-access journal of the European Geosciences Union

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**Research article**
02 Dec 2020

**Research article** | 02 Dec 2020

Effects of 3D electric field on saltation during dust storms: an observational and numerical study

^{1}Department of Mechanics and Engineering Science, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China^{2}Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education of China, Lanzhou 730000, China

^{1}Department of Mechanics and Engineering Science, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China^{2}Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education of China, Lanzhou 730000, China

**Correspondence**: You-He Zhou (zhouyh@lzu.edu.cn)

**Correspondence**: You-He Zhou (zhouyh@lzu.edu.cn)

Abstract

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Particle triboelectric charging, being ubiquitous in nature and
industry, potentially plays a key role in dust events, including the lifting and transport of sand and dust particles. However, the properties of the electric field (** E** field) and its influences on saltation during dust storms remain obscure as the high complexity of dust storms and the existing numerical studies are mainly limited to the 1D

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How to cite.

Zhang, H. and Zhou, Y.-H.: Effects of 3D electric field on saltation during dust storms: an observational and numerical study, Atmos. Chem. Phys., 20, 14801–14820, https://doi.org/10.5194/acp-20-14801-2020, 2020.

1 Introduction

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Contact or triboelectric charging is ubiquitous in dust events (Schmidt et
al., 1998; Zheng et al., 2003; Kok and Renno, 2008; Lacks and Sankaran,
2011; Harrison et al., 2016). The pioneering electric field (** E** field)
measurements in dust storms by Rudge (1913) showed that the vertical
atmospheric

The significant influences of the ** E** field on pure saltation (that is, in the absence of suspended dust and aerosol particles) have been verified, both
numerically (e.g. Kok and Renno, 2008; Zhang et al., 2014) and
experimentally (e.g. Rasmussen et al., 2009; Esposito et al., 2016). The
effects of the

Most field observations, such as Schmidt et al. (1998) and Bo et al. (2014),
have studied the electrical properties of sand particles in dust events.
However, many environmental (lurking) factors, such as relative humidity,
soil moisture, surface crust, etc., cannot be fully controllable (recorded)
in these field observations. The uncertainties in the field observations
provide the motivation for numerical studies of the particle triboelectric
charging in saltation. In addition, unlike pure saltation, the dust storm is
a very complex dusty phenomenon that is made up of numerous polydisperse
particles embedded in a high Reynolds number turbulent flow. Such a high
complexity of dust storms challenges the accurate simulation of the 3D ** E** field in dust storms. It is therefore more straightforward to characterize the 3D

In this study, we evaluate the effects of the 3D ** E** field on saltation during dust storms by combining measurements and modelling. To reveal the
properties of the 3D

2 Field campaign

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We performed 3D ** E** field measurements at the Qingtu Lake Observation Array (QLOA) site (approximately 39

A detailed description of VREFM can be found in the Supplement of Zhang et
al. (2017), but we briefly describe it here. The working principle of VREFM
is based on the dynamic capacity technique, as illustrated in the inset of
Fig. 1b. Unlike a traditional atmospheric electric field mill, VREFM is
composed of only one vibrating electrode. As the electrode oscillates, it
charges and discharges periodically. The magnitude of the induced electric
current *i*(*t*) is proportional to the ambient ** E** field intensity

$$\begin{array}{}\text{(1)}& i\left(t\right)\propto E\mathit{\omega}\mathrm{cos}\left(\mathit{\omega}t\right),\end{array}$$

where *ω* is the vibration frequency of the electrode. The induced
electric current is then converted to an output voltage signal, which is
linearly proportional to the ambient ** E** field through functional modules within VREFM. In addition, the length and diameter of the VREFM sensor are approximately 2.5 and 7 cm, respectively. This small-sized sensor allows us to measure

The measurement uncertainties in our field campaign are threefold, namely wind velocity (CSAT3B), particle mass flux (SPC-91), and ** E** field (VREFM). The CSAT3B is factory calibrated with an accuracy of ±8 cm s

In general, the actual wind direction exits at a specific angle to the
prevailing wind direction. A projection step is therefore needed to obtain
the streamwise ** E** field,

After completing the projection step, we then perform the following steps
sequentially to reveal the pattern of 3D ** E** field in the sub-metre layer: (1) estimating the time-varying mean values of

The DWT uses a set of mutually orthogonal wavelet basis functions, which are
dilated, translated, and scaled versions of a mother wavelet, to decompose
an ** E** field series

$$\begin{array}{}\text{(2)}& E=\sum _{i=\mathrm{1}}^{N}{\mathit{\psi}}_{i}+{\mathit{\chi}}_{N},\end{array}$$

where *N* is the total number of decomposition levels, *ψ*_{i} denotes
the *i*th level wavelet detail component, and *χ*_{N} represents the
*N*th level wavelet approximation (or smooth) component. As *N* increases,
the frequency contents become lower, and thus, the *N*th level approximation
components could be regarded as the time-varying mean values (e.g. Percival
and Walden, 2000; Su et al., 2015). In this study, the DWT decomposition is
performed with the Daubechies wavelet of the order of 10 (db10) at level 10, and thus, the 10th order approximation component can be defined as the
time-varying mean as follows:

$$\begin{array}{}\text{(3)}& \stackrel{\mathrm{\u203e}}{E}={\mathit{\chi}}_{\mathrm{10}},\end{array}$$

which reflect the averages of the *E* series over a scale of 2^{10} s
(Percival and Walden, 2000).

On the other hand, according to the empirical mode decomposition (EMD)
method, the time series *E* can be decomposed as follows (Huang et al., 1998):

$$\begin{array}{}\text{(4)}& E=\sum _{i=\mathrm{1}}^{N}{\mathit{\xi}}_{i}+{\mathit{\eta}}_{N},\end{array}$$

through a sifting process, where *ξ*_{i} are the intrinsic mode
functions (IMFs), and *η*_{N} is a residual (which is the overall trend
or mean). To reduce the end effects and mode mixing in EMD, the EEMD method
is proposed by Wu and Huang (2009). In EEMD, a set of white noise series,
${w}_{j}\left(j=\mathrm{1},\mathrm{2},\mathrm{\dots}{N}_{\mathrm{e}}\right)$, are added to the
original signal *E*. Then, each noise-added series is decomposed into IMFs
followed by the same sifting process as in EMD. Finally, the *i*th EEMD
component is defined as the ensemble mean of the *i*th IMFs of the total of
*N*_{e} noise-added series (see Wu and Huang, 2009, for details).

In this study, the time-varying mean values $\stackrel{\mathrm{\u203e}}{E}$ can be
alternatively defined as the sum of the last four EEMD components, *ξ*_{10} to *ξ*_{13}, and the residual, *η*_{13}, as follows:

$$\begin{array}{}\text{(5)}& \stackrel{\mathrm{\u203e}}{E}=\sum _{i=\mathrm{10}}^{\mathrm{13}}{\mathit{\xi}}_{i}+{\mathit{\eta}}_{\mathrm{13}}.\end{array}$$

According to the above definitions, the time-varying mean can be
synchronously obtained by the DWT and EEMD methods. As an example, Fig. 2
shows the results of the DWT analysis (Fig. 2b) and EEMD decompositions (Fig. 2c) for an ** E** field time series

Since the 3D ** E** fields are measured at five heights in our field campaign, we thus define the height-averaged time-varying mean values as follows:

$$\begin{array}{}\text{(6)}& \u2329\stackrel{\mathrm{\u203e}}{{E}_{i}}\u232a=\left|{\displaystyle \frac{\mathrm{1}}{\left(\mathrm{0.7}-\mathrm{0.05}\right)}}\underset{\mathrm{0.05}}{\overset{\mathrm{0.7}}{\int}}\stackrel{\mathrm{\u203e}}{{E}_{i}}\mathrm{d}z\right|,\end{array}$$

in the range of 0.05 to 0.7 m height in order to normalize the ** E** field data by a unified quantity. Furthermore, the

$$\begin{array}{}\text{(7)}& {E}_{i}^{\ast}={\displaystyle \frac{{E}_{i}}{\u2329\stackrel{\mathrm{\u203e}}{{E}_{i}}\u232a}}.\end{array}$$

Additionally, to obtain the dimensionless vertical profile of the 3D ** E** field, the height

$$\begin{array}{}\text{(8)}& {z}^{\ast}={\displaystyle \frac{z}{{\overline{z}}_{\mathrm{salt}}}},\end{array}$$

where the saltation height *z*_{salt} during a certain time interval is defined as the height below which 99 % of the total mass flux is present and can be estimated based on the measured SPC-91 data (see Text S1 in the Supplement for more details).

Finally, the dimensionless vertical profiles of the 3D ** E** field at different periods are fitted together by the third-order polynomial functions as follows:

$$\begin{array}{}\text{(9)}& {E}_{i}^{\ast}\left({z}^{\ast}\right)={a}_{\mathrm{0},i}+{a}_{\mathrm{1},i}{z}^{\ast}+{a}_{\mathrm{2},i}({z}^{\ast}{)}^{\mathrm{2}}+{a}_{\mathrm{3},i}({z}^{\ast}{)}^{\mathrm{3}},i=\mathrm{1},\mathrm{2},\mathrm{3},\end{array}$$

where *i*=1, 2, and 3 correspond to the streamwise, spanwise, and
vertical components, respectively.

3 Saltation model

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For modelling steady state saltation, there are four primary processes including (1) particle saltating motion, (2) particle–particle midair collisions, (3) particle–bed collisions, and (4) particle–wind momentum coupling (Dupont et al., 2013; Kok and Renno, 2009). Also, the changes in both the momentum and electrical charge of each particle are taken into account in the particle–particle midair and particle–bed collisions. To avoid overestimating midair collisions in the 2D simulation (Carneiro et al., 2013), we simulate saltation trajectories in a real 3D domain. We use the discrete element method (DEM), which explicitly simulates each particle motion and describes the collisional forces between colliding particles encompassing normal and tangential components, to advance the evaluation of the effects of particle midair collisions. In a steady state saltation, the mean streamwise wind speed is statistically stationary and statistically 1D so that the mean wind flow can be modelled as a 1D field. In other words, in this study the numerical simulation is a 3D DEM model for particle motion but a 1D model for wind field. In the following subsections, we will describe each process in detail.

Granular materials in natural phenomena, such as sand, aerosols, pulverized
material, seeds of crops, etc., are made up of discrete particles with a
wide range of sizes ranging from a few micrometres to millimetres. The
log-normal distribution is generally used to approximate the size
distribution of the sand sample (Marticorena and Bergametti, 1995; Dupont et
al., 2013). Thus, the mass distribution function of a sand sample with two
parameters, average diameter *d*_{m}, and geometric standard deviation *σ*_{p} can be written as follows:

$$\begin{array}{}\text{(10)}& {\displaystyle \frac{\mathrm{d}M\left({d}_{\mathrm{p}}\right)}{\mathrm{d}\mathrm{ln}\left({d}_{\mathrm{p}}\right)}}={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi}}\mathrm{ln}\left({\mathit{\sigma}}_{\mathrm{p}}\right)}}\mathrm{exp}\left\{-{\displaystyle \frac{{\left[\mathrm{ln}\left({d}_{\mathrm{p}}\right)-\mathrm{ln}\left({d}_{\mathrm{m}}\right)\right]}^{\mathrm{2}}}{\mathrm{2}{\left[\mathrm{ln}\left({\mathit{\sigma}}_{\mathrm{p}}\right)\right]}^{\mathrm{2}}}}\right\}.\end{array}$$

The total force acting on a saltating particle consists of three distinct
interactions (Minier, 2016). The first one refers to the wind–particle
interaction, which is dominated by the drag force with lifting forces, such
as the Saffman force and Magnus force, being of secondary importance (Kok and
Renno, 2009; Dupont et al., 2013). The second interaction refers to the
particle–particle collisional forces or cohesion caused by physical contact
between particles. Such interparticle collisional forces can be described as
a function of the overlaps between the colliding particles. The third
interaction refers to the forces due to external fields such as gravity and the ** E** field. In this study, in addition to the drag force, we also take into account the Magnus force because of the remarkable rotation of saltating particles of the order of 100–1000 rev s

$$\begin{array}{}\text{(11a)}& {\displaystyle}\begin{array}{rl}{m}_{\mathrm{p},i}{\displaystyle \frac{\mathrm{d}{\mathit{u}}_{\mathrm{p},i}}{\mathrm{d}t}}& ={\mathit{F}}_{i}^{\mathrm{d}}+{\mathit{F}}_{i}^{\mathrm{m}}+\sum _{j}({\mathit{F}}_{ij}^{n}+{\mathit{F}}_{ij}^{\mathrm{t}})\\ & +{m}_{i}\mathit{g}+{\mathit{\zeta}}_{\mathrm{p},i}\mathit{E}\end{array}\text{(11b)}& {\displaystyle}{I}_{i}{\displaystyle \frac{\mathrm{d}{\mathit{\omega}}_{\mathrm{p},i}}{\mathrm{d}t}}={\mathit{M}}_{i}^{\mathrm{w}-\mathrm{p}}+\sum _{j}({\mathit{M}}_{ij}^{\mathrm{c}}+{\mathit{M}}_{ij}^{\mathrm{r}}),\end{array}$$

where *m*_{p,i} is the mass of the *i*th particle, *u*_{p,i} is the
velocity of the particle, ${\mathit{F}}_{i}^{\mathrm{d}}$ is the drag force,
${\mathit{F}}_{i}^{\mathrm{m}}$ is the Magnus force, ${\mathit{F}}_{ij}^{\mathrm{d}}$ and ${\mathit{F}}_{ij}^{\mathrm{t}}$ are the normal and tangential collisional forces from the *j*th particle, respectively. ** g** is the gravitational acceleration, and

In the absence of saltating particles, the mean wind profile over a flat and homogeneous surface is well approximated by the log law (Anderson and Haff, 1988) as follows:

$$\begin{array}{}\text{(12)}& {u}_{\mathrm{m}}\left(z\right)={\displaystyle \frac{{u}_{\ast}}{\mathit{\kappa}}}\mathrm{ln}{\displaystyle \frac{z}{{z}_{\mathrm{0}}}},\end{array}$$

where *u*_{m} is the mean streamwise wind speed, *z* is the height above the surface, and *u*_{∗} is the friction velocity. *κ*≈0.41 is the von Kármán constant, and *z*_{0} is the aerodynamic roughness, which varies substantially from different flow conditions and can be approximately estimated as *d*_{m}∕30 for the aeolian saltation on Earth (e.g. Kok et al., 2012; Carneiro et al., 2013). In the presence of saltation, due to the momentum coupling between the saltating particles and wind flow, the modified wind speed gradient can be written as follows for steady state and horizontally homogeneous saltation (e.g. Kok and Renno, 2009; Pähtz et al., 2015):

$$\begin{array}{}\text{(13)}& {\displaystyle \frac{\mathrm{d}{u}_{\mathrm{m}}\left(z\right)}{\mathrm{d}z}}={\displaystyle \frac{{u}_{\ast}}{\mathit{\kappa}z}}\sqrt{\mathrm{1}-{\displaystyle \frac{{\mathit{\tau}}_{\mathrm{p}}\left(z\right)}{{\mathit{\rho}}_{\mathrm{a}}{u}_{\ast}^{\mathrm{2}}}}},\end{array}$$

where *ρ*_{a} is the air density, and *τ*_{p}(z) is the particle momentum flux and can be numerically determined by (Carneiro et al., 2013; Shao, 2008) the following:

$$\begin{array}{}\text{(14)}& {\mathit{\tau}}_{\mathrm{p}}\left(z\right)=-{\displaystyle \frac{\sum {m}_{\mathrm{p},i}{u}_{\mathrm{p},i}{w}_{\mathrm{p},i}}{{L}_{x}{L}_{y}\mathrm{\Delta}z}},\end{array}$$

with *L*_{x}, *L*_{y}, and Δ*z* being the streamwise- and
spanwise-width of the computational domain and the vertical grid size,
respectively. *u*_{p,i} and *w*_{p,i} are the streamwise and vertical components of the particle velocity. The summation in Eq. (14) is performed on the particles located in the range of $\left[z,z+\mathrm{\Delta}z\right]$. Once the saltating particle trajectories are known, the wind profile can be determined through integrating Eq. (13) with the no-slip boundary condition *u*_{m}=0 at *z*=*z*_{0}.

Since sand particles are much heavier than the air and are far smaller than the Kolmogorov scales, the drag force is the dominant force affecting the particle motion, which is expressed by (Anderson and Haff, 1991) the following:

$$\begin{array}{}\text{(15)}& {\mathit{F}}_{i}^{\mathrm{d}}=-{\displaystyle \frac{\mathit{\pi}{d}_{\mathrm{p}}^{\mathrm{2}}}{\mathrm{8}}}{\mathit{\rho}}_{\mathrm{a}}{C}_{\mathrm{d}}{\mathit{u}}_{\mathrm{r}}\mathrm{|}{\mathit{u}}_{\mathrm{r}}\mathrm{|},\end{array}$$

where *d*_{p} is the diameter of the particle, *C*_{d} is the drag coefficient, and ${\mathit{u}}_{\mathrm{r}}={\mathit{u}}_{\mathrm{p}}-{\mathit{u}}_{\mathrm{w}}$ is the
particle-to-wind relative velocity. The drag coefficient *C*_{d} is a function of the particle Reynolds number, ${\mathit{Re}}_{\mathrm{p}}={\mathit{\rho}}_{\mathrm{a}}\mathrm{|}{\mathit{u}}_{\mathrm{r}}\mathrm{|}{d}_{\mathrm{p}}/\mathit{\mu}$, where *μ* is the dynamic viscosity of the air. We calculate the drag coefficient with an empirical relation ${C}_{\mathrm{d}}={\left[{\left(\mathrm{32}/{\mathit{Re}}_{\mathrm{p}}\right)}^{\mathrm{2}/\mathrm{3}}+\mathrm{1}\right]}^{\mathrm{3}/\mathrm{2}}$, which is
applicable to the regimes from Stokes flow *Re*_{p}≪1 to high Reynolds number turbulent flow (Cheng, 1997).

Additionally, we also account for the effects of particle rotation on particle motion using the Magnus force expressed as follows (White and Schulz, 1977; Anderson and Hallet, 1986; Loth, 2008):

$$\begin{array}{}\text{(16)}& {\mathit{F}}_{i}^{\mathrm{m}}={\displaystyle \frac{\mathit{\pi}{d}_{\mathrm{p}}^{\mathrm{2}}}{\mathrm{8}}}{\mathit{\rho}}_{\mathrm{a}}{C}_{\mathrm{m}}\left({\mathit{\omega}}_{\mathrm{p},i}\times {\mathit{u}}_{\mathrm{r}}\right),\end{array}$$

where *C*_{m} is a normalized spin lift coefficient, depending on the particle Reynolds number and the circumferential speed of the particle. The torque acting on a particle caused by wind flow is calculated from (Anderson and Hallet, 1986; Shao, 2008; Kok and Renno, 2009) the following:

$$\begin{array}{}\text{(17)}& {\mathit{M}}_{i}^{\mathrm{w}-\mathrm{p}}=\mathit{\pi}\mathit{\mu}{d}_{i}^{\mathrm{3}}\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{\mathrm{d}{u}_{\mathrm{m}}}{\mathrm{d}z}}-{\mathit{\omega}}_{i}\right).\end{array}$$

Under moderate conditions, saltation is a dilute flow in which the particle–particle collisions are negligible. However, as wind velocity increases, midair collisions become increasingly pronounced, especially in the near-surface region (Sørensen and McEwan, 1996). Previous studies found that the probability of midair collisions of saltating particles almost increased linearly with wind speed (Huang et al., 2007), and such collisions indeed enhanced the total mass flux substantially (Carneiro et al., 2013). For spherical particles, one of the most commonly used collisional force models is the non-linear viscoelastic model, consisting of two components, i.e. elastic and viscous forces (Haff and Anderson, 1993; Brilliantov et al., 1996; Silbert et al., 2001; Tuley et al., 2010).

Considering that two spherical particles *i* and *j*, with diameters
*d*_{i} and *d*_{j} and position vectors *x*_{i} and *x*_{j},
are in contact with each other, the relative velocity *v*_{ij} at the
contact point and its normal and tangential components, ${\mathit{v}}_{ij}^{\mathrm{n}}$ and ${\mathit{v}}_{ij}^{\mathrm{t}}$, are respectively defined as follows (Silbert et al., 2001; Norouzi et al., 2016):

$$\begin{array}{}\text{(18a)}& {\displaystyle}{\mathit{v}}_{ij}={\mathit{u}}_{\mathrm{p},i}-{\mathit{u}}_{\mathrm{p},j}+\mathrm{0.5}\left({d}_{i}{\mathit{\omega}}_{\mathrm{p},i}+{d}_{j}{\mathit{\omega}}_{\mathrm{p},j}\right)\times {\mathit{n}}_{ij}\text{(18b)}& {\displaystyle}{\mathit{v}}_{ij}^{n}=\left({\mathit{v}}_{ij}\cdot {\mathit{n}}_{ij}\right){\mathit{n}}_{ij}\text{(18c)}& {\displaystyle}{\mathit{v}}_{ij}^{t}={\mathit{v}}_{ij}-{\mathit{v}}_{ij}^{n},\end{array}$$

where ${\mathit{n}}_{ij}=\left({\mathit{x}}_{j}-{\mathit{x}}_{i}\right)/\mathrm{|}{\mathit{x}}_{i}-{\mathit{x}}_{j}\mathrm{|}$ is the unit vector in the direction from the
centre of particle *i* point towards the centre of particle *j*. Supposing that these colliding particles have identical mechanical properties to
Young's modulus *Y*, shear modulus *G*, and Poisson's ratio *ν*, the normal collisional force can thus be calculated by (Brilliantov et al., 1996; Silbert et al., 2001) the following:

$$\begin{array}{}\text{(19)}& {\mathit{F}}_{ij}^{\mathrm{n}}=-{\displaystyle \frac{\mathrm{4}}{\mathrm{3}}}{Y}^{\ast}\sqrt{{R}^{\ast}}{\mathit{\delta}}_{\mathrm{n}}^{\mathrm{3}/\mathrm{2}}{\mathit{n}}_{ij}-\mathrm{2}\sqrt{{\displaystyle \frac{\mathrm{5}}{\mathrm{6}}}{m}^{\ast}{S}_{\mathrm{n}}}\mathit{\beta}{v}_{\mathrm{n}}{\mathit{n}}_{ij},\end{array}$$

where ${Y}^{\ast}=Y/\mathrm{2}/\left(\mathrm{1}-{\mathit{\nu}}^{\mathrm{2}}\right)$ is the equivalent Young's
modulus, ${\mathit{\delta}}_{\mathrm{n}}=\mathrm{0.5}\left({d}_{i}+{d}_{j}\right)-\left|{\mathit{x}}_{i}-{\mathit{x}}_{j}\right|$ is the normal overlap, ${m}^{\ast}={m}_{i}{m}_{j}/\left({m}_{i}+{m}_{j}\right)$ is the equivalent particle mass,
${S}_{\mathrm{n}}=\mathrm{2}{Y}^{\ast}\sqrt{{R}^{\ast}{\mathit{\delta}}_{\mathrm{n}}}$ is the normal contact
stiffness, ${R}^{\ast}={d}_{i}{d}_{j}/\mathrm{2}/\left({d}_{i}+{d}_{j}\right)$ is the
equivalent particle radius, and *β* is related to the coefficient of
restitution *e*_{n} by the relationship $\mathit{\beta}=\mathrm{ln}{e}_{\mathrm{n}}/\sqrt{(\mathrm{ln}{e}_{\mathrm{n}}{)}^{\mathrm{2}}+{\mathit{\pi}}^{\mathrm{2}}}$; and ${v}_{\mathrm{n}}={\mathit{v}}_{ij}\cdot {\mathit{n}}_{ij}$. The first term on the right-hand side of Eq. (19) represents
the elastic force described by Hertz's theory, and the second term
represents the viscous force reflecting the inelastic collisions between
sand particles. Similarly, the tangential collisional force, which is
limited by the Coulomb friction, is given as follows (Brilliantov et al., 1996; Silbert et al., 2001):

$$\begin{array}{}\text{(20)}& {\mathit{F}}_{ij}^{t}=\left\{\begin{array}{l}-\mathrm{8}{G}^{\ast}\sqrt{{R}^{\ast}{\mathit{\delta}}_{\mathrm{n}}}{\mathit{\delta}}_{\mathrm{t}}{\mathit{t}}_{ij}-\mathrm{2}\sqrt{\frac{\mathrm{5}}{\mathrm{6}}{m}^{\ast}{S}_{\mathrm{t}}}\mathit{\beta}{v}_{\mathrm{t}}{\mathit{t}}_{ij},\\ \mathrm{if}\phantom{\rule{0.25em}{0ex}}\left|{\mathit{F}}_{ij}^{\mathrm{t}}\right|\le {\mathit{\gamma}}_{\mathrm{s}}\left|{\mathit{F}}_{ij}^{\mathrm{n}}\right|\\ -{\mathit{\gamma}}_{\mathrm{s}}\left|{\mathit{F}}_{ij}^{\mathrm{n}}\right|{\mathit{t}}_{ij},\\ \mathrm{if}\phantom{\rule{0.25em}{0ex}}\left|{\mathit{F}}_{ij}^{\mathrm{t}}\right|>{\mathit{\gamma}}_{\mathrm{s}}\left|{\mathit{F}}_{ij}^{\mathrm{n}}\right|,\end{array}\right.\end{array}$$

where ${G}^{\ast}=G/\mathrm{2}/\left(\mathrm{2}-\mathit{\nu}\right)$ is the equivalent shear modulus,
*δ*_{t} is the tangential overlap,
${\mathit{t}}_{ij}={\mathit{v}}_{ij}^{\mathrm{t}}/\left|{\mathit{v}}_{ij}^{\mathrm{t}}\right|$ is the tangential unit vector at the contact point, ${S}_{\mathrm{t}}=\mathrm{8}{G}^{\ast}\sqrt{{R}^{\ast}{\mathit{\delta}}_{\mathrm{n}}}$ is the tangential stiffness, ${v}_{\mathrm{t}}={\mathit{v}}_{ij}\cdot {\mathit{t}}_{ij}$, and *γ*_{s} is the coefficient of static friction. The torque on the *i*th particle arising from the *j*th particle collisional force is defined as (Haff and Anderson, 1993) the follows:

$$\begin{array}{}\text{(21)}& {\mathit{M}}_{ij}^{\mathrm{c}}=\mathrm{0.5}{d}_{i}{\mathit{n}}_{ij}\times {\mathit{F}}_{ij}^{\mathrm{t}}.\end{array}$$

To account for the significant rolling friction, we apply a rolling resistance torque (Ai et al., 2011) as follows:

$$\begin{array}{}\text{(22)}& {\mathit{M}}_{ij}^{\mathrm{r}}=-{\mathit{\gamma}}_{\mathrm{r}}{R}^{\ast}\left|{\mathit{F}}_{ij}^{\mathrm{n}}\right|{\mathit{\omega}}_{ij},\end{array}$$

on each colliding particle, where *μ*_{r} is the coefficient of rolling friction, and ${\mathit{\omega}}_{ij}=\left({\mathit{\omega}}_{\mathrm{p},i}-{\mathit{\omega}}_{\mathrm{p},j}\right)/\left|{\mathit{\omega}}_{\mathrm{p},i}-{\mathit{\omega}}_{p,j}\right|$ is the unit vector of relative angular velocity.

As a saltating particle collides with the sand bed, it not only has a chance to rebound but may also eject several particles from the sand bed. For simplicity, we use a probabilistic representation, termed as the “splash function”, to describe the particle–bed interactions quantitatively (Shao, 2008; Kok et al., 2012). Currently, the splash function is primarily characterized by wind-tunnel and numerical simulations (e.g. Anderson and Haff, 1991; Haff and Anderson, 1993; Rice et al., 1996; Huang et al., 2017). The rebounding probability of a saltating particle colliding with the sand bed is approximately (Anderson and Haff, 1991) the following:

$$\begin{array}{}\text{(23)}& {P}_{\mathrm{reb}}=\mathrm{0.95}\left[\mathrm{1}-\mathrm{exp}\left(-{v}_{\mathrm{imp}}\right)\right],\end{array}$$

where *v*_{imp} is the impact speed of the saltating particle. The kinetic energy of the rebounding particles is taken as 0.45±0.22 of
the impact particle (Kok and Renno, 2009). The rebounding angles *θ*
and *φ*, as depicted in Fig. 4a, obey an exponential distribution
with a mean value of 40^{∘}, i.e. $\mathit{\theta}\sim \text{Exp}\left(\mathrm{40}{}^{\circ}\right)$, and a normal distribution with the parameters 0±10^{∘}, i.e. *φ*∼*N*(0^{∘},10^{∘}), respectively (Kok and Renno, 2009; Dupont et al., 2013).

It is reasonable to assume that the number of ejected particles depends on
the impact speed and its cross-sectional area. Thus, the number of ejected
particles from the *k*th particle bin is (Kok and Renno, 2009) as follows:

$$\begin{array}{}\text{(24)}& {N}_{k}={\displaystyle \frac{\mathrm{0.02}}{\sqrt{g{D}_{\mathrm{250}}}}}{\displaystyle \frac{{D}_{\mathrm{imp}}}{{D}_{\mathrm{eje}}^{k}}}{p}_{k}{v}_{\mathrm{imp}},\end{array}$$

where ${D}_{\mathrm{250}}=\mathrm{0.25}\times {\mathrm{10}}^{-\mathrm{4}}$ m is a reference diameter, *D*_{imp} and ${D}_{\mathrm{eje}}^{k}$ are the diameter of the impact and ejected particles, respectively, and *p*_{k} is the mass fraction of the *k*th particle bin. The speed of the ejected particles obeys an exponential distribution, with the mean value taken as $\mathrm{0.6}\left[\mathrm{1}-\mathrm{exp}\left(-{v}_{\mathrm{imp}}/\mathrm{40}/\sqrt{g{D}_{\mathrm{250}}}\right)\right]$ (Kok and Renno, 2009). Similar to the rebound process, the ejected angles *θ* and *φ* are assumed to be $\mathit{\theta}\sim \text{Exp}\left(\mathrm{50}{}^{\circ}\right)$ and
$\mathit{\phi}\sim N\left(\mathrm{0}{}^{\circ},\mathrm{10}{}^{\circ}\right)$.

In this study, the calculation of the charge transfer between sand particle
collisions is based on the asymmetric contact model, assuming that the
electrons trapped in high-energy states on one particle surface can relax on
the other particle surface (Kok and Lacks, 2009; Hu et al., 2012). Thus, the
net increment of the charge of particle *i* after colliding with particle
*j*, Δ*q*_{ij}, can be determined by the following:

$$\begin{array}{}\text{(25)}& \mathrm{\Delta}{q}_{ij}=-e\left({\mathit{\rho}}_{\mathrm{h}}^{j}{S}_{j}-{\mathit{\rho}}_{\mathrm{h}}^{i}{S}_{i}\right),\end{array}$$

where $e=\mathrm{1.602}\times {\mathrm{10}}^{-\mathrm{19}}$ C is the elementary charge, ${\mathit{\rho}}_{\mathrm{h}}^{i}$ is the density of the electrons trapped in the high energy states on the surface of particle *i* (assuming that all particles have an identical initial value, i.e., ${\mathit{\rho}}_{\mathrm{h}}^{i}={\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$), which is modified as follows (Zhang et al., 2013):

$$\begin{array}{}\text{(26)}& {\mathit{\rho}}_{\mathrm{h},i}^{\mathrm{after}}={\mathit{\rho}}_{\mathrm{h},i}^{\mathrm{before}}-{\displaystyle \frac{\mathrm{\Delta}{q}_{ij}}{e\mathit{\pi}{d}_{i}^{\mathrm{2}}}},\end{array}$$

due to collisions between particle *i* and *j*. *S*_{i} is the particle
contact area, which can be approximately calculated as a line integral along
the contact path *L*_{i} of particle *i* as follows:

$$\begin{array}{}\text{(27)}& {S}_{i}=\mathrm{2}\underset{{L}_{i}}{\int}\sqrt{{R}^{\ast}{\mathit{\delta}}_{\mathrm{n}}}\mathrm{d}{l}_{i},\end{array}$$

where d*l*_{i} is the differential of the contact length. In general, when two particles are in contact with each other, the relative sliding motion between the two particles results in two unequal contact areas, *S*_{i} and *S*_{j}, thus producing a net charge transfer Δ*q*_{ij} between the two particles. If the particle's net electrical charge is known, its charge-to-mass ratio can be easily determined.

Similar to particle momentum flux (i.e. Eq. 14), the particle horizontal mass flux *q*, total mass flux *Q*, mean particle mass concentration *m*_{c} (Carneiro et al., 2013; Dupont et al., 2013), and mean particle charge-to-mass ratio 〈*ζ*_{p}〉 can be numerically determined by the following:

$$\begin{array}{}\text{(28a)}& {\displaystyle}q\left(z\right)={\displaystyle \frac{\sum {m}_{\mathrm{p},i}{u}_{\mathrm{p},i}}{{L}_{x}{L}_{y}\mathrm{\Delta}z}}\text{(28b)}& {\displaystyle}Q={\displaystyle \frac{\sum {m}_{\mathrm{p},i}{u}_{\mathrm{p},i}}{{L}_{x}{L}_{y}}}\text{(28c)}& {\displaystyle}{m}_{\mathrm{c}}\left(z\right)={\displaystyle \frac{\sum {m}_{\mathrm{p},i}}{{L}_{x}{L}_{y}\mathrm{\Delta}z}}\text{(28d)}& {\displaystyle}\langle {\mathit{\zeta}}_{\mathrm{p}}\rangle \left(z\right)={\displaystyle \frac{\sum {\mathit{\zeta}}_{\mathrm{p},i}{m}_{\mathrm{p},i}{u}_{\mathrm{p},i}}{\sum {m}_{\mathrm{p},i}{u}_{\mathrm{p},i}}},\end{array}$$

where the summation ∑ is performed over the saltating particles
located in the range of $\left[z,z+\mathrm{\Delta}z\right]$ for *q*, *m*_{c}, and 〈*ζ*_{p}〉, but it is performed over all saltating particles for *Q*. Here, we define
the 〈*ζ*_{p}〉 as the ratio of charge flux and mass flux in the range of $\left[z,z+\mathrm{\Delta}z\right]$.

We consider polydisperse soft-spherical sand particles to have a log-normal
mass distribution in a 3D computational domain $\mathrm{0.5}\phantom{\rule{0.125em}{0ex}}\mathrm{m}\times \mathrm{0.1}\phantom{\rule{0.125em}{0ex}}\mathrm{m}\times \mathrm{1.0}\phantom{\rule{0.125em}{0ex}}\mathrm{m}$ (as shown in Fig. 4a), with periodic boundary conditions in the *x* and *y* directions. Here, the upper boundary is set to be high enough so that the particle escapes from the upper boundary can be avoided. To reduce the computational cost, the spanwise dimension is chosen as *L*_{y}=0.1, since the saltating particles are mainly moving along the streamwise direction.

As shown in Fig. 4b, the model is initiated by randomly releasing 100
uncharged particles within the region below 0.3 m, and then, these released
particles begin to move under the action of the initial log-law wind flow,
triggering saltation through a series of particle–bed collisions. We use
cell-based collision-searching algorithms, which perform a collision search
for particles located in the target cell and its neighbouring cells, to find
the midair colliding pairs. The random processes, particle–bed collisions
described previously, are simulated using a general method called the
inverse transformation. The particle motion and wind flow equations are
integrated by predictor-corrector method AB3-AM4; that is, the third-order
Adams–Bashforth method is used to perform the prediction and fourth-order Adams–Moulton method is used to perform the correction. One of the main advantages of using such a multi-step integration method is that the accuracy of the results is not sensitive to the detection of exact moments of collision (Tuley et al., 2010). The charge transfer between the colliding pairs is caused by their asymmetric contact and can be determined by Eqs. (25)–(27). When calculating the particle–bed charge transfer, the bed is regarded as an infinite plane. According to the law of charge conservation, the surface charge density of the infinite bed plane and the newly ejected particles, *σ*, is (Kok and Renno, 2008; Zhang et al., 2014) as follows:

$$\begin{array}{}\text{(29)}& \mathit{\sigma}=-\underset{{z}_{\mathrm{0}}}{\overset{+\mathrm{\infty}}{\int}}{\mathit{\rho}}_{\mathrm{c}}\left(z\right)\mathrm{d}z,\end{array}$$

where *ρ*_{c} is the space charge density. To model pure
saltation, the ** E** field is calculated by Gauss's law (e.g. Zhang et al., 2014). To model saltation during dust storms, the 3D

4 Results

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On 6 May 2014, field measurements began at ∼ 12:00 UTC+8 due to the
limited power supply by solar panels. As shown in Fig. 5, although the early
stage of the dust storm has not been observed completely, we successfully
recorded data of about 8 h, which is substantial enough to reveal the
pattern of the 3D ** E** field. From Fig. 5, it can be seen that the relative magnitudes of

As shown in Fig. 6a–c, in different periods, each component of the
normalized 3D ** E** field roughly collapses on a single third-order polynomial curve (with

Before quantifying the effects of the 3D ** E** field on saltation by our numerical model, we draw a comparison of several key physical quantities between the simulated results and measurements in the case of pure saltation in order to ensure the convergence and validity of our numerical code, as shown in Fig. 7a–c. It is clearly shown that the saltation eventually reaches a dynamic steady state after ∼4 s. The number of the
impacting particles (∼72 grains) is equal to the sum of the
rebounding (∼50 grains) and the ejected particles
(∼22 grains) during the time interval of 10

In addition to affecting sand transport, midair collisions also affect charge exchanges between saltating particles. When considering midair collisions, the charge-to-mass ratio distribution shifts slightly towards zero as the wind velocity increases, as shown in Fig. 8a–c. As the wind speed increases, the difference in the charge-to-mass ratio distribution between the cases with and without midair collisions is increasingly notable. This is because the probability of midair collisions becomes more significant for larger wind speed (Sørensen and McEwan, 1996; Huang et al., 2007).

By substituting the formulations of the 3D ** E** field (i.e. $\u2329{\stackrel{\mathrm{\u203e}}{E}}_{i}\u232a{E}_{i}^{\ast}$, $i=\mathrm{1},\mathrm{2},\mathrm{3}$) into our model (i.e. Eq. 11a), we then properly evaluated the effects of the 3D

Additionally, we also explore how the key parameter, the density of charged
species ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$, affects saltation, as shown in Fig. 11a–c. Since the height-averaged time-varying mean is strongly dependent on the ambient conditions, such as temperature and RH, the height-averaged time-varying mean is set at two different levels. The predicted results show that, at each height-averaged time-varying mean level, the magnitude of the mean charge-to-mass ratio increases with increasing ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$ and then reaches a relative equilibrium value at approximately ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}={\mathrm{10}}^{\mathrm{18}}$ m^{−2} (Fig. 11a), thus leading to the constant enhancement of total mass flux *Q* and saltation height *z*_{salt} (Fig. 11b and c). From Eqs. (25)–(26), it can be seen that the net charge transfer Δ*q*_{ij} is proportional to the initial density ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$, so that 〈*ζ*_{p}〉 increases rapidly with increasing ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$ for small ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$. However, for larger ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$, Δ*q*_{ij} is no longer proportional to ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$ because, in this case, the difference in the number of trapped electrons between two colliding particles (i.e. ${\mathit{\rho}}_{\mathrm{h}}^{j}{S}_{j}-{\mathit{\rho}}_{\mathrm{h}}^{i}{S}_{i}$) has the same value, and ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$ is not the key parameter for determining the mean charge-to-mass ratio (Kok and Lacks, 2009). Figure 11c shows a peak of increase in *z*_{salt} at ${\mathit{\rho}}_{\mathrm{h}}^{\mathrm{0}}$ of about 10^{16}–10^{17} m^{−2} because
〈*ζ*_{p}〉 also exhibits a peak in the same range of ${\mathit{\rho}}_{h}^{\mathrm{0}}$. In addition, the peak is more apparent in Fig. 11c. This is because *z*_{salt} is very sensitive to the mass flux profile. A little change in mass flux profile can lead to an apparent change in *z*_{salt} (see Text S1 in the Supplement). For the larger height-averaged time-varying mean, the enhancements in the total mass flux *Q* and saltation height *z*_{salt} could exceed 20 % and
15 %, respectively.

5 Discussion

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To determine the effects of particle triboelectric charging on saltation
precisely, 3D ** E** field measurements in the saltation layer (i.e. sub-metre above the ground) are required. Although the

In ** E** field data analysis, the

Like many previous studies, the ** E** field can be simplified to 1D (i.e.
vertical component) in pure saltation (e.g. Kok and Renno, 2008) since, in
such cases, the magnitude of the streamwise and spanwise components is much
less than that of the vertical component (Zhang et al., 2014). However, during dust storms, the magnitudes of the streamwise and spanwise components of the

Additionally, one possible explanation for the intense streamwise and
spanwise ** E** fields is that large- and very large-scale motions exist in atmospheric surface flows, leading to a large-extent charge segregation in
the streamwise and spanwise directions. In atmospheric surface layer flows,
the largest vortices or coherent motions of the wind flows are found to be
compared to the boundary layer thickness (∼60–200 m; Kunkel
and Marusic, 2006; Hutchins et al., 2012). This may lead to a phenomenon
in which the charged particles are more non-uniformly distributed (over a larger spatial scale) in the streamwise and spanwise directions than in the
vertical direction. Accordingly, the intensities of the streamwise and
spanwise

Although most physical mechanisms, such as asymmetric contact, polarization
by external ** E** fields, statistical variations in material properties, and shift of aqueous ions, are responsible for particle triboelectric charging, contact or triboelectric charging is the primary mechanism (e.g. Lacks and Sankaran, 2011; Zheng, 2013; Harrison et al., 2016). In the previous model, however, the charge-to-mass ratios of the saltating particles are either assumed to be a constant value (e.g. Schmidt et al., 1998; Zheng et al., 2003; Zhang et al., 2014) or are not accounted for in the particle–particle midair collisions (e.g. Kok and Renno, 2008). In this study, by using DEM together with an asymmetric contact electrification model, we account for the particle–particle triboelectric charging during midair collisions in saltation. The DEM implemented by cell-based algorithms is effective for detecting and evaluating most of the particle–particle midair collisional dynamics (Norouzi et al., 2016). Meanwhile, the charge transfer between colliding particles can be determined by Eqs. (25) and (26). Compared to the previous studies (e.g. Kok and Lacks, 2009), the main innovation of this model is that the comprehensive consideration of the particle collisional dynamics affecting particle charge transfer is involved. In summary, the present model is a particle–particle midair collision resolved model, and the predicted charge-to-mass ratio agrees well with the published measurement data (see Fig. 7c). These findings indicate that midair
collisions in saltation are important, both in momentum and charge
exchanges.

One limitation of our model is that the effects of turbulent fluctuations on particle charging and dynamics are not explicitly accounted for. In actual conditions, saltation is unsteady and inhomogeneous at small scales, and the wind flow is mathematically described by the continuity and Navier–Stokes equations. However, in many cases, wind flow is statistically steady and homogeneous over a typical timescale of 10 min (Durán et al., 2011; Kok et al., 2012). For example, in the relatively stationary period in Fig. 5, all long period-averaged statistics become independent of time. In this case, the governing equations of the wind flow can be reduced to a simple model described by equation Eq. (13). There is no doubt that 3D turbulent fluctuations could affect particle charging and dynamics considerably (e.g. Cimarelli et al., 2014; Dupont et al., 2013). Further work is therefore needed to incorporate turbulence into the numerical model.

It is generally accepted that the ** E** field could considerably affect the lifting and transport of sand particles. As in the findings of previous 1D

However, a remaining critical challenge is still to simulate particle
triboelectric charging in dust storms precisely. The driving atmospheric
turbulent flows, having a typical Reynolds number of the order of 10^{8},
cover a broad range of length and timescales, which need huge
computational cost to resolve (e.g. Shao, 2008). On the other hand, particle
triboelectric charging is so sensitive to the particle's collisional dynamics
that it needs to resolve each particle's collisional dynamics (e.g. Lacks and
Sankaran, 2011; Hu et al., 2012). To model the particle's collisional
dynamics properly, the time steps of DEM are generally from 10^{−7} to
10^{−4} s (Norouzi et al., 2016). However, steady state saltation motion
often requires several seconds to several tens of seconds to reach the
equilibrium state. In this study, when ${u}_{\ast}=\mathrm{0.5}$ m s^{−1} and the computational domain is $\mathrm{0.5}\times \mathrm{0.1}\times \mathrm{1.0}$ m^{3}, the total number of saltating particles exceeds 7×10^{4} (Fig. S6 in
the Supplement). Consequently, the triboelectric charging in saltation is
currently very difficult to simulate, where a large number of polydisperse
sand particles, the high Reynolds number turbulent flow, and the
interparticle electrostatic forces are mutually coupled. In the present
version of the model, we neither consider the particle–particle interactions, such as particle agglomeration and fragmentation, during particle collision, frictional contact, nor the particle–turbulence interaction that is the effect of turbulent fluctuations on the triboelectric charging and dynamics of particles. Further studies require considerable effort to incorporate these interactions into a tractable numerical model, especially turbulence, which is very important for large wind velocity.

6 Conclusions

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Severe dust storms occurring in arid and semiarid regions threaten human
lives and result in substantial economic damages. An intense ** E** field up to ∼100 kV m

We have also performed discussions about various sensitive parameters such
as the density of charged species, the coefficient of restitution, and the
height-averaged time-varying mean of the 3D ** E** field. These results
add significant new knowledge on the role of particle triboelectric
charging in determining the transport and lifting of sand and dust
particles. A great effort is further needed to understand the
interactions such as particle agglomeration and fragmentation and
the effects of the turbulence on the triboelectric charging and dynamics of
particles.

Data availability

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Data availability.

The ** E** field data recorded in our field campaign are provided as a CSV file in the Supplement.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/acp-20-14801-2020-supplement.

Author contributions

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Author contributions.

HZ performed the field observations, numerical simulation, and data analyses and wrote the paper, which was guided and edited by YHZ. Both authors discussed the results and commented on the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We thank the editor and anonymous reviewers for their insightful comments that greatly improved the final paper. This work was supported by the National Natural Science Foundation of China (grant no. 11802109), the Young Elite Scientists Sponsorship Program by the China Association for Science and Technology (CAST; grant no. 2017QNRC001), and the Fundamental Research Funds for the Central Universities (grant no. lzujbky-2018-7).

Financial support

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Financial support.

This research has been supported by the National Natural Science Foundation of China (grant no. 11802109), the Young Elite Scientists Sponsorship Program by CAST (grant no. 2017QNRC001), and the Fundamental Research Funds for the Central Universities (grant no. lzujbky-2018-7).

Review statement

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Review statement.

This paper was edited by Ulrich Pöschl and reviewed by four anonymous referees.

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Short summary

We assess the effects of triboelectric charging on wind-blown sand via observations and a numerical model. The 3D electric field within a few centimetres of the ground is characterized for the first time. By using the discrete element method together with the particle-charging model, we explicitly account for the particle–particle charging during collisions. We find that triboelectric charging could enhance the total mass flux and saltation height by up to 20 % and 15 %, respectively.

We assess the effects of triboelectric charging on wind-blown sand via observations and a...

Atmospheric Chemistry and Physics

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