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Special issue: Dust aerosol measurements, modeling and multidisciplinary...

**Research article**| 07 Apr 2022

# Large-eddy-simulation study on turbulent particle deposition and its dependence on atmospheric-boundary-layer stability

Xin Yin Cong Jiang Yaping Shao Ning Huang and Jie Zhang

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^{1},

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^{2,3},

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**Xin Yin et al.**Xin Yin Cong Jiang Yaping Shao Ning Huang and Jie Zhang

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^{1}Institute for Geophysics and Meteorology, University of Cologne, 50969 Cologne, Germany^{2}Key Laboratory of Mechanics on Disaster and Environment in Western China, Lanzhou University, 730000 Lanzhou, China^{3}College of Civil Engineering and Mechanics, Lanzhou University, 730000 Lanzhou, China

^{1}Institute for Geophysics and Meteorology, University of Cologne, 50969 Cologne, Germany^{2}Key Laboratory of Mechanics on Disaster and Environment in Western China, Lanzhou University, 730000 Lanzhou, China^{3}College of Civil Engineering and Mechanics, Lanzhou University, 730000 Lanzhou, China

**Correspondence**: Jie Zhang (zhang-j@lzu.edu.cn)

**Correspondence**: Jie Zhang (zhang-j@lzu.edu.cn)

Received: 29 Sep 2021 – Discussion started: 29 Nov 2021 – Revised: 16 Feb 2022 – Accepted: 28 Feb 2022 – Published: 07 Apr 2022

It is increasingly recognized that atmospheric-boundary-layer stability (ABLS) plays an important role in eolian processes. While the effects of ABLS on particle emission have attracted much attention and been investigated in several studies, those on particle deposition have so far been less well studied. By means of large-eddy simulation, we investigate how ABLS influences the probability distribution of surface shear stress and hence particle deposition. Statistical analysis of the model results reveals that the shear stress can be well approximated using a Weibull distribution, and the ABLS influences on particle deposition can be estimated by considering the shear stress fluctuations. The model-simulated particle depositions are compared with the predictions of a particle-deposition scheme and measurements, and the findings are then used to improve the particle-deposition scheme. This research represents a further step towards developing deposition schemes that account for the stochastic nature of particle processes.

Dry deposition is the removal of particulates and gases at the air–surface interface by turbulent transfer and gravitational settling (Sehmel, 1980; Droppo, 2006; Hicks et al., 2016). Because it is the only process for the removal of particles from the atmosphere in the absence of precipitation, developing reliable methods for estimating dry deposition of particles has attracted much interest since the early 1940s (Gregory, 1945; Chamberlain, 1953; Slinn and Slinn, 1980; Slinn, 1982; Walcek et al., 1986; Zhang et al., 2001; Petroff and Zhang, 2010; Zhang and Shao, 2014; Seinfeld et al., 2016). Several particle-deposition schemes have been proposed (Slinn, 1982; Walcek et al., 1986; Zhang and Shao, 2014; Zhang et al., 2001) for regional and global models, which are driven using several environmental parameters, including the Reynolds surface shear stress (typically averaged over 15–30 min). However, field observations indicate that the use of Reynolds stress as the only wind-related parameter in such schemes may not be sufficient to achieve accurate estimates of particle deposition because of the nonlinear relationship between deposition velocity and wind shear. Observations using the eddy correlation method show that particle-deposition velocity has strong spatiotemporal variations associated with the fluctuations of wind speed (Connan et al., 2018; Damay et al., 2009; Lamaud et al., 1994; Wesely et al., 1983, 1985). It is also observed that when the background wind speeds are similar, dry deposition velocities under convective conditions are larger than under neutral and stable conditions (Fowler et al., 2009). Pellerin et al. (2017) suggested that cospectral similarities exist between heat and particle-deposition fluxes and that atmospheric turbulence plays a role in particle deposition. It is therefore necessary to find a link between instantaneous wind and particle deposition and to correctly represent this link in particle-deposition schemes, i.e., to introduce and account for the effect of turbulence on particle deposition.

Some eolian processes, e.g., turbulent particle emission (Klose and Shao, 2012, 2013) and intermittent saltation (Li et al., 2020; Liu et al., 2018; Rana et al., 2020), have been under development. To the best of our knowledge, although turbulent particle deposition is now perceived to be important, a scheme is yet to be constructed for its quantitative estimate.

The turbulent wind flow in a particle-deposition scheme is reflected in the turbulent shear stress (or vertical momentum flux) (Fowler et al., 2009; Zhang and Shao, 2014). It is well known that apart from gravitational settling, particle deposition is driven by turbulent diffusion, which is intimately related to the vertical momentum transfer in the atmospheric boundary layer (ABL) (Wyngaard, 2010). Based on the Prandtl mixing-length theory, the shear stress can be parameterized in neutral conditions. However, it is known that for a given mean wind speed (at a reference height) in the ABL, both the mean value and the perturbations of shear stress depend on the atmospheric-boundary-layer stability (ABLS); for instance, shear stress shows generally larger fluctuations in convective ABLS. Klose and Shao (2013) pointed out the following:

In a convective atmospheric boundary layer, large eddies have coherent structures of dimensions comparable to boundary-layer depth, which are efficient entities in generating localized momentum fluxes to the surface. Although the eddies only occupy fractions of time and space, the momentum fluxes to these fractions can be many times the average. (p. 49)

Hicks et al. (2016) mentioned that ABLS is of immediate concern in the micrometeorological community because of its influences on the intermittency, gustiness and diurnal cycle of particle deposition. Similar to turbulent dust emission and intermittent sand saltation, intermittent particle deposition also occurs as a result of fluctuating surface shear stress. The current particle-deposition schemes only consider the mean behavior of wind (Slinn, 1982; Zhang and Shao, 2014; Zhang et al., 2001) and how this mean behavior varies with ABLS via the Monin–Obukhov similarity theory (Monin et al., 2007; Monin and Obukhov, 1954) but not the fluctuations of the associated shear stress and how they vary with ABLS.

We argue that focusing only on the effects of ABLS on mean wind is insufficient to accurately model particle deposition. In this study, we explore the influences of ABLS on the turbulent behavior of particle deposition and attempt to improve an existing particle-deposition scheme. A large-eddy-simulation (LES) model is used here to simulate turbulence and particle deposition under various ABLS conditions, and parts of the study design follow Klose and Shao (2013). The particle depositions simulated using the LES model and predicted using the particle-deposition scheme of Zhang and Shao (2014, ZS14 hereafter) are compared with each other and with measurements. Specifically, we address the following three issues: (1) how ABLS affects the probability distribution of surface shear stress, (2) how ABLS impacts particle deposition and (3) how the ZS14 scheme can be improved to account for the ABLS effect. On this basis, an improvement to the ZS14 scheme (also applicable to other schemes) is proposed. The remaining part of the paper is organized as follows: Sect. 2 gives a brief description of the Weather Research and Forecasting – Large-Eddy-Simulation Model with Dust module (WRF-LES/D), the ZS14 scheme and the design of the numerical experiments. Section 3 discusses the findings of the numerical simulations and the improvement to the ZS14 scheme. The concluding remarks are given in Sect. 4.

## 2.1 WRF-LES/D

The WRF-LES/D used here is initially developed by
Shao et al. (2013) and
Klose and Shao (2013) by coupling the WRF-LES
(Moeng et al., 2007; Skamarock et al., 2008) with a
land-surface module and dust module. As demonstrated in the earlier studies,
WRF-LES/D is a well-established system for applications of simulating
turbulence, turbulent particle emission and transport for various ABLS
conditions. WRF-LES is a three-dimensional and non-hydrostatic model for
fully compressible flow. The model separates the turbulent flow into a
grid-resolved component and a subgrid component. The *k*−*l* subgrid closure
(Deardorff, 1980) together with the turbulent kinetic
energy (TKE) equation (Skamarock et al., 2008) is used
here. The governing equations in WRF-LES/D include the equations of motion,
continuity equation, enthalpy equation, equation of state and the particle
conservation equation, as shown below:

where ${u}_{i}(u,v,w)$ is the grid-resolved flow velocity along *x*_{i} (*x*,
*y*, *z*), referring to the streamwise, spanwise and vertical directions,
respectively; *g* is the acceleration due to gravity; *ρ*_{a} is the air
density; *f* is the Coriolis parameter; *p* is the air pressure; *τ*_{ij} is
the subgrid stress tensor modeled using an eddy viscosity approach, where the
eddy viscosity is represented as the product of a length scale and a
velocity scale characterizing the subgrid-scale (SGS) turbulent eddies
(Dupont et al., 2013), with the
velocity scale being derived from the SGS TKE and the length scale from the
grid spacing (Skamarock et al., 2008); *ν* is the
kinematic viscosity; *δ*_{i3} is the Kronecker operator; *ε*_{i3} is the alternating operator; *c*_{p} is the specific heat
of air at constant pressure; *T* is air temperature; *H*_{j} is the *j*th component
of subgrid heat flux; *R*_{a} is the specific gas constant of air; *c* is
particle concentration; *w*_{t} is the particle terminal velocity; *F*_{j} is
the *j*th component of subgrid particle flux; and *s*_{T} and *s*_{r} are the
source or sink terms for heat and particles, respectively. The subgrid eddy
diffusivity is set to subgrid eddy viscosity divided by the Prandtl number.
For the surface layer, an important parameterization to solve the governing
equations for high-Reynolds-number turbulence is embedded in the surface
boundary condition, which computes the instantaneous local surface shear
stress using the bulk-transfer method
(Kalitzin
et al., 2008; Kawai and Larsson, 2012; Piomelli et al., 2002; Zheng et al.,
2020) as follows:

with

where *K*_{m} is the eddy viscosity, *φ*_{m} is the Monin–Obukhov similarity theory (MOST) stability
function, $V=\sqrt{{u}^{\mathrm{2}}+{v}^{\mathrm{2}}}$, and *κ* is the von Karman constant.
Even though Shao et al. (2013)
questioned the application of the MOST in LES, it is still used here, as our
emphasis is on the variance of shear stress in the simulation domain.
Several land-surface models (LSMs) can be selected (e.g.,
Chen and Dudhia, 2000; Pleim and Xiu, 2003)
in WRF-LES/D, and the 5-layer thermal diffusion (Dudhia, 1996) is
used in this study. Furthermore, the surface heat flux, denoted *H*_{0}, is
specified. The dry deposition flux to the ground for each grid, denoted
*F*_{d}, is obtained by multiplying the deposition velocity *V*_{d} and
particle concentration *c* in the lowest layer, and *V*_{d} is estimated using
the ZS14 deposition scheme.

## 2.2 Particle-deposition scheme of ZS14

The particle deposition on the surface is more complicated than the momentum
flux as the change of particle concentration close to the surface is
unclear. To solve the particle conservation equation (Eq. 5), the emission
and deposition fluxes at the surface need to be specified. The problem of
particle emission has been dealt with elsewhere (e.g.,
Shao, 2004, focuses on the particle emission without
turbulence effects; Klose
et al., 2014, and Klose and Shao, 2012, emphasize the turbulent particle
emission) and is not considered here. For our purpose, particle emission is
assumed to be zero. This section gives the parameterization scheme proposed
by ZS14. The details of the scheme are described in ZS14; only the main
results are given here for completeness. In general, we can express the particle-deposition flux *F*_{d} as

where *K*_{p} and *k*_{p} are the eddy diffusivity and the molecular
diffusivity, respectively. By analogy with the bulk-transfer formulation of
scalar fluxes in ABL, *F*_{d} can be parameterized as

where *c*(*z*) is the particle concentration at height *z* (the center height of the
lowest model level in this study), and *V*_{d}(*z*) is the corresponding dry
deposition velocity.

The surface layer is divided into an inertial layer and a roughness layer.
Integrating Eq. (8) into the inertial layer and substituting Eq. (9) into it,
*V*_{d}(*z*) is obtained as follows:

with *r*_{g} being the gravitational resistance, *r*_{s} being the collection
resistance, and *r*_{a} being the aerodynamic resistance for the inertial
layer.

The gravitational resistance *r*_{g} is defined as the reciprocal of the
gravitational settling velocity *w*_{t} and depends mainly on particle size
and density. A free-falling particle is subject to gravitational and
aerodynamic drag forces. When these forces are in equilibrium, the
gravitational settling velocity of the particle smaller than 20 µm can
be reasonably accurately calculated according to the Stokes formula
(Malcolm and Raupach, 1991; Seinfeld and Pandis, 2006):

where *D*_{p} is the particle diameter, *ρ*_{p} is the particle
density, *μ*_{a} is the air dynamic viscosity, and *C*_{u} is the Cunningham
correction factor that accounts for the slipping effect affecting the fine
particles.

Using the MOST, the aerodynamic resistance is calculated as

where *z*_{d} is the displacement height, *h* is the height of roughness element,
*ψ*_{m} is the integral of stability function in the inertial layer, and $S{c}_{T}={K}_{m}/{K}_{p}$ (Csanady, 1963).

In the roughness layer, the collection process is reflected in collection resistance, defined by ${r}_{s}=-\frac{c\left(h\right)}{{F}_{\mathrm{d}}}$, with an assumption that particle concentration is zero on roughness elements or the ground. In addition to the meteorological factors and land-use category, Zhang and Shao (2014) established a relationship between aerodynamic and surface collection processes using an analogy between the drag partition and deposition flux partition, which can describe surface heterogeneity.

where *R* is the reduction in collection caused by particle rebound; *u*_{h} is
the wind speed at the top of roughness layer; *E* is the collection coefficient
of roughness elements, and it includes the collection efficiency from
Brownian motion (*E*_{B}), impaction (*E*_{im}), and interception
(*E*_{in}); *C*_{d} is the drag coefficient for isolated roughness element;
${\mathit{\tau}}_{c}/\mathit{\tau}$ describes the drag partition, with *τ*_{c} being
the pressure drag (the force exerted on roughness elements); *Sc* is the
Schmidt number, which is the ratio of air viscosity to molecular diffusion; and
${\mathrm{10}}^{-\mathrm{3}/\widehat{T}}{T}_{\mathrm{p},\mathit{\delta}}^{+}$ represents
the turbulent impaction efficiency, with $\widehat{T}$ being the dimensionless
particle relaxation time. *E*_{B}, *E*_{im}, *E*_{in}, and *R* are expressed as

where *Re* is the roughness element Reynolds number, *C*_{B} and *n*_{B} are
parameters depending on *Re*, *d*_{c} is the diameter of the roughness
element, and *St* is the Stokes number.

The ratio ${\mathit{\tau}}_{c}/\mathit{\tau}$ can be calculated according to Yang and Shao (2006), as follows:

and

with *β*_{1}(= 200) being the ratio of the drag coefficient for
isolated roughness element to that for bare surface, *λ* being the
frontal area index of the roughness elements, and *η* being the basal
area index or the fraction of cover.

From Eqs. (10) to (19), it can be seen that *V*_{d} and *τ* are
nonlinearly related. For example, for particles with a diameter of 1 µm,
analysis shows that *V*_{d} is dominated by *w*_{t} when *τ* is small. As
*τ* increases, *w*_{t} and *τ* are both important to *V*_{d}. With
*τ* increasing further, the effect of *τ* becomes much greater than
gravitational settling; thus the *V*_{d} is mainly determined by *τ*.

## 2.3 Simulation setup

Numerical experiments are carried out with WRF-LES/D for various atmospheric
stability and background wind conditions for two different roughness lengths
(Table 1). Similar to Klose and Shao (2013),
the domain of the simulation is $\mathrm{2000}\times \mathrm{2000}\times \mathrm{1500}$ m^{3}, and the
number of grid points is $\mathrm{200}\times \mathrm{200}\times \mathrm{90}$, corresponding to a
horizontal resolution $\mathrm{\Delta}x=\mathrm{\Delta}y=\mathrm{10}$ m. The Arakawa-C
staggered grid is used. The depth of the lowest model layer is 1 m, and the
grid above is stretched following a logarithmic function of *z*. The simulation
time is 90 min, with a time step of 0.05 s, and the output interval is 10 s. The first 30 min of the simulation is the model spin-up time, and the
data of the remaining 60 min are used for the analysis.

For model initialization, the wind and particle
(*ρ*_{p}=2650 kg m^{−3}) concentration
(Chamberlain, 1967;
Monin, 1970; Kind, 1992) are assumed to be logarithmic in the vertical and
uniform in the horizontal direction. For each experiment, a constant surface
heat flux is specified. A 300 m deep Rayleigh damping layer is used at the
upper boundary with a damping coefficient of 0.01. The wind speed at the top
boundary, *U*, is given in Table 1. The surface heat flux, *H*_{0}, increases
from −50 to 600 W m^{−2}, and for each surface heat flux, the top wind
speed increases from 4 to 16 m s^{−1} in Exp (1–20) and from 5.44 to 18.12 m s^{−1} in Exp (21–35). The roughness length *z*_{0} for sand surface used
in Exp (1–20) is 0.153 mm following wind tunnel experiments
(Zhang and Shao, 2014) but 0.76 mm in Exp (21–35)
according to field observations
(Bergametti et al., 2018). The
lateral boundary conditions are periodic, which allows for the simulation of a
well-developed boundary layer. The vertical scaling velocity is estimated
using the heat flux, ${w}_{\ast}={\left(\frac{g}{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}\frac{{H}_{\mathrm{0}}}{{\mathit{\rho}}_{\mathrm{a}}{c}_{\mathrm{p}}}\phantom{\rule{0.125em}{0ex}}{z}_{l}\right)}^{\frac{\mathrm{1}}{\mathrm{3}}}$, with
$\stackrel{\mathrm{\u203e}}{\mathit{\theta}}$ being the mean potential temperature and *z*_{l}=1000 m the boundary-layer inversion height. Usually, *w*_{∗} is not used for
stable ABLS, but it is used here as an indicator for the suppression of turbulence
by negative buoyancy.

## 3.1 Turbulent shear stress

In the first set of analyses, we examine the impact of atmospheric stability
on shear stress fluctuations. Early particle-deposition studies considered
only the time average of surface shear stress, *τ*_{r}, with the
assumption that shear stress is horizontally homogeneous. In WRF-LES/D, the
corresponding mean resultant shear stress *τ*_{r} can be obtained as

The shorthand notation $\stackrel{\mathrm{\u203e}}{f}=\frac{\mathrm{1}}{{N}_{x}{N}_{y}{N}_{t}}\sum _{{n}_{x}{n}_{y}{n}_{t}}f({n}_{x},{n}_{y},{n}_{t})$ is introduced
to represent the space and time average over the simulation domain and time
period (hereafter ensemble mean) with *N*_{x} (i.e., 200) and *N*_{y} (i.e., 200)
being the numbers of grid points in the *x* and *y* direction, respectively, and
*N*_{t} (i.e., 360) the time steps of model output.

Figure 1a–c show the instantaneous shear stress, *τ*, of a sample grid
(*n*_{x}=198, *n*_{y}=41) over a 1 h period for the runs with
*z*_{0}=0.153 mm, *U*=4 m s^{−1} and various ABL stabilities
(*H*_{0}=0, 200, 600 W m^{−2}). Figure 1d–f are the same as Fig. 1a–c but
for *U*=16 m s^{−1}. The panel shows that *τ* is not a constant, and
the mean resultant shear stress, as well as the shear stress fluctuations,
increases with increasing atmospheric instability. In addition, the inset
plots in Fig. 1 show that the autocorrelation function (ACF) oscillates
during the decrease. The oscillation periodicity is longer under weak wind
conditions (Fig. 1a–c) than strong wind (Fig. 1d–f). The ACF in neutral
conditions decreases more rapidly than in convective conditions. Recall the
definition of coherent motion given by Robinson (1991) –
the correlation of variables over a range of long time larger than the
smallest scales of flow is evidence of coherent oscillating motion. Thus,
the regular oscillation and a long-time correlation of *τ* are closely
related to the evolvement of the coherent structure. This indicates that in
a convective ABL, stronger large-scale coherent structures exist, even under
weak wind conditions.

To gain insight into the behavior of the unsteady shear stress field, we
introduce the turbulence intensity of surface shear stress (TI-S), defined as
the ratio of the standard deviation of fluctuating surface shear stress,
*σ*_{τ}, to the mean resultant stress *τ*_{r}, i.e., ${\mathit{\sigma}}_{\mathit{\tau}}/{\mathit{\tau}}_{\mathrm{r}}$. Analysis shows that ${\mathit{\sigma}}_{\mathit{\tau}}/{\mathit{\tau}}_{\mathrm{r}}$
increases as atmospheric conditions become more unstable and decreases with
increasing wind speed (e.g., Fig. 1). High wind speeds tend to force the
ratio to be more similar to that in neutral ABLS, as the mean-wind-induced
shear stress becomes dominant over the large-eddy-induced shear stress
fluctuations. For a weak TI-S, *τ* is dominated by *τ*_{r}, and the
stress fluctuations are small compared to *τ*_{r}. As TI-S increases,
the contribution of momentum transport by large eddies becomes significant
because in unstable ABLS, buoyancy-generated large eddies penetrate to high
levels and intermittently enhance the momentum transfer to the surface.

The intermittent surface shear stress can directly cause localized particle
deposition. Therefore, particle deposition is also intermittent in space and
time. However, to the best of our knowledge, in existing particle-deposition schemes
(e.g., ZS14 used here), the particle-deposition velocity is calculated using
only the mean resultant shear stress *τ*_{r} instead of the
instantaneous shear stress. We denote this deposition velocity as ${V}_{\mathrm{d},{\mathit{\tau}}_{\mathrm{r}}}$. The mean deposition velocity simulated by WRF-LES/D, denoted as
*V*_{d,LES}, is estimated via the ratio of the ensemble mean of particle-deposition flux and the ensemble mean of particle concentration:

which is consistent with the methods commonly used in field observations and wind tunnel experiments.

Figure 2a and b, with the same wind conditions and surface heat fluxes as
in Fig. 1c and f, show the time evolution of the instantaneous deposition
velocity *V*_{d} for particles with a diameter of 1.46 µm. This size is
chosen because it is the most sensitive to turbulent diffusion compared to
the other four sizes (2.8, 4.8, 9, 16 µm) used in Exp (1–20). As shown,
the fluctuating behavior of *V*_{d} is consistent with that of *τ*.
Moreover, Fig. 2a shows a substantial difference between *V*_{d,LES} and
${V}_{\mathrm{d},{\mathit{\tau}}_{\mathrm{r}}}$, while Fig. 2b shows ${V}_{\mathrm{d},{\mathit{\tau}}_{\mathrm{r}}}$ is similar with
*V*_{d,LES}. This suggests that the ZS14 scheme can more accurately estimate
the deposition velocity for weak TI-S but underestimates the deposition
velocity for strong TI-S. The reason for this is that in the case of strong
TI-S, particle deposition caused by the gusty wind plays an important role
as *V*_{d} and *τ* are nonlinearly related, which is not reflected in
${V}_{\mathrm{d},{\mathit{\tau}}_{\mathrm{r}}}$. Since *τ* fluctuates and sometimes strongly, a bias
always exists in conventional particle-deposition schemes, and the magnitude
of the bias depends on turbulence intensity. Therefore, in order to estimate
particle deposition accurately, we need to first describe and parameterize
the shear stress.

As a main predisposing factor for eolian processes, turbulent shear stress
has attracted increasing attention in recent years
(e.g.,
Klose et al., 2014; Li et al., 2020; Liu et al., 2018; Rana et al., 2020;
Zheng et al., 2020). Similar to previous studies, we use the probability density function (pdf) *p*(*τ*) to characterize the stochastic variable *τ*.
Figure 3 shows that the variability of *τ* increases as atmospheric
instability increases in different wind conditions. The statistic moments of
*τ*, including its mean resultant value *τ*_{r}, standard deviation
*σ*_{τ}, and skewness *γ*_{1} of Exp (1–20) are listed in
Table 2. *σ*_{τ} and *τ*_{r} increase with increased
instability, and the distribution is positively skewed. Positive skewness is
characterized by the distribution having a longer positive tail as compared
with the negative tail, and the distribution appears as a left-leaning (i.e.,
tends toward low values) curve. This indicates that large negative
fluctuations are not as frequent as large positive fluctuations. The data
also show that *γ*_{1} generally shows a downward trend as TI-S
decreases, which is consistent with Monahan (2006); i.e., as TI-S decreases, *p*(*τ*) becomes increasingly Gaussian.

The parameterization of surface shear stress has attracted intense
interest; for example, Klose et al. (2014)
reported that *τ* in unstable conditions is Weibull-distributed based on
large-eddy simulations. Shao et al. (2020) found that *p*(*τ*) is skewed to small *τ* values (i.e.,
positively skewed) based on field observations.
Li et al. (2020) suggested that *τ* in
neutral conditions is Gauss-distributed based on a wind tunnel experiment.
Colella and Keith (2003) explained that in
turbulent shear flows, the nonlinear interaction between the eddies gives
rise to a departure from Gaussian behavior. Our results show that the
Gaussian approximation is inadequate in representing the skewed *p*(*τ*),
especially for the conditions of strong turbulence intensity (e.g., unstable
cases in Fig. 3a). Therefore, *p*(*τ*) here is approximated using a Weibull
distribution, i.e.,

where *α* and *β* are the shape and scale parameters,
respectively. The values of *α* and *β* for the numerical
experiments Exp (1–20) are listed in Table 2. It can be seen that both
*α* and *β* depend on wind speed and atmospheric stability.
However, *β* is mainly determined by wind conditions when the wind is
strong, while it is affected by ABL stability when the wind is weak. The
behavior of *α* and *β* is shown in Fig. 4. $\left|\mathrm{1}/{L}_{\mathrm{o}}\right|$ is the absolute value of the reciprocal of the Obukhov length
*L*_{o}, which can be calculated using

In both stable and unstable atmospheric conditions, analysis shows that the
scale parameter *α* is related to ABL stability as the power of
$\left|\mathrm{1}/{L}_{\mathrm{o}}\right|$. Figure 4a shows that *α*
decreases with the $\left|\mathrm{1}/{L}_{\mathrm{o}}\right|$, satisfying Eq. (24) approximately. For neutral conditions, *L*_{o} goes to infinity, and Eq. (24) no
longer applies. Therefore, the shape parameter obtained by the fitting was
directly used for pdf reproduction for the neutral cases instead of the
approximated *α* used for stable and unstable conditions. As Fig. 4b
shows, the *β* parameter increases almost linearly with ${u}_{\ast r}^{\mathrm{2}}+\mathrm{0.001}\cdot {w}_{\ast}^{\mathrm{2}}$ but can be best approximated using Eq. (25), with ${u}_{\ast r}=\sqrt{{\mathit{\tau}}_{\mathrm{r}}/{\mathit{\rho}}_{\mathrm{a}}}$.

Using Eqs. (22)–(25), we can approximately describe the turbulent surface shear stress in non-neutral cases.

## 3.2 Improvement to particle-deposition scheme

Figure 5a shows the performance of WRF-LES/D by comparing the simulated
deposition velocity, *V*_{d,LES}, with wind tunnel experiments
(Zhang and Shao, 2014) and field observation
(Bergametti et al., 2018). The
observed data are measured under neutral conditions and similar wind flow.
As shown, the simulation results agree well with the observed data. On this
basis, by further evaluating the performance of the ZS14 scheme, we found
that the accuracy of the ZS14 scheme decreases with increasing instability.
For example, Fig. 5b compares the deposition velocities of Exp (5, 9, 17)
and Exp (24, 27, 33), *V*_{d,LES}, with those calculated by the ZS14 scheme
using *τ*_{r} from the corresponding experiments, ${V}_{\mathrm{d},{\mathit{\tau}}_{\mathrm{r}}}$. It
shows that under weak wind conditions, ${V}_{\mathrm{d},{\mathit{\tau}}_{\mathrm{r}}}$ predicts the
deposition well under neutral conditions and underestimates the deposition
under convective conditions, especially for particles that are not dominated
by molecular diffusion and gravity, and the underestimation increases with
the atmospheric instability. To predict the deposition velocity more
accurately for convective conditions, we need to account for the effect of
shear stress fluctuations, i.e., the instantaneous shear stress
distribution. Thus, the dry deposition scheme can be improved as

with *p*(*τ*) as given by Eqs. (22)–(25). As Fig. 5c shows, the improved
scheme results *V*_{d,τ} and the simulation value *V*_{d,LES} show
a remarkable congruence.

To make the comparison more clear, the relative errors (REs) of the predicted deposition velocity by the ZS14 scheme and improved scheme are compared with the WRF-LES/D simulation value and are calculated as below:

Analysis shows that the value of relative error (RE) depends on surface
conditions, wind conditions, atmospheric stabilities, and particle sizes. It
increases obviously with increased atmospheric instability under weak wind
conditions, while it becomes less sensitive to stability when the wind is
strong. Through the analysis, we find that the RE of the ZS14 scheme generally
increases with the shear stress turbulence intensity, TI-S, and the value
depends on particle size, as shown in Fig. 5d (left). Thus, we compared the
RE of some different sized particles to investigate the particle in which size
range is strongly affected (Fig. A2). The result shows that the RE first increases
and then decreases with increasing particle size, and the particles with
size normally in the range of 0.01 to 5 µm are strongly affected by
turbulent shear stress, and *p*(*τ*) needs to be considered. After
modification, the errors are limited to less than 10 % or approximately 10 %. For example,
the relative error of Exp (17; i.e., *U* =4 m s^{−1} and *H*_{0}=600 W m^{−2}) for particles of 1.46 µm is reduced from ∼25 *%* to ∼3 *%*. The relative error of Exp (33; i.e., *U* =5.44 m s^{−1} and *H*_{0}=600 W m^{−2}) for particles of 0.5 µm
is reduced from ∼50 *%* to ∼12 *%*.

To further analyze if the RE of ZS14 in unstable conditions is dominated by
kinetic instability or dynamic instability, the Richardson number is
calculated. Analysis shows that TI-S is positively correlated to gradient
Richardson number *Ri* (Eq. A1). Under unstable conditions associated with
strong vertical motion and weak winds, the RE of ZS14 increases with the
increasing magnitude of Richardson number *Ri* (Fig. A3). The relationship
between *Ri* and TI-S needs further study. Consequently, the results
illustrate that the modified scheme *V*_{d,τ} tends to be more accurate
than the unmodified scheme ${V}_{\mathrm{d},{\mathit{\tau}}_{\mathrm{r}}}$.

The present study was designed to determine the effect of ABL stability on particle deposition. For this purpose, the WRF-LES/D was used to model atmospheric-boundary-layer turbulence under the presence of atmospheric stability effects to recover statistics of shear stress variability. We then presented an improved particle-deposition scheme with the consideration of turbulent shear stress. While ABLS can broadly represent levels of atmospheric turbulence, its effect on particle deposition is wind-speed-dependent. Through a series of numerical experiments, we have shown the turbulent characteristics of particle-deposition velocity caused by the turbulent wind flow and pointed out the shortcomings of the ZS14 scheme in representing particle deposition under convective conditions. The relative error (RE) increases as the ABL instability increases for low wind conditions; i.e., the RE increases with shear stress turbulence intensity, especially for a certain size range of particles.

Since the dependency of particle deposition on micrometeorology is embedded
in the application of the surface shear stress, we believe that the
dependency of particle deposition on ABL stability is ultimately attributed
to the statistical behavior of shear stress *τ*. Therefore, in this
study, a model including the effects of surface shear fluctuations is
proposed and validated by numerical experiments. Additionally, the
fluctuations of surface shear caused by turbulence can be approximated with
a Weibull distribution. The shape parameter decreases exponentially with the
reciprocal of Monin–Obukhov length, and the scale parameter increases
linearly with ${u}_{\ast r}^{\mathrm{2}}+\mathrm{0.001}{w}_{\ast}^{\mathrm{2}}$. After statistically
revising the original scheme, an improved model is obtained. Using the
modified model, the deposition velocity tends towards numerical experimental
results.

The project is the first comprehensive investigation of the turbulent
characteristics of particle deposition, and the findings will be of interest to
improve the accuracy of particle-deposition predictions on regional or
global scales. One source of weakness in this study is that the variation of
*τ* may be changed by surface roughness and needs further study, as the
roughness length does not fully reflect the effect of the surface topography
on the turbulence structure. In spite of this limitation, the study adds to
our understanding of the influence caused by ABLS on particle deposition.

Figure A1 shows the probability density distribution of surface shear stress
for experiments (21–35); Fig. A2 shows the changing of relative error with
particle size; Fig. A3 shows the variation of relative error (RE) of the
ZS14 scheme (Eq. 10) and improved scheme (Eq. 26) with gradient
Richardson number *Ri*.

where *z* is the center height of the lowest layer, and $\stackrel{\mathrm{\u203e}}{\mathit{\theta}}$ is the
potential temperature of the lowest layer.

The source code used in this study is the WRF-Chem version 3.7 in the LES mode coupled with a new deposition scheme. WRF-LES model can be downloaded at https://www2.mmm.ucar.edu/wrf/users/download/get_sources.html (WRF Users page, 2022). The code of the coupled deposition scheme and data set obtained by the simulation are available online at https://doi.org/10.5281/zenodo.6390432 (Yin et al., 2022).

XY, YP, and JZ were responsible for the formal analysis and methodology. XY and CJ were responsible for the data curation, software, validation, and visualization. YP, JZ, and NH were responsible for the supervision, project administration, and funding acquisition. XY was responsible for investigation and original draft preparation. XY, YP, JZ, and CJ were responsible for the review and editing of the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Dust aerosol measurements, modeling and multidisciplinary effects (AMT/ACP inter-journal SI)”. It is not associated with a conference.

We thank the Second Tibetan Plateau Scientific Expedition and Research Program (grant no. 2019QZKK020611) the State Key Program of the National Natural Science Foundation of China (grant no. 41931179), the China Scholarship Council (201606180041), the Fundamental Research Funds for the Central Universities (grant no. lzujbky-2020-cd06), and the Major Science and Technology Project of Gansu Province (grant no. 21ZD4FA010).

This research has been supported by the Second Tibetan Plateau Scientific Expedition and Research Program (grant no. 2019QZKK020611), the State Key Program of the National Natural Science Foundation of China (grant no. 41931179), the China Scholarship Council (grant no. 201606180041), the Fundamental Research Funds for the Central Universities (grant no. lzujbky-2020-cd06), and the Major Science and Technology Project of Gansu Province (grant no. 21ZD4FA010).

This paper was edited by Stelios Kazadzis and reviewed by Gilles Bergametti and one anonymous referee.

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