Dry Deposition Methods Based on Turbulence Kinetic Energy: Part 2. Extension to Particle Deposition Using a Single-Point Model

Abstract

Magnitude of atmospheric turbulence, a key driver of several processes that contribute to aerosol (i.e., particle) deposition, is underrepresented in current models. Various formulations have been developed to model particle dry deposition; all these formulations typically rely on friction velocity and some use additional ad hoc factors to represent enhanced impacts of turbulence. However, none were formally linked with the three-dimensional (3-D) turbulence. Here, we propose a set of 3-D turbulence-dependent resistance formulations for particle dry deposition simulation and intercompare the performance of new resistance formulations with that obtained from using the existing formulations and measured dry deposition velocity. Turbulence parameters such as turbulence velocity scale, turbulence factor, intensity of turbulence, effective sedimentation velocity, and effective Stokes number are newly introduced into two different particle deposition schemes to improve turbulence strength representation. For an assumed particle size distribution, the newly proposed schemes predict stronger diurnal variation of particle dry deposition velocity and are comparable to corresponding measurements while existing formulations indicate large underpredictions. We also find that the incorporation of new turbulence parameters either introduced or added stronger diurnal variability to sedimentation velocity and collection efficiencies values, making the new schemes predict higher deposition values during daytime and nighttime when compared to existing schemes. The findings from this research may help improve the capability of dry deposition schemes in regional and global models.

Plain Language Summary

Aerosols, also known as particulate matter, in the atmosphere can affect ecosystem health through a process called dry deposition, which is a helpful process that can reduce human exposure to air pollutants. There are several processes involved in particle dry deposition and one of the most important processes is the chaotic motions of the atmosphere, which is known as turbulence. However, turbulence strength is underrepresented in mathematical modeling of particle dry deposition. In this study, we introduced several turbulence parameters to improve the representation of turbulence effects on deposition and introduced new formulations. These new formulations are tested in a simple mathematical model and then field measurements are used to evaluate the performance of new formulations as well as existing formulations. Results indicate that the new formulations largely improved results, which are closer to measurements, while existing formulations showed large underestimations. This research offers improved capability of models in estimating particle deposition, and, in turn, hopefully leads to better estimation of particle pollution and related exposures.

1. Introduction

Atmospheric aerosols (liquid and solid), hereafter referred to as particles, have substantial influences on the air quality, radiative balance of the earth, climate, and ecosystem (e.g., Fowler et al., 2009; Lee et al., 2013). Dry deposition is the process that removes gases and aerosols from the air (e.g., Pryor et al., 2008) and is a source of many nutrients to natural ecosystems. Large uncertainties exist in the modeling of particle dry deposition leading to more uncertainties in the estimation of (1) aerosol direct, semi-direct, and indirect effects in climate models; (2) damages to environment via impairment and shifts in biodiversity and health of vegetation; (3) critical loads of nutrients affecting terrestrial ecosystem, and (4) human health impacts via exposure to pollutants. Thus, refined deposition formulations and updated comprehensive measurements are needed to improve estimation of particle dry deposition to address these important topics (Farmer et al., 2021).

Particle dry deposition is driven by atmospheric turbulence and exhibits a significant relationship with particle size distribution (PSD) as well as strong dependence on land use/land cover types (e.g., Slinn, 1982; Wesely, 1985; Riemer et al., 2019). Particle dry deposition is controlled by various processes including Brownian diffusion, gravitational settling, interception, and impaction, while the contributions of these deposition processes to particle dry deposition velocities vary substantially (Zhang and Shao, 2014; Hicks et al., 2016). Saylor et al. (2019) reported that depending upon the type of dry deposition scheme used, particle dry deposition predictions for forest land may differ by over 200%. The PSD also determines the characteristics of particle deposition dynamics where deposition of smaller particles (<300 nm) tended to be driven by Brownian diffusion, while interception, impaction, and gravitational settling have greater impacts on larger particles (Erisman and Draaijers, 1995; Emerson et al., 2020). In various models, many distinct methods were employed to represent the PSD characteristics, e.g., single particle diameter, discrete size bins, and continuous lognormal distributions (Seinfeld and Pandis, 2016; Riemer et al., 2019), and these different approaches also added some uncertainties to the estimation of particle dry deposition velocity (Shu et al., 2017). Moreover, deposition surfaces such as terrestrial and hydrological surfaces remove particles at different rates due to the various characteristics of the surfaces, and land cover with larger surface area with more collectors tended to induce greater deposition rates (e.g., forest > grassland > lakes) (Farmer et al., 2021).

Farmer et al. (2021) provided a comprehensive review of dry deposition of aerosols that include various approaches used, measurements available as of present, and modeling outcomes. They listed size-resolved particle flux measurements that are available to researchers for four different types of land use categories (two vegetative, one water, and one snow/ice) as well as instrumental methods used along with particle size range and associated deposition velocity. For grass and forests, it was shown that the average lower and upper bounds of measured dry deposition velocity can be around ~0.01 to ~10 cm s⁻¹, which is about three orders of magnitude range, depending upon the particle diameter (0.001 to 100 μm). The strong association between deposition velocity and particle diameter signifies the importance of accurately estimating particle diameter. These very limited measurements reflect the scarcity and huge data gaps for particle deposition modeling and evaluation. They also mentioned that friction velocity plays an important role in modulating particle flux while more turbulent conditions (e.g., convective conditions) induced stronger flux. While consistent meteorological and particle measurements are much needed for a long-term period to evaluate existing formulations establishing statistical significance of their accuracy, there is also a need to realistically represent turbulence effects on particle deposition in the existing formulations.

Essential resistances included in a particle dry deposition model are aerodynamic resistance and surface resistance (e.g., Saylor et al., 2019), both strongly influenced by the strength of the atmospheric turbulence and thus more turbulent conditions may lead to higher particle deposition fluxes (Sievering, 1987; Ahlm et al., 2010; Saylor et al., 2019). Recognizing the inadequacy of using friction velocity alone for representing strong turbulence effects, Wesely et al. (1985) have introduced an empirical formulation to represent the enhanced role of turbulence, thereby estimated higher deposition rates were found to be comparable to observed deposition rates. Saylor et al., (2019) showed that depending on the value of the convective velocity (w_∗), estimated dry deposition can differ by as much as about a factor of 10. This result signified the uncertainty arising from using empirical formulations to enhance turbulence effects on particle deposition that were not rigorously evaluated. Many particle deposition formulations and related sub-formulations (e.g., Zhang et al., 2001) used in regional and global air quality models are still based on solely friction velocity, offering an opportunity to improve the representation of turbulence effects in such schemes, which is the focus of this research. This paper is a companion paper to the Part-1 by Alapaty et al., (2022) that focused on gas deposition and here we adopt the major findings from that paper to improve turbulence representations for particle deposition.

None of the deposition schemes for gases and particles utilize the three-dimensional (3-D) turbulence (i.e., variances of velocity fluctuations) to represent effects of turbulence on deposition. In the companion paper (Part-1, Alapaty et al., 2022), we have developed a new velocity scale and validated it for use in dry deposition of ozone at a decadal timescale using a single-point model. This velocity scale, known as turbulence velocity scale, was derived from using the surface turbulence kinetic energy equation that includes 3-D aspects of turbulence. In this study, we extended our previous research by proposing and validating new resistance formulations for particle dry deposition by introducing the few turbulence parameters for better representation of turbulence effects.

Thus, the objectives of this research are to (1) improve the turbulence representation in particle deposition formulations and related sub-formulations, and (2) evaluate the performance of these new formulations using available measurements and a single-point model.

2. Methods

There are several processes involved in the particle deposition modeling. Before describing particle deposition schemes used in this study, we begin with the presentation of some of the processes (methods) in which turbulence representation can be improved. First, we describe the estimation of turbulence velocity while its evaluation details can be found in the Part-1 paper (Alapaty et al., 2022). Second, we show how turbulence effects can be improved in some of the particle deposition processes. This way, all new developments can be easily found in this section rather than scattering them in various subsections.

2.1. Turbulence velocity scale

Almost all gas and particle deposition formulations use friction velocity (u_∗) for all stability regimes in the planetary boundary layer (PBL). Since u_∗ is only applicable for neutral conditions, different stability correction parameters are used for each stability regime of PBL to account for turbulence generated by buoyancy and/or shear production. Thus, a large number of different stability functions appeared in the literature suiting the needs of an atmospheric model, which led to different modeling outcomes (Liu et al., 2007; Toyota et al., 2016). To alleviate this type of issue, Alapaty et al. (2022) have proposed a new approach where resistance formulations are functions of turbulence generated by shear and buoyancy production for different PBL stability regimes. It was achieved by using surface turbulence kinetic energy (TKE) approximations so that a single velocity scale will be suitable for different stability conditions in the PBL. In that methodology (Alapaty et al., 2022) the mean TKE of eddies near the surface was written as:

$$TKE=\overline{{e^{\prime }}^2}=\frac{1}{2}\left(\overline{{u^{\prime }}^2}+\overline{{v^{\prime }}^2}+\overline{{w^{\prime }}^2}\right)$$ (1)

where e is mean velocity, u, v, and w is eastward, northward, and vertical components of wind, superscript prime denotes fluctuations. Then, terms on the right-hand side of the Eq. (1) were rewritten in terms of respective variance of velocity fluctuations as:

$$\overline{{e^{\prime }}^2}=\frac{1}{2}\left({\sigma }_u^2+{\sigma }_v^2+{\sigma }_w^2\right)$$ (2a)

where σ is standard deviation. Alapaty et al., (2022) defined the turbulence velocity scale (e_∗) representative of turbulence created by mechanical and buoyant forces at the surface as:

$$e_{\ast }=\sqrt{\overline{{e^{\prime }}^2}}=\sqrt{\frac{1}{2}\left({\sigma }_u^2+{\sigma }_v^2+{\sigma }_w^2\right)}$$ (2b)

As documented in the literature (e.g., Hicks, 1985), velocity variance is a result of independent contributions associated with surface momentum flux and the surface vertical heat flux. Following the findings of Hicks, (1985), Wyngaard and Cote, (1974), and Wyngaard, (1975), these surface velocity variances are related to friction and convection velocity scales. Accordingly, velocity variances can be written as:

$${\sigma }_{u,v,w}^2=a^2{u}_{\ast }^2+b^2{w}_{\ast }^2+c^2{u}_{\ast }^2{(-z/L)}^{2/3}$$

where a, b, and c, are constants, u_∗ is friction velocity and w_∗ is convection velocity in boundary layer, z is altitude (usually taken as thickness of a model’s lowest layer or altitude at which measurements are made), and L is Monin-Obukhov length. From the studies of Hicks (1985) based on the Minnesota Turbulence Experiment (Izumi and Caughey, 1976) and Coral Sea (Warner, 1972) observational data over land and tropical ocean (surfaces with a wide range of roughness) respectively, the above relations are found to be fairly applicable for a set of values for a, b, and c. However, different studies, e.g., Deardorff (1974), Wyngaard and Cote (1974), Wyngaard (1975), and Mailhot and Benoit (1982) have used different values for these three constants. One set of values that were used and successfully tested was by He and Alapaty (2018) improving precipitation predictions with their cumulus convection parameterization scheme (Multi-Scale Kain-Fritsch, MSKF scheme) in a regional meteorological modeling study. Following that study the new turbulence velocity scale, e_∗ from the above equations can be rewritten for unstable conditions (i.e., when surface sensible heat flux > 0) in PBL as:

$$e_{\ast }=\sqrt{3.8u_{\ast }^2+0.22w_{\ast }^2+1.9u_{\ast }^2{(-z/L)}^{2/3}}$$ (3a)

where z is the measurement height and L is Monin-Obukhov length, and for stable conditions in PBL (surface sensible heat flux < 0):

$$e_{\ast }=\sqrt{3.8u_{\ast }^2}$$ (3b)

An advantage of the above equations is that the parameterized e_∗ transitions smoothly from one stability regime to another since the 2^nd and 3^rd terms on the right side of Eq. 3a drop out for stable conditions. For neutral conditions that exist infrequently, e_∗ can be made equal to u_∗ as a transition point. By using 3-D variances measured by the 3-D sonic anemometer for a decadal period, Alapaty et al. (2022) hypothesized and verified that 3-D sonic anemometer measurements of friction velocity (u_∗c) (that includes contributions by vertical heat flux to the vertical transport of horizontal momentum) can be approximated as the product of the von Karman constant (k) and turbulence velocity scale, e_∗, as:

$$u_{\ast c}=k{e}_{\ast }$$ (4)

Surface wind estimation using friction velocity alone is applicable only for neutral conditions, but for other PBL conditions the current modeling approaches heavily depend on different stability functions to account for differing strengths of turbulence. The main advantage of this new velocity scale is that it includes turbulence contribution from buoyancy production as well as shear production, making it suitable for use for stable and unstable conditions in the PBL. Thus, with the usage of turbulence velocity scale there is no need to use any explicit stability functions in representing turbulence effects. That work has opened up doors to avoid the usage of a variety of stability functions reported in the literature that contributed to differences in modeling results. Our methodology based on e_∗ can open doors to the concept of community dry deposition model. Evaluation of Eq. 4 using decadal measurements can be found in SI (see Figure S1) that was already reported in Alapaty et al. (2022). For more details, readers are referred to the companion paper (Part-1) by Alapaty et al. (2022).

2.2. Introducing generalized Turbulence Factor (T_f)

To account for increased particle deposition under convective conditions in the PBL, Wesely et al. (1985) have suggested an empirical equation to increase the deposition velocity of sulfate particles. This empirical equation was written as:

$$W_f=0.24\frac{w_{\ast }^2}{u_{\ast }^2}$$ (5)

To the best of our knowledge, the above empirical equation was neither validated with any other measurements nor compared with other formulations, but it has been widely used in many studies (e.g., Binkowski and Shanker, 1995). However, our study offers such an intercomparison and thus we propose a generalized turbulence factor, T_f, a ratio of the terms on the right-hand side of the TKE equation shown in Eq. 2. The proposed T_f can be written as:

$$T_f=\frac{\left({\sigma }_v^2+{\sigma }_w^2\right)}{{\sigma }_u^2}$$ (6)

Using the Eq. 2 and 3, we write:

$$\begin{gathered} \frac{1}{2}({\sigma }_u^2+{\sigma }_v^2)=3.8u_{\ast }^2+0.22w_{\ast }^2 \\ \text{or}\hspace{1em}({\sigma }_u^2+{\sigma }_v^2)=2\left(3.8u_{\ast }^2+0.22w_{\ast }^2\right)=7.6u_{\ast }^2+0.44w_{\ast }^2 \end{gathered}$$ (7)

$$\begin{gathered} \frac{1}{2}{\sigma }_w^2=1.9u_{\ast }^2{\left(-z/L\right)}^{2/3} \\ \text{or}\hspace{1em}{\sigma }_w^2=3.8u_{\ast }^2{\left(-z/L\right)}^{2/3} \end{gathered}$$ (8)

The turbulence factor equation shown above (Eq. 6) is rooted in physical realism, and it is conceptually similar to ad-hoc relation, Eq. 5, suggested by Wesely et al. (1985). They stated that convective motions in PBL have nonlinear influence on particulate sulfate deposition velocity. There seems to be a physical connection between convective velocity in the PBL and the friction velocity in the surface layer influencing particle deposition velocity. Also, Wesely et al. (1983) suggested that this connection might lead to rapid multidirectional flow around surface elements and thus a connection between buoyancy and horizontal flows. They also stated that σ_v is more strongly correlated to enhanced particle deposition than σ_u, and accordingly we gave more weight to σ_v coefficients. Thus, the nonlinear connection between buoyancy and mechanical forces can result in wind gusts at the surface and such conditions can be responsible for enhanced particle deposition. Also, based on the turbulence data over land and ocean, Hicks (1985) documented that contribution by mechanical and buoyant forces to the total variance came from σ_v and σ_w for convective boundary layers. From the literature we found that magnitude of coefficients differs from one study to the other (e.g., Wyngaard and Cote, 1974; Hicks, 1985) and thus there is no universal agreement on the magnitude of these coefficients. In this study, for the u_∗ coefficient, we assigned 70% weight to σ_v and 30% to σ_u, consistent with above described qualitative findings of Wesely et al. (1985). Accordingly, the numerator and denominator components are devised in the proposed T_f equation containing velocity variances ratios, i.e., $\left({\sigma }_v^2+{\sigma }_w^2\right)/{\sigma }_u^2$. Thus, we propose individual variances of velocity fluctuations as:

$${\sigma }_u^2=2.28u_{\ast }^2$$ (9a)

$${\sigma }_v^2=5.32u_{\ast }^2+0.44w_{\ast }^2$$ (9b)

$${\sigma }_w^2=3.8u_{\ast }^2{(-z/L)}^{2/3}$$ (9c)

Then the turbulence factor Eq. 6 can be rewritten as:

$$T_f=\frac{{\sigma }_v^2+{\sigma }_w^2}{{\sigma }_u^2}=\frac{\left(5.32u_{\ast }^2+0.44w_{\ast }^2\right)+\left(3.8u_{\ast }^2{(-z/L)}^{2/3}\right)}{2.28u_{\ast }^2}$$ (10)

Note that the above equation works for all stability regimes in the PBL while Eq. 5 was designed to work only for convective conditions. For stable conditions (and neutral conditions if they exist) T_f value reduces to 2.33. At this stage we do not have any observational evidence to support this constant value, but it is based on the variance equations that were derived from several field measurements (see references cited in the Section 2.1).

2.3. Introducing effective sedimentation velocity (V_ge)

One of the processes responsible for particle deposition is gravitational settling velocity and is also known as terminal velocity or popularly known as sedimentation velocity (V_g). Traditionally, the V_g is estimated only for still air conditions. Thus, only two forces are acting on a particle, i.e., aerodynamic drag and gravity (Hinds, 1999). Then, for a solid spherical particle vertical drag force will be balanced by gravity force and thus there is no acceleration of the particle. However, turbulent eddies present in the PBL will be another force acting on a particle and turbulence is not commonly accounted for in the estimation of sedimentation velocity. For still air, V_g can be written as (see supplemental information Text S1 for derivation):

$$V_g=\frac{gC{\rho }_p{d}_p^2}{18\mu }$$ (11)

where g is acceleration due to gravity, C is the Cunningham correction factor, ρ_p is density of particle, d_p is diameter of the particle, and μ is temperature dependent viscosity of air. Since the above equation is only good for neutral conditions, we propose to introduce the effects of turbulent flows on V_g in the boundary layer. In particular, for convective conditions, many particles would be brought down to the surface much faster by the convective downdrafts in the PBL. Thus, to account for turbulent processes, we propose to use the intensity of turbulence (I_t) in the estimation of sedimentation velocity, and we refer to it as the effective sedimentation velocity and it can be written as:

$$V_{ge}=V_g(1+I_t)=\left(\frac{gC{\rho }_p{d}_p^2}{18\mu }\right)(1+I_t)$$ (12)

where I_t can be estimated as

$$I_t=\frac{e_{\ast }}{U}$$

In the above equation U is horizontal windspeed. See Alapaty et al. (2022) for more details about the I_t parameter where it was estimated and analyzed for a decadal time period. Since intensity of turbulence is applicable for all stability regimes in the PBL, Eq. 12 works for all conditions in the PBL. Intercomparison results of both V_g and V_ge will be presented in the Results section.

2.4. Introducing effective Stokes number (St_e)

Stokes number, also known as reference Stokes number, is an important non-dimensional parameter that characterizes particle motion in atmospheric flows. It is defined as ratio of particle response timescale to the characteristic timescale of the flow. In estimating these respective timescales, traditionally sedimentation velocity and friction velocity along with respective characteristic length scale are used. For smooth and vegetated surfaces, reference Stokes number can be, respectively, written as:

$$St=\frac{(V_g{u}_{\star }^2)}{g\nu }\hspace{1em}\text{and}\hspace{1em}St=\frac{(V_g{u}_{\star })}{gA}$$ (13)

where g is acceleration due to gravity, ν is kinematic viscosity of air, and A is characteristic radius of collectors. Note that friction velocity and hence reference Stokes number by themselves are only applicable for neutral conditions in the PBL while e_∗ is a better representation of the flow field’s turbulence. Thus, we propose a new Stokes number, referred to as effective Stokes number (St_e), that includes the new velocity scale that works for different stability regimes in PBL. Then, St_e can be estimated by using Eq. 13 as:

$$S{t}_e=\frac{V_{ge}{k}^2{e}_{\star }^2}{g\nu }=\frac{\left(1+I_t\right)V_g{k}^2{e}_{\star }^2}{g\nu }\hspace{1em}\text{and}\hspace{1em}S{t}_e=\frac{V_{ge}k{e}_{\star }}{gA}=\frac{\left(1+I_t\right)V_{g}k{e}_{\star }}{gA}$$ (14)

It is important to note that the new effective Stokes number works for different stability conditions in the PBL and is identically same as Stokes number for neutral conditions. Intercomparison results of Eq. 13 and 14 will be presented in the Results section.

2.5. Particle dry deposition formulations

In this section we describe two different particle deposition schemes and also respective new schemes that include 3-D turbulence effects. These two particle deposition schemes are suggested by (1) Zhang et al. (2001): referred to as Z01 scheme and the newly proposed scheme based on it is referred to as C01, and (2) Shu et al. (2022): referred to as S21 and the newly proposed scheme based on it is referred to as C21.

2.5.1. The Z01 and C01 schemes

The particle deposition scheme proposed by Zhang et al. (2001) has been used by regional and global chemical transport models (e.g., CAMx and GEOS-Chem). Here we present both the original scheme formulations as well as the proposed new resistance formulations that include the new turbulence parameters. Z01 writes particle size-dependent deposition velocity (V_d) as:

$$V_d=V_g+\frac{1}{R_a+R_b}$$ (15)

where V_g is sedimentation velocity, R_a and R_b are the aerodynamic resistance for heat and boundary layer (quasi-laminar sublayer) resistance, respectively. The aerodynamic resistance (R_a) for heat in the Z01 scheme is written as:

$$R_a=\frac{1}{k{u}_{\ast }}\left(\text{ln}\left[\frac{z_r}{z_0}\right]-{\Psi }_H\right)$$ (16)

where k is von Karman constant, u_∗ is friction velocity, z_r is the height at which dry deposition velocity is evaluated, z₀ is surface roughness length, and Ψ_H is stability function for heat. Since buoyancy and shear production forces are included in the estimation of turbulence velocity, Alapaty et al. (2022) has proposed and validated a new formulation for R_a as:

$$R_a=\frac{1}{e_{\ast }{k}^2}ln\left[\frac{z_r}{z_0}\right]$$ (17)

Since the above equation will work seamlessly for different stability regimes in the PBL, there is no need for explicitly using a stability function, a unique feature to avoid the options to choose from a wide selection of stability functions reported in the literature.

Resistance to transport through the very thin viscous sub-layer at the surface, R_b, is referred to as the quasi-laminar layer, laminar deposition layer or boundary layer (e.g., Pleim and Ran, 2011) where transport is fundamentally characterized by molecular diffusion for gases. For this reason, u_∗ is used traditionally in all such formulations without any stability correction parameters, like in Z01 scheme (Eq. 18). However, this is valid only for neutral conditions. During turbulent periods, such as daytime with surface heating, the literature indicates the presence of convective plumes at the leaf scale under still and windy air conditions since plants are subjected to heat load or thermal stress. Convection phenomena from plants in calm and windy air was observationally studied firstly by Gates and Benedict (1962). Using Schlieren photography and other instruments along with an infrared radiation gun, they quantitatively estimated the amount of energy convected away from a leaf under free and forced convection for broad-leaved and coniferous tree needles. Schlieren photography movies showed distinct convective plumes moving away from leaf surface indicating turbulence. In another experimental study, Wigley and Clark (1974) determined the heat transfer from model leaves heated by a constant energy flux under forced convection. In another experimental and field study, Brenner and Jarvis (1995) found that for forced convection conditions, at wind speeds above 2.5 m s⁻¹ leaf boundary layer conductances were higher than those for a laminar leaf boundary layer. As mentioned above, Wesely et al. (1985) have already introduced convection factor to improve turbulence effects. Considering results from these studies, we justify using the product ke_∗ in the place of u_∗ in the boundary layer resistance formulation. The boundary layer resistance (R_b) in the Z01 scheme can be written as:

$$R_b=\frac{1}{{\epsilon }_0{u}_{\star }\left(E_B+E_{IM}+E_{IN}\right)R_1}$$ (18)

where ε₀ is empirical constant set to 3, E_B, E_IM, and E_IN are the collection efficiencies from Brownian diffusion, impaction, and interception, respectively, and R₁ is correction factor to represent fraction of particles that stick to the underlying surface and is a function of Stokes number (St), which is also a function of land use. Here, R₁ is estimated as:

$$R_1=e^{-\sqrt{St}}$$ (19)

The collection efficiencies are estimated as:

$$E_B=S_c^{-\gamma }$$ (20)

$$E_{IM}={\left(\frac{St}{\alpha +St}\right)}^2$$ (21)

$$E_{IN}=\frac{1}{2}{\left(\frac{d_p}{A}\right)}^2$$ (22)

where S_c is ratio of kinematic viscosity of air to the Brownian diffusivity and γ is constant that varies as a function of land use and α is a constant for each land use, d_p is particle diameter, and A is characteristic radius of collectors. Since we have defined the effective Stokes number, E_IM will also become a new equation because of the usage of Eq. 14. For finer details of original equations, readers are referred to Zhang et al. (2001).

In Z01 scheme, the R_b formulation (Eq. 18) has a global constant parameter (ε₀), which was set to 3. However, this formulation originated from Slinn (1982) in which ε₀ was defined as the ratio of friction velocity to the surface wind speed (U). Though it was neither stated nor recognized, this ratio is the intensity of turbulence (I_t) as defined earlier, and was written as ε₀ = u_∗/U. Then, the product of ε₀ and u_∗ in the denominator of Eq. 18 becomes aerodynamic resistance for neutral conditions. When we tested this original R_b formulation from Slinn (1982) in our single-point model, it resulted in unrealistically small (large) boundary layer conductances (resistances), particularly during daytime while aerodynamic conductance (resistance) is usually larger (smaller) during daytime (see Figure S2 in supplementary information). Potentially, for this reason Z01 might have introduced a global constant factor 3 in the place of ε₀ to achieve better performance. To relax this issue with the R_b estimation, we propose a two-step revision: (1) replace u_∗ with ke_∗, and (2) replace the ε₀ with the turbulence factor, (1+T_f), along with replacing u_∗ with ke_∗. Then, using Eq. 4 and 6 in Eq. 18, the new R_b formulations can be written as:

$$R_b=\frac{1}{{\epsilon }_{0}k{e}_{\star }\left(E_B+E_{IMe}+E_{IN}\right)R_{1\text{e}}}$$ (23a)

$$R_b=\frac{1}{\left(1+T_f\right)k{e}_{\star }\left(E_B+E_{IMe}+E_{IN}\right)R_{1\text{e}}}$$ (23b)

where

$$R_{1e}=e^{-\sqrt{S{t}_e}}$$ (24)

$$E_{IMe}={\left(\frac{S{t}_e}{\alpha +S{t}_e}\right)}^2$$ (25)

2.5.2. The S21 and the new C21 Schemes

Formulations of Shu et al. (2022), referred to as S21 (See Text S2 in supplementary information), are based on Pleim and Ran (2011) where S21 added a new term to R_b to calculate leaf area index (LAI)-dependent vegetative surface uptake. S21 showed that the new formulation introduced a vegetation dependence that is directionally consistent with the observed impact of vegetation on particle dry deposition and this update has resulted in relatively better results in the estimation of particle dry deposition. The R_a, and R_b equations used in S21 are written as:

$$R_a=0.95\frac{\text{ln}\left(\frac{Z_R}{Z_0}\right)-{\Psi }_H}{k{u}_{\star }}$$ (26)

$$\begin{gathered} R_b={\left[\left(1+f_{veg}\left(max\left(LAI-1,0\right)\right)\right)\left(1+W_f\right)\cdot u_{\star }(E_B+E_{IM})\right]}^{-1} \\ \text{and}\hspace{1em}{W}_f=0.24\frac{w_{\ast }^2}{u_{\ast }^2} \end{gathered}$$ (27)

where f_veg is the fractional area of vegetation surface in a grid cell, which was introduced in the original R_b equation of Pleim and Ran (2011) in which they ignored the interception term (E_IN) in the Eq. 27 since its magnitude is very small (~10⁻⁶). In their scheme, dry deposition is estimated as:

$$V_d=\frac{V_g}{1-e^{-\left(R_a+R_b\right)V_g}}$$ (28)

and sedimentation velocity equation used in Eq. 28 is same as that in Eq. 11.

Using the new velocity scale (Eq. 4) and turbulence factor (Eq. 10), we propose that the new equations for R_a, R_b, and V_d are written as:

$$R_a=0.95\frac{\text{ln}\left(\frac{Z_R}{Z_0}\right)}{{\kappa }^2{e}_{\star }}$$ (29)

$$R_b={\left[\left(1+f_{veg}\left(max\left(LAI-1,0\right)\right)\right)\left(1+T_f\right)\cdot k{e}_{\star }(E_B+E_{IMe})\right]}^{-1}$$ (30)

$$V_d=\frac{V_{ge}}{1-e^{-\left(R_a+R_b\right)V_{ge}}}$$ (31)

As mentioned earlier, the T_f was estimated using bulk boundary layer parameters that are related to micrometeorological variables, 3-D variances of velocity fluctuations.

3. Model, Measurements, Simulations, and Metrics

3.1. Single-point model

Numerical model used in this study is the same single-point model (DepoBoxToolv1.0) that was used by Shu et al. (2022) to test different particle dry deposition schemes for different land use. DepoBoxToolv1.0 was configured as an open-source Python tool, which can be easily modified to incorporate updates on the dry deposition schemes. All the deposition schemes were built into Models.py file, basic functions were configured in functions.py file, and land category definition and parameterization were set in eval_luc.py file. DepoBoxToolv1.0 has been freely available to the research community and the source code and required model inputs can be downloaded from the GitHub source (https://github.com/shumarkq) and Zenodo open data repository (http://doi.org/10.5281/zenodo.4749636).

3.2. Measurements

As mentioned in the Introduction section, there are huge data gaps for particle measurements that include gaps in meteorological measurements. Thus, given the requirements for model inputs to perform single-point model simulations, we could find only three measurement data sets in the literature. Of these three data sets, dry deposition measurements documented by Vong et al. (2004) indicated a bi-modal distribution for deposition velocity and such a variability cannot be well simulated by a single-point model since such these models typically do not consider advective processes or cloud processes affecting deposition velocity. The second data set documented by Lamaud et al. (1994) for a coniferous site has data less than 12 hours and thus may not provide a statistically meaningful analysis and thus we excluded these two data sets. Thus, we are left with using only one data set, the third data set, a field experiment at a forest site of the foothills of Mt. Asama in central Japan that was conducted during July 2–8, 2009 (5.5 days -132 data points) and this site is characterized as alpine forest. Its canopy is dominated by the birch and alder tree species. During the study PM_2.5 sulfate fluxes were estimated as product of four-hour averaged transfer velocity between two measurement heights (21 and 27 m) and concentration difference at these heights. For this reason, estimated deposition velocities represent averaged values for six time periods: 0600–1000,1000–1400, 1400–1800, 1800–2200, and 2200–0200 local time (LT) and they also assumed nighttime fluxes do not vary. Meteorological instruments and ultrasonic anemometer were placed at 28 m AGL using which various measurements were made. Here we briefly mention a few details and for specific details, readers are referred to Matsuda et al. (2010). All measurements included local time, air temperature, pressure, relative humidity, leaf area index, horizontal wind speed, friction velocity, canopy height, zero-plane displacement height, roughness length, measurement height, Monin-Obukhov length, and deposition velocity. The convective velocity was unavailable and thus was extracted from Weather Research and Forecasting (WRF) Model as documented in Shu et al. (2022). These measurements are used to provide needed inputs to the single-point model to simulate various deposition processes used in the estimation of particle dry deposition velocity.

3.3. Numerical simulations

Sulfate particle density was set as ρ = 1500 kg m⁻³, and since measured PSD were not available we characterized PSD by mass median diameter (d_pg) = 0.48 μm, and geometric standard deviation (σ_g) = 1.7. As the PSD parameterization in the model may significantly impact the simulation results of particle deposition velocity, we followed the methods used by Shu et al. (2022) where the PSD was assumed to be discrete size bins for Z01, C01, S21, and C21 schemes with 100 bins (See Text S3 in supplementary information) and continuous log-normal distribution for S21 and C21 schemes. The log-normal distribution of the particle size as defined by d_pg and σ_g and the single-point model integrates across the whole particle size range using the modal integration formulations (See Text S4 in supplementary information). We reiterate that PSD data was not available and thus the chosen PSD may not be representative of actual size distribution that was not measured. Three schemes that used sectional approach to perform numerical simulations are referred to as: Z01-B, C01-E, and C01-ETF, where Z01-B refers to base scheme, C01-E represents base scheme using e_∗ (turbulence velocity scale), and C01-ETF represents base scheme using e_∗ as well as the turbulence factor. The S21 and C21 schemes are configured to use sectional as well as modal approaches and thus S21-BS and S21-BM represent base scheme using sectional and modal approaches, while C21-ETFS and C21-ETFM represent base scheme using e_∗ as well as the turbulence factor with sectional and modal approaches, respectively. Detailed information about all equations used in Z01, S21, C01, and C21 can be found in the Table 1. A list of all simulations performed and differences between each simulation is shown in Table 2. Model simulations were performed for 132 hours for each of the seven cases described above, shown in Table 2.

Table 1.

Equations used to represent several processes in each of the schemes (Zhang et al., 2001 and Shu et al., 2022)

Single Diameter									Modal approach	Sectional approach
Scheme	Vd	Vg or Vge	Ra	Rb	EB	EIM	EIN	St
Z01-B	${\text{V}}_{\text{g}}+\frac{1}{R_a+R_b}$	$\frac{\rho d_p^{2}gC}{18\eta }$	$\frac{\text{ln}\left(\frac{Z_R}{Z_0}\right)-{\Psi }_H}{k{u}_{\star }}$	$\frac{1}{{\epsilon }_0{u}_{\star }\left(E_B+E_{IM}+E_{IN}\right)R_1}$	Sc−γ	${\left(\frac{St}{\alpha +St}\right)}^{\beta }$	$\frac{1}{2}{\left(\frac{d_p}{A}\right)}^2$	$\frac{V_g{u}_{\star }}{gA}$	NA	Yes
C01-E	${\text{V}}_{\text{ge}}+\frac{1}{R_a+R_b}$	$\frac{\rho d_p^{2}gC}{18\eta }\left(1+I_t\right)$	$\frac{\text{ln}\left(\frac{Z_R}{Z_0}\right)}{k^2{e}_{\star }}$	$\frac{1}{{\epsilon }_{0}k{e}_{\star }\left(E_B+E_{IMe}+E_{IN}\right)R_{1\text{e}}}$	Sc−γ	${\left(\frac{S{t}_e}{\alpha +S{t}_e}\right)}^{\beta }$	$\frac{1}{2}{\left(\frac{d_p}{A}\right)}^2$	$\frac{V_{ge}k{e}_{\star }}{gA}$	NA	Yes
C01-ETF	${\text{V}}_{\text{ge}}+\frac{1}{R_a+R_b}$	$\frac{\rho d_p^{2}gC}{18\eta }\left(1+I_t\right)$	$\frac{\text{ln}\left(\frac{Z_R}{Z_0}\right)}{k^2{e}_{\star }}$	$\frac{1}{\left(1+T_f\right)k{e}_{\star }\left(E_B+E_{IMe}+E_{IN}\right)R_{1\text{e}}}$	Sc−γ	${\left(\frac{S{t}_e}{\alpha +S{t}_e}\right)}^{\beta }$	$\frac{1}{2}{\left(\frac{d_p}{A}\right)}^2$	$\frac{V_{ge}k{e}_{\star }}{gA}$	NA	Yes
S21-B (S21-BS, S21-BM)	$\frac{V_g}{1-e^{-V_g\left(R_a+R_b\right)}}$	$\frac{\rho d_p^{2}gC}{18\eta }$	$0.95\frac{\text{ln}\left(\frac{Z_R}{Z_0}\right)-{\Psi }_H}{k{u}_{\star }}$	${\left[\left(1+f_{veg}\left(max\left(LAI-1,0\right)\right)\right)\left(1+0.24\frac{w_{\ast }^2}{u_{\ast }^2}\right)\cdot u_{\star }(E_B+E_{IM})\right]}^{-1}$	$S{c}^{-\frac{2}{3}}$	$\left(\frac{S{t}^2}{1+S{t}^2}\right)$	NA, assume 0	$\frac{V_g{u}_{\star }}{gA}$	$\widehat{D}\widehat{V_g}\widehat{E_{IM}}$ $\widehat{E_{IM}}=\frac{{\widehat{St}}^2}{1+{\widehat{St}}^2}$ $\widehat{St}=\frac{\widehat{V_g}{u}_{\ast }}{gA}$	Yes
C21-ETF	$\frac{V_{ge}}{1-e^{-V_{ge}\left(R_a+R_b\right)}}$	$\frac{\rho d_p^{2}gC}{18\eta }\left(1+I_t\right)$	$0.95\frac{\text{ln}\left(\frac{Z_R}{Z_0}\right)}{{\kappa }^2{e}_{\star }}$	${\left[\left(1+f_{veg}\left(\text{max}\left(LAI-1,0\right)\right)\right)\left(1+T_f\right)\cdot k{e}_{\star }(E_B+E_{IMe})\right]}^{-1}$	$S{c}^{-\frac{2}{3}}$	$\left(\frac{S{t_e}^2}{1+S{t_e}^2}\right)$	NA, assume 0	$\frac{V_{ge}k{e}_{\star }}{gA}$	$\widehat{D}\widehat{V_{ge}}\widehat{E_{IM}}$ $\widehat{E_{IM}}=\frac{{\widehat{St}}^2}{1+{\widehat{St}}^2}$ $\widehat{St}=\frac{\widehat{V_{ge}}k{e}_{\ast }}{gA}$	Yes

Table 2.

Description of schemes used in model simulations

Scheme	Description
Z01-B	Base Z01 scheme (Zhang et al., 2001)
C01-E	Base Z01 scheme using e∗
C01-ETF	Base Z01 scheme using e∗ and Tf
S21-BS	Base S21 scheme (sectional) (Shu et al., 2022)
S21-BM	Base S21 scheme (modal)
C21-ETFS	Base S21 scheme using e∗ and Tf (sectional)
C21-ETFM	Base S21 scheme using e∗ and Tf (modal)

3.4. Metrics used for evaluation

The evaluation of the newly proposed particle dry deposition schemes was performed via comparison to the measured particle deposition velocities. Following the approaches by Chang and Hanna (2004) and Emery et al. (2017), three metrics including fractional bias (FB), normalized mean error (NME), and Pearson correlation coefficient (R) were used to evaluate the performance of the model formulations. Associated parameter descriptions are as follows: the overbar means averaging, the bracket means averaging, V_o and V_p denote measured and predicted deposition velocities, and subscript i indicates the paired V_o and V_p.

$$\text{FB}=\frac{2\times \left(\left[V_p]-[V_o\right]\right)}{\left[V_o]+[V_p\right]}$$

$$\text{NME}=\frac{\sum \left|V_{p,i}-V_{o,i}\right|}{\sum V_{o,i}}$$

$$\text{R=}\frac{\sum \left[\left(V_{p,i}-\overline{V_p}\right)\times \left(V_{o,i}-\overline{V_o}\right)\right]}{\sqrt{\sum {\left(V_{p,i}-\overline{V_p}\right)}^2\times \sum {\left(V_{o,i}-\overline{V_o}\right)}^2}}$$

4. Results and Discussions

First, we present results for new factors introduced in Section 3, including the estimation of intensity of turbulence, effective sedimentation velocity, effective Stokes number, and turbulence factor. We then present results obtained from Z01 and C01 schemes and also from S21 and C21 formulations. As mentioned earlier, fluxes are estimated for four-hour time intervals to be consistent with the analysis of Matsuda et al. (2010), and we present medians for those time intervals in the following figures. We also present hourly averages for some parameters to analyze diel variations.

4.1. Characterization of turbulence parameters

Figure 1 shows diurnal variation of median e_∗, horizontal wind speed (U), and intensity of turbulence, I_t. Error bars indicate variability within ± 1 standard deviation. Since e_∗ includes the contribution by the surface sensible heat flux to the turbulence, during daytime it shows a variability typical of heat flux and thus usage of e_∗ should have an impact on the resistances estimation equations used to estimate deposition velocity. Since I_t varies from 0 to about 1, usage of I_t in equations for sedimentation velocity and Stokes number will introduce diurnal variability and also an increase in magnitudes of such estimations. It can be seen that trends in measured wind speeds are higher during nighttime than during daytime reflecting trends in e_∗ during nighttime, as expected as per the e_∗ formulation.

Figure 1.

Diurnal variation of median intensity of turbulence (I_t) (blue curve), turbulence velocity scale, e_∗ (m s⁻¹) (red curve), and horizontal wind speed, U (m s⁻¹) (black curve) for six time periods.

Diurnal variation of median observed u_∗ and the estimated e_∗ is shown in Figure 2. Reported values of u_∗ are confined between 0.2 and 0.3 m s⁻¹ because the original research only reported a single daytime and nighttime u_∗ value (Matsuda et al., 2010). Numerous studies documented in the literature have indicated a strong diurnal variation of u_∗ unlike reported in the Matsuda et al (2010) study. Since measured u_∗ is constrained, it does not closely follow the ke_∗ temporal variability (Fig. 2). Thus, usage of such u_∗ without utilizing a stability function can underrepresent turbulence magnitude in process formulations and thus will lead to underestimation of dry deposition velocity. This is where the advantage of using e_∗ lies in, i.e., no need to use a stability function since it includes the effects of surface sensible heat flux on the turbulence and thus responsible for large diurnal variation. We have demonstrated in our companion paper (Part-1, Alapaty et al., 2022) and also can be found in SI (see Figure S1) that the product of k and e_∗ can successfully replace u_∗ measured by 3-D sonic anemometer (that also includes surface heat flux contribution) for use in several resistance formulations, producing realistic ozone deposition fluxes and deposition velocities. In this study also, temporal variation of the product k and e_∗, shown in Figure 2, also has a typical diurnal variation which is a signature of many boundary layer parameters. Error bars (e.g., Fig. 2) indicate variability within ± 1 standard deviation.

Figure 2.

Diurnal variation of median friction velocity, u_∗ (m s⁻¹), and product of von Karman constant and turbulence velocity scale, ke_∗ (m s⁻¹) for six time periods.

Figure 3 shows diurnal variation of median sedimentation velocity and effective sedimentation velocity using Eq. 11 and 12. As expected, for the chosen particle diameter (0.48 μm), sedimentation velocity is constant and is independent of time as reflected in Figure 3. However, with the usage of I_t, turbulence effects are reflected in the effective sedimentation velocity estimation, showing a diurnal variation typical of certain boundary layer parameters during warmer days. The third force, namely, the turbulence is included in Eq. 12 and thus effective sedimentation velocity seems to be a good choice representing effects of turbulence on particle deposition. We find that the maximum median V_ge is about two times higher than that of V_g and for all days of simulation V_ge has been always higher than V_g irrespective of PBL stability regimes.

Figure 3.

Diurnal variation of median sedimentation velocity (V_g) and effective sedimentation velocity (V_ge) for six time periods. Units: cm s⁻¹

Figure 4 shows diurnal variation of estimated median reference Stokes number (St) and median effective Stokes number (St_e) using Eqs. 13 and 14 for each time interval. Variability for the St is solely dictated by the variation of u_∗ since other parameters used in Eq. 13 are constant in time. However, St_e estimation is controlled by both the V_ge and e_∗ and thus the net impact of these parameters has introduced increased diurnal variability accounting for diurnally varying turbulence. As a result, the maximum median St_e is about three times higher than that of St. Further, St_e is always higher than that St for all PBL conditions. Since boundary layer resistance estimation depends on the magnitude of Stokes number, its magnitude will also be affected.

Figure 4.

Diurnal variation of median reference Stokes number (St) and effective Stokes number (St_e) for six time periods.

Figure 5 shows diurnal variation of median and hourly averaged turbulence factors (T_f and W_f) estimated using Eq. 5 and 10. To the best of our knowledge, temporal variation, and magnitude of W_f was neither compared with nor evaluated against other similar turbulence factor estimates and thus our new formulation fills this scientific gap. Saylor et al. (2019) have shown that depending up on the magnitude of convective velocity (w_∗), estimated dry deposition can differ by as much as about 10 times when using Eq. 5. Though our new turbulence factor (Eq. 10) also uses w_∗, our e_∗ equation (Eq. 3) that we proposed was validated using the 3-D micrometeorological data for a decadal period (see Figure S1 for more information) and thus offers a good level of confidence in using Eq. 10. Both equations (Eq. 5 and 10) differ in complexity and Eq. 5 only works for convective boundary layers while Eq. 10 works for all stability regimes in the PBL. Thus, for time intervals 0200–0600, 1800–2200, and 2200–0200 and nighttime (Figure 5) W_f is set to zero while T_f is non-zero by design. Thus, during nighttime, R_b estimations using T_f will be lower as compared to that using W_f. During convective conditions, maximum difference between T_f and W_f occurs at time interval 6–10 and that difference reduces to about 0.2 until PBL regime switches to stable conditions at time interval 18–22. This feature of T_f is similar to generally observed rapid growth of PBL during morning hours in warmer days. Finally, T_f is always higher than W_f and thus turbulence effects are stronger in formulations where T_f is used as compared to that with W_f. In particular, turbulence effects will be relatively stronger with using T_f during stable conditions. In this case study, two very different formulations yielded very similar results for convective conditions. However, when we estimated T_f and W_f using the Harvard Forest site data for a decadal period (not shown), we found that T_f is always higher by about two times than the W_f with magnitude ranges between about 1 to 10.

Figure 5.

Diurnal variation of (a) median and (b) hourly averaged turbulence factors T_f and W_f for six time periods.

4.2. Intercomparison of results from Z01 and C01 formulations

Single-point model simulation results from Z01-B, C01-E, and C01-ETF are presented in this section, which are based on sectional mode of PSD. In addition to the temporal distributions of V_d, its components’ distribution will also be shown here to explore the difference between these cases. Figure 6 shows the median deposition velocities (V_d), estimated (referred to as measured and OBS) derived from using 4-hour averaged aerosol flux and transfer velocity measurements (Matsuda et al., 2010), and corresponding model simulations for Z01-B, C01-E, and C01-ETF. V_d for OBS ranges from about 0.2 to 1 cm s⁻¹ with maximum occurring at time interval 3 (1000 to 1400 LT). In general, all schemes underestimated V_d when compared to corresponding measured V_d values, which are significantly higher during daytime than nighttime due to convective conditions in PBL. Both Z01-B and C01-E didn’t capture strong diurnal variation found in measured V_d though they both have a very weak diurnal variability. However, C01-ETF predicted relatively stronger diurnal variation with higher deposition velocities during daytime, but maximum median V_d is about 50% smaller to that of measured maximum. During nighttime C01-ETF has slightly higher V_d compared to other two cases because the turbulence factor T_f works for both day and night, however, V_d is still lower than measured values. Thus, even with increased turbulence representation in C01-ETF, estimated V_d is much underpredicted and thus it seems that there are opportunities to improve processes representation in the Z01 scheme. One type of improvement was demonstrated by Emerson et al. (2020), based on measurements, revised coefficients used in the Zhang’s formulations for Brownian motion, interception, and impaction to achieve observed deposition velocities for different size distributions.

Figure 6.

Diurnal variation of estimated median dry deposition velocity from measurements and model simulations for Z01-B, C01-E, and C01-ETF for six time periods.

Modeled V_d is the sum of V_g and the conductance through R_a and R_b, thus, these parameters control the diurnal variations of V_d. In the Z01-B parameterization there is no diurnal variation for V_g (Figure 3) meaning the diurnal variation of the modeled V_d can only be attributed to conductance through R_a and R_b. However, total conductance (Figure 7a) has a negligible diurnal variation in the Z01-B parameterization because of its dependence on the u_∗ variation. In contrast, for both the C01-E and C01-ETF parameterizations diurnal variation from V_ge also contributes to the trends seen in V_d. Additionally, since C01-E uses e_∗, its diurnal trend for conductance was stronger than that of Z01-B. The measured u_∗ has only dual values with a single daytime and nighttime value. This is clearly an artifact of measurement frequency or issues with measurements because many measurements of u_∗ documented in the literature shows a diel variation like, for example, surface heat flux. Usage of dual vale of u_∗ limits the accuracy of results that used such u_∗. On the other hand, our e_∗ methodology overcomes such artifacts or limitation of using binary value of measured u_∗.

Figure 7.

Diurnal variation of (a) median total conductance and (b) boundary layer conductance for Z01-B, C01-E, and C01-ETF for six time periods.

Figure 7b shows temporal variation of median boundary layer conductance, and it is largely similar to the total conductance shown in Figure 7a. This result implies that the boundary layer resistance is controlling the pattern and magnitude of V_d. It is interesting to note that the introduction of turbulence factor (T_f) has increased boundary layer conductance in C01-ETF by about five times that of the C01-E and thus have contributed to the improved estimation of V_d. Thus, the incorporation of e_∗ and T_f into the Z01 scheme equations resulted in the better model performance of estimating V_d. Comparison of simulated aerodynamic conductance is presented in the next section along with other cases.

4.3. Intercomparison of results from S21 and C21 formulations

In addition to S21 and C21 schemes’ performance analysis, we also include the results from C01-ETF since it has the best performance among all other schemes considered in the previous Section. Figure 8 shows temporal variation of median deposition velocities from OBS and corresponding simulations by C01-ETF, S21-BS, C21-ETFS, S21-BM, and C21-ETFM. At the outset, it can be seen that the C21 scheme (i.e., C21-ETFS and C21-ETFM) performed better than all other 3 schemes (i.e., C01-ETF, S21-BS, and S21-BM). It is interesting to note that there exist minor differences between simulated V_d for sectional and modal approaches of S21 and C21 schemes. It is also interesting to note that C01-ETF performed better than the S21-BS and S21-BM and these improvements in C01-ETF may be attributed to the fact that it includes newly introduced turbulence parameters. All schemes underestimated V_d during nighttime while C21 schemes performed marginally better than other schemes.

Figure 8.

Diurnal variation of estimated median dry deposition velocity from measurements and model simulations for C01-ETF, S21-BS, C21-ETFS, S21-BM, and C21-ETFM for six time periods.

To further analyze contributions by various processes in the estimation of V_d, we show temporal variation of median total conductance for all cases in Figure 9. Since contribution by sedimentation velocity is smaller compared to other terms in V_d estimation, total conductance magnitude and its temporal variation is very similar to that in V_d. To further probe into the relative contributions by each conductance, we show median boundary layer and aerodynamic conductances temporal variation in Figures 10 and 11. It can be seen that boundary layer conductance for C01-ETF is slightly higher than that for S21 schemes while the aerodynamic conductance for C01-ETF is slightly lower that for S21. Effectively, V_d for C01-ETF has ended up slightly higher than that in S21 schemes. On the other hand, maximum median aerodynamic conductance (Figure 11) for Z01 scheme is higher than that for all other schemes and it is directly attributed to the usage of a type of stability function for heat in the Z01 scheme (Eq. 16) while a different type of stability function is used in S21 while no stability function was used in C21 resulting in different estimations. This variable outcome caused by the choice of stability function used is the reason that we argue against the usage of stability functions in dry deposition modules. This is one of the causes of differing outcomes from using different stability functions in the dry deposition modeling. Truly, it is one of the objectives of the Part-1 paper where we have introduced the e_∗ formulations to avoid using such stability functions and help to mobilize scientists towards the development of the community dry deposition modeling. The minor differences in the estimated median aerodynamic resistance in C01 and C21 are attributed to the usage of factor 0.95 in C21, since both use our e_∗ methodology.

Figure 9.

Diurnal variation of median total conductance for C01-ETF, S21-BS, C21-ETFS, S21-BM, and C21-ETFM for six time periods.

Figure 10.

Diurnal variation of median boundary layer conductance for C01-ETF, S21-BS, C21-ETFS, S21-BM, and C21-ETFM for six time periods.

Figure 11.

Diurnal variation of median aerodynamic conductance (cm s⁻¹) for C01-ETF, S21-BS, C21-ETFS, S21-BM, and C21-ETFM for six time periods.

Another characteristic feature present in the aerodynamic conductance estimations using Z01 and S21 schemes is that the maximum median aerodynamic conductance occurs at time interval 0600 to 1000 LT while for the C01 and C21 formulations the maximum occurs at time interval 1000 to 1400 LT. Usually, such maximum values, for clear sky warmer days like in this study, happen during mid-day but not in the morning hours (Figure 11). This result also points to the potential source of differing V_d estimations, which can result from using different stability functions in aerodynamic conductance. Long-term measurements-based estimation of aerodynamic conductance for heat at canopy and leaf scale (Kumagai et al., 2004; Mallick et al., 2018) indicated a general maximum value of 20 cm s⁻¹ with a maximum value of about 12 cm s⁻¹ at the highest probability density for a forest site (Panwar et al., 2020). Z01 median maximum value is similar to that was infrequently observed in the estimations based on log-term measurements, its maximum value is much higher than its median value. Thus, Z01 overestimates the most observed maximum of about 12 cm s⁻¹. On the other hand, though S21 median maximum occurs at a wrong time, its magnitude is comparable to frequently measured values like in the C01 and C22. Different types of aerodynamic resistance formulations were evaluated (e.g., Liu et al., 2007) and it was found that the most crucial parameters were roughness lengths for momentum and heat transfer as well as wind speed. In those formulations, different stability functions and constants were used that control the heat transfer estimations, and thus stability functions partly contributed to the differences in the aerodynamic resistance estimations. As an alternative method, to evaluate the accuracy of several stability functions used in aerodynamic resistance formulations, is the estimation of latent heat fluxes using different stability functions and comparison with long term latent heat flux measurements (such as those available for Harvard Forest site, Alapaty et al., 2022) that would shed light on the accuracy of such stability functions. This will be our near future work on highlighting how our proposed stability-functions-free formulation fares well compared to those that use such functions.

To examine the differences in V_d simulated by various schemes with differing particle size distributions, we simulated V_d curves integrated either by sectional (S) or modal (M) approaches from seven deposition schemes (described previously) with three σ_g values (1.01, 1.7, and 2.5) and these are shown in the Figures 12a, b, c. It is apparent that inter-comparison of simulated V_d across schemes is not σ_g-dependent because all schemes exhibit consistent patterns as σ_g changes. However, V_d trends for all schemes start to shift to the left bringing the minimum V_d from about 2 to 0.1 μm as σ_g is increased from 1.01 (Figure 12a) to 2.5 (Figure 12c), showing that predicted V_d is sensitive to σ_g. This characteristic change is expected, as reported in Shu et al. (2022) because when bigger particles (D_p > 10 m) affect the integration method (S or M), where V_d is orders of magnitude larger, the scheme with a larger σ_g (1.7 or 2.5) tends to predict a higher V_d at a mid-range particle size (0.2 < D_p < 1 m) than the scheme with a lower σ_g (1.01). Meanwhile, at smaller sizes (D_p < 0.1 m), the scheme (1.7 or 2.5) predicts a lower V_d than that with a lower σ_g (1.01) as they are influenced by particles of midsize (0.2 < D_p < 1 m) where V_d is orders of magnitude smaller. Also shown in the Figure 12 are various measurements available from the literature. For σ_g = 1.01 and 1.7 (Figures 12a and 12b) all schemes fail to reproduce the observed minimum V_d that occurs at about 0.1 mm particle size. For σ_g = 2.5 (Figure 12c) three schemes Z01-B, S21-BM, and C0–1E underpredict V_d for particle sizes 0.01 to 0.1 microns while three other schemes S21-BS, C01-ETF, and C21-ETFM closely follow the measurements with C21-ETFS ending up with slight overpredictions. For the particle sizes from 0.1 to 0.5 S21-BS closely follows median values while C21-ETFM follows maximum values in the measurements while for smaller particle sizes less than 0.1 microns, both the S21-BS and C21-ETFM simulated almost same V_d values. Measurements by Hofken and Gravenhorst (1982) (red filled triangles) indicate a plateaued V_d for the particle sizes between 0.5 to about 2 microns. Though C21-ETFM did not show such plateaued values, but it does have a feeble indication of a such signature. Proper way to replicate this missing plateaued feature is to improve formulations for collection efficiencies by interception process and to the best of our knowledge no studies exist in the literature that targeted improvement of interception process. It is worth noting that the V_d predicted by the same scheme differs substantially after applying alternative particle size distribution approaches (S or M). In general, we noticed that the sectional approach leads to higher V_d values as compared to modal approach. It is also important to note that these measured V_d values are relevant for certain meteorological conditions, different land use types, and particle size distributions and thus intercomparisons with measurements have to be made with a caution and thus the outcome may not be used to tune or redesign parameterization schemes. For example, in this study, when evaluating diurnal variability of schemes discussed above is performed, all available measurements were pre-screened and suitable field experimental data (Matsuda et al., 2010) were selected as described in the introduction section. However, this analysis can serve as a yardstick for the reasonableness of the performance of formulations in their functionality. This brings out an important point that there is a real shortage of consistent meteorological and chemical long-term measurements to understand aerosol deposition that will aid a statistically meaningful evaluation of various parameterization schemes.

Figure 12.

Variation of deposition velocity from measurements and model simulations for each dry deposition scheme with three geometric standard deviations (σ_g): (a) σ_g = 1.01, (b) σ_g = 1.7, and (c) σ_g = 2.5. D_p is median particle diameter (μm).

To quantitatively show the differences in model performance for all cases, fractional bias, (FB), Pearson correlation coefficient (R), and normalized mean error (NME) results are shown in Table 3.

Table 3.

Model performance evaluation for all cases using median values as well as entire simulation data

Schemes	Median FB	Whole data FB	Whole data R	Whole data NME
Z01-B	−1.52	−1.61	0.67	0.89
C01-E	−1.42	−1.47	0.68	0.85
C01-ETF	−1.04	−0.84	0.91	0.59
S21-BS	−1.32	−1.07	0.94	0.70
C21-ETFS	−0.79	−0.43	0.88	0.43
S21-BM	−1.32	−1.05	0.94	0.69
C21-ETFM	−0.80	−0.48	0.89	0.44

Incorporation of turbulence velocity, intensity of turbulence, and new turbulence factor into resistances estimation, has improved the model performance in the estimation of particle deposition velocity and is reflected in the FB and NME estimations while R has shown slightly different metrics. Consistent with the results presented earlier, Table 3 indicates that all schemes underestimated deposition velocity leading to negative FB values. Median values as well as all modeled values used to estimate FB and NME indicate that C21-ETFS has performed the best followed by C21-ETFM while Z01-B the least. In the next section, we present results from a sensitivity simulation.

4.4. Sensitivity study

Based on the results presented in the above sections, it will be interesting to perform one more simulation to study impacts of using T_f in the place of W_f in S21-BS in the estimation of particle deposition and we refer to it as S21-BSTF. Since in this case study, in general, T_f is very similar to W_f in its magnitude and temporal variation during daytime (Figure 5), one would expect that switching to T_f in S21-BS would not result in major improvements in V_d estimation by S21-BS. Figure 13 shows temporal variation of median V_d from OBS and S21-BS, C21-ETFS, and S21-BSTF. As expected, S21-BSTF has a minor improvement during convective conditions as compared to stable conditions, but in general there is no significant improvement when compared to C21-ETFS and OBS. This result confirms a fact that major improvements in C21-ETFS came from improved representation of turbulence from introducing intensity of turbulence and turbulence velocity scale into the respective processes, i.e., effective sedimentation velocity and effective Stokes number.

Figure 13.

Diurnal variation of median dry deposition velocity (cm s⁻¹) from measurements and model simulations for S21-BS, C21-ETFS, and S21-BSTF for six time periods.

5. Conclusions

We proposed four new deposition formulation schemes (i.e., C01-E, C01-ETF, C21-ETFS, and C21-ETFM) to model particle dry deposition with an aim to improve representation of turbulence in various deposition processes and inter-compare model results with three existing schemes (i.e., Z01-B, S21-BS, and S21-BM). All these formulations were applied at a site characterized by deciduous forest using a single-point model. Numerical simulations were performed for 5.5 days, and model evaluation was performed using particle deposition velocities estimated from measurements available at the site. Notably, we improved turbulence representation in process via introducing three turbulence parameters: (1) a new turbulence velocity scale (e_∗) in the place of friction velocity; (2) intensity of turbulence (I_t); and (3) a new turbulence factor (T_f). As a consequence, we have introduced effective sedimentation velocity and effective Stokes number that account for realistic representation of turbulence for conditions in the PBL.

Results indicate that new schemes performed better than existing schemes. Introduction of three turbulence parameters into the Z01 scheme significantly improved the model performance and predicted stronger diurnal variation of V_d though it still underpredicted V_d when compared to that estimated from measurements. We found that these three parameters either introduced or added more diurnal variability to processes and subprocesses during daytime due to the strong diurnal variability of these three turbulence parameters. In this case study, the best performing scheme was found to be C21-ETFS followed by C21-ETFM. As expected, results are barely sensitive to aerosol size distribution methodologies used in the study.

Though we improved representation of turbulence in these schemes, each scheme resulted in differing estimates of V_d for particle sizes that are of health concerns (i.e., PM_2.5). This result highlights an urgent need for required long term and comprehensive meteorological and particle measurements for a systematic evaluation of each scheme that would help to recognize best performing formulations which in turn open doors for laying the foundations for the community particle deposition modeling framework. Without such comprehensive measurements, accurate particle deposition modeling will continue to be an elusive goal. In addition, fundamental understanding of particle dry deposition is still needed in the future to improve our ability to accurately model it. At least, findings from this research may help improve the capability of dry deposition schemes for improved estimation of particle dry deposition. Further testing of the new schemes in 3-D air quality model is needed, which will be the focus of our future studies.

Key Points:

• New turbulence parameters are introduced to improve particle deposition estimation

• New dry deposition formulations largely reduce biases in deposition velocity estimations

Data and code availability

Model code and all the data used to generate figures and tables shown in this article can be freely downloaded at http://doi.org/10.5281/zenodo.5874973.