Review and uncertainty assessment of size-resolved scavenging coefficient formulations for snow scavenging of atmospheric aerosols

Introduction Conclusions References

An assessment of uncertainties on size-resolved Λ for aerosols scavenged by rain (Λ rain ) was recently conducted by Wang et al. (2010).The present study follows a similar approach to assess uncertainties of size-resolved Λ for aerosols scavenged by snow (Λ snow ).Such a study is needed given that current knowledge of snow scavenging is considerably more limited than that for rain scavenging.One reason is that scavenging by snow is more complicated due to the wide variety of snow particle shapes, sizes, and densities, which results in different fall speeds, cross-sectional areas, and flow patterns around snow particles (Pruppacher and Klett, 1997;Jylhä, 1999).On the other hand, snow scavenging is an important removal mechanism in mid-latitude and polar regions in the winter and in mountainous areas and in the upper troposphere at all times of year.One study estimated that roughly 30 % of below-cloud scavenging of sulphate particles by precipitation is due to snow (Croft et al., 2009) Current treatments of snow scavenging of atmospheric aerosol particles in CTMs vary substantially, ranging from using a bulk Λ parameterized as a function of snowfall intensity (as liquid water equivalent) without considering the sizes of either aerosol or snow particles (Baklanov, 1999;Sofiev et al., 2006) to using the same size-resolved Λ formula as that for rain scavenging to using a size-resolved Λ formula specifically developed for snow conditions (e.g., Gong et al., 2006;Croft et al., 2009;Feng, 2009).Past reviews have documented these various approaches (Rasch et al., 2000;Textor et al., 2006;Sportisse, 2007;Zhang, 2008;Gong et al., 2011).The present study, however, attempts to quantify the uncertainties related to various parameters chosen for the existing size-resolved Λ snow formulas developed specifically for snow conditions.
In the following sections, a brief overview of current size-resolved Λ snow parameterizations, including their component parameters, is first given (Sect.2); next, a summary of the results of sensitivity tests that were conducted to investigate uncertainties in Λ snow induced by these various parameters is provided (Sect.3).The uncertainties of existing theoretical size-resolved Λ snow parameterizations is then assessed further by using various combinations of the component parameter formulas (Sect.4.1) and by comparing with an available empirical Λ snow parameterization derived directly from fits to field measurements (Sect.4.2).The impact of different Λ snow formulas on predicted aerosol concentrations is then briefly discussed (Sect.4.3) and a comparison of uncertainties between Λ snow and Λ rain is presented (Sect. 4.4).Lastly, some conclusions are given in Sect. 5.

Theory of size-resolved snow scavenging coefficient Λ snow
The terminology of ice or snow particles reflects the greater physical variability of frozen or solid hydrometeors vs. liquid hydrometeors (rain drops).As discussed by Pruppacher and Klett (1997), small ice particles that have grown only by water vapour diffusion are called ice crystals or snow crystals.These crystals have different shapes or habits, including plates, columns, stars, needles, dendrites, spheres, and bullets.Ag-Introduction

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Full gregates of snow crystals are called snowflakes.Individual snow crystals usually have a maximum dimension D m of less than 5 mm whereas snowflakes may have a maximum dimension of several cm.Snow crystals can also grow by collisions with cloud drops, which is called riming.Depending upon the degree of riming, these snow particles may be referred to as rimed snow crystals or graupel particles or ice pellets.All of these rimed snow particles usually have D m values of less than 5 mm; heavily-rimed larger particles are called hailstones.
In CTMs that simulate aerosol particle number concentrations, the below-cloud scavenging of aerosol particles by snow is commonly described as (Seinfeld and Pandis, 2006) where n(d p , t) is the number concentration of aerosol particles with diameters d p at time t, and Λ snow (d p ) is the scavenging coefficient for aerosol particles of size d p and can be calculated based on the concept of collection efficiency between falling hydrometeors and aerosol particles (e.g., Slinn, 1984).The size-resolved scavenging coefficient is parameterized as where size distribution; (iii) the snow-particle terminal velocity (assuming V D v d ); and (iv) the snow-particle effective cross-sectional area.Available formulas for these four parameters are reviewed and discussed below.All symbols used in this study are defined in Table A1.

Snow particle-aerosol particle collection efficiency E(d p , D p )
E (d p , D p ), the collection efficiency for aerosol particles of diameter d p of a snow particle of melted diameter D p , gives the rate of collection of aerosol particles of diameter d p by the falling snow particle normalized by the number of upstream particles of diameter d p swept across an area equal to the effective cross-sectional area of the snow particle (e.g., Slinn, 1984).The collection efficiency is the most important parameter in the calculation of Λ snow in Eq. ( 2).There are considerably fewer studies on E for snow particles and aerosol particles than there are for rain drops and aerosol particles.However, there are a few studies that describe E based on rigorous theoretical models involving (i) a particle trajectory model under the influence of the flow field of falling ice crystals and (ii) a convective diffusion model for small aerosol particles.For example, Martin et al. (1980) studied E for planar ice crystals (approximated as hexagonal plates) at low-to-intermediate Reynolds numbers and Miller and Wang (1989) studied E for columnar ice crystals using a theoretical model.Several field measurements and laboratory experiments under controlled conditions have also been conducted to study and verify theoretical results (e.g., Knutson et al., 1976;Sauter and Wang, 1989;Murakami et al., 1985).These studies suggest that a complete theoretical model for E would be too complex to be implemented in CTMs.Three different size-resolved semiempirical formulas for E have thus been developed for CTM applications (Slinn, 1984;Murakami et al., 1985;Dick, 1990) as listed in Table 1.Some of these formulas for E have been used to parameterize Λ snow in current CTMs (e.g., Gong et al., 2006;Croft et al., 2009;Feng, 2009).Introduction

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Full Λ snow also depends on the number size spectrum of snow particles.Various microphysical and dynamical processes inside and below cloud layers modify snow-particle size spectra.Other factors affecting snow-particle size spectra include ambient temperature, particle habit, precipitation intensity, and the stage of cloud and precipitation development (e.g., Harimaya et al., 2004;Woods et al., 2008).In practical applications, empirical mathematical formulas derived from the observed size spectra have been used to approximate natural snow-particle size distributions (e.g., Marshall and Palmer, 1948;Gunn and Marshall, 1958;Sekhon and Srivastava, 1970;Scott, 1982;Smith, 1984;Mitchell, 1991;Heymsfield, 2003;Field et al., 2005;Woods et al., 2008).
For example, the exponential Marshall-Palmer size distribution (Marshall and Palmer, 1948), originally proposed for raindrop size distribution, was also found to describe snow particle size distribution reasonably well (Passarelli, 1978).Gunn and Marshall (1958) reported another exponential size distribution function for aggregate snowflakes, the first one to be derived directly from ground observations of snow, following an assessment method similar to that used for raindrop size distributions by Marshall and Palmer (1948).By reanalyzing the dataset of Gunn and Marshall (1958) as well as analyzing additional snowflake size distribution measurements, Sekhon and Srivastava (1970) suggested an updated exponential formula.Scott (1982) modified the parameters in the Marshall-Palmer distribution based on results from Passarelli (1978) and Houze et al. (1979) so the modified exponential function can be applied to large spatial scales.To date, exponential distributions have been widely used in various cloud microphysics to represent snow size spectra (e.g., Cotton et al., 1982;Lin et al., 1983;Rutledge and Hobbs, 1983;Reisner et al., 1998;Thompson et al., 2004;Croft et al., 2009;Feng, 2009;Solomon et al., 2009).
The basic form of the exponential function for snow particle number size distribution is written as (3) 14829 Introduction

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Full where N 0e is the intercept parameter and β e is a slope parameter.Different researchers, however, have treated N 0e and β e in different ways, some have adopted a fixed N 0e whereas others have varied N 0e according to precipitation intensity (Table 2).Note that the parameters in Scott (1982) are based on actual snow particle size D m whereas the other three distributions listed in Table 2 are for equivalent drop sizes D p .A conversion of snow particle size to equivalent melted drop size is needed for the Scott (1982) formula (see Appendix A) to allow a direct comparison of these distributions (see Sect. 3.2).

Snow particle terminal velocity V D
Terminal velocities for various ice particle types have been studied both experimentally and theoretically, and corresponding empirical parameterizations have been developed (e.g., Langleben, 1954;Pruppacher and Klett, 1997;Mitchell, 1996;Mitchell and Heymsfield, 2005).Early formulas for snow particle terminal velocity were derived directly from fall speed measurements (i.e., experimentally based) and treated the terminal velocity V D as a power-law function of the ice particle maximum dimension D m : that is, m , where a v and b v are empirical constants but varying with ice crystal habit (e.g., Langleben, 1954;Starr and Cox, 1985).However, the application of most experimentally-based empirical formulas is limited to the particle shape for which the measurements were conducted (see Table 3).More recently developed parameterizations are theoretically-based formulas.A power-law relationship is first determined between the Reynolds number (Re; dimensionless) and the Best or Davies number Aµ 2 a ; dimensionless) (Bohm, 1989(Bohm, , 1992;;Mitchell, 1996); the terminal velocity is then derived from Re that is determined in terms of X .The detailed description of generating X and the empirical relationship of Re-X were given in Mitchell (1996) and Mitchell and Heymsfield (2005).Since X is a function of the ice particle mass (m) and the cross-sectional area (A), both of which are parameterized as a power-law function of the maximum dimension of the ice particle (D m ), the selection of different power-law Introduction

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Full functions for m and A may lead to large differences in the X value, and thus to large errors in V D (Mitchell, 1996).The advantage of the theoretically-based parameterizations, however, is that they can be applied to any particle shape (Table 3).

Snow particle cross-sectional area A
Knowledge of the cross-sectional area of a snow particle is essential for accurate calculation of Λ snow and for the estimation of snow particle terminal velocity.Snow particles can have dozens of irregular shapes and it is not realistic to represent the A of all particle shapes accurately using one single theoretical formula.A common approach associating A of a snow particle and its mass (m) is through the definition of a parameter: the particle's maximum dimension, D m .Both m and A are parameterized as power-law functions of D m : m = αD β m and A = γD σ m , where α, β, γ, and σ are empirical constants developed from measurements of natural snow particles (e.g., Locatelli and Hobbs, 1974;Mitchell et al., 1990;Mitchell and Arnott, 1994;Mitchell, 1996;Pruppacher and Klett, 1997;Woods et al., 2008).The detailed empirical expressions and related parameters for various snow types were reviewed by Mitchell (1996).
In the present study, four habit types of snow crystals -spherical ice crystal, dendrite snow plate, columnar ice crystal, and graupel particle -were chosen for analysis and discussion (Table 4).These are the four habits of snow crystals that occur most frequently as revealed by ground observations (Hobbs et al., 1972); they are believed to be the main habits of ice crystals based on the classification of habit composition as determined from the airborne 2D-C probe imagery and ground-based stereomicroscope observations (Woods et al., 2008).As well, current cloud-scale CTMs and numerical weather prediction models only explicitly distinguish and predict a few types of ice crystals, including dendrite snowflake, columnar crystal, and graupel (hail) (e.g., Field and Heymsfield, 2003;Thompson et al., 2008;Morrison et al., 2009).
Note that the particle size D m used in the diameter-based mass and area power law formulas shown in Table 4 is the maximum dimension for a frozen particle.These relationships can also be represented in terms of D p , the equivalent drop diameter of Introduction

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Full a snow particle when it melts.The melted mass of a snow particle can be expressed in terms of the diameter of its equivalent water drop as where ρ water is the water density.The cross-sectional area of a falling snow particle can then be written as (5) 3 Sensitivity of theoretical size-resolved Λ snow to input-parameter selections From Sect. 2 we know that four component parameters determine Λ snow values and that different formulas have been proposed in the literature for these parameters (see Tables 1-4).The sensitivity of Λ snow to the choice of one of these different formulas for each of these component parameters are discussed below.Note that for all of the sensitivity tests, the temperature and pressure were assumed to be −10 • C and 1013.5 hPa, respectively.certainly caused by the size-dependence of the collection mechanisms, namely Brownian diffusion, interception, and inertial impaction, considered in the formulas in Table 1.

Sensitivity of Λ snow to E
The contribution of Brownian diffusion to E dominates for the ultrafine particles but decreases rapidly as particle size increases; the contribution of inertial impaction becomes significant when the diameter of an aerosol particle is larger than a few microns; and the contribution of the interception mechanism increases with increasing particle size and appears to be important for particles in the diameter range from 1.0 µm to a few microns.The combined contributions of the three mechanisms lead to low E values for particles in the size range 0.01 µm < d p < 1.0 µm.Note that other potential collection mechanisms such as diffusiophoresis, thermophoresis, and electric charges are not included in these formulas.For rain scavenging of atmospheric aerosols, these several mechanisms are less important than the three major mechanisms discussed above and are only significant for particles in the size range of 0.01 µm-1.0 µm (Wang et al., 2010;Santachiara et al., 2012).This is also expected to be the case for snow scavenging of aerosols.
It is evident from Fig. 1 that the E (d p ) profiles for fixed D m from the Murakami et al. (1985) and Dick (1990) formulas are not very sensitive to the snow particle shapes.The four E (d p ) profiles for four snow particle shapes based on the same formula are similar, e.g., all have a minimum E value at the same particle diameter.E (d p ) values for these two formulas also differ only by a factor of 2 to 3 between different snow particle shapes across the entire aerosol particle size range.Note that all of the formulas in Table 1 depend on snow particle terminal velocity V D either directly or through the Reynolds and Stokes numbers.In the sensitivity tests presented in Fig. 1, V D values were calculated for all snow-particle habits based on the theoretical formula developed by Mitchell and Heymsfield (2005) (see Table 3; the details of the V D calculation will be discussed later in Sect.showed a different pattern.The E (d p ) profiles for the dendrite and column snow particle shapes are basically the same and the E (d p ) profiles for the sphere and graupel particle shapes are also similar.However, the E (d p ) profiles between these two groups differ significantly, especially for the aerosol particle sizes where the minimum E value occurs.This is due to values specified for two of the parameters used in Slinn's formula (see Table 1); λ and α were given as 10.0 µm and 1.0, respectively, for dendrite and column shapes but 100.0 µm and 2/3, respectively, for sphere and graupel shapes.
Differences in E between the Murakami et al. (1985) and Dick (1990) formulas are significant for all aerosol particle sizes and for all snow particle sizes and shapes considered here.The largest differences occur for particle diameters around 0.1 µm, for which the difference can be larger than one order of magnitude.E decreases significantly with increasing collector (i.e., snow particle) size in these two formulas.The difference in E between these two formulas also decreases with increasing collector size.The dependence of E on collector size is because larger collectors have larger V D values, and thus larger Re values, which results in smaller E values (see formulas in Table 1).Comparing E values for the Slinn (1984) formula with those from the Murakami et al. (1985) and Dick (1990) formulas, the differences are even larger, especially for smaller collectors (e.g., Fig. 1a).Differences up to nearly three orders of magnitude can be seen for aerosol particle sizes from 0.1 µm to 2 µm.It should be pointed out that the E values for the Slinn (1984) formula do not change much with collector size because λ and α values are fixed for all collector sizes, a different behaviour from the other two formulas discussed above.
The sensitivity of Λ snow , calculated using Eq. ( 2), to the choice of the three different formulas for E (Table 1) is illustrated in Fig. 2 for two snowfall intensity conditions (as liquid water equivalent): 0.1 mm h −1 (solid line) and 10 mm h −1 (symbol line).The snow particle terminal velocity used for Fig. 2 was that of Mitchell and Heymsfield (2005) (see Table 3) and the snow particle size spectrum followed Sekhon and Srivastava (1970) (see Table 2).Figure 2 indicates that the differences in Λ snow due to the different E (d p , D p ) formulas vary with aerosol particle size, snow particle shape, and snowfall Introduction

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Full intensity.For the largest aerosol particles (i.e., d p > 10.0 µm), the differences in Λ snow are small (e.g., a factor of 2) for both snowfall intensities and all snow particle shapes due to the very similar E values in this aerosol particle size range (close to unity; not shown in Fig. 1).For aerosol particles smaller than 10 µm, a difference of one order of magnitude or larger is seen under all snowfall intensity and snow particle shape conditions.It can also be seen that the differences in Λ snow are smaller for aerosol particles smaller than 0.01 µm than for particles between 0.01 µm-10.0µm, consistent with the differences in E profiles shown in Fig. 1.
Figure 2 also suggests that Λ snow values derived from the Murakami et al. (1985) and Dick (1990) formulas agree well (e.g., within a factor of 2) for aerosol particles larger than 1.0 µm and differ by a factor of 3 to 4 for aerosol particles smaller than 0.1 µm for all snow shapes and snowfall intensities.In contrast, Λ snow values from the Slinn (1984) formula shows a large deviation from those of the other two formulas, in particular for the aerosol particle size range of 0.1 µm < d p < 10.0 µm, except for the case for dendrites and a light snowfall intensity.Again, this can be explained by the E pattern shown in Fig. 1.These results suggest that the formulation used to describe the collection efficiency is a very important source of uncertainty in estimating Λ snow .

Sensitivity of Λ snow to N(D p )
Snow particle size distributions (N(D p )) generated from the four widely used exponential formulas listed in Table 2 are shown in Fig. 3 for two snowfall intensities (as liquid water equivalent): 0.1 and 1.0 mm h −1 .The Gunn-Marshall (GM) and the Sekhon-Srivastava (SS) N(D p ) profiles are quite close due to their similar values for the intercept parameter N 0e and slope parameters β e (see Table 2).The Marshall-Palmer (MP) distribution differs significantly from those of GM and SS, and the Scott (SC) distribution is even more different.All four exponential distributions yield large numbers of small snow particles (< 0.1 mm).This is due to the limitation in the definition of the exponential formula, which generally predicts maximum number concentration for particle sizes approaching zero (see Eq. 3).

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Full The percentages of snow particle number concentrations in different size ranges are shown in Table 5 for three of the four snow-particle size distributions and four snowfall intensities.Note that N 0e is fixed for the MP and SC distributions but decreases with increasing snowfall intensity for the SS distribution (see Table 2).Thus, the total snow particle number concentrations from the MP and SC distributions increase and those from the SS decrease with increasing snowfall intensity (Table 5).The total number concentrations from different size distributions can differ from less than one order of magnitude to more than two orders of magnitude, depending on snowfall intensity.For all of the size distributions, however, the percentages of the smallest snow particles (< 0.1 mm) decrease and those of the largest snow particles (> 1 mm) increase with increasing snowfall intensity.This can also be seen from Fig. 3, in which all of the snow-particle size distribution profiles shift to larger snow particle sizes with increasing snowfall intensity.
Figure 4 shows the sensitivity of Λ snow to the four different snow particle number size distributions N(D p ) considered in Fig. 3 for four snow particle shapes and two snowfall intensities (as liquid water equivalent): 0.1 mm h −1 and 10 mm h −1 .V D and E (d p , D p ) were assumed to follow the theoretical formulas of Mitchell and Heymsfield (2005) and Murakami et al. (1985), respectively.Differences in Λ snow values derived from these different N(D p ) formulas are up to one order of magnitude for all aerosol particle sizes under light snowfall intensity (0.1 mm h −1 ).The differences in Λ snow also increase with increasing snowfall intensity and can be larger than one order of magnitude for very strong snowfall intensity (e.g., 10 mm h −1 ).The dependence of Λ snow on snowfall intensity is also greater for some N(D p ) formulas than others.Based on the Λ snow profiles shown in Fig. 4, we can conclude that in general different assumptions for N(D p ) contribute an uncertainty to the Λ snow profile of about one order of magnitude for all aerosol particle sizes under all snow particle shape and snowfall intensity conditions.Introduction

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Full Figure 5a shows the terminal velocities V D of snow particles with four different shapes calculated from empirical and theoretical formulas selected from Table 3.Each colour represents one particle shape and each symbol represents one formula.Note that the theoretical formula of Mitchell and Heymsfield (2005) was considered to apply to any kind of snow particle shape.Results from the theoretical formula of Mitchell (1996) are not shown in the figure because the calculated values are quite close to those from Mitchell and Heymsfield (2005).For snow particles larger than 0.2 mm, V D values for the same particle shape but based on different formulas are generally within a factor of 5; however, the differences can be larger than a factor of 10 if considering both different particle shapes and different formulas.For snow particles smaller than 0.2 mm, the differences in V D can be up to two orders of magnitude and generally increase rapidly with decreasing snow particle size.As well, V D values from all of the empirical formulas are larger than those from the theoretical formula of Mitchell and Heymsfield (2005) for all particle shapes.The best agreement between the empirical and theoretical formulas is for the dendrite shape and snow particles larger than 0.2 mm. Figure 5b shows the cross-sectional area A of a snow particle versus its maximum dimension for four different snow particle shapes based on the power-law formulas listed in Table 4.The differences in A between different snow particle shapes increases from a factor of 3 to a factor of more than 10 as snow particle size increases from 0.1 to 10 mm.
The results of sensitivity tests conducted to investigate the influence of V D and A on Λ snow for four different snow particle shapes and three different snowfall intensities (0.1, 1.0 and 10 mm h −1 as liquid water equivalent) are shown in Fig. 6.Λ snow profiles were calculated for the nine V D profiles shown in Fig. 5a and the four A profiles shown in Fig. 5b for each snowfall intensity.All of the sensitivity tests shown in this figure used the snow particle size spectrum formula of Sekhon and Srivastava (1970) (Table 2) and the collection efficiency formula of Murakami et al. (1985) (Table 1).As in Fig. 5a, Introduction

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Full each colour in Fig. 6 represents one snow particle shape and each symbol represents one V D formula.It is also evident from Fig. 5b that each snow particle shape only has one formula available for A. Thus, the influence of V D on Λ snow can be identified by comparing Λ snow profiles for the same snow particle shape (i.e., same coloured lines), while the influence of A on Λ snow can be identified by comparing Λ snow profiles based on the same V D formula (e.g., the four lines using the formula of Mitchell and Heymsfield, 2005).The overall uncertainty in the Λ snow profile shown in Fig. 6 is thus due to the combination of influences from both V D and A.
Figure 6 shows that Λ snow may vary by a factor of 2 to 3 for the same snow particle shape for all aerosol particle sizes if different V D formulas are used, and it may also vary by a factor of 2 to 3 for different snow particle shapes even for the same V D formula.The combined uncertainties from both V D and A can thus be as high as a factor of 10.Λ snow values also increase with increasing snowfall intensity, as do the uncertainties in Λ snow values.While the uncertainty in Λ snow caused by uncertainties in either V D or A are smaller than those associated with the representation of E (d p , D p ) or N(D p ), the combined uncertainty due to V D and A can be comparable to the other two factors in some cases, e.g., for large aerosol particles and for strong snowfall intensity (e.g., compare Fig. 6c with Fig. 2).It is also worth noting that the uncertainties in Λ caused by V D are larger for the snow conditions discussed here than for the rain conditions discussed in Wang et al. (2010), and the largest uncertainties under snow conditions are for large aerosol particles vs. submicron aerosol particles under rain conditions.Thus, significant differences exist in the uncertainties associated with Λ between rain and snow conditions.Introduction

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Full As discussed in the previous sections, theoretically-based parameterizations of Λ snow require knowledge of E (d p , D p ), N(D p ), V D , and A. However, due to the natural variability of snow particle shapes and densities, the limited experimental evidence, and the complexity of microphysical collection processes, there has not been any agreement or consensus in the modelling community as to which formulas should be used for the above-mentioned component parameters in the calculation of Λ snow .For example, Feng (2009) proposed a size-resolved model for below-cloud snow scavenging, in which E (d p , D p ) was based on a combination of schemes by Martin et al. (1980), Miller and Wang (1989), and Murakami et al. (1985), N(D p ) followed Sekhon and Srivastava (1970), and V D and A followed Mitchell (1996).Croft et al. (2009) also proposed a sizeresolved parameterization for below-cloud snow scavenging, in which E followed Dick (1990) or Slinn (1984), but all snow particles were assumed to be 30 µg in mass and 0.5 mm in radius and to fall at 80 cm s −1 .Gong et al. (2006) parameterized aerosol scavenging by snow based on the Slinn (1984) formula for E and assuming a stellar shape for snow crystals when −25 • C < T < 0 • C and a graupel shape when T < −25 • C. In this section, the uncertainties in existing theoretical size-resolved Λ snow parameterizations were investigated using various combinations of the available formulas for the above-mentioned component parameters.Three semi-empirical formulas for E (d p , D p ) (Slinn, 1984;Murakami et al., 1985;and Dick, 1990; see Table 1 and Sect.2.1) and three formulas for N(D p ) (SS - Sekhon and Srivastava (1970); SC -Scott (1982); and MP - Marshall and Palmer (1948); see Table 2 and Sect.2.2) were combined together to generate nine sensitivity tests for each of four snow particle shapes (Fig. 7).The V D formula of Mitchell and Heymsfield (2005) was used in every sensitivity test because this is the only formula applicable to all snow particle shapes.This formula is a physically-based parameterization and it seems to predict more reasonable V D values for small snow particles (i.e., D m < 0.5 mm) than empirically-based formulas 14839 Introduction

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Full (see Sect. 3.3).Besides, the uncertainty in Λ snow values due to the specification of V D is much smaller than those introduced by the specification of E (d p , D p ) and N(D p ) (Sect.3.3).Note that uncertainties from various A formulas are implicitly included in different snow particle shapes, as can be seen by comparing the four panels in both Fig. 7 and Fig. 8.
Under light snowfall intensities (e.g., 0.1 mm h −1 in Fig. 7), the uncertainties in the calculated Λ snow are generally in the range of one to two orders of magnitude for very small (e.g., < 0.01 µm) and very large (e.g., > 10 µm) aerosol particles.The uncertainties are much larger for the median size aerosols, i.e., two orders of magnitude or more.The largest uncertainty occurs at an aerosol particle size of around 0.1 µm for dendrite and column habits and at an aerosol particle size of around 1 µm for sphere and graupel habits.This difference is largely associated with snow particle shape caused by the differences in E (d p , D p ) profiles for different snow particle shapes as shown in Fig. 2. The ranges of Λ snow values for any aerosol particle size are also different for different snow particle shapes as can be seen by comparing the four Fig. 7  Figure 8 shows a similar comparison to Fig. 7 for a snowfall intensity of 1 mm h −1 .
When snowfall intensity increases, Λ snow values also increase for all aerosol particle sizes (compare Fig. 8 with Fig. 7; note the different scales for the y-axes), as do uncertainties in the Λ snow values.The increases in uncertainty are larger for small aerosol particles (0.001-0.1 µm) than for large particles.Apparently, some formulas are more Introduction

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Full sensitive to snowfall intensity than others are for smaller aerosol particles.The uncertainties in Λ snow can be as high as two orders of magnitude even for very small aerosol particles (e.g., 0.001 µm).From Sect.3.2 it is known that the differences in the total number of snow particles between different N(D p ) formulas increase with increasing snowfall intensity.This behaviour at least partly explains the increased uncertainties in Λ snow with decreasing aerosol particle size.

Comparison between theoretically-and empirically-estimated Λ snow profiles
Λ snow values calculated using the empirical formula of Paramonov et al. (2011) (Appendix B) are also shown in Figs.7 and 8 (pink curves).The formula was developed based on the empirical fit to four years of field measurements in an urban environment in Finland and applies to a variety of different snow particle shapes (e.g., snowflakes, snow grains, ice crystals, ice pellets, and mixed snow and rain).Although the urban field data covered a range of snowfall intensities (as liquid water equivalent) from 0.1 to 1.2 mm h −1 , the Λ snow values in the formula of Paramonov et al. (2011) do not depend on snowfall intensity or snow particle shape.Therefore, the same pink curve is plotted in each panel of Figs. 7 and 8.Note that this empirical formula is only valid for aerosol particle sizes of 0.01-1.0µm.It should also be noted that there was another formula in the literature which was developed by Kyrö et al. (2009) based on four years of field data collected in a rural background environment; but the formula was only applicable to light snow intensities and thus was not compared here.
Λ snow values from this empirical formula are several times larger than the upper range of the theoretically-estimated values for a light snowfall intensity (Fig. 7) but are within the upper range of the theoretically-estimated values for a strong snowfall intensity (Fig. 8) for aerosol particles of diameter 0.01-1.0µm.As mentioned above, the empirical formula was based on measurements spanning snowfall intensities from 0.1 to 1.2 mm h −1 .If the snowfall intensities during the experiment were equally likely, then since Λ snow should vary directly with snowfall intensity (e.g., a factor of 5 larger for 1.0 mm h −1 than for 0.1 mm h −1 as shown from the theoretically-based Λ snow profiles Introduction

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Full in Figs.7 and 8), the empirical Λ snow formula should overpredict Λ snow for a snowfall intensity of 0.1 mm h −1 but slightly underpredict Λ snow for a snowfall intensity of 1.0 mm h −1 .Thus, the empirical profiles should shift down to smaller values in Fig. 7 and shift up to larger values in Fig. 8.This adjustment suggests that the empiricallyestimated Λ snow profiles are in the upper range of, or are just slightly larger than, the theoretically-estimated values for all aerosol particle sizes and snowfall intensities for which the empirical formula applies.

Impact of Λ snow uncertainties on predicted aerosol concentrations
Following Wang et al. (2010), two aerosol particle size distributions, representing marine and urban aerosol populations, respectively, were taken as examples to investigate the impact of different Λ parameterizations on predicted aerosol particle concentrations under two snowfall intensities (0.1 mm h −1 and 1.0 mm h −1 , as liquid water equivalent).
The initial size distribution for each aerosol type was described as a sum of three lognormal functions.Three Λ snow parameterizations shown in Figs.7 and 8 (MP+MH+SL representing lowest theoretical Λ snow , SC+MH+MU representing highest theoretical Λ snow , and the empirical Λ snow of Paramonov et al., 2011) were chosen to be applied to Eq. ( 1).The time evolution of the particle number and mass concentrations was then calculated by integrating Eq. ( 1) with very small time steps (10 s) and a large number of size bins (100) to a time of reaching a total precipitation amount of 5 mm (Fig. 9).As shown in Fig. 9, significant differences in the bulk number and mass concentrations were found from using different Λ snow formulas.In less than one hour of a typical snowfall intensity (e.g., 1.0 mm h −1 as liquid water equivalent, which is approximately 1 cm h −1 of snow depth, second row in Fig. 9), a factor of 2 differences were found in both number and mass concentrations for both marine and urban aerosol distributions.
It is also clear from Fig. 9 that the impacts of using different Λ snow parameterizations are quantitatively different for the bulk number and mass concentrations.This is because the bulk number concentration is associated with small particles whereas the Introduction

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Full bulk mass concentration is generally associated with large particles, as can be seen from the initial particle size distributions shown in Fig. 10 in Wang et al. (2010).

Comparison between Λ snow and Λ rain
Comparing the uncertainties for Λ snow that have been reviewed in this study with those for Λ rain that were reviewed in a previous study (Wang et al., 2010), both similarities and differences were found in terms of the uncertainties caused by various input parameters.For both Λ snow and Λ rain , the formulation of the collection efficiency E between hydrometeors and aerosol particles is the largest source of uncertainty amongst all of the input parameters.The uncertainties in Λ snow and Λ rain caused by E can be more than one order of magnitude for almost all aerosol particle sizes.Uncertainties in Λ snow caused by other parameters (snow particle number size spectrum, terminal velocity, and shape) can also be as large as one order of magnitude, whereas the corresponding uncertainties for Λ rain are all smaller than a factor of 5.0.The combined uncertainty from all sources is thus larger for Λ snow than for Λ rain .
It has been speculated that snow particles might scavenge more aerosol particles than rain drops do for an equivalent precipitation amount given the larger surface areas and various shapes of snow particles (Reiter and Carnuth, 1969;Magono et al., 1975;Graedel and Franey, 1975;Murakami et al., 1985;Sparmacher et al., 1993;Croft et al., 2009;Kyrö et al., 2009).However, this hypothesis has not yet been verified by either field or theoretical studies.To shed some light on this issue, one simple approach would be to compare directly the magnitude of Λ snow and Λ rain profiles generated for the same precipitation amount.One challenge to this approach, though, is that both Λ snow and Λ rain have a large range of values and very large uncertainties.
A typical snowfall intensity (e.g., 1 cm h −1 of snow, which is approximately equivalent to 1 mm h −1 of liquid water) is chosen below as an example to compare the relative magnitudes of Λ snow and Λ rain .The minimum and maximum Λ snow values (two blue lines) shown in Fig. 10 were extracted from all panels of Fig. 8 in the present study while those for Λ rain (two red lines) were obtained from Fig. 8a of a previous study (Wang 14843 Introduction

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Full  , 2010).Λ snow from the empirical formula of Paramonov et al. (2011) (shown in Fig. 8) and Λ rain from an empirical formula plotted in Fig. 8a of Wang et al. (2010) are also depicted in Fig. 10 (two dashed lines).
It can be seen in Fig. 10 that uncertainties in (or ranges of) Λ snow are up to two orders of magnitude for small (< 0.01 µm) and large aerosol particles (> 10 µm) and up to three orders of magnitude for median size aerosol particles.In comparison, uncertainties in Λ rain are smaller than one order of magnitude for small and large aerosol particles and mostly smaller than two orders of magnitude for median size aerosol particles; the only exception for rain is for aerosol particles of 1-3 µm, for which the uncertainties are slightly higher than two orders of magnitude.It should be pointed out that part of the large range of Λ snow values will be due to real variability (e.g., different snow particle shapes and related properties affecting Λ snow ) while the other part will be due to errors (e.g., improper formulation of related parameters).The median Λ snow value seems to be a factor of 5-10 higher than the median Λ rain value for most aerosol particle sizes, which suggests the possibility of faster removal of atmospheric aerosols by snow than by rain for an equivalent precipitation amount.However, almost all Λ rain values lie within the range of Λ snow values, which suggests that snow removal of aerosol particles may not always be faster than rain removal.The relative magnitudes of Λ snow and Λ rain should also depend on snow particle shape (see the minimum Λ snow profiles in Fig. 8a  and d) and other conditions that may not be explicitly considered in either Λ snow or Λ rain (e.g., Wang et al., 2011;Paramonov et al., 2011).
As discussed in Sect.4.2, Λ snow from the field-based empirical formula shown in Fig. 10 should be adjusted upwards to reflect the case of 1 mm h −1 precipitation intensity.This adjustment to the empirical Λ snow profile would make it higher than the corresponding empirical Λ rain profile for submicron aerosols.Thus, it is likely that snow removal is more effective than rain removal in many situations, although this conclusion may not apply to all snow particle shapes, to all aerosol particle sizes, or under all other conditions.A firm conclusion thus cannot be drawn at this stage due to the Introduction

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Full limited number of field and laboratory studies that are available as well as the large uncertainties in theoretical studies.

Conclusions
A review of current knowledge about Λ snow , the size-resolved scavenging coefficient for atmospheric aerosols scavenged below cloud by falling snow, was conducted in this study.The four component parameters needed for theoretical formulations of Λ snow all contribute significant uncertainties to the estimated Λ snow values.As expected, the formulation of the collection efficiency E between snow particles and aerosol particles contributes the largest uncertainty to Λ snow amongst all of the component parameters.However, uncertainties caused individually by the other parameters were also up to one order of magnitude, which was unexpectedly large in contrast to values obtained in an uncertainty analysis for Λ rain presented in a previous study by Wang et al. (2010).For a typical snowfall intensity of 1 mm h −1 (as liquid water equivalent, or approximately 1 cm h −1 of snow), the uncertainty associated with theoretically estimated Λ snow profiles spans nearly three orders of magnitude, in contrast to the one to two orderof-magnitude range for Λ rain .Moreover, most Λ rain values lie in the lower end of the range of Λ snow values, which suggests that snow scavenging of atmospheric aerosol particles is likely more effective than rain scavenging in many cases for an equivalent precipitation intensity.However, under certain circumstance (e.g., aerosol particle size, snow particle shape, snowfall and rainfall intensity), removal by snow might be slower than removal by rain.A complete picture cannot be drawn at the present time due to our limited knowledge.
Because of the large range of estimated Λ snow and Λ rain values, their close magnitudes, and their similar aerosol-particle-size dependent profiles, a simple semiempirical formula for size-resolved Λ as a polynomial function of precipitation intensity might be appropriate for both Λ snow and Λ rain .Such a formula could be developed through curve-fitting over a wide range of precipitation conditions using the set of ex-Introduction

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Full where d p is particle diameter (in m), a 1 = 28.0, a 2 = 1550.0,a 3 = 456.0,g = 0.00015, h = 0.00013, and RH is relative humidity.The formula is only valid for aerosol particles of 0.01-1.µm in diameter and snowfall intensities of 0.1 to 1.2 mm h −1 (as liquid water equivalent).Nevertheless, the formula is applicable to snowfall episodes of snowflakes, snow grains, ice crystals, ice pellets, as well as mixed with rain.Introduction

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Full Pe (1 + 0.4Re 1/6 Pe 1/3 ) a λ is the characteristic capture length and depends on the shape of snow particles (e.g., sleet/graupel, rimed crystals, powder snow, dendrite, tissue paper, and camera film).Re λ is the Reynolds number corresponding to the specific λ.Sc is the Schmidt number: Sc = µ a /ρ a D diff , where µ a is the dynamic air viscosity, ρ a is the air density and D diff is the aerosol-particle diffusion coefficient.St is the Stokes number and St * is the critical Stokes number: . b The formula is for snow aggregates.D diff is the aerosol-particle diffusion coefficient, Re is the Reynolds number of a snow particle: Re = D m V D ρ a /µ a , where ρ a is the air density and µ a is the dynamic air viscosity.Sc is the Schmidt number: Sc = µ a /ρ a D diff , and I is the size ratio d p /D c , with D c the characteristic length of the snow particle.The third term is the theoretical solution of a simplified flow model by Ranz and Wong (1952), involving parameters S 1 , S 2 and t , and can be simplified to exp   .m and A are the mass and cross-sectional area of a snow particle, respectively.α, β, γ and σ are constants (see discussion in Sect.2.1), a 1 and b 1 are described as functions of X (see Mitchell and Heymsfield, 2005).Introduction

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Full Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | N(D p )dD p is the number of snow particles with a melted diameter between D p to D p + dD p in a unit volume of air (m −3 ), V D and v d are the terminal velocities (m s −1 ) of snow particles and aerosol particles, respectively, E (d p , D p ) is the collection efficiency (dimensionless) between an aerosol particle of size d p and a snow particle of size D p , and A is the effective cross-sectional area of a snow particle projected normal to the fall direction (m 2 ).According to Eq. (2), four parameters determine the value of Λ snow (d p ): Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Figure 1
Figure 1 compares collection efficiencies E (d p , D p ) based on the three formulas listed in Table1across the aerosol particle diameter range from 0.001 µm to 10 µm for collection by monodisperse snow particles with four different shapes and three different maximum sizes D m .Each colour in Fig.1represents one formula listed in Table1, and the different symbols on the lines distinguish the four different snow particle shapes (Table4).A strong dependence of E on aerosol particle size is found for all cases.The ultrafine particles (d p < 0.01 µm) and large particles (d p > 3 µm) have the largest E values while particles with d p around 0.1 µm have the smallest E values.This variation is 14832 3.3).Since different snow particle shapes have different A and m values (Table 4), this leads to different Reynolds number Re and different Best or Davies number values, and thus to different V D values, which caused the small differences in E (d p ).In contrast, the E (d p ) profiles for fixed D m from the Slinn (1984) formula Introduction Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 3.3 Sensitivity of Λ snow to V D and A Discussion Paper | Discussion Paper | Discussion Paper | in theoretical estimates of size-resolved Λ snow profiles Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | panels, which is due in part to the impact of different A formulas on the calculated Λ snow values.It was shown in Sect. 3 that, for a snowfall intensity (as liquid water equivalent) of 0.1 mm h −1 , different E (d p , D p ) formulas can cause uncertainties in Λ snow of one to two orders of magnitude and different N(D p ) formulas can cause uncertainties in Λ snow of one order of magnitude, depending on aerosol particle size.As shown in Fig. 7, the combined uncertainties from both E (d p , D p ) and N(D p ) are larger than those caused by the individual parameters.Thus, the uncertainties in Λ snow values from each individual parameter can either cancel each other (i.e., profiles close together) or enhance each other (i.e., profiles further apart).
Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | et al.
Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | The number of snow particles with a diameter between D m to D m +dD m in a unit volume of air, N(D m )dD m can be expressed N(D m )dD m = N 0e exp −β N 0e = 5.0 × 10 7 (m −4 ), M = 0.37R 0.94 (g m −3 ), and β e = 2072M −0.33 (m −1 ).Appendix B An empirical Λ snow formula from Paramonov et al. (2011) Paramonov et al. (2011) proposed a Λ snow parameterization from the empirical fit of four years of field measurements in an urban environment in Helsinki, Finland: Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | if St ≥ 1/16, or to 0 if St < 1/16(Feng, 2009), where St is the Stokes number.c m is the aerosol particle mass, µ a is the dynamic air viscosity, and Pe is the Peclet number: Pe = D m V D /D diff , where D diff is the aerosol-particle diffusion coefficient.Re is the Reynolds number: Re = D m V D ρ a /2µ a , where ρ a is the air density and µ a is the dynamic air viscosity.Discussion Paper | Discussion Paper | Discussion Paper |

,
N(D p ) = N 0e exp(−β e D p ) Source N 0e [cm −4 ] β e [cm −1 (1970) N 0e = 0.025R −0.94 β e = 22.9R −0Discussion Paper | Discussion Paper | Discussion Paper |Table 3. List of empirical and theoretical snow particle terminal velocity (cm s −1 ) formulas.e= a 1 X b1 , m = αD β m , A = γD σ b v = b 1 (β − σ + 2) − 1Here D p (cm) is the equivalent diameter of a melted snow particle and D m (cm) is the maximum dimension of the frozen snow particle.X is the Best number, X = Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Table A1.Nomenclature.a v , b v empirical constants in V D power-law relationships A snow-particle effective cross-sectional area projected normal to the fall direction (m 2 ) d p aerosol particle diameter (m) D p melted diameter of a snow particle (m) D m maximum dimension of a snow particle (m) D c snow-particle characteristic length used in E expression of Murakami et al. (1985) (m) D diff aerosol-particle diffusivity coefficient (m 2 s −1 ) E (d p , D p ) snow particle-aerosol particle collection efficiency g acceleration of gravity (m s −2 ) M precipitation water concentration (g m −3 ) m particle mass (kg) n(d p , t) aerosol number concentration with diameters d p at time t N 0e intercept parameter for exponential size distribution (m −4 ) N(D p ) snow particle number size distribution (m −4 ) N total total number concentration of snow particles (m −3 terminal velocity (m s −1 ) V D snow-particle terminal velocity (m s −1 ) X Davies number α, β empirical constants in mass-diameter power-law relationships β e slope parameter for exponential size distribution γ, σ empirical constants in Area-diameter power-law relationships λ snow-particle characteristic capture length used in E expression of Slinn (1984) (m) Λ(d p ) size-resolved scavenging coefficient (s −1 ) µ a dynamic air viscosity (kg m −1 s −1 ) ρ a air density (kg m −3 ) ρ water water density (kg m −3 )

Table 1 .
List of semi-empirical formulas for E (d p , D p ).

Table 2 .
Scott (1982)nential snow particle number size distributions.Actual snow particle size was used inScott (1982)(see TableA1) whereas melted snow particle sizes were used in other formulas.R is precipitation intensity (mm h −1 ) and M is precipitation water concentration (g m −3 ).