Influence of common assumptions regarding aerosol composition and mixing state on predicted CCN concentration

A 4-year record of aerosol size and hygroscopic growth factor distributions measured at the Department of Energy’s SGP ARM site in Oklahoma, U.S. were used to estimate supersaturation (S)-dependent cloud condensation nuclei concentrations (NCCN). Baseline or reference NCCN(S) spectra were estimated by using the data to create a matrix of sizeand 10 hygroscopicity-dependent number concentration (N)and then integrating for S > critical supersaturation (Sc) calculated for the same size and hygroscopicity pairs using -Kohler Theory. The accuracy of those estimates was assessed through comparison with the directly measured NCCN at the same site. Subsequently, NCCN was calculated using the same dataset but with an array of simplified treatments in which the aerosol was assumed to be either an internal or an external mixture and the hygroscopicity either assumed or based on averages derived from the growth factor distributions. The CCN spectra calculated using the 15 simplified treatments were compared with those from the baseline approach to evaluate the impact of commonly used approximations. Among the simplified approaches, assuming the aerosol is an internal mixture with size-dependent hygroscopicity parameter () resulted in estimates closest to those from the baseline approach over the range in S considered.


Introduction
Indirect forcing by aerosols involves complex interactions between the aerosol and clouds that affect the droplet number concentration, albedo, precipitation efficiency, and lifetime of clouds and the chemical processing and scavenging of aerosol particles.Mediating this interaction are cloud condensation nuclei (CCN), which are those particles that activate to form droplets at a given supersaturation (S), known as the critical supersaturation (S c ).In part because of the high temporal and spatial variability of aerosol and clouds, climate model predictions of indirect forcing have a high uncertainty.Estimates of CCN concentration (N CCN ) using detailed size-dependent aerosol chemical composition are computationally intensive and simplifications used to minimize computation introduce additional uncertainty.
Methods of determining aerosol activation have evolved to consider complex aerosol composition and structure.Particle S c can be determined for inorganic species using Köhler Theory, provided the physico-chemical properties of the solutes are known [1].However, atmospheric aerosols frequently contain a significant amount of organic material as well [2,3].Compared with soluble inorganic particles, those composed entirely of organic species have higher S c and are often comparatively inefficient in droplet formation [4,5].But the solubility and surface tension-reducing properties of the organic component can sometimes have significant influence on S c [6][7][8][9][10][11][12][13].Extended Köhler Theory can predict CCN concentration of a multi-component aerosol reasonably well from a description of its size distribution and chemical composition [9,10,14 -16].Introduction of a single hygroscopicity parameter by Petters and Kreidenweis [17] has simplified description and comparison of hygroscopicity and CCN activity for particles composed of single or multiple inorganic and organic species [18].
CCN concentration is also dependent upon aerosol mixing state.Previous closure studies have shown that assumption of an internal mixture generally results in an overestimate of the CCN concentration and assumption of an external mixture in an underestimate [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].However, assumption of either mixing state leads to reasonable results for aged aerosols [35].While inclusion of mixing state [36][37][38][39][40] and chemical composition [37,38,[41][42][43][44] can increase the accuracy with which CCN concentration can be estimated, both can be highly variable with time and with particle size, and are often unavailable with current measurement techniques and not easily incorporated into aerosol descriptions used in models.Moreover, chemical composition and mixing state are greatly simplified in large scale models, with simple categorizations such as inorganic/organic and internal/external mixture.Supporting such treatment, several studies have shown that N CCN is most sensitive to the aerosol size distribution [45,46], and the assumption of internal mixing has resulted in fairly accurate predictions [42,[47][48][49][50][51][52].Ervens et al. [35,53] reported that description of the mixing state is relatively more important than that of the size-resolved chemical composition.Other studies suggest detailed information of size distribution, chemical composition, and mixing state is important for achieving closure among aerosol and CCN measurements [38,43,44,[54][55][56][57][58].
Here we use size-resolved aerosol concentration and subsaturated hygroscopicity measurements to estimate CCN concentration using an array of assumptions for composition and mixing state.The data were collected by the U.S. Department of Energy's (DOE) Atmospheric Radiation Measurement (ARM) program at the Southern Great Plains (SGP) site in rural Oklahoma.The study evaluates N CCN estimation methods of varying degrees of complexity with respect to size, chemical composition, and mixing state.Baseline CCN spectra (N CCN vs. S) were first derived by using (i) each combined set of size and hygroscopicity distributions and (ii) κ-Köhler Theory to create pairs of matrices describing (i) N and (ii) S c , respectively, as a function of particle dry diameter, D d , and hygroscopic growth factor, GF.The N CCN (S) for each measurement set was calculated by integrating the N matrix over all elements for which S > S c .Those results were compared with direct measurements of N CCN made at the site to confirm that the calculated spectra were reasonably accurate.The objective was not to perform a rigorous closure analysis with consideration of measurement uncertainties but rather to simply evaluate the consistency of the datasets.Spectra calculated using a number of alternate approaches and assumptions were then compared to those from the baseline approach and the results were used to consider the scatter and bias introduced with simplifications commonly employed in large scale models.

Site Description and Measurements
The data were recorded at the Southern Great Plains (SGP) central facility (CF1) (36 • 36 18.0" N, 97 • 29 6.0" W), located in a mixed land use area of cattle pastures and agricultural fields (mainly wheat, hay and corn) near Lamont, OK, USA.The climate at the site is continental with hot and humid summers and cool winters.The site is impacted by air masses originating from several regions, with accompanying diversity in aerosol concentration and properties.The chemical composition of the aerosol found at the site is complex and highly variable with time (evident in Figures S1 and S2) and with particle size.The size dependence of aerosol composition as reflected in that of GF was described by Mahish and Collins [59].Table 1 lists the routine aerosol measurements at the site that were used for the analysis presented here.All datasets used for this analysis are available for download from the ARM archive.Data from the scanning mobility particle sizer (SMPS)/hygroscopicity tandem differential mobility analyzer (HTDMA) system were used for most of the analyses described here.
That instrument sequentially measures a size distribution and then a set of hygroscopic growth factor distributions at 7 dry particle sizes every ~45 min.Nafion tube bundles are used to dry the sample flow prior to entering the upstream DMA and to humidify it to 90% RH between the two DMAs when measuring hygroscopicity.The sheath and excess flows are recirculated using a variable speed blower.The sample flow rate is controlled by varying the amount of dilution flow that is introduced upstream of the 3.0 L min −1 TSI 3762 condensation particle counter (CPC).Details of the SMPS/HTDMA system and processing of the data it generates are available in the Tandem Differential Mobility Analyzer/Aerodynamic Particle Sizer handbook posted on the ARM archive site [60] and in the work of Gasparini et al. [61].As noted above, the calculated CCN spectra were compared with direct measurements at the site made with a DMT CCN-100 counter (CCNc) [62].The CCNc cycles through 7 fixed supersaturations every half hour.A description of the instrument and data processing is available in the Aerosol Observing System (AOS) handbook [63].Non-refractory chemical composition and black carbon content measured with an Aerosol Chemical Speciation Monitor [64] and a Particle Soot Absorption Photometer [65] were used in the selection of a representative hygroscopicity of the organic component for use in some of the calculations.Details of those instruments are available in the ACSM [66] and AOS [63] handbooks.

Screening and Time Interval Selection
Data from each instrument were validated separately and periods having erroneous data or no data were excluded from analysis.Data from time periods during which instrument problems or failure was evident or when one of the following occurred were not used: a.
The total particle concentration (N CN ) calculated by integrating the SMPS size distribution differed significantly from that directly measured with the CPC b.
The N CCN measured with the CCNc exceeded the N CN measured by the CPC, possibly due to malfunction of the CCNc c.
The sample flow entering the upstream (1st) DMA had an RH > 30%, or d.
The sample flow entering the downstream (2nd) DMA had an RH < 85%.
The categorized data quality during the period of analysis is shown in Figure 1.Periods during which data are available and no significant problems were identified are colored green, those during which confidence in at least some subset of the data is low are yellow, and those during which data are unavailable or thought to be erroneous are red.Data used to generate this graph are available at the site identified in the Acknowledgments section below.

Use of Κ-Köhler Theory
For each of the approaches used to estimate NCCN, the calculation of Sc from measured particle size and measured or assumed hygroscopicity employed κ-Köhler Theory [17].The κ values of aerosol constituents were assumed to be the same in the aqueous particles as measured in the HTDMA at 90% RH and in the more dilute solutions at the point of activation to form cloud droplets.This has the same effect as assuming that the water activity coefficient in the solutions is constant and that all soluble species are fully dissolved at the RH in the HTMDA.A more accurate treatment for which activity coefficients or κ are calculated as a function of RH (and solution concentration) is not justified because size-resolved composition measurements are not available and the bulk submicron measurements made with the ACSM show the aerosol composition is complex and varies considerably during the year.Specifically, there is a strong seasonality in the soluble inorganic content, with sulfate dominant from roughly April through October and nitrate dominant from November through March, as is evident in the sulfate:nitrate ratio shown in Figure 2. Thus, the choice of soluble inorganic component(s) needed to model the extent and effect of variation in κ between 90% RH and at Sc would vary by month over the 4-year period of this analysis, as well as over shorter periods of days or even hours accompanying changes in the origin and processing of the sampled aerosol.Furthermore, any attempt to estimate RH-dependent changes in κ would require consideration of the influence of the significant organic content at the site.As shown in Table 2, averaged throughout the year organics contribute over 50% to the total submicron mass concentration.Periods during which data are available and no significant problems were identified are colored green, those during which confidence in at least some subset of the data is low are yellow, and those during which data are unavailable or thought to be erroneous are red.Data used to generate this graph are available at the site identified in the Acknowledgments section below.

Use of K-Köhler Theory
For each of the approaches used to estimate N CCN , the calculation of S c from measured particle size and measured or assumed hygroscopicity employed κ-Köhler Theory [17].The κ values of aerosol constituents were assumed to be the same in the aqueous particles as measured in the HTDMA at 90% RH and in the more dilute solutions at the point of activation to form cloud droplets.This has the same effect as assuming that the water activity coefficient in the solutions is constant and that all soluble species are fully dissolved at the RH in the HTMDA.A more accurate treatment for which activity coefficients or κ are calculated as a function of RH (and solution concentration) is not justified because size-resolved composition measurements are not available and the bulk submicron measurements made with the ACSM show the aerosol composition is complex and varies considerably during the year.Specifically, there is a strong seasonality in the soluble inorganic content, with sulfate dominant from roughly April through October and nitrate dominant from November through March, as is evident in the sulfate:nitrate ratio shown in Figure 2. Thus, the choice of soluble inorganic component(s) needed to model the extent and effect of variation in κ between 90% RH and at S c would vary by month over the 4-year period of this analysis, as well as over shorter periods of days or even hours accompanying changes in the origin and processing of the sampled aerosol.Furthermore, any attempt to estimate RH-dependent changes in κ would require consideration of the influence of the significant organic content at the site.As shown in Table 2, averaged throughout the year organics contribute over 50% to the total submicron mass concentration.For all CCN concentration estimate approaches for which the HTDMA data were used, GF was related to κ using the following expression from Petters and Kreidenweis [17].
where M w and ρ w are the molecular weight and liquid density of water, respectively.The solution surface tension, σ sol , was assumed to be that of pure water, 0.072 J m −2 .Because it is assumed that κ values were the same for the more concentrated and the more dilute aqueous solutions in the HTDMA and at activation, respectively, it is not necessary to know the contributions of different aerosol components to the overall hygroscopicity for the internal mixture calculations.But for assumed external mixtures the hygroscopicity of two or more particle types must be determined or assumed.For this analysis, particle types assumed to be present in external mixtures were (i) particles composed of soluble inorganics; (ii) particles composed of soluble organics; and (iii) particles composed of insoluble (and non-hygroscopic) components.The hygroscopicity parameter of the soluble inorganic particles, κ inorg , was assumed to be 0.6, which is similar to that of ammonium sulfate and ammonium nitrate.As aerosol organic components at the SGP site are not well characterized, direct derivation of κ org is not possible.Here, κ org was estimated using the mixing rule [17].
size and measured or assumed hygroscopicity employed κ-Köhler Theory [17].The κ values of aerosol constituents were assumed to be the same in the aqueous particles as measured in the HTDMA at 90% RH and in the more dilute solutions at the point of activation to form cloud droplets.This has the same effect as assuming that the water activity coefficient in the solutions is constant and that all soluble species are fully dissolved at the RH in the HTMDA.A more accurate treatment for which activity coefficients or κ are calculated as a function of RH (and solution concentration) is not justified because size-resolved composition measurements are not available and the bulk submicron measurements made with the ACSM show the aerosol composition is complex and varies considerably during the year.Specifically, there is a strong seasonality in the soluble inorganic content, with sulfate dominant from roughly April through October and nitrate dominant from November through March, as is evident in the sulfate:nitrate ratio shown in Figure 2. Thus, the choice of soluble inorganic component(s) needed to model the extent and effect of variation in κ between 90% RH and at Sc would vary by month over the 4-year period of this analysis, as well as over shorter periods of days or even hours accompanying changes in the origin and processing of the sampled aerosol.Furthermore, any attempt to estimate RH-dependent changes in κ would require consideration of the influence of the significant organic content at the site.As shown in Table 2, averaged throughout the year organics contribute over 50% to the total submicron mass concentration.For all CCN concentration estimate approaches for which the HTDMA data were used, GF was related to κ using the following expression from Petters and Kreidenweis [17].
where Mw and• ρ are the molecular weight and liquid density of water, respectively.The solution surface tension, σsol, was assumed to be that of pure water, 0.072 J m −2 .Because it is assumed that κ values were the same for the more concentrated and the more dilute aqueous solutions in the HTDMA and at activation, respectively, it is not necessary to know the contributions of different aerosol components to the overall hygroscopicity for the internal mixture calculations.But for assumed external mixtures the hygroscopicity of two or more particle types must be determined or assumed.For this analysis, particle types assumed to be present in external mixtures were (i) particles composed of soluble inorganics, (ii) particles composed of soluble organics, and (iii) particles composed of insoluble (and non-hygroscopic) components.The hygroscopicity parameter of the soluble inorganic particles, κinorg, was assumed to be 0.6, which is similar to that of ammonium sulfate and ammonium nitrate.As aerosol organic components at the SGP site are not well characterized, direct derivation of κorg is not possible.Here, κorg was estimated using the mixing rule [17].
For these calculations the total particle volume concentration (Voverall) was determined from the For these calculations the total particle volume concentration (V overall ) was determined from the measured size distribution.The submicron average hygroscopicity parameter (κ overall ) was calculated from the measured size and hygroscopicity distributions as the volume concentration-weighted average κ.The mass concentration of non-refractory organics (m orgNR ) was measured by the ACSM and that of black carbon, BC, was estimated as the ratio of measured submicron particle light absorbance (b absBC ) and an assumed absorption efficiency at 0.55 µm wavelength of 7.5 m 2 g −1 [67].The density, ρ, of non-refractory organics and BC were both assumed to be 1.3 g cm −3 [68].
The resulting seasonal profiles of κ overall and κ org in 2011 are shown in Figure 3. Unlike the κ org profile, κ overall was highest in the winter and lowest in the summer.The high wintertime κ overall is a result of high concentrations of inorganic compounds, especially nitrate, while the relatively low summertime κ overall is caused by higher organic mass concentrations, as shown in Figure 4. Positive Matrix Factorization (PMF) analysis used for the Organic Aerosol Component (OACOMP) ARM Value Added Product (VAP) [69] indicates that less-hygroscopic biomass burning organic aerosol (BBOA) was prevalent from February through April in 2011, thus lowering κ org in winter and spring.Aged SOA (MO-OOA), which is moderately hygroscopic, was more abundant in summer, and thus raised κ org .The κ org in the spring and fall lies between that of the winter and summer.The 2011 average of 0.1 was used for this study, while the sensitivity of the results to the selected value was assessed by repeating the calculations for the seasonal minimum (κ org = 0.06, winter) and maximum (κ org = 0.16, summer).
Atmosphere 2018, 9, x FOR PEER REVIEW 6 of 18 Value Added Product (VAP) [69] indicates that less-hygroscopic biomass burning organic aerosol (BBOA) was prevalent from February through April in 2011, thus lowering κorg in winter and spring.Aged SOA (MO-OOA), which is moderately hygroscopic, was more abundant in summer, and thus raised κorg.The κorg in the spring and fall lies between that of the winter and summer.The 2011 average of 0.1 was used for this study, while the sensitivity of the results to the selected value was assessed by repeating the calculations for the seasonal minimum (κorg = 0.06, winter) and maximum (κorg = 0.16, summer).

Description of Models Used for Estimating NCCN
To simplify both the comparison of the varied approaches used to estimate NCCN and the comparison of the results, a common framework will be used to describe all of the approaches even though more straightforward descriptions would suffice for many of them.For all approaches a CCN spectrum, NCCN(S), was calculated for each size distribution measured by the SMPS.For the simplest approaches, the HTDMA data were not considered and a fixed hygroscopicity parameter was assumed.For all others the GF distributions were interpolated and extrapolated to each of the 90 size bins in the size distribution measurements and then converted to κ distributions using Equation (1).The differences among the approaches arise from the use and (any) averaging of the κ distributions, as summarized in Figure 5.  Value Added Product (VAP) [69] indicates that less-hygroscopic biomass burning organic aerosol (BBOA) was prevalent from February through April in 2011, thus lowering κorg in winter and spring.Aged SOA (MO-OOA), which is moderately hygroscopic, was more abundant in summer, and thus raised κorg.The κorg in the spring and fall lies between that of the winter and summer.The 2011 average of 0.1 was used for this study, while the sensitivity of the results to the selected value was assessed by repeating the calculations for the seasonal minimum (κorg = 0.06, winter) and maximum (κorg = 0.16, summer).

Description of Models Used for Estimating NCCN
To simplify both the comparison of the varied approaches used to estimate NCCN and the comparison of the results, a common framework will be used to describe all of the approaches even though more straightforward descriptions would suffice for many of them.For all approaches a CCN spectrum, NCCN(S), was calculated for each size distribution measured by the SMPS.For the simplest approaches, the HTDMA data were not considered and a fixed hygroscopicity parameter was assumed.For all others the GF distributions were interpolated and extrapolated to each of the 90 size bins in the size distribution measurements and then converted to κ distributions using Equation (1).The differences among the approaches arise from the use and (any) averaging of the κ distributions, as summarized in Figure 5.

Description of Models Used for Estimating N CCN
To simplify both the comparison of the varied approaches used to estimate N CCN and the comparison of the results, a common framework will be used to describe all of the approaches even though more straightforward descriptions would suffice for many of them.For all approaches a CCN spectrum, N CCN (S), was calculated for each size distribution measured by the SMPS.For the simplest approaches, the HTDMA data were not considered and a fixed hygroscopicity parameter was assumed.For all others the GF distributions were interpolated and extrapolated to each of the 90 size bins in the size distribution measurements and then converted to κ distributions using Equation (1).The differences among the approaches arise from the use and (any) averaging of the κ distributions, as summarized in Figure 5.
Atmosphere 2018, 9, x FOR PEER REVIEW 6 of 18 Value Added Product (VAP) [69] indicates that less-hygroscopic biomass burning organic aerosol (BBOA) was prevalent from February through April in 2011, thus lowering κorg in winter and spring.
Aged SOA (MO-OOA), which is moderately hygroscopic, was more abundant in summer, and thus raised κorg.The κorg in the spring and fall lies between that of the winter and summer.The 2011 average of 0.1 was used for this study, while the sensitivity of the results to the selected value was assessed by repeating the calculations for the seasonal minimum (κorg = 0.06, winter) and maximum (κorg = 0.16, summer).

Description of Models Used for Estimating NCCN
To simplify both the comparison of the varied approaches used to estimate NCCN and the comparison of the results, a common framework will be used to describe all of the approaches even though more straightforward descriptions would suffice for many of them.For all approaches a CCN spectrum, NCCN(S), was calculated for each size distribution measured by the SMPS.For the simplest approaches, the HTDMA data were not considered and a fixed hygroscopicity parameter was assumed.For all others the GF distributions were interpolated and extrapolated to each of the 90 size bins in the size distribution measurements and then converted to κ distributions using Equation (1).The differences among the approaches arise from the use and (any) averaging of the κ distributions, as summarized in Figure 5.As a baseline for comparison with estimates from other approaches and with direct measurements, N CCN was calculated using the full GF distributions without any averaging.Because all of the Atmosphere 2018, 9, 54 7 of 18 information in the GF distributions is retained, these estimates are expected to be more accurate than those from any of the other approaches, all of which rely on averaged or assumed hygroscopicity.For each measurement sequence the interpolated κ distributions were combined with the size distribution to create a matrix of number concentrations as a function of D d and κ, such as that shown graphically in Figure 6a.A Köhler curve relating equilibrium S to droplet diameter, D, was calculated for each (D d , κ) pair using Equation (3) below.The S c for each pair was calculated as the maximum value of equilibrium S along the curve.
As with calculations using Equation (1), σ sol was assumed to be that of water, 0.072 J m −2 .The result can be viewed as a matrix with elements of S c and the same D d and κ arrays as used in the number concentration matrix described above.N CCN was estimated for a prescribed S by integrating the number concentration N(D d , κ) for which S > S c .This is presented graphically in Figure 6b, with N CCN calculated by summing the N elements (whole or part) above and to the right of one of the four curves, each of which connects the elements having the same S c .The resulting N CCN estimates were first compared with direct measurements made with the CCNc and then with the results of the other estimate approaches outlined below.
For all other N CCN estimates the aerosol was assumed to be either an internal mixture or an external mixture, as is generally required for regional and global scale climate models.The goals here were to assess the error introduced when making these simplifying assumptions and to identify the approach(es) most suitable for an aerosol similar to that found at SGP.
To treat the aerosol as an internal mixture the D d -dependent κ distributions described above were replaced with a single κ value that is either dependent on D d (κ = κ(D d )) or the same for all D d (κ = κ(D d )).The former comes simply from the number concentration-weighted average of the κ distributions at each D d , with the result depicted in Figure 6c in the same manner as for the baseline approach matrix.For approaches for which κ is assumed to be size independent, it was calculated either as the average of κ(D d ) (Figure 6d) or, neglecting the hygroscopicity measurements, as that of particles composed of 20%, 50%, or pure soluble inorganics (~ammonium sulfate, AS) by volume (κ = 0.12, 0.30, and 0.60, respectively, with only κ = 0.30 shown in Figure 6g for clarity).
External mixtures were assumed to be comprised of two of the three particle types considered: insoluble, κ = 0.0, organic, κ org = 0.1, and inorganic, κ inorg = 0.6.As with the assumed internal mixture approaches, both size dependent and size independent scenarios were considered.For both, independent size distributions of the different particle types were calculated from the average κ(= f(D d ) or = f(D d )) using Equations ( 4) and ( 5 The results for κ(D d ) and κ( = D d ) are depicted graphically in Figure 6e,f, respectively, where the lower and higher horizontal lines represent the organic and inorganic particle types, respectively.As with the other approaches, N CCN was calculated by summing N elements above and to the right of the constant S c curves.Contributions from the inorganic and organic particle types can also be calculated separately and then added to determine the total N CCN (i.e., N CCN = N CCN.inorg + N CCN.org ).
Atmosphere 2018, 9, x FOR PEER REVIEW 8 of 18 The results for κ(Dd) and κ(≠ Dd) are depicted graphically in Figure 6e,f, respectively, where the lower and higher horizontal lines represent the organic and inorganic particle types, respectively.As with the other approaches, NCCN was calculated by summing N elements above and to the right of the constant Sc curves.Contributions from the inorganic and organic particle types can also be calculated separately and then added to determine the total NCCN (i.e., NCCN = NCCN.inorg+ NCCN.org).

Comparison between Measured and Baseline N CCN Estimate
The concentration measured by the CCNc was compared to that calculated using the baseline approach for all available data from 2009-2012.The results for May 2011 are shown in Figure 7 and for all of 2011 in Figure S3.

Comparison between Measured and Baseline NCCN Estimate
The concentration measured by the CCNc was compared to that calculated using the baseline approach for all available data from 2009-2012.The results for May 2011 are shown in Figure 7 and for all of 2011 in Figure S3.

Comparison between Measured and Baseline NCCN Estimate
The concentration measured by the CCNc was compared to that calculated using the baseline approach for all available data from 2009-2012.The results for May 2011 are shown in Figure 7 and for all of 2011 in Figure S3.x i − y i x i 2 0.5 (6) where x i is the measured N CCN at a given S, y i the estimated N CCN at the same S, and n the number of concentration pairs compared.The average NRMSE between the measured and calculated concentrations for May 2011 and for each of the four years analyzed are summarized in Table 3.
The measured N CCN at 0.25% S during 2012 was very noisy and was excluded from the analysis.In addition to measurement error, some possible reasons for deviations between the measured and calculated concentrations are (i) differences in assumed actual properties of aerosol chemical species; (ii) interactions among components not captured by κ-Köhler Theory; (iii) the presence of low solubility organics that dissolve under the dilute conditions with S ~Sc but not in the more concentrated solution in the HTDMA at 90% RH; (iv) the presence of particles that contain slowly dissolving compounds or that are in an amorphous/glassy state for which hygroscopic growth and activation timescales may be comparable to or greater than the HTDMA and CCNc residence times; and (v) the presence of surface tension-reducing species, which influence S c much more than GF.But as noted above, the primary goal here was not to assess closure but to confirm that the baseline spectra calculated from the size and GF distributions were reasonably accurate and were suitable to serve as the reference for comparison with the results from the simplified treatments of composition and mixing state.

Comparison of N CCN Calculated from Different Approaches
Estimates of N CCN assuming the aerosol is an internal or external mixture relative to those of the baseline approach for which no assumption about mixing state is made are presented in Figures 8 and 9 for 0.25% and 0.85%, respectively, and in Figures S4 and S5 for 0.45% and 0.65% S, respectively.Deviations from the 1:1 lines are interpreted as error introduced by the averaging or approximations required for all but the baseline calculations.Best fits through the data were assumed to be linear and were forced through the origin to facilitate interpretation of the results and simply because of the apparent linear correlations with minimal offset in the figures.Table 4 summarizes the slope (m = N CCN from alternate approach N CCN from baseline approach ) and goodness of fit (r 2 ) for each of the approaches with respect to the baseline estimate.Values of m above (below) 1 indicate that the simplified treatment results in N CCN greater (less) than that from the baseline estimate for a given S. Treating the aerosol as an internal mixture with size dependent hygroscopicity resulted in the best agreement with the baseline estimates, with the average m closest to 1.0 and the highest r 2 (Table 4).Though there is considerably more scatter in the results for the internal and external mixtures with κ( = D d ) than in those with κ(D d ), the deviations are found to be tightly correlated.The relationships between the results for the assumed internal and external mixtures with κ( = D d ) are shown in Figure 10e,f.The strong correlation in N CCN (S) obtained for those assumed internal and external mixtures could be interpreted as reflecting a general lack of sensitivity to the mixing state as has been reported in some closure studies.But, at least for these data, both estimates simply differ in the same direction from what is believed to be the most accurate estimate.For both mixing state assumptions, the estimated N CCN with κ( = D d ) is higher than that with κ(D d ), as the κ averaged over the complete size range is typically higher than the κ(D) over the ~0.05-0.20 µm size range that dominates N CCN .
estimate.For both mixing state assumptions, the estimated NCCN with κ(≠ Dd) is higher than that with κ(Dd), as the κ averaged over the complete size range is typically higher than the κ(D) over the ~0.05-0.20 µm size range that dominates NCCN.estimate.For both mixing state assumptions, the estimated NCCN with κ(≠ Dd) is higher than that with κ(Dd), as the κ averaged over the complete size range is typically higher than the κ(D) over the ~0.05-0.20 µm size range that dominates NCCN.A correction of 1/m could be applied to results using any of the simplified treatments to increase accuracy.Of course, that sort of annual average correction would not reduce error that arises from variability in aerosol characteristics on seasonal or shorter time scales and is reflected in the r 2 values.For example, assuming the aerosol has a fixed κ of 0.12 (20% AS) results in average best fit slopes between 1.00 and 1.06, indicating that, on average, an error of only a few percent would be introduced if both the composition and hygroscopicity were not considered at all in the calculations.However, that represents the average of comparably large negative and positive errors for individual measurements or days, as is shown in Figure 11 that compares the 2011 daily averages of the results from the internal mixture with κ(D d ) and the fixed 20% AS treatments relative to the baseline calculations.And as is shown in Figure 10, similar deviations from the 1.0 line are evident in the results for which the hygroscopicity was considered, but not its size dependence.Furthermore, the utility of any of the simplifying approaches is limited if not applicable over a wide range in S. The S dependence of the best fit slopes quantified as the fractional change over the full range in S considered is included in the 6th column in Table 4.That S dependence is much lower for the two cases for which the size dependence of κ is considered than for any of the others, which is expected because the change in N CCN with S largely reflects inclusion or exclusion of particles within a rather narrow slice of the full size range.A correction of 1/m could be applied to results using any of the simplified treatments to increase accuracy.Of course, that sort of annual average correction would not reduce error that arises from variability in aerosol characteristics on seasonal or shorter time scales and is reflected in the r 2 values.For example, assuming the aerosol has a fixed κ of 0.12 (20% AS) results in average best fit slopes between 1.00 and 1.06, indicating that, on average, an error of only a few percent would be introduced if both the composition and hygroscopicity were not considered at all in the calculations.However, that represents the average of comparably large negative and positive errors for individual measurements or days, as is shown in Figure 11 that compares the 2011 daily averages of the results from the internal mixture with κ(Dd) and the fixed 20% AS treatments relative to the baseline calculations.And as is shown in Figure 10, similar deviations from the 1.0 line are evident in the results for which the hygroscopicity was considered, but not its size dependence.Furthermore, the utility of any of the simplifying approaches is limited if not applicable over a wide range in S. The S dependence of the best fit slopes quantified as the fractional change over the full range in S considered is included in the 6th column in Table 4.That S dependence is much lower for the two cases for which the size dependence of κ is considered than for any of the others, which is expected because the change in NCCN with S largely reflects inclusion or exclusion of particles within a rather narrow slice of the full size range.The results for the assumed external mixtures were found to be quite sensitive to the κorg used.Specifically, substituting the seasonal minimum (0.06) for the annual average (0.10) of the estimated κorg results in an average decrease in NCCN of 8% when the size dependence of κ is considered and an increase of 10% when it is not.Similarly, use of the seasonal maximum of 0.16 results in an average decrease of 1% with κ(Dd) and increase of 15% with κ(≠ Dd).(The results for SGP are likely especially sensitive to the assumed value because it is similar to that average κ of 0.12 that was found to result in reasonable estimates averaged throughout the year).For any measurement a change in κorg will cause a change in partitioning between the three particle types, with the net influence on NCCN depending on the coupling of that change and the shape of the size distribution.Because the approach The results for the assumed external mixtures were found to be quite sensitive to the κ org used.Specifically, substituting the seasonal minimum (0.06) for the annual average (0.10) of the estimated κ org results in an average decrease in N CCN of 8% when the size dependence of κ is considered and an increase of 10% when it is not.Similarly, use of the seasonal maximum of 0.16 results in an average decrease of 1% with κ(D d ) and increase of 15% with κ( = D d ).(The results for SGP are likely especially sensitive to the assumed value because it is similar to that average κ of 0.12 that was found to result in reasonable estimates averaged throughout the year).For any measurement a change in κ org will cause a change in partitioning between the three particle types, with the net influence on N CCN depending on the coupling of that change and the shape of the size distribution.Because the approach used allows for at most two particle types, an increase in κ org that crosses the average κ of the aerosol will cause a shift in the assumed mixture from consisting of organic and inorganic (κ = 0.6) particles to organic and insoluble (κ = 0.0) particles, with a resulting increase in N CCN due to the increased hygroscopicity of the organic particles, a decrease due to the loss of the inorganic particles, and a net change that could be positive or negative.To evaluate the sensitivity of the calculated N CCN to the choice in κ inorg , the analysis was repeated using values of 0.64 and 0.67 (instead of 0.6).The average change in N CCN was about 0.6% when size-dependence is considered and about 1.0% when it is not.The results for the assumed external mixtures were found to be quite sensitive to the κorg used.Specifically, substituting the seasonal minimum (0.06) for the annual average (0.10) of the estimated κorg results in an average decrease in NCCN of 8% when the size dependence of κ is considered and an increase of 10% when it is not.Similarly, use of the seasonal maximum of 0.16 results in an average decrease of 1% with κ(Dd) and increase of 15% with κ(≠ Dd).(The results for SGP are likely especially sensitive to the assumed value because it is similar to that average κ of 0.12 that was found to result in reasonable estimates averaged throughout the year).For any measurement a change in κorg will cause a change in partitioning between the three particle types, with the net influence on NCCN depending on the coupling of that change and the shape of the size distribution.Because the approach used allows for at most two particle types, an increase in κorg that crosses the average κ of the aerosol will cause a shift in the assumed mixture from consisting of organic and inorganic (κ = 0.6) particles to organic and insoluble (κ = 0.0) particles, with a resulting increase in NCCN due to the increased hygroscopicity of the organic particles, a decrease due to the loss of the inorganic particles, and a net change that could be positive or negative.To evaluate the sensitivity of the calculated NCCN to the choice in κinorg, the analysis was repeated using values of 0.64 and 0.67 (instead of 0.6).The average change in NCCN was about 0.6% when size-dependence is considered and about 1.0% when it is not.

Summary and Conclusions
Size distributions and hygroscopic growth factor distributions measured from 2009 to 2012 at the SGP ARM site were used to estimate CCN concentrations over a range in supersaturation.An initial estimate of NCCN that served as a basis for comparison used all of the information in the combined distributions without any averaging.For those estimates, matrices of N(Dd, κ) and Sc(Dd, κ) were calculated from the measured distributions and from κ-Ko hler Theory, respectively, and NCCN(S) was calculated by integrating all of the N elements for which the corresponding Sc element < S. Comparisons of those estimates with direct measurements from a collocated CCN counter show that this baseline approach can reasonably predict NCCN over a range of S.
The baseline spectra were then compared with those calculated using the same dataset but with aerosol treatments that are more commonly used for more efficient computation or simply because size-dependent composition or hygroscopicity distributions are not available.These included approximating the aerosol as an internal mixture with fixed inorganic volume fraction, as an internal mixture with size-dependent or size-independent hygroscopicity, and as an external mixture with size-dependent or size-independent hygroscopicity.Bias and variance relative to the baseline estimates were described with best fit slopes, m, and coefficients of determination, r 2 , respectively, with both calculated from the thousands of NCCN pairs over the period of analysis.
Though the results presented here are most directly applicable for the aerosol at SGP, that aerosol is quite diverse, with a range in sources, origins, and atmospheric processing prior to sampling.Thus, at least the overall ~10% uncertainty range accompanying assumption of either internal or external mixtures reported here is likely relevant for NCCN prediction in other regions as well and can provide some guidance for interpretation of model results.And for prediction of NCCN for an aerosol known to be similar to that at SGP the bias and scatter associated with each of the simplified treatments can be weighed together with computational cost for selection of the most

Summary and Conclusions
Size distributions and hygroscopic growth factor distributions measured from 2009 to 2012 at the SGP ARM site were used to estimate CCN concentrations over a range in supersaturation.An initial estimate of N CCN that served as a basis for comparison used all of the information in the combined distributions without any averaging.For those estimates, matrices of N(D d , κ) and S c (D d , κ) were calculated from the measured distributions and from κ-Köhler Theory, respectively, and N CCN (S) was calculated by integrating all of the N elements for which the corresponding S c element < S. Comparisons of those estimates with direct measurements from a collocated CCN counter show that this baseline approach can reasonably predict N CCN over a range of S.
The baseline spectra were then compared with those calculated using the same dataset but with aerosol treatments that are more commonly used for more efficient computation or simply

Figure 1 .
Figure 1.Data quality during the analysis period.Periods during which data are available and no significant problems were identified are colored green, those during which confidence in at least some subset of the data is low are yellow, and those during which data are unavailable or thought to be erroneous are red.Data used to generate this graph are available at the site identified in the Acknowledgments section below.

Figure 1 .
Figure 1.Data quality during the analysis period.Periods during which data are available and no significant problems were identified are colored green, those during which confidence in at least some subset of the data is low are yellow, and those during which data are unavailable or thought to be erroneous are red.Data used to generate this graph are available at the site identified in the Acknowledgments section below.

Figure 2 .
Figure 2. Sulfate to nitrate equivalent concentration ratio based on ACSM chemical composition data from (a) 2011, (b) 2012, and (c) 2013.Equivalent concentration = , where the equivalence factor for sulfate and nitrate are 0.5 and 1, respectively.

Figure 2 .
Figure 2. Sulfate to nitrate equivalent concentration ratio based on ACSM chemical composition data from (a) 2011, (b) 2012, and (c) 2013.Equivalent concentration = molar concentration equivalence factor , where the equivalence factor for sulfate and nitrate are 0.5 and 1, respectively.

Figure 4 .
Figure 4. Seasonal average of (a) organic and inorganic mass concentration and (b) the inorganic mass fraction (%) in 2011.

Figure 5 .
Figure 5. Overview of the NCCN calculation approaches.

Figure 4 .
Figure 4. Seasonal average of (a) organic and inorganic mass concentration and (b) the inorganic mass fraction (%) in 2011.

Figure 5 .
Figure 5. Overview of the NCCN calculation approaches.

Figure 4 .
Figure 4. Seasonal average of (a) organic and inorganic mass concentration and (b) the inorganic mass fraction (%) in 2011.

Figure 4 .
Figure 4. Seasonal average of (a) organic and inorganic mass concentration and (b) the inorganic mass fraction (%) in 2011.

Figure 5 .
Figure 5. Overview of the NCCN calculation approaches.Figure 5. Overview of the N CCN calculation approaches.

Figure 5 .
Figure 5. Overview of the NCCN calculation approaches.Figure 5. Overview of the N CCN calculation approaches.

Figure 6 .
Figure 6.A graphical illustration of the approaches used to estimate NCCN.(a) Number concentration matrix calculated from a set of measured size and GF distributions.The size distribution is partitioned among the κ bins according to the GF distributions interpolated to each Dp without any averaging or assumptions about mixing state or composition.The same matrix is shown in (b), with lines of constant Sc added.As with all of the estimation techniques, NCCN(S) is estimated by summing the concentration elements (whole or part) above and to the right of the constant Sc (= S) lines.The graphs in (c-g) are from the same set of size and GF distributions as those used for (a,b), but with various assumptions for mixing state and composition that collapse the κ distribution at each bin in the size distribution to one or two values.(c) internal mixture with κ(Dd), (d) internal mixture with κ(≠ Dd), (e) external mixture with κ(Dd), (f) external mixture with κ(≠ Dd), and (g) internal mixture with an assumed inorganic volume fraction of 0.5.

Figure 6 .
Figure 6.A graphical illustration of the approaches used to estimate N CCN .(a) Number concentration matrix calculated from a set of measured size and GF distributions.The size distribution is partitioned among the κ bins according to the GF distributions interpolated to each D p without any averaging or assumptions about mixing state or composition.The same matrix is shown in (b); with lines of constant S c added.As with all of the estimation techniques, N CCN (S) is estimated by summing the concentration elements (whole or part) above and to the right of the constant S c (= S) lines; The graphs in (c-g) are from the same set of size and GF distributions as those used for (a,b); but with various assumptions for mixing state and composition that collapse the κ distribution at each bin in the size distribution to one or two values; (c) internal mixture with κ(D d ); (d) internal mixture with κ( = D d ); (e) external mixture with κ(D d ); (f) external mixture with κ( = D d ); and (g) internal mixture with an assumed inorganic volume fraction of 0.5.

Figure 8 .
Figure 8. NCCN estimated from simplified approaches vs. that from the baseline approach at 0.25% S for all 2011 data.

Figure 9 .
Figure9.NCCN estimated from simplified approaches vs. that from the baseline approach at 0.85% S for all 2011 data.

Figure 8 .
Figure 8. N CCN estimated from simplified approaches vs. that from the baseline approach at 0.25% S for all 2011 data.

Figure 8 .
Figure 8. NCCN estimated from simplified approaches vs. that from the baseline approach at 0.25% S for all 2011 data.

Figure 9 .
Figure9.NCCN estimated from simplified approaches vs. that from the baseline approach at 0.85% S for all 2011 data.

Figure 9 .
Figure9.N CCN estimated from simplified approaches vs. that from the baseline approach at 0.85% S for all 2011 data.

Table 1 .
List of instruments, measured quantities, manufacturer, and year installed.

Table 2 .
Mass concentration fraction from ACSM measurements at SGP.

Table 3 .
NRMSE between measured and estimated N CCN at 4 different S from 2009-2012 and for May 2011.

Table 4 .
Fit parameters of N CCN estimate approaches for 2011 data

Table 4 .
Fit parameters of NCCN estimate approaches for 2011 data