Explaining Variance in Black Carbon's Aging Timescale

The size and composition of particles containing black carbon (BC) are modified soon after emission by condensation of semivolatile substances and coagula-tion with other particles, known collectively as " aging " processes. Although this change in particle properties is widely recognized, the timescale for transformation is not well constrained. In this work, we simulated aerosol aging with the particle-resolved model PartMC-MOSAIC (Particle Monte Carlo – Model for Simulating Aerosol Interactions and Chemistry) and extracted aging timescales based on changes in particle cloud condensation nuclei (CCN). We simulated nearly 300 scenarios and, through a regression analysis, identified the key parameters driving the value of the aging timescale. We show that BC's aging timescale spans from hours to weeks, depending on the local environmental conditions and the characteristics of the fresh BC-containing particles. Although the simulations presented in this study included many processes and particle interactions, we show that 80 % of the variance in the aging timescale is explained by only a few key parameters. The condensation aging timescale decreased with the flux of condensing aerosol and was shortest for the largest fresh particles, while the coagulation aging timescale decreased with the total number concentration of large (D > 100 nm), CCN-active particles and was shortest for the smallest fresh particles. Therefore, both condensation and coagulation play important roles in aging, and their relative impact depends on the particle size range.


Introduction
Particles containing black carbon (BC) alter the Earth's energy balance by scattering and absorbing solar radiation (Mc-Cormick and Ludwig, 1967;Rosen et al., 1978;Schulz et al., 2006), by interacting with clouds (Twomey, 1977;Twomey et al., 1984;Lohmann et al., 2005;Albrecht, 1989;Ackerman et al., 2000), and by decreasing the albedo of ice and snow (Hansen and Nazarenko, 2004;Jacobson, 2004).Each 35 of these climate effects depends on the properties of individual BC-containing particles and their atmospheric residence time.The dominant removal mechanism of BC mass from the atmosphere is wet deposition (Cozic et al., 2007), with one important pathway being the activation of BC-containing 40 particles into cloud condensation nuclei (CCN) and their subsequent removal if the cloud precipitates.Although freshly emitted BC-containing particles are too small and hydrophobic to activate (Maricq, 2007;Weingartner et al., 1997), their morphology and chemical composition are altered soon after 45 emission by condensation of semi-volatile gases and coagulation with pre-existing particles (Johnson et al., 2005;Oshima et al., 2009;Zaveri et al., 2010).
These changes in particle characteristics, termed "aging", often increase the particles' susceptibility to cloud droplet 50 nucleation and wet removal (Furutani et al., 2008;Cantrell et al., 2001;Zuberi et al., 2005), so these processes must be included in global models.However, a complex aerosol population that evolves with time is not easily simulated in climate models, so even sophisticated aerosol schemes do not 55 fully resolve aerosol properties on a per-particle level (Jacobson, 1997;Wexler et al., 1994;Bauer et al., 2008;Binkowski and Roselle, 2003;McGraw, 1997;Jacobson, 2002;Aquila et al., 2011;Matsui et al., 2013).The simplest representation of aging classifies BC mass as either hydrophobic or 60 hydrophilic, such that hydrophilic BC is susceptible to removal by wet deposition and hydrophobic BC is not.In this framework, BC is transferred from the hydrophobic (fresh) category to the hydrophilic (aged) category according to a first-order aging timescale (Cooke and Wilson, 1996;Croft 65 et al., 2005;Koch, 2001).Global models apply a fixed aging timescale of 1-3 days.Global modeling studies have shown that estimates of BC's climate forcing are sensitive to the assumed aging timescale (Koch et al., 2009), but its value is not well constrained.While some climate models have been 2 L. Fierce et al.: Explaining variance in BC aging timescales moving toward aerosol modules that represent aerosol aging using several interacting modes (Aquila et al., 2011;Bauer et al., 2008;Wilson et al., 2001;Matsui et al., 2013), the practice of using a fixed aging timescale is still widespread (Jo et al., 2013;Chin et al., 2014;Schmidt et al., 2014).
To improve upon using one constant value for the aging timescale, several studies have developed parameterizations of BC's aging timescale that vary with environmental conditions.Liu et al. (2011) developed a parameterization of black carbon aging by condensation that depended on the conden-80 sation rate of sulfuric acid and overall BC surface area.They showed that, by allowing for slower aging in the winter, their parameterization was better able to represent seasonal variability in black carbon transport to the Arctic.Oshima and Koike (2013) extended this approach and developed a parameterization of aging timescales based on simulations with a box model.Their parameterization predicted the rate for BC to transition from a hydrophobic class to a hydrophilic class, expressed as a function of the mass-normalized coating rate and on parameters of the fresh BC size distribution.Riemer et al. (2004) showed that timescales for aging by coagulation decrease with the overall aerosol number concentration, which they parameterized using a simple power law, and this parameterization was applied by Croft et al. (2005).Pierce et al. (2009) parameterized size-resolved coagulation 95 rates as a first-order loss process that depends on the overall size distribution.In an analysis of aging timescales in a specific urban environment using a particle-resolved model, Riemer et al. (2010) showed that timescales for particles to transition from CCN-inactive to CCN-active varied diurnally due to variations in condensation aging rates.Because the timescale from Riemer et al. (2010) is based on changes in particle CCN activity, it quantifies changes in particle characteristics that these first-order aging models are meant to represent.
This study builds on the work of Riemer et al. (2010) to generalize how the CCN-based aging timescale varies with scenario-specific properties.Unlike other aerosol schemes, which simplify the representation of particle composition, the particle-resolved model tracks the composition of each 110 simulated particle and is, therefore, uniquely suited to study the impact of aging on per-particle CCN activity.The focus of this paper is to identify the set of independent variables that best explain variance in BC's aging timescale for a large collection of simulations.Using the independent variables identified in this study, we will later introduce a simple aging parameterization for use in global models.

Extracting aging timescales from particle-resolved model output
In a first-order model of aging, particles transition from fresh to aged according to an aging timescale, τ aging .In this framework, a criterion must be applied to distinguish fresh and aged particles.Particle CCN activity at a specified environmental supersaturation is the aging criterion applied in this work, so the aging timescale indicates changes in particles' 125 susceptibility to removal by wet deposition.We define aged particles as those that are activated at a specified environmental supersaturation, and fresh particles are CCN-inactive at that supersaturation threshold.The first-order aging model is given by: where N fresh is the number concentration of fresh particles.Before discussing the full set of sensitivity simulations in Section 2.3, we describe the particle-resolved simulation of aerosol dynamics in a baseline scenario and show how the ag-135 ing timescale is used to quantify changes in per-particle CCN activity.The particle-resolved model is described in Section 2.1, and the κ-Köhler model (Petters and Kreidenweis, 2007) for computing CCN activity from the PartMC-MOSAIC data is discussed in Section 2.2.We discuss methods for comput-140 ing aging timescales from the particle-resolved model output in Section 2.3.

Particle-resolved simulation of aerosol aging
The Particle Monte Carlo model (Riemer et al., 2009) coupled to the Model for Simulating Aerosol Interactions and 145 Chemistry (Zaveri et al., 2008), PartMC-MOSAIC, is a Lagrangian box model that simulates gas and aerosol chemistry, gas-aerosol mass transfer, aerosol coagulation, gas and aerosol emissions, and dilution with background air.The boundary layer height varies temporally according to 150 a prescribed profile.The treatment of dilution is the same as in (Riemer et al., 2009); a constant dilution rate of 1.5×10 −5 s −1 is applied and additional dilution with background air occurs when the boundary layer height increases.Coagulation events, particle emissions, and dilution with 155 background air are simulated stochastically by PartMC.Gasand aerosol-phase chemistry and gas-aerosol mass transfer are simulated deterministically by MOSAIC.MOSAIC includes modules for gas-phase photochemistry (Zaveri and Peters, 1999), particle-phase thermodynamics (Zaveri et al.,160 2005b,a), and gas-particle mass transfer (Zaveri et al., 2008).MOSAIC treats secondary organic aerosol formation based on the SORGAM scheme (Schell et al., 2001).The coupled model represents all atmospherically important aerosol species, including sulfate (SO 4 ), nitrate (NO 3 ), chloride 165 (Cl), carbonate (CO 3 ), ammonium (NH 4 ), sodium (Na), calcium (Ca), methanesulfonic acid (MSA), black carbon (BC), primary organic aerosol (POA), and eight secondary organic aerosol (SOA) species.A full description of the coupled model can be found in Riemer et al. (2009).PartMC-170 MOSAIC represents changes in particle composition by condensation and coagulation; we do not consider changes in particle shape or aging by photochemical oxidation.PartMC Version 2.1.4was used to generate the results in this paper.Simulations were performed at a time step of 60 seconds, with approximately 10 5 computational particles.
We simulated 288 plume scenarios, varying meteorological conditions, emissions of gases and particles, and the background number concentration, with further description given in Section 2.3.The atmospheric composition and environmental conditions differed between the scenarios, but the general structure of all simulations was the same.In each case, we simulated a well-mixed air parcel that is advected over and away from large urban area.All scenarios started at 06:00 LST, at which time the parcel contained only background gas and aerosol without any freshly emitted particles.During transport over the urban area, the parcel received gas and aerosol emissions from 06:00 LST until 18:00 LST, after which all emissions ceased.In these scenarios, we simulate a well-mixed boundary layer during the day, and the parcel is assumed to be in the residual layer at night.The temperature, mixing height, and relative humidity were held constant.Before discussing the full set of scenarios in Section 2.3, we show changes in CCN activity and the diurnal evolution of aging timescales in a baseline scenario.For this baseline scenario, Table 1 outlines background aerosol number concentration, aerosol emission intensity, and size distribution and composition information for both background and emitted aerosols.The background concentration and emission intensity of gas-phase species are provided in Table 2.

κ-Köhler model for computing CCN activity
We determined aging timescales from the particle-resolved results by tracking changes in CCN activity over two consecutive time steps.A particle's ability to activate cloud formation depends on its dry diameter D dry,i and its hygroscopicity parameter κ i .The equilibrium saturation ratio (S i ) over an aqueous droplet is computed through the κ-Köhler model (Köhler, 1936;Petters and Kreidenweis, 2007) as: where σ w is the surface tension of water, M w is the molecular weight of water, R is the universal gas constant, T is the ambient temperature, ρ w is the density of water, D i is the particle wet diameter, D dry,i is the particle dry diameter, and κ i is the hygroscopicity parameter introduced by Petters and Kreidenweis (2007).All other factors being equal, particles with a greater κ i are more hygroscopic and more easily activated.The parameter κ has been determined empirically for a number of aerosol species (Table 3), and the effective hygroscopicity parameter κ i for each particle is the volume-weighted average of κ for its constituent aerosol species.If a particle grows to its critical wet diameter (D c,i ), it will continue to grow without bound at that supersaturation and is said to be "activated".We denote the critical sat-uration ratio at which a particle activates and forms a cloud droplet with S c,i = S i (D c,i ) and the critical supersaturation 225 as s c,i = S c,i − 1 × 100.
Figure 1 shows the two-dimensional number density distribution as a function of the particle dry diameter (D dry,i ) and the particle hygroscopicity parameter (κ i ).Only particles containing BC are shown in this figure.In the scenar-230 ios presented in this study, all BC originated from diesel or gasoline exhaust.Choosing a certain environmental supersaturation threshold allows us to classify the particles as fresh or aged.For example, all particles to the left of the line for s c,i = 1% are considered "fresh" for environmental supersat-235 urations of 1% or lower, and all particles to the right of the line for s c,i = 1% are considered "aged" at supersaturations above 1%.
The number distributions corresponding to fresh emissions, prior to any aging, are shown in Figure 1.a, and 240 changes in the distribution during two time periods are shown in Figures 1.b and 1.c.Freshly emitted combustion particles are small and hydrophobic, with geometric mean diameter D dry,gm = 0.5 µm and with a hygroscopicity parameter of κ = 3 × 10 −4 or κ = 8 × 10 −4 for particles from diesel or 245 gasoline, respectively.Therefore, most BC-containing particles are initially unable to activate at any environmental supersaturation s < 1% (lines in Figure 1.a).As D dry,i and κ i for individual particles increase by condensation and coagulation, their critical supersaturation s c,i for CCN-active 250 decreases, shown by particles crossing the lines of constant critical supersaturation in Figure 1.Secondary aerosol forms through photochemical reactions during the day, causing rapid changes in particles' size and hygroscopicity.At night aging by condensation rates are slow, so coagulation 255 is the dominant aging mechanism.This diurnal variation in aging rates is consistent with observations (Rose et al., 2011;Cheng et al., 2012).We define particles that "age" over a specific time period as those that transition from CCN-inactive to CCN-active, that is the particles that move from below a 260 supersaturation line (CCN-inactive) at time t to above supersaturation line (CCN-active) at t + ∆t.

CCN-based aging timescale
For the entire particle population, this change in the particle properties is quantified using the first-order aging timescale 265 defined in Equation 1.Because the time period ∆t is short relative to the timescale τ aging , Equation 1 can be approximated as: where ∆N f→a is the number of discrete particles that transi-270 tion from fresh at time t to aged at time t + ∆t, calculated from changes in the number of fresh particles.In this study, aging timescales are computed over a time step ∆t = 10 minutes.Combining Equations 1 and 3, the aging timescale is We refer to this aging timescale as the "bulk aging timescale" because it corresponds to the entire fresh particle population, and the term ∆N f→a includes all particles that transition from fresh to aged, regardless of their size.Later, we will introduce an analogous "size-resolved aging timescale".Further details on the derivation of the bulk aging timescale, including number balances for all processes affecting aging, are given in Riemer et al. (2010).
The temporal evolution of the timescale is shown for the baseline scenario in Figure 2.a at s = 0.1%, s = 0.3%, and s = 1%.The aging timescale is a simple metric for quantifying the effects of changes in per-particle size and hygroscopicity that are shown in Figure 1, and the gray shading in Figure 2.a corresponds to the time periods shown in Figure 1.
Particles must become highly hygroscopic to activate into cloud droplets at low s (e.g.s = 0.1%) but require less processing to become CCN at higher values of s (e.g.s = 1%), so the aging timescale tends to be shorter for higher values of s.

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Any particle that transitions from fresh at t to aged at t + ∆t does so either by coagulation with a large, hygroscopic particle or by accumulating sufficient condensing material to become hygroscopic.The overall aging timescale τ aging can be represented as the combination of separate timescales for aging by condensation τ cond and by coagulation τ coag : The contribution of condensation and coagulation to the overall aging timescale is shown by separate timescales for aging by condensation (τ cond ) and coagulation (τ coag ) at s = 0.3% in Figure 2.b.We computed the coagulation and condensation aging timescales by counting the number of particles that transition from fresh to aged after participating in a coagulation event, ∆N f→a,coag , or that age only by condensation, ∆N f→a,cond .Then, we applied Equation 4 to find the corresponding condensation and coagulation aging timescales.Figure 2.b shows that the overall aging timescale is shortest during the day (e.g. 1 h at s = 0.3%) due to rapid condensation of semi-volatile substances, and it is considerably longer at night (e.g.24 h at s = 0.3%), when coagulation is the dominant aging mechanism.The temporal evolution of τ aging and τ cond are shown for multiple supersaturation levels in Riemer et al. (2010).

Ensemble of particle-resolved model scenarios 320
The aging timescales shown in Figure 2 are limited to only one scenario, and aging rates vary with local conditions.For example, the number concentration and size distribution of background particles affect coagulation rates and, thereby, the coagulation aging timescale.In order to identify the set 325 of independent variables that best explain variance in BC's aging timescale under a range of atmospheric conditions, we simulated aerosol dynamics in a series of plume scenarios and extracted aging timescales for each scenario.As we will show, the environmental properties that affect aerosol 330 dynamics varied diurnally and differed between scenarios, causing the aging timescale to range from less than an hour (a large portion of particles age per time interval) to longer than a week (few particles age per time interval).
The input parameters that were varied between the scenar-335 ios are shown in Table 4.These input parameters were selected to produce a range of environmental conditions, consistent with observations described by Jimenez et al. (2009) and references therein.Simulations were performed using every combination of input parameters given in Table 4, lead-340 ing to a total of 288 scenarios.In each scenario, aerosol concentrations and particle characteristics varied throughout the 24-hour simulation.The conclusions in this study are based on these simulations of urban air masses.
Figure 3 shows the distribution of aerosol mass concentra- Variance in the aging timescale is shown by the probability density distribution in Figure 5, which includes each 10-minute time interval in each of the 288 simulations.Distributions are shown for timescales computed at s = 0.1%, 365 s = 0.3%, and s = 1%.The supersaturation threshold s specifies the degree of change in particle properties required to classify a particle as aged, and timescales tend to decrease as s increases.In the following sections, we show that most of the variance in black carbon's aging timescale at a specific s 370 is explained by only a few key variables.4 Nonparametric regression analysis to quantify explained variance Black carbon's aging timescale ranges from minutes to weeks (Figure 5), depending on local conditions and charac-different combinations of independent variables explain variance in black carbon's aging timescale by comparing predictions of aging timescales from nonparametric regression with exact aging timescales from PartMC-MOSAIC.A nonparametric regression was chosen, rather than a parametric regression, because we do not know a priori the shape of the predictor surface.
The procedure in applying a nonparametric regression is as follows: 1) select a set of candidate independent variables to test; 2) use most (90% of simulations) of the data as the training set to find the expected value of the aging timescale as a function of the independent variables, as will be explained below; 3) evaluate this expected aging timescale using the rest of the data (10% of simulations), called the testing set.
The timescale from the regression is assessed by how well it predicts the values of the aging timescale in the testing set, represented by R 2 .The purpose of this exploration is to find the independent variables that explain most of the variance in the aging timescale, indicated by the largest value of R 2 .
To ensure that our conclusions did not depend on the choice of scenarios, we repeated the analysis several times with randomly chosen testing and training sets and verified that R 2 was insensitive to the specific choice of testing and training sets.

Kernel density regression applied to particleresolved model data
Figure 6 shows how the regression analysis is applied in this study.For all times in all simulations in the testing set, a particle that is fresh at time t may age between t and some later time t + ∆t or it may remain fresh over that time period.Because these two events are mutually exclusive, this aging behavior in PartMC-MOSAIC may be represented by a binary variable Y age,j (t, t + ∆t, s), where Y age,j = 1 if the particle ages between t and t+∆t and Y age,j = 0 if it remains fresh.The aging timescale at each model time step can then be computed as the average of Y age,j across all fresh BCcontaining particles: which is equivalent to Equation 4, computed from N p,fresh individual particles over a specific model time step.Alternatively, the expected probability that a fresh particle will age, given its characteristics or the aging conditions that it experiences, can be estimated from a nonparametric regression.We applied the kernel density regression introduced by Watson (1964) and Nadaraya (1964).The expected value of Y age,j for a specific particle in the testing set is predicted using the kernel density regression, using information about the candidate variable x j only.The candidate variable x j may be a particle-level characteristic, which varies between particles and, for a specific particle, varies over time (e.g.particle wet diameter).The candidate variable x i may also be a charac-teristic of the environment, which varies over time but, at a specific time, is the same for all particles (e.g.aerosol number concentration).All candidate variables explored in this 430 study are outlined in Table 5.In this section, we show how the nonparametric regression can be applied to evaluate variance explained by a single candidate variable at a time.Later, we show how this analysis can be extended to evaluate combinations of independent variables.
where x i is the value of the independent variable for each In Section 4.1, we provide further explanation on the in-

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clusion of particle-level characteristics in the prediction of τaging .
In this study we used a Gaussian kernel function with standard deviation h: 460 The kernel function K h (x j (t) − x i ) defines the weight applied to each model timescale τ age,i to compute the expected timescale τage , such that timescales for conditions similar to the conditions of the target point are weighted most heavily in the regression.The regression function predicted by 465 the kernel regression depends on the prescribed value for h, where larger h results in smoother regression functions.We applied Silverman's rule of thumb to select the value for h (Silverman, 1986), such that h depends on the number of independent variables, the standard deviation of each inde-470 pendent variable, and the total number of data points in the testing set.
If the candidate variable x strongly affects the value of the aging timescale, the expected aging timescale τaging,j will accurately represent the actual aging timescale τ aging,j , assuming a suitable kernel bandwidth h is applied in the kernel regression.Aging rates scale with the inverse of the aging timescale, so we quantified the variance explained by the regression function, R 2 , in terms of 1/τ aging : where τ aging,j is the timescale from PartMC-MOSAIC for each data point in the testing set, τaging,j is the expected timescale from the regression for each data point in the testing set, τaging is the harmonic mean of the aging timescales across all data points in the testing set, and N test is the num-485 ber of data points in the testing set, where the data points include all time steps in all scenarios.

Inclusion of particle-level variables in the kernel regression
To illustrate our approach for including particle-level variables, we demonstrate the regression procedure using the wet diameter as the independent variable x.The resulting regression surface is a size-dependent timescale, which gives insight into the importance of aging processes as a function of particle wet diameter.For a given set of environmental conditions, some particles are more likely to age than others, and we find that a particles' tendency to age depends on their characteristics just prior to the aging period.We evaluated how aging rates vary with a number of per-particle characteristics, such as particles' diameter at emission, their dry diameter at the time when aging is evaluated, or their hygroscopicity parameter when aging is evaluated.We found that for given environmental conditions, per-particle aging rates were most correlated with the wet diameter of fresh (CCN-inactive) particles; that is, values of R 2 were greatest 505 for regression functions that included the time-varying wet size distribution of fresh BC-containing particles.
It is therefore useful to introduce a size-resolved aging timescale that accounts for differences in aging rates between particles of different sizes.Size-resolved aging timescales 510 were computed at each time t and supersaturation s using the kernel regression described in Section 4.1.The expected value of Y age for a particle in the testing set with wet diameter D j (t) was computed as the weighted average of Y age,i for rest particles the training set i = 1, ..., N p,train , computed at a 515 specific t and s: for a particle in the testing set of wet diameter D j was where the kernel weighting function K h (D j (t) − D i ) is a Gaussian (Equation 9), such that fresh particles with D i sim-520 ilar to the target diameter D j are weighted most heavily in the regression.The size-resolved aging timescale can also be defined for a continuous size distribution of fresh particles n fresh (t, D, s).Similar to Equation 8, the size-resolved aging timescale, τ aging (D), is computed as a function of E Y age |D] 525 and the time step ∆t: For a particle-resolved population of fresh particles j = 1, ..., N p,fresh , where each particle has a unique wet diam-530 eter D j , the bulk, population-level aging timescale can be estimated as the average of E Y age |D j (t), ∆t, s across all N p,fresh particles, as given in Equation 8 using x j = D j .Equivalently, the population-level aging timescale can be computed through the average of the continuous size-535 resolved aging timescale τaging (D, t, s), weighted by the size distribution of fresh particles n fresh (D, t, s): integration of the continuous size-resolved aging timescale τaging (D, t, s) over all D, continuous size distribution of fresh particle n fresh (D, t, s) as the average of τaging (D, t, s) across 540 all D, weighted by the size distribution of fresh particles n fresh (D, t, s): A comparison between Figures 7.a and 7.c shows that con-555 densation was the dominant process driving diurnal variation in the size-resolved aging timescale.This diurnal pattern in condensation aging conditions is reflected in the bulk aging timescale shown in Figure 2. The bulk condensation aging timescale was shorter than 4 h during the day for this 560 scenario, and this was the dominant process affecting aging rates at this time.However, Figure 7.c shows that these rapid transitions from CCN-inactive to CCN-active occurred only for the largest (D > 50 nm) fresh particles, although condensation also caused an increase in D for smaller fresh particles.The coagulation aging timescale, on the other hand, was short for the smallest fresh particles and varied only slightly over the course of the simulation.

Combining particle-level and population-level variables in kernel regression
In this study, we performed a series of multivariate kernel regressions to identify the combination of independent variables that best explain variance in black carbon's aging timescale.In many cases, we extracted aging timescales that depend both on characteristics of individual particles, such as D, and on properties of entire particle populations or the environment, such as the overall aerosol number concentration N .One advantage of this approach is that both particle-level variables and population-level variables can be included in the prediction of Y age,j .For example, the expected value of Y age,j for a particle with diameter D j that is exposed to an aerosol number concentration N j is computed with the bivariate kernel regression: The overall aging timescale for a particular size distribution exposed to a specific number concentration is the computed as the sum across individual particles (Equation 8) or, equivalently, by integrating over the size distribution (Equation 13).
Equation 14 can easily be generalized to three or more independent variables.

Independent variables that best explain variance in aging timescales
We found that most variance in the aging timescale is explained by only a few independent variables.Explained variance R 2 is shown different combinations of independent variables as a function of the environmental supersaturation s at which CCN activity is evaluated.For all supersaturation levels, 90% of variance in the coagulation aging timescale (Figure 8.a) was explained by regression predictions that included the size distribution of fresh BC-containing particles (n fresh (D)) and the number concentration of large, CCNactive particles (N CCN,large ).Three variables were needed to explain 85% of variance in the condensation aging timescale (Figure 8.b): the size distribution of fresh BC-containing particles (n fresh (D)); the flux of secondary aerosol ( ḟcond ), defined as the volume condensation rate of semi-volatile substances per particle surface area density; and the effective hygroscopicity parameter of secondary aerosol (κ cond ), where κ cond is the volume-weighted average of κ for condensing semi-volatile species.The size distribution of fresh BC was included in each case by determining a regression for the size-resolved aging timescale before computing the bulk aging timescale according to Equation 13.Only 10-15% of 615 variance remains unexplained, indicating that variables other than n fresh (D), ḟcond , κ cond , and N CCN,large also weakly affect the value of the aging timescale.

dependent variables
Figure 8 shows the explained variance R 2 as a function of s for the independent variables that best explain variance in the coagulation and condensation aging timescales.Approximately 90% of variance in the coagulation aging 625 timescale was explained by regressions in terms of n fresh (D) and N CCN,large (black line of Figure 8.a).Brownian coagulation events are most likely to occur between large and small particles, so the coagulation aging timescale decreases when there are more particles that are CCN-active and are also 630 large enough to be good coagulation partners.The smallest fresh particles are likely to coagulate with large background particles, where we found the threshold for "large" to be D > 100 nm by identifying the threshold that resulted in the highest R 2 .A regression computed in terms of the of n fresh (D), ḟcond , and κ cond (black line of Figure 8.b) explained greater than 80% of variance in the condensation aging timescale, R 2 was less than 60% for regressions that did not include n fresh (D) (red line).Only 10-30% of variance was explained if κ cond was not included in the regres-660 sion (grey line).If ḟcond was not included, R 2 ≈ 0% for all s, regardless of the other variables included in the regression (not shown).This suggests, not surprisingly, that the conden-sation rate is the key variable driving aging by condensation, but the condensation aging timescale also depends strongly on the hygroscopicity of condensing aerosol κ cond and on the size distribution of fresh particles n fresh (D).

Sensitivity of aging timescale to aging conditions
In this section we apply the regression surfaces shown in Figure 9 to selected example cases to demonstrate how aging conditions and the fresh particle size distribution affect particle aging rates.For these two size distributions (Figure 10.a) and different combinations of aging conditions (Figure 10.b), bulk aging 715 timescales at s = 0.3% were computed according to Equation 13, and the results are given in Table 6.The combinations of environmental conditions are as follows: 1) rapid condensation aging (yellow lines in Figure 10.b) and slow coagulation aging (red lines), 2) slow condensation aging (green lines) and rapid coagulation aging (blue lines), 3) slow aging by both condensation and coagulation, and 4) rapid aging by both condensation and coagulation.
The sensitivity of the bulk aging timescales to ḟcond and N CCN,large depends strongly on the environmental supersatu-725 ration s, as shown in Figure 11.At each supersaturation, sensitivities are quantified as a logarithmic derivative, or relative change in τ age to a relative change in ḟcond or N CCN,large .Negative values of this metric indicate that increasing N CCN,large or ḟcond corresponds to a decrease in τ aging .

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While τ aging is most sensitive to N CCN, large at low supersaturation levels, τ aging shows the greatest sensitivity to ḟcond at high supersaturation levels.At low supersaturation levels, τ aging is insensitive to ḟcond if the distribution contains a higher fraction of small particles (D gm = 30 nm), regard-735 less of conditions for aging by coagulation.If particles are large (D gm =60 nm), τ aging at this low s is τ aging sensitive to ḟcond only under conditions of slow aging by coagulation.At s = 1%, τ aging is sensitive to ḟcond in all cases, regardless of the fresh particle size distribution or conditions for aging 740 by coagulation.Coagulation aging is relatively more important at low supersaturation compared to high supersaturation thresholds.Consistent with this fact, Figure 11a shows that the sensitivity of the aging time scale to N CCN,large generally decreases as s increases.The magnitude of the condensa-745 tional flux ḟcond impacts the sensitivity towards N CCN,large .
Environments with lower ḟcond result in a larger sensitivity to N CCN,large .

Discussion
Global models that employ first-order aging models assume 750 a fixed timescale of 1-3 days, but observations show that aging timescales can be as short as a few hours in polluted areas (Zhang et al., 2008).Other modeling studies have suggested parameterizations that account for this variation in aging conditions.Riemer et al. (2004) evaluated aging timescales in a 755 mesoscale model and parameterized timescales for aging by coagulation as a function of the overall number concentration.Pierce et al. (2009) developed an analytical expression that accounts for decreases in the number concentration of primary aerosol through coagulation events; for emitted par-760 ticles of a specific size, the coagulation loss rate was computed by integrating the coagulation kernel over the entire background size distribution.However, the regression analysis applied in the current study reveals that 90% of the variance in the coagulation aging timescale can be explained using a relatively simple representation of the background size distribution.We showed that the variation in the sizeresolved aging timescales can be attributed to the number concentration of particles that are both large (D > 100 nm) and CCN-active.Other characteristics of the background size distributions are not needed.Oshima and Koike (2013) developed a parameterization of the condensation aging timescale based on results from a box model, and, similar to the present study, computed aging timescales based on changes in CCN activity.However, unlike the present study, Oshima and Koike (2013) did not consider differences in the hygroscopic properties of the condensing material, and their aging timescale varied with the mass condensation rate per total BC mass concentration.In contrast, the regression analysis in the present study reveals that the volume condensation rate per overall aerosol surface area is the variable that best explains variance in BC's condensation aging timescale, which is consistent with laboratory studies (Zhang et al., 2008;Khalizov et al., 2009).The present work also differs from Oshima and Koike (2013) in the representation of the aerosol size distribution.Whereas Oshima and Koike (2013) parameterized bulk aging timescales for lognormal size distributions, we presented a size-resolved aging timescale that can be applied to any arbitrary size distribution.
As in all relationships for BC's aging timescale, the value of the aging timescale depends strongly on the criterion used to distinguish fresh and aged particles.Particle activation at a specific environmental supersaturation is the aging criterion applied in this study, representing changes in particle characteristics that most affect their susceptibility to wet deposition.Table 6 shows that the value of the aging timescale depends strongly on the criterion supersaturation at which CCN activation was evaluated, consistent with Riemer et al. (2010) and Oshima and Koike (2013).Further, the relative importance of condensation versus coagulation as aging processes also depends on the supersaturation threshold.

Conclusions
This study identifies the minimal set of independent variables needed to explain variance in black carbon's aging timescale.
We simulated the evolution of gases and aerosols in a series of urban scenarios with the particle-resolved model PartMC-MOSAIC and extracted time-dependent aging timescales based on the rate at which individual particles transition from CCN-inactive to CCN-active at a specified environmental supersaturation.The value of the aging timescale spanned orders of magnitude, depending on local environmental conditions and the supersaturation threshold at which CCN activity was evaluated.Aging timescales were shorter than an hour under conditions of rapid secondary aerosol formation, but on the order of days in the absence of secondary aerosol precursors.Condensation aging timescales exhibited more variation than coagulation aging timescales, and the relative importance of each aging mechanism depended on the size range of particles to be aged.We performed a non-parametric 820 regression analysis on model data from 288 scenarios in order to identify the independent variables with which aging timescales are best correlated and quantified the portion of variance explained by regressions in terms of these variables.This paper is the groundwork for the development of aging 825 parameterizations suitable for use in global models.
To our knowledge, this is the first study to apply a regression analysis to identify the minimal set of parameters needed to explain variance in black carbon aging rates.After evaluating a number of independent variables, we found 830 that the flux of secondary aerosol, the hygroscopicity of secondary aerosol, and the size distribution of CCN-inactive (fresh) BC-containing particles were the minimal set of parameters needed to explain 80% of variance in the condensation aging timescale.On the other hand, 90% of variance in 835 the coagulation aging timescale was explained by only two variables: the size distribution of fresh BC-containing particles and the number concentration of particles that are both large (D > 100 nm) and CCN-active.This work distills the complex interactions captured by the particle-resolved model 840 to a few input variables, all of which are tracked by existing global climate models, and is a first step toward developing physically-based parameterizations of aerosol aging.
Acknowledgements.This project is funded by NASA.N. Riemer also acknowledges US EPA grant 835042.Its contents are solely 845 the responsibility of the grantee and do not necessarily represent the official views of the US EPA.Further, US EPA does not endorse the purchase of any commercial products or services mentioned in the publication.The authors are grateful to Matthew West and Peter Maginnis for their suggestions in the early stages of this work.

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We also thank the two anonymous reviewers for their helpful comments.

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tion for selected aerosol species for all scenarios simulated with PartMC-MOSAIC (black lines) and corresponding ambient observations compiled by Jimenez et al. (2009) (vertical colored lines).The range of conditions simulated in the ensemble of scenarios is representative of the distribution in 350 concentrations observed in these urban areas.The distribution in the number concentration of all particles and of BCcontaining particles are shown in Figure 4.a and 4.b, respectively.The size and composition of BC-containing particles also varied over the course of an individual simulation, as we 355 showed in Figure 1.Variation in the geometric mean diameter and in the geometric mean hygroscopicity parameter of BC-containing particles across all simulations are shown in Figures 4.c and 4.d, respectively.Figures 3 and 4 were constructed from data at 10-minute intervals in each of the 288 360 simulations, corresponding to 41,000 time steps.

∞ 0 n
fresh (t, D, s)dD .(13) By this relationship, the bulk aging timescale under a specific set of environmental conditions also varies with the size 545 distribution of CCN-inactive (fresh) BC.The temporal evolution of the size-resolved aging timescale is shown for the baseline scenario in the middle column of Figure 7 for s = 0.3%.The contributions of coagulation (Figure 7.b) and condensation (Figure 7.c) to the 550 overall aging timescale (Figure 7.a) are shown by the separate size-resolved timescales for each process.The dominant mechanism driving the aging timescale depends on the time of day and the particle size.

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number concentration of large particles (green line of, rather than the number concentration of large and CCN-active particles, gave R 2 ≈85% at high supersaturation thresholds (s > 0.8%) but R 2 < 10% at low supersaturation thresholds (s < 0.1%).This is because not all particles with 640 D > 100 nm are CCN-active at s = 0.1%, but nearly all particles that are CCN-active at this low s have D > 100 nm.On the other hand, if the independent variable was the number concentration of CCN-active particles (blue line of, rather than the number concentration of large and 645 CCN-active particles, R 2 ≈ 90% for timescales at low supersaturation thresholds (s < 0.1%) and R 2 ≈ 70% at high supersaturation thresholds (s > 0.8%).Only by considering the number concentration of particles that are both CCN-active and large, N CCN,large were we able to explain variance in the 650 coagulation aging timescale at all supersaturation levels. I the size distribution of fresh BC was neglected, R 2 ranged from 40% to 60%, depending on the supersaturation threshold (yellow line of Figure8.a).While the expected aging timescale computed in terms 655 CCN,large , D) and τcond ( ḟcond , κ cond , D) are shown in Figures 9.a and 9.b, respectively.Figure9.ashows that timescales for aging by coagulation range from hours to weeks.The coagulation aging timescale decreases with the number concentration of "large", CCN-active particles (N CCN,large ) and, for a given 675 N CCN,large , small BC-containing particles are more likely to age by coagulation than large BC-containing particles.On the other hand, condensation aging timescales are shortest for the largest fresh particles and, for these particles, the condensation aging timescale tends to decrease as ḟcond or κ cond increase.The two panels of Figure9.b show τcond as a function of ḟcond and D for secondary aerosol with differing hygroscopicity, κ cond = 0.65 on the left, representing secondary inorganic aerosol, and κ cond = 0.1 on the right, representing secondary organic aerosol. 685 Figure 10 shows how aging rates by condensation and coagulation can be reconstructed as a function of the size distribution of fresh particles (Figure 10.a) and the size-resolve aging timescale (Figure 10.b).We compare lognormal size distributions with geometric mean diameter (D gm ) of 30 nm (dashed line of Figure10.a)and 60 nm (solid line of Figure10.a).Timescales were computed for limiting environmental conditions, indicated by line colors in Figure10.b):slow coagulation aging (N CCN,large = 500 cm −3 , red line) or fast coagulation aging (N CCN,large = 10, 000 cm −3 , blue line) and slow condensation aging ( ḟcond = 0.01 nm h −1 , green line) or fast condensation aging ( ḟcond = 1 nm h −1 , yellow line).Size-resolved aging timescales are taken from the regression surfaces in Figure9for these values of ḟcond and N CCN,large , assuming κ cond = 0.65 in both cases.Then, the rate at which particles of a given size transition from fresh to aged (Figure10.c) is computed as the product of n f (D) and 1/τ age (D).Figure10.cshows aging rates for particle distributions with D gm = 30 nm (dashed lines) and D gm = 60 nm (solid lines) under these limiting environmental conditions that promote rapid (blue lines) or slow (red lines) aging by coagulation and rapid (yellow lines) or slow (green lines) aging by condensation.

Fig. 1 .
Fig. 1.Two-dimensional probability density distribution shows changes in particle properties.As particles increase in size (horizontal axis) and hygroscopicity (vertical axis), they are able to activate at lower critical supersaturation thresholds (superimposed lines).a) Freshly emitted particles are hydrophobic, with κ = 3 × 10 −4 and κ = 8 × 10 −4 for diesel and gasoline, respectively.b) During the daytime, particles age rapidly by condensation of semi-volatile substances that are produced through photochemical reactions.c) At night, condensation aging is slow, and particles age only by coagulation.

Fig. 2 .
Fig. 2. For a single scenario, overall aging timescale for s = 0.1%, s = 0.3%, and s = 1% in Figure 2.a and the overall, condensation, and coagulation aging timescales for s = 0.3% in Figure 2.b.The shaded regions show how the value of the aging timescale reflects changes in per-particle characteristics, which correspond to Figure 1.Short aging timescales correspond to rapid increases in particle size and hygroscopicity (Figure 1.b), and long aging timescales correspond to slow changes in particle properties (Figure 1.c)

Fig. 3 .
Fig. 3. Probability density function of aerosol mass species in simulations (black line in each graph) show that model cases represent variation in atmospheric conditions from ambient observations (vertical colored lines).Probability density functions include all output time steps in the full ensemble of sensitivity simulations.

Fig. 4 .
Fig. 4. Probability density function of a) total aerosol number concentration, b) total number concentration of BC-containing particles, c) geometric mean diameter of BC-containing particles, and d) geometric mean hygroscopicity parameter of BC-containing particles.

Fig. 6 .
Fig.6.Procedure for applying kernel regression to predict black carbon's aging timescale and quantifying the portion of variance explained by that prediction, shown for a hypothetic input variable x.

Fig. 8 .
Fig. 8. Coefficient of determination R 2 for a) coagulation and b) condensation timescales as a function of supersaturation for selected combinations of independent variables, where the combination of variables that explain most of the variance are shown by the black lines in each graph.Regression analyses on the coagulation aging timescales are shown for four combinations of independent variables: (I) including wet diameter, D, of fresh BC-containing particles and the number concentration of large (D > 100 nm), CCN-active particles, NCCN,large, (II) including D of fresh BC-containing particles and the number concentration of large particles, Nlarge, rather than NCCN,large, (III) including D of fresh particles and the number concentration of CCN-active particles, NCCN, rather than NCCN,large, and (IV) including NCCN,large but without including D of fresh BC-containing particles.Regression analyses on the condensation aging timescale are shown for three combinations of independent variables: (V) including secondary aerosol flux, ḟcond, the hygroscopicity of secondary aerosol, κcond, and D of fresh BCcontaining particles, (VI) including ḟcond and κcond but without including D of fresh BC-containing particles, and (V) including ḟcond and D of fresh BC-containing particles but without including κcond.For all s, approximately 90% of variance in coagulation aging timescale is explained by two independent variables (black line in Figure 8.a), and 80% of variance in condensation aging timescale is explained by three independent (black line in Figure 8.b).

Fig. 9 .
Fig. 9. Coagulation aging timescale as a function of wet diameter and number of large, CCN-active particles (Figure 9.a) and condensation aging timescale as a function of wet diameter, secondary aerosol flux, and hygroscopicity of secondary aerosol (Figure 9.b).Results are shown for a threshold supersaturation s of 0.3%.

Fig. 10 .
Fig. 10.Rate at which particles of specific size transition from fresh to aged (Figure 10.c) depends on size distribution of fresh BC (Figure 10.a) and size-resolved aging timescale (Figure 10.b).Results are shown at s = 0.3%, where the size-resolved aging timescale under different conditions are determined from the regression function in Figure 9.The line colors in Figure 10.c correspond to the aging conditions shown in Figure 10.b, and the line style in Figure 10.c correspond to the size distributions shown in Figure 10.a.

Fig. 11 .
Fig. 11.Sensitivity of aging timescale to a) NCCN,large and b) ḟcond as a function of supersaturation level, expressed as the logarithmic derivative of the timescale with respect to each variable.The value of d log τage/d log NCCN,large, for example, indicates the relative change in τage to a relative change in NCCN,large.Shown for the size distributions and aging conditions in Figure 10.

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At each time step in each simulation of the testing set, the expected value of Y age (t, t + ∆t, s) for each particle was computed as a weighted average of Y age,i (t, t + ∆t, s) for millions of individual particles in the training set.Values for Y age,i (t, t + ∆t, s) in the training set are weighted 440 according to the kernel function K h (x − x i ), where x is the independent variable of interest.The expected value of Y age (t, t + ∆t, s) is given by:

Table 1 .
Aerosol Emissions and Initial Conditions for Baseline Simulation

Table 2 .
Gas-phase Initial Conditions and Emissions 1 for Baseline Simulation

Table 3 .
Hygroscopicity parameter assigned to aerosol species

Table 4 .
Input parameters varied in the ensemble of sensitivity simulations.Scenarios corresponding to the baseline conditions are indicated in bold.All combinations of scenarios were included in the full ensemble of 288 simulations.

Table 5 .
Candidate variables included in the regression analysis.CCN,large num.conc. of CCN-active particles with D > 100 nm population-level

Table 6 .
Bulk aging timescale for two fresh particle size distributions under different aging regimes.Condensation, coagulation, and overall aging timescales are given for s = 0.3%.We assumed lognormal size distributions of fresh BC with a geometric standard deviation of 1.7.