A semi-analytical expression has been developed that accurately models the population dynamics of an aerosol growing from the detection limit (3 nm) to a characteristic CCN size (100 nm), quantifying the contributions of size and time-dependent source and sink terms such as coagulation of smaller particles and scavenging by the pre-existing aerosol. These model inputs were calculated from measured aerosol size distributions and growth rates acquired during intensive measurement campaigns in Boulder, CO, Atlanta, GA, and Tecamac, Mexico. Twenty CCN formation events from these campaigns were used to test the validity of this model. Measured growth rates ranged from 3–22 nm/h. The modeled and measured CCN production probabilities agreed well with each other, ranging from 1–20%. The pre-existing CCN number concentration increased on average by a factor of 3.8 as a result of new particle formation.
 Atmospheric aerosols exert a significant impact on global climate by affecting the earth's radiation balance directly through the scattering and absorption of solar radiation and indirectly through their role as cloud condensation nuclei (CCN) [Albrecht, 1989; Charlson et al., 1992]. This indirect effect of aerosols contributes the largest uncertainty to estimates of global radiative forcing [Intergovernmental Panel on Climate Change, 2007]. Accurate assessment of the relationship between CCN and forcing in global climate models requires understanding processes that determine CCN concentrations. Several field campaigns [Kerminen et al., 2005; Laaksonen et al., 2005] have implicated newly formed particles from atmospheric nucleation events as an important source of CCN.
 New particle formation in the atmospheric boundary layer and subsequent growth to sizes that are large enough to serve as CCN (100 nm) have been observed in the continental troposphere [Stolzenburg et al., 2005]. As these newly formed particles grow, they undergo processes that enhance and deplete the growing population, such as coagulation production and scavenging by the pre-existing aerosol, respectively [Stolzenburg et al., 2005]. Reducing the uncertainty in the CCN number population due to the growth of newly formed particles depends on accurately accounting for these sources and sinks, which depend on particle size and growth rate.
 Recent modeling efforts incorporating aerosol microphysics have studied the effect of boundary layer new particle formation on CCN concentrations using an off-line chemical transport model [Spracklen et al., 2008] and a particle growth model [Pierce and Adams, 2007]. The model inputs included parameterized new particle formation rates [Spracklen et al., 2008] and simulated size distributions and growth rates [Pierce and Adams, 2007]. In the present work, a model for CCN formation was developed based on measured new particle formation events, yielding an analytical expression for the number distribution of nucleated particles that grew to 100 nm, an assumed CCN-active diameter for soluble particles at 0.2% supersaturation [Pierce and Adams, 2007], based on measured aerosol size distributions and growth rates. The model was applied to twenty CCN formation events measured in three North American locations: Boulder, CO [Iida et al., 2006]; Atlanta, GA [McMurry et al., 2005]; and Tecamac, Mexico [Iida et al., 2008]. The results of these calculations are compared with observations. Enhancements to pre-existing CCN number concentrations due to new particle formation were also calculated.
2. Measurements and Techniques
 Data from the three measurement campaigns were acquired by researchers from the University of Minnesota and the National Center for Atmospheric Research. Detailed descriptions of the physical and meteorological conditions at each site as well as a summary of pertinent aerosol instrumentation are given by Kuang et al. . This analysis utilized measurements of aerosol size distributions.
 For a measured CCN formation event, the size distribution of 100 nm particles was modeled by following a population of newly formed particles as they grew from 3 to 100 nm, and accounting for how various aerosol sources and sinks added to and depleted the population during growth. While loss processes such as wet deposition and air mass dilution are important aerosol removal processes, they are not included in this analysis since there was no measured precipitation associated with the observed events and the effect of dilution would only deplete the aerosol concentration by at most a factor of two. An example of such an event is shown in Figure 1, where new particle production occurred just before 12:00 and was followed by nearly continuous particle growth approaching 100 nm in diameter over the next 33 hours. It is the goal of this work to develop a simple analytical expression that accurately models the size distribution of the nucleated particles as they grow to 100 nm using measured size distributions and growth rates to account for sources and sinks.
 The dynamics of an aerosol population growing by condensation and coagulation are described by the particle size distribution n evolving through size and time according to the general dynamic equation (see auxiliary material) [Seinfeld and Pandis, 1998]
where GR(Dp, t) is the particle diameter growth rate, ψ = (Dp3 - p3)1/3, and K(ψ, Dp) is the coagulation coefficient for particles of diameter ψ and Dp. There are well-established analytical [Ramabhadran et al., 1976] and numerical [Gelbard and Seinfeld, 1978] methods of solving equation (1) and obtaining n. In this work however, equation (1) is solved only for a subset of the aerosol population n† that follows a diameter trajectory Dp†, defined as the path through diameter space that a growing particle follows according to the measured growth rate, a representative example of which is shown in Figure 1. After expanding the growth term in equation (1) and grouping similar terms, equation (1) is reduced to an ordinary differential equation along Dp† by the method of characteristics:
which defines the size and time-dependent sources and sinks of n† as it grows along the measured diameter trajectory Dp†. On the RHS of equation (2), the first term defines contributions to n† from coagulation of smaller particles that yield a larger particle of size Dp†, the second term defines losses due to scavenging by the pre-existing aerosol, and the third term defines losses due to size-dependent growth. For the CCN formation events analyzed, the loss from self-coagulation of n† was calculated to be negligible. Equation (2) is then integrated along Dp†, yielding an analytical expression for n†:
where n3 is the value of the measured distribution function at the start of growth (nominally 3 nm), τloss is a sink term characterizing the various loss mechanisms that deplete n†, and Fcoag is a source term representing all collisions of smaller particles that yield larger particles with sizes equal to the diameter trajectory Dp†. To determine the size distribution for 100 nm particles (n† = n100), equation (3) is evaluated at a time t along the diameter trajectory where Dp† = 100 nm. This approach is an extension of earlier methods [Weber et al., 1997; Kerminen and Kulmala, 2002; McMurry et al., 2005; Lehtinen et al., 2007] where particle growth from 3 to 100 nm is now examined and where time and size-dependent particle sources, sinks, and growth rates are now included. This is particularly important during periods of substantial new particle formation, where the aerosol number concentration and surface area (from which sources and sinks are calculated) can change significantly. In this analysis, the percent contributions to the population of particles (Dp > 100 nm) from coagulation of the pre-existing aerosol with n† and self-coagulation of n† are relatively small (<5%) compared to the contribution from growth of n† through 100 nm.
 A natural product of this analysis that can be obtained from equation (3) is the CCN production probability defined as n100/n3, which is the ratio of the size distribution of 100 nm particles at the end of the diameter trajectory to the size distribution of 3 nm particles at the beginning of the trajectory. For the case of constant particle growth rate, the value of n100/n3 is equivalent to the ratio of the particle production rate at 100 nm to that at 3 nm. For CCN formation events where there is negligible enhancement from coagulation of smaller particles, n100/n3 represents the survival probability of a population of 3 nm particles growing to 100 nm. For this case, the aerosol population n† only undergoes loss as it grows and therefore only contains particles that were originally present at 3 nm. The ratio n100/n3 is then only a function of τloss, a dimensionless particle lifetime that captures the competing interactions between loss and growth as the particles approach 100 nm. For CCN formation events characterized by a fast growth rate, there is a relatively shorter time over which the various loss mechanisms can act, resulting in a relatively larger CCN population. This dimensionless lifetime is conceptually similar to the L parameter of McMurry et al. , which accounts for the survival probability of clusters growing from 1 to 3 nm while being depleted by coagulation.
 A related quantity of interest is the enhancement to the pre-existing number concentration of CCN-active particles N100 (Dp > 100 nm) due to new particle formation, defined as the ratio of the peak N100 after new particle formation to the initial, pre-existing N100. Enhancements to N100 due to condensational growth of the pre-existing aerosol (Dp < 100 nm), condensational growth of the nucleated aerosol, and coagulation of smaller particles and depletions to N100 due to self-coagulation are determined by solving equation (1) for N100 from the start of new particle formation to when the peak value of N100 occurs.
 Analysis of a given CCN formation event begins by identifying the initial distribution of the nucleated aerosol population, which was defined in this study as the peak value of the distribution function of 3–6 nm particles during a new particle formation event. The diameter width for this initial distribution was small enough to be considered newly formed but large enough to achieve good particle counting statistics. The diameter trajectory Dp† of this initial population was then defined by the time evolution of the peak of the measured distribution which, to a first approximation, ensures that the same aerosol population is being modeled during growth. The diameter trajectory defines both the size and time-dependent sinks and sources (τloss and Fcoag, which are integrated numerically from measured size distributions) that deplete and increase the particle population as it grows to 100 nm, as well as the length of time over which the effects of these sinks and sources are integrated. Model sensitivity to the initial condition n3 was explored by starting individual trajectory calculations over an interval of 15 minutes before and after the peak value of n3 during the period of new particle production.
3. Results and Discussion
 The diameter trajectory Dp† is a critical quantity in this analysis, defining the various sink and source terms that control the dynamics of n† as it grows. A distinguishing feature of this model is the use of particle trajectories determined by measured growth rates (3–22 nm/h for the three measurement campaigns included in this analysis). Previous studies modeling CCN formation from ultrafine particle emissions [Pierce and Adams, 2007] assume sulfuric acid vapor as the only condensing species contributing to particle growth and use either measured or modeled sulfuric acid vapor concentration to model growth rates. Studies have shown that growth rates due solely to measured sulfuric acid vapor condensation can significantly underestimate the measured growth rate [Weber et al., 1997], largely because organic compounds are responsible for up to 90% of the growth [Mäkelä et al., 2001; O'Dowd et al., 2002; Iida et al., 2008; Smith et al., 2008]. The increase in growth rate due to the condensation of these organic compounds is particularly important as it enhances CCN production due to the reduced particle lifetime and corresponding losses as the aerosol grows up to 100 nm.
 The modeled and measured values of n100/n3 range from 1–20% across the three measurement campaigns and are plotted versus the calculated loss parameter τloss in Figure 2. These measured and modeled production probabilities are several orders of magnitude larger than the corresponding probabilities calculated by Pierce and Adams . Their models were initialized with simulated stationary pre-existing aerosol size distributions and particle growth rates based on sulfuric acid condensation, rates which are an order of magnitude smaller than the measured growth rates used in this analysis. The results in Figure 2 apply only to those events where growth was strong enough for newly formed particles to reach 100 nm, which represent half of the observed new particle formation events from the three measurement campaigns. For the remaining events, the growing particles did not reach 100 nm in size. Also included on the plot is the model prediction of n100/n3 versus τloss assuming no coagulation enhancement (loss-only solution, Fcoag = 0).
 For those events where the modeled and measured n100/n3 lie on or close to the loss-only solution, n100/n3 approximates the survival probability of newly formed 3 nm particles growing to 100 nm. For these events, approximately 1–10% of newly formed particles survive to 100 nm. It is primarily the competition between loss and growth rates contained within τloss that controls the CCN survival probability. For those events where the modeled and measured n100/n3 deviate significantly (at least an order of magnitude) from the loss-only solution, n† contains both particles from the initial nucleated population and particles formed by coagulation of particles beneath the growing nucleated mode. These types of events were observed in Tecamac and Atlanta and were characterized by sustained periods of particle production with large total aerosol number concentrations (>1 × 105 cm-3).
 The pre-existing number concentrations of CCN-active particles (N100) are increased due to new particle formation by factors of 1.6–9.1 with a mean value of 3.8, which are plotted as a histogram of enhancement factors in Figure 3a. The mean and upper range of these enhancement factors are comparable to the maximum CCN concentration enhancements reported by Spracklen et al.  when modeled growth rates were increased to match observed growth rates, emphasizing the importance of using measured growth rates when simulating CCN populations. The percent contributions of self-coagulation loss, coagulation production, condensation on pre-existing aerosol, and condensation on nucleated aerosol to the N100 enhancements are shown in Figure 3b along with the pre-existing and peak values of N100, averaged over each measurement campaign. Condensational growth of the nucleated aerosol past 100 nm contributes more than 80% to the observed N100 enhancement in Tecamac and Atlanta, while the contribution of condensational growth of the pre-existing aerosol is comparable to that of the condensational growth of the nucleated aerosol in Boulder.
 The relatively narrow distributions of CCN survival probabilities (1–10%) and CCN enhancement factors (>50% are between 2–3) suggest a self-regulating process in the boundary layer, as was observed by Spracklen et al. . CCN enhancement due to high particle growth rates tends to be mitigated by rapid depletion due to a correspondingly large pre-existing aerosol surface area as in Tecamac, and vice versa as in Boulder.
 A semi-analytical model for CCN production was developed by simulating the growth of a subset of an aerosol population from 3 to 100 nm and accounting for various source and sink processes constrained by measured size distributions and growth rates. Modeled production probabilities agreed well with measured values, ranging from 1–20%. The model enabled a quantitative comparison of loss processes (scavenging and size-dependent condensation) with coagulation production, which was shown to be significant in Tecamac and Atlanta. For events with relatively little coagulation production, survival probabilities ranged from 1–10%. New particle formation increased pre-existing CCN number concentrations by factors of 1.6–9.1 with a mean enhancement of 3.8. These enhancements were dominated by contributions from condensational growth of the nucleated aerosol.
 This work was supported by the NSF IGERT Program (award DGE-0114372), NSF award ATM-050067 and DOE grant DE-FG02-05ER63997.