Estimating atmospheric nucleation rates from size distribution measurements: Analytical equations for the case of size dependent growth rates
Introduction
Atmospheric new particle formation via nucleation and growth is known to contribute significantly to total aerosol number as well as to cloud condensation nuclei concentration (Kerminen et al., 2012). However, despite recent progress in controlled laboratory experiments (Kirkby et al., 2011), field observations (Kuang et al., 2012) and theoretical studies (Chen et al., 2012, Loukonen et al., 2010), many details of the initial steps of atmospheric nucleation still remain unknown (Zhang et al., 2012, Kulmala et al. 2013). The first direct observations of atmospheric nucleation have confirmed that new particle formation initiates at a cluster size of about 1–2 nm in diameter (Jiang et al., 2011, Kulmala et al. 2013). Unfortunately instruments capable of measuring down to such small sizes are not yet in wide use, and the vast majority of measurement sites still employ techniques that have a lower cut-off diameter in the range between 3 and 10 nm. Therefore, obtaining a global picture of the first steps of atmospheric new particle formation requires reliable ways to estimate the initial nucleation rate at 1–2 nm based on the size distribution of larger particles.
To this end, methods to estimate the initial nucleation rate have been developed by assuming a constant (Weber et al., 1996, Kerminen and Kulmala, 2002, Lehtinen et al., 2007, Kuang et al., 2008) or partially constant (Kerminen et al., 2004) growth rate between the initial nucleation size (in this study d1.5=1.5 nm) and the detectable size (dx, typically 3 nm), despite observations that the growth rate below 3 nm is typically clearly smaller than above this threshold (Hirsikko et al., 2005, Yli-Juuti et al., 2011, Häkkinen et al., 2013). However, the most recent atmospheric measurements, which can directly detect particles as small as 1 nm in diameter, indicate that the particle growth rates below 3 nm may be strongly size dependent. Analysing new particle formation events in Atlanta and Boulder, Kuang et al. (2012) found that the growth rate increased approximately linearly with particle diameter below 3 nm. This behaviour was attributed to unidentified condensing species (i.e., other than sulphuric acid) which accounted for up to 95% of the observed growth. Similarly, Kulmala et al. (2013) observed a strongly size dependent growth rate below 3 nm diameter at the Hyytiälä boreal station and interpreted it as contribution of organic vapours to initial particle growth.
To incorporate these new findings into future analyses of atmospheric nucleation events, we present analytical formulations for calculating the initial nucleation rate (hereafter: J1.5) from the “apparent” nucleation rate at a detectable size in two special cases: when the growth rate below the detectable size follows a (1) linear, or (2) power-law dependence on the particle size. We then test the derived formulae against an ensemble of model-generated nucleation events, for which the initial nucleation rates as well as particle growth rates are known exactly.
Section snippets
Theory
The evolution of the particle flux J for a growing nucleation mode (i.e. the number of nucleation mode particles reaching a certain size per unit volume and time) is described by (Lehtinen et al., 2007)
Here is the size-dependent growth rate and the coagulation sink of particles of diameter dp is given bywhere β is the coagulation coefficient between particles of diameters dp and , and is the particle size
Set-up
The new analytical formulae were tested with model-generated new particle formation events. We used a modified version of UHMA model (Korhonen et al., 2004) to simulate nucleation, growth and coagulation of a particle population. Table 1 summarises the simulated background particle concentrations as well as the maximum initial nucleation rates, and nucleation mode growth rates at 1.5 and 3 nm sizes. In 189 simulations the particle growth rate below 3 nm was forced to follow a linear size
Discussion
In the analysis above, we have used the new analytical equations to estimate the initial nucleation rate from “observations” of larger particles. However, if Eqs. (5), (6) are rearranged, they can be readily used to predict the “apparent” formation rate of larger particles when the initial nucleation rate is known or taken from a nucleation parameterisation. This need commonly arises in large scale models which aim to limit the number of model tracers in order to reduce the computational cost,
Summary and conclusions
We have presented new analytical equations to connect the real and “apparent” nucleation rates in two special cases, i.e. when the cluster growth rate follows a (1) linear, or (2) power-law dependence on the particle size. Applying these equations to numerical nucleation events reveals that especially the latter equation is very accurate, at least if the time evolution of all relevant input parameters is known. This power-law formulation performs reasonably well also for the simulated events
Acknowledgements
This work has been supported by the Academy of Finland by the Computational Science Research Programme (decision: 135199) and an Academy Research Fellow position (decision: 250348).
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