Estimating atmospheric nucleation rates from size distribution measurements: Analytical equations for the case of size dependent growth rates

While laboratory and field measurements indicate that atmospheric nucleation most likely initiates with the formation clusters of ~1.0-1.5 nm diameter, most atmospheric observations to date measure only particles larger than 3-10 nm in size. Because of this, several analytical formulations have been developed to estimate the real nucleation rate at the initial cluster size from the “apparent” nucleation rate at the measured larger sizes. All previous analytical formulations have assumed a constant particle growth rate below the instrument detection limit; however, recent atmospheric measurements have shown that the growth rate is often strongly size dependent. This study presents 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. The accuracy of these equations is tested with an ensemble of numerical model simulations of new particle formation events. Both new formulations are capable of estimating the nucleation rate at 1.5 nm fairly accurately (largest normalised mean bias −1.4% for the power-law and −23% for the linear events). We find, however, that the power law formulation gives a more accurate estimate of the nucleation rate even for a majority of the events with linear growth rate dependence. Further analysis indicates that previous studies of atmospheric nucleation events, which have assumed a constant cluster growth rate, may have clearly underestimated the real nucleation rate.