Revisiting Twomey's approximation for peak supersaturation

Abstract

Abstract. Twomey's seminal 1959 paper provided lower and upper bound approximations to the estimation of peak supersaturation within an updraft and thus provides the first closed expression for the number of nucleated cloud droplets. The form of this approximation is simple, but provides a surprisingly good estimate and has subsequently been employed in more sophisticated treatments of nucleation parametrization. In the current paper, we revisit the lower bound approximation of Twomey and make a small adjustment that can be used to obtain a more accurate calculation of peak supersaturation under all potential aerosol loadings and thermodynamic conditions. In order to make full use of this improved approximation, the underlying integro-differential equation for supersaturation evolution and the condition for calculating peak supersaturation are examined. A simple rearrangement of the algebra allows for an expression to be written down that can then be solved with a single lookup table with only one independent variable for an underlying lognormal aerosol population. While multimodal aerosol with N different dispersion characteristics requires 2N +1 inputs to calculate the activation fraction, only N of these one-dimensional lookup tables are needed. No additional information is required in the lookup table to deal with additional chemical, physical or thermodynamic properties. The resulting implementation provides a relatively simple, yet computationally cheap, physically based parametrization of droplet nucleation for use in climate and Numerical Weather Prediction models. BackgroundAtmospheric aerosols are fundamental to the formation of clouds. They provide the substrate onto which liquid droplets can form, overcoming the energy barriers associated with clustering of water molecules (see Mason, 1971; Pruppacher and Klett, 1997, hereafter PK97). Not only do they facilitate the formation of cloud, but they also determine the initial sizes and numbers of droplets as the cloud forms. Variations in the underlying aerosol population; through numbers, sizes, dispersion characteristics and chemical composition; combined with variations in the dynamic and thermodynamic properties of the ambient air; lead to wide ranging differences in the number concentrations of droplets within clouds. Differences in number further impact the physical evolution of the clouds, their lifetime and their interaction with radiation. These impacts are most famously seen through the Twomey (1977) and Albrecht (1989) effects, but also through other mechanisms once feedbacks with the dynamics (e.g. Ackerman et al., 2004;Jiang et al., 2006;Wang and Feingold, 2009) or the ice phase (Lohmann and Feichter, 2005) are taken into account. Collectively, the interactions between aerosol and clouds represent a significant source of uncertainty in our ability to model the climate system and, moreover, to estimate the impacts of anthropogenic climate change (IPCC, Summary for Policymakers, 2007).The approach generally used to ascertain how many of the to...

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