Revisiting Twomey's approximation for peak supersaturation

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 which 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 which can then be solved with a single lookup table with only one independent variable for an underlying lognormal aerosol population. Multimode aerosol with only N different dispersion characteristics require only N of these one-dimensional lookup tables. 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 and very accurate physically-based parametrization of droplet nucleation for use in climate and NWP models.