Abstract
Abstract. Analytical solutions for the critical radii and supersaturations of the cloud condensation nuclei (CCN) with insoluble fractions were derived by Khvorostyanov and Curry (2007, hereafter KC07). These solutions generalize Köhler's solutions for an arbitrary soluble fraction of CCN, and have two limiting cases: large soluble fraction (Köhler's original solution); and a new "low soluble fraction" limit. Similar solutions were found subsequently by Kokkola et al. (2008, hereafter Kok08); however, Kok08 used the approximation of an ideal and dilute solution, while KC07 used more accurate assumptions that account for nonideality of solutions. Kok08 found a large discrepancy with KC07 in the critical supersaturations. It is shown that the major discrepancy with KC07 found in Kok08 was caused by the simple mistake in Kok08, where comparison was made not with the general solution from KC07, but with the Köhler's solution or with some unknown quantity, not even with the "low soluble fraction" limit. If general solutions from the two works are compared, the equations from Kok08 mostly repeat the equations from KC07, except that Kok08 use the ideal dilute solution approximation. If the mistake in Kok08 is corrected, then the differences in the critical radii and supersaturations do not exceed 16–18%, which characterizes the errors of the ideal dilute solution approximation. If the Kok08 scheme is modified following KC07 to account for the non-ideality of solution, then the difference with KC07 does not exceed 0.4–1%.
Highlights
A theoretical basis for consideration of hygroscopic growth of atmospheric aerosols or cloud condensation nuclei (CCN) and their activation into cloud drops was provided by the Kohler (1936) equation that enabled prediction of the CCN critical radii rcr and supersaturations scr for drop activation
Kok08 arrived at the similar cubic equation (Eq 4 in Kok08) identical in form to Eq (7); the difference is in the particle size term Dp,0 instead of rd in KC07 The term rd in Eq (7) is the dry radius of a CCN that includes soluble and insoluble fractions, while Dp0 is the equivalent diameter of the insoluble fraction, which arises from using a dilute solution approximation
The analytical solutions for the critical radii rcr and supersaturations scr for CCN activation derived in Khvorostyanov and Curry (2007) for the arbitrary CCN soluble fraction εm are correct
Summary
A theoretical basis for consideration of hygroscopic growth of atmospheric aerosols or cloud condensation nuclei (CCN) and their activation into cloud drops was provided by the Kohler (1936) equation that enabled prediction of the CCN critical radii rcr and supersaturations scr for drop activation. Simple analytical solutions for rcr and scr are desirable for understanding the parametric dependencies and for developing cloud activation parameterizations for cloud and climate models. Such analytical solutions were derived by Khvorostyanov and Curry (2007, hereafter KC07) with sufficiently general assumptions and for arbitrary mass soluble fractions. It was shown in KC07 that the new equations for rcr and scr transform into the classical Kohler’s equations for sufficiently high soluble fraction, and yield a new analytical “low soluble fraction limit” for very small masses of soluble fractions where the classical equations fail. The accuracy of an ideal dilute solution approximation used in Kok is estimated, and it is found that the use of a non-ideal solution approximation as in KC07 improves the accuracy
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