## 1. Introduction

[2] The autoconversion process whereby cloud droplets grow into embryonic raindrops is a key microphysical process that needs to be parameterized in atmospheric models such as cloud resolving models and global climate models [*Kessler*, 1969; *Manton and Cotton*, 1977; *Liou and Ou*, 1989; *Baker*, 1993; *Liu and Daum*, 2004]. Accurate parameterization of the autoconversion process is especially important for estimating the second indirect aerosol effect [*Boucher et al.*, 1995; *Lohmann and Feichter*, 1997; *Rotstayn*, 2000; *Rotstayn and Liu*, 2005].

[3] All the autoconversion parameterizations that have been developed so far can be generically written as

where P is the autoconversion rate; P_{0} is the rate function describing the conversion rate after the onset of the autoconversion process, and T is the threshold function describing the threshold behavior of the autoconversion process. The rate function P_{0} has been the primary focus of previous studies, and great progress has been made over the last few decades [*Kessler*, 1969; *Manton and Cotton*, 1977; *Liou and Ou*, 1989; *Baker*, 1993; *Liu and Daum*, 2004; *Chen and Liu*, 2004; *Wood*, 2005]. The threshold function, however, has received little attention, and the commonly used threshold functions are ad hoc in nature [*Kessler*, 1969; *Sundqvist*, 1978; *Del Genio et al.*, 1996; *Liu et al.*, 2006a].

[4] We have recently derived a theoretical threshold function by truncating the collection equation at the critical radius (LDM threshold function) [*Liu et al.*, 2005]. Although the LDM threshold function provides a firm physical basis for the threshold behavior of the autoconversion process, it only considers the liquid water content (L) and the droplet concentration (N) as independent variables, and implicitly assumes a constant relative dispersion (ɛ, defined as the ratio of standard deviation to the mean radius of the cloud droplet size distribution). The assumption of a constant ɛ is a drawback of the LDM threshold function, because the spectral shape of the droplet size distribution is expected to vary in ambient clouds and to have a significant effect on rain initiation [*Hudson and Yum*, 1997]. Furthermore, both observational and theoretical evidence indicates that increasing aerosols concurrently increase N and ɛ, and the enhanced ɛ leads to a warming dispersion effect on climate [*Liu and Daum*, 2002; *Rotstayn and Liu*, 2003; Peng and Lohmann, 2003; *Liu et al.*, 2006b]. Without explicit specification of ɛ, the LDM threshold function is handicapped in applications such as investigating rain initiation and the second indirect aerosol effect [*Rotstayn and Liu*, 2005].

[5] The primary objective of this work is to generalize the LDM threshold function to account explicitly for ɛ in addition to L and N, and to use this new generalized threshold function to examine commonly used ad hoc threshold functions.