The recently derived theoretical threshold function associated with the autoconversion process is generalized to account for the effect of the relative dispersion of the cloud droplet size distribution. This generalized threshold function theoretically demonstrates that the relative dispersion, which has been largely neglected to date, essentially controls the cloud-to-rain transition if the liquid water content and the droplet concentration are fixed. Comparison of the generalized threshold function to existing ad hoc threshold functions further reveals that the essential role of the spectral shape of the cloud droplet size distribution in rain initiation has been unknowingly buried in the arbitrary use of ad hoc threshold functions in atmospheric models such as global climate models, and that commonly used ad hoc threshold functions are unable to fully describe the threshold behavior of the autoconversion process that likely occurs in ambient clouds.
 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].
 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.
2. LDM Threshold Function and Its Generalization
 According to Liu et al. , the threshold function can be generally described by
where r is the droplet radius, n(r) is the cloud droplet size distributions, and rc is the critical radius. Under the assumption that the cloud droplet size distribution is described by
where Γ(t, z) = xt−1e−xdx is the incomplete Gamma function, and γ() denotes the incomplete Gamma function normalized by the corresponding complete Gamma function [Press et al., 1992]. Equations (7a) and (7b) show that Tq is determined by xc and q, and reduces to the LDM threshold function when q = 3.
 Although Tq as given by a combination of equations (6) and (7) quantifies the dependence of the threshold behavior on ɛ, the dependence has to be determined by repeating the procedure of calculating Tq and ɛ for different values of q. This is not an ideal feature for application in atmospheric models, which prefer simple relationships.
It is noteworthy that equation (8) actually gives the exact results for ɛ = 0, 1, and ∝, which corresponds to q = ∝, 1, and 0, respectively. Substitution of equation (8) into equation (7) yields
 The above theoretical analysis shows that the threshold function is determined by two dimensionless quantities: xc and ɛ. Figure 1 shows Tq (solid lines) as a function of the mean-to-critical mass ratio, i.e., the reciprocal of xc, at different values of ɛ. Also shown is Tɛ (dotted lines) to compare its performance with Tq. Two points are evident from this figure. First, the dependence of the threshold behavior on the mean-to-critical mass ratio (xc−1) gradually changes from a constant of T = 1 to a discontinuous δ-function as ɛ decreases from ∝ to 0. When xc−1 < 1, a larger ɛ leads to a larger value of the threshold function, and the same amount of water converted from cloudwater to rainwater requires a smaller xc (higher liquid water content and/or lower droplet concentration) for a smaller ɛ. This behavior is largely due to the enhanced collection process resulting from a broader droplet size distribution. These results highlight the importance of ɛ in rain initiation, and are consistent with the microphysical theory that droplet collision requires relative velocities resulting from droplets of different sizes [Pruppacher and Klett, 1997]. Second, Tɛ is an excellent approximation of Tq for virtually all the combinations of ɛ and xc and can be used as a substitute for Tq in practice (note that all the dashed Tɛ lines but that for ɛ = 1.76 overlap with the solid Tq lines).
3. Application of Tq to Examining Ad Hoc Threshold Functions
 To obtain physical understanding of the commonly used ad hoc threshold functions, this section examines them by comparing to the theoretically derived generalized threshold function. Traditional ad hoc threshold functions can be generally classified as Kessler-type, Berry-type or Sundqvist-type according to their mathematical form. Briefly, the Kessler-type threshold function is a Heaviside step function [Manton and Cotton, 1977; Liou and Ou, 1989; Baker, 1993; Liu and Daum, 2004]
where rm and rc denotes the driving and critical radii, respectively. The other extreme is often associated with empirical autoconversion parameterizations obtained by fitting simulations from detailed microphysical models, which generally have no threshold functions, or implicitly assume a constant threshold function [Berry, 1968; Beheng, 1994; Khairoutdinov and Kogan, 2000], i.e.,
According to the discussion in Section 2, the Kessler-type and Berry-type threshold functions in fact only represent monodisperse and extremely broad cloud droplet size distributions, respectively. Furthermore, because the two extreme spectral shapes are unlikely to occur in ambient clouds [Liu and Daum, 2000], neither the Kessler-type nor the Berry-type threshold functions are generally applicable.
 The Sundqvist-type threshold function lies between the two extremes [Sundqvist, 1978; Del Genio et al., 1996]. However, traditional Sundqvist-type parameterizations account only for L, and are therefore unsuitable for studies of the second indirect aerosol effect. Liu et al. [2006a] has recently proposed a generalized Sundqvist-type threshold function
where μ ≥ 0 is an empirical exponent. The generalized Sundqvist-type threshold function not only exhibits a smooth threshold behavior, but also encompasses virtually all the ad hoc threshold functions: it reduces to the traditional Sundqvist-type threshold functions proposed by Sundqvist and Del Genio when μ = 2 and 4, respectively, approximately becomes the Berry-type (T = 0.63 not 1) when μ = 0, and approaches the Kessler-type when μ approaches ∝ (see Liu et al. [2006a] for detailed discussion). Evidently, Ts is an improvement over the traditional ad hoc threshold functions that only work for some special spectral shapes of the droplet size distribution. Nevertheless, there is no physical basis for Ts, and especially, the physical meaning of μ is elusive.
 Careful comparison of Ts to Tq further indicates that Ts approximately describes the overall threshold behavior of Tq, if μ is a decreasing function of ɛ [A trial-and-error analysis suggests that μ = ɛ−1 is not a bad approximation]. Unfortunately, Despite the improvement over the Kessler-type and Berry-type threshold functions, Ts can accurately describe the threshold behavior only for narrow droplet size distributions, and is not suitable for broad droplet size distributions in ambient clouds.
 Furthermore, because ɛ likely varies from 0.1 to 10 in ambient clouds [Liu and Daum, 2000; Wood, 2000; McFarquhar and Heymsfield, 2001], the above analyses suggest that all the commonly used ad hoc threshold functions cannot adequately describe the real threshold behavior of the autoconversion process that occurs in nature.
4. Concluding Remarks
 The theoretical threshold function associated with the autoconversion process presented by Liu et al.  is generalized to account explicitly for the effect of the relative dispersion of the cloud droplet size distribution. The generalized threshold function theoretically shows that the relative dispersion, which has been largely neglected to date, essentially controls the initial transition from cloudwater to rainwater for fixed liquid water contents and the droplet concentrations. Comparison of the generalized threshold function with existing ad hoc threshold functions shows that the threshold behavior of the autoconversion process changes from the Berry-type to the Sundqvist-type to the Kessler-type as the relative dispersion decreases, and that the empirical parameter μ in the generalized Sundqvist-type threshold function is a decreasing function of the relative dispersion, providing physical explanations for these strikingly different ad hoc threshold functions. The comparison also suggests that commonly used ad hoc threshold functions only well describe the threshold behavior for some special spectral shapes of the cloud droplet size distribution, and cannot fully describe the threshold behavior that likely occurs in ambient clouds. These results indicate that the important role of ɛ in rain initiation has been unknowingly buried in the traditional practice of arbitrarily choosing ad hoc threshold functions.
 It is noted that although the importance of the spectral shape of the cloud droplet size distribution in rain initiation has been long recognized, the spectral shape effect has been poorly understood and quantified in atmospheric models. A recent study has demonstrated that modeling results are very sensitive to the treatment of the threshold function [Rotstayn and Liu, 2005]. It will be interesting to examine the effect of the relative dispersion on modeling results using this generalized threshold function. The explicit consideration of the relative dispersion in the generalized threshold function also allows for evaluation of the effect of the relative dispersion caused by anthropogenic aerosols on rain initiation, and the second indirect aerosol effect. Furthermore, the high sensitivity of the threshold behavior to the relative dispersion reinforces the need to account explicitly for the relative dispersion in the parameterization of the autoconversion process, which is still in its infancy [Liu and Daum, 2000; Liu et al., 2006b].
 The authors thank Y. Peng at Max Planck Institute of Meteorology and the anonymous reviewers for their insightful comments. This research was supported by Atmospheric Radiation Measurements Program and Atmospheric Sciences Program of the U.S. Department of Energy. The United States Government retains, and the publisher, by accepting the article for publications, acknowledges, a world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes.