An analytical expression for the critical radius associated with Kessler-type parameterizations of the autoconversion process is derived. The expression can be used to predict the critical radius from the cloud liquid water content and the droplet number concentration, eliminating the need to prescribe the critical radius as an empirical constant in numerical models. Data from stratiform clouds are analyzed, indicating that on average continental clouds have larger critical radii than maritime clouds. This work further suggests that anthropogenic aerosols affect the autoconversion process by increasing the critical radius and decreasing the characteristic radius, which in turn inhibits the initiation of embryonic raindrops, and by decreasing the autoconversion rate after the initiation process. The potential impact of this work on the evaluation of the second indirect aerosol effect is discussed.
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Kessler  proposed a parameterization that linearly relates the autoconversion rate to the cloud liquid water content (L), and assumes a critical value for L below which no autoconversion occurs. Later Kessler-type parameterizations explicitly account for the droplet concentration (N) as well as L [Manton and Cotton, 1977; Tripoli and Cotton, 1980; Liou and Ou, 1989; Baker, 1993; Liu and Daum, 2004]. The inclusion of N allows for studying the second indirect aerosol effect. It has also been recognized that the threshold process should be determined by a critical radius (rc) rather than by a critical L as conceived by Kessler. Considering autoconversion as a threshold process is a distinctive feature that sets Kessler-type parameterizations apart from other types of autoconversion parameterizations [e.g., Berry, 1968; Beheng, 1994; Khairoutdinov and Kogan, 2000].
 All the later Kessler-type parameterizations can be generically written as
where P is the autoconversion rate, f a generic function, and rm the characteristic radius. The Heaviside function H(rm − rc) is introduced to describe the threshold process such that there is no autoconversion when rm < rc. The characteristic radius is the volume-mean radius given by Manton and Cotton , Tripoli and Cotton , Liou and Ou , and Baker , the mean radius of the 4th moment given by Boucher et al. , and the mean radius of the 6th moment in the parameterization that we have recently derived [Liu and Daum, 2004]. The function f is also different for different parameterizations.
 Although model results are sensitive to rc [Boucher et al., 1995; Rotstayn, 1999], the idea of threshold process embedded in Kessler-type parameterizations has been loosely used, and rc has been largely prescribed as an empirical parameter that is arbitrarily tuned to match model simulations with observations. Here we derive an analytical expression for rc by coupling the Liu-Daum parameterization with a new theory on rain formation that we have recently formulated [McGraw and Liu, 2003]. The expression can be used to predict rc from L and N. Data from continental and marine stratiform clouds are examined, and implications for the evaluation of the autoconversion rate and the second indirect aerosol effect are discussed.
2. Kinetic Potential Theory, Threshold Process and Critical Radius
 Although it has been well established that three physical processes (condensation, evaporation and collection) are involved in the formation of warm rain, many issues regarding the initiation of warm rain remain unsolved [Beard and Ochs, 1993; Telford, 1996]. Recently, we developed a new theory on rain formation by extending the theory of statistical crossing of a kinetic potential barrier in nucleation to the processes of condensation, evaporation and collection occurring in warm clouds [McGraw and Liu, 2003].
 Briefly, by analogy to the kinetic theory on nucleation, the kinetic potential Φ(j) for a droplet consisting of j water molecules is given by
where βcon (s−1), βcol (s−1), and γeva (s−1) denote the condensation, collection, effective evaporation rate constants, respectively, for a droplet consisting of g water molecules; ν = 3.0 × 10−23 (g) is the mass per water molecule; κ = 1.1 × 1010 cm−3 s−1 is a constant in the Long collection kernel [Long, 1974]; ρw is the water density (g cm−3). The kinetic potential as a function of droplet radius (r) can be calculated by applying the relation g = r3 to equations (2a)–(2c). Figure 1 shows a typical example. The kinetic potential first increases with increasing droplet radius because (βcon + βcol) < γeva, and then decreases after reaching a peak because (βcon + βcol) > γeva [Note that = −ln].
 The point where the kinetic potential reaches its maximum is worth emphasizing because it physically defines a critical point. As in nucleation theory, the maximum kinetic potential is referred to as the “barrier”; the corresponding droplet radius defines rc. Before reaching the critical point, the droplet system is in a stable state because more potential is needed to climb the “hill”. Once the barrier is passed, the system becomes unstable down the “hill”, and embryonic raindrops spontaneously form. Therefore, the idea of a threshold process and rc inherent in Kessler-type parameterizations emerges naturally from the kinetic potential theory.
Because (νN/L) ≪ 0, and exp ≈ 1 + , equation (4a) can be simplified to
where rc is in μm, L in g m−3, N in cm−3, and βcon in s−1.
 In general, βcon is a function of turbulence that is unknown at present [McGraw and Liu, 2003]. Nevertheless, it can be estimated from microphysical measurements in drizzling clouds as follows. According to Liu and Daum , autoconversion starts when the mean radius of the 6th moment reaches rc. The mean radius of the 6th moment (r6) is given by
where α = 1.12. Equating r6 with rc, we obtain
where the subscript “*” denotes the corresponding variables at threshold where the two radii are equal; L* is in g m−3, N in cm−3, and βcon in s−1.
Equation (6) is used to estimate βcon from concurrent measurements of L and N in drizzling clouds reported by Yum and Hudson [2001, 2002]. As shown in Figure 2, the majority of βcon values fall between 1022 and 1023 (s−1); the mean, minimum and maximum of βcon are 1.15 × 1023, 1.02 × 1023 and 1.67 × 1024 (s−1), respectively.
4. Critical Radius of Ambient Clouds and Important Implications
Equation (4b) indicates that rc is a function of L and N, varying from cloud to cloud, even from place/time to place/time in the same cloud. To demonstrate this, Figure 3 shows rc calculated from equation (4b) using the mean βcon and the data on L and N from stratiform clouds given by Miles et al. . It is clear that rc varies significantly, from circa 6 μm to 40 μm. Note that since each point in Figure 3 actually represents an average of many samples, variation in rc is expected to be even larger for individual clouds. This suggests that prescribing rc as a constant is more troublesome in small-scale models than in GCMs.
 It is interesting to compare r6 with rc because r6, or its “equivalent”, is often used in studies of both the indirect aerosol effects [Twomey, 1991; Albrecht, 1989]. The equation relating rc to r6 can be derived by using equation (5) to eliminate N in equation (4b),
where L and the radii are in g m−3 and μm, respectively. Equation (7) indicates that rc decreases with increasing r6 when L remains constant. This is illustrated by the dot and dash lines in Figure 4, which correspond to the lower and upper limits of L = 0.001 g m−3 and L = 2 g m−3 in the Miles et al. data, respectively. It is well known that an increase in aerosol loading leads to an increase in N and a decrease in the mean radius. Accordingly, anthropogenic aerosols tend to move clouds up the line, diminishing r6 but increasing rc. The data presented in Figure 4 support this notion. Because of higher droplet concentrations, continental clouds have an average rc (12.71 μm) larger than that of marine clouds (10.32 μm). These values are close to those prescribed in cloud-scale models [Tripoli and Cotton, 1980], but larger than those assigned in GCMs [Boucher and Lohmann, 1995; Rasch and Kristjansson, 1998; Rotstayn, 1999]. A smaller rc for GCMs is often attributed to a coarser model resolution and subgrid variabilities of cloud properties [Rotstayn, 2000; Pincus and Klein, 2000; Zhang et al., 2002]. The nonlinear dependence of rc on L and N revealed by equation (4b) supports this argument.
Figure 4 also shows that on average, continental clouds are more likely to have r6 < rc than marine clouds, suggesting that rain onset is inhibited in continental clouds by increased anthropogenic aerosols. This result is consistent with the assumption of the second indirect aerosol effect [Albrecht, 1989]. However, some continental clouds have r6 > rc whereas some marine clouds have r6 < rc, suggesting the complexity of the second indirect aerosol effect. It is known that anthropogenic aerosols affect precipitation by decreasing the characteristic radius (e.g., r6) and by decreasing the conversion rate from cloud water to rain water after the onset of the autoconversion process. This study suggests that anthropogenic aerosols also inhibit the initiation of embryonic raindrops by increasing rc.
5. Concluding Remarks
 An analytical expression for rc is derived by coupling the kinetic potential theory on rain formation with the Liu-Daum autoconversion parameterization. The expression can be used to predict rc from L and N, eliminating the need to prescribe rc in models. Examination of data from stratiform clouds indicates that rc varies from cloud to cloud, and that on average, continental clouds have larger rc than maritime clouds. It is shown that anthropogenic aerosols have the effect of increasing rc and decreasing the mean radius concurrently, inhibiting the onset of embryonic raindrops. Use of this new expression will improve the representation of precipitation in numerical models and our ability to model the second indirect aerosol effect.
 Three points are worth noting. First, there is uncertainty in our estimate for βcon (hence rc) due to limited measurements of L and N used in the calculation, and to the uncertainty as to if the measurements were taken at threshold points as required by equation (6). Furthermore, βcon is expected to depend on cloud turbulence and droplet radius. It would be desirable to derive it from the first principles. Second, relative dispersion of the cloud droplet size distribution is not considered an independent variable in the current formulation. However, this quantity is also crucial for evaluating cloud radiative properties and indirect aerosol effects [Liu and Daum, 2000, 2002; Peng and Lohmann, 2003; Rotstayn and Liu, 2003]. Further progress requires extending the formulation to explicitly account for the relative dispersion. Finally, the expression for rc is essentially local. The effect of subgrid variabilities of cloud properties on rc needs to be addressed in GCMs. As a first order approximation, this effect could be simply accounted for by adjusting βcon because rc is proportional to βcon1/6 (equation (4b)).
 This research was supported by the Atmospheric Radiation Measurements Program under contract DE-AC03-98CH10886, a LDRD grant from the BNL, and the Atmospheric Chemistry Program under contract DE-AC02-98CH10886. The authors thank Dr. Schwartz at BNL for stimulating discussions, and Dr. Ghan at PNL and an anonymous reviewer for their insightful comments.